Knowledge Area Module 2:

Principles of Human Development

by

Joseph Michael Dillon joseph.dillon@waldenu.edu

Student ID #A00074456

Program: PhD in Education

Specialization: Educational Technology

KAM Assessor: Dr. William A. Sugar william.sugar@waldenu.edu

Faculty Mentor: Dr. William A. Sugar william.sugar@waldenu.edu

Walden University

November 22, 2010

Abstract

Breadth

The breadth portion of the KAM describes the basic tenets of the educational,

intelligence, and learning theories developed by John Dewey, Howard Gardner, Michael

Martinez, and Lev Vygotsky. The nature of reflective thought and the role of experience

in education are highlighted in a review of Dewey’s work. Gardner’s theory of Multiple

Intelligences is described. The premise of cultivating intelligence through education is

the key idea addressed by Martinez. Vygotsky’s “zone of proximal development” is also

discussed. These theoretical ideas are analyzed and blended into a framework for

assessing learning activities with the goal of promoting positive changes in classroom

instruction. The role of technology is also integrated into the framework to provide a

comprehensive model for analyzing instruction.

Abstract

Depth

A review of the current literature on educational theory, mathematics instruction, and

technology-based activities is conducted to identify the characteristics of effective

instruction. Effective technology-based instruction promotes higher-level thinking, leads

to the independent use of technology, and incorporates relevant contexts. The main

theme underlying these characteristics is that the use of technology alone cannot

guarantee improvements in student learning. Therefore, educators must carefully design

technology-based instruction. The literature also points to the significant impact of

learning style on student success. To bring about positive social changes in the

classroom, the roles of effective instruction and learning style will be used to address the

obstacles that hinder the implementation of technology-based activities. Ideas for further

research are also presented.

Abstract

Application

The application project is a fully developed unit designed for an Algebra I course. The unit

addresses the multiple ways that mathematical content can be represented. Presented as a wiki,

the unit incorporates an array of learning activities and technology-based instruction. The unit

design is influenced by the concepts gleaned from the analysis of Dewey, Gardner, Martinez, and

Vygotsky and from the review of current literature. The unit is critically analyzed using the

model designed in the breadth and is also evaluated using the characteristics of effective

instruction and the role of learning styles as noted in the depth. The analysis is concluded with a

self-critique of the unit design and the potential impact for change in the mathematics classroom.

Table of Contents

List of Figures.....................................................................................................................iv

Breadth.................................................................................................................................1

Description of the Theories............................................................................................2

Educational Theories of John Dewey......................................................................2

Howard Gardner and the Theory of Multiple Intelligences...................................12

Social Modeling and Vygotsky’s Zone of Proximal Development.......................19

Martinez, Learning Intelligence, and the 3E Model..............................................23

Synthesis of an Evaluation Framework.......................................................................27

Planning for Instruction.........................................................................................30

Implementing Instruction.......................................................................................32

Evaluating Instruction............................................................................................37

Incorporation of Technology into Evaluation Framework..........................................40

Technology and Planning for Instruction..............................................................42

Technology and Implementing Instruction............................................................44

Technology and Evaluating Instruction.................................................................46

Conclusion...................................................................................................................48

From Theory to Research.............................................................................................48

Depth..................................................................................................................................50

Annotated Bibliography...............................................................................................50

Literature Review Essay............................................................................................123

i

Effectiveness of Technology-Based Instructional Strategies..............................124

Alignment of Technology-Based Instruction and Student Learning Styles........137

Obstacles to Technology Integration and Synthesis of Recommendations.........149

Analysis of, Suggestions for, and Applications of Current Research..................154

Conclusion.................................................................................................................159

From Research to Practice.........................................................................................160

Application.......................................................................................................................162

Description of the Application Project......................................................................163

Structure of the Daily Lesson Plan Outline.........................................................164

About the Wiki Page............................................................................................165

Description of Mathematical Content..................................................................173

Learning and Instructional Activities...................................................................174

Integration of Technology....................................................................................175

Possible Adaptations and Modifications..............................................................176

Application Discussion and Critique.........................................................................177

Analysis Based on Current Research...................................................................185

Critical Considerations.........................................................................................190

Ethical Considerations.........................................................................................192

Potential for Social Change.................................................................................194

Conclusion.................................................................................................................196

References........................................................................................................................198

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Appendix..........................................................................................................................205

iii

List of Figures

Figure 1. Summary of Dewey’s learning theories and educational applications............................3

Figure 2. Summary of Gardner’s theory of intelligence and educational applications................13

Figure 3. Summary of Vygotsky’s theory of social learning and educational applications.........20

Figure 4. Summary of Martinez’s theory of intelligence and educational applications...............24

Figure 5. Visual representation of rubric for developing and assessing instruction.....................29

Figure 6. Role of technology in rubric for developing and assessing instruction.........................41

Figure 7a. Technology considerations during the planning phase................................................43

Figure 7b. Technology considerations and the zone of proximal development...........................44

Figure 7c. Technology considerations and instructional design...................................................45

Figure 7d. Technology considerations and entry point activities.................................................46

Figure 7e. Technology considerations and Martinez’s 3E model of intelligence.........................47

Figure 7f. Technology considerations, future experiences, and assessment.................................47

Figure 8. Wiki screenshot of sample assignments for the unit...................................................164

Figure 9. Wiki screenshot of homepage.....................................................................................166

Figure 10a. Wiki screenshot of Unit 1 page with video.............................................................168

Figure 10b. Wiki screenshot of Unit 1 page with section links..................................................168

Figure 10c. Wiki screenshot of Unit 1 page with post-test link..................................................169

Figure 11. Screenshot of example question from Unit 1 pretest.................................................169

Figure 12a. Wiki screenshot of unit subsection page with DQs.................................................171

Figure 12b. Wiki screenshot of unit subsection page with video tutorial...................................171

iv

Figure 12c. Wiki screenshot of unit subsection page with assignments.....................................172

Figure 13. Wiki screenshot of discussion question page example..............................................172

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1

Breadth

SBSF 8210: Theories of Human Development

Suppose that a mathematics teacher is standing in a room full of students introducing a

complex problem. Although the educator may have an advanced knowledge of the content and

understands how a particular problem applies to real world situations, many of the students may

lack the necessary background knowledge to effectively use given problem-solving strategies to

develop relevant solutions. Math instruction often consists of activities that only encourage rote

learning. This type of classroom interaction does not necessarily assist the students to truly

reflect upon what they are learning, why it is important, or how it may apply to them. Dewey

(1910) proposed that learners will engage in the process of reflection when they are faced with

dilemmas for which they are seeking an answer. In many cases, math instruction does not

always create opportunities for this to occur. Thus, a majority of students learn enough

superficial information to regurgitate decontextualized patterns on an assessment but fail to

internalize the underlying processes that form the foundation for more generalized problem-

solving situations. The question that remains is what happens (or does not happen) during

instruction to result in varying degrees of true student learning?

The answer to this question is complex, but a search must begin by exploring the theories

of those educators that focused their professional lives on understanding cognitive development,

learning, and intelligence. Thus, a critical analysis of the learning, intelligence, and educational

theories of John Dewey, Howard Gardner, Michael Martinez, and Lev Vygotsky will be

conducted. This analysis will lead to the synthesis of a framework by which instructional

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techniques, particularly in mathematics, can be evaluated. Criteria based on the classic theories

will also be developed for effectively incorporating the use of technology into the learning

process and for infusing it into the evaluation rubric. Critically analyzing the classic theories of

learning and intelligence with the goal of improving mathematics instruction will serve as a

reflective opportunity for assessing current instructional practices and promoting positive

changes in the learning process.

Description of the Theories

The complex nature of learning, intelligence, and education cannot be delineated in a

single, narrow description. Over time, theorists, scholars, and practitioners have contributed to

the greater understanding of what constitutes human cognition, learning, and intelligence as well

as how educational practices influence what people learn. Although each theoretical perspective

addresses human cognition in a unique way, each theory can distinctly contribute to how the

general understanding of learning can be translated into effective practice. Thus, it is essential to

examine the key characteristics of each foundational theory.

Educational Theories of John Dewey

John Dewey (1859 to 1952) was an American educator and philosopher who developed

influential theories about learning, intelligence, and educational practice. His philosophies were

rooted in the pragmatic notion that learning needed to be intimately tied with the experiences and

applications of concepts in the real world (Saettler, 2004). Saettler noted that Dewey’s ideas had

a significant impact on the American education system which can still be felt today. Of the

concepts promoted by Dewey’s conceptualization of learning and education, several key themes

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(see Figure 1) are important to note: the definition, role, and support of reflective thought; the

necessity of experience (expressed as the concepts of continuity and interaction) in the learning

process; and classroom practices that promote real learning.

Reflective thought. An important pillar in the educational theories of John Dewey is the

notion of reflective thought and the educational nature of this type of mental activity. Dewey

(1910) stated that “reflective thinking…means judgment suspended during further inquiry…” (p.

Figure 1. Summary of Dewey’s learning theories and educational applications.

4

13). Dewey (1910) noted that thoughts exist on a continuum, but only those thoughts that lead

individuals to reflection are really educative in nature. Truly reflective thought requires a person

to be disciplined enough to effectively utilize time and resources in consideration of how new

information influences the problem-solving process. In turn, the reflection leads an individual to

take action based on the information and conclusions that are considered. The nature of

reflection is an inherent characteristic of human thinking; and the ability to reflect allows people

to transcend thinking in concrete, time-dependent terms. However, learning how to actively

reflect is not something that comes easily to every person. Therefore, Dewey (1910) noted the

need for individuals to receive training on reflection in the thinking process. Everybody has

access to a variety of tools that aid in effective reflection and can utilize those tools to further

develop their skills as active, engaged learners, problem-solvers, and thinkers.

In order to help a person develop his or her abilities as a reflective, active thinker, Dewey

(1910) emphasized that the “training of the mind” (p. 28) must be based on the individual. He

stated: “...even more truly one can teach others to think only in the sense of appealing to and

fostering powers already active in them” (p. 30). Two resources directly accessible to an

individual during the process of developing the skills of reflection are his or her curiosity and

prior experiences.

Curiosity is the level of interest a person takes in exploring new concepts, information,

ideas, and/or stimuli from the environment and is the driving force that guides that person to use

his or her energy and intelligence to pursue further investigation, understanding, and answers

(Dewey, 1910). Dewey conceded that there are relatively few individuals that embody a level of

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curiosity that requires little, if any, encouragement from outside sources. These individuals

naturally explore their surroundings, ask questions and seek out answers, and actively engage in

reflection. Every individual has interests that spark curiosity, but educators face the challenge of

helping individuals to tap their curious natures with regard to concepts that are not inherently

interesting or immediately relevant. According to Dewey, curiosity is vital in order for

individuals to actively engage in thinking that is educational, purposeful, and reflective.

Another resource that acts as a catalyst for the training of thought is personal experience.

Experiences exist on a continuum with respect to how they influence the behavior and thought

processes of an individual (Dewey, 1938). Furthermore, an inherent connection exists between

what an individual experiences and what that individual actually learns (Dewey, 1938). Dewey

(1938) stated that “…every experience lives on further in further experiences” (p. 27). An

individual brings a history of experiences, attitudes, and ideas to each new situation he or she

encounters. That history will impact how that person proceeds through the situation. Dewey

(1938) emphasized that “…what he has learned in the way of knowledge and skill in one

situation becomes an instrument of understanding and dealing effectively with situations which

follow” (p. 44). The degree of reflective thought that the person engages in will depend on the

level of, type of, and reaction to previous experiences; and this store of information is vital to

thinking in a manner that is truly educative.

The individual learner provides the primary resources necessary to engage in reflective

thought. However, there are additional mental habits and processes that can support reflective

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thinking. These include: the use of the scientific method, critical thinking, and inductive and

deductive reasoning.

The scientific method is a strategy that allows an individual to systematically process

information that may otherwise appear to that person in a seemingly random way. Dewey (1910)

noted that the scientific method allows a person to effectively and efficiently draw conclusions

and logically order information by breaking down larger amounts of empirical information into

smaller pieces. A person’s interactions within the environment do not necessarily lend

themselves to clear understandings of a particular phenomenon or general concept; yet reflective

thought requires that person to manage the information that is uncovered through those

interactions. Dewey (1910) relied on the scientific method as a foundational approach to support

reflective thought through an orderly process of organizing, synthesizing, and analyzing

information.

Whereas the scientific method is a strategy employed by a person to manage information

associated with experiences, reflection that allows a person to educationally progress also

requires a level of critical thought. Critical thinking is the ability to delay the need to jump to an

immediate conclusion (Dewey, 1910). To critically analyze a situation and reflect on the deeper

nature of a particular concept necessitates the continued search for additional evidence to further

mold a conclusion. Critical thinking also implies the willingness to adapt existing judgments in

light of new information. As Dewey (1910) noted, one of the key goals of critical thinking is to

improve “schema quality” (p. 180) in order to have the most accurate conceptualization of one’s

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surroundings. Employing critical thinking skills is essential for fostering a clear understanding

of one’s environment.

In conjunction with the scientific method and critical thinking, the process of reasoning

involves systematically identifying and structuring the relationships between various pieces of

information. The route to developing these relationships is a two-way street and involves both

deductive and inductive reasoning. Deductive reasoning involves beginning with generalizations

and moving toward particular details or concepts whereas inductive reasoning utilizes details to

work toward generalizations (Dewey, 1910). Reflective thought depends on a person’s ability to

freely move in both directions as information is synthesized into a greater understanding of the

world. Dewey (1910) stated that: “…every complete act of reflective inquiry makes provision

for experimentation—for testing suggested and accepted principles by employing them for the

active construction of new cases, in which new qualities emerge” (p. 99). Reflective thought

requires an individual to engage in both inductive and deductive reasoning in order to maintain

the continual growth of knowledge and understanding.

Reflective thinking is the foundation for true learning. Reflection is a conscious, active

process. Although it builds upon the curiosity and experiences of the individual, it does require

practice and opportunities to utilize and hone critical thinking, scientific inquiry, and reasoning

skills. Dewey (1938) argued that educators play a significant role in developing these

opportunities and in providing situations where students can engage in reflective thinking. To

this end, Dewey advocated for experience-based learning.

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Experience-based learning. Dewey (1938) posited a fundamental educational question:

“…How shall the young become acquainted with the past in such a way that the acquaintance is

a potent agent in appreciation of the living present?” (p. 23). For Dewey, the answer to this

question was rooted in the progressive philosophy of experience-based learning. Although

Dewey did not discard the types of experiences had by students in more traditional academic

settings, he saw value in the educational gains (as measured by an ability to engage in reflective

thought) that stemmed from the personal experiences of directly interacting with the phenomena

in question. Dewey was careful to point out that individuals are continually having new

experiences, but only certain experiences prove to be truly educative. Therefore, he developed a

“theory of experience” (Dewey, 1938, p. 30) that can be used in educational settings to

characterize experiences.

In order to identify experiences that guide reflective thought and positive learning

activities, Dewey (1938) envisioned a theory of experience based on two key principles—

continuity and interaction. Continuity is the characteristic of experience that links a present

experience to both previous and future experiences. As Dewey noted, an experience is built

upon previous experiences had by an individual and will, in turn, influence future experiences.

This notion places a premium on identifying the types of experiences an individual has had in the

past in order to assess how those prior experiences will guide reflective thought in the present.

Interaction, the second principle of experience, is the characteristic of experience that is based on

how an individual’s experience relates to the immediate environment. A person’s surroundings,

the stimuli they receive, and the people present will all influence how an experience is shaped for

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an individual. Dewey (1938) stated that continuity and interaction are “…the longitudinal and

lateral aspects of experience” (p. 44). These characteristics are what determine the value of an

experience from an educational point of view and are essential considerations in the development

of a program for assisting the development and refinement of reflective thought.

Experience-based learning that leads to positive educational gains must be tied to the

needs and characteristics of the individuals. The experiences, attitudes, learning characteristics,

and skills that learners bring to the classroom must be considered in instructional planning

(Dewey, 1938). These plans should also balance the individual natures of the students with the

development of continuous experiences that move toward the achievement of the overarching

learning goals (Dewey, 1938). Establishing a learning environment that meets these criteria and

leads to effective learning places a significant amount of responsibility on the educator.

Role of educators and classroom practices. In outlining the factors that influence

reflective thought and define experienced-based learning, Dewey provided several

recommendations for educators with regard to their roles in the classroom. Primary goals of

every teacher should be: to help students develop their abilities to engage in reflective thought;

to utilize their experiences, curiosity, and reasoning to foster new ideas and continued

exploration; and to develop positive attitudes and habits of inquiry and investigation (Dewey,

1910). In order to achieve these goals, the teacher must carefully arrange for experiences that

engage the students, reflect the characteristics of truly educative experiences, and carry the

potential for sparking future experiences (Dewey, 1910, 1938). Dewey (1910) stated that it is

the job of the teacher “…to keep alive the sacred spark of wonder…[and]…to protect the spirit

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of inquiry…” (p. 34). The ideals that underpin this philosophy of educational practice were

translated by Dewey into tangible factors that teachers can address.

First, educators must be acutely aware of the individual characteristics of every student

(Dewey, 1910). Not only does the teacher need to understand the natural strengths of each

student, but he or she also needs to uncover the underlying attitudes and learning routines that

students bring to the classroom (Dewey, 1910). The teacher should “have that sympathetic

understanding of individuals as individuals which gives him an idea of what is actually going on

in the minds of those who are learning” (Dewey, 1938, p. 39). Second, teachers can utilize their

understanding of the immediate environment to develop learning activities that will help

establish meaningful connections between the content and the experiences of the students

(Dewey, 1938). Finally, teachers need to have an understanding of pedagogical strategies that

can be employed to modify how the learning experiences of the students evolve over the course

of instruction (Dewey, 1910). The goals of a learning experience for students in conjunction

with factors that teachers can address while developing and guiding instruction can then be

translated into classroom practices that reflect Dewey’s notion of experienced-based learning.

The types of activities that are utilized in the classroom must be carefully chosen to

ensure that they produce the types of experiences that promote continued intellectual growth.

Dewey (1938) stated:

…it is part of the educator’s responsibility to see equally to two things: First, that the

problems grow out of the conditions of the experience being had in the present, and that it

is within the range of the capacity of the students; and, secondly, that it is such that it

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arouses in the learner an active quest for information and for production of new ideas.

The new facts and new ideas thus obtained become the ground for further experiences in

which new problems are presented. The process is a continuous spiral. (p. 79)

Experience-based learning has the potential to lead students down a road of discovery where

relevant connections between content and application can be established. However, successful

instruction in this setting requires significant thought, preparation, and implementation on the

part of the teacher. Dewey (1910) identified three generic stages that can frame the design of

instruction: 1) the apprehension of facts; 2) generalization; and 3) application and/or verification.

Through theses stages, careful attention must be given to provide students with opportunities to

experience phenomena, build a personal connection with the content, and apply the knowledge in

a meaningful way. More importantly, the educator must provide sufficient direction to link the

present experiences to previous learning and to promote the exploration of expanded ideas in the

future (Dewey, 1938). In the end, the classroom practices represent how the ideals of reflective

thought and experienced-based learning are converted into tangible learning opportunities for

students. These opportunities must be attended to with care, for they are what impact the growth

and intellectual development of each learner.

Dewey (1938) advocated for the need to provide students with educational opportunities

that promote true learning and that, more importantly, lead individuals to develop the desire to

continue learning. These opportunities could be developed through experience-based learning

and through the training of reflective thought. Although Dewey’s theories were geared toward

how every person acquires knowledge, he always noted the importance of the individual and

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how he or she uniquely experienced the world. Along these lines, Howard Gardner took the

notion of individuality with regard to intelligence to another level with his theory of Multiple

Intelligences.

Howard Gardner and the Theory of Multiple Intelligences

Howard Gardner (1943 – ) is a psychologist who is best known for his theory of Multiple

Intelligences (MI theory). To develop his theory, Gardner drew upon a wide range of knowledge

from scientific fields and information sources including psychology, biological evidence,

neuroscience, cultural information, and so on. MI theory presented a drastically new perspective

on cognitive ability by broadening the conceptualization of which skills, talents, and/or abilities

constituted intelligence. Gardner’s theory (see Figure 2) can be better understood by exploring

the criteria used to identify an intelligence, outlining the eight intelligences currently recognized,

and analyzing the implications of the theory in educational settings.

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Criteria used to define an intelligence. Prior to the work of Gardner, the notion of

intelligence was traditionally focused on a person’s IQ (or intelligence quotient). It was assumed

that IQ could be measured through the use of paper and pencil tests that primarily focused on

verbal and mathematical skills. Gardner (1983) believed that this conceptualization of

intelligence was too narrow and did not account for the various biological and environmental

factors that influence intelligence. Therefore, Gardner identified a wider range of intelligences

Figure 2. Summary of Gardner’s theory of intelligence and educational applications.

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that he believed were shared among every person. In order to support this claim, he utilized a

variety of different information sources and developed a set of criteria by which to judge a

particular skill (or trait) as an independent intelligence. These criteria included:

possible isolation by brain damage….existence of idiot savants, prodigies, and other

exceptional individuals….an identifiable core operation or set of operations….a

distinctive developmental history, along with an definable set of end-state

performances….an evolutionary history….support from experimental psychological

tasks….support from psychometric findings….[and] susceptibility to encoding in a

symbol system. (Gardner, 1983, pp. 63-66)

The use of the criteria set forth by Gardner served as a way to ensure that identified intelligences

functioned in a manner that aligned with the basic premise of what intelligence was defined to

be:

…a set of skills of problem solving—enabling the individual to resolve genuine

problems or difficulties that he or she encounters and, when appropriate, to create

an effective product—and…the potential for finding or creating problems—

thereby laying the groundwork for the acquisition of new knowledge. (Gardner,

1983, pp. 60-61)

To date, Gardner has identified eight intelligences: linguistic, musical, mathematical-logical,

spatial, bodily-kinesthetic, interpersonal, intrapersonal, and naturalist.

Eight intelligences. The linguistic intelligence involves an ability to work with

words and language (Gardner, 1999). Although Gardner (1983) used the example of a

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poet as an individual who has strengths in the linguistic area, this intelligence involves

working with both written text as well as spoken language. Gardner (1983) also noted

that various tasks that are linguistic in nature may require different types of skills.

However, the common denominator among all of the products that are related to the

linguistic intelligence involves the use of words to convey some type of message.

Individuals with strengths in the musical intelligence have an aptitude with regard to

playing, performing, and/or composing music (Gardner, 1999). In addition, an appreciation for

the qualities of music also represents a potential strength in the musical intelligence (Gardner,

1999). Playing an instrument, using one’s voice, performing, writing music, and appreciating

music all require different types of basic skills. Although the specific skills may vary, each of

these musical areas requires a strong sense of pitch, rhythm, and tone—the underlying features of

music (Gardner 1983).

Like the linguistic intelligence, the mathematical-logical intelligence is an intelligence

that has been traditionally valued in education and measured through standardized testing

(Gardner, 1999). This intelligence involves abilities to work with numbers, to solve problems

mathematically and/or logically, and to process problems scientifically (Gardner, 1999).

Moreover, individuals with a propensity for the mathematical-logical tend to move away from

the concrete and search for more abstract or general relationships that exist (Gardner, 1983).

Patterns, chains of reasoning, deductive and inductive processes, and symbols serve as the basic

components of this intelligence.

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In its most basic sense, the spatial intelligence deals with the abilities to perform spatial

operations and to work with patterns or problems that exist in space (Gardner, 1999). Some of

the skills associated with this intelligence include: identifying differences and similarities

between objects; performing transformations of objects (physically, graphically, or mentally);

visualizing images mentally; reproducing physical objects in a sketch or other media; and so on

(Gardner, 1983). The spatial intelligence can be applied in various situations from having a

sense of where one is located directionally to creating artistic renderings of objects or mental

images.

As the name suggests, the bodily-kinesthetic intelligence focuses on the ability to utilize

the body to express oneself. Gardner (1983) noted that dance is one of the most common

examples of an activity that utilizes the bodily-kinesthetic intelligence. However, he noted that

the use of fine-motor skills to design, construct, invent, and/or manipulate objects also

exemplifies the bodily-kinesthetic intelligence (Gardner, 1983). Working with one’s hands,

displaying athletic skills, purposefully moving one’s body, and so forth all illustrate the use of

the bodily-kinesthetic intelligence.

Of the eight identified intelligences, the personal intelligences—interpersonal and

intrapersonal—are the only ones that were described by Gardner simultaneously. The

interpersonal intelligence deals with a person’s ability to understand others and to function

effectively in groups (Gardner, 1999). On the other hand, the intrapersonal intelligence involves

an understanding of oneself. Although these intelligences are distinct from one another based on

the criteria set forth by Gardner, they are often closely related. In many cases, an understanding

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of oneself often allows a person to better understand others and vice versa (Gardner, 1983). In

addition, the skills associated with each of these intelligences are often developed through social

interactions. The personal intelligences encompass the skills necessary to effectively interact

with others while successfully managing one’s own life.

In his original work, Gardner (1983) identified seven intelligences; however, he

acknowledged that the number of intelligences would never be completely known and that the

potential existed for the identification of additional intelligences. The naturalist intelligence has

such been identified since the original publication of MI theory. This intelligence deals with the

ability to identify, discriminate, and categorize various elements of the surrounding environment

(Gardner, 1999). Although elements of the naturalist intelligence have often been related to the

mathematical-logical and spatial intelligences, Gardner (1999) believed that the skills associated

with categorization and identification met the criteria for a unique intelligence and warranted the

addition of an eighth intelligence to the existing list.

Gardner’s supposition of eight distinct intelligences has led to new perspectives on what

intelligence actually is and how an individual’s cognitive ability can (and should) be assessed.

The identification of intelligences beyond linguistic and mathematical-logical skills has also

impacted the traditional notion of education and what it means to be intelligent. Although MI

theory was not necessarily created as an educational philosophy, Gardner’s work has had a

significant influence in the educational setting.

Educational implications. A model of cognition that recognizes a wide range of

intellectual competences leads to a perspective of teaching and learning that requires educators to

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rethink approaches to schooling. MI theory brings to light the implication that the traditional

focus on linguistic and mathematical-logical skills tends to relegate the development of the

remaining intelligences to other settings (Gardner, 1983). Gardner (1983) noted that the

importance of matching how students learn to how content is presented is an issue that educators

must continually grapple with in order to maximize learning. The upshot is that meeting the

unique learning needs of the students requires educators to recognize each of the eight

intelligences in the classroom setting. Gardner (1983) noted that no single intelligence is

superior to another; therefore, there is a valid place for each intelligence in the classroom.

Instructionally accounting for the various intellectual skills and strengths that each

student brings to the classroom can be beneficial for all students in the class. This is because a

second implication of MI theory is the premise that every typical individual possesses a distinct

combination of each of the intelligences (Gardner, 1999). Although the descriptive structure of

MI theory separately defines each of the eight intelligences, Gardner (1983) did not suggest that

the intelligences operate in isolation from one another. For example, an individual that plays the

piano has strengths in the musical intelligence. However, that individual must also have a

certain degree of strength in the bodily-kinesthetic intelligence in order to physically play the

instrument. It could also be argued that the individual may use skills associated with the

personal intelligences in order to perform a piece that emotionally resonates within or with an

audience. In most cases, individuals will draw from a variety of intelligences when asked to

complete more complex tasks and activities. Ultimately, instruction that taps into each of the

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intelligences can help more individuals connect to the content through channels that align with

their cognitive strengths.

In order to help educators conceptualize how to incorporate the various intelligences into

instructional activities, Gardner (1999) provided a list of “entry points” (p. 169) that can be used

to engage students in material in ways that reflect the various intelligences. These entry points

include: narrational, quantitative, logical, existential, aesthetic, hands-on, and social (Gardner,

1999). Gardner noted that the entry points are roughly aligned with the intelligences. For

instance, a story, anecdote, or movie utilizes the narrational entry point to engage the students in

the topic. The use of data, syllogisms, what-if questions, works of art or music, manipulatives,

and group-work represent some instructional techniques that exemplify the other six entry points,

respectively. The entry points to a lesson act as a pathway for naturally linking all types of

content to student interests and learning needs.

Gardner’s theory of Multiple Intelligences emphasizes that there are different ways to

characterize intelligence. Gardner used evidence from biology, psychology, and neuroscience as

well as cultural information to identify eight different intelligences. Like Dewey, Gardner

recognized that the traditional model of intelligence did not account for the various educational

needs of the students. Whereas MI theory presented a perspective on cognition that broadened

what it meant for an individual to be considered intelligent, Vygotsky’s notion of intelligence

provided insight regarding how intellect is developed and how it can be effectively nurtured in

educational settings.

Social Modeling and Vygotsky’s Zone of Proximal Development

20

Lev Vygotsky (1896-1934) was a psychologist who was best known for his theories

regarding social learning. Vygotsky suggested that individuals (children, in particular) learn

through their social interactions with others. Children are especially subject to the influences of

adult behavior and often learn through social modeling (Vygotsky, 1978). Vygotsky’s work (see

Figure 3) brought three important concepts to the forefront with regard to learning and classroom

instruction: the role of language in learning, social modeling, and the zone of proximal

development.

Role of language in learning. Language and speech act as tools in the learning process.

Vygotsky (1978) emphasized that language serves a purpose for transferring concrete

experiences into mental conceptualizations of experiences in the environment. An individual is

able to use the symbolization of language to create and internalize meaning which can be drawn

Figure 3. Summary of Vygotsky’s theory of social learning and educational applications.

21

upon in future situations. For Vygotsky, the nature of learning in a developing individual

transitions from reaction to the environment and the behaviors of others to interaction with the

environment and the behaviors of others. Language serves as the vehicle for this transition; and

“…the most significant moment in the course of intellectual development…occurs when speech

and practical activity, two previously completely independent lines of development, converge”

[italics by author] (Vygotsky, 1978, p. 24). The ability to use language significantly enhances the

learning process and leads individuals to more opportunities to acquire and construct new

knowledge. Whether or not an individual has reached the point where language and behavior

blend to redirect cognitive development, Vygotsky contended that learning always has a social

component.

Social modeling. The social nature of learning has a significant impact on the

development of an individual as an independent thinker. Vygotsky (1978) noted that younger

children react to their environment, and learning is constituted by effectively reacting to the

stimulus. Therefore, the role of others as behavior models is significant in helping younger

individuals make successful choices. As an individual continues to mature, the tools of speech,

written language, and memory allow the individual to think in abstract terms, plan ahead for

stimuli, and interact with the environment rather than simply react. Vygotsky (1978) stated that

“for the young child, to think means to recall; but for the adolescent, to recall means to think”

[italics by author] (p. 51). Throughout the developmental process, the social interactions that an

individual has with others (particularly those that are older and more experienced) are vital to the

cognitive maturation of the person.

22

Zone of proximal development. The social nature of learning has significant

implications for the development of classroom instruction. First, Vygotsky (1978) emphasized

that every individual has unique learning needs that must be attended to by educators. Second,

he also noted that educators must recognize that students come to the classroom setting having

acquired a vast amount of information through previous experiences and learning activities. It is

important to note that the accuracy, breadth, and applicability of that information may vary from

student to student. Nonetheless, the information that a student possesses will influence his or her

experience in the classroom. Finally, the role of the educator as well as student peer groups can

impact what an individual learns, how that individual learns, and at what pace the individual

learns. These influential factors can be addressed by recognizing what Vygotsky (1978) referred

to as “the zone of proximal development” (p. 85).

The zone of proximal development (ZPD) is the region of learning between what an

individual can accomplish successfully on his or her own and what that person can accomplish

with the assistance of someone who is more competent (Vygotsky, 1978). Recognition of each

child’s ZPD provides a great deal of information for teachers with regard to structuring

educational activities for students. Not only can teachers use the ZPD to see what has been

learned, but they can also see where the child is developmentally and can determine the direction

that the child is heading in cognitively. In addition, identification of a child’s ZPD can help an

educator determine the degree of support (or scaffolding) that he or she must provide in order to

help the student successfully navigate the ZPD toward independent completion of the given task.

The ZPD can serve as a guidepost for helping educators to effectively match the unique learning

23

needs of the students with activities and content that will serve as catalysts for continued learning

and cognitive growth.

