Problem-Solving Strategies in the Applied Mathematics Classroom

Deborah RobertsNovember 4, 2003

Committee Members:Dr. Richard Tondra, Major Professor

Dr. Irvin HentzelSarah Lubienski

Table of Contents

Chapter OneIntroduction 3

Chapter TwoLiterature Review 8

Chapter ThreeModifications 20

Chapter FourAnalysis and Results 25

Chapter FiveDiscussion 29

Appendix 32

Bibliography 43

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CHAPTER ONE

INTRODUCTION

Ever since I began my teaching career, the question that arose most often

in my classroom was: “Where will I ever use this?” It has often been my

response that whether or not the student uses a particular skill, the logic and

problem-solving skills they are learning will have lifelong applications. Eventually

I began to take a closer look at my glib answer and to analyze just how serious I

was about my comments. Was I really providing my students with the skills that

would help them to be successful in the future? I began to examine more closely

how mathematics education may have changed over the past thirty years, and

how this relates to the general focus of my curriculum and teaching style.

Background

My first recollections of my own math education experiences are positive

ones. I remember races on the board to see who could do multi-digit

multiplication problems the fastest, or verbal quizzes where again speed was the

key. Since these things came easily for me, I continued to gain confidence in my

math abilities. In recent years, I find myself reflecting on the effect these types of

competitions had on the class as a whole. The same two or three students were

always vying for the championship, and I wonder how those who understood the

algorithmic process perfectly, but were unable to work at that top-level speed,

viewed these whole proceedings.

By the time I tackled algebra and geometry in high school, I found that my

mathematics courses took more effort than most of my other studies. However, I

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was willing to work hard enough to get that ‘A’. I began to regard math as my

most challenging subject and envied those for whom it seemed to come more

‘naturally’. My last two years of high school, I had an instructor who would get

very animated when explaining a new concept, and I enjoyed his excitement in

the subject, but still felt somehow that I was missing a piece of the puzzle.

Although I enjoyed finding the pieces to make everything work out logically, I still

felt that I was barely grasping the basic essentials and was always on the verge

of being lost.

When I began to look at career choices, I found that education seemed

like a good fit for me. However, I didn’t even consider the field of mathematics—

after all, it was my hardest subject! Instead, I considered English, social studies

—there were so many interesting topics out there. Along the way, I was required

to take a mathematics class, College Algebra and Trigonometry (pre-calculus),

as part of my class work. I was rather surprised to find myself intrigued all over

again. I enjoyed that class so much that I decided to tackle calculus, as well as

to consider picking up a minor in mathematics.

I had several lengthy discussions with my college professor about this,

and she gave me a piece of advice that I’ve always remembered. She said that

for most mathematicians, (those who aren’t incredibly naturally talented),

knowledge and problem-solving skills come as a result of being a ‘plodder’. This

idea began to simmer in my mind as I considered the implications. Was it

possible that I had been looking for memorizable skills and handy little algorithms

that I could apply at will rather than digging to the heart of problem-solving

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concepts? If I had some of those ‘plodder’ tools, would I be able to tackle more

difficult concepts in ways that would allow me to break them down into something

more familiar or relate them to something I already knew? Did I possess the

skills to learn the language and logic of mathematics in a way that would enable

me to look at this as a viable career choice?

I began to pursue my mathematics studies in earnest and decided to

make this my major degree area. Calculus seemed fairly logical, as did discrete

mathematics and most of my other classes. There were a few, like abstract

algebra, that really taxed my self-confidence. Fortunately, it seemed like most of

the class was in the same quandary, and we found ourselves meeting almost

daily to discuss homework assignments or new concepts. I began to see a real

benefit in the sharing of ideas and approaches as we struggled with the more

difficult material.

About this time I took a course on teaching methods for mathematics. I

am embarrassed to admit that I was totally unfamiliar with the National Council of

Teachers of Mathematics or the National Math Standards. Suddenly, the

emphasis was not just on the curriculum itself, but also on the thinking processes

involved in attaining mathematical knowledge, as well as how students could

apply this knowledge to real-world applications! One of the basic principles of

the NCTM standards (p. 21) states: “Students must learn mathematics with

understanding, actively building new knowledge from experience and prior

knowledge.” How much more exciting it would be if students were confident that

they had the tools to not only master the skills presented to them in a math

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classroom, but also to apply these same skills in their everyday life. This was

what my college professor was trying to explain to me about being a ‘plodder’. I

vowed to be the kind of teacher who challenged and inspired her students to new

levels of achievement.

Overview

I am currently in my tenth year of teaching high school mathematics and

science. Somewhere along the way, the real world has intruded and I find myself

weighed down with six different classes to prepare for, papers to grade, and

other life responsibilities all demanding attention. I have begun to take a closer

look at my teaching styles as well as the strengths and weaknesses of my

students, and am determined to prioritize a little better.

For several years now, I have taught classes in applied mathematics.

These students are for the most part non-college-bound individuals. Many go on

to attend vocational or technical schools, and some will probably enter the work

force directly after high school. I’m concerned that these students are often

under-achieving in their academic pursuits and have low motivation for scholastic

endeavors. Several of them are resourced for learning or behavior disabilities.

Short attention spans for more formal types of instruction are common and the

students tend to prefer active types of learning activities.

