Technology*Enhanced.Authentic.Professional.Development.Framework.for.STEM.Educators.Using.Inquiry*Based.Activities!

Jhodi!Leong!ETEC!5330!Spring!2014!

!Title: Technology-Enhanced Authentic Professional Development Framework for STEM Educators Using

Inquiry-Based activities

Target Audience (Community): STEM Educators in K-12 classrooms

Context: Research has identified the insufficient amount of students adequately prepared for future careers in

STEM fields to be a result of a lack of authentic preparation in STEM K-12 education (Nadelson et al, 2012).

Although technology has been shown to improve STEM education through opportunities of exploration,

visualization, and authentic experiences, a lack of teacher confidence and self-efficacy integrating technology

into STEM classrooms is preventing effective, increased use by STEM educators (Wachira & Keengwe, 2011).

Professional development (PD) opportunities dedicated to the growth of technological, pedagogical, content

knowledge (TPCK) and the development of technology-enhanced context-specific activities has been identified

to improve teacher confidence and self-efficacy, increasing the implementation of technologies into STEM

classrooms. Based on several key characteristics identified by research of effective PD to meet teacher-reported

needs, this PD opportunity will be based on the following characteristics:

Teacher Needs Characteristic of PD Opportunity

Opportunities to collaborate with other STEM

educators on the development of curriculum-specific

content

Educators will work in small cohorts on the

development of STEM activities focused on a specific

concept for a desired curriculum

Opportunities to collaborate with working

professionals in STEM fields

The development of a partnership with STEM

professionals employed at the local university will

provide access to authentic STEM experts

Development of knowledge related to specific STEM

concepts with curricular ties

STEM knowledge will be expanded through

collaborations and discussions with colleagues and

STEM professionals, designing of inquiry-based

STEM activities, participation as a learner to complete

inquiry-based STEM activities, and opportunities to

observe group presentations on STEM content

presented by experts in the field.

Exposure to available technologies relevant to STEM An online database of STEM-specific technology tools

education will be available for reference by educators.

Development of technological skills and effective

methods of technology implementation into the STEM

classroom

Educators will have access to online tutorials about

how to use specific technologies, examples of

implementation, participation in thorough analysis of

case studies to develop an understanding of effective

methods for implementation, and opportunities to use

a variety of technologies themselves in exploratory

tasks.

Development of inquiry-based pedagogy and

implementation

Educators will have access to digital resources

highlighting the educational benefits of inquiry-based

activities, rationale of inquiry-based methods, and

effective strategies for implementation, as well as

opportunities for discussion regarding its impact on

student learning and impacts for teaching practice.

Access to continued technology support and methods

of problem-solving

Educators will form a learning community, sharing

their expertise as well as learning from that of others.

Educators will support each other in the use of

technology as well as receive support from

administration, board technology support, and online

troubleshooting resources.

Continuation of PD opportunity (ie. not limited to a

one-day session)

The scope of this PD opportunity is continual,

emphasizing continual commitment to further develop

databases and provide online support. Access to

online forums and resources can be accessed at any

time.

This program will be provided mainly through an online website hosting a variety of forums, technological

tools, and resources to develop and support a community of practice amongst STEM educators. Additionally,

this program will utilize system-wide organizational days (teachers are present at school, but students are not)

as opportunities for system-wide, face-to-face meetings amongst STEM educators. This online collaboration

will foster STEM teachers’ PD by offering continual access to and support from other STEM educators and

professionals as well as a bank of resources. The focus of these collaborations will be specifically related to the

participants’ teaching assignment, providing opportunities to develop tools and resources for concrete use in the

classroom. Teachers will also have access to online learning activities to place them in the role of a student to

analyze the learning processes of students. This will help educators develop an accurate view of concept

formation from a relational view of learning in context (Confrey, 1990). Organizational days will also allow for

experts to present learning tasks to educators to encourage them to think deeply about the content and possibly

experience subject matter from a different perspective, experiencing an inquiry-based activity first hand.

Role of Educators

Educators will consist of K-

12 STEM educators

• Educators will actively participate in online discussions regarding

pedagogy, technology, literature, effective strategies, student learning,

etc.

• Educators will actively make use of the online databases containing

educational resources such as text resources, videos, tutorials, digital

links, etc. to further develop knowledge and understanding.

• Educators will contribute to online databases with educational resources.

• Educators will collaborate with other educators and STEM professionals

to design curriculum-specific learning tasks.

• Educators will implement specific activities into their classrooms and

provide feedback to the learning community to promote further

refinement.

• Educators will provide assistance in areas of expertise to other

educators.