Like Dewey and Gardner, Vygotsky recognized that each individual is unique and has

specific educational needs that must be met in order to maximize learning. Dewey approached

this fact by identifying the role of meaningful experiences in learning whereas Gardner

developed a theory that expanded the notion of intelligence to encompass a wider range of skills

and talents rooted in both the biological and cultural characteristics of people. Vygotsky’s

notion of learning was based on the social nature of learning and the importance of social

interactions in the cognitive transition toward becoming an independent thinker. Although

Michael Martinez’s viewpoint on intelligence and learning has commonalities with Dewey,

Gardner, and Vygotsky, his conceptualization emphasizes that intelligence can be learned.

Martinez, Learning Intelligence, and the 3E Model

As a more current theorist, Michael E. Martinez (1957 – ) has developed a model for

intelligence that builds upon and blends many of the ideas described by past theorists. Martinez

(2000) agreed with Dewey’s perspective on the role of experience in learning, and he supported

the broader inclusion of abilities that denote intelligence as proposed in Gardner’s theory of

Multiple Intelligences. Martinez (2000) also noted the social nature of learning as well as the

role of language in cognitive growth as described by Vygotsky. However, Martinez introduced

some new themes and conceptualizations of intelligence including: the 3E model of intelligence,

the ability for intelligence to be learned, the existence of a creative intelligence, and the

significant role of the environment in the development of intelligence (see Figure 4).

24

3E model of intelligence. Although the nature of intelligence can be conceptualized in

different ways, the 3E model of intelligence highlights three key characteristics: “intelligence is

entelic…, intelligence is efficient…, and intelligence is evaluative” [italics by author] (Martinez,

2000, p. 57). The entelic nature of intelligence is rooted in the social value placed on

intelligence and one’s ability to solve problems within the parameters of a given setting. The

Figure 4. Summary of Martinez’s theory of intelligence and educational applications.

25

efficient aspect of intelligence addresses a person’s ability to move information within the basic

structures of the cognitive system (i.e. quickly retrieving information from one’s memory). The

evaluative component of intelligence is based on an individual’s ability to apply his or her

existing cognitive schemas in thoughtful and proactive ways in a given situation (Martinez,

2000). The structure of the 3E model emphasizes that the level of one’s intelligence is defined

by a wide array of characteristics. Cognitive processing, the development and use of quality

mental schemas that lead to effective actions in novel settings, and the cultural value (including

values from other cultures) placed on particular problem-solving skills all influence how

intelligence is perceived and judged. Most importantly, though, the 3E model suggests that

intelligence is dynamic and can be developed through education.

Ability for intelligence to be learned. The 3E model depicts intelligence as a set of

“cognitive functions” (Martinez, 2000, p. 1). Martinez (2000) referred to these functions as the

“intelligence repertoire” [italics by author] (p. 57). By characterizing intelligence as a diverse

set of skills (problem-solving, information processing, and evaluation), Martinez suggested that

intelligence can be learned and developed through practice rather than simply being viewed as an

inborn, static ability. For the author, education serves as the vehicle for practicing those skills

that increase intelligence. This perspective creates a new role for the concept of intelligence in

the educational setting. Martinez (2000) stated that “intelligence is not just an input to

education, but also an output, or product, of educational experience” [italics by author] (p. 3).

The learnability of intelligence in conjunction with the skills, experiences, and cultural needs that

foster this learning suggest that there are different forms of intelligence that individuals employ.

26

Creative and other types of intelligence. Martinez identified several types of

intelligences that characterize different components of the 3E model. First, Martinez (2000)

distinguished between fluid intelligence and crystallized intelligence. Fluid intelligence relates

to one’s ability to adapt to new situations and process novel information being presented in those

situations (Martinez, 2000). Crystallized intelligence relates to the information that is transferred

to long term memory and the ability of an individual to draw upon that information in a given

situation (Martinez, 2000). Essentially, fluid and crystallized intelligences are descriptors of the

underlying processes of the efficient and evaluative portions of the 3E model. Martinez (2000)

also identified the practical and creative intelligences. Practical intelligence is generally drawn

upon when a person is functioning in everyday, real-world situations. Creative intelligence,

however, is utilized when a person is generating new ideas or products in problem-solving

situations. Martinez (2000) stated that “creative intelligence entails skills and attitudes that

enable a person to transcend the existing order to produce something new and culturally

significant…” [italics by author] (p. 39). Ultimately, the creative and practical intelligences

reflect a combination of the entelic and evaluative natures of intelligence outlined in the 3E

model. The types of intelligences identified as part of the 3E model suggest that the general

nature of intelligence is actually a diverse combination of skills and cognitive functions.

However, the development and use of those skills and cognitive functions are significantly

influenced by environmental factors.

Environmental influence. Martinez (2000) stated that “unequal achievement among

population groups is a source of unrelenting frustration to all who care about education and

27

social equity…” (p. 98). For Martinez, the educational achievement gap that has developed

among different cultural and ethnic groups in America is evidence that intellectual differences

are more related to environmental factors than to genetic predispositions. Thus, the traditional

conceptualization of IQ as well as traditional assessment methods do not adequately account for

environmental influences and only assess a narrow portion of intelligence. Due to the significant

role that the environment plays in cognitive development, a huge responsibility is placed on

educators to provide the necessary opportunities to help all students meet their potential.

Martinez (2000) emphasized that the right type of education—one that fosters higher-order

thinking and promotes relevant problem-solving for all students—is essential for “cultivating

intelligence” (p. 173).

Each of the theories developed by Dewey, Gardner, Vygotsky, and Martinez present a

different perspective on the nature of intelligence, learning, and educational practice. The

individual ideas do have some limitations with regard to putting theory into practice. However,

combining the strengths of each approach can lead to a structured model for evaluating

instructional strategies.

Synthesis of an Evaluation Framework

The theories of Dewey, Gardner, Vygotsky, and Martinez provide different perspectives

on intelligence, cognition, and learning. Since the nature of human cognition is complex, a

single theory may not sufficiently describe every aspect (or individual case) of intelligence and

may not provide enough information to develop comprehensive practical applications of the

theory in the classroom. However, combining the strengths of individual theories can lead to a

28

more robust foundation for designing, implementing, and evaluating instructional strategies.

Based on the strengths of each theoretical perspective, a framework consisting of three distinct

(yet interconnected) components—planning, instruction, and evaluation—for formulating and

assessing instructional activities will be developed (see Figure 5).

29

Figure 5. Visual representation of rubric for developing and assessing instruction.

30

Planning for Instruction

As Dewey (1938) noted, a truly educative experience must be one that provokes

reflection on the part of the student; and the teacher bears the responsibility of planning and

implementing classroom activities that provide opportunities for such experiences. Therefore,

planning for instruction must entail careful consideration of all aspects of the learning process.

This begins by identifying the learning goals for a particular lesson. Identifying goals is essential

to minimizing potential problems that may arise during the course of instruction (Gardner, 1983).

The learning goals frame the new knowledge that is intended for students to acquire by way of

instruction. Martinez (2000) described this new knowledge as the intelligence that is output via

education. However, Martinez also emphasized that intelligence has a dual nature as both a

product and a resource for learning. In addition, Dewey (1938) stressed that individuals may

learn other unintended things based on their unique natures. Therefore, it is essential to consider

the individual characteristics, intelligences, and learning needs of the students during the

planning phase.

Every learner brings a unique combination of prior knowledge, prior experience,

intellectual strengths (and weaknesses), and level of curiosity that will impact an educational

activity. Gardner (1999) identified eight “intelligences” (p. 3) that most individuals possess to

varying degrees of strength and/or development. An understanding of this profile of intellectual

skills can help an educator identify the types of activities and educational strategies that could

potentially prove to be the most effective for student learning (Gardner, 1983). At the same

time, this insight allows a teacher to make determinations about whether to capitalize on existing

31

intellectual strengths or to work toward developing less utilized intelligences. Martinez (2000)

referred to this development as the “cultivation of intelligence” (p. 112). An individual’s

cognitive predisposition is also complemented by his or her previous knowledge and

experiences. Dewey (1938) theorized that experience is both interactive and continuous. In

other words, every experience is intimately influenced by prior experiences and will, in turn,

influence future experiences. Those experiences also represent a direct interaction between an

individual and his or her surroundings. A combination of these experiences, intellectual

proclivities, and the natural curiosity of a student represents the intelligence input that Martinez

suggested is a necessary resource to promote further development of intelligence. Gardner

(1983) also emphasized the importance of determining which methods will best assist a student

to acquire the intended skill or knowledge. Ultimately, identifying what the students bring to the

classroom provides the necessary point of departure for instruction.

By identifying both the learning objectives for a lesson and the intellectual characteristics

of the students, it becomes possible to identify each student’s “zone of proximal development”

(ZPD) (Vygotsky, 1978, p. 85). A student’s ZPD is the gap between what that student can

already complete or understand independently and what that student can do with the assistance of

someone who is more knowledgeable given a particular concept (Vygotsky, 1978). In other

words, it is the span between the intellectual resources, experiences, and curiosity that the

student brings to the classroom and the intellectual growth that is anticipated as a result of

instruction. When an educator determines the learning objectives for a lesson, an anticipated

ZPD is established based on what he or she intends to have the students learn. Dewey (1910)

32

noted that teachers understand the purpose of the objective since they already understand the

meaning associated with that objective. However, the true ZPD for the students can only be

clarified (and maximized) as the unique intellectual characteristics are acknowledged and

factored into the choice of and implementation of instructional strategies. Vygotsky (1978)

emphasized that the ZPD for each student provides the pathway for gearing instruction to help

students effectively expand their knowledge about the concepts being addressed during an

educational activity.

The planning phase for designing instructional activities involves significant reflection

regarding the learning characteristics of students. Although the educational objectives for a

lesson define the intelligence “output” (Martinez, 2000, p. 3) that is anticipated for the activity,

the learning needs of the students play the crucial role in determining how to achieve those

objectives. The intelligence profile of the student, the level of curiosity, and previous

experiences all influence the student’s ZPD and define the parameters for the effectiveness of

instruction. Thus, the efforts made during the planning phase are essential in realizing effective

instruction in the classroom.

Implementing Instruction

Educational instruction comes in a plethora of forms. There are many additional factors

that influence how an activity is designed and, more importantly, implemented. The catch is that

instruction can lead (directly or indirectly) to a variety of different results. Dewey (1910) stated:

Some [teachers] succeed in arousing enthusiasm, in communicating large ideas, in

evoking energy. So far, well; but the final test is whether the stimulus thus given to

33

wider aims succeeds in transforming itself into power, that is to say, into the attention to

detail that ensures mastery over means of execution…. Other teachers succeed in training

facility, skill, mastery of the technique of subjects. Again it is well—so far. But unless

enlargement of mental vision, power of increased discrimination of final values, a sense

for ideas—for principles—accompanies this training, forms of skill ready to be put

indifferently to any end may be the result. (p. 220)

As Dewey further noted, the challenge for educators is finding the balance between creativity

and excitement for a subject and the ability to successfully apply skills in a meaningful way. For

educators, that challenge lies in carefully choosing and aligning instructional techniques with

learning needs of the students.

Gardner’s theory of Multiply Intelligences (MI theory) provides a structure for

identifying, organizing, and planning instruction. The eight intelligences—linguistic,

mathematical-logical, spatial, musical, bodily-kinesthetic, interpersonal, intrapersonal, and

naturalist—represent the routes that most people follow to process information, to interpret their

surroundings, and to solve problems. Therefore, instructional strategies that align with these

intelligences can provide more effective opportunities for students to learn the intended material

(Gardner, 1983). Gardner conceded that the demands of a typical classroom require educators to

make decisions about the use of instructional strategies that may not always match the exact

needs of every student. He further noted that teachers must decide whether the goals of a

particular lesson are better suited for playing to the strengths of the students or for promoting the

34

use of weaker intellectual capacities. In the end, making every effort to match teaching strategies

to the intellectual strengths of the students is important (Gardner, 1983).

MI theory can be used in several ways to structure educational activities. The eight

intelligences can be used as both the medium for instruction and as a source of content for

instruction (Gardner, 1983). Gardner (1999) did caution educators about how MI theory fits into

instructional design:

…I regard MI theory as a ringing endorsement of three key propositions: We are not all

the same; we do not all have the same kinds of minds; and education works most

effectively if these differences are taken into account rather than denied or ignored. (p.

91)

He went on to suggest that there are certain misconceptions about how MI theory can and should

be implemented in the classroom. For example, creating lessons that attempt to address all of the

intelligences at once, using intelligences as only pneumonic devices, engaging in trivial actions

or activities to trigger certain intelligences, and so forth are strategies that miss the fundamental

meaning behind the theory (Gardner, 1999). Instead, teachers must make careful choices about

how to design learning activities in order to create good matches between the needs of the

students and the instructional approach as often as possible. The set of eight intelligences can be

used a guide for assessing how a particular activity addresses various intellectual capacities.

Furthermore, the use of the eight intelligences as a design guide can allow educators to evaluate

the extent to which they are differentiating instruction throughout a course.

35

Gardner (1999) also identified “entry points” (p. 169) that educators can use in a lesson to

introduce concepts in ways that align with the various intelligences. Narratives, movies, art and

music, syllogisms, hands-on activities, pictures, analogies, data, philosophical and rhetorical

questions, social interactions, and so on are all strategies that align with the various intelligences

and can engage students in a topic along an intellectual avenue that matches with their particular

strengths and learning needs (Gardner, 1999). By having an understanding of the individual

learning characteristics of the students, teachers can select to use various activities as entry

points to a particular concept that will engage the students in the material whether or not that

material happens to align with an intellectual strength of the student. For instance, a math

teacher may use a hands-on approach to describe the characteristics of an ellipse by having the

students trace out the shape using paper, string, and some push pins. The teacher could also use

historical narratives to share how scientists like Johannes Kepler determined that the orbits of the

planets were elliptical in nature rather than circular. Furthermore, the teacher could rely on the

mathematical formula for an ellipse to explore the graphical nature of the shape. Ultimately, the

teacher must make careful choices about which strategies will help the students connect with the

intended content; and the appropriate selection of entry point activities can spark the necessary

interest in the student to make subsequent instruction successful.

In conjunction with the use of the eight intelligences as a guide for developing

instruction, there are also several tools that teachers can employ to direct student thinking.

Dewey (1910) advocated for the use of the scientific method to drive how students approach the

process of analyzing and synthesizing information. Analytical thinking allows an individual to

36

break down information, scrutinize details, and corroborate (or counter) what is assumed to be

true while synthesizing involves applying what is known to new and different situations (Dewey,

1910). The basic nature of the scientific method also fosters the use of inductive and deductive

reasoning. For Dewey, inductive reasoning involves assimilating details and smaller pieces of

information to identify more general, overarching themes and principles. Deductive reasoning

moves in reverse as one uses broader concepts to better understand and uncover details.

Dewey’s (1910) emphasis on the role of reflective thought in truly educative experiences hinges

on an individual’s ability to successfully navigate between inductive and deductive reasoning.

Opportunities for students to develop their reasoning skills are crucial in the development of

reflective thought. Learning activities that are rooted in relevant experiences can help students

engage in active reflection while working with content in a meaningful way (Dewey, 1938). As

teachers design instruction, emphasis should be placed on opportunities to utilize the scientific

method through experienced-based activities in order to foster reasoning that leads to truly

reflective thought.

Classroom instruction can take on many different forms. Lectures, films, group

activities, projects, and so on can all provide different opportunities for students to engage

material and expand their knowledge. However, an instructional technique may not always work

for every student every time it is used. Therefore, it is important for educators to consider the

diverse learning needs of the students as they plan instruction and implement classroom

activities. Gardner (1999) stated: “…if personalization is fused with a commitment to achieving

educational understandings for all children, then the cornerstone for a powerful education has

37

indeed been laid” (p. 92). Consideration of the tenets of MI theory, the strategic use of various

entry points for engaging students, and the role of scientific thinking in promoting reflective

thought can help guide the development of effective instruction. As Dewey (1938) noted, not

every experience is educational despite the initial intent. Therefore, it is essential that educators

take time to honestly evaluate instruction to determine if the intended learning goals were truly

achieved.

Evaluating Instruction

The ultimate goal of instruction is to help move students through their relative zones of

proximal development toward higher levels of knowledge and understanding. In order to make

this transition, educators must carefully consider the factors that influence the learning that

occurs. This begins with the characteristics of the students. Prior knowledge, past experiences,

intellectual strengths and needs, levels of curiosity, and so forth all influence the process of

developing intelligence; and these factors must influence the design and delivery of instruction.

As Dewey (1910) noted, though, reflective thinking is a spiraling process that brings an

individual to new knowledge and to new questions that foster new opportunities for acquiring

more knowledge. This same cycle must be echoed in instructional design. The efforts put into

planning, lesson design, and implementation will only be realized if the opportunity is taken to

evaluate the learning process.

No matter what level of formality is utilized to perform an evaluation of the learning

process, the key to success lies in honestly answering a series of questions:

38

Were the choices for entry points, intelligences (either as a medium for instruction or

as the content for instruction), and instructional tools appropriate? Gardner (1999)

emphasized the importance of finding good matches between instructional techniques

and the cognitive needs of the students.

What means were used to assess learning and were those means sufficient to

adequately judge student learning?

Were there opportunities for students to engage in reflective thought? Dewey (1910)

suggested reflective thought must occur in order for experiences to be truly educative.

The process of reflective thought also takes a person through the stages outlined in

Martinez’s 3E model for intelligence (2000).

Based upon the extent to which students were able to reflect, was the instructional

activity truly educative and geared toward the intended learning outcomes? Did the

student progress through his or her respective ZPD?

What types of intelligences were engaged by the students? Based upon Martinez’s

(2000) conception of intelligence, were the students asked to engage their fluid

intelligence (i.e. assimilating knowledge from new situations) or their crystallized

intelligence (i.e. drawing upon existing knowledge)? Did the students apply their

creative or practical intelligences?

What types of future experiences could (or will) this instructional activity lead to?

Dewey (1938) theorized that experiences are continuous and, therefore, influence

39

future experiences. It is essential for educators to consider how today’s instructional

activities have and will fit with previous and future experiences.

Careful reflection on the answers to these questions allows a teacher to assess the degree to

which a lesson meets the learning objectives that were initially set. In addition, this evaluation

can lead an instructor back to the planning phase for the next activity. The answers to these

questions can provide further clarity about the individual learning characteristics of the students

and can provide further knowledge about the experiences that the students bring to the classroom

as future activities are planned. Gardner (1999) noted the importance of knowing individual

students; and the evaluation process results in a better understanding of their learning needs. If

one of the primary goals of an educational activity is to promote reflective thought, then an

instructional design process rooted in the same features that define reflective thought will allow

educators and students to reach this goal.

The theories of Dewey, Gardner, Vygotsky, and Martinez provide a solid foundation for

structuring a rubric to evaluate an instructional activity. A blend of the strengths of each

theoretical perspective allows an educator to assess the degree to which a teaching strategy leads

to a truly educative experience for the student. The basic components of the various theories

have withstood the test of time and still prove relevant in today’s classroom. However, the

dynamic nature of society and the advances in technology have led to vast changes in the

classroom. Therefore, it is essential to examine how educational technologies fit into the

development of relevant instructional strategies.

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Incorporation of Technology into Evaluation Framework

The definition of educational technology can be difficult to clearly pinpoint. Saettler

(2004) noted that the terms that define educational technology come from theory and practice,

physical tools and conceptual ideas, and cultural needs and priorities. However, a basic

definition, developed by Heinrich (as cited in Saettler, 2004), that can be applied to general

situations is “…the application of our scientific knowledge about human learning to the practical

task of teaching and learning” (p. 5). Therefore, the role of technology in the learning process

can take on a variety of forms and can be used at all levels of planning, instructing, and

evaluating. The framework designed above to evaluate the effectiveness of a lesson incorporated

the strengths of the theories developed by Dewey, Gardner, Vygotsky, and Martinez. In order to

enhance the descriptive and evaluative capabilities of the framework, the role of educational

technology will be integrated into the process (see Figure 6).

41

Figure 6. Role of technology in rubric for developing and assessing instruction.

42

Technology and Planning for Instruction

Educational technology can play a variety of different roles in preparing and planning for

instructional activities. Various technologies can be used in ways ranging from tools for

organizing and analyzing information about students to the media for delivering instruction to

the focus of the lesson itself. Therefore, it is essential for educators to determine what types of

technologies are available and how those technologies will be used during teaching. Gardner

(1999) stressed that technology should not necessarily drive the discussion about learning goals,

but the resources that are available to teachers and students could influence how, when, to what

extent learning goals are met.

During the planning phase, there are a variety of considerations that should be addressed

to determine how to effectively integrate technology into the learning process (see Figure 7a):

What resources are available to both the teacher and the student and how will the

availability of those resources influence instructional decisions? Gardner (1999)

emphasized that the availability of technology itself does not guarantee effective

learning. The question of purpose for that technology must be answered.

How can technology be used to assess the prior knowledge and experiences of the

students?

Do the students have previous knowledge/experience working with a proposed

technological application? How will that previous knowledge (or lack thereof)

influence instruction and the nature of the learning goals for a lesson?

43

Does a purposed technology relate to the particular intelligences or learning needs of

the students?

Can the proposed use of technology spark curiosity in the students? Dewey (1938)

noted that experiences that spark curiosity are essential for building inertia in a

student to continue exploring concepts.

Another important consideration is related to the zones of proximal development for the

students. Martinez (2000) concluded that “…the cognitive proficiencies associated with

intelligence can be enhanced through direct intervention” (p. 167). Therefore, the instructional

Figure 7a. Technology considerations during the planning phase.

44

strategies and educational technologies chosen by the teacher will directly influence student

learning. The ZPD for a student provides an indication of what that student is capable of

learning with the assistance of others who are more knowledge; and effective use of technologies

can help scaffold student learning during the progression through his or her ZPD and beyond

(see Figure 7b). During the planning phase, it is important for educators to assess how the

various types of educational technologies available to them will most effectively enhance

instruction and aid student learning.

Technology and Implementing Instruction

What role does educational technology play in instruction? Gardner’s theory of Multiple

Intelligences provides a framework for designing learning activities and classroom instruction.

A fundamental tenet of instructional design based on MI theory is the assumption that students

bring diverse intellectual strengths and capacities to the table (Gardner, 1999). In addition,

Martinez (2000) emphasized that well-chosen interventions can help students develop their

intelligence. Therefore, instruction must be differentiated in a manner that reflects student needs.

The further infusion of elements endorsed by Dewey (1910)—the scientific method, deductive

Figure 7b. Technology considerations and the zone of proximal development.

45

and inductive reasoning, and experienced-based learning—into this process can lead to truly

educative experiences for the students. The use of educational technologies can significantly

enhance the delivery and impact of instruction (see Figure 7c). Gardner (1999) wrote: “…we

have in our grasp today technology that should allow a quantum leap in the delivery of

individualized services for both students and teachers” (p. 179). In order to determine how

technology can aid in instruction, the role of each chosen technology must be clearly defined for

each aspect of instruction from the initial entry point (see Figure 7d) into a particular concept to

the learning activity to the opportunities for experiences and reflective thinking on the part of the

students.

Figure 7c. Technology considerations and instructional design.

46

Technology and Evaluating Instruction

As with the planning and instruction phases of a lesson, technology can be both a tool for

and the focus of the evaluation of that lesson. From Vygotsky’s (1978) theoretical point of view,

an important question addresses whether or not the instruction promoted development in the

student. Dewey (1910) emphasized the importance of determining whether or not the experience

of the student was truly educative as well as the importance of reflective thought in leading to

further educative experiences For Martinez (2000), an effective educational experience should

lead to the “cultivation” (p. 6) of the student’s intelligence. A corollary to Gardner’s (1999)

theory addresses how well instruction aligns with the individual needs of students. In evaluating

the answers to these questions, the role of technologies must be considered (see Figure 7e and

Figure 7f). Did the use of technology make it possible to meet the learning goals of the lesson?

Did the technology serve as a useful mechanism or support for assessment? Did the use of a

technology meet the goals behind the intended purpose of that use? Gardner (1999) noted that

technology cannot replace the intellectual work performed by the student. Therefore, did the use

Figure 7d. Technology considerations and entry point activities.

47

of technology support or supplant student thinking? Ultimately, the potential of various

educational technologies is limitless. However, careful evaluation of the role of that technology

in instruction is essential to ensure that student learning is enhanced and not detracted from.

Figure 7e. Technology considerations and Martinez’s 3E model of intelligence.

Figure 7f. Technology considerations, future experiences, and assessment.

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Conclusion

The learning, intelligence, and educational theories of John Dewey, Howard Gardner,

Michael Martinez, and Lev Vygotsky provide in-depth and diverse perspectives on human

cognition. Whether one chooses to focus on the nature of experience in learning, the gamut of

intellectual capabilities inherent in the human experience, the learnability of intelligence, or the

role of social interactions in the advancement of knowledge, the common thread appears to lead

back to the unique cognitive needs of individuals. This theme has a significant impact when it

comes to educational design; and a blend of the strengths of each theoretical perspective can

provide educators with the necessary strategies and guidance to plan, design, implement, and

evaluate instruction. In addition, the advancement of educational technologies brings additional

tools to the table that educators can draw from to maximize the learning potential for students in

the classroom. Gardner (1999) stated: “…when it comes to learning, using our minds well, and

informing others and being informed by others, there need be no limitations. Knowledge need

not be competitive; we can all increase our own knowledge and the knowledge of others…” (p.

218). The realization of this vision is nurtured in the individual and can be brought to fruition

through the careful, thoughtful, and purposeful design of educative experiences.

From Theory to Research

The original intelligence, learning, and educational theories of John Dewey, Howard

Gardner, Michael Martinez, and Lev Vygotsky provide a strong framework for designing

curriculum and for approaching classroom instruction. The framework built based upon these

theories highlights many of the important considerations that an educator must address while

49

preparing for instruction. In addition, the framework also integrates key questions about the role

of current technology with the fundamental theories about learning. However, simply asking the

questions about the infusion of modern technology into current pedagogy does not necessarily

guarantee sufficient answers. Therefore, reviewing current research on the role and use of

technology in the classroom (and particularly the mathematics classroom) can shed light on how

to adequately balance what has been known for decades about learning with the latest

technologies that are sparking the evolution of the 21st century classroom.

50

Depth

SBSF 8220: Current Research in Human Development

Annotated Bibliography

Abramovich, S. (2005). Early algebra with graphics software as a type II application of

technology. Computers in the Schools, 22(3/4), 21-33. doi: 10.1300/J025v22n03_03

Summary. The article addressed the use of graphics software as a tool for developing

algebraic thinking in elementary students. In the study, students working in groups of two were

presented with age-appropriate problems that were linked to the mathematical skills associated

with solving systems of linear equations. They also were given access to KidPix™, a graphics

program, to aid in visualizing the problems. During the first session, students were provided

assistance by a tutor who could answer questions, address technical issues, and ask guiding

questions to support critical thinking. During the second session, the students were given similar

questions and were asked to solve the problems with minimal support from the tutor.

Abramovich (2005) found that the graphics software served as a medium for the students to

convert their intuitive ideas about the problems into concrete visualizations. More importantly,

observations made regarding student interactions and the problem-solving steps taken by the

students led to a better understanding of the mathematical reasoning utilized by the children.

Ultimately, the appropriate use of graphics software in the problem-solving process proved to be

an effective tool for students in developing, visualizing, and communicating mathematical

reasoning.

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Critical analysis. Two key ideas were mentioned in the description of the study: Type II

applications of technology and Vygotsky’s concept of the zone of proximal development. First,

a Type II application is a situation where students are mainly responsible for their interactions

with the technology (Abramovich, 2005). It was suggested that these types of applications can

enhance and shed light upon the nature of student thinking during an activity. In this situation,

KidPix™ served as a tool that helped students communicate their reasoning in the problem-

solving process. The use of Type II applications help to elevate the use of technology to a level

where students utilize the tools to enhance their own cognitive processes rather than to rely upon

those tools as crutches.

The second concept emphasized in the article dealt with the zone of proximal

development—the cognitive zone where students are ready to move from scaffolded support for

learning to independent thought (Abramovich, 2005). The effective use of appropriate

technological applications can aid students in the journey to independent understanding of a

concept. In addition, observations of a student’s progress can aid educators in determining when

that student is ready to tackle more advanced aspects of the concept. Abramovich noted that

mathematics curricula is often repetitive with regard to fundamental ideas; and the identification

of the a child’s zone of proximal development for a given topic can help guide progress from

concrete to more abstract lines of thought.

Statement of value. This article highlights several important issues being address in this

KAM. First, the study highlights the role of theory, particularly the concepts described by

Vygotsky, in assessing the cognitive development of students. Second, identifying how a

52

technology-based application is utilized by the student (i.e. a Type II application) is essential in

order to effectively and appropriately enhance student learning. Finally, technology can play an

important role at all age levels in promoting advanced mathematical reasoning.

Ali, R. M., & Kor, L. K. (2007, May). Association between brain hemisphericity, learning styles

and confidence in using graphics calculator for mathematics. Eurasia Journal of

Mathematics, Science & Technology Education, 3(2), 127-131.

Summary. The basic premise of the study conducted by Ali and Kor was to investigate

the relationship, if any such relationship existed, between brain hemisphericity, individual

learning styles, and the level of confidence associated with using technological applications in

the mathematics classroom (2007). The participants were pre-service teachers and mathematics

students who enrolled in a college course that dealt with the role of technology in mathematics

curricula. Data was collected through separate questionnaires that addressed brain-dominance

and confidence using graphing calculators. Although the preliminary findings did not suggest

anything conclusive about a relationship between confidence in using graphing calculators and

brain hemisphericity or learning style, the authors concluded that the results do show some

support for the characteristics often associated with left- or right-brained individuals (Ali & Kor,

2007). For example, left-brained individuals are often characterized by sequential thinking

whereas the thought processes of right-brained people tend to be more global. Ultimately, the

authors advised that more research can and should be conducted to explore how an

53

understanding of learning styles and brain hemisphericity can enhance learning in the math

classroom.

Critical analysis. The authors stated: “that there were no significant differences in GC

[graphing calculator] confidence across brain hemisphericity as well as learning styles” (Ali &

Kor, 2007, para. 10). The lack of a significant, statistical correlation actually provides insight

into how mathematics educators can incorporate graphing calculator technology into the learning

process. The results seem to show that a student with any type of learning style or dominant

approach to thinking through a problem can effectively learn how to utilize graphing calculators

in the problem-solving process. In addition, the study provided additional support for potential

ways to identify and define how individuals think. Mathematics teachers can utilize this insight

to effectively design instruction.

Statement of value. The article contributes to this project by addressing additional

forms of technology that can be utilized in the mathematics classroom. The study also provides

support for developing connections between mathematical thinking and the various learning

styles that individuals possess. Most importantly, the data seems to suggest that individuals with

various learning styles can effectively utilize technology to enhance and promote mathematical

learning and reasoning.

Berry, J., Graham, E., & Smith, A. (2006). Observing student working styles when using graphic

calculators to solve mathematics problems. International Journal of Mathematical

Education in Science & Technology, 37(3), 291-308. doi: 10.1080/00207390500322009

54

Summary. Graphing calculators are an innovative, hand-held technology that has the

potential to change mathematics education. Although the technology has been available for

many years, there are still many questions regarding the effective use of graphing calculators in

instruction and learning. The article written by Berry, Graham, and Smith (2006) described the

development of a software program, key-recorder, that could be used to further analyze how

students use graphing calculators in a variety of settings.

The goal of the research study was to develop an unobtrusive technique that could be

used to gather data about student calculator use. The authors asked: “…how can we effectively

observe student’s working in a naturalistic way with graphic calculators in mathematical

problem-solving situations?” (Berry, Graham, & Smith, 2006, para. 20). In order to answer this

question, three pilot studies were conducted. The goal of these studies was to provide evidence

as to whether or not the software would yield quality data about student use of the calculators. In

short, the authors concluded that the key-recorder software could be used in a variety of different

settings to gain insight into how students used graphing calculators. However, it was noted that

data obtained with this tool would be only one piece of the puzzle in analyzing the mathematical

reasoning used by students in the problem-solving process.

Critical analysis. Although the primary focus of the article involved a description of a

new technique that can be used to obtain data regarding how students use graphing calculators,

there were several additional concepts that evolved out of the article. First, the authors noted the

importance of identifying the mathematical “working styles” (Berry et al., 2006, para. 4) of

students. In the problem-solving setting, students may come up with a correct answer; however,

55

it may be difficult to pin down the reasoning process that they employed. The ability to

determine how a student thinks through a problem can help educators better understand the

mathematical development of that student. Second, the authors posed a variety of questions that

must be considered when addressing the effective integration of technology in the classroom.