Problem

My curriculum includes many word problems that are applications of

vocational fields: business, agriculture, health/medicine, or mechanical. I began

to ask myself several questions:

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What mathematics skills do high school students not interested in a four-

year college degree need before entering the workplace?

What skills are our local industries looking for in their employees?

Is the applied/practical mathematics curriculum we offer in Corning High

School successful in teaching these skills?

As I researched this topic, I began to see some consistent themes emerge

from interviews and studies on the issue. One of the concerns of many

employers was that new staff often lacks the skills to deal competently with

problem-solving situations that arise on the job. They felt that high school

mathematics preparation for job skills should provide students with the ability to

“use logic to draw conclusions from available information. Students must

problem solve, apply rules and principles to new situations, and use questioning,

inquiry, and the scientific process” (Day, 1997, p. 37).

I began to seriously assess my teaching curriculum and techniques to see

how I could find new ways to motivate my students while also enhancing their

problem-solving skills. This is a rather broad scope, and so I will not try to

encompass all of these aspects of my classroom in this paper. More specifically,

I have decided to focus on the issue of teaching problem-solving skills,

recognizing at the same time that part of the battle comes from sparking student

motivation.

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CHAPTER TWO

Assessing the Need

Before deciding what types of curriculum changes to implement into the

classroom, it would be in order to assess more specifically the concerns and

needs of the business community. There seems to be a common consensus

from many sources that skill requirements in the labor market have escalated

much faster than the schools have improved (Murnane and Levy, 1997). The

problem is not so much that workers are less educated than they have been in

previous generations, but that the demands of the current workplace are more

sophisticated than in the past. To compete effectively today, American workers

must employ skills at a ninth-to-twelfth-grade level (Scott and Span, 1996). The

concern here is that there is increasing disparity between the skills students learn

in the classroom and those that are applicable on the job site. There is also

evidence to show that those jobs requiring a higher level of math skills also tend

to yield a higher financial return as opposed to salaries provided by lower-skilled

jobs. So what are the math skills that would better prepare our high school

students for success in their future career choices?

The SCANS report issued by the U.S. Department of Labor in June 1991

specifically dealt with the question of what skills they wanted schools to provide

their students with for success in the work place. This commission spent twelve

months talking to business owners, public employers, and the people who

manage employees daily in an effort to answer this question. Their first

assessment was that good jobs depend on people who can put knowledge to

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work. New workers must “be creative and responsible problem solvers and have

the skills and attitudes on which employers can build” (SCANS, 1991, p. 3).

Traditional jobs are changing and new jobs are created everyday. High paying

but unskilled jobs are disappearing. Employers and employees share the belief

that all workplaces must "work smarter" (SCANS).

Although the SCANS report included the need for basic competencies in

reading, writing, arithmetic and mathematics, speaking, and listening, they also

reported the need for solid thinking skills. These skills include thinking creatively,

making decisions, solving problems, and reasoning. One of the problems, as the

writers of the SCANS report see it, is that employers have never clearly told

educators what students need to know and be able to do in order to succeed.

Therefore, this commission strove to put together a report that would define the

skills needed for employment, propose acceptable levels of proficiency as well as

ways to assess these proficiencies, and develop a strategy to implement these

skills into our nation’s school curriculum. The commission maintained that

SCANS research verifies that what we call workplace know-how defines effective

job performance today. This know-how has two elements: competencies and

foundation (SCANS).

The basic competencies that would be expected from any high school

degree program would include the ability to perform basic computations. The

problem is that students may have difficulty transferring the algorithms taught in

the classroom to real-world problems in the workplace. Employers expect

students to also have the ability to solve practical problems by choosing from a

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variety of mathematical techniques. The concern arises as to whether schools

have focused too much on the algorithms, and not enough on the more

fundamental issues of quantities and their relationships (Bracey, 2001).

This leads to an assessment of thinking skills, and a look at whether

schools are equipping their graduates with the ability to combine ideas and

information in new ways in order to achieve goals. One of the thinking skills that

the SCANS report specifically targets is that of problem solving. The commission

suggested that students’ problem-solving skills needed to include “the ability to

recognize problems and devise and implement a plan of action to resolve the

problem” (SCANS, p. 35).

The National Council of Teachers of Mathematics has also targeted the

issue of problem solving as worthy of being included in their standards that were

published in 2000. They suggest that teachers should “enable all students to

apply and adapt a variety of appropriate strategies to solve problems while

monitoring and reflecting on the process of mathematical problem solving”. The

goal is to equip students with the knowledge and tools that enable them to

approach and solve problems beyond those they have studied in their curriculum.

In other words, teachers should strive to help students “develop a broad

repertoire of problem-solving (or heuristic) strategies” (NCTM Standards, 2000,

p.335).

Many researchers have concurred with the SCANS report’s stand on the

importance of problem solving in today’s workplace. Although a few will warn

against teaching problem-solving skills at the expense of a solid knowledge base

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(Sweller, 1989), most commentaries support the agenda that the

mathematician’s main reason for existence is to solve problems (Halmos, 1980).

Lawson breaks problem solving strategies into three main categories:

task orientation strategies which influence the dispositional state of the student,

executive strategies which are concerned with planning and monitoring cognitive

activity, and domain-specific strategies with include heuristics and other

procedures developed for organizing and transferring knowledge (Lawson,

1989). Lawson maintains that there is evidence to suggest that training in the

use of different types of general problem solving strategies will positively affect

performance in both mathematics and other curriculum areas. He then goes on

to suggest that this training will also lead to a greater ability to transfer similar

problem solving strategies to other areas. In summary, Lawson advises that

general problem-solving strategies can have a powerful influence on an

individual’s success in a variety of situations as long as that individual is also in

possession of a well-organized knowledge base.