• Educators will maintain a contributory role as an active member of the

learning community.

Role of Facilitators

Facilitators will consist of

STEM educators considered

experts in a specific STEM

field, STEM professionals,

and school board

administration

• Facilitators will oversee the overall workings of the learning community

to ensure that contributions are valuable and representative of a

technology-enhanced inquiry-based method, as well as ensure the

appropriate use of netiquette by participants.

• Facilitators will identify the strengths of community members to use

expert resources to the fullest extent in community formation and

development (Baker-Doyle & Yoon, 2011).

• Facilitators will contribute valuable resources to online databases.

• Facilitators will contribute to the design of curriculum-specific learning

tasks.

• Facilitators will provide technological and pedagogical support to

educators.

• Facilitators will actively contribute and instigate meaningful discussion.

• Facilitators will keep up-to-date on effective TPCK impacting STEM

education and share their knowledge in the online forums.

• Facilitators will provide necessary feedback to administration and the

school board to identify teachers’ needs.

Role of Technology

Technology will consist of a

variety of digital

technologies that promote

authentic experiences in

STEM fields

• Online forums will provide the necessary collaboration space for

educators.

• Educational online databases such as Google Scholar will provide

educators with a source of scholarly reference.

• Digital technologies will be used in learning tasks performed by

educators to further develop their understandings of learning and

teaching with a tool.

• Online databases and reference tools will foster distance, anytime,

anywhere learning.

• Online design allows educators to participate from wherever and

whenever they choose.

• Digital technologies will be used in the development of authentic

activities to promote deeper understanding of STEM fields. These

technologies, technical support, examples of implementation, and user

feedback will provide valuable information to educators for effective

use.

Pedagogical Goals:

1. Participating educators will develop a greater understanding of inquiry-based instruction and learning

methods in STEM education.

a. Educators will focus on developing their skills around designing an inquiry-based activity to

achieve specific learning outcomes and effective methods for implementation using the

assignment of student and teacher roles.

b. An online forum via a website hosting blogs for discussion will promote continual access to

teachers to participate in discussion. This online forum will also be able to host a database of

resources (ie. collection of readings, studies, links to other online discussions, or access to

experts in the field) accessible by educators for further reference.

2. Participating educators will acquire knowledge about specific technologies supporting STEM education

and develop the skills required for implementation.

a. Educators will have access to a database of technologies supporting STEM education and be

encouraged to participate in discussion on each technology, activities using a given technology,

educational benefits, methods of effective implementation, and opportunities to receive/give

support for troubleshooting problems.

b. Educators will also participate in smaller projects around the development of a specific

curriculum-related task integrating a specific technology to further develop a database of STEM

activities ready for use in the classroom.

c. Teachers will actively participate in discussion on the educational benefits and affordances of

technology implementation in STEM fields.

d. Educators will extend their knowledge of available technologies and have the opportunity to

explore the usage and implementation of specific technologies individually, as well as with

colleagues, fostering opportunities for the sharing of expertise amongst professionals.

3. Participating educators will further develop their knowledge and understanding of direct links between

school-based curriculum and authentic application in STEM fields.

a. Educators will have access to online connections with STEM professionals for consultation as

well as open discussion about the development of tasks to provide authentic experiences for

students.

b. Collaboration with other educators on the development of curriculum-specific, inquiry-based

activities integrating a specific piece of technology is supported through the formation of online

communities, facilitating easy access to collaboration and shared knowledge of technologies.

c. The specific technologies to be used will be applicable to STEM fields of education, geared

towards providing authentic experiences to students by providing links to real-life applications.

Technology to be Used: The structure of this PD opportunity will be the participation in an online learning

community. A website or online learning management system, preferably hosted by the school board, will

provide an online forum for collaboration. This online forum will need to include online blogs for discussion,

allowance for uploading content, online databases of resources, and the ability to link to external online

resources.

Open databases of scholarly reference such as Google Scholar will be used to provide access to scholarly

references in the development of a collection of resources, readings, case studies, etc. that can be accessed by

educators to further develop their knowledge of pedagogy, STEM fields, implementation of technologies and

inquiry-based instruction, etc. Educators can contribute resources as well as participate in discussions about

these resources through the online forum. This will help build upon current knowledge and understandings of

pedagogy, teaching and learning strategies, previous research identifying key characteristics of effective efforts

at the implementation of technology in STEM fields, and effective strategies for implementation. Ultimately,

this knowledge will help build valuable educators that can contribute effectively to the online learning

community.