Third, the pilot studies revealed a variety of issues that educators might encounter including:

problem-solving based on trial and error, dependence on technology, lack of critical reflection,

and so on. Finally, educators must develop new ways to encourage the effective use of graphing

calculators in the problem-solving process.

Statement of value. By identifying some of the key issues associated with the effective

use of calculators in the problem-solving process, the article provided support for identifying and

capitalizing on the learning styles of students in the mathematics classroom. In addition, the use

of quality data can aid in the process determining the most effective means for integrating

technology in mathematics instruction. The article also pointed out potential gaps in existing

research dealing with the role of technology in the process of learning and developing

mathematical skills.

Bruce, B. (1998, November). Dewey and technology. Journal of Adolescent & Adult Literacy,

42(3), 222-227.

Summary. Technology is quickly evolving and is changing the educational landscape.

Although different types of technological applications are providing new, unprecedented

opportunities in learning, Bruce (1998) addressed how the classical educational philosophy of

56

John Dewey can still be applied to today’s classrooms. The basic philosophy of Dewey is

centered on the experiences of the students; and learning tools including subject matter, books,

lab equipment, and/or technologies that are employed in the learning process need to be used in a

manner that enhances the quality and understanding of personal experience (Bruce, 1998). In

order to better understand the role of technology in learning, Bruce suggested several questions

that should be asked by teachers. These questions include: “In what ways is the experience

afforded by interaction with a computer a substitute for other modes of learning? Does it

[technology] provide new avenues for experience and the means to access previously

inaccessible realms?” (para. 18). Ultimately, the author argued that the foundation of Dewey’s

philosophy should lead educators to use technology as a means for helping students learn

through experience rather than as a learning end in itself.

Critical analysis. In order to create learning environments that truly benefit student

learning, educators must find a way to balance the ideals of educational philosophy with the

practical concerns of classrooms that exist in the 21st century. It can often be difficult to make

time to read the works of theorists written nearly a century ago. However, Bruce (1998)

delineated how the ideas promoted by Dewey can still be applied in today’s educational setting.

With regard to technology, a critical question that teachers must ask involves whether or not the

use of technology is truly aiding in the learning process. Although the assumption tends to be

that use of technology will automatically lead to quality learning experiences, educators must

still reflect upon how the use of technology is effectively integrated into the learning process.

Bruce (1998) stated that “we need to learn technology, to learn through technology, and to learn

57

about technology” (para. 5). In the end, though, technology must be infused in the educational

process in a manner than effectively contributes to each student’s specific learning experiences.

Statement of value. The issues addressed by Bruce (1998) help to bridge the gap

between educational philosophy and the role of rapidly changing technological applications in

today’s learning environments. The author provided examples of important questions rooted in

the philosophy of John Dewey that educators must ask in order to ensure that the use of

technology is relevant, appropriate, and useful. The information in this article can be used to

link the theoretical foundation developed by classic theorists with the current research on

educational technology.

Cohen, V. L. (1997, Summer). Learning styles in a technology-rich environment. Journal of

Research on Computing in Education, 29(4), 338-351.

Summary. An exploratory study was conducted by Cohen (1997) to look at the

connection between individual learning styles and the role of technology in the learning process.

A small group of ninth-grade students attending a magnet school were observed, interviewed,

and administered a learning style inventory in order to assess how a technology-rich learning

environment impacted the learning process. The magnet school centered on several important

themes/philosophies: 1) the school was focused on math, science, and technology content; 2)

technology was to be infused into almost every part of the learning experience; 3) students were

expected to work in team-oriented, group situations; and 4) constructivism was the educational

philosophy that drove the instructional process.

58

Although the researcher acknowledged the limitations of the study, several important

conclusions were drawn. First, the use of technology as part of instruction has an impact on how

the curriculum is presented as well as the hidden curriculum that is infused throughout all aspects

of the educational setting. Cohen (1997) noted several positive examples including the

integration of new ways of viewing and analyzing the subject matter, new presentation

techniques, and enhanced social interactions between both teachers and students. Second, the

use of technology in the classroom was not an end in itself. Many students still wanted to be

challenged by the content and problems associated with that content. Finally, the learning styles

of students and the teaching styles of the educators must be considered carefully as instructional

activities (technology-based or otherwise) are developed.

Critical analysis. The study conducted by Cohen (1997) provides a starting point for

exploring the impact of both the learning styles of students and the infusion of technology in the

classroom. The study did have several inherent limitations such as the size of sample group (15

students), the composition of the sample, and the specified nature of the learning environment.

However, several key themes emerged that educators must continue to explore as they develop

instructional activities in the classroom. First, the characteristics of each student’s learning style

must be addressed, acknowledged, and considered in instructional planning. Each student has

learning preferences that will influence his or her performance in various learning situations

(Cohen, 1997). Second, the changing nature of the global community requires that technology-

use becomes an integral part of the learning process. However, the use of technology is not an

end itself. Technology is a tool that aides the cognitive processes associated with learning and

59

problem-solving (Cohen, 1997). Student learning styles and the role of technology in the

classroom are important considerations for educators to address; however, simply

acknowledging student differences and/or the random use of technology is not enough. Careful

thought must be used in developing curriculum, classroom activities, and instructional methods

to ensure that the individual strengths of students and the use of technology as a “cognitive tool”

(Cohen, 1997, para. 4) will maximize student learning.

Statement of value. This article provides support for addressing the role of learning

styles in the development of classroom activities in the mathematics classroom. In addition, the

observations made by the author suggest that technology-use can have a significant impact on

how and what students learn. The key lies in determining how to effectively apply both the

knowledge of the students’ learning needs as well as the best ways to integrate technology into

the learning process in order to develop instruction that will enhance learning. The information

in the article not only suggests that these two characteristics of the classroom setting impact the

learning process, but it also establishes a solid link between them in that the selection of

technology applications must reflect the learning needs of students and vice versa.

Cohen, V. L. (2001, Summer). Learning styles and technology in a ninth-grade high school

population. Journal of Research on Computing in Education, 33(4), 355-367.

Summary. The study conducted by Cohen (2001) utilized data from Dunn and Dunn’s

Learning Style Inventory (LSI) in conjunction with interviews to assess the impact of the

classroom environment on student learning styles. The reactions of students from two high

60

school environments—a regular, traditional setting and a magnet school focusing on

mathematics and science—were compared in an effort to determine how the classroom setting

affected learning styles. Students who attended the magnet school were exposed to project-based

learning that emphasized the constructivist approach to learning. The projects were problem-

based, required the use of technology, and focused primarily on math and science topics. In

general, the setting provided students with more freedom yet required them to take on more of

the responsibility for learning the content. The students who attended the regular high school

followed more of a traditional routine with regard to the daily schedule, content and course-load,

participation in extracurricular activities, and so on.

The interview data referenced in the study demonstrated that both groups of students

expressed pros and cons regarding their respective learning environments. However, students

who attended the magnet school tended to find the content and learning activities to be more

relevant to real-world applications (Cohen, 2001). In addition, they valued the role that

technology played in the learning process. Cohen noted that students in the regular high school

expressed greater concerns over managing the daily schedule and interacting with the various

teachers. The students also mentioned that they hoped for more technology usage in the

classroom. Ultimately, the author concluded that the school environment can have an impact on

shaping how students learn.

Critical analysis. Cohen (2001) presented evidence to support the idea that the learning

environment can influence the learning process. Although the authored conceded that there

exists evidence that different individuals have certain strengths and weaknesses, likes and

61

dislikes, and/or preferences when it comes to content and the process of learning that content, the

setting can influence those preferences. Cohen suggested that technology usage, motivational

factors, certain instructional strategies, and so on can have an impact on student learning. The

article does not present a significant amount of data to support the direct connection between the

learning environment and the learning styles of students; however, the author raised interesting

points regarding the interconnectedness of learning styles, content, relevance to everyday life,

and technology. Development of strategies that reflect these concepts in addition to the use of

problem-based learning can help students find relevance in the material and develop a sense of

how today’s content is linked to what they will learn and see in the future.

Statement of value. This article demonstrated the connection between learning styles,

technology, and instructional methodology. One goal of this KAM involves developing

strategies that will enhance learning in the algebra classroom. The article suggested that the

appropriate use of technology and problem-based learning can help students find relevance in the

content being presented. In order for learning to be effective, students must have opportunities

to work with concepts that are tied to real-life applications and the relevance they hold in

everyday settings.

Dahl, B. (2006, Spring). Analyzing cognitive learning processes through group interviews of

successful high school pupils: Development and use of a model. Educational Studies in

Mathematics, 56(2/3), 129-155.

62

Summary. The primary focus of the study conducted by Dahl (2006) involved the

development of a model that could be used to assess the cognitive processes of students who are

learning mathematics. The CULTIS (Conscious-Unconscious-Language-Tacit-Individual-

Social) model was a conglomeration of classic theories that incorporated the ideas developed by

Piaget, Vygotsky, Hadamard, and others. The author’s description of the study included a

summary of the key components of each theory and a description of how the theories fit into a

dynamic model for describing the cognitive aspects of learning. In addition to the development

of an analytical model, the author also presented data and conclusions regarding the students

involved in the study.

Dahl (2006) identified several key themes that were highlighted through observations and

interviews including: the role of planning in preparing to learn new mathematics concepts; the

need for opportunities to think about concepts and to reflect on the process; the importance of

language in communicating and processing new ideas; the function of schematic learning in

assimilating and accommodating new information; and the role of social interactions in the

learning process. In addition, a new concept was derived outside of the framework of the model

which the author identified as the “zone of proximal teaching (ZPT)” (Dahl, 2006, para. 39).

ZPT aligns with the zone of proximal development (ZPD) developed by Vygotsky. Essentially,

Dahl noted that ZPT provides an indication that if a teaching style is too unfamiliar to a student,

learning may be difficult. Finally, the role of metacognition was emphasized in the research. If

students are able to think about their cognitive activities, strengths, and needs, then they will be

more likely to find success in the learning process.

63

Critical analysis. The research conducted by Dahl provides insight into identifying how

students learn mathematics. First, the CULTIS model is a potential tool that both students and

teachers can use to gain a better understanding of how students learn (Dahl, 2006). Dahl noted

that if teachers can clearly identify how students learn, they will be able to provide training about

useful strategies to students and/or incorporate applicable techniques into their own instruction.

Second, the results of the research pointed to the important role of metacognition in grasping

new mathematical concepts. Students need opportunities to reflect upon their own processes in

order to solidify their conceptualization of new ideas. Finally, the research draws on a wide

range of classic learning theories which allows educators to incorporate and apply a wide range

of ideas to instruction, assessment, and reflection.

Statement of value. An important aspect of this research is the connection that is made

between current issues in mathematics instruction and the foundational theories that drive

educational philosophy. The analytical framework developed by Dahl (2006) provides a

structure for assessing how students learn mathematics. Although the study has limitations with

regard to scope and size of the sample population, the results provide a significant amount of

food for thought as educators address current needs in mathematics instruction and student

learning in a challenging content area.

Debevec, K, Shih, M., and Kashvap, V. (2006, Spring). Learning strategies and performance in

a technology integrated classroom. Journal of Research on Technology in Education,

38(3), p. 293-307.

64

Summary. The primary goal of the research study conducted by Debevec, Shih, and

Kashvap (2006) was to determine the role of technology-based applications in the learning

process. Since the process of developing multimedia learning activities requires a significant

amount of time, educators need to determine if that time is well-invested and enhances student

learning (Debevec, Shih, & Kashvap, 2006). The researchers developed several questions to

address this problem. The researchers tried to determine how much students utilized the

technology-based resources, the impact of the availability of these resources outside of classes on

attendance, the influence on tests scores, and the relationship between technology-based

strategies and more traditional techniques (Debevec et al., 2006).

The study was conducted on a college campus. Seventy-nine participants from two

sections of the same business class were used in the study. The data that was collected included

attendance records for the semester, exams scores, and an online survey conducted at the end of

the course. Students were provided extra credit as an incentive for completing the final survey.

Researchers used the survey to determine how the applications were used by the students in

order to categorize them based on preferred learning strategies. The authors used a variety of

statistical methods to summarize and analyze the data. Based on the results of the study, the

authors derived several conclusions. First, they confirmed that attendance is an essential

component of success in the classroom. Second, the results seemed to indicate that students

tended to use study notes, PowerPoint presentations, and online quizzes to prepare for class and

exams, occasionally to the extent of not utilizing the information provided in textbooks. Finally,

65

the researchers suggested that there are a variety of strategies that can successfully be used to

enhance student learning.

Critical analysis. One goal of this KAM is to analyze the role of technology in the

learning process. Specifically, a question that arises addresses whether or not the use of

technology has a greater impact on learning than more traditional techniques. The study

conducted by Debevec, Shih, and Kashvap attempts to answer this question. This study builds

upon previous research that suggests that technology has a positive impact on more traditional

approaches to teaching and learning. However, earlier research, as noted by the authors, also

suggests that technology use does not necessarily work better than more lecture-based strategies.

The authors used an appropriate quantitative approach to collect, summarize, and analyze the

data. In addition, the authors clearly outlined the method they followed to conduct the study

which makes it possible to replicate. One limitation of the study that may affect the ability to

generalize the conclusions is the relatively small sample size. The authors do provide a limited

description of the demographics of the participants, but it may be difficult to generalize the

conclusions to other students, other age groups, or other learning settings.

That being said, the conclusions from this study seem to corroborate previous study

results. In particular, students learn in a variety of ways. Some individuals learn better in more

traditional settings, whereas others prefer to use different techniques. Educators that tend to use

a variety of teaching techniques can help a majority of students find success in the classroom.

The information from the study seems to suggest that the pursuit of more research on technology

use in the classroom is important.

66

Statement of value. This article is useful for building upon the existing body of research

with respect to technology use in the classroom. Although the data corroborates previous results,

the authors of the study recognize the limitations of the sample size and the types of applications

that were used in the experimental study. Continued research is essential to determine how

technology can be successfully used in the learning process.

Dunn, R., Beaudry, J., & Klavas, A. (1989). Survey of research on learning styles. Educational

Leadership, 46(6), 50-58.

Summary. Dunn, Beaudry, and Klavas (1989) reported on the wide range of research

dealing the role of learning styles in the educational process. The authors commented primarily

on studies conducted during the 1970’s and 1980’s. In general, the authors highlighted several

key themes. First, every person has a unique learning style (Dunn, Beaudry, & Klavas, 1989,

¶2). This learning style is a combination of biological, environmental, and social factors.

Second, research has provided evidence of the existence of learning styles and the importance of

aligning instructional methods with the learning strengths of students. The authors stated: “…

when youngsters were taught with instructional resources that both matched and mismatched

their preferred modalities, they achieved statistically higher test scores in modality-matched,

rather than mismatched, treatments” (Dunn et al., 1989, para. 16). Third, there are many

perspectives from which to view learning styles, but no learning style is better, more productive,

or more effective than another (Dunn et al., 1989). Finally, an emphasis was placed on the

necessity for educators to find ways to adapt classroom instruction to meet the learning needs of

67

the students. Although diverse groups of students can make this process challenging, the use of

multiple instructional strategies that tap into various learning styles can help all students be more

successful in any classroom setting.

Critical analysis. The information presented by the authors provides a good summary of

research dealing with the influence of learning styles on achievement in the classroom.

However, a limitation of this article is the age of the research. The studies surveyed by the

authors were between 25 and 40 years old. Although the specific data may be outdated, the

themes derived by Dunn et al. (1989) are still relevant in today’s classrooms. These studies

provide a bridge between the classic theories and current research dealing with mathematics

instruction, learning styles, and technology. In addition, stability of the educational themes

associated with learning styles, instructional strategies, multiples modalities, and so on provide

support for continued investigation into the connections between the unique learning needs of

students, teaching techniques utilized by educators, and the role of current

technologies/resources in the classroom.

Statement of value. Although the study itself is somewhat outdated, the key issues

outlined in the summary demonstrate the significance of identifying the best methods through

which each student learns. The studies surveyed by the authors also establish the connection

between classic learning theories and more modern views on intelligence and learning. The

themes establish a framework for continued research and investigation regarding how the

resources, issues, and professional demands of today align with the learning strength and needs

of students.

68

Evuleocha, S. U. (1997, June). The effect of interactive multimedia on learning styles. Business

Communication Quarterly, 60(2), 127-129.

Summary. Meta-learning is the primary idea emphasized in the article. Meta-learning

involves two important activities: (1) the use of various effective teaching techniques that allow

for the alignment of instruction and student learning styles; and (2) the sharing of skills between

both the instructor and the students (Evuleocha, 1997). The author noted that the changing role

of technology in the educational process is making the premise of meta-learning even more

important. In order to make classroom instruction relevant to what students will experience in

the real-world, students and teachers need to work alongside each other to learn how to use,

apply, and integrate the latest technology. Evuleocha emphasized that meta-learning in

conjunction with the incorporation of new technologies in the classroom can strengthen the

significant role of the individual learning styles of students on achievement. Ultimately, the

author noted that teachers should embrace the notion of meta-learning in order to make sure that

classrooms can continue to adapt to the changing demands of today’s society.

Critical analysis. The idea of meta-learning is an important concept that can serve as a

framework for blending a wide variety of demands placed on educators in today’s classrooms.

As noted by the author, meta-learning allows educators to incorporate the diverse learning styles

of students into the development of instruction (Evuleocha, 1997). In addition, the philosophy of

meta-learning can also allow educators to make the necessary transition to multimedia-based

instruction in a smooth, yet quick manner. The current and future demands of a technology-

69

based global community require education in today’s classroom to prepare students for what they

will encounter in the real world. Evuleocha (1997) stated that “a meta-learning environment

bolstered by interactive multimedia is better suited for teaching and learning in classrooms of the

twenty-first century because of the mutual exchange that occurs” (para. 6). The type of

cooperation between teachers and students that is exemplified by meta-learning can allow

educators to develop instructional activities that capitalize on the learning styles of students, the

advantages of technological resources, and the sharing of ideas among those involved in the

learning process.

Statement of value. The concept of meta-learning adds an additional piece to the

framework that can be used to design instruction that builds on the unique abilities and needs of

the students. In addition, the use of multimedia techniques can be integrated into the overall

learning process. The effective use of technology can not only prepare students for what they

will experience after school, but it will also provide a way to enhance and support the individual

learning styles of each student.

Forster, P. A. (2006). Assessing technology-based approaches for teaching and learning

mathematics. International Journal of Mathematical Education in Science & Technology,

37(2), 145-164. doi: 10.1080/00207390500285826

Summary. The use of technological applications to assist in the educational process

continues to expand throughout classrooms worldwide (Forster, 2006). A question, though, that

requires continued study involves the effective use of that technology in learning. The study

70

conducted by Forster assessed how technologies, particularly Java applets, spreadsheets, and

graphing calculators, could be used to enhance instruction in the area of statistics. The literature

review conducted by the author revealed that the use of technology has pros and cons. Several

important considerations related to these advantages and disadvantages involved: the

instructional methods utilized by the teacher; the background knowledge of the students

regarding both the use of the technology as well as the content; the logistical parameters of the

learning environment such as time and the availability of technology; and so on.

The observations and analysis made by Forster appeared to support many of the findings

in the literature. The author found that educators must find appropriate ways to integrate

technology into the learning process. The use of technology does require that time and attention

are paid to teaching students the necessary skills for inputting information, carrying out

computations, and interpreting the results generated by the technology (Forster, 2006). In

addition, there are situations were direct instruction, completing work by-hand, and teacher-

centered approaches are necessary. However, the use of open-ended questioning techniques,

problems set in relevant, real-world situations, and visual aides to complement more abstract

ideas enhance the understanding of mathematical concepts (Forster, 2006). Ultimately, the

author concluded that technology will aid in the instructional process assuming that the

considerations dealing with effectiveness, appropriateness, logistics, and student needs are

addressed.

Critical analysis. The conclusions reached by the author support the general theme that

the role of technology-use in instruction requires educators to carefully assess how that

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technology will enhance student learning. In the mathematics classroom (as well as other

classrooms), technological applications provide a plethora of avenues for presentation,

interaction with content, analysis of data, and the sharing of ideas. For example, Forster (2006)

noted how the graphics capabilities of calculators and/or spreadsheets can help students visually

understand the relationships among various representations of data. However, technology alone

will not account for improved instruction or enhanced learning. Technology must be integrated

with clear purposes, knowledge of student needs, ties to sound educational pedagogy, and

appropriate consideration of the parameters of the learning environment. The study adds

continued support for the use of technology in the classroom setting and emphasizes the

important connection between the capabilities of various applications and sound educational

philosophy in various content areas.

Statement of value. Assessing the role of technology in the classroom requires

an educator to consider a variety of different issues that will influence the effectiveness of

that technology as a learning tool. Forster (2006) described three aspects of integrating

technology into a lesson: the input, the calculation, and the interpretation. Each of the

these phases of technology use requires students to think carefully and critically about the

underlying mathematical processes; however, time is also necessary to teach students the

procedures associated with the technology in order to ensure that the students are truly

grasping the central concepts. The benefits of technology use can be somewhat tempered

by the trade-offs required to make the learning tools effective. The research conducted

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by Forster contributes to the general understanding of what educators must consider as

they make important decisions about instruction in their classrooms.

Glover, D., Miller, D., Averis, D., & Door, V. (2007, March). The evolution of an effective

pedagogy for teachers using the interactive whiteboard in mathematics and modern

languages: An empirical analysis from the secondary sector. Learning, Media, &

Technology, 32(1), 5-20. doi: 10.1080/17439880601141146

Summary. The study conducted by Glover, Miller, Averis, and Door (2007) involved

observations of the instructional use of interactive whiteboards (IWBs) in mathematics and

foreign language classes in order to assess how the technology enhanced the learning process.

Based on the data collected through observations, videotaped lessons, and interviews, the

researchers developed three classifications to categorize teaching styles that incorporated IWBs.

These categories were: supported didactic, interactive, and enhanced interactivity (Glover,

Miller, Averis, & Door, 2007). The supported didactic approach involved the use of technology

as a novelty but did not result in any significant changes in pedagogic applications (Glover et al.,

2007). In other words, the teachers at this level used the IWBs to support a traditional, teacher-

centered style. At the other end of the spectrum, teaching styles that reflected enhanced

interactivity utilized the IWBs to promote critical-thinking, to accommodate various learning

styles, to encourage student activity and interaction, to provide feedback, and to structure the

lessons (Glover et al., 2007). These teachers not only incorporated technology into instruction,

but they also changed the manner in which they taught. Lessons that were characterized as

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interactive reflected the attempts of those teachers that were making the transition from

traditional to more technology-enhanced instruction. Ultimately, the authors concluded that

instruction that exemplifies enhanced interactivity reflects both changes in technology use as

well as changes in pedagogic ideals.

Critical analysis. The authors of the study provide a solid rationale for the relationship

that must exist between pedagogy and technology implementation in order to ensure that the use

of various technological applications will truly support the learning process. Glover et al. (2007)

noted that the goal for instruction that incorporates technology (particularly IWBs) is “enhanced

interactivity” (para. 11). When lessons are designed at this level, the technology implementation

and instruction have several key characteristics: 1) technology applications (i.e. IWBs) provide

the basis of the lesson structure; 2) instruction involves multiple representations and

visualizations of the concepts; 3) activities encourage active thinking; 4) the lessons logically

and sequentially progress from simple to more complex ideas; 5) activities provide quick

feedback for both students and the instructor; and 6) recall is used to tie one lesson to the next

(Glover et al., 2007). The main theme that emerges from the study is the notion that effective

instruction designed for the 21st century rests upon the recognition of the fundamental connection

between teacher practice and technology implementation.

Statement of value. The research conducted by Glover et al. provides a solid structure

for evaluating the implementation of technology into classroom instruction. Based on the

researchers’ conclusions, enhanced interactivity characterizes lessons that effectively weave

relevant pedagogy with available technology in order to raise the quality of instruction and

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student learning. Although the study was focused primarily on IWBs, the concepts described by

the authors can be easily applied to other technology-based applications.

Graf, S., Viola, S. R., Leo, T., & Kinshuk. (2007, Fall). In-depth analysis of the Felder-

Silverman learning style dimensions. Journal of Research on Technology in Education,

40(1), 79-93.

Summary. The Felder-Silverman learning style model (FSLSM) is a framework that can

be used to assess the learning styles of students. The FSLSM is a model that assesses learning

styles based on four different dimensions: active/reflective, sensing/intuitive, verbal/visual, and

sequential/global (Graf, Viola, Leo, & Kinshuk, 2007). Through a questionnaire, the Index of

Learning Styles (ILS) developed by Felder et al., the preferences of students can be determined;

and the discernment of these preferences can help educators tailor instruction to meet the unique

needs of those students. Graf, Viola, Leo, and Kinshuk (2007) analyzed the results of the

administration of the ILS to a group of online learners in an attempt to further refine the

characteristics of the learning dimensions defined by Felder and Silverman. The results of the

study found that although the dimensions described in the FSLSM were mostly accurate, there

were further details that could be assessed to provide a more accurate understanding of a

student’s learning style. For example, the FSLSM distinguishes between active and reflective

learners. However, Graf et al. (2007) found that the opportunity to experiment with concepts

was more important to active learners as compared to the social components of the activity. In

the end, the researchers found that further refinement of an already tried and true method of

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assessing learning styles can continue to help educators more appropriately gear instruction to

the needs of unique students.

Critical analysis. The manner in which an individual best learns new concepts has a

significant impact on the success or failure of formal learning experiences. Therefore, it is

essential for those responsible for developing, implementing, delivering, and assessing

instruction to have a solid understanding of how individual students learn. Graf et al. (2007)

noted that a lack of alignment between learning styles and instructional techniques can lead to

difficulties in the learning process. The FSLSM in conjunction with the ILS are valid, reliable

tools for addressing the learning needs of students; and the authors of the study sought to further

refine the capabilities of the model. Ultimately, the four dimensions of

learning—active/reflective, sensing/intuitive, visual/verbal, and sequential/global—provide

significant amounts of information about the students and provide a basis for assessing the

capabilities of various instructional techniques, particularly those that depend on technological

applications.

Statement of value. The FSLSM involves another viewpoint from which to assess the

role of technology in the process of learning mathematics. It allows educators to draw

connections between the use of educational technology and unique learning needs of the

individuals students. Not only does this study further solidify the connection between the

appropriate alignment of instructional techniques with individual learning needs, but it also

outlines a way to assess that alignment based on the various ways that individuals interact with

the content, their peers, and the learning environment.

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Henningsen, M. & Stein, M. K. (1997) Mathematical tasks and student cognition: Classroom-

based factors that support and inhibit high-level mathematical thinking and reasoning.

Journal for Research in Mathematics Education, 28, 524–549.

Summary. Although mathematics is often viewed as a structured, fixed body of

knowledge that only requires individuals to learn facts, procedures, and rules, the educational

perspective on mathematical learning is shifting toward a more dynamic approach to

understanding the content (Henningsen & Stein, 1997). The goal for learners is to develop a

“mathematical disposition” that involves critical thinking, self-reflection, problem-solving, and a

deep understanding of mathematical concepts (Henningsen & Stein, 1997, para. 3). In order for

this transition to take place, both teachers and students need to transform what happens in the

classroom, and the focus of the learning process needs to be high-level thinking. Teachers play

the biggest part in this process by paying close attention to how instruction unfolds in the

classroom. The authors of this article identified several key factors that influence learning in the

classroom and encourage high-level, mathematical thinking on the part of students.

Instructional factors that tend to engage students and nurture effective mathematical

reasoning and thinking include: building on the knowledge and experiences of the students, the

use of scaffolding to support learning, designating appropriate amounts of time to complete

activities, modeling the essential skills, and continually seeking student explanations and

interpretations of meaning (Henningsen & Stein, 1997). On the other hand, several factors can

lead to a decline in the effectiveness and “cognitive demands” (Henningsen & Stein, 1997, para.

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32) of instruction and classroom activities. Henningsen and Stein noted that removing the

challenging aspects of a task, focusing on the answer rather than the process, and setting aside

inappropriate amounts of time for an activity can often lead to a decline in the mathematical

activity of the students.

Critical analysis. The authors of the study utilized observations in a variety of different

mathematics classrooms in order to determine the types of factors that either promote or lead to a

decline in mathematical thinking. Through observations, the researchers were able to identify

four types of learning situations: maintaining cognitive demands, decline into procedural

thinking, decline into unsystematic exploration, and decline to no mathematical activity

(Henningsen & Stein, 1997). These profiles each have unique combinations of the factors

identified by the researchers that promote high-level thinking; however, they each illustrate how

well-intended instruction and/or activities can deteriorate into activities that do not support high-

level mathematical thought. Overall, the authors provide an excellent argument for identifying

the classroom and teaching factors that either foster or deter true mathematical reasoning on the

part of students, for these factors contribute to the success (or failure) of students acquiring a

deep understanding of mathematical concepts and applications.

Statement of value. A goal of the depth portion of the KAM involves identifying

technology-based instructional activities that are truly effective in promoting learning in the

algebra classroom. The factors identified by Henningsen and Stein (1997), in addition to the

descriptive profiles outlined in the research, provide a standard by which to ascertain the quality

of mathematical instruction and learning associated with those technology-based activities. Any

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instructional activity (whether or not it utilizes technology) should lead students to a better

understanding of the concepts; and the research conducted by Henningsen and Stein provides a

method for characterizing the pedagogical quality of an activity.

Holahan, P. J., Jurkat, P. M., & Friedman, E. A. (2000, Spring). Evaluation of a mentor teacher

model for enhancing mathematics instruction through the use of computers. Journal of

Research on Computing in Education, 32(3), 336-351.

Summary. Holahan, Jurkat, and Friedman (2000) presented a summary of a training

model that was designed by CIESE (Center for Improved Engineering and Science Education) to

support the integration of technology into the mathematics classroom. Initiating institution-wide

change, however, can be a difficult challenge and “…CIESE sought to develop a comprehensive

model that could adequately deal with the complexity of the issues that influence the technology

integration process” (Holahan, Jurkat, & Friedman, 2000, para. 3). The Mentor Teacher Model

(MTM) was based on the premise of fostering systemic change through a phased approach to

professional development and staff training. By training a small core of mentor teachers and

utilizing the support of administrators, infusing technology into the pedagogical philosophy of

the school would become a much easier undertaking.

The authors of the study found that the success behind using the MTM for integrating

technology into mathematics classroom hinged on several variables. Holahan et al. (2000) noted

that the most success occurred when: there was administrative follow-through in reaching project

goals; supportive management on the part of building administrators; consistency among mentor

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assignments; long-term commitment; and a general perception that the program fits with district

goals. The integration of technology as a ubiquitous and effective educational tool ultimately

requires a long-term commitment to a process that is linked to all aspects of the educational

setting.

Critical analysis. The research conducted by Holahan et al. (2000) supports the notion

that the MTM can be used as successful model for training educators and infusing various

technologies into the mathematics classroom. The primary benefit of using such a model lies in

the fact that change can permeate the various facets of an educational setting and can lead to

long-lasting change. However, the use of the model itself does not guarantee success. Many

factors including administrative support, availability of resources, teacher buy-in, professional

atmosphere, and long-term commitment are necessary ingredients for fostering change. There

are many factors that can deter technology integration and polarize the debate among educators;

but given the appropriate educational conditions, the use of the MTM can bridge the gap and

lead to improved, technology-infused mathematics instruction.

Statement of value. Although the article focuses primarily on the organizational aspects

of integrating technology into mathematics instruction, the general framework of the MTM can

be used as a model for implementing technology-use on a classroom level. For example, student

buy-in to the instructional process as well as the choice of various instructional tools can

significantly enhance the positive effects of the chosen learning activities. In addition, long-term

commitment to effective technology integration on the part of the teacher plays a key role in

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developing a classroom environment and routine conducive to high-quality mathematics

learning.

Hoyer, J. (2005/2006). Technology integration in education. International Journal of Learning,

12(6), 1-8.

Summary. A meta-analysis of recent research on educational technology was conducted

by Hoyer (2006) in order to assess the nature and focus of that research. Hoyer noted that a

transition has appeared to take place over the past 30 years in regard to the type of research being

developed. During the 1970s and 1980s, many of the studies dealing with technology in the

educational setting were geared toward assessing the delivery aspects of the technology (Hoyer,

2006). Hoyer emphasized, though, that current research seems to be focused on how technology

and technology-based instruction are linked to the learning process. The main theme outlined by

Hoyer (2006) dealt with developing a “research agenda” (para. 13) that addressed how

technology can be used to improve instruction and learning rather than being simply viewed as

means of information delivery.