Problem-Solving Strategies as Related to Learning Taxonomies

Before developing problem-solving strategies for the classroom, it would

be advantageous to review some of the more popular theories on learning

taxonomies. An understanding of how students learn would provide an instructor

with a better foundation upon which to base an instructional problem-solving

strategy.

Piaget divided a child’s learning into stages of development, regarding the

student’s intellectual development as an outgrowth of their physical age. He

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maintained that by the age of 11 to 15, a child was ready to begin the period of

formal operations. During this time, he determined that children begin “to reason

realistically about the future and to deal with abstractions…or ideas about

qualities and characteristics viewed apart from the objects that have them”

(World Book Encyclopedia, 1986). Since the age of students commonly found in

the high school classroom ranges from about 14 to 18, for the purposes of this

paper, we will assume the students have all reached the period of formal

operations.

Bloom’s taxonomy divides learning into three domains: cognitive, affective,

and psychomotor. The cognitive domain looks at an individual’s knowledge and

intellectual skill development, and is divided into six major categories. Bloom

maintained that each category must be mastered before an individual could

move on to the next level. These six categories include: knowledge,

comprehension, application, analysis, synthesis, and evaluation (Bloom, Mesia,

and Krathwohl, 1964). Many of the concerns put forth from employers is that

while workers may have the knowledge and comprehension of a concept, they

often fall short of the ability to apply this concept to a new situation, or even

more, to analyze the scenario, making inferences from the present situation to

use in future situations. This reinforces Lawson’s concern about transfer of

knowledge.

The SOLO taxonomy (Structure of Observed Learning Outcomes) is

somewhat similar to Bloom’s taxonomy. It was developed by Biggs and Collis in

1982 and is also based on the concept that students learn with advancing

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degrees of complexity, dividing this learning framework into five stages. The

problem-solving competencies that we are concerned about would fall under the

last two stages. The first of these is the rational level, where students are able to

see the significance of the parts of their knowledge base as it relates to the entire

concept. The last stage is the extended abstract level. This is the most

important stage in terms of competencies in the modern workplace as it deals

with an ability to make connections and transference of previously learned ideas

and principles to new situations that arise (Atherton, 2002).

All of the aforementioned taxonomies have a common theme. It seems

obvious that the concerns expressed by the SCANS report could be narrowed

down to a concern that we are not preparing our students to enter the world of

work at the more advanced end of the learning spectrum. Rather than stopping

with rote memorization of common mathematical algorithms, we need to be

focusing on guiding students to the upper levels of learning, which allows for the

synthesis of ideas, and the transfer and application of a sound knowledge base

to new situations and problems.

Span and Overtoom-Corsmit performed an investigation to assess the

ways that gifted children solved mathematical problems as opposed to the

average student. Their conclusion was that the more talented students

immersed themselves in the details of the problem, using their analytical abilities

to form hypotheses and a systematic approach to finding the solution (Span &

Overtoom-Corsmit, 1986). Average students took a more random approach to

looking for possible solutions, and seemed to lack the tools to tackle the problem

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in a higher-level manner. The question becomes whether there are specific

strategies than can be taught to students that will enable them to use some of

these higher-level skills that more advanced students already possess.

Problem-Solving Strategies

I began to study various approaches on implementing problem-solving

strategies into the classroom. Burton (1987) suggests the use of an

investigational, enquiry-based style of teaching. Even in my college methods

classes I recalled the emphasis on using open-ended questions and similar

discussion tools to generate higher-level thinking processes. What I was looking

for, however, was a comprehensive set of strategies designed to encourage

students to synthesize prior knowledge with their own creativity in order to allow

them to successfully find solutions to new problem situations. In other words, I

wanted to help provide my students with the tools and confidence to deal with

new situations that might occur in future real-world situations, whether in the

workplace or otherwise.

Lynn Steen addresses this question in an article in Educational

Leadership in which he discusses the necessity for a broader curriculum focus

that puts greater emphasis on patterns and relationships, spatial reasoning,

observation and conjectures, and genuine problems, rather than spending so

much time on more common algorithms like fractions, graphing by hand, or two-

column proofs. He pushes for a larger picture of mathematics from kindergarten

through high school, with more experiences on the themes of “chance and

change, shape and dimension, and quantity and variable” (Steen, 1989, p.19).

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Halmos suggests that the answer is in a more in-depth approach to curriculum

work, emphasizing the quality of the exploration over the quantity of material

covered (Halmos).

A change in curriculum focus would necessitate a change in teaching

practices. Steen suggests the use of more active learning processes that

incorporate real-life problem-solving situations presented with instructional

variety, as well as some means of assessing the success of the various

approaches (Steen). Susan Forman suggests that while real-life problems are

usually fairly concrete, they are not always as straightforward as one might think.

She also concurs that the use of this type of problem can challenge students to

use higher-level thinking skills while making connections between the

mathematics they’ve previously learned and the world around them (Forman,

1995-96). She reminds us that workers in what used to be called “blue-collar”

jobs are now expected to have the skills to use their own problem-solving

strategies while working in an environment that incorporates a high level of

technology dealing with large amounts of data.