Additionally, resources pertaining to the identification of specific technologies for implementation in STEM

fields to effectively promote understanding in learners will provide educators with a variety of technologies to

instigate designs of technology-enhanced STEM learning activities. This will help educate teachers on

available technologies as well as provide examples of usage and processes in effective implementation as well

as strategies that were not effective and need refining.

Technology Integration Plan:

Goals:

1. Provide STEM educators with an online forum for collaboration and discussion.

2. Provide an online space for collaboration on curriculum-specific projects.

3. Foster online connections with STEM professionals.

4. Provide online resources to technologies relevant to STEM education.

5. Provide a forum for discussion about inquiry-based educational practices and

resources for additional reference.

6. Use expertise present in the community of practice amongst teachers and

professionals to provide a forum for problem-solving and technological assistance.

Format:

1. Online website: An online website will host a variety of forums, technological tools,

and databases of resources to develop and support a community of practice amongst

STEM educators. This will provide educators with forums for discussion,

collaboration, and support.

2. Face-to-Face Collaboration Opportunities and Learning Experiences: This

program will utilize system-wide organizational days (teachers are present at school,

but students are not) as opportunities for system-wide, face-to-face meetings amongst

STEM educators.

Framework:

The Learning For Use framework as presented by Daniel Edelson (2001) will be used to

inform the design of this PD experience as its four main principles:

1. Learning takes place through the construction and modification of knowledge

structures.

2. Knowledge construction is a goal-oriented process that is guided by a combination of

conscious and unconscious understanding goals.

3. The circumstances in which knowledge is constructed and subsequently used

determine its accessibility for future use.

4. Knowledge must be constructed in a form that supports use before it can be applied.

align with methods and processes for addressing the needs stated by STEM teachers for PD.

(See Appendix for rationale).

Step of LfU

Framework

Description of LfU Step

(Edelson, 2001)

Description of LfU Applied to PD

Experience

Motivation:

Experiencing

the Need for

New

Knowledge

Recognizing a limitation in

one’s knowledge creates a

desire to acquire new

knowledge to address an

issue or gap or to extend

one’s thinking to further

develop current

understanding. This

recognition fosters the

development of a context

for which the development

of new knowledge was

pursued, creating a memory

in one’s mind and

understanding for why the

new knowledge is relevant

and understood.

Activities must create a

demand for knowledge and

create a desire in the

learner to obtain new

knowledge. This can be

promoted through the

presentation of a task that

exposes a limitation of

one’s current knowledge

that requires new

knowledge in order to be

Experience Demand: The recognition of

insufficient numbers of students pursuing

careers in STEM fields and an inadequate

level of operational skill learned in STEM

classrooms from K-12 will motivate

educators to seek new instructional

approaches to better meet the needs of their

students, more adequately preparing them

for careers in STEM fields. The recognition

of potentially lacking skills and TPCK to

implement technology into STEM education

to further develop understanding, leading to

low confidence levels and technology

avoidance will motivate educators to seek

meaningful PD opportunities such as this to

further develop their skills and knowledge.

Additionally, more technologically

confident educators may desire greater

technology integration and a forum for

collaboration with other educators to

develop meaningful tasks. Desire for PD

opportunities to collaborate with other

STEM educators on designing curriculum-

specific learning tasks and access to

educational resources will be seen as a

valuable resource, leading to the further

development of one’s teaching practice.

Experience Curiosity: Being presented

explained. with the possibility of alternative

instructional methods such as inquiry-based

activities and instructional methods will

identify potential areas of development in

teaching practice. The identification of

these methods as beneficial to student

learning and understanding will instigate

interest about instructional approaches and

learning benefits. Educators will see this PD

opportunity as a chance to further develop

their knowledge in this area.

Knowledge

Construction:

Building

New

Knowledge

Structures

During this step, new

knowledge structures are

developed through

experiences that allow for

the addition of new

concepts to memory, divide

existing concepts into

subsections for further

analysis and development,

or create new connections

between new and/or

existing concepts. This can

be accomplished through

hands-on activities, direct

experiences, interaction

with or communication

from others, or a

combination of these

processes.

Activities must present

learners with opportunities

for direct interaction with

Observe: Educators will have opportunities

to experience inquiry-based activities from

the perspective of a learner as well as an

educator. Educators will also be provided

with opportunities to experiment with novel

technologies and/or novel strategies for

technology implementation. Experiencing

STEM-related activities and technology

integration, educators will further explore

STEM content and analyze various aspects

of relationships to real-life connections,

further developing their understandings of

the content and knowledge of the process

that learners go through in creating

knowledge structures related to the specific

content. Educators will also experience

applications of these strategies and

technologies, deepening their understanding

of the design of these tasks and effective

strategies for implementation.