Critical analysis. This article provides a brief overview of the nature of research in

educational technology over the past 30 years. In general, the research cited by the author shows

that the current understanding of the role of technology in the educational setting encompasses a

wide range of perspectives. However, the general trend in research has transitioned from a focus

on the role of technology as a delivery system to a focus on the role of technology as an integral

part of instructional design as well as the learning process itself (Hoyer, 2006). Due to the

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complex nature of the educational system and the variety of factors that influence how and what

students learn, there remains a vast amount of research territory to be covered in the field of

educational technology. Hoyer’s analysis demonstrates the need for continued research in order

to effectively understand the role of technology in the learning process.

Statement of value. The article provides a basis for assessing how technology fits into

the learning process and into instructional design. Based on the analysis of Hoyer, the role of

technology in education has been elevated to being more than just a means of delivering

information. Technology can have a significant impact on how students learn, on how students

interact with information and data, and on how students apply the content in other situations.

Therefore, the process of designing activities for the classroom should include an opportunity for

reflection on how that technology is being used and whether or not it is effectively enhancing

both instruction and learning.

International Society for Technology in Education. (2005). A new theory of learning. In Multiple

Intelligences & Instructional Technology (2nd ed.) (3-9). Eugene, OR: International

Society for Technology in Education.

Summary. The introductory chapter of this book established the connections between

Gardner’s theory of Multiple Intelligences (MI), the role of technology in the learning process,

and the skills that today’s students must acquire in order to be successful in the real world.

According to the authors, MI theory established a definition of intelligence that encompasses the

various skills, techniques, and knowledge that people use to interact with each other, to interact

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with their surroundings, and to communicate the significance of those interactions (ISTE, 2005).

The previous limitations of the definition of intelligence as a purely linguistic, logical, and

mathematical property are giving way to other characteristics that define intelligent thought. The

authors noted that technology can be a tool that helps educators and students integrate all

intelligences into the learning process. In addition, the recognition of each student’s unique

intelligence profile in conjunction with the effective use of technology in the classroom can

ensure that students have the opportunity to acquire the necessary skills demanded by today’s

competitive, global community.

Critical analysis. Six skills were emphasized by the authors that students must have in

order to function in today’s competitive, information-based climate: information technology (IT)

skills, information literacy skills, the ability to solve problems, collaboration skills, flexibility,

and creativity (ISTE, 2005). These skills can be developed in the classroom setting through the

design and delivery of instruction that successfully balances the various intelligences that

students possess with the effective use of technology to enhance unique combinations of

intelligences. Although sound educational philosophy and pedagogy must always establish the

foundation of classroom instruction, Gardner’s theory and appropriate applications of technology

can establish a learning environment that values each student’s potential and unique skills while

preparing them with the skills necessary to succeed in the “Information Age” (ISTE, 2005, para.

7).

Statement of value. The six needs—IT skills, literacy skills, problem-solving,

collaboration, flexibility, and creativity—outlined in the article are essential skills that students

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must develop in order to thrive in today’s society. Regardless of the current social demands,

these skills would still be necessary in order for an individual to be successful in a learning

environment, particularly the mathematics classroom. Therefore, an assessment of the

technology-based classroom activities measured against the presence (or lack thereof) of

opportunities to develop these skills could be used to determine the value of those activities.

Kahveci, M., & Imamoglu, Y. (2007). Interactive learning in mathematics education: Review of

recent literature. Journal of Computers in Mathematics & Science Teaching, 26(2), 137-

153.

Summary. In a review of recent literature on mathematics instruction, Kahveci and

Imamoglu (2007) concluded that mathematical achievement and the development of higher-order

mathematical skills “require the students to communicate mathematically” (para. 35); and

“interaction with peers, teachers, and any other media plays an essential role” (para. 35) in the

process. The analysis addressed different situations in which students interact in the

mathematics classroom including: working with technology applications, functioning in

cooperative groups, and operating within the whole-class setting. In order to maximize the

educational benefits of these settings, educators must pay close attention to the quality and type

of interactions taking place.

Although an interaction can be any event that occurs in the classroom, Kahveci and

Imamoglu (2007) identified an “instructional interaction” (para. 1) as an event that produces a

desirable change in a student that is directed toward a specified educational goal. In order for a

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technology-based interaction to be productive, educators must assess how the students are

expected to respond, the manner in which the technology processes a student’s response, and the

feedback that is presented to the student (Kahveci & Imamoglu, 2007). With regard to

cooperative learning groups, the nature of the task, the type of questions and feedback, the

composition and interdependence of the group, and interactions among the students all influence

the instructional quality of small-group interactions (Kahveci & Imamoglu, 2007). The authors

also noted that in order to promote student participation in whole-class settings, teachers must

take care to promote positive social interactions, to build upon the motivational goals of the

students, and to create an environment that supports learning.

Critical analysis. The significant role of classroom interactions in the learning process

was highlighted through the authors’ review of current literature in mathematics education. The

descriptions of the various avenues that students use to learn the content, establish connections

among various concepts, communicate ideas, and interact with teachers and with each other

demonstrated the importance of ensuring that the these interactions are instructional and

supportive of quality learning. Kahveci and Imamoglu (2007) also noted the importance that

educational interactions must have a “two-way effect” (para. 1) in order to truly benefit student

learning and the classroom dynamic. In general, the meta-analysis of the research illustrated the

need for educators to attend to the influence of interactivity on the learning process.

Statement of value. Kahveci and Imamoglu (2007) outlined several educational

implications of their research including: the use of multiple representations in class; setting goals

of mastery of content; the development of problem-solving skills; encouraging student

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participation; setting expectations for the use of mathematical reasoning; promoting

metacognition; and so forth. In order to meet these criteria, the interactions that students have

during various learning activities (including the use technology) must be of a high quality and

grounded in sound instruction. The information delineated by the authors provides additional

criteria for critically analyzing the activities that are developed for and implemented into the

learning environment.

Kumar, P., Kumar, A., & Smart, K. (2004). Assessing the impact of instructional methods and

information technology on student learning styles. Issues in Informing Science &

Information Technology, 1, 533-544.

Summary. The primary goal of the study conducted by Kumar, Kumar, and Smart

(2004) was to determine if the appropriate use of technology in instruction can influence

students’ learning styles. The framework of the study was built upon the types of learning styles

developed by Grasha. Grasha’s definition of learning style is based on the characteristics of

individual learners that influence how they interact with information, peers and teachers, and the

environment (Kumar, Kumar, & Smart, 2004). Based upon this definition, Grasha indentified

six learning style categories: Independent, Dependent, Competitive, Collaborative, Avoidant, and

Participant. The unique aspect of these particular learning styles is that they were developed

with consideration of the classroom environment in addition to individual, personal, and

cognitive traits. Based upon the characteristics of these learning styles, the authors studied two

classrooms that utilized various educational technologies that were geared toward supporting the

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Independent, Participant, and Collaborative learning styles. Classroom observations as well as a

pre- and post-test form of a learning style assessment were used to determine the affect of the

technology-based instructional techniques. Kumar et al. (2004) concluded that the appropriate

use of technology in instruction can help students enhance, improve, and/or change their

preferred learning styles.

Critical analysis. The results of this study provide support for the need to develop

instructional methods that align with the learning styles of the students. In addition, Kumar et al.

(2004) suggested that the development of “creative mismatches” (para. 13) can also help students

further develop characteristics of non-preferred learning styles. The authors further noted that

technology-based applications can be used as tools for creating instructional activities that are

individualized to meet the needs of diverse students with diverse learning styles. Although the

study was conducted in a relatively narrow setting, the conclusions of the authors have

significant potential for generalizability to other educational settings. In particular, the algebra

classroom could prove to be a setting where the need for adapting content and instruction to meet

individual learning styles is necessary.

Statement of value. The categories of learning styles utilized by the authors provide

ways to identify how students function within the mathematics classroom. The recognition of

these learning styles also provides educators with criteria for developing and implementing

instructional activities that can accommodate the various ways in which students interact in the

classroom. Technology also plays a vital role in incorporating activities that are individualized,

dynamic, and supportive of student learning.

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Laborde, C. (2007, January). The role and uses of technologies in mathematics classrooms:

Between challenge and modus Vivendi. Canadian Journal of Science, Mathematics, &

Technology Education, 7(1), 68-92.

Summary. This article addressed the role of technology in the mathematics classroom

and examined how technology can be used to enhance the learning of mathematics. Laborde

(2007) identified four types of functions that technology can play in the classroom: technology

can speed up a particular activity but not influence the nature of the activity (i.e. using a

calculator to complete basic computations); technology can provide opportunities for students to

explore concepts (i.e. assessing changes in graphs when certain values are changed); technology

can provide alternative methods for solving beyond paper-and-pencil strategies (i.e. graphical

interpretations verses algebraic solutions); and technology can itself be the focus of learning (i.e.

assessing the logic behind various commands or syntax). The author emphasized that

“technologies mediate mathematics and are mediated by mathematical thinking…” (Laborde,

2007, para. 10).

A key theme of the article focused on the need for consistency between technologies

utilized by students both in and out of school. Laborde (2007) noted that educators cannot

ignore the fact that technology pervades almost every aspect of the real world and must not

exclude that technology when it comes to designing mathematics instruction. Mathematics and

technology are intimately related; and students must have opportunities to develop their

mathematical skills through the use of technology.

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Critical analysis. In order to establish the critical connection between mathematics,

technology, instruction, and learning, Laborde (2007) assessed a variety of technical applications

including dynamic geometry environments (DGEs), computer algebra systems (CAS), and so

forth. Not only do these applications provide opportunities to integrate technology into the

mathematics classroom, but they also can have a significant impact on how mathematical

concepts are taught and learned. The use of technology can facilitate opportunities for students

to explore and experiment with mathematical concepts. The limitations of paper-and-pencil

techniques can be overcome by the interactivity offered by various applications. In turn, the

symbiotic relationship of mathematics and technology can deepen student understanding and

problem-solving skills (Laborde, 2007).

Statement of value. On the surface, technology may be used as a novel way to present

information and work with mathematical concepts. However, the connection between

technology and mathematics extends much deeper. The research conducted by Laborde (2007)

provides support for the necessity to integrate technology into mathematics instruction. In

addition, learning activities in the math classroom must be transformed from traditional

techniques into opportunities for students to explore concepts, to develop reasoning skills, and to

solve problems. Technologies provide an avenue for making this possible.

Martin, G. P., & Burnette, C. (2000, October). Maximizing multiple intelligences through

multimedia: A real application of Gardner’s theories. Multimedia Schools, 7(5), 28-33.

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Summary. Martin and Burnette (2000) addressed a technique for combining the learning

theories of Howard Gardner with the advantages of multimedia. In order to accurately determine

the learning progress of students, it is essential that the assessment goes beyond the traditional

focus on the verbal and logical-mathematical intelligences (Martin & Burnette, 2000). The

authors believed that an electronic portfolio can incorporate a variety of multimedia applications

that could allow students to demonstrate their knowledge through each of the eight intelligences

identified by Gardner. In order to integrate an electronic portfolio into existing curricula and

classroom practices so that learning is efficiently enhanced, Martin and Burnette noted the

importance of assessing current instructional activities within the framework of MI theory and

the advantages of multimedia tools. The authors developed a formula for identifying the

effectiveness of an activity. In essence, an activity is considered to be effective if it utilizes more

of Gardner’s intelligences (through multimedia applications) while decreasing the time necessary

to complete the instructional process. Using these parameters, an educator can determine how to

transform current activities into ones that will translate into artifacts that truly represent student

learning.

Critical analysis. The article addresses the importance of identifying the unique learning

needs of students by providing them with opportunities to tap into the various intelligences that

characterize them. This can occur by using current technologies as a foundation for allowing

students to demonstrate what they have learned. This article outlines a potential bridge between

theoretical understanding of learning and the current technologies at the disposal of teachers and

students. However, it was emphasized that educators must be critical of the types of activities

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that are employed and their connection to both the application of MI theory and the use of

multimedia tools. The hypothesized formula proposed by the authors suggested that educators

must factor in how the various intelligences can be drawn upon while managing the logistics of

time, materials, and technology. The authors provide a case for the use of an electronic portfolio

as a means for providing students with a real opportunity for demonstrating what they have

learned.

Statement of value. An important aspect of this KAM involves developing the

connections between theoretical understanding of knowledge and learning with the technological

tools available to today’s students. In addition to an electronic portfolio, Martin and Burnette

(2000) suggested the use of a matrix as a means for assessing the effectiveness of activities that

are integrated into the classroom. This type of analysis can allow teachers, particularly in the

math classroom, to determine what types of activities will enhance learning while simultaneously

providing a way to accurately assess student learning.

Mayer, R. E. (2003). The promise of multimedia learning: Using the same instructional design

methods across different media. Learning and Instruction, 13, 125–139. doi:

10.1016/S0959-4752(02)00016-6

Summary. Multimedia has significant potential to enhance student learning due to the

fact that the inherent nature of multimedia leads to the presentation of information that accesses

different processing channels. Mayer (2003) maintained that in order to reach this potential,

educators must design instructional strategies that incorporate multiple representations and allow

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students to go through the process of selecting, organizing, and integrating information . Mayer

also noted that instructional strategies that prove effective in one setting also tend to be effective

with the use of other forms of media.

Four particular methods that teachers should attend to during the design phase include:

“…the multimedia effect, the coherence effect, the spatial contiguity effect, and the

personalization effect” (Mayer, 2003, para. 18). According to Mayer, the multimedia effect

deals with the research that demonstrates that students tend to learn more when visual and verbal

cues are used together. The coherence effect suggests that extra, non-essential information

should be eliminated from an explanation so that students can efficiently process the relevant

information. The spatial contiguity effect is the idea that students will better understand

information if the verbal and visual cues are presented in close proximity to one another.

Finally, the personalization effect refers to the fact that students tend to integrate information

when it is presented in a “conversational style” (Mayer, 2003, para. 35). With these concepts in

mind, educators can create multimedia activities that maximize the learning potential of the

chosen media.

Critical analysis. The fundamental theme underlying the research conducted by Mayer

is the necessity to strive for meaningful student learning by aligning instructional activities to the

natural way that humans tend to process information. Multimedia applications have the potential

to tap into the various channels through which individuals gather information; and it is essential

to effectively develop pedagogical practices that match the advantages of multimedia with the

natural ways that people learn. Mayer (2003) stated that “redesigning multimedia explanations

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to mesh with the way humans learn enabled students to generate more creative solutions to

problem-solving transfer questions…” (para. 45). The multimedia effect, the coherence effect,

the spatial contiguity effect, and the personalization effect are research-based principles that can

guide the development of effective instruction rooted in various types of media.

Statement of value. This article provides additional factors to consider in the

development of instructional methods that build upon the unique learning characteristics of

students while addressing the need to incorporate various technologies in the learning process.

The four principles identified by Mayer can contribute to a framework that can be used to assess

the effectiveness of a given strategy. The concepts can also lead to potential avenues of

improvement for transforming existing activities into more student-centered, technology-based

learning situations.

Mok, I. A., Johnson, D. C., Cheung, J. Y. H., & Lee, A. M. S. (2000, July/August). Introducing

technology in algebra in Hong Kong: Addressing issues in learning. International

Journal of Mathematical Education in Science & Technology, 31(4), 553-567. doi:

10.1080/002073900412660

Summary. A significant weakness that has plagued many students, on global scale, is

the ability to think algebraically and to grasp more abstract mathematical concepts. Mok,

Johnson, Cheung, and Lee (2000) suggested that these weaknesses are often perpetuated by

teaching practices—teacher-centered instruction and a focus on procedures—rather than by

student ability. The authors also noted that more effective teaching and learning strategies such

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as group-based learning and hands-on experiences could be enhanced by the use of various

technologies. Computers, software programs, graphing calculators, and so on could all prove

useful in promoting student interest, inquiry, and achievement (Mok, Johnson, Cheung, & Lee,

2000). In particular, the effective use of technology during instruction could lead students

through the “hypothesizing-verifying cycle” (Mok et al., 2000, para. 18) where students are able

to engage in “cognitive conflict, metacognition, and construction” (para. 33). Ultimately, the

goal of instruction is to raise the level of student thinking and processing in the mathematical

setting. Technology can be used as tool to transform less effective, traditional teaching methods

into activities that promote student-centered, active learning.

Critical analysis. Although the authors of the study do not go into great detail regarding

how various technologies can specifically be used to promote higher-order, algebraic thinking,

the emphasis on the processes of cognitive conflict, metacognition, and construction outline the

goals that educators can aim for during the design and implementation phases of instruction.

These processes characterize the hypothesizing-verifying cycle which can lead students to

internalize more abstract concepts. In addition, the appropriate use of technology and

collaborative learning opportunities can bring the characteristics of these processes to the

forefront and avoid the pitfalls in learning associated with more traditional instructional methods.

Statement of value. The primary value of the research conducted by Mok et al. (2000)

involves the recognition of the importance of making instruction learner-centered. With the

concepts of cognitive conflict, metacognition, and construction in mind, educators can design

instructional activities that promote higher-order thinking. The characteristics of these processes

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can also be used to assess (and potentially enhance) the nature of how technologies are integrated

into the learning process.

Moreno, R. (2006, April). Learning in high-tech and multimedia environments. Current

Directions in Psychological Science, 15(2), p. 63-67. doi: 10.1111/j.0963-

7214.2006.00408.x

Summary. Two seemingly opposing ideas tend to dominate the debate regarding the

role of technology in education: the “media-affects-learning hypothesis” and the “methods-

affect-learning hypothesis” (Moreno, 2006, para. 2). Moreno indicated that the media-affects-

learning idea suggests that the simple use of better, more advanced technologies will directly

improve learning while the methods-affect-learning concept implies that learning will be

improved if effective instructional techniques are used. The research conducted by Moreno

(2006), however, suggests that these perspectives can be combined and the use of various media

will facilitate effective instructional methods.

In order to effectively integrate media and methodology, Moreno (2006) proposed a

cognitive theory of learning with media (CTLM) which outlines how media can be developed to

effectively streamline the cognitive processes of students and establish alignment between

instruction and student thinking. The CTLM is based on 10 principles for developing media-

based instruction. The first five principles—modality, redundancy, temporal-contiguity, spatial-

contiguity, and coherence—are necessary for streamlining how students can effectively process

information. These principles allow for multiple opportunities for students to process

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information without wasting time or energy dealing with unimportant details (Moreno, 2006).

The second five principles—multimedia, personalization, interactivity, guidance, and reflection

—are intended to provide students with opportunities to actively engage the material and

concepts being presented. Essentially, the CTLM delineates the characteristics of media-based

instruction that must be present in order to maximize the cognitive processes of students.

Critical analysis. When integrating media and technology into classroom instruction,

Moreno (2006) indicated that there are two main considerations: 1) assessing what methods align

with the use of a particular technology; and 2) determining how those methods fit into the

cognitive processes of students. By addressing these issues, educators can structure media-

enhanced instruction to maximize students learning. The cognitive theory developed by the

Moreno addressed how technology and media tie in with the cognitive processes of students.

The author noted, though, that technology alone cannot guarantee improved learning, and other

factors including the classroom environment, the role of the teacher, student interactions, student

characteristics, and so on must also be accounted for. Consideration of the 10 principles of the

CTLM will provide teachers with an increased opportunity to deliver instruction that

accommodates diverse learners in diverse classroom settings.

Statement of value. In order to ensure that technology-based instruction will effectively

meet the learning needs of students, the 10 principles of the CTLM can be used as a checklist to

assess learning activities in the mathematics classroom. The CTLM is based on a cognitive

perspective and can be used to assess instruction according to how students process information.

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This type of assessment can complement other research-based perspectives on the role of

educational technology in order to maximize the learning potential of each classroom activity.

Nelson, G. (1998, June). Internet/web-based instruction and multiple intelligences. Educational

Media International, 35(2), 90-94.

Summary. One of the significant themes found in the article written by Nelson is the

convergence of new technological applications and modern theories of learning. In particular,

the author highlighted Gardner’s theory of Multiple Intelligences (MI theory) and the underlying

notion that individuals have unique cognitive strengths that influence how they learn and process

information. Nelson (1998) argued that the emerging capabilities of the Internet and instruction

rooted in web-based activities provide an avenue for aligning instructional practice with the

unique learning/thinking characteristics of each student. Although it is essential to critically

evaluate a learning tool and how that tool is used in the educational process, the availability of

new technologies make it possible to capitalize on the strengths of learning theories that

effectively describe the cognitive processes of learners.

Critical analysis. This article draws an important connection between the availability of

technology and the role of learning theories in educational practice. The recognition of the

unique learning needs of students is essential for tailoring learning activities that align with those

strengths and needs; and advancing technologies are making it possible to effectively and

efficiently improve the instructional design process. That being said, the article does not provide

a strong researched-based foundation for the presented interpretation. In addition, the age of the

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article does not necessarily factor in the quantum leaps that have been made in technology over

that past decade. The assertions and conclusions draw by Nelson (1998) seem to be logical and

applicable to today’s classrooms, but further support must be supplemented in order to

corroborate the author’s statements.

Statement of value. The primary value of this article is the connection that is drawn

between the theoretical conceptualizations behind Gardner’s MI theory and types of technology

that are currently available to teachers and students. The Internet, web-based materials, and

other resources are excellent tools for designing instructional activities that reflect the diverse

cognitive needs of individual students as described by Gardner. Developing the bridge between

theory and current research/practice is essential for ensuring that practical applications in the

classroom will be effective.

Norton, S., McRobbie, C. J., & Cooper, T. J. (2000, Fall). Exploring secondary mathematics

teachers' reasons for not using computers in their teaching: Five case studies. Journal of

Research on Computing in Education, 33(1), 87-110.

Summary. Although a vast amount of technology is currently available to educators,

there is resistance to the implementation of that technology, particularly in mathematics

classrooms. This resistance has consequences regarding the understanding that students develop

regarding the intimate connection between mathematics and technology. Norton, McRobbie, and

Cooper (2000) studied several mathematics classrooms in a single school district in an effort to

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determine why secondary mathematics teachers generally tend to resist the implementation of

technology into the teaching/learning process.

During the observation and data collection process, the authors identified three types of

teaching profiles evidenced in the math classrooms: calculation-based instruction with a focus on

learning algorithms and decontextualized rules; a conceptual approach that was teacher-centered

and focused on explanation-based instruction; and a student-centered approach rooted in

constructivist ideas that promoted investigation, problem-solving, and reflection (Norton,

McRobbie, & Cooper, 2000). The last profile was considered to be a strong model for effective

instruction, but the authors concluded that the teacher-centered profiles tended to dominate and

reflected attitude’s that resisted change and the implementation of new technology. Teacher

beliefs, resource and time issues, lack of professional development and dialogue, and many other

issues were cited as potential reasons why mathematics teachers were reluctant to adapt their

traditional routines. In addition, innovative teachers who were willing to try new things were

often stifled by their surroundings and lack of professional support. Ultimately, the authors

found that the attitudes and beliefs (valid or otherwise) were significant obstacles facing the

integration of technological applications in the mathematics classroom.

Critical analysis. The case study approach utilized by Norton et al. (2000) provided

evidence how the characteristics, mindset, and beliefs of the classroom teacher have a significant

impact on the types of instructional activities that are implemented in the learning process. In

this situation, the authors sought to determine why math teachers have been reluctant to integrate

technology in instruction. Although the authors were careful not to over generalize their

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findings, the notion that the teachers themselves are one obstacle to technology integration was

prevalent. The authors also subtly noted that the attitudes against the use of technology were not

often rooted in a particular theory or based on valid research. An additional point that was

emphasized was the fact that the mere presence of technology was not sufficient to ensure that

the technology was used in an effective manner. In the end, the attitudes and practices of

mathematics teachers must be carefully considered during the process of technology integration.

Statement of value. One goal of the KAM is to assess obstacles to the implementation

of technology in the mathematics classroom. The study conducted my Norton et al. (2000)

provides a solid reason for exploring how to help teachers overcome existing attitudes that

prevent the use of potentially successful technology-based instructional strategies. The study

also provides specific areas to address when it comes to synthesizing recommendations for

positive change in the mathematics classroom.

Passey, D. (2006, June). Technology enhancing learning: Analyzing uses of information and

communication technologies by primary and secondary school pupils with learning

frameworks. Curriculum Journal, 17(2), p. 139-166. doi: 10.1080/09585170600792761

Summary. The effective use of technology in the classroom has the potential to

significantly enhance the learning process; but in order to do so, these technologies need to be

used appropriately (Passey, 2006). The author of this study addressed how information and

communication technologies (ICT) were used in the classroom by assessing their use with regard

to several established frameworks including Gardner’s theory of Multiple Intelligences, Bloom’s

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taxonomy, and additional learning models. By using a wide range of frameworks, cognitive

activities as well as instructional and assessment strategies could be evaluated with regard to the

role of ICT. Based on observations in the field, Passey (2006) found that most uses of ICT led to

learning situations that emphasized lower-level cognitive skills and/or narrow views of how

students process (i.e. only verbal) and relate information. The author emphasized three key

points: the need to utilize the features of ICT that make it possible to promote higher-order

thinking; the importance of using diverse problem-solving strategies; the need to recognize the

unique learning characteristics of each student.

Critical analysis. This study provides a good model for blending a variety of different

frameworks in the analysis of classroom practices and the use of technology in the learning

process. Passey (2006) acknowledged the importance of drawing from a wide range of ideas in

order to clearly identify the shortcomings (and/or strengths) of existing teaching practices. The

use of ICT is especially important in this analysis. Various technologies are often underutilized

for many different reasons, but it is essential for educators to determine how to reverse that trend

in order to make use of the quality learning opportunities afforded by technology. The author

drew from a wide range of existing literature in an effort to synthesize a super-framework that

educators can use as a reliable evaluation tool. If available technologies are to be effectively

integrated into today’s instruction, it is important to determine where the obstacles to their use

are located and to determine how to utilize the resources to raise the level of cognitive processing

and communication in all students.

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Statement of value. This article provides a foundation for blending existing theories into

a new framework for assessing classroom activities, particularly those that incorporate ICT. By

tying learning theories with the role of modern technology in education, the importance of

recognizing how the cognitive process of students are directly linked to the resources, activities,

and instruction found in the classroom. Building a bridge between theory and practice is

essential to ensure that positive changes occur in education and to ensure that the technology

available today can be effectively used in the learning process.

Reid-Griffin, A., & Carter, G. (2004, December). Technology as a tool: Applying an

instructional model to teach middle school students to use technology as a mediator of

learning. Journal of Science Education & Technology, 13(4), 495-504. doi:

10.1007/s10956-004-1470-2

Summary. In an effort to explore how technology could be successfully used as a

learning tool in the science classroom, the authors developed a course rooted in scientific

investigation which was designed to gradually scaffold the use of technology in the learning

process. It was noted by Reid-Griffin and Carter (2004) that students must be prepared to

properly use technological tools in problem-solving situations in order to effectively function in

real world settings. The nine-week technology course was implemented in three phases where

students were trained on the technology and eventually handed more responsibility in the inquiry

process. By gradually removing teacher support, the students were ultimately able to utilize the

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available technologies as tools (a.k.a. as a means to a learning end verses an end in itself) in their

own scientific inquiries.

Critical analysis. The study conducted by Reid-Griffin and Carter (2004) involved an

investigation of the process of scaffolding the use the technology in an effort to help students

learn how to use various technologies as support tools in the learning process. The authors

designed the structure of the course using Vygotsky’s zone of proximal development (ZPD) and

Bruner’s notion of scaffolding. The course was split into three phases—teacher directed

instruction, teacher-student directed inquiry, and student-directed investigation—whereby the

students were eventually held responsible for their own learning (Reid-Griffin & Carter, 2004).

The author’s found that the gradual removal of teacher support helped students to develop their

own skills in using technology to solve problems and to seek answers to scientific questions. In

addition, the authors found that the technological resources were used to support learning rather

than being the center of learning.

Statement of value. The research conducted by Reid-Griffin and Carter (2004) further

established the connections between learning theory and the role of technology in the

instructional process. The author’s found that the use of scaffolding and the awareness of the

students’ ZPD were highly useful in creating a classroom environment that effectively utilized

available technologies. The authors also noted that the blind use of technology alone cannot

ensure that students will learn the content or develop higher-order thinking skills (Reid-Griffin &

Carter, 2004). The evidence presented by the authors demonstrates the need for educators to

thoughtfully consider how technology can and should be used in the learning process.

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Ross, J., & Schultz, R. (1999, January). Can computer-aided instruction accommodate all

learners equally? British Journal of Educational Technology, 30(1), 5-25.

Summary. The role of technology in the learning process is an issue that continues to be

researched and better understood. Ross and Schultz (1999) utilized the Gregorc Style

DelineatorTM to assess the various learning styles of students and then analyzed pre- and post-test

data to determine if computer-aided instruction (CAI) was an effective learning strategy for all of

the students. Four main learning styles were identified by Gregorc: concrete sequential (CS),

concrete random (CR), abstract sequential (AS), and abstract random (AR) (Ross & Schultz,

1999). Through the analysis process, Ross and Schultz determined that CAI was not necessarily

the most effective learning/teaching strategy for all students. In general, the authors found that

students who were identified as sequential tended to have the most success with CAI. The

authors also concluded that factors such as motivation, level of user control, and attitudes about

computers also influenced student success in using CAI. Ultimately, the study results led to the

conclusion that educators should make every attempt to match teaching strategies with the

learning styles of students in order to maximize the educational effects.

Critical analysis. Although the article is somewhat dated, the general findings of the

authors are still valid in today’s learning environment. The primary conclusion suggests that the

maximum learning potential of an activity occurs when the teaching strategy employed by the

educator is aligned with the preferred learning style of the student. The author’s do concede,

though, that logistics, classroom parameters, and so on must be considered in lesson planning,

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and students must develop the ability to adapt to less-than-ideal learning situations (Ross &

Schultz, 1999). However, this notion emphasizes that the non-thoughtful use of technology does

not necessarily guarantee that students will learn. Despite the significant potential associated

with CAI, educators must acknowledge that CAI is not necessarily the best match for every

student or for every learning style. Making every effort to match teaching and learning styles is

an essential task for educators if they are truly attempting to maximize student learning.

Statement of value. The article provides additional criteria for analyzing the

relationship between the learning needs of the students and the use of technology in the

classroom. By identifying students as concrete sequential, concrete random, abstract sequential,

or abstract random, teachers can determine better matches between teaching and learning styles.

In addition, this information about the students can help teachers determine when and where

technology resources can be effectively utilized. Ross and Schultz (1999) noted that technology

alone is not the answer to all learning needs. Ultimately, the alignment of instructional methods

with students’ learning needs is the most important consideration to address.

Ruthven, K., & Hennessy, S. (2002). A practitioner model of the use of computer-based tools

and resources to support mathematics teaching and learning. Educational Studies in

Mathematics, 49(1), 47-88.

Summary. Ruthven and Hennessy (2002) conducted a study in an effort to develop a

model for assessing information and communication technology (ICT) use in the mathematics

classroom. The data that served as the foundation for the model was gathered through interviews

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with students and teachers in a variety of different schools. The authors derived various themes

from an analysis of the data in order to generate the overarching concepts that seemed to support

the use of technology in the classroom.

Ten themes were identified during the analysis. These themes were: ambience enhanced,

restraints alleviated, tinkering assisted, motivation improved, engagement intensified, routine

facilitated, activity effected, features accentuated, attention raised, and ideas established

(Ruthven & Hennessy, 2002). Educators who participated in the study found these general

concepts to be important outcomes that resulted from the use of ICT in the classroom.

Ultimately, the authors concluded that the development of curriculum and instructional practices

should account for the themes deemed important by practitioners in the field in order to yield

positive learning results for students.

Critical analysis. A unique feature of the study conducted by Ruthven and Hennessy

(2002) is that the data was based on the perceptions of classroom educators using the various

forms of ICT as part of their instructional practice. Although the authors conceded that the

general themes are not necessarily comprehensive, they noted that it is essential to address the

perceptions of those that are expected to implement instruction that utilizes ICT. By attending to

the way that those in charge of classroom implementation conceptualize the learning tools

available to them, a more cohesive process can be developed as curriculum is created,

implemented by the teacher, and understood by the student.