Benander, Cavanaugh, and Rubenzahl developed a group of problem-

solving activities for their classrooms at a community college in Massachusetts.

Their first focus was on developing a sense of cooperation and confidence

among the students in approaching problems. They used both group-building

exercises as well as initiative problems to accomplish these goals (Benander,

Cavanaugh, & Rubenzahl, 1990). These activities included games that involved

the students physically while at the same time developing reasoning skills. The

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activities also included student self-evaluations of both their success and their

sense of confidence at the end of the activities. They reported that students

expressed a reduction in anxiety levels in approaching math problems, a higher

confidence level in their abilities, and a more positive attitude towards math

education.

Barba and Rubba approach the incorporation of problem-solving

strategies as a procedural task analysis whereby tasks or problems are broken

down into pieces. These pieces are then analyzed to determine which bits of

information are the most important and what the relationship is between them

(Barba & Rubba, 1992). They suggested the use of audio or videotape so that

the teacher can then analyze the student’s declarative, procedural, and structural

knowledge base. The teacher can then look for misconceptions or areas in the

student’s problem-solving skills that need remediation.

The Workplace Literacy Project funded by the U.S. Department of

Education specifically targeted lower-achieving math students. Rather than

concentrating on applied math problems only, however, this program “blended

conceptual approaches to mathematics with problem-solving exercises that were

anchored in the students’ world” (Woodward, 1999, p.75). One of the instructors

involved in the pilot project, Terry Wilson, focused on using common

technologies such as calculators and software programs to collect and analyze

real data, rather than pencil and paper skills more often seen in the mathematics

classroom. The students used simple calculators for all the fraction work, and

instead put their time into learning to use spreadsheets and computer graphics to

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make pie charts and other pictorial representations to display their results. The

students then presented their results orally to the class, verbally expressing their

understanding of the problem and the results. Students were excited to

approach their math in a manner that they felt closely mirrored skills they would

use in the workplace later on.

Another discrepancy in approaches to problem-solving strategies is

whether we are teaching students particular rules or algorithms that they learn by

rote, or whether we encourage them to take commonsense approaches to

analyzing new problems. The challenge appears to be how to integrate these

two facets, so that students have a strategy in mind for where they are headed in

their problem-solving efforts, and then have the algorithms in their repertoire of

skills that they can use to finish the job. Booth suggests that if we teach students

how to switch back and forth between strategies, so that they are comfortable

implementing the most practical system for the problem rather than the most

basic or elementary strategy they know, we can help them increase their level of

cognitive functioning greatly (Booth, 1981). She maintains that many students

cling to their earliest problem-solving strategies basically because they are

comfortable with them and haven’t compared them to the efficiency of more

recent and advanced methods they now know. If we can use the students’ own

valid commonsense approaches and integrate them with more ‘mathematical’ or

logical approaches, we can effectively broaden their problem-solving capabilities.

Ruth Parker maintains that a good portion of the difficulties of students

(and later adults) in mathematics is that too often teachers view mathematics as

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a “collection of abstract skills or concepts to be mastered” rather than regarding

mathematics as an understanding of patterns and relationships (Parker, 1991, p.

443). She suggests that we must make a clear connection between classroom

mathematics and real world mathematics. In other words, as the NCTM

standards state: students must see mathematics as “an integrated whole,

explore problems and describe results using graphical, numerical, physical, and

algebraic and verbal mathematical models or representations” (NCTM

Standards, p.84). Ms. Parker encourages teachers to develop classrooms where

students learn to work both cooperatively and independently on real-world based

problems, stressing that students should be assured that there are many

approaches to any problem rather than just one right strategy. In effect, the math

classroom should resemble the environment one might encounter in the

workplace.

Parker had several recommendations to implement these concepts into

the mathematics curriculum. First of all, she suggested presenting open-ended

problems or situations that would encourage the students’ independent

mathematical reasoning without leading them to a particular solution. Secondly,

she asked students (and collaborative groups) to share their ideas with others in

the class, verbalizing their thoughts and reasoning. Lastly, she encouraged

students to try more than one approach to the problem, and stressed that there

could be several correct ways of solving a problem (Parker).

Parker also had specific suggestions for staff development. She

recognized that it is difficult to change the goals in mathematics curriculum

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without providing teachers with new methods to do so. She encouraged

professional development projects that would last from three to five years and be

available to all mathematics faculty. She also recommended that new

assessments be written that would allow teachers to evaluate the broader goals

of the current NCTM standards. According to the NCTM (1991, p. 1),

“Mathematical power includes the ability to explore, conjecture, and reason

logically; to solve non-routine problems; to communicate about and through

mathematics; and to connect ideas within mathematics and between

mathematics and other intellectual activity. Mathematical power also involves the

development of personal self-confidence and a disposition to seek, evaluate, and

use quantitative and spatial information in solving problems and in making

decisions.”

It would seem imperative, therefore, to implement specific strategies into

the curriculum that would encourage and develop these problem-solving skills.

Ina Miller suggests some techniques in an article for Lifelong Learning that

summarize many of the ideas presented previously by other authors. These

include acting out the problem, constructing a table, making a model, or drawing

a picture, looking for a pattern, making an organized list, working backwards, and

writing an equation to represent the problem (Miller, 1986). I decided to try some

of these specific strategies with the students in my classroom to see if I could

increase their level of comprehension and competency in working with problem

solving.