Receive Communication: Educators will

have several opportunities for direct and

novel ideas and tasks that

present opportunities for

detecting relationships that

can be reinforced through

communication with others

to verify understandings

and build new knowledge

structures.

indirect communication to further develop

their understanding and knowledge. Direct

communication will come from

collaborations and discussions with other

STEM educators and working professionals.

This communication will happen at several

stages- the creation of learning activities,

and discussion of pedagogy, best practices,

effective strategies, etc., as well as

discussions about their experiences

completing inquiry-based tasks and their

assessment of the process for learners and

impact of teaching and learning. Indirect

communication will come from the

effectiveness of the learning activity in

creating meaningful learning in students and

through observations of engagement during

the implementation of a learning activity.

Knowledge

Refinement:

Organizing

and

Connecting

Knowledge

Structures

This step provides the

learner with an opportunity

to reorganize their

understanding, develop

meaningful connections

between new knowledge

and prior knowledge, and

promotes future recognition

by situating the learning

experience in a relevant

context. The acquisition of

knowledge must be situated

within a context of

application, utilizing

previous knowledge in the

formation of new

Apply: Educators will have the opportunity

to implement the activities they create in

collaboration with colleagues in their own

STEM classrooms. This will provide

educators with an opportunity to assess the

fundamental components of designing

technology-enhanced inquiry-based

activities and an opportunity to observe the

effectiveness of their ability to implement

these learned strategies effectively. Through

direct observation, educators will receive

valuable feedback about their

understandings of effective strategies for

implementation and the effect of different

strategies and design on student learning.

knowledge to strengthen

connections and foster

future recollection and use.

An opportunity for

reflection helps the learner

reinforce their

understanding and

promotes the development

of useful knowledge.

Activities must provide

opportunities for the

relevant application of

knowledge to reinforce

understanding and promote

recollections. In

conjunction with an

opportunity for reflection,

learners organize new

knowledge structures in

meaningful ways and are

encouraged to consider the

process of learning and the

links formed with prior

knowledge and

understanding. The

reflection process can help

refine understanding and

appropriately organize new

knowledge amongst prior

knowledge.

Reflect: Educators will have an opportunity

to individually reflect by assessing the

effectiveness of an implemented learning

task to achieve desired outcomes and

learning in their own classroom.

Additionally, Educators can reflect using

direct observations as well as

feedback/discussion with students,

colleagues, and professionals post-task.

This will provide educators with an

opportunity to assess and reflect upon their

level of understanding and knowledge

regarding task design, their role as a teacher,

effective strategies to elicit desired learning

outcomes for students, and the identification

of potential areas for further development.

Post-task, educators will be given an

opportunity to continue collaborations with

colleagues on the same task, using their

reflections to guide further developments

and the improvement of implementation

strategies, tasks design, or any other areas

identified for further development. Further

explorations and research into effective

strategies and developing a deeper

understanding of design, pedagogy,

authentic application, etc. will be

encouraged through access to additional

resources focused on identified areas of

improvement. Opportunities for alterations

and improvements will be followed with

additional opportunities for implementation

and further reflection and refinement.

Example of

Collaboration

Project for

Educators:

I have provided an example of one LfU-framed STEM activity using Geometer’s Sketchpad

(GSP) Software as an example of an activity that a group of educators could collaborate on to

create, implement, reflect, and refine.

(See Appendix for rationale).

Description of task: This activity will initiate student thinking about the meaning of degrees

and the potential need for an alternative form of angular measure. The design of this activity

focuses on the development of an understanding of the radian as a unit of angular measure

that can be described as a ratio between arc length and radius. A focus on developing a

definition and the ability to apply skills within an authentic context afford students with an

opportunity for knowledge creation grounded in real-life experience requiring strong

connections to previous knowledge. This activity should be completed in pairs to promote

collaboration and discussion.

Learning Objectives:

• Students will be able to define and understand the meaning of a radian as a ratio

between the radius of a circle and an arc length of equal measure.

• Students will articulate advantages to the use of radian measure for angles.

• Students will be able to solve for the radius of a circle given an angular measure and

are length.

• Students will be able to solve for an arc length given the radius of a circle and an

angular measure.

• Students will be able to solve for angular measure given the radius of a circle and an

arc length.

• Students will develop and use a method of converting between angular measures in

degrees and radians.

• Students will be able to solve application problems involving radian measure.

• Students will be able to solve application problems involving arc length.

Teacher Role:

• Motivation