Statement of value. There are many different perspectives that must be considered when

looking at how technology is implemented into the classroom. Students, teachers, parents,

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community members, and so on each bring a unique point-of-view to the table when looking at

the needs, benefits, and costs of ICT-use in the classroom. In order to develop the most effective

pedagogical practices, the perceptions of teachers—those responsible for the implementation of

curricular and pedagogical routines—must be considered during the development process. The

themes unveiled by Ruthven and Hennessy (2002) provide criteria that can be used to assess how

technology is being used in the mathematics classroom. By synthesizing the themes presented in

this study with additional themes from other research projects, it becomes possible to develop a

thorough framework for evaluating technology use in the classroom.

Solvie, P., & Kloek, M. (2007, April). Using technology tools to engage students with multiple

learning styles in a constructivist learning environment. Contemporary Issues in

Technology & Teacher Education, 7(2), 7-27.

Summary. The study conducted by Solvie and Kloek (2007) addressed the connections

between constructivism, student learning styles, and technology-based instruction. Using the

model developed by Kolb, the authors delineated four categories of learning styles: concrete

experience, reflective observation, abstract conceptualization, and active experimentation (Solvie

& Kloek, 2007). Kolb’s model suggests that students cycle through these stages during the

learning process. Thus, the authors developed technology-based activities and instructional

techniques that would align with each of the four modes of learning in the hope that all students

would find an avenue for successfully accessing and processing the content of the course. The

main focus of the study attempted to determine whether or not “technology-enhanced learning

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experiences aligned to learning styles of students [would] support a constructivist setting and

students’ understanding of course content” (Solvie & Kloek, 2007, para. 15).

The results of the study revealed that there were a variety of factors that influenced the

success of the students. First, the learning styles of the students had an impact on the extent to

which the learning strategies allowed the students to construct meaning from the content. In

general, learners with more flexible styles tended to have better success with the various

technology-enhanced activities as compared to those with more static styles (Solvie & Kloek,

2007). Second, “metacognition” (para. 64) and “agency” (para. 67) shaped the extent to which

students processed, integrated, and applied the information. These reflective skills allow the

learner to move through the learning cycle as described by Kolb. Finally, the role of the

instructor with respect to effectively guiding students by scaffolding and planning had an impact

on constructing knowledge through the use of technology-based learning strategies.

Critical analysis. Technology can play a vital role in the learning process. However, the

use of technology alone does not guarantee that quality learning will occur. When used

appropriately, technology-based learning activities can provide opportunities for students to

construct meaning. The process, though, must be supported by additional factors. These factors

include, but are not limited to, the strategic guidance of the instructor, the learning styles of the

students and the alignment of technology applications to those learning styles, appropriate use of

technology with respect to the content and learning goals, metacognition, and so on (Solvie &

Kloek, 2007). This research adds additional support for the need to integrate technology into the

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instructional process in a thoughtful, purposeful manner that reflects the necessary alignment

among the various facets that influence how and what students learn.

Statement of value. Although learning styles can be categorized in a variety of ways,

the four steps in the learning cycle outlined by the authors provides a framework for assessing

how various technology-based activities fit into how information is processed. In addition, the

framework can also provide a standard for determining whether or not various technologies

provide different students with dynamic learning styles access to the content being presented.

Ultimately, the necessity for alignment among content, learning styles, and technology must be

addressed in the development of classroom activities.

Spector, J. M. (2001, July/September). Philosophical implications for the design of instruction.

Instructional Science, 29(4/5), 381-402. doi: 10.1023/A:1011999926635

Summary. Psychological theories and philosophy have both informed the process of

instructional design. In this article, Spector (2001) presented a learning model that was rooted in

a philosophical perspective on learning. Model facilitated learning (MFL) is built upon the

notion that the basic units of instruction should be the complex systems found in real-world

situations (Spector, 2001). In other words, learners need opportunities to investigate the dynamic

systems that exist in reality and involve multiple variables, multiple representations, complex

relationships, and so on. The author noted that once a particular system is identified as a context

for learning, the individualized characteristics of the system can be broken down and analyzed.

Ultimately, MFL is intended to promote high-level thinking and the development of an

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understanding that views complex systems as “interconnected” (Spector, 2001, para. 37) parts.

The framework encourages learners to build models, to progress through well-articulated

instructional sequences, and to apply concepts in the context of the reality they live in (Spector,

2001).

Critical analysis. Spector’s (2001) analysis of model facilitated learning (MFL)

provides a distinctively philosophical approach to the learning process. By looking at learning

contexts from an epistemological perspective that acknowledges the complex world in which

students live, implications are uncovered for instructional design. In particular, the authors

emphasize a shift in focus from discrete pieces of information or skills to the complex systems

that exist in reality. In other words, problem-based learning that is based in a real-world context

provides opportunities for students to truly learn. Specific algorithms, problem-solving

strategies, technological applications, and so on can be presented in a supporting role to help

students develop a broader understanding of the larger picture. Modeling, developing multiple

representations, and analyzing relationships among dynamic variables become the primary focus

of the learning process.

Statement of value. The analysis found in the article provides support for developing

learning strategies that are rooted in problem-solving. The unique contribution of the article is

the philosophical perspective that is employed to demonstrate the importance of problem-based,

collaborative learning. Model facilitated learning (MFL) is intended to guide students from

concrete learning experiences through more abstract conceptualizations all within in the

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parameters of complex, reality-based situations. This framework provides additional criteria to

assess the effectiveness of technology-based instruction in the mathematics classroom.

Vincent, A., & Ross, D. (2001, Fall). Learning style awareness. Journal of Research on

Computing in Education, 33(5), 1-10.

Summary. The classroom setting introduces a wide variety of variables that influence

the learning process of all students. One such variable is the learning style that is inherent to

each student. Vincent and Ross (2001) conducted a research study to develop a comprehensive

list of characteristics that generally define three primary learning styles—auditory, visual, and

kinesthetic. For example, students with strengths as visual learners tend to be more successful in

the traditional classroom settings since they learn best through reading text, viewing pictures,

and creating mental images of information (Vincent & Ross, 2001). The authors provided

similar lists for the other learning styles. They also provided suggestions for both learners and

teachers for coping in the classroom where the teaching and learning styles may not always

match. Typically, teachers can develop their understanding of student learning styles through

questioning and observation; and knowledge of how students learn best is an important key to

success in the classroom.

Critical analysis. The authors of the study utilized the LSI (Learning Style Inventory) as

the primary tool for assessing the learning styles of 177 students. Based on the profiles

developed using the LSI, the authors presented an assessment of the learning characteristics of

the sample. This assessment was then followed by suggestions for working with students

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identified as having the various learning styles. The research conducted by Vincent and Ross

supports the case for understanding how each student learns best in the classroom. However, the

drawback of the study is the lack of evidence that demonstrates whether or not instructional

strategies adapted for the various learning styles did, in fact, produce improved learning results.

Although the study provides a strong foundation for identifying the learning styles of students,

additional research would be helpful in determining how best to put the theory into practice.

Statement of value. A theme that underscored the research conducted by Vincent and

Ross (2001) is that “a better understanding of learning styles can benefit not only educators but

also their students” (para. 16). Although the authors focused on three particular learning styles,

the general characteristics and suggestions presented provide a checklist for assessing classroom

practices and for ensuring that instructional activities are being tailored to the varied needs of the

students. Each student enters with classroom with unique learning needs. Educators have the

responsibility to attempt to match instruction with as many of those needs as possible. The

characteristics found in the article provide a basis for determining what types of activities will

best serve a given group of students.

Wang, K. H., Wang, T. H., Wang, W. L., & Huang, S. C. (2006, June). Learning styles and

formative assessment strategy: Enhancing student achievement in web-based learning.

Journal of Computer Assisted Learning, 22(3), 207-217. doi: 10.1111/j.1365-

2729.2006.00166.x

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Summary. The authors stated that “…given that students have diverse backgrounds,

abilities, and knowledge bases, teachers who are able to use various instructional strategies have

been shown to be more effective than those who just use single teaching strategies” (Wang,

Wang, Wang & Huang, 2006, para. 39). Based on this premise, Wang, Wang, Wang, and Huang

(2006) studied learning in Web-based environments. Using Kolb’s Learning Style Inventory

(LSI), the authors associated students with four learning styles: Diverger, Assimilator,

Converger, and Accommodator (Wang et al., 2006, para. 3). Groups of the middle school

students were also given different types of formative assessments. The results suggested that

paying attention to the learning styles of students plays an important role in their success with

various activities in the classroom. In addition, the authors found that effective formative

assessments can also serve to support student learning. The authors conceded that there are

diverse perspectives on the characterization of learning styles. However, the existence of such

styles is generally accepted and educators can capitalize on their knowledge of student learning

styles in order to effectively design classroom instruction.

Critical analysis. Wang et al. (2006) studied classroom situations by focusing on three

important aspects of learning and teaching—learning styles and the alignment of those styles

with teaching strategies; the role of formative assessment in instruction; and the nature of

technology as a learning tool. The conclusions of the authors tended to support the notion that

the unique learning needs of students play a critical role in the effectiveness of teaching

strategies, particularly when those strategies are technology-based. It was also concluded that

properly designed formative assessments that provide beneficial feedback serve as constructive

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learning supports for students. The use of technology-based formative assessment strategies has

the added benefits of flexibility, repetition, and immediacy (Wang et al., 2006). The research

presented in the article supported the importance of understanding the intimate connections that

support the balance among student needs, teacher practices, and technological learning tools.

Statement of value. This article addresses the important role of using formative

assessments in the learning process. Various forms of information technology (IT) can be

effectively used as tools for providing students with the important feedback necessary to

successfully progress through the content. It is essential, though, that the assessment activities

build upon on the strengths associated with IT in order to enhance the learning process. In

addition, the alignment of teaching strategies with the learning styles of the students must also be

considered in order to ensure that individual students will benefit from the learning activity. The

conclusions of the authors can serve as a sounding board for educators as they evaluate and

reflect on the effectiveness of the learning and assessment activities utilizes in the classroom.

Watts, M. (2003, October). The orchestration of learning and teaching methods in science

education. Canadian Journal of Science, Mathematics, & Technology Education, 3(4),

451-464.

Summary. A large amount of research supports the view that the inherent learning styles

of students play a significant role in how and what students learn in the classroom (Watts, 2003).

In addition, the strategy of differentiation has been promoted as a classroom practice that

teachers can use to align teaching strategies/styles with the individualized learning needs of

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students. Although the goal of individualized instruction is the ideal that many educators strive

for, Watts (2003) acknowledged that the parameters and logistics of the typical classroom setting

make total differentiation nearly impossible. First, the identification of a student’s learning style

encompasses a wide range of complex, interrelated variables that are dynamic in nature. Second,

the teaching styles of educators reflect the same level of complexity and can be difficult to

strictly define. Finally, the nature of a given concept or learning task influences both how a

student might react as well as how a teacher may present the material. Watts’ conclusion is that

teachers need to strive to optimize learning as much as possible in the differentiation process

with the understanding that total differentiation is not necessarily a realistic goal. In addition,

students need to learn (or be educated about) learning strategies that can help them cope during

times when instructional activities are not necessarily aligned with their preferred learning styles.

Critical analysis. Watts (2003) synthesized information from a variety of different

resources and theories to support the point that differentiated instruction is important for

optimizing student learning. However, the author acknowledged that the complexity of

identifying learning styles makes it difficult to completely differentiate instruction for every

student in a given classroom. Learning is a dynamic process that involves a unique combination

of both static and fluid characteristics. The author attempted to adapt the metaphor of teaching

as “orchestration” (Watts, 2003, para. 39) from that of the harmonious tones of a single, unified

band playing the same song to the balance and blend of the unique characteristics brought to the

group by the conductor (teacher) and musicians (students) alike.

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Statement of value. Learning style is an important characteristic of every student that

has a significant impact on the educational dynamic in the classroom. Although Watts (2003)

acknowledged the importance of understanding student learning needs and developing

instructional routines that accommodate those needs, he conceded that total differentiation is

difficult given the complex nature of how individuals learn. The main conclusion, though, is that

educators must strive to optimize learning for all students within the parameters of given

situation. By developing an understanding of the various learning needs of students, by

employing different teaching strategies, by educating students about various coping mechanisms,

and by working to minimize the misalignment of teaching and learning styles, educators can

anticipate greater success in the learning process.

Weiss, I., Kramarski, B., & Talis, S. (2006, March). Effects of multimedia environments on

kindergarten children’s mathematical achievements and style of learning. Educational

Media International, 43(1), 3-17. doi: 10.1080/09523980500490513

Summary. At what age should cooperative and computer-based learning be introduced

to students? Weiss, Kramarski, and Talis (2006) stated that “ the growing use of computers in

offices, factories, homes and schools is often cited as a reason for introducing computers to

children at an early age” (para. 2) and that research is showing that “…the social effects of using

computers in the classroom are ‘overwhelmingly positive’” (para. 37). Therefore, Weiss et al.

(2006) conducted a study at the kindergarten level in order to determine if cooperative learning

as well as instruction rooted in technology-based formats would be valuable for young students.

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The authors concluded that both effectively-designed collaboration and technology-based

learning were effective for young children. Although the strategies were not perfect and the age

(both chronological and cognitive) of the students presented some unique challenges, the authors

found that the opportunities to work in groups and to use computers as learning tools not only

helped a majority of students learn but they also helped students become more comfortable with

these types of learning situations. The changing nature of society is demanding that individuals

are competent in both mathematics and technology use; and the research presented in the article

suggests that the process can begin at early ages.

Critical analysis. Weiss et al. (2006) utilized a pre-test/post-test design to determine the

effectiveness of collaborative groups and computer-based, multimedia instruction on learning

and on developing the learning styles of kindergarten students. The authors found that both

techniques led to positive learning results. In particular, the research results suggested that the

use of collaborative groups, though not perfect, helped students become more comfortable with

this type of setting. The demands of a global society require students to work effectively in

collaboration with others. Therefore, it is essential for students to develop their skills for

working in small groups as early as possible. The data presented by Weiss et al. (2006) adds

support for the importance of utilizing teaching strategies that incorporate collaboration as well

as multimedia tools; and they suggest that implementing these strategies is possible at very

young ages.

Statement of value. Although the study was conducted with young students, the results

promote the need to incorporate group work as well as instruction built around multimedia in the

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classroom. These concepts should be reflected in the design and implementation of instructional

strategies at any grade level. In addition, Weiss et al. (2006) demonstrated that these methods

can be used successfully with young people. Therefore, it seems plausible that these strategies

can also prove effective with older students working with higher level mathematical concepts.

Weiss, R. P. (2000, September). Howard Gardner talks about technology. Training &

Development, 54(9), 52-57.

Summary. In this article, Weiss (2000) summarized an interview with Howard Gardner

that dealt with the role of technology in the educational process. Several key themes emerged

from Gardner’s comments. First, technology should be regarded as a tool for enhancing

education and should not be regarded as an end in itself (Weiss, 2000). Second, Gardner’s

theory of Multiple Intelligences suggests that people learn in a variety of different ways and have

unique combinations of strengths that impact how they interact with their surroundings (Weiss,

2000). Technology can be used to underscore those strengths and maximize the learning

potential in classroom activities. Finally, the educational goals that drive the learning process

must always be at the forefront when making classroom and instructional decisions; and the role

of technology in those decisions must be determined based on sound educational principles.

Ultimately, “technology…is here to stay” (Weiss, 2000, para. 3) and it is essential for educators

to carefully assess how that technology can enhance the learning process.

Critical analysis. Gardner’s theory of Multiple Intelligences provides an important

perspective on how people interact with their surroundings, learn and process new information,

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and transform cognitive development into external action. New forms of technology have

changed how the role of formal education influences individual learning. The perspective of

Gardner suggests that the use of technological applications in education can serve to support and

enhance the unique abilities and skills of individuals. However, the use of technologies alone

cannot change how learning occurs in the classroom. The goals of each instructional activity

must be carefully considered and technology must be integrated only when it can serve to truly

help students learn. In other words, the many advantages of technology can only be maximized

if the applications are used appropriately and effectively.

Statement of value. The themes outlined in the article provide general criteria for

assessing the role of technology integration in the mathematics classroom. First and foremost,

technology use can only be justified if it truly aids in helping students learn a given concept.

Secondly, recognition of the unique learning styles of students can provide a basis for

determining how technology can best be used. Howard Gardner highlighted the fact that

technology can be effectively used to support various intelligences and learning styles (Weiss,

2000). Finally, technology is not necessarily a panacea for all mathematics instruction, and

teachers must carefully weigh the pros and cons of technology use for various mathematical

concepts.

Wiest, L. R. (2001). The role of computers in mathematics teaching and learning. Computers in

the Schools, 17(1/2), 41-56.

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Summary. The use of computers in the mathematics classroom has had a significant

impact on instruction and learning. Wiest (2001) suggested that technology has influenced

mathematics instruction by reducing the importance of certain skills, changing pedagogical

philosophies, and “making some mathematical topics and skills…more accessible” (para. 2). To

this end, there are many different ways that technology can be integrated into the classroom, and

there are many different applications that have been developed for use in mathematics. Tool

software is designed to support other learning goals, and instructional software is intended to

teach students various concepts (Wiest, 2001). Categories of instructional software include drill-

and-practice, problem-solving programs, simulations, games, concept instruction, assessment

tools, remediation, and tutorials. In addition to applications geared specifically toward

mathematics, Wiest noted that the Internet, general purpose software (i.e. Word™, Excel™,

etc.), and computer programming tools can be used to support mathematics instruction. A

plethora of technologies are available to enhance learning in mathematics; however, these

applications must complement the role of the teacher and can only alter the nature of instruction

if they help promote higher-order thinking and mathematical reasoning.

Critical analysis. The survey of computer use in mathematics instruction provided a

general overview of how technology is permeating the numerous facets of education and

applications of mathematical concepts in the real world. Wiest (2001) emphasized that

mathematics instruction must reflect what is happening outside of the classroom. In order to

make sure that the use of technology is effective, educators must carefully consider how various

applications can be infused into instruction, must address issues of equity for all students, and

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must ensure that technology use does not replace the essential skills that students need to

develop. The research conducted by the author reflected the dynamic nature of technology and

the importance of promoting the continued evolution of mathematics instruction.

Statement of value. This article emphasized several key points about the role of

technology in the mathematics classroom. First, there are many different types of applications

available to enhance learning and instruction. The categories outlined by Wiest (2001) can be

used to identify where various applications are best integrated into learning activities. Second,

the author emphasized the significant role that the teacher plays in selecting, implementing, and

integrating technology into instruction. Finally, the effective use of technology hinges on

attaining the goal of higher-order thinking on the part of the students. Therefore, the use of

technology can be assessed based on how learning is enhanced for the students.

Williamson-Schaffer, D., & Kaput, J. J. (1998/1999). Mathematics and virtual culture: An

evolutionary perspective on technology and mathematics education. Educational Studies

in Mathematics, 37(2), 97-119.

Summary. The role of technology in mathematics education was addressed from the

point of view of the evolutionary changes in thought and communication experienced by

humankind throughout history. Williamson-Schaffer and Kaput (1998/1999) reviewed the

theories of Merlin Donald. According to Donald, humans have experienced evolutionary

changes in cognition that have significantly influenced human culture. The early stages of

human cognition revolved around “episodic” (Williamson-Schaffer & Kaput, 1998/1999, para.

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6) thought and eventually transformed into “mimetic” (para. 7) cognition. The development of

verbal communication led to a “mythical” (para. 8) culture. Finally, the creation of symbols and

written language transformed the culture into a “theoretic” (para. 10) culture that allowed people

to store information in external forms. This transformed how people were able to process and

connect information.

The authors of the article theorized that modern technology has led to a new evolutionary

stage in human cognition. Williamson-Schaffer and Kaput (1998/1999) referred to this stage as

a “virtual” culture (para. 34). The characteristic of modern technology that led the authors to this

conclusion is the ability of various technological devises to perform certain cognitive processes

for people. By removing the necessity of processing certain algorithms, pieces of information,

and so on, the cognitive demands for individuals have changed. The implication for the

mathematics classroom is that the need to learn theoretical processes and computational

algorithms should no longer be the focus of instruction. Rather, the authors argue that problem-

solving, exploration, and concept-based understanding of mathematical ideas should be the

primary focus with computational processes (which can be performed by available technologies)

taught as secondary, supportive structures (Williamson-Schaffer & Kaput, 1998/1999).

Although knowledge of abstract, theoretical mathematics should not be completely disregarded,

individuals living in a virtual culture require a different focus in mathematics in order to

successfully function in the modern world.

Critical analysis. Williamson-Schaffer and Kaput (1998/1999) provide an evolutionary-

based argument for making changes in how instruction occurs in the mathematics classroom.

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Current technologies, according to the authors, are supporting an evolutionary change in human

cognition that requires educators to adapt how concepts are presented in the classroom.

Calculators, computers, and other software serve roles to both store information and perform

various mathematical processes. By moving these processes to an “external field” (William-

Schaffer & Kaput, 1998/1999, para. 22), the internal cognitive demands change. In the area of

mathematics, the implication is that the focus must change from a study of computational

algorithms and theoretical concepts to applications for problem solving. This paradigmatic shift

requires educators to rethink mathematics instruction in order to meet the evolving needs of

individuals living in a virtual culture.

Statement of value. The article written by Williamson-Schaffer and Kaput (1998/1999)

provides a basis for addressing the role the technology and problem-solving in curricular and

instructional decisions. The authors argued that the traditional focus on computation and

algebraic algorithms must shift to general concepts, applications, and problem-solving

techniques. With this in mind, educators can assess how various instructional activities will best

serve the needs of students who have technologies available to them that can perform routine

processing tasks. Although the authors do not suggest that the teaching of basic arithmetic or

algebraic skills should be abandoned, their conclusion suggests that the mathematical

connections to problem-solving and applications should be moved to the forefront of

instructional design.

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Literature Review Essay

In 1938, John Dewey advocated for an educational philosophy that emphasized the need

to incorporate personal experience and real-world applications into the formal education of

students. Dewey noted the direct link between how and what students learn and the natural,

social, and technological aspects of each student’s surroundings. Despite the decades that have

passed, Bruce (1998) noted that the underlying philosophy championed by Dewey still has

significant influence in today’s educational setting. The only differences are that the experiences

of contemporary students are influenced by changing social perspectives and dramatic

technological advances. In many ways, the same notion holds true for the learning and

intelligence theories endorsed by Gardner, Martinez, and Vygotsky. In order to successfully

bridge the gap between theory and classroom practice, it has become essential for educators,

researchers, and scholars to explore the impact of today’s technologies on educational pedagogy

(Abramovich, 2005). This is particularly true in the mathematics classroom (Berry & Smith,

2006). Therefore, a review of current literature will explore: the effectiveness of instructional

methods that incorporate technology into student learning in mathematics classrooms; the

alignment of technology-based instruction in the mathematics classroom and student learning

styles within the framework of the theories of Dewey, Gardner, Martinez, and Vygotsky; the

obstacles that hinder technology integration in the math classroom; and the potential

recommendations for overcoming those barriers. An analysis of the research methods utilized in

the studies highlighted in this review as well as suggestions for further research and applications

of the synthesized theories and research will also be presented.

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Effectiveness of Technology-Based Instructional Strategies

When it comes to assessing how technology-based instructional strategies are utilized in

the classroom, particularly in the mathematics classroom, the term effectiveness provides only a

broad notion of how a particular technique worked and may leave many unanswered questions.

In what ways was the instruction effective? For which students did the activity work effectively?

Was the use of a particular technology more effective than an alternative method of instruction?

Did the use of a technology-based application effectively maximize student learning while

minimizing costs, time factors, and/or other resources? Abramovich (2005) emphasized that the

use of technology can only be justified if that use leads to “a qualitative change in how we teach

or learn” (para. 2). Therefore, it is essential to establish tangible criteria for characterizing

effective technology-based instruction. Current research on this topic provides a wide variety of

strategies, models, identifiers, and so forth that illustrate the effective use of technology in the

learning process. In general, though, the common themes that surface in the literature include:

promoting higher-level thinking through the use of relevant applications and contexts; moving

students toward the independent use of technology in problem-solving; identifying the critical

considerations in the design of technology-based activities; using technology in the assessment

process; identifying the key characteristics of the effective use of technology; and defining the

connection between technology and improved learning outcomes.

Higher-level thinking. During the design and implementation of instructional activities,

educators must have a clear sense of the intended learning goals; and a primary goal that is

emphasized in the literature is the promotion of higher level thinking by the students (Wiest,

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2001). This is particularly true in the mathematics classroom where Henningsen and Stein

(1997) suggested that instruction needs to help students develop a “mathematical disposition”

(para. 3). The use of technology in this process simply adds a new dimension to what must be

considered during instructional design. Fortunately, the plethora of applications afforded by

current technology has made it possible to shift the instructional focus away from routine

algorithms and/or the theoretical treatment of mathematics and more toward problem-solving,

exploration, and conceptual understanding (Williamson-Schaffer & Kaput, 1998/1999).

Therefore, the successful implementation of technology-based instruction must allow students to

engage in higher-level thinking and problem-solving. Two approaches that can make this

happen include opportunities for students to explore concepts and to engage in metacognitive

activities.

Technological applications can be used in a variety of different ways in the classroom.

Laborde (2007) stated that the flexibility of technology makes it possible for students to explore

information and experiment with mathematical concepts. In addition, the clever use of

technologies by educators can also challenge students to further explore their own assumptions

and mathematical conceptions (Passey, 2006). The process of exploring ideas promotes the

basic tenets of the constructivist philosophy (Solvie & Kloek, 2007) and engages students in

what Mok, Johnson, Chueng, and Lee (2000) referred to as the “hypothesizing-verifying cycle”

(para. 5). In their study, Mok et al. found that the students who went through this cycle tended to

demonstrate higher levels of reasoning. The effective infusion of technology into learning

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activities makes it possible for educators to encourage students to actively engage in material

rather than just passively receive information.

Another key feature of instruction that leads to higher-level thinking is metacognition.

Both Solvie and Kloek (2007) and Mok et al. (2000) define metacognition as the process of

consciously reflecting on the thinking process itself. Metacognition requires students to do more

than simply regurgitate a process. Students need to take the time to reflect on the meaning of

that process. For Mok et al., higher-level processing of a concept occurs as students struggle

through a “cognitive conflict” (para. 33), engage in metacognition, and construct meaning.

Metacognition has the potential to occur in any learning situation, so Evuleocha (1997) took the

concept one-step further and promoted the use of “meta-learning” (para. 2). Meta-learning

involves building opportunities into a learning activity for students to actively reflect on concepts

as well as the methods being utilized in the activity to promote learning. Not only do students

engage in the curricular concepts, but they also reflect on how a given technology, application, or

teaching technique influences their construction of knowledge. Meta-learning, in conjunction

with the utilization of the benefits of technology, creates a “two-way effect” (Kahveci &

Imamoglu, 2007, para. 1) that allows students to effectively construct lasting meaning behind a

given concept. Like any other learning tool, the use of technology can only be useful if it

promotes student learning. Metacognitive activities promote higher-level thinking and allow

students to actively reflect on the content and the role of technology in their own learning.

Relevant applications and contexts. In order to engage students in the general content of

a mathematics course, current research points to the need to help students make relevant

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connections to real-world situations where the content can be applied. Technology is one avenue

that can lead to these connections. In a study of traditional versus technology-based instructional

pedagogy, Cohen (2001) found that students who were given real opportunities to integrate the

use of technology in their learning found more relevance in the content. Laborde (2007) also

emphasized the inherent relationship that exists between mathematics and technology; and the

author goes on to note that technology permeates almost every aspect of daily life. Therefore,

technology-based instruction in the mathematics classroom can enhance the learning process as

well as lay the foundational links between content and application.

A fundamental tenet of Dewey’s progressive educational philosophy was the role of

experience in learning. The vital role of student experience in today’s classroom is still

supported by current research. Abramovich (2005) noted the necessity for presenting students

with non-routine problems while Bruce (1998) emphasized the importance of using available

technologies as aides in learning and in the problem-solving process. Technology can serve

multiple roles by providing tools for exploring information, analyzing data, and generally

“bringing to life mathematical objects” (Forster, 2006, para. 2). One approach to providing

students with opportunities to solve complex, real-world problems is the use of “model

facilitated learning” (MFL) (Spector, 2001, para. 34). This approach is intended to promote

higher-level thinking through problem-solving situations that require students to apply content in

a novel manner. No matter what approach is actually employed, students require opportunities to

practice thinking outside of the box as they prepare for entrance into the real world.

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The International Society for Technology in Education (ISTE) (2005) proposed that

individuals need the following skills in order to successfully compete in today’s world:

information technology (IT) skills, information literacy skills, problem-solving skills,

collaboration skills, flexibility, and creativity. The integration of technology-based instruction in

the mathematics classroom makes it possible for students to begin the development of these

skills. In addition, the incorporation of relevant applications helps students to build the

connections between what they are learning and how that content is essential as they become

members of the modern workforce.

Independent use of technology. An additional goal associated with the use of

technology in the classroom involves moving students toward the independent use of technology

in both learning and in the problem-solving process. Like any other curricular area, students

must be supported as they learn how to use technology and apply the functionality of given

technologies in various situations. For this reason, educators must pay close attention to the

students’ relative zones of proximal development (ZPDs) (Reid-Griffin & Carter, 2004). By

providing the necessary scaffolding during instruction, students can gradually develop their

abilities to carefully choose which technologies to employ and to effectively use them to achieve

desired learning outcomes (Reid-Griffin & Carter, 2004; Solvie & Kloek, 2007). In order to

ensure that designed instruction is effectively guiding students, Dahl (2006) suggested that

educators attend to the “zone of proximal teaching” (ZPT) (para. 39). In other words, teachers

need to make sure that their teaching methods are within reach of the students and can effectively

guide them through the learning process. The role of technology can make it possible to tailor

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the ZPT in order to effectively support student learning, but the function and use of that

technology must be deliberately considered to ensure that student learning is maximized.

In assessing the effectiveness of technology in mathematics instruction, educators need to

consider a variety of factors regarding the overall learning goals of a lesson. In particular, an

effective instructional method will promote higher-order thinking, will establish a relevant

context for the application of content, and will guide the students toward the independent use of

technology as a resource to support greater learning outcomes. With these ideas in mind, it

becomes possible to identify more specific characteristics of effective technology-based

instruction.

Considerations for instructional design. It is essential that instruction be designed and

implemented with the learning goals for the students continuously in mind (Solvie, 2007).

Instruction is the period of time when the educator is directly engaged in students’ learning

processes; and the various factors that impact the classroom environment can alter the actual

learning outcomes as compared to the initial intent behind a learning activity. In addition, the

planned role of technological applications in an activity can influence how the learning goals are

(or are not) met. Ultimately, the educator must consider a variety of things while implementing

technology-based instruction including: personal perspectives/biases toward the use of

technology; the intended purposes behind the implementation of a particular technology and the

manner in which the technology facilitates learning in the classroom; and the relationship

between the technology and the learning needs of the students.

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Ruthven and Hennessy (2002) conducted a study to evaluate the role of technology in the

mathematics classroom. Specifically, the authors were exploring the questions of why

technology-use in the mathematics classroom tended to be limited and/or sluggishly integrated

despite the intimate connection that exists between mathematics and technology. Ruthven and

Hennessy (2002) stated the results of the study were intended to be a “starting point for further

development” (para. 14); and ten major themes in the data that described mathematics teachers’

perspectives on the purpose of technology in instruction were identified. These themes were:

ambience enhanced, restraints alleviated, tinkering assisted, motivation improved, engagement

intensified, routine facilitated, activity effected, features accentuated, attention raised, and ideas

established. It is important for classroom teachers to question their own perceptions of the role

of technology in instruction. Is a particular application being used as a novel way to simply

reinforce traditional approaches to teaching the content or is the activity promoting more student-

centered learning? Does the use of a technology reflect a pedagogical shift that can lead to a

more constructivist approach to instruction? Is an educational technology being incorporated in

a manner that reflects what students will encounter in real-world situations? Hoyer (2005/2006)

emphasized that the use of technology in the learning process can (and should) go well beyond

the mere presentation of material. Therefore, the model created by Ruthven and Hennessey

provides a basis for educators to assess their positions on technology-use in the mathematics

classroom.