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CHAPTER THREE

Classroom Strategies

The class that I had targeted for my research was my Applied

Mathematics I classroom. I chose to specifically strive to improve the students’

self-reliance and competency as problem-solvers. As mentioned in my

introduction, these students often lack self-confidence in their mathematics

abilities and are sometimes difficult to motivate. If the research I had studied

holds true, it seems of utmost importance to provide these students with both the

confidence and the skills to enable them to be successful in their future

workplaces. I decided to implement the changes over a period of two units of

study. I would then compare the results from the unit tests from this year’s class

to those from last year’s to see if there were measurable signs of improvement.

Unfortunately, my class sizes range from fifteen to twenty students, so it may be

difficult to call this a viable sample group.

The units that I focused on for my classroom implementations were

entitled “Working with Shapes in Two Dimensions” and “Working with Shapes in

Three Dimensions”. These materials are part of a CORD series for applied

mathematics. The first unit dealt with identifying common geometric figures and

calculating their perimeters and areas. The students also were challenged to find

solutions to work-related problems that involved these figures (Appendix A). The

second unit worked with geometric shapes in three dimensions: cylinders,

rectangular solids, cones, and spheres. They learned to calculate the surface

area and volume of these figures as well as solve problems involving them.

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The strategies that I implemented into my classroom came in a couple of

different forms. The first strategy I used was to challenge the students to do

some problem-solving activities in an informal setting or as more of a game-type

activity. The purpose was to encourage them to use a variety of problem-solving

strategies in a non-threatening environment. The second approach I used was to

give them a list of strategies to help them break down problems into workable

pieces, and then to require them to demonstrate the steps that they followed

when they used these strategies as part of their required work.

While looking for activities to supplement into my curriculum, I came

across a text written by Dolan and Williamson entitled Teaching Problem-Solving

Strategies. It contained several fun activities that were designed to challenge

and strengthen students’ logic abilities with more informal activities. One of the

activities that I used was entitled “Selecting a Model” (Appendix B). This activity

involved using either drawings or actual physical objects to come up with models

for the situations presented. I gave this activity to my students individually, but

encouraged them to discuss options or ideas openly within their table groups.

Assessment for this activity was pass/fail. I wanted to see the results of their

brainstorming regardless of accuracy. I observed that the students were quite

involved and animated in their discussions and efforts on this activity, and

several times we stopped to discuss questions or approaches as a class.

The second activity I used from this text was called “Regular Polygons in a

Row”, and was designed to help students explore the concept of perimeter

(Appendix C). I had the students work in pairs on this activity to encourage

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further team building, and again graded this activity on a pass/fail basis. I was

surprised by how intense the students got by the end of the activity, as they tried

to find the correct pattern that would give them an algorithm for finding the

perimeter of any number of polygons placed end to end.

Another adaptation I made to my curriculum was to write a list of

strategies on the board that I left up for the entire time that we spent on these two

units (several weeks). These strategies were designed to help the students

become comfortable with organizing their own logical thinking processes as new

problems or situations arose. These strategies included:

1) What does the problem want to know?2) What information do you have?3) Can you draw a picture to represent the problem?4) What mathematical processes will you need to do to get where you

want to go?5) Can you write a formula to present your problem?6) Solve!!!

We spent quite a lot of time modeling these strategies by doing problems as a

class using this approach. I was amazed to realize that many times students

were off track because they got caught up in the language of the problem and

forgot to look at what exactly the problem was asking for. I encouraged them to

write down information as they encountered it, making a list of all data whether

they knew if it would be relevant to the solution or not. I also strongly

recommended a picture wherever it was applicable. When I stopped to help a

student with a question, I often responded first with: “Where’s your picture?”.

Then we coud begin a discussion of what they were trying to do.

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We also spent a great deal of time discussing the nuances of language

and what that means in terms of mathematics. If they are to find the price per

square foot of carpet, what mathematical operation does that represent? If they

only sell one-third of their tickets, what does that mean in terms of numbers?

What clues will help them distinguish between perimeter, area, or volume

problems?

Once they had assessed what they needed to do, I asked them to write

down the formula or expression that they will use. Many in this class struggle

with pencil and paper mathematics, but I wanted to emphasize the problem-

solving and logic aspect of the process rather than the ability to crunch numbers.

Therefore, I allow full use of calculators, while also spending time discussing how

to decide if an answer is reasonable. I encourage the students to look at the

calculator as a tool to supplement their own logic and skills. Every time they

reach a solution, I ask them to look at it and decide if it seems reasonable. If

lumber costs $3.30 per board foot, would it make sense that fifteen board feet

would cost $.22? Once the students got in the habit of checking their own work

over, it was amazing how many times I would hear them say: “Well, that can’t be

right!”, and back to the beginning they would go.

Another change I made was in the way I presented some of our lab

activities (Appendix D). I began to ask more specific strategy-oriented questions

on the lab papers that I handed out at the beginning of the lab and we would go

through the first few questions as a class. This seemed to help the students get

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a better overall understanding of what the activity was about so that they could

develop their own plan of how they wanted to solve the problem.