Technology applications vary in terms of their forms, intended purposes, and uses in the

classroom. Types of educational technologies include (but are not limited to): tool software like

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calculators, drawing programs, and gradebooks; instructional software such as drill-and-practice

programs, problem-solving programs, simulations, games, concept-focused programs,

assessment tools, remediation applications, and tutorials; the Internet; general purpose software

such as Word™, Excel™, PowerPoint™, and so forth; and computer programming tools (Wiest,

2001). Each of these applications has an intended purpose and can be incorporated into

instructional design in unique ways. However, a delicate balance between planned outcomes,

implementation, and the actual learning results must be attended to in order to effectively utilize

technology.

A primary goal for students in the mathematics classroom is developing a “mathematical

disposition” (Henningsen & Stein, 1997, para. 3). According to Henningsen and Stein, achieving

this way of thinking involves nurturing the underlying impulse in students to explore patterns, to

be flexible and think outside of the box, to communicate ideas, and to thoughtfully reason at a

higher level. Carefully designed instruction can foster a learning environment that promotes

higher level thinking and encourages students to “do mathematics” (Henningsen and Stein, 1997,

para. 4). However, Henningsen and Stein identified four scenarios that can result during the

course of instruction based on the alignment (or lack thereof) of various factors that influence the

classroom: maintaining cognitive demands, decline into procedural thinking, decline into

unsystematic exploration, and decline to no mathematical activity. Unfortunately, the latter three

of the four scenarios result in learning outcomes that do not achieve the intended learning goals.

Educators are often challenged to facilitate a classroom environment where students are

continually pushed to think and work at high cognitive levels. The use of technology does have

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the potential to support this type of learning. Glover, Miller, Averis, and Door (2007) found that

technology can be used at three essential levels. The “supported didactic” (para. 11) level is the

case where teachers utilize technology to simply augment traditional, teacher-centered

instruction. The “interactive” (para. 11) level is the case where technology is used to engage

students and promote different ways to explore information, but the application is not used to the

full potential for transforming instructional practice. The level of “enhanced interactivity” (para.

11) is the case where the technology is utilized in a manner that transforms the pedagogical

foundation behind instruction. Ultimately, educators should strive to incorporate technology-

based instruction that maintains high cognitive demands for the students through opportunities to

work interactively with carefully chosen and integrated resources and tools.

A key component to Glover’s et al. (2007) conception of enhanced interactivity is the

capability of technological applications to be differentiated to accommodate the various learning

needs of the students. Henningsen and Stein (1997) concluded that effective instruction in the

mathematics classroom includes: building on the knowledge and experiences of the students, the

use of scaffolding to support learning, designating appropriate amounts of time to complete

activities, modeling the essential skills, and continually seeking student explanations and

interpretations of meaning. The use of technology makes it possible to differentiate these aspects

of effective instruction to meet the individual needs of students by presenting information through

a wide array of channels (ISTE, 2005). In addition, different forms of technology allow students

to access, process, and synthesize information in ways that align with individual intellectual and

learning profiles (Cohen, 1997). Kahveci and Imamoglu (2007) also noted that different types of

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interactions among students and teachers can be facilitated through the use of technology. In the

end, educators should consider how the role of technology can impact the differentiation of

instructional methods in order to meet the dynamic needs of the students.

Assessment and feedback. The role technology must be considered on multiple levels

as teachers work to design and implement instructional activities in the classroom. Teacher

perspectives, the expectations for student cognition, the advantages and limitations of

technology, and the unique characteristics of each student all factor into how and what the

students learn. In order to complete the overall picture, the use technology in the assessment of

student learning must also be addressed. Wang, Wang, Wang, and Huang (2006) emphasized the

importance of formative assessments throughout the learning process. Various technology-based

applications can be used to help the teacher formally and informally assess the progress that

students are making. Martin and Burnette (2000) also made a strong case for the use of

electronic portfolios as a means for recording, organizing, and tracking student learning.

Electronic portfolios make it possible for teachers and students to manage different types of

artifacts that reflect student learning; and this use of current technology provides a method of

communication between students, teachers, and parents (Martin and Burnette, 2000). Although

assessment is generally considered a process driven by teachers, Martin and Burnette suggest

that electronic portfolios make it possible for students to reflect on their own learning and engage

in the metacognitive process endorsed by Mok et al. and others.

Characteristics of effective technology use. The key criteria that educators should

address in order to ensure that technology-based instruction benefits students can be readily

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gleaned from the current literature. Promoting the use of higher-level thinking, engaging

students in relevant problem solving activities, encouraging the independent use of technology as

a learning tool, considering the diverse learning styles and needs of students, and so forth all

reflect sound goals and generalizations for effective instruction. However, concrete

characteristics of effective technology use are also necessary in order for educators to have

reliable indicators for judging the success (or lake thereof) of technology-infused instruction.

Several key characteristics and themes emerge from current research.

First, educators should have clearly identified and stated learning goals for the students as

well as a means for accurately assessing student progress. In order for a technology-based

interaction to be productive, educators must identify how the students are expected to respond,

the manner in which the technology handles a student’s response, and the feedback that is

presented to the student (Kahveci & Imamoglu, 2007). Wang et al. (2006) also noted that

student-technology interactions must also be accompanied by useful feedback that will help

students continue to progress. Without explicitly stated learning goals and objectives, the

potential associated with technology-enhanced activities may be compromised.

Second, the function of a particular technology in a given activity needs to be understood.

The chosen technology may be used to: speed up a particular activity, provide opportunities for

students to explore, provide alternative methods for solving beyond traditional strategies, and/or

serve as a context or focus for learning (Laborde, 2007). These functions (and potentially others)

can be used to design new instructional techniques or simply enhance activities that have already

been utilized in the classroom (Mok, Johnson, Cheung, & Lee, 2000). Various technologies can

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be used to meet a variety of functionality requirements and can be used to create multiple

representations of the content (Mayer, 2003). Identifying the purpose and the function of

technological applications is essential for establishing the context and priorities for instruction.

Third, activities that involve instruction should reflect elements of quality design. Mayer

(2003) identified four “effects” (para. 18) or design methods that should be addressed. The

multimedia effect refers to the fact that students tend to respond more efficiently to information

if it is presented in multiple formats such as text, audio, and visual aids. The coherence effect

suggests that students will learn more from a presentation or activity if extra, nonessential

material is minimized. The spatial contiguity effect emphasizes that the multiple representations

(i.e. text, pictures, etc.) of a given concept should be physically oriented in close proximity to

one another. The fourth consideration, the personality effect, deals with the fact that information

presented in a conversational style tends to be better comprehended by learners. A theoretical

model developed by Moreno (2006) referred to as the CTLM (Cognitive Theory of Learning

with Media) includes several principles that complement the design elements proposed by

Mayer. Moreno proposed that both the methods employed by a teacher as well as the chosen

technologies will affect the learning process. Based on this notion, Moreno (2006) identified 10

principles to guide technology-enhanced instructional design: modality, redundancy, temporal-

contiguity, spatial-contiguity, coherence; multimedia, personalization, interactivity, guidance,

and reflection. The first five principles align with Mayer’s conceptualization of the coherence

effect. The last five principles are intended to promote students’ abilities to process information

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(Moreno, 2006). In order for the potential of technology-based activities to be maximized, basic

design elements must be considered by the teacher.

Finally, the role of a particular technology must be integrated into the core structure of a

lesson plan. Glover et al. (2007) found that the effective use of technology in the classroom

setting meets the following criteria: 1) technology applications provide the basis of the lesson

structure; 2) instruction involves multiple representations and visualizations of the concepts; 3)

activities encourage active thinking; 4) the lessons logically and sequentially progress from

simple to more complex ideas; 5) activities provide quick feedback for both students and the

instructor; and 6) recall is used to tie one lesson to the next. The characteristics put forth by

Glover et al. illustrate that the incorporation of technology in instruction must be woven into the

fabric of the lesson and cannot be simply viewed as an add-on to existing pedagogical practices.

Using technology to improve learning. In an effort to assess the effectiveness of an

educational technology, an important theme permeates the literature—namely, the mere presence

of technology in the classroom or the inclusion of the technology-based application in a lesson

does not guarantee that student learning will be enhanced (Forster, 2006; Reid-Griffin & Carter,

2004; Solvie & Kloek, 2007). In other words, the use of technology must be carefully and

deliberately considered during the planning phases of instructional design. Additionally, that

consideration must continue during the implementation of the activity in the classroom and into

the evaluation process after a lesson has been completed. Debevec, Shih, and Kashvap (2006)

stressed that different types of activities (traditional, constructivist, technology-based, and so

forth) will work differently depending on the students, the teacher, the content area, and the

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available resources. The parameters of a particular learning environment will also influence the

effectiveness of a chosen strategy (Forster, 2006). In addition, the educational technology that is

employed is generally not a learning end in itself but is an additional tool to assist in achieving

some other outcome (Cohen, 1997). The presence of technology alone cannot guarantee a

positive improvement in student learning as compared to other instructional methods. Only the

careful consideration, planning, and execution of technology-based instruction can lead to new,

effective, and relevant shifts in pedagogical paradigms.

Labeling a technology-based instruction as effective requires an educator to assess a

broad array of considerations. The goals of a lesson must be clearly understood and the role of

technology needs to have a clear purpose in aiding the achievement of those goals. Expectations

for cognitively demanding activities must be set and maintained during the implementation of a

lesson. The characteristics of quality design and the use of multiple representations should

surface during instruction as well. All planning aside, a truly effective lesson can only be labeled

as such if it meets the learning needs of the students. Vincent and Ross (2001) stated: “…

learning styles function as teaching blueprints in some respects. They indicate a student’s

preferred method of learning and guide the development of instructional strategies that

incorporate the appropriate content and context” (para. 2). The use of technology in instructional

activities must be steered by the unique learning styles and needs of the students who will engage

in those activities.

Alignment of Technology-Based Instruction and Student Learning Styles

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The learning and intelligence theories promoted by John Dewey, Howard Gardner, Lev

Vygotsky, and Michael Martinez have withstood the test of time with regard to their applicability

to educational practice. This is particularly true even in an age where the newest technologies

have transformed the landscape of the world. Current research on the use of various

technologies in the educational setting has led to new understandings of how pedagogy can (and

should) change in order to meet the modern needs of today’s students. A significant theme that

has emerged in the literature that closely ties classic learning theories with modern technology is

the role of student learning styles in the design, implementation, and ultimate success of

educational strategies. Dunn, Beaudry, and Klavas (1989) stated that “learning style is a

biologically and developmentally imposed set of personal characteristics that make the same

teaching method effective for some and ineffective for others” (para. 1). Technology provides an

opportunity to apply the essential characteristics of educational theory to classrooms that house

diverse groups of students with diverse learning needs (ISTE, 2005). Therefore, the connections

between the fundamental characteristics of the learning theories of Dewey, Gardner, Vygotsky,

and Martinez will be applied to the basic concept of learning style, the various models that can

be used to describe and identify student learning styles will be explored, and the role of

technology in maximizing student learning based on their individualized needs will be

delineated.

Theoretical perspectives and learning style. Despite the fact that John Dewey wrote

about his educational philosophy during the mid-1900’s, Bruce (1998) pointed out that the basic

tenets of Dewey’s theory of experience are as prevalent in today’s classroom as ever before. An

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important component of the theory is that the experiences of the students can and should inform

how instruction is delivered (Bruce, 1998). The individualized experiences that students bring to

the classroom contribute to unique perspectives on content and to differentiated learning needs

for each person. Therefore, the basic construct purposed in Dewey’s pragmatic approach to

education establishes the importance of attending to the learning styles of the students when

developing learning activities.

Dewey’s theory calls for educators to not only consider the unique learning needs of the

students, but it also requires teachers to reflect on how the students are asked to assimilate the

content. The constructivist approach to teaching and learning embodies the notion that students

need to assume a certain level of control in developing their understanding of concepts and

applications (Solvie & Kloek, 2007). Furthermore, constructivism assumes that students will

internalize information and construct it in a manner that aligns with how they learn best (Cohen,

1997). Results from current research suggest that the constructivist approach proves beneficial

to students; and technological applications can facilitate a learning environment built on

constructivist and experiential practices. For instance, Abramovich (2005) promoted the use of

“Type II applications of technology” (para. 2) where students are given nearly complete control

of how the application is used to assist in the construction of knowledge. Abramovich goes on to

note that these types of applications allow teachers to get a better understanding of how

individual students process information. In the end, the ideas promoted by Dewey over 60 years

ago still provide a strong argument for taking into account the significant role of individual

learning styles and experiences in the educational process.

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Although Dewey’s theory of experience suggested the importance of the individual in

education, Howard Gardner was one of the first scholars to question the traditional notion of

intelligence and to raise the awareness of the various abilities and contributions that qualify a

person as intelligent. Although Gardner was careful to state that the eight intelligences—

linguistic, logical-mathematical, spatial, musical, bodily-kinesthetic, interpersonal, intrapersonal,

and naturalist—are different than conceptions of a person’s learning style, his theory of Multiple

Intelligences (MI theory) can inform educators and students about the potential avenues that can

be taken in the classroom to maximize learning (Nelson, 1998). As Weiss (2000) noted, “it’s

people’s individual strengths (and weaknesses) that Gardner is talking about when he proposes

these separate intelligences that all human beings have…” (para. 11). MI theory brings to the

forefront the fact that each individual has a unique combination of intelligences that influences

how he or she learns, interprets the surrounding world, and contributes to society.

One of the most relevant locations for the application of MI theory is in the classroom

(International Society for Technology in Education, 2005). Martin and Burnette (2000) found

that the framework of MI theory allowed them to “work smarter” (para. 6) in the classroom. By

designing activities that resonated with several of the intelligences identified by Gardner, the

activities proved to be more simultaneously effective for a wider array of students with diverse

learning styles (Martin & Burnette, 2000). Furthermore, the use of an electronic portfolio in

conjunction with activities rooted in MI theory provided opportunities to assess the true nature of

the students’ understanding of the material (Martin & Burnette, 2000). Technologies like

electronic portfolios or other web-based applications make it possible to design instruction that

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accommodates the diverse learning needs of students (Nelson, 1998). The ISTE (2005) stressed

that “there is no longer a one-size-fits-all solution for providing instruction” (para. 10); and

Gardner’s theory establishes a framework for tailoring instruction to fit the various needs of

students.

The basic educational theories of Dewey and Gardner provide a general model for what

needs to happen in the classroom in order to help students successfully (and efficiently) achieve

success, but Lev Vygotsky put forth a teaching construct—the zone of proximal development

(ZPD)—that aids in transforming theory into practice. A student’s ZPD represents the gap

between what that student already knows or can perform and the next stage of development

where the student needs assistance to accomplish the given task (Dahl, 2006). Like the theories

of Dewey and Gardner, Vygotsky’s notion of the ZPD is unique for each person and requires

educators to pay close attention to the individual learning needs of each student.

From a pedagogical standpoint, Dahl (2006) put forth the concept of the zone of proximal

teaching (ZPT) to parallel the student’s ZPD. In order to ensure that instructional activities can

help students move through their respective ZPDs, the ZPT must be carefully considered. In

addition, the types of technology or other resources must be chosen so that the ZPD and the ZPT

can align as much as possible. Abramovich (2005) suggested that Type II applications of

technology are flexible enough to allow students to manipulate information in a manner that

matches their ZPDs. Various technologies can enhance a teacher’s ZPT and can serve as

supports as students navigate their ZPDs to reach the intended learning goals.

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In a study conducted by Kumar, Kumar, and Smart (2004), the results suggested that the

learning styles of the students could be influenced and even potentially changed based upon

various classroom conditions as well as the types of instructional activities employed by the

teacher. Cohen (2001) found a similar result in a separate study conducted several years earlier.

The conclusions drawn by these researchers tend to support the basic premise of Martinez’s

notion of “cultivating” (Martinez, 2000, p. 147) intelligence through education. Martinez’s

theory emphasizes that intelligence is neither predetermined nor static and can be developed

through learning. Current research seems to suggest that addressing the unique learning styles of

students and establishing learning environments that are accommodating to their needs

capitalizes on the flexibility of intelligence and the ability for individuals to continually develop

and refine their cognitive skills.

Students entering the classroom bring with them certain skills, attitudes, and experiences

which shape how they learn. Educators have the task of developing lessons that align to and help

strengthen the particular learning styles of the students. They also have the challenge, though, of

implementing instructional activities that push the development of weaker skills and/or coping

mechanisms to accommodate “mismatched” learning situations (Kumar, Kumar, & Smart, 2004,

para. 11). Ever-expanding technologies are providing more opportunities for educators to

individualize instruction for students so that they are given opportunities to learn through their

preferred learning styles (Cohen, 2001). Cohen (2001) stated:

This study suggests that an environment that is actively engaged in many of the reform

efforts promulgated in the literature—such as establishing a technology-rich school,

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using constructivist methods of instruction, employing project-based teams that solve

problems, and discouraging the use of lecture—can have an even greater effect on student

learning style. (para. 48)

Proper consideration of the unique learning styles of students and the basic premises of the

foundational intelligence theories that describe cognition can aid in creating modern classroom

environments that can effectively and efficiently maximize student learning.

The theories of Dewey, Gardner, Vygotsky, and Martinez were instrumental in changing

educational philosophy and pedagogy. At present, technology is serving as the catalyst for

sweeping social changes, yet current research continues to produce results that support the basic

premises of the theorists. In particular, recognition of the impact of the dynamic learning styles

that students bring to the table is leading to the development of models that characterize what

students need as they progress through their unique educational journeys.

Learning style models and descriptions. A plethora of models that attempt to

characterize learning styles can be found throughout the current literature on intelligence,

learning, and education. As Dunn et al. (1989) pointed out, there are many different ways to

define, assess, and identify a learning style, but the key lies in recognizing that every student has

a unique style which influences how he or she functions in a learning setting. Dunn et al. (1989)

stated: “Every person has a learning style—it’s as individual as a signature” (para. 2). The

models for learning style vary in terms of focus and complexity.

One of the most basic models for identifying learning styles is based on brain

hemisphericity. Ali and Kor (2007) stated that “left hemispheric dominant learners are

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analytical, verbal, linear, and logical, whereas those right-hemispheric dominants are highly

global, visual, relational, and intuitive” (para. 4). Vincent (2001), on the other hand, divides

learning styles into three main categories: auditory, visual, and kinesthetic. Many other models

tend to utilize dichotomous pairings to characterize students. For instance, the CULTIS model

identifies the pairings of conscious and unconscious, language and tacit, and individual and

social (Dahl, 2006). Kumar et al. (2004) endorsed the categories of independent and dependent,

competitive and collaborative, and avoidant and participant. Research conducted by Ali and Kor

(2007), Graf, Viola, Leo, and Kinshuk (2007), and Watts (2003) supported models that utilized

characteristics including: active and reflective, sensing and intuitive, verbal and visual, and

sequential and global. Solvie and Kloek (2007) and Wang et al. (2006) cited the work of Kolb in

identifying the learning styles of Diverger, Assimilator, Converger, and Accommodator. Finally,

Ross and Schultz (1999) adopted the descriptors outlined by Gregorc—concrete sequential (CS),

concrete random (CR), abstract sequential (AS), and abstract random (AR)—as the model for

characterizing the learning styles of students.

Despite the specific terminology that is chosen to describe learning styles, many of the

models point to the same general characteristics. For example, the sequential learner tends to

like learning activities that are orderly, linear, and well-organized (Ross & Schulz, 1999). This

is similar to Ali and Kor’s (2007) description of a left-brained individual and to Kolb’s notion of

an “assimilator” as described by Wang et al. (2006, para. 3). Another example of the similarities

across the models is the conception of an intuitive learner as noted by Graf et al. (2007). The

intuitive learner tends to “prefer to learn abstract learning material…. They are more able to

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discover possibilities and relationships and tend to be more innovative and creative…” (Graf,

Viola, Leo, & Kinshuk, 2007, para. 12). The characterization aligns with Wang’s et al. (2006)

description of a diverger and Ross and Shultz’s (1999) identification as a random learner.

Comparing and contrasting the various models that exist in the literature illustrates that many of

the models tend to be more similar than different.

An underlying theme that exists across the spectrum of learning style conceptualizations

is that recognizing a learner’s style and educational needs is essential for effectively and

efficiently maximizing what a student learns (Vincent & Ross, 2001). More importantly,

educators must make every attempt to align how a particular concept is taught with the preferred

learning styles of their students (Dunn, Beaudry, & Klavas, 1989). Furthermore, instruction that

is geared to present information in a variety of ways and through a variety of different learning

channels tends to benefit all students and leads to better outcomes on assessments (Dunn et al.,

1989; Mayer, 2003).

Research suggests that the learning style of an individual student has a significant impact

on his or her success in the classroom. Attempts by teachers to align how they teach with the

learning preferences of their students tend to lead to better learning outcomes. In addition, the

alignment of teaching and learning styles tends to make almost any content area more accessible

to students. The role of educational technology in that process is showing a significant positive

impact on learning now and is offering continued promise in the future.

Technology and maximizing student learning. Dunn et al. (1989) stated that

“identifying learning styles as a basis for providing responsive instruction has never been more

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important than now, as educators meet the needs of a diverse student population (para. 33). Both

Cohen (1997) and Ross and Schultz (1999) dovetailed this research and noted that educational

technologies can significantly influence student achievement if it is utilized in a manner that

reflects the learning needs of the students. The notion of learning styles raises questions about

the individualized nature of instruction and the role of technology in making instruction more

accessible to students. In a time where student-centered instruction (Mok et al., 2000) and

concepts such as “meta-learning” (Evuleocha, 1997, para. 2) are being brought into the spotlight,

the potential of educational technology is as important as ever.

One of the most important benefits of technology is the capability of various technologies

to align to different learning and teaching styles (Nelson, 1998). For instance, multimedia

presentations illustrate information through several different means such as auditory and visual

channels. Vincent and Ross (2001) suggested that auditory learners tend to learn best through

listening, working in groups, and/or asking questions while visual learners tend to respond to

written text, graphic and pictures, videos, and so forth. Multimedia capitalizes on many of these

suggestions and makes it possible for diverse learners to assimilate the information presented.

Mayer (2003) stated that: “redesigning multimedia explanations to mesh with the way humans

learn enabled students to generate more creative solutions to problem-solving transfer

questions…” (para. 45). Not only does the use of multimedia make it possible for information to

be presented in a manner that accommodates different (and preferred) learning styles, but Dunn

et al. (1989) emphasized that content is reinforced when presented to students through additional

channels.

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In order for a technology to truly be an effective instructional tool, it must be used in an

appropriate manner (Weiss, 2000). This is particularly true in the mathematics classroom. In a

study of computer-aided instruction (CAI), the authors found that the CAI program was not

necessarily the best for all types of learners (Ross & Schultz, 1999). They also concluded that

other factors such as the level of control, attitudes about technology, and motivation influenced a

student’s success in a class. Another study conducted by Ali and Kor (2007) investigated the

connection between the use of graphing calculators in a mathematics classroom and the learning

styles of the students. The results did not suggest that graphing calculators necessarily work

better for one type of learner but emphasized that learning style could influence how the

technology is used. Berry, Graham, and Smith (2006) also raised the important issue of a

student’s “working style” (para. 4) in the mathematics classroom. They asked four important

questions:

1) How do students work with new technologies? 2) How do their working styles change

in comparison with traditional paper and pencil work? 3) How do students’ working

styles compare with the approach of their teacher? 4) What messages may be inferred

from student use with respect to their understanding of mathematics? (Berry et al., 2006,

para. 8)

Berry et al. found that students did not always use the given technology in a manner the fostered

higher-level thinking or meaningful progression toward a solution to the given problem. In the

end, technology in and of itself cannot guarantee student learning; but the careful integration of

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that technology in the learning process can provide students with opportunities and tools that can

help them meaningfully connect to the content.

Alignment of teaching style, technology, and learning style is essential in order to

promote high-quality educational experiences for the students (Dunn et al., 1989). Overall, the

benefits of technology continue to prove positive as research is conducted. Cohen (1997) found

that various technologies built into instruction helped to create more fluid interactions between

students, teachers, and other resources. Technological applications make it possible to conduct

formative assessments and to provide students with useful and immediate feedback (Wang,

Wang, Wang, & Huang et al., 2006). The use of technology to differentiate instruction can also

provide opportunities for students to develop learning strategies to cope with situations when a

particular teaching style does not align with their own (Watts, 2003). When used appropriately,

the latest technologies can enhance instruction and can make it possible for teachers to

differentiate instruction to simultaneously meet the unique learning needs of the students in their

classrooms.

The classical learning and intelligences theories of Dewey, Gardner, Vygotsky, and

Martinez laid the foundation for educational practices that are appearing in today’s classroom.

In particular, the notions of multiple intelligences, experiential learning, and so forth have been

synthesized into the concept of learning styles; and the significant role that the unique learning

style of a student has on classroom instruction repeatedly surfaces throughout the research.

Moreover, the infusion of technology into educational pedagogy has made the promise of

individualized instruction more plausible in that educators have tools that allow them to present

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information (and allow students to seek out information) through a wide range of channels. The

question that remains involves determining how best to go about making the potential associated

with individualized learning and technology a reality given the constraints within which many

schools must operate.

Obstacles to Technology Integration and Synthesis of Recommendations

Despite the promise that is associated with the integration of technology into the

classroom and the potential of that technology to make the ideals of educational theory a reality,

school districts, communities, and educators are forced to work within certain parameters.

Financial issues, the political tenor in the community, region, state, and nation, and so forth all

influence what happens in the classroom. In order to successfully bring technology into the

classroom, obstacles and solutions to overcome those obstacles must be identified. Four areas

that can be addressed include: teacher perspectives, lesson planning and classroom structure,

professional development, and district-level support and implementation.

Teacher perspectives. Norton, McRobbie, and Cooper (2000) suggested that one of

most significant barriers (or supports) to the integration of technology into the classroom—

particularly, the mathematics classroom—may be the teachers themselves. Non-constructive

behaviors of teachers can surface in a variety of different forms. Berry et al. (2006) cited a lack

of enthusiasm and missing pedagogical knowledge regarding the effective use of technology as

potential reasons for a teacher’s hesitance to use technology. Teacher-centered approaches to

instruction also tended to hinder the integration of technology (Ruthven & Hennessy, 2002).

“Latent competition among teachers has been previously cited as a hindrance to innovation”

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(Norton, McRobbie, & Cooper, 2000, para. 23). Norton et al. concluded that the individual

beliefs and educational philosophies of the teachers tend to have the biggest impact on whether

or not technology is effectively incorporated into the classroom structure.

How can the obstacles related to the attitudes of teachers be addressed? The first step lies

in identifying the perceptions of teachers with regard to educational technology. Ten themes

identified in a study conducted by Ruthven and Hennessy (2002) that describe teacher

perspectives on the role of technology ranged from facilitating routine and engaging students to

influencing the classroom atmosphere and cementing conceptualizations. Passey (2006) noted

that the teacher plays a critical role in implementing technology-based instruction. Therefore,

the next step involves convincing teachers of the instructional and learning benefits that can be

realized with technology (Glover, Miller, Averis, & Door, 2007; Holahan, Jurkat, & Friedman,

2000; Norton et al., 2000). Once teacher buy-in is established, more concrete steps can be taken

to facilitate the effective use of technology.

Lesson planning and classroom structure. A common theme associated with

educational technology is the basic understanding that the mere presence of technology itself will

not guarantee improvements in student learning (Norton et al., 2000). Technology needs to be

used strategically and with purpose in order for students to experience the learning benefits

(Berry et al., 2006). Berry et al. (2006) suggested that educators must ask important questions

about how the students will use technology. In addition to addressing these types of questions,

other issues may also be raised. Wiest (2001) suggested that there is often concern about what

students might not learn as instructional time is devoted to technology use. Passey (2006) also

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acknowledged that there are times where technology is often used to simply reinforce lower-

level thought processes. This reflects the pedagogical struggle that exists in mathematics

education between traditional approaches to teaching and more innovative strategies (Ruthven &

Hennessy, 2002). Ultimately, the structure of a lesson needs to fundamentally change in order to

effectively incorporate technology in the learning process.

Structural change in lesson planning and instruction involves a variety of considerations.

First, higher-order thinking and reasoning on the part of the students must become a high priority

on the list of learning goals (Henningsen & Stein, 1997). Second, educators need to move away

from teacher-centered routines (Glover et al., 2007). Third, Glover et al. (2007) emphasized that

the role of technology must be directly incorporated into the structure of the lesson. Among

other suggestions, the use of multimedia as well as opportunities for students to work

collaboratively with a given technology can address these issues and result in fundamental

changes (Weiss, Kramarski, & Talis, 2006). In order to promote the types of pedagogical shift

that is suggested in the literature, an assessment of the effectiveness of professional development

must be undertaken.

Professional development. Although educational technology takes on many different

forms, recent advances in computer-based applications have exploded onto the educational

scene. However, many teachers lack the “pedagogical knowledge of how to teach effectively

with technology” (Berry et al., para. 55). In addition, classroom teachers often do not have (or

choose not to) access the available research about how to effectively incorporate technology-

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based applications into their instruction (Passey, 2006). Professional development provides one

avenue for remedying this gap in teacher knowledge.

First and foremost, educators need training and updated knowledge in order to change

how they use technology in their classrooms (Glover et al., 2007). This training must be

substantial, must be long-term, and must be supported by administrators (Wiest, 2001; Holahan

et al., 2000). Glover et al. (2007) also pointed out that teachers must be given time to develop

their technology literacy and to make the necessary changes to curricula and materials. If

professional development is ineffective, ill-focused, or limited, it can become an obstacle to

integrating technology into the classroom. However, if professional development is

implemented in a strategic and well-planned manner, it can become a significant resource for

fostering change in pedagogy, student learning, and school climate.

District-wide implementation. In a study conducted by Norton et al. (2000) dealing

with teacher perceptions of technology use in the mathematics classroom, the authors concluded

that “unsupported reform driven by individuals who lack status and support is likely to fail”

(para. 56). The reasons cited by the authors for this lack of success included extensive time

demands placed on the individuals trying to make changes as well as the fear of putting student

learning behind during the integration phase of a new technology. For these reasons among

others, the integration of technology use in instruction (and specifically in the mathematics

classroom) continues to progress slowly (Ruthven & Hennessy, 2002). The literature suggests

that in order to promote the significant, effective use of technology in the classroom, reform

must be driven on district-wide level.

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The first piece that is necessary to promote successful technology integration is a long-

term commitment by the district to implement change (Holahan et al., 2000). A critical feature

of this long-term investment in change involves the sustained support by district administrators

and a commitment by those administrators to achieve stated goals (Holahan et al., 2000; Norton

et al., 2000). The second piece to the puzzle involves the infusion of technology throughout all

areas of the educational setting (Holahan et al., 2000). Glover et al. (2007) referred to this

process as changing the “culture” (para. 13) of the district. They also pointed out that the four

factors that need to be addressed in order to bring about systemic change include pedagogical

issues, the social context of learning, the technology itself, and the nature of student learning.

Weiss, Kramarski, and Talis (2006) found that the use of technology needs to begin at early ages

in order for students to learn how technology can be used in the learning process. A third

component to district-wide support for technology involves creating a professional environment

where partnerships can be developed among teachers, administrators, students, and community

members (Holahan et al., 2000). Finally, the necessary resources must be made available for

bringing technology to the classroom. Although Holahan et al. (2000) specifically proposed the

use of “phased approach” (para. 5) to implementing technology, any approach must involve the

allocation of time, the implementation of training opportunities for teachers, the development of

an appropriate budget, and the inculcation of the benefits of technology use within the

community (Henningsen & Stein, 1997; Holahan et al., 2000; Norton et al., 2000; Wiest, 2001).

Although many current studies have pointed to the benefits of and the need for

technology applications in the classroom, many obstacles have slowed the overall

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implementation of technology into the classroom. Some of these barriers include the

preconceived notions of educators regarding technology, traditional instructional methods,

insufficient and/or ineffective professional development for teachers, and the lack of district-

wide investment (both philosophically and economically) in technology. However, these same

areas that hinder technology growth in the educational setting can also serve as places to begin

the process of change. Identifying the current perspectives of teachers and working to convince

them of the advantages of technology in instruction is an important first step. Promoting student-

centered approaches to instruction and pushing for higher-order thinking on the parts of the

students can maximize the potential of various technologies as learning tools. Implementing

relevant, consistent, and long-term professional development is also essential for keeping

educators current on technology uses in the classroom. Finally, district-wide support for

technology use is essential in order for significant philosophical and systemic change to be

sustained.

Analysis of, Suggestions for, and Applications of Current Research

As evidenced by the amount of information that has surfaced in the current literature, the

use of technology in classroom is a significant issue being addressed in the field of education.

The various applications of technology, the pedagogical impact on learning, the direct

connections between content and technology, the obstacles that impede the integration of

technology, and so forth are important, albeit challenging, issues that educators need to be

addressing. Although the research is extensive, rich with innovation and ideas, and continually

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expanding, it is essential to analyze how existing research has been conducted, identify any

significant gaps in that research, and explore practical applications of what has been learned.

Analysis of research. Thirty-nine research studies and scholarly articles published

between 1989 and 2007 (31 of which were published between 2000 and 2007) were reviewed

and synthesized in order to determine how technology could best be utilized to enhance teaching

and learning in the mathematics classroom. In addition, the role of technology as presented in

the current research was linked to the educational perspectives of classic theorists in order to

establish a foundation for effectively infusing solid pedagogical strategies into technology-

enhanced instruction. Although the key themes of the educational theories of Dewey, Gardner,

Vygotsky, and Martinez have withstood the potential eroding factors of time, new research,

social and technological change, and so forth, current research must continue to be evaluated

through a critical lens.

The educational studies reviewed here took on a wide variety of forms. For example, the

studies conducted by Ali and Kor (2007) and Debevec et al. (2006) were quantitative in nature

whereas the research developed by Norton et al. (2000) and Ruthven and Hennessy (2002) was

qualitative. Other studies utilized mixed methods while some of the articles provided meta-

analyses and/or simple reviews of existing research. Despite the various types of studies and

levels of analysis, a key theme emerged from a critical evaluation of research. The

individualized nature of learning and the unique characteristics of every learning environment

present obstacles to deriving broad generalizations from individual studies. Gardner (1999)

noted on several occasions that the unique learning profiles of students create variables for

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classroom instruction that do not guarantee that a successful instructional strategy used in one

setting will prove effective in another. In many instances, the conclusions drawn by the authors

of each study fit the data collected in the research, complemented and/or supplemented existing

research, and made educational sense. Most of the authors also recognized the limitations

associated with their studies and suggested caveats for further research. Although, many of the

results seemed to have appropriate implications across various educational settings, directly

applying the results of one study in a different setting should be considered with caution. In

order to utilize educational research in a valid and reliable manner, it is essential to view that

research as a larger whole and to consider the unique characteristics of the learning environment

where the research will be implemented. The individualized nature of learning and the special

circumstances of the individuals (i.e. children) who are being serviced in the educational setting

require educators and scholars to thoughtfully, deliberately, and carefully apply research findings

in the learning process.

Although the difficulties for generalization are an inherent characteristic of educational

research, this same difficulty also keeps the door open to large amounts of fertile territory for

continued research. Each study tends to beg the question regarding how the conclusion could

potentially be applied in new settings or with different students or in other areas of the

curriculum. The synthesis of the research presented here is no different.

Suggestions for future research. The major themes that emerged from the review of the

current literature on technology integration and the role of learning style in the mathematics

classroom included: 1) promoting higher-level among students; 2) moving students toward the

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independent use of technology; 3) identifying the key considerations for technology-based

instructional design; 4) using technology-based assessments; 5) identifying the key

characteristics of effective technology use; 6) defining how technology can lead to improved

learning outcomes; 7) establishing the significance of student learning styles in classroom

success; 8) exploring the role of technology in maximizing the benefits of the alignment between

learning styles and teaching strategies; and 9) recognizing the role of teacher perspectives, lesson

planning and classroom structure, professional development, and district-level support for

overcoming the barriers to successful technology implementation. Based on the nature of these

key themes, several potential suggestions for further research emerged:

How can specific types of technology be used to enhance learning in the math

classroom? For instance, Glover et al. (2007) explored how the use of interactive

whiteboards (IWBs) could enhance learning in the mathematics classroom. How

might other technologies (i.e. tablet PCs, document projectors, virtual reality

equipment, etc.) fare in improving mathematics instruction?

A variety of different models for characterizing learning styles were presented in the

literature. Are certain models more useful in characterizing learners in the

mathematics classroom? Can a more general model that incorporates the advantages

of each model be developed?

Are the results of studies conducted in particular areas of the mathematics curriculum

generalizable to other areas of the curriculum? For example, could the scaffolding

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approach used by Reid-Griffin and Carter (2004) in a middle school, science- and

math-oriented class be applied in a high school algebra course?

Given the importance that a classroom teacher’s disposition toward and experience

with technology plays in bringing technology-based instruction to the students, what

strategies, professional development activities, and/or learning experiences would

work best for training teachers in the use of technology? From a technological point-

of-view, how can teachers best be served and how can educators be convinced of the

importance of integrated technology in instruction?

These suggestions for further research provide only a small sample of the potential research

questions that can and should be addressed in the future. As new ideas are explored and as

research continues to be developed, educators must also be willing to develop new applications

that utilize both the fundamental tenets of educational theory and the current understanding of

technology-based instruction.

An application of current research. An important step in bridging the gap between

theory, research, and practice involves the development of theory- and research-based

applications that educators can use in the classroom. Models for curriculum and instruction that

not only illustrate how research can inform practice but that also provide educators with tangible

products that can be directly implemented in the classroom are essential in making the ideals of

theory a practical reality. Therefore, the application in the following section is a fully-developed

algebra unit that is structured on the key points outlined in the reviewed theories and research.

Presented in the form of a wiki and accompanied by a tentative lesson plan outline, the

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application builds upon the essential elements promoted by learning theorists—namely, John

Dewey, Howard Gardner, Lev Vygotsky, and Michael Martinez—and current researchers

exploring the role of educational technology in the classroom.

Conclusion

Howard Gardner (1999) suggested that current technologies can make it possible to

individualize instruction and to enhance the learning experiences of students with diverse

learning profiles. However, Gardner (1999) went on to raise a more important point:

…any consideration of education cannot remain merely instrumental: If we get

more computers, what do we want them for? More broadly, what do we want

education for? I have taken here a strong position: Education must ultimately

justify itself in terms of enhancing human understanding… (p. 180)

Although technology has the potential to significantly change how educators teach and how

students learn, technology alone will not foster change. The effective use of that technology

must be considered. This is particularly true in the mathematics classroom where the cultural,

economic, and political importance of the content has been elevated into the spotlight and the

intimate connection between the content and technology itself has been socially magnified.

Debevec et al. (2006) noted that there are many ways that students can effectively learn.

They emphasized that “it is the instructors’ challenge to adopt appropriate technology to support

and create different types of learning environments that replicate and expand the traditional

classroom to enhance students’ learning experiences and maximize their performance” (para.

33). Therefore, educators must be willing to ask themselves many different questions. How can

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a given application be used to improve instruction? How can a particular educational technology

promote higher-order thinking? How can a teacher utilize a technology to create more student-

centered learning activities? In addition to considerations of how technology can be used, the

unique learning needs of individual students must be addressed. Learning style has been shown

to have a significant impact on the success of students in the learning process; and various types

of technology could be used to match students with appropriate strategies in order to maximize

learning. Finally, every effort must be made to overcome the barriers to effectively using

technology as an educational tool. Every teacher, administrator, school, and district has to work

within a certain set of economic, political, and social parameters. These parameters are often

obstacles to realizing what research has shown to be effective educational practice. However,

creative solutions to these issues as well as attention paid to pedagogical shifts that are

influenced by technology-based instruction can make it possible to bring learning and teaching—

and specifically instruction in the mathematics classroom—into the 21st century.

From Research to Practice

The current literature about the role of educational technology in the mathematics

classroom suggests that modern technologies have significant potential to enhance the learning

of every student. When coupled with the knowledge gleaned from classic theories about the

individualized nature of learning and intelligence, the latest technologies have the power to

transform pedagogy and contribute to learning environments that can successfully meet the

unique learning needs of each student. However, the presence of technology alone will not

necessarily yield fundamental changes. Educators must find ways to effectively integrate what

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has been discovered through research into their everyday routines. Therefore, curricula,

instructional techniques, and classroom activities must be designed, implemented, and evaluated

in order to bring what is understood in both theory and research to the level of practical

application.

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Application

SBSF 8230: Professional Practice and Human Development

In a study conducted by Abramovich (2005), the researchers explored how the use of

graphics software could be used to assist young students in solving challenging, yet grade-

appropriate mathematical problems. The authors found that the employed technology made it

possible to identify the initial characteristics of higher-level algebraic reasoning in the children.

Abramovich’s study provided an insight into how the effective use of technology in instruction

can help to elicit high levels of mathematical thinking in students of all ages. More importantly,

the implications of the research potentially suggest that the appropriate use of technology can

mediate student learning and pedagogical practices so that learners can master content that has

traditionally appeared elusive to many. This example of current research complements many of

the key ideas about intelligence, learning, and educational practice emphasized by theorists like

John Dewey, Howard Gardner, Michael Martinez, and Lev Vygotsky. Gardner’s theory of

Multiple Intelligences, for instance, provides a basis for identifying the various strengths and

abilities that learners embody. Meanwhile, Dewey’s notion of experiential learning and

Martinez’s perspective on learned intelligence place an emphasis on understanding the role of

the chosen learning activity in student success. In addition, Vygotsky’s conceptualization of the

“zone of proximal development” (Vygotsky, 1978, p. 85) lays the foundation for the alignment

between instruction and the students’ learning needs. Ultimately, it is essential that current

educational practice is rooted in the fundamental theories of learning and intelligence and is

informed by the latest research.

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In order to demonstrate the process of bringing to fruition educational materials and

instructional methods tied directly to theory and current research, a sample unit has been

developed for an Algebra 1 course. Through the use of the framework developed in the breadth

section, the unit demonstrates how the theories of Dewey, Gardner, Martinez, and Vygotsky can

be applied in the practical setting. The unit also demonstrates how current research on

mathematics learning, intelligence, and technology can be used to inform the design process and

to aid in the creation of learning activities and materials that provide students with diverse and

effective learning opportunities. Following a general description of the project and its

components, the project will be critiqued with regard to: 1) the role that both theory and research

informed the design, development, and practical application; 2) the ethical implications

associated with continued research and implementation in the classroom setting; and 3) the

potential for catalyzing social change.

Description of the Application Project

The application project—a fully developed curricular unit entitled “Multiple

Representations”—is a unit that could be integrated into a typical high school or remedial level

college Algebra 1 course. The academic content in the unit involves a basic introduction to the

variety of ways that mathematical information can be expressed, manipulated, and applied in

problem-solving situations. The information for the project is presented in two complementary

pieces: 1) a tentative lesson plan outline (see Appendix) that describes daily activities,

assignments, assessments, and due dates; and 2) a wiki page (http://algebra1online.pbworks.com)

that supports the unit activities, stores and provides access to handouts, worksheets, audio visual

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material, and curricular content, hosts space for online class discussions, and outlines the general

structure for the unit. The textbook assignments that are specifically referenced in the

assignments section (see Figure 8) are taken from “Algebra: Structure and Method – Book 1” by

Brown, Dolciani, Sorgenfrey, and Cole (2000). However, the assignments could be easily

modified to align with most algebra textbooks. In order to provide a more detailed description of

the project, the structure of the tentative lesson plan outline, the characteristics of the wiki page,

the nature of the mathematical content, the types of activities, the role of technology, and

potential adaptations or modifications will now be discussed.

Structure of the Daily Lesson Plan Outline

The tentative lesson plan outline (see Appendix) is designed to provide a general

overview of the timeline associated with the unit as well as the main topics and activities found

Figure 8. Wiki screenshot of sample assignments for the unit.

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throughout the unit. The lesson plans are laid out by day of the unit (rather than day of the week)

for a 34-day period. For each day, a brief description of the classroom and/or online activities is

provided. The assignments that should be assigned for the day as well as the potential due dates

are also outlined. Specific details about the assignments, however, can be found on the wiki

page. Finally, the assignments that should be collected and/or graded are also identified for each

day. Although some details about potential activities or discussion topics are found in the

outline, the lesson plans are left somewhat vague in order to allow for flexibility and other

variables that influence each classroom differently.

About the Wiki Page

The wiki page (http://algebra1online.pbworks.com) serves as the primary online resource

for both teacher and student use for the unit. PBWorks™ was chosen as the wiki provider since

it offers features including easy navigation, threaded comments for promoting online

discussions, capabilities for limiting access and/or editing privileges from page to page,

relatively simple editing controls, and so forth. The use of a wiki (as opposed to some other

webpage design program or format) was chosen because it allows for both a combination of

webpage design (with easy tools for updating) and online collaborative space for the students.

The pages that have limited access are generally used more as a traditional website whereas the

pages that are opened for logged-on users serve as collaborative forums and design spaces. The

online structure for the unit also serves as the template for the development and implementation

of additional units for the course.

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The homepage (see Figure 9) is the starting point for students to access the digital content

of the unit. From this page, students are able to access each unit for the course. In addition, the

“navigator” and “sidebar” areas (both standard features associated with PBWorks™) found on

the right side of the screen allow for students to quickly access individual pages and other

general features of and information about the course. By default, these areas always appear no

matter what page a user is on; and this allows for a standard navigational template for the entire

wiki. Clicking on a link for a particular unit will take a student from the homepage to the main

page of the unit where he or she can access specific content and materials.

Figure 9. Wiki screenshot of homepage.

Access to Each Unit

Chat ForumNavigator and Primary

Navigation Links

Sidebar for General Course Information

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The main page for Unit 1 (see Figures 10a, 10b, and 10c) includes a variety of features

that are standard for each subsequent unit in the course. An introductory video is used to

welcome students and to delineate the general nature of the unit. Students also access the online

pre- and post-tests which were created using resources at www.quizegg.com (see Figure 11 for a

sample question). The pre- and post-tests are intended to provide some basic data for tracking

students’ academic growth. The resource guide is also accessed from the main page. This

document can be downloaded and/or printed off by the students and includes notes, examples,

journal questions, KWL activities, and so forth for the entire unit. The main page also features:

links to the specific content, assignments, and audio-visual materials for each subsection of the

unit; a link to a general questions forum; access to the group workspaces as well as a link to the

group assignment rubric; an overview of the summative assessments; and a checklist of the

standards being addressed in the unit. Through the individual section links, students are able to

view more detailed content information.

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The Resource Guide Contains Notes for Unit

Access to Online Pre-Test

Figure 10a. Wiki screenshot of Unit 1 page with video.

Link for Posting General Questions

during Unit

Access to Individual Sections

Links to Group Workspaces for Unit

Project

Figure 10b. Wiki screenshot of Unit 1 page with section links.

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Access to Online Post-Test

ICC and NCTM Standards Addressed in

Unit

Figure 10c. Wiki screenshot of Unit 1 page with post-test link.

Figure 11. Screenshot of example question from Unit 1 pretest.

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The individual subsections for the unit are designed to break down the general theme of

the unit into smaller, more manageable concepts. Each subsection is comprised of several

different learning tools, activities, and assignments (See Figures 12a, 12b, and 12c). In addition

to the information contained in the resource guide for the unit, each subsection includes video

tutorials, PowerPoint™ presentations, and links to websites and other online resources. Each

subsection also includes several activities including: discussion questions, homework

assignments, word problems, an application assignment, and a brief online quiz (which is similar

to the pre- and post-tests). The homework activities available in each subsection are digital and

can function as either online activities or as more traditional assignments. For example, the

worksheets and handouts can be printed off and completed by hand or can be downloaded and

completed in a paperless fashion. Furthermore, the web-based nature of the wiki makes it

possible for students to access all components of the course at any time as long as they have an

Internet connection. The discussion questions (see Figure 13) are also an essential component of

the course and are intended to support both in class discussions as well as continued

asynchronous conversations. The comments feature of the wiki page makes it possible for

students to easily participate in a threaded discussion. The students are made aware (through in

class conversations and through written copies) of the expectations for posting initial responses

and substantive replies to their classmates. The subsection areas of the wiki page encompass the

major portion of the mathematical content associated with the unit.

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Links to the Discussion Question Forums

PowerPoint Presentations for Each Main Topic

Figure 12a. Wiki screenshot of unit subsection page with DQs.

Digital Videos for Various Problem-Solving

Situations

Figure 12b. Wiki screenshot of unit subsection page with video tutorial.

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Assignments and Links to Worksheets

Real-World Application

Access to Online Quiz

InternetResources

Textbook References

Figure 12c. Wiki screenshot of unit subsection page with assignments.

Comments Section for Discussion

DQ Expectations

Figure 13. Wiki screenshot of discussion question page example.

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Description of Mathematical Content

Multiple representations is the overarching theme for the unit. The National Council for

Teachers of Mathematics (NCTM) identified several key process standards that students should

be able to meet as a result of their mathematics education. According to the NCTM (2010):

Instructional programs from prekindergarten through grade 12 should enable all students

to—create and use representations to organize, record, and communicate mathematical

ideas; select, apply, and translate among mathematical representations to solve problems;

[and] use representations to model and interpret physical, social, and mathematical

phenomena. (para. 5)

The NCTM (2010) also noted that students should be able to develop their mathematical

knowledge through problem solving, to effectively communicate mathematical ideas through a

variety of media, and to develop connections among various mathematical ideas. In accordance

with the standards established by the NCTM, one general goal of the unit is to introduce students

to the various forms that mathematical ideas can take. An additional goal of the unit is to help

students develop problem-solving skills. The third goal of the unit is to help students begin to

develop connections among mathematical ideas and between mathematical content and real-

world applications.

The first subsection of the unit deals with symbolic forms of mathematical ideas.

Variables, expressions, and equations are introduced. The use of order of operations for

simplifying and evaluating expressions and the role of the basic rules for solving equations are

also introduced. The second subsection addresses how to work with mathematical ideas that are

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presented verbally or in a written form. The section also introduces a problem solving plan that

can be used as a guide for solving word problems and for attacking non-routine problems. The

third subsection focuses on the nature of functions. Not only is the basic definition of a function

examined, but the various ways that functional relationships are depicted (i.e. mapping diagrams,

tables and charts, graphs, equations, and so forth) are also explored. A wide variety of

instructional strategies and learning activities are utilized in this unit to help the students learn

the mathematical content.

Learning and Instructional Activities

As noted in the lesson plan outline (see Appendix), the unit incorporates a combination of

both online and in class learning activities. In the online setting, students have access to many

different presentations of the mathematical content. These learning tools include written notes

and examples, PowerPoint™ presentations, video tutorials, Internet-based resources, and so on.

The students also have opportunities to work with the content through written homework

assignments, problem-solving activities, and application projects rooted in real-world contexts.

The students are given additional opportunities to collaborate with one another (and the

instructor) via asynchronous discussion forums and a collaborative project developed using a

wiki. The students are also asked to take several quizzes throughout the unit to aid in the overall

assessment of their learning. The online learning activities are complemented by in class

learning activities. The in class activities include: small and large group discussions, teacher-led

instruction and examples, journaling and KWL activities, one-on-one instruction with the

teacher, review games, opportunities for questions and answers, written and verbal assessments,

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and so forth. Although many of the online activities can be transferred to the classroom setting

(and vice versa), the unit is intended to promote a hybrid style of learning where the online and

in class learning activities can be combined synergistically to maximize student learning. To this

end, the role of technology plays a significant role in promoting the goals of the unit.

Integration of Technology

Technological applications have been infused throughout the unit as both a means for

presenting information and managing the course and as a means for supporting and driving many

learning activities. The computer, Microsoft Office™ software, and access to the Internet are the

primary technological applications utilized by students in this project. The online learning

environment for the unit is managed on a wiki. The wiki is used not only for managing content

and files, but it is also used a place for students to communicate and collaborate. Microsoft

Word™, PowerPoint™, and Excel™ are the main applications that students use to complete

assignments and review mathematical content. For instance, almost all of the handouts are

Word™ documents. These documents can be printed out and completed in hard copy form.

However, the documents are also designed to be completed digitally and submitted via email or

server folder. Potentially, the unit could be completed in a paperless manner. An example of the

use of Excel™ spreadsheets is found in the baseball statistics activity. Students are asked to use

an Excel™ spreadsheet to explore information about how batting statistics are calculated. Other

applications utilized throughout the unit include movie software, email, Google™ documents,

Paint™, and the Internet. The activities in the classroom are also supported by the use of

different technologies including laptop computers, projection equipment, an overhead projector

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and document camera, a scanner and printer, a mimeo™ whiteboard, audio equipment,

calculators, and so forth. The integration of technology throughout the unit not only enhances

the delivery of instruction, but it is also intended to provide students with additional tools for

supporting and maximizing their learning experiences.

Possible Adaptations and Modifications

As presented, the unit could be integrated directly into an Algebra 1 course. The details

provided in the lesson plans and the information shared on the wiki page are intended to allow an

educator to begin instruction without the need for significant additional preparation. However,

the unit can (and should) be adapted and modified as necessary to accommodate the variables of

a particular classroom setting. For instance, the unit could potentially be taught under either a

strictly traditional or strictly online model as opposed to the hybrid model described. Most of the

assignments and assessments throughout the unit can be completed either by hand or digitally.

As an example, the worksheet accompanying the Drug Elimination Project calls for the

generation of graphs. These graphs could be drawn by hand or could be constructed in Excel™

or other graphing program. The various activities in the unit can also be modified as necessary

to accommodate the availability of hardware and software applications. Finally, the lesson plan

outline allows room for classroom teachers to choose how to deliver classroom instruction based

on one’s teaching style and/or the needs of the students. Ultimately, the unit illustrates how

varied learning activities and technological applications can be brought together to present

mathematical content in a meaningful and applicable manner and to promote high-level learning

for the students.

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The application project described above presents a curricular unit for an Algebra 1

course. The unit, an introduction to the multiple representations used to depict mathematical

ideas, incorporates a wide range of presentation techniques, learning activities, collaborative

opportunities, and assessment tools. The ultimate goal of the unit is to increase the students’

understanding of algebra and its applications to the real world. In order, though, for a project

like this to contribute significantly to student learning and to a greater understanding of

mathematics teaching and learning, it must be rooted in both educational theory and current

research.

Application Discussion and Critique

Michael Martinez (2000) stated that the “features that define intelligence—the capacity to

think flexibly, to welcome complexity, and to bring knowledge to bear on important problems—

are precisely the kinds of cognition that are needed to untangle the world problematique” (p.

192). He went on to note that intelligence is not only a necessary ingredient for learning but is

also a product of learning. Classroom opportunities for students to engage in problem-solving

activities and learning experiences that promote abstract thinking help to nurture the growth of

intelligence (Martinez, 2000). However, these opportunities must be thoughtfully designed and

implemented; and the tools that are utilized to aid the learning process must be carefully chosen

and integrated into the lesson plan. For example, a common theme that emerges through the

current literature suggests that the mere presence of technology in the classroom cannot alone

guarantee that students will achieve the intended learning goals. Ultimately, curricula and the

associated instructional activities must reflect—both theoretically and practically— what is

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understood about the nature of student learning. Therefore, the application project presented

here will be critiqued based upon: 1) the framework developed from the theories of Dewey,

Gardner, Martinez, and Vygotsky; and 2) the review of current literature regarding intelligence,

technology, and mathematics education.

The general framework developed in the breadth portion incorporates the key elements of

the learning, educational, and intelligences theories of Dewey, Gardner, Martinez, and Vygotsky.

The model serves as guide for designing and assessing instructional activities. In addition, the

model incorporates important considerations that should be addressed when implementing

technological applications into the learning activities. Much like the learning process and growth

of intelligence, the model is cyclical in nature. The planning phase addresses both the initial

considerations associated with designing instruction and the intended learning outcomes. The

planning phase then leads to the actual development and implementation of instruction. This is

followed by an evaluation phase where both the instructional process and student learning are

assessed. This phase moves toward identifying the actual learning outcomes which, in turn,

leads to planning for the next learning activity. With the nature of the framework in mind, the

application project can be analyzed and critiqued.

Planning. A variety of considerations must be addressed while planning instructional

activities. The most important of these considerations centers on the students who will engage in

the activities. According to both Dewey (1938) and Martinez (2000), the previous experiences

and knowledge of the students will significantly influence the educational experience associated

with the unit. In addition, the students’ intelligence profiles influence how they learn and play a

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role in the types of activities that will most effectively engage them in the content. Gardner’s

(1983) theory of Multiple Intelligences highlights the unique characteristics of each individual.

For this reason, several preliminary strategies (as well as activities for each subsection of the

unit) are called for in the project. These activities include a brief intelligence inventory, a pre-

test for the unit, periodic journaling, discussion questions, and KWL activities. These

instructional strategies are intended to gauge what knowledge, experiences, and attitudes the

students bring with them to class. Furthermore, these activities along with personal interactions

with the students can be used to identify what might help to spark their curiosity in the content.

All of this preliminary information helps the teacher to identify the initial boundary for each of

the students’ respective “zones of proximal development” (ZPD) (Vygotsky, 1978, p. 85). The

ZPD represents the metaphorical distance between what the students currently know about a

concept and what they should know at the conclusion of instruction. The targeted learning goals

and/or incremental steps toward those goals define where the ending point of the ZPD is located.

Martinez (2000) referred to that end point as the intelligence output of the learning activity. In

this unit, the intelligence output is for students to recognize the multiple ways that mathematical

information can be represented and to begin to develop the necessary skills to build, integrate,

and utilize connections between the various representations. Assessing the learning needs of the

students and formulating the goals for the unit are essential characteristics of planning phase.

Initial planning also includes addressing any technology related issues.

There are two general themes under which the role of technology should be addressed in

the planning phase. The first theme deals with the logistics associated with implementing

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various forms of technology. For example, this unit is built upon the assumption that students

will have routine access to computers both in class and outside of class. Another example

involves the use of various online resources for completing activities. The group project that is

completed at the conclusion of the unit requires the students to utilize a variety of programs in

order to develop a wiki page that illustrates multiple representations of the data. A question that

may arise involves how easily the wiki pages can be accessed and managed by the students. A

final example of the logistics associated with technology addresses what knowledge the students

bring with them with regard to using the technology. For instance, Excel™ can be used to build

a variety of graphical representations that are used while discussing functions. Will the students

need additional instruction when it comes to utilizing the program? The second theme involves

the relationship between the students and the technology. Will the use of technology spark the

curiosity of the students or increase their motivation in class? For instance, the online discussion

forums may allow students who are more introverted in the classroom to find their voices in a

discussion setting. As noted by Gardner (1999), the use of technology should enhance the

learning experience in a meaningful way rather than simply be used as novelty or high-tech

mechanism for delivering traditional instruction. The questions raised about the learning needs

of the students, the goals of the instruction, and the role of technology naturally lead into the

design and delivery of instruction.

Instruction. In the transition from planning to the design and implementation of

instruction, the acknowledgement of the unique learning needs of students is pivotal. Gardner

(1999) stated:

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…I regard MI theory as a ringing endorsement of three key propositions: We are not all

the same; we do not all have the same kinds of minds; and education works most

effectively if these differences are taken into account rather than denied or ignored. (p.

91)

In other words, instruction must be designed to provide students with as many opportunities as

possible for the learning activities to align with individual strengths. Gardner did note that the

practicalities associated with typical classroom settings require educators to make (sometimes

difficult) choices about how to implement instruction; however, making every attempt to attend

to individual differences is critical. He went on to state:

…I would happily send my children to a school that takes differences among children

seriously, that shares knowledge about differences with children and parents, that

encourages children to assume responsibility for their own learning, and that presents

materials in such a way that each child has the maximum opportunity to master those

materials and to show others and themselves what they have learned and understood.

(Gardner, 1999, pp. 91-92)

In light of the premises of Gardner’s MI theory as well as the roles of experience-based learning,

deductive and inductive reasoning, and scientific inquiry promoted by Dewey, the activities

developed for this unit attempt to accommodate a diverse set of learner needs. For example,

discussions are incorporated in large group settings as well as in online, asynchronous forums.

These opportunities along with journaling activities, KWL reflections, large and small group

collaboration, and so forth are designed to mesh with the “personal” strengths (i.e. Gardner’s

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interpersonal and intrapersonal intelligences) of the students as well as the linguistic intelligence.

As an additional example, the students are asked to engage in problem solving activities rooted

in real world applications. These activities draw from a variety of different contexts (i.e. code

breaking, sports statistics, and drug interactions in the human body) and ask the students to solve

problems requiring inductive and deductive reasoning and spatial, logical-mathematical,

naturalist, and verbal skills. Gardner (1999) also advocated for engaging students by using

various “entry points” (p. 169) that align with the different intelligences. In the unit, written text,

video tutorials, real-world problems, exploratory activities (i.e. baseball statistics assignment),

group collaboration, and so forth provide a wide range of contexts through which to learn and

apply the mathematical content. In order to make it possible to accommodate the diverse

learning needs of the students, technology is an essential feature of the instructional design.

Based on the educational theories described in the breadth, there are multiple roles for

technology in the design and implementation of instruction. Technology could be used for

delivering instruction, for engaging the students, for providing alternative instruction, and so

forth. The instructional activities in this unit utilize technology for each of these purposes. First,

the wiki page itself serves as a medium for managing and presenting information. The wiki page

also hosts different instructional tools. Second, the use of technology for communication, for

assessment, for exploring content, for creating representations of mathematical information, and

for completing assignments can help to engage students in the learning activities. Finally,

applications including PowerPoint™, Microsoft Movie Maker™, and mimioStudio™ were

utilized to develop audio-visual presentations to supplement and enhance written material and to

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provide alternative instructional opportunities for the students. The delivery of instruction can

occur in many different ways. However, instruction is not (and should not be) the end of the

process. In order to determine if the instruction was effective, educators must take time to reflect

upon what occurred during the learning process.

Evaluation. To complete the cyclical process of effectively helping students to move to

higher levels of knowledge and understanding, reflection on learning and teaching is an essential

step. Reflecting requires educators to answer many questions. These questions range from

addressing the alignment of instructional activities and student needs to analyzing how and to

what extent various intelligences were engaged. One of the most important questions to address

involves asking whether or not an activity or learning experience was truly educative in nature

(Dewey, 1938). Although assessments can serve to determine if students learned content

material, the educative nature of an activity also involves the students’ experiences with the

materials or learning strategies. For instance, the application problems for each subsection of the

unit are challenging activities. In the first section, students are asked to explore the process of

code-breaking. Although these problems are intended to engage the students, to challenge their

logical reasoning skills, and to illustrate real-world uses of variable expressions and equations,

the problems could also be potentially frustrating for students. In this situation, the educator

must reflect upon how the students reacted and must determine what course of action should be

taken to move forward. Reflecting on student learning with regard to content is critical, but the

use of technology and instructional tools must also be evaluated.

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In the algebra unit, technological applications are used in many different ways. One

theme regarding the use of technology that must be addressed centers around how the technology

was used and whether or not that technology served a specific instructional purpose. One

example of this is the use of an Excel™ spreadsheet in the baseball statistics activity in

subsection two of the unit. In order to complete the activity, students are asked to explore how

various entries in the spreadsheet impact a particular batting statistic. The use of the spreadsheet

makes it possible for the students to conveniently explore and deduce how each batting statistic

is calculated. Without the use of technology, the constructivist nature of the exploration activity

would not be readily accessible. Technology is also used to assess students at the end of each

subsection. The use of an online quiz makes the process of completing, evaluating, and grading

the assessment relatively easy. However, it is essential to reflect upon the reliability and validity

of the quizzes. Was the format of the questions clear for the students? Were the students

comfortable taking a quiz online versus a more traditional approach? How do the online quizzes

factor into the overall assessment of student learning? Dewey (1938) emphasized the fact that

experiences interact and are continuous. Therefore, evaluation and reflection are essential to

ensuring that a given learning experience will lead to continued, effective learning.

The framework developed in the breadth portion provides a template for guiding the

design, implementation, and evaluation of learning activities. The framework was rooted in the

theoretical ideas promoted by Dewey, Gardner, Martinez, and Vygotsky; and the theories

provide a solid basis for designing quality instructional activities. However, bridging the gap

between theory and practice requires additional research about the modern, practical applications

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and implications of theory in classroom settings. Therefore, a review of the current literature can

provide added insight into the design of instructional activities.

Analysis Based on Current Research

During the review of current research on educational technology, mathematics

instruction, and intelligence, several key concepts emerged to aid in the process of designing

high-quality, effective instructional activities. One concept dealt with defining what an

“effective” strategy truly entails. Another concept addressed the role of a student’s learning style

in the educational process. The current research also addressed the role of technology in creating

effective instruction and in meeting the unique learning needs of individual students. The major

themes that surfaced in the literature review helped to shape the design of the application project.