Finally, I adjusted the way I assessed these activities. Although for my

final research analysis I went back to unit test scores for comparison, as we

engaged in some of these new activities I found myself using a more holistic

approach to assessing the students’ work. I found a text published by the

National Council of Teachers of Mathematics entitled How to Evaluate Progress

in Problem Solving that outlined some suggestions. I tended to use a

generalized holistic approach that was less time-consuming than some other

forms of assessment and took into account my considerable class load. The

holistic approach allowed me to take into consideration other aspects of the

students’ solutions than just the correct answer, while still being efficient enough

to be manageable on a day-to-day basis. Since I wasn’t using a more specific

rubric, I made it a point to make comments or suggestions on students’ papers

when applicable. I also revisited the more challenging problems in class

discussions, both during and after homework assignments.

Overall, I made a concerted effort during this time to conduct my

classroom in a manner that continually encouraged the students to be creative,

but yet logical, in their problem-solving approaches. I made a conscious effort to

use open-ended questions in our discussions, to encourage multiple approaches

to the same problem, and to provide an atmosphere where the students felt

comfortable sharing their thoughts and ideas.

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CHAPTER FOUR

Assessing the Modifications

For a formal comparison of the effect of the modifications I implemented

into my curriculum for this quarter, I used only the unit test scores as compared

to test scores from the same unit from the class the year before. The second unit

was rather long, so in the past I have tested over this unit once mid-chapter, and

then a final time at completion of the unit.

I took a look at comparisons of both the test scores from the two units in

which I had made modifications, as well as the previous six unit tests (Figure 1).

In this manner, I could better ascertain whether class score discrepancies were

related to the differences in teaching methods or just different student abilities.

Figure 1

At first glance, it would appear that there was some improvement after the

modifications were implemented. Although the 2002-03 class also scored better

on the unit 6 test, which was before I had implemented the modifications, I was

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encouraged by the fact that they consistently continued to score better than the

class from the year before on the remaining three unit tests, the ones after I’d

implemented the modifications. This class had scored slightly better than the

previous year’s class on the first four unit tests on the graph, but the discrepancy

between the two classes scores increased in the later units, and continued to rise

until the final unit test showed an improvement of over ten percent from the

previous year.

I decided to run the statistical test for two independent populations to see

if it would support my claim that the test scores had improved. I first needed to

perform an F-distribution test to see if the variances were equal. The

assumptions made for this test are that the populations are independent and

approximately normal. I charted my data for the previous year’s class as well as

the recent year’s class to see if they approximated a normal curve (Appendix E

and Appendix F). I thought the first year’s data was very close to a normal curve,

but I was somewhat concerned with whether the second chart could be

considered normal or if it was slightly skewed to the right. I did perform the equal

variance test on these sets of data. My calculations were as follows:

Ho: variances are equal (s12 = s2

2)

H1: variances are not equal (s12 s2

2)

N1 = 15 N2 = 15

x1 = 74.378 x2 = 81.933

s1 = 12.147 s2 = 11.932

F = s12 / s2

2 = 1.0364

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In a two-tailed F test with 14 and 14 as my degrees of freedom and using

= .05, the rejection region is F > 2.9829. Since my test statistic does not fall in

the critical region, I do not need to reject my hypothesis and I can consider my

variances equal. I went on to do a two-tailed t-test analysis. For this test my

data was as follows:

Ho: 1 2

H1: 1 < 2

sp2 = 144.961

t = -1.6602

With the two-tailed t-test with 28 degrees of freedom, I will reject my null

hypothesis if t < -2.048. Since my test value did not fall within the critical region, I

could not reject my null hypothesis and therefore could not statistically support

my assertation that my modified curriculum increased the test scores of my class.

Before I quit with the statistical analysis, I decided to also compare my

modified curriculum’s scores with those from the same class for the first part of

the year before I had implemented the modifications. I again ran an F-

distribution test to see if my variances were equal. My data was as follows:

Ho: variances are equal (s12 = s2

2)

H1: variances are not equal (s12 s2

2)

N1 = 15 N2 = 15

x1 = 77.2933 x2 = 81.933

s1 = 11.99862 s2 = 11.932

F = s12 / s2

C = 1.0111

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In a two-tailed F test with 14 and 14 as my degrees of freedom and using

= .05, the rejection region is F > 2.9829. Since my test statistic does not fall in

the critical region, I do not need to reject my hypothesis and I can consider my

variances equal. I went on to do a two-tailed t-test analysis. For this test my

data was as follows:

Ho: 1 2

H1: 1 < 2

sp2 = 144.961

t = -1.0260

With the two-tailed t-test with 28 degrees of freedom, I will reject my null

hypothesis if t < -2.048. Since my test value did not fall within the critical region, I

could not reject my null hypothesis and therefore, again, I cannot prove that my

class showed improvement.

Due to the fact that I teach in a small school, my test sample was very

small. Even if the statistical analysis had been in favor of my hypothesis, it would

probably not be valid to draw too many conclusions from this. I turned then to

other observations to continue my assessment of the modifications I was

implementing.

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CHAPTER FIVE

Further Discussion

There were several observations that I think bear mentioning. First of all,

the fact that the data from the scores after the curriculum modifications seemed

skewed slightly to the right indicated to me that there were fewer students in the

lower skills area. In other words, although the overall scores only showed a

slight improvement, the growth in the lower third of the class was more marked. I

would speculate that perhaps those that scored in the top third from the

beginning of the year already possessed competent problem-solving skills,

whereas those who struggled with these concepts reaped greater benefits.