Effective instructional strategies. The literature review brought to light several key

components to effective instruction, particularly in the mathematics classroom. The first of these

components involves engaging students in higher-level thinking. Higher-level thinking requires

students to move beyond rote activities that center on basic skills or decontextualized problem-

solving. In order to raise the bar, several problem-based activities were incorporated into the

unit. The code-breaking activity, baseball statistics assignment, and drug elimination

investigation were designed to push the students to utilize (but move beyond) the basic skills

they learned in prior learning activities. The assignments were designed to support the students

while pushing them to think about concepts at a deeper level. The final wiki group project also

asked the students to synthesize the material from the entire unit into a presentation of their

understanding of the multiple representations of mathematical information. Relevant contexts

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coupled with challenging mathematics were intended to push students to a level beyond the rote

manipulation of algebraic expressions and equations.

The second theme that underscores the effectiveness of instruction involves promoting

the independent use of technology by the students as a tool for learning and problem-solving.

Although some instructional time may be necessary to teach students how to initially use a

particular technology, the goal in the learning process is for students to develop their abilities to

carefully and appropriately choose tools that will aid in learning, completing a task, and/or

creating a product. Several technologies were incorporated into the unit with the expectation that

students would eventually be able to independently use them. One of these technologies is the

discussion feature on the wiki page. The discussion questions are intended to provide a forum

for continued dialogue about concepts addressed in the class. Given some degree of initial

instruction, the students are expected to post responses and to reply to one another. Although the

instructor can and should be involved in the discussion, the ultimate goal is for the conversation

to be student-driven. The use of a given technology in the learning process should not only serve

as an instructional tool, but it should also become a resource that students can eventually utilize

on their own in subsequent settings.

As part of both the planning and evaluation phases of instructional design, educators must

continually address the role of assessment in the learning process. As noted in the literature

review, formative assessment is an important ingredient in guiding the learning process (Wang,

Wang, Wang, & Huang, 2006). Formative assessment helps both the teacher and the student to

know where they stand with regard to the learning objectives for an activity, lesson, or unit. In

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addition, formative assessments also help the teacher to make appropriate adjustments to

instruction in order to best meet the needs of the students. In the application project, formative

assessments are embodied in a variety of activities including: discussion and journaling,

classroom interactions and direct questioning, subsection quizzes, homework assignments

involving more knowledge-based material as well as problem-solving activities, KWL

reflections, and so forth. These formative assessments are intended to inform both the teacher

and the student about the students’ progression through their respective ZPDs toward the

anticipated new knowledge.

In order to evaluate the use of technology as part of the instructional design of a unit,

lesson, or activity, several key considerations and characteristics of effective implementation

surfaced in the review of the literature. Effective technology-based instruction should: have

clearly defined goals and outcomes, have a clear purpose for the particular use of a chosen

technology, reflect elements of quality design, and permeate the structure of the lesson.

Throughout the unit, technological applications are integrated into the activities for specific

purposes. For instance, Excel™ is used as a spreadsheet to explore the nature of baseball

statistics as well as a tool for constructing graphs. A wiki page is used by students as a

collaborative space for completing the group project. Internet-based quizzes are utilized to

administer pre- and post-tests. These are only few examples of the use of technology beyond the

level of a novel instructional tool. In the same vein, several activities are completed using more

traditional approaches. For instance, many of the homework assignments from the textbook are

expected to be completed using paper-and-pencil (however, the assignments could be completed

188

digitally if necessary and appropriate). The overall designs of all of the resources in the unit

reflect the key elements of visual design, are created digitally, and are generally appealing to the

eye. Most importantly, the use of technology is directly incorporated into the design of each

activity. The use of technology is purposeful and essential to the overall learning goals of the

unit.

A critical theme that resounds throughout the current literature on the role of technology

in the learning process is the fact that the use of technology alone does not guarantee that student

learning will improve. To this end, it is essential for educators to take a hard look at instructional

activities, particularly those that incorporate technology, and reflect upon the actual learning of

the students. In the algebra unit, all of the activities were designed with the goal of enhancing

the learning of students. As the unit is implemented, however, it becomes critical to determine

the true effectiveness of each activity and lesson. If a particular activity does not work as well as

expected, it is important to determine why that was the case and then work to make the necessary

adjustments. All of the activities presented as part of the algebra unit have the potential to be

highly successful, but they could also result in outcomes below what is expected. Ultimately,

though, each activity can also be tweaked and adjusted; and future iterations of the unit can lead

to positive learning outcomes for the students.

While assessing the effectiveness of an instructional strategy, it is important to

consider one variable that is often beyond the control of the educator—the students.

Each student brings unique learning needs to the classroom. Each student will react

differently to the same learning activity. Each student is also dynamic and can

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potentially change from week to week, class to class, and learning strategy to learning

strategy. For this reason, the learning styles of each student must be considered.

Alignment of instructional strategies and learning styles. The role of learning style is

a prominent issue that surfaces throughout the current literature on education. Learning style has

its roots in the work of the classic theorists; but current research provides ideas for aligning

instruction and learning style. One of the most important reasons for considering the impact of

student learning style on instruction is that effectiveness is maximized when students have the

chance to learn in a manner that is optimal for meeting their individualized needs. Although

creating the optimal learning situation for every student all of the time is not always possible,

providing opportunities for students to utilize their strengths in learning and to develop their

lesser abilities is essential.

Throughout the application project, a variety of different teaching strategies are

employed. Audio-visual materials and teaching aides are incorporated to complement written

material so that students can interact with content through a variety of channels. Introspective

activities are balanced out by opportunities to communicate in large and small group settings.

Social interactions between students as well as more anonymous dialogues in the discussion

forums provide different avenues for students to contribute to the ongoing discussion of the

content. The skills developed in rote practice activities are applied in problem-based settings.

Assessments take on a variety of forms including online multiple choice quizzes, written tests,

group activities, and verbal discussions with the teacher. Although no single lesson addresses

every unique learning style, the unit as a whole is designed with the goal of providing students

190

with activities that will align (at some point) with their particular learning needs. Content is

presented in a repeated fashion and through multiple channels so that students have the

opportunity to connect with the mathematics in some manner.

When all is said and done, the students themselves are the key pieces that bring the

educational process puzzle together. This notion is supported by the prominence that the

learning needs of students take in both educational theory and current research. Aligning

instructional practice with individual student needs is an essential, albeit challenging, pursuit.

Therefore, critical reflection must be a continual component of the design of curricula and

instructional strategies.

Critical Considerations

The application project involves an algebra unit that utilizes various technologies and

pedagogical strategies to deliver instruction to students regarding the multi-dimensional

characteristics of mathematical information. The unit was designed based upon educational

ideas developed both in theory and in research. However, a self-critique of the unit provides an

opportunity to promote the further expansion of ideas and applicability in the classroom. Several

comments and potential questions (in no particular order) can be addressed regarding the project:

Does the unit provide access to the mathematical content through all of the

intelligences identified by Gardner and/or other theorists and researchers? Gardner

(1999) continually noted that no single intelligence is more or less important than

another. In this unit, for example, the musical intelligence does not surface as much

191

as the others. Is it necessary to either add activities or reformulate existing ones to

reach this particular intelligence?

Are the goals and objectives outlined in the resource guide for the unit specific

enough to guide the various activities and uses of technologies? Explicit, clear goals

are paramount in an effort to help students achieve identified levels of learning.

Should the goals and objectives be stated more clearly and in additional areas of the

course? Would it be appropriate to clearly state to students why a particular

technology is being used in a given situation?

Does the design of the unit provide enough flexibility to meet the learning needs of

the students who engage in the material? The unit does incorporate several routines

in order to create a consistent structure. However, do these routines allow for

students to effectively learn in a manner that works well for them? Is it possible, if

necessary, to adjust a routine or add new elements in order to meet a need that is not

anticipated?

What types of assumptions are made in the design process with regard to technology?

Are those assumptions realistic with regard to access to technology, the students’

abilities to work with given applications, and the role of a chosen technology in

completing a certain goal? What level of training might the students need in order to

effectively and efficiently utilize the technology built into the unit structure?

In general, the unit appears to meet the overall ideals outlined in the reviews of both

educational theory and current research. However, do (or should) the individual

192

activities also meet those ideals? Can the framework developed in the breadth be

applied to individual instructional strategies in addition to the unit as a whole?

This set of questions represents only a small number of potential considerations that could be

raised with the application project. However, many of the answers can only be uncovered as the

project is implemented in a classroom setting and evaluated with regard to the actual learning of

the students. As with any educational endeavor, the implementation in a real classroom is

necessary to bridge the gap between educational theory and practice. The implementation of a

project in the classroom will lead back to continued research and theoretical analysis; and this

cycle leads to the continued development of the field’s knowledge of the best educational

practices for yielding high-level student learning. In order for the research cycle, though, to

produce the most accurate results, ethical considerations associated with the project or proposed

research must be addressed.

Ethical Considerations

The theoretical foundations of educational practice provide a basis for assessing and

describing what occurs in the different facets of learning. Theory can provide insights about

student cognition, effective instructional practice, intelligence, the use of technology, and so

forth. However, theory can only be substantiated through implementation and research in the

classroom. The catch is that research requires educators to test strategies, curriculum designs,

and projects with actual students. Given the fact that students under the age of 18 are often the

targeted population of educational research, the ethical implications associated with conducting

research with a special population must be carefully weighed. In order for the application project

193

to be implemented and researched, several ethical considerations/questions (in no particular

order) should be addressed:

Although the mathematical content addressed in the unit aligns with state and national

standards, will the instructional strategies associated with the project provide the

students with adequate and fair opportunities to meet those standards?

Given the important role that technology plays in the project design, do the students

have sufficient access to the programs and hardware required to successfully meet the

expectations of the unit? If students do not have access to technology outside of the

classroom, will they be given adequate opportunities to utilize required equipment

and software during school hours? Are there measures in place to provide alternative

instruction or assignments in the event that accommodations cannot be made?

If the implementation of the project requires all students to participate, do special

subgroups (i.e. handicapped students, students with IEPs, ELL students, low SES

students, minorities, and so on) within the student population have equitable access

and accommodations to meet the requirements of the unit as well as the overall

requirements set by the school district?

Are all stakeholders including students, parents, educators, and administrators

informed about the nature of the project? Are they given the opportunity to give

appropriate consent for participation?

Are all of the participants—especially the students and their parents/guardians—made

aware of the potential risks associated with the implementation of the project in the

194

classroom setting? Is the project designed in order to avoid any undue risk or

potential harm to the students who participate in the instruction?

The overarching goals of the unit are intended to help students learn the mathematical content in

an effective manner. The features of the project are also intended to make sure that the learning

experiences of the students are truly educative and lead to a better overall educational

experience. In addition, the integration of technology and problem-based activities are intended

to help students prepare for and make connections to real-life mathematical situations that they

will experience outside of the classroom. Despite the best of intentions, though, the ethical

considerations raised here must be addressed in order to ensure that potential research activities

benefit the participants involved in the development of that research. In turn, the benefits and

new knowledge that are realized through this process can lead to positive social changes locally

as well as throughout the educational field.

Potential for Social Change

In 2000, the National Council of Teachers of Mathematics (NCTM) published an updated

version of “Principles and Standards for School Mathematics.” The authors stated:

Principles and Standards calls for a common foundation of mathematics to be learned by

all students. This approach, however, does not imply that all students are alike. Students

exhibit different talents, abilities, achievements, needs, and interests in mathematics.

Nevertheless, all students must have access to the highest-quality mathematics

instructional programs. Students with a deep interest in pursuing mathematical and

scientific careers must have their talents and interests engaged. Likewise, students with

195

special educational needs must have the opportunities and support they require to attain a

substantial understanding of important mathematics. A society in which only a few have

the mathematical knowledge needed to fill crucial economic, political, and scientific roles

is not consistent with the values of a just democratic system or its economic needs.

(NCTM, 2010, p.4)

The NCTM recognized that students are unique and contribute to a dynamic learning setting

that educators must accommodate. Through the exploration of the intelligence and learning

theories of Dewey, Gardner, Martinez, and Vygotsky, the characteristics of human cognitive

development can be directly tied to the responsibilities placed on educators to assist students in

learning mathematical content. Furthermore, current research suggests that technological

applications can be used to help students access that mathematical content through channels that

resonate with their particular learning needs. The application project presented here is a

demonstration of how the existing knowledge of cognitive development, human intelligence,

educational technology, and educational pedagogy can be brought together to yield innovative

instructional activities in the mathematics classroom.

Despite the rapid advancement of educational technology and the close connection

between technology and mathematical knowledge, mathematics educators have been one of

slowest subgroups of teachers to implement technology into the classroom and to adapt new

instructional practices to accommodate the changing needs of today’s students (Norton,

McRobbie, & Cooper, 2000). Evuleocha (1997) also suggested that it can be difficult to bring

about the types of fundamental changes in the traditional classroom culture that are necessary in

196

order to keep pace with the ever-changing social and technological demands placed on students.

Regardless of the challenges, change that encompasses quality mathematical content,

applicability to the real world, the development of problem-solving skills, and the utilization of

the latest technologies must occur in mathematics education in order for student learning to

remain germane and for content to remain accessible. This type change can be promoted through

research and the development of projects such as the one presented here. Not only does this

algebra unit illustrate how the knowledge of student learning needs as well as the use of various

technologies can be applied first-hand in a mathematics setting, but it also provides an

opportunity for continued research to further influence social change by exploring how to best

meet the educational needs of students.

Conclusion

The application project consisted of a fully developed algebra unit that addressed the

multiple representations of mathematical content. Through the use of a wiki and a variety of

other resources, the unit presents mathematical content in many ways. These methods included

written text, audio-visual materials in the form of PowerPoint™ presentations and video

tutorials, online and classroom-based discussions, problem-solving applications, and so forth.

The algebra unit also engages the students in the materials through different types of contexts

and activities such as writing, online asynchronous discussions, group collaboration, individual

reflection, rote practice, and non-routine problem-solving. The design of the application project

was rooted in theoretical ideas promoted by Dewey, Gardner, Martinez, and Vygotsky.

Simultaneously, the findings of current research on mathematics instruction, learning styles, and

197

educational technology were used to create design elements that reflect the needs of students in

today’s classrooms. Debevec, Shih, and Kashvap (2006) noted that:

…there is more than one path to optimize student learning and performance. It is the

instructors’ challenge to adopt appropriate technology to support and create different

types of learning environments that replicate and expand the traditional classroom to

enhance students’ learning experiences and maximize their performance. (para. 33)

Ultimately, the application project is an attempt to bring what is known about the best practices

in education to fruition in a realistic classroom setting.

198

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Appendix

Lesson Plan Outline (tentative schedule)(see Wiki page for more details: http://algebra1online.pbworks.com)

Day 1 Day 2Introduction to the Unit Unit 1 – Section 1:

Variables, Expressions, and Equations Classroom ActivitiesAdminister the Unit 1 pre-test (assessment). Watch the introduction video and take a brief “tour” of the unit (go through the available resources, locate the discussion forums, assign small groups, and so forth).

Classroom ActivitiesComplete KW portion of the KWL for Section 1 in the resource guide. Go through notes/PowerPoint dealing with variables and order of operations (PowerPoint #1 and #2). Begin working on the journal responses for Section 1 in the resource guide.

Online ActivitiesComplete the Unit 1 post-test online. Begin exploring the unit website and post any initial questions.

Online ActivitiesBegin initial responses to DQs for Section 1. Review PowerPoint presentations, video examples, and additional Internet resources.

Assignments Unit 1: Pre-Test (completed in class)

Assignments KW(L) for Section 1 (completed in class) Journal Responses for Section 1 (due Day

5) Text: Section 1-1 and 1-2 (due Day 4) Initial Responses: Section 1 – DQ #1 / DQ

#2 (due Day 5) Replies: Section 1 – DQ #1 / DQ #2 (due

Day 9)

Assignments Due Today Unit 1: Pre-Test

Assignments Due Today KW(L) for Section 1

206

Day 3 Day 4Unit 1 – Section 1:Variables, Expressions, and Equations

Unit 1 – Section 1:Variables, Expressions, and Equations

Classroom ActivitiesGo through information regarding exponents in expressions. Clarify steps in more complex order of operations problems (i.e. multiple grouping symbols, etc.). Have students work in pairs/small groups to complete multi-step order of operations problems.

Classroom ActivitiesDiscuss solving one-step equations. Review the “Three Golden Rules” for solving equations. Utilize “pan balance” analogy for keeping the equation balanced.

Online ActivitiesContinue posting initial responses/replies to DQs for Section 1. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Online ActivitiesContinue posting initial responses/replies to DQs for Section 1. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Assignments Text: Section 4-1 (due Day 5)

Assignments Text: Section 1-3 (due Day 6)

Assignments Due Today NONE

Assignments Due Today Text: Section 1-1 and 1-2

207

Day 5 Day 6Unit 1 – Section 1:Variables, Expressions, and Equations

Unit 1 – Section 1:Variables, Expressions, and Equations

Classroom ActivitiesReview day to wrap-up Sections 1-1, 1-2, 4-1, and 1-3. Complete journal responses. Complete L portion of the KWL for Section 1 in the resource guide. Play review game in class to review material for Section 1.

Classroom ActivitiesBegin working on word problems for Section 1 as well as the application assignment (Code Breaking Worksheets). Class time will be dedicated to beginning the assignments and providing students with time to work on problems and ask questions. The Coding Breaking assignments should be completed digitally, if possible.

Online ActivitiesFinish posting initial responses and continue posting replies to DQs for Section 1. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Online ActivitiesContinue posting replies to DQs for Section 1. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources. Download Code Breaking worksheets from website.

Assignments (KW)L for Section 1 (completed in class)

Assignments Section 1 – Word Problem Applications

(due Day 8) Code Breaking Worksheet #1 (due Day 9) Code Breaking Worksheet #2 (due Day 9)

Assignments Due Today Initial Responses: Section 1 – DQ #1 / DQ

#2 (kw)L for Section 1 Journal Responses for Section 1 Text: Section 4-1

Assignments Due Today Text: Section 1-3

208

Day 7 Day 8Unit 1 – Section 1:Variables, Expressions, and Equations

Unit 1 – Section 1:Variables, Expressions, and Equations

Classroom ActivitiesStudent will have class time to continue working on assignments, code-breaking worksheets, DQ replies, prepping for the Section 1 quiz, etc. Opportunity will be given for questions with large group and for work time individually and/or in small groups.

Classroom Activities

Online ActivitiesContinue posting replies to DQs for Section 1. Conduct Internet searches to complete questions from the code-breaking worksheets.

Online Activities

Assignments NONE

Assignments NONE

Assignments Due Today NONE

Assignments Due Today Section 1 – Word Problem Applications

209

Day 9 Day 10Unit 1 – Section 1:Variables, Expressions, and Equations

Unit 1 – Section 2:Words to Symbols (and vice versa)

Classroom ActivitiesComplete (in class) the online quiz (assessment) for Section 1. Student will have remaining class time to finish working on code-breaking worksheets and DQ replies. Opportunity will be given for questions with large group and for work time individually and/or in small groups.

Classroom ActivitiesComplete KW portion of the KWL for Section 2 in the resource guide. Go through notes/PowerPoint dealing with converting from words into algebraic expressions and equations (PowerPoint #1). Begin working on the journal responses for Section 2 in the resource guide.

Online ActivitiesComplete (in class) the online quiz for Section 1. Finish posting replies to DQs for Section 1. Finish conducting Internet search to complete questions from the code-breaking worksheets.

Online ActivitiesBegin initial responses to DQs for Section 2. Review PowerPoint presentations, video examples, and additional Internet resources.

Assignments Section 1 Quiz (completed in class

online)

Assignments KW(L) for Section 2 (completed in class) Journal Responses for Section 2 (due Day

14) Text: Section 1-4 and 1-5 (due Day 12) Initial Responses: Section 2 – DQ #1 / DQ

#2 (due Day 14) Replies: Section 2 – DQ #1 / DQ #2 (due

Day 18)

Assignments Due Today Section 1: Quiz Replies: Section 1 – DQ #1 / DQ #2 Code Breaking Worksheet #1 Code Breaking Worksheet #2

Assignments Due Today KW(L) for Section 2

210

Day 11 Day 12Unit 1 – Section 2:Words to Symbols (and vice versa)

Unit 1 – Section 2:Words to Symbols (and vice versa)

Classroom ActivitiesStudents will conduct an Internet (or library) search of recent periodicals (preferably news magazines or newspapers) and locate 5 articles or advertisements that include math related verbal phrases. They will then need to translate those phrases into expressions or equations. Additional class time will be used to review translating words to expressions/equations.

Classroom ActivitiesGo through PowerPoint presentation on the “Problem Solving Plan.” Utilize several word problems to illustrate how to incorporate the problem solving plan into the solving process. Provide remaining class time for students to begin working on assignment.

Online ActivitiesInternet search for math phrases in periodicals. Continue posting replies to DQs for Section 2. Continue reviewing PowerPoint presentations, videos, and additional resources.

Online ActivitiesContinue posting replies to DQs for Section 2. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Assignments Internet Search for Math Phrases

(completed in class)

Assignments Text: Section 1-6 and 1-7 (due Day 14)

Assignments Due Today Internet Search for Math Phrases

Assignments Due Today Text: Section 1-4 and 1-5

211

Day 13 Day 14Unit 1 – Section 2:Words to Symbols (and vice versa)

Unit 1 – Section 2:Words to Symbols (and vice versa)

Classroom ActivitiesIntroduce “Consecutive Integer” word problems. Discuss how to utilize the problem solving plan in solving these types of problems.

Classroom ActivitiesReview day to wrap-up Sections 1-4, 1-5, 1-6, 1-7, and 2-7. Complete journal responses. Complete L portion of the KWL for Section 2 in the resource guide. Play review game in class to review material for Section 2.

Online ActivitiesDownload Consecutive Integer Worksheet from website. Continue posting replies to DQs for Section 2. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Online ActivitiesFinish posting initial responses and continue posting replies to DQs for Section 1. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Assignments Text: Section 2-7 worksheet (due Day 15)

Assignments NONE

Assignments Due Today NONE

Assignments Due Today (kw)L for Section 1 Journal Responses for Section 1 Initial Responses: Section 2 – DQ #1 / DQ

#2 Text: Section 1-6 and 1-7

212

Day 15 Day 16Unit 1 – Section 2:Words to Symbols (and vice versa)

Unit 1 – Section 2:Words to Symbols (and vice versa)

Classroom ActivitiesBegin working on word problems for Section 2 as well as the application assignment (Baseball Statistics). Class time will be dedicated to beginning the assignments and providing students with time to work on problems and ask questions. The Baseball Statistics assignment should be completed digitally, if possible.

Classroom ActivitiesStudent will have class time to continue working on assignments, baseball statistics worksheet/spreadsheets, DQ replies, prepping for the Section 2 quiz, etc. Opportunity will be given for questions with large group and for work time individually and/or in small groups.

Online ActivitiesContinue posting replies to DQs for Section 2. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources. Download Baseball Stats Worksheet and Spreadsheet from website. Conduct Internet search (as necessary) to define the meanings of the various baseball statistics.

Online ActivitiesContinue posting replies to DQs for Section 2. Conduct Internet search (as necessary) to define the meanings of the various baseball statistics.

Assignments Section 2 – Word Problem Applications

(due Day 17) Baseball Stats Worksheet (due Day 18)

Assignments NONE

Assignments Due Today Text: Section 2-7 worksheet

Assignments Due Today NONE

213

Day 17 Day 18Unit 1 – Section 2:Words to Symbols (and vice versa)

Unit 1 – Section 2:Words to Symbols (and vice versa)

Classroom Activities Classroom ActivitiesComplete (in class) the online quiz (assessment) for Section 2. Student will have remaining class time to finish working on baseball statistics worksheet and DQ replies. Opportunity will be given for questions with large group and for work time individually and/or in small groups.

Online Activities Online ActivitiesComplete (in class) the online quiz for Section 2. Finish posting replies to DQs for Section 2.

Assignments NONE

Assignments Section 2 Quiz (completed in class

online)

Assignments Due Today Section 2 – Word Problem Applications

Assignments Due Today Section 2: Quiz Replies: Section 2 – DQ #1 / DQ #2 Baseball Stats Worksheet

214

Day 19 Day 20Unit 1 – Section 3:Functions

Unit 1 – Section 3:Functions

Classroom ActivitiesComplete KW portion of the KWL for Section 3 in the resource guide. Go through notes to introduce the concept of functions. Focus on the “multiple representations” that can be used to define a function. Collect data from the group about a favorite “something” to generate a bar/line graph as an example. Begin working on the journal responses for Section 3 in the resource guide.

Classroom ActivitiesContinue to review the concept of a function. Reiterate the terms: function, domain, and range. In addition, highlight to notion that every member of domain matches up with one “answer” in the range. Have the students generate questions and gather answers from classmates. Using the data, the students will develop functions and present table in table and graph form.

Online ActivitiesBegin initial responses to DQs for Section 3. Review PowerPoint presentations, video examples, and additional Internet resources.

Online ActivitiesContinue posting initial responses/replies to DQs for Section 3. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Assignments KW(L) for Section 3 (completed in class) Journal Responses for Section 3 (due Day

23) Text: Section 8-6 (due Day 21) Initial Responses: Section 3 – DQ #1 / DQ

#2 (due Day 23) Replies: Section 3 – DQ #1 / DQ #2 (due

Day 27)

Assignments Create graphs of Class Data (due Day 21)

Assignments Due Today KW(L) for Section 3

Assignments Due Today NONE

215

Day 21 Day 22Unit 1 – Section 3:Functions

Unit 1 – Section 3:Functions

Classroom ActivitiesInvestigate the nature of function notation. Describe the relationship between function notation (i.e. “f(x)”) and regular variables (i.e. “y”). Go through the process of evaluating functions.

Classroom ActivitiesExplore the relationships between functions written in equation form and their corresponding graphs. Use “GCalc” website to investigate what types of graph are functions and which ones are not. Discuss how to use the vertical line test to determine if a graph is the graph of a function.

Online ActivitiesContinue posting initial responses/replies to DQs for Section 3. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Online ActivitiesAccess the “GCalc” website to utilize the graphing capabilities and explore graphs of functions. Continue posting initial responses/replies to DQs for Section 3. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Assignments Text: Section 8-7 (due Day 23)

Assignments Vertical Line Test Worksheet (due Day 24)

Assignments Due Today Text: Section 8-6 Create graphs of Class Data

Assignments Due Today NONE

216

Day 23 Day 24Unit 1 – Section 3:Functions

Unit 1 – Section 3:Functions

Classroom ActivitiesReview day to wrap-up Sections 8-6 and 8-7. Complete journal responses. Complete L portion of the KWL for Section 3 in the resource guide. Play review game in class to review material for Section 3.

Classroom ActivitiesBegin working on word problems for Section 3 as well as the application assignment (Drub Elimination Worksheet). Class time will be dedicated to beginning the assignments and providing students with time to work on problems and ask questions. The Drug Elimination assignment should be completed digitally, if possible.

Online ActivitiesFinish posting initial responses and continue posting replies to DQs for Section 3. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources.

Online ActivitiesContinue posting replies to DQs for Section 3. Continue reviewing PowerPoint presentations, video examples, and additional Internet resources. Download Drug Elimination worksheets from website.

Assignments (KW)L for Section 3 (completed in class)

Assignments Section 3 – Word Problem Applications

(due Day 26) Drug Elimination Worksheet (due Day 27)

Assignments Due Today (kw)L for Section 3 Journal Responses for Section 3 Initial Responses: Section 3 – DQ #1 / DQ

#2 Text: Section 8-7

Assignments Due Today Vertical Line Test Worksheet

217

Day 25 Day 26Unit 1 – Section 3:Functions

Unit 1 – Section 3:Functions

Classroom ActivitiesStudent will have class time to continue working on assignments, drug elimination worksheet, DQ replies, prepping for the Section 3 quiz, etc. Opportunity will be given for questions with large group and for work time individually and/or in small groups.

Classroom Activities

Online ActivitiesContinue posting replies to DQs for Section 3.

Online Activities

Assignments NONE

Assignments NONE

Assignments Due Today NONE

Assignments Due Today Section 3 – Word Problem Applications

218

Day 27 Day 28Unit 1 – Section 3:Functions

Unit 1 – Section 4:Review

Classroom ActivitiesComplete (in class) the online quiz (assessment) for Section 3. Student will have remaining class time to finish working on drug elimination worksheet and DQ replies. Opportunity will be given for questions with large group and for work time individually and/or in small groups.

Classroom ActivitiesIntroduction to the review process. Introduce the group project (see rubric). Assign review assignments. Begin working on Wrap-up DQ. Students will be given time to meet in small groups to begin planning/working on group wiki project.

Online ActivitiesComplete (in class) the online quiz for Section 3. Finish posting replies to DQs for Section 3.

Online ActivitiesGroups will work on wiki pages. Begin initial responses to Wrap-up DQ for Section 4. Review PowerPoint presentations, video examples, and additional Internet resources to prep for the assessments.

Assignments Section 3 Quiz (completed in class

online)

Assignments Problem Solving Practice (due Day 30) Review Worksheet for Unit 1 (due Day 31) Initial Responses: Wrap-up DQ (due Day

30) Replies: Wrap-up DQ (due Day 33) Group Project (due day 33)

Assignments Due Today Section 3: Quiz Replies: Section 3 – DQ #1 / DQ #2 Drug Elimination Worksheet

Assignments Due Today NONE

219

Day 29 Day 30Unit 1 – Section 4:Review

Unit 1 – Section 4:Review

Classroom ActivitiesContinue working on review assignments and DQ responses. Students will be given time to meet in small groups to continue planning/working on group wiki project. Opportunity for questions about assignments will be provided.

Classroom ActivitiesReview day to wrap-up chapter. Students will use “Webspiration” to develop a concept map activity based on the review “summary” sheet for the chapter. Assign “Notes Quiz” (assessment) for the Unit. This will be a take home, open book/note quiz.

Online ActivitiesGroups will work on wiki pages. Continue posting responses to Wrap-up DQ for Section 4. Review PowerPoint presentations, video examples, and additional Internet resources to prep for the assessments.

Online ActivitiesAccess “Webspiration” to develop concept maps. Continue group work on wiki pages. Continue posting responses to Wrap-up DQ for Section 4. Review PowerPoint presentations, video examples, and additional Internet resources to prep for the assessments.

Assignments NONE

Assignments Concept Map Review Activity (due Day 33) Notes Quiz for Unit 1 (due Day 33)

Assignments Due Today NONE

Assignments Due Today Problem Solving Practice Initial Responses: Section 4 – Wrap-up DQ

220

Day 31 Day 32Unit 1 – Section 4:Review

Unit 1 – Final Assessment

Classroom ActivitiesReview day to wrap-up Unit 1. Answer any remaining questions from students. Go over review worksheet for the unit. Play review game in class to review material for the unit.

Classroom ActivitiesComplete the following assessment activities: 1) Problem Solving Activity (assessment); and 2) Oral/Written Problem with Instructor (assessment).

Online ActivitiesContinue group work on wiki pages. Continue posting responses to Wrap-up DQ for Section 4. Review PowerPoint presentations, video examples, and additional Internet resources to prep for the assessments.

Online ActivitiesContinue group work on wiki pages. Continue posting responses to Wrap-up DQ for Section 4. Review PowerPoint presentations, video examples, and additional Internet resources to prep for the assessments.

Assignments NONE

Assignments Problem Solving Activity (completed in

class) Oral/Written Problem with Instructor

(completed in class)

Assignments Due Today Review Worksheet for Unit 1

Assignments Due Today Problem Solving Activity Oral/Written Problem with Instructor

221

Day 33 Day 34Unit 1 – Final Assessment Unit 1 – Final Assessment

Unit 2 – IntroductionClassroom ActivitiesComplete the written test (assessment) for Unit 1.

Classroom ActivitiesAdminister the Unit 1 post-test (assessment). Administer Unit 2 pre-test (assessment).

Online ActivitiesFinish posting responses to Wrap-up DQ for Section 4. Finish group work on wiki pages.

Online ActivitiesComplete the Unit 1 post-test and Unit 2 pre-test online.

Assignments Unit 1: Final Written Test (completed in

class)

Assignments Unit 1: Post-Test (completed in class) Unit 2: Pre-Test (completed in class)

Assignments Due Today Replies: Section 4 – Wrap-up DQ Concept Map Review Group Project (wiki page) Notes Quiz for Unit 1 Unit 1: Final Written Test

Assignments Due Today Unit 1: Post-Test Unit 2: Pre-Test