Another observation I would like to point out is that I felt that the class as a

whole exhibited a greater willingness to participate in activities, classroom

discussions, or other challenges presented to them. They were more willing to

‘risk’. I felt they were gaining confidence that they had some tools with which to

break down situations into workable pieces, and did not let a fear of failure keep

them from tackling new situations. I tried to reinforce this behavior by positively

responding to all efforts and by giving credit for exhibiting sound problem-solving

logic rather than just a correct answer.

The modifications that I was making in my applied math classroom carried

over into other areas of my teaching. I also teach chemistry and physics, and I

found many times I would use the same problem-solving approaches to their

curriculum as I did to the applied mathematics. I soon came to the conclusion

that no matter what the level or topic of study, these same skills were important

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for the students to find success. My upper classmen, juniors and seniors, were

often dealing with the same confidence issues as my freshmen and sophomores.

Many of them will soon be going to college or out into the workplace, and it is my

job to see that they have the skills to be successful.

One of the areas that I would like to explore further is the use of rubrics to

assess these problem-solving skills. I found several interesting approaches in

my research, and I plan to try other methods in addition to the generalized

holistic approach I used most often during my study. This past summer I helped

evaluate ICAM math tests that our area high school juniors had taken as part of

their state assessment requirements. The evaluation involved the use of an

analytical rubric that divided the problem into several phases and assigned points

according to the level of understanding the student exhibited in solving the

problem. I found I did not always agree with the rubric we were required to follow

in our assessment, and thought it would be interesting to devise my own for

some of my classroom testing to see if I could do better!

I find myself recalling one of those defining points in my teaching career

when I had a student who was helping his dad and uncle harvest come excitedly

to class one day. He shared with us that they were hauling the corn to the bins,

and were debating how many more bushels of corn they could get into the bin

where they were currently unloading. They knew about how many cubic feet it

took per bushel, but were estimating how much space was presently left. My

student had just finished studying cylinders and cones, and recognized the

shapes in the bin. He quickly pencil and papered out the correct volume for the

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remaining space. He was quite proud of making the connection and actually

being able to use something he had learned, and of course, his teacher was

exceedingly proud of him also!

The point is this: We have to make math education applicable to daily life

and then insure that the students have the skills and confidence to transfer their

mathematics knowledge to these real-life situations. Carmel Schettino (2003) put

this very well in a recent article for the Mathematics Teacher journal. She holds

that “an educator must first commit to the premise that helping students develop

their ability to solve problems independently is the major goal of mathematics

education”. She maintains that this “entails teaching students that they have the

freedom to solve problems with a set of given tools and knowledgeable guidance

and that the goal is to further develop their mathematical toolkit”.

I completely concur with Ms. Schettino’s position. As the National Council

of Mathematics reminds us: “A major goal of high school mathematics is to equip

students with knowledge and tools that enable them to formulate, approach, and

solve problems beyond those that they have studied” (NCTM, p.335). This is the

same theme that appears in the Department of Labor’s SCANS report. My goals

as a mathematics instructor must include the desire to provide my students with

the skills and confidence to meet this challenge. This entire project has

broadened the scope of how I perceive myself as a mathematics/science

teacher. I would now venture to say that the curriculum itself is not the main

focus of my instruction, but rather the desire to provide my students with the skills

and assurance to approach future undertakings with confidence.

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Appendix A

Area Problem Examples (from the CORD curriculum)

1) Mary trains for cross-country by running around the block 5 times each evening. The block is a rectangle 500 feet long and 250 feet wide.

a) How far does she run when she runs ONCE around the block?b) Find the total distance that she runs each evening.c) Convert the distance she runs to miles. (5280 ft. = 1 mile)

2) A furniture factory builds rectangular tables that are 5 feet long and 2 feet 8 inches wide. They put oak trim all around the edge of the table. This month they will build 85 tables, and the trim costs 56 cents per foot. How much will they spend on trim this month? Ask if you need help understanding the steps of this problem.

3) A biologist is studying insect populations in a field. In a test square 10 feet on a side, he finds 30 greenbugs. The whole field is 450 feet long and 300 feet wide. He wants to estimate the total number of greenbugs in the field. Follow these steps:

a) Find the area of the test square.b) Find the area of the field.c) How many of the test squares would fit in the entire field?d) Approximately how many greenbugs are in the field?

4) A gardener has a rectangular garden that is 120 feet long and 80 feet wide. She is planting the whole garden with a cover crop of rye grass, and she is supposed to use one pound of seed for every 1000 square feet. Find the area of the garden, and then find how many pounds of seed she should use.

5) A map of Ms. Brown’s land shows a large pond (almost circular) inside a rectangular field. She wants to plant the field to pasture, so she needs to know the area of the field, not counting the area of the pond.

a) Measure, and use the scale given to find the lengths she needs to know. Scale: 1 inch = 300 feet

b) Find the area of the field and pond.c) Find out how many square feet of grass she will plant.

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Volume Problem Examples (from the CORD curriculum)

1) A loading chute brings grain into a bin as shown in the drawing. The bin is a cone. How many bushels of wheat will the bin hold? (There are 0.8 bushels per cubic foot.)

2) A spherical water tank has a radius of 10 feet. It lets the water flow out of the spigot at 2.5 ft.3/sec. How many minutes will it take to drain the tank?

3) A company has a shipment of 100 mirrors arriving in boxes which are 3 feet long, 15 inches wide, and 6 inches tall. They want to store them in a space that is 8 feet long, 6 feet wide, and 5 feet tall. They want to know if all the boxes will fit. Follow these steps.

a) Change all the units of the dimensions of the box to feet, and find the volume of one box.

b) Find the volume of all 100 boxes.c) Find the volume of the storage space.d) Answer clearly: will all the boxes fit in the storage space?

4) A construction company is pouring a sidewalk that is 120 yards long, 6 feet wide, and 9 inches thick. Concrete will cost $35 per cubic yard. They want to know how much all the concrete will cost. Follow these steps.

a) Change all the units to yards.b) Find the volume of the sidewalk.c) Find the cost of the concrete.

5) A backhoe has a bucket that holds about 5 cubic feet. The backhoe operator knows that he can dig about 30 buckets an hour. He needs to dig a trench that is 400 feet long, 6 feet wide, and 2 ½ feet deep. He wants to estimate how long it will take him to dig the trench.

a) How many cubic feet per hour does he dig if he digs 30 buckets per hour at 5 cubic feet per bucket?

b) What is the volume of the trench?c) How long will it take him to dig the trench?

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Appendix B

34

35

36

Appendix C

37

38

39

Appendix D

Measuring Soft Drink Cans and Cartons

Problem:How much wasted space is there in a 12-pack carton of pop?

Procedure:What will you need to know to figure out this problem?

What formulas will you use?

Data:

Calculations:

Solution:

Critical Thinking:Can you think of a design that would make better use of the carton’s

shape? (Draw a sketch to demonstrate.)

40

Appendix E

41

Appendix F

42

Bibliography

Atherton, J S (2003) Learning and Teaching:  Solo taxonomy [On-line] UK: Available: http://www.dmu.ac.uk/~jamesa/learning/solo.htm

Barba, Roberta H. & Rubba, Peter A. (1992). Procedural Task Analysis. School Science and Mathematics, 92, 188-92.

Benander, Lynn & others (1990). Ideas in Practice: A New Equation in Developmental Mathematics. Journal of Developmental Education, 14, 26-28.

Bloom, Bengamin S., Mesia, Bertram B., & Krathwohl, David R. (1964). Taxonomy of Educational Objectives (two vols: The Affective Domain & The Cognitive Domain). New York: David McKay.http://www.nwlink.com/~donclark/hrd/bloom.html

Booth, Lesley R. (1981). Child-Methods in Secondary Mathematics. Educational Studies in Mathematics, 12, 29-41.

Bracey, Gerald W. (2001). Does Higher Tech Require Higher Skills?, Phi Delta Kappan, 82, 715-716.

Burton, L. (1987). From failure to success: Changing the experience of adult learners in mathematics, Educational Studies in Mathematics.

Center for Occupational Research and Development. (1988). Cord Applied Mathematics. Texas: Author.

Charles, Randall, Lester, Charles, & O’Daffer, Phares (1987). How to Evaluate Progress in Problem Solving. Virginia: National Council of Teachers of Mathematics.

Day, Sheryl L. & Koorland, Mark A. (1997). The Future Isn’t What it Used to Be: Student Competencies for the 21st Century. Contemporary Education, 69, 34-40.

Dolan, Daniel T. & Williamson, James (1983). Teaching Problem Solving Strategies. Menlo Park, California: Addison-Wesley Publishing Company, Inc.

Forman, Susan (1995-96). Preparing a Mathematically Fit Workforce, Community College Journal, 66, 40-43.

43

Halmos, P.R. (1980). The Heart of Mathematics, American Mathematical Monthly, 87, 519-524.

Lawson, Michael J. (1989). The Case for Instruction in the Use of General Problem-Solving Strategies in Mathematics Teaching: A Comment on Owen and Sweller, Journal for Research in Mathematics Education, 21, 403-410.

Miller, Ina (1986). Techniques for Teaching Word Problems, Lifelong Learning, 9, 29-30.

Murnane, Richard J. & Levy, Frank (1997). A Civil Society Demands Education for Good Jobs, Educational Leadership, 54, 34-46.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Parker, Ruth E. (1991). Implementing the Curriculum and Evaluation Standards: What Will Implementation Take?, Mathematics Teacher, 84, 442-449, 478.

Schettino, Carmel (2003). Transition to a Problem-Solving Curriculum, Mathematics Teacher, 96, 534-537.

Scott, Brenda (1996). Are We Teaching the Mathematics Skills Students Will Need for Work in the Twenty-First Century?. Clearing House, 69, 354-357.

Span, Pieter & Overtoom-Corsmit, Ruth (1986). Information Processing by Intellectually Gifted Pupils Solving Mathematical Problems, Educational Studies in Mathematics, 17, 273-295.

Steen, Lynn Arthur (1989). Teaching Mathematics for Tomorrow’s World, Educational Leadership, 47, 18-22.

Sweller, John (1990). On the Limited Evidence for the Effectiveness of Teaching General Problem-Solving Strategies, Journal for Research in Mathematics Education, 21, 411-415.

United States Dept. of Labor. (1991). A SCANS Report for America 2000. Washington, D.C.: U.S. Government Printing Office.

Woodward, John (1999). Redoing the Numbers: Secondary Math for a Postsecondary Work World, Teaching Exceptional Children, 31, 74-79.

The Worldbook Encyclopedia. 22 vols. Chicago: World Book, Inc., 1986.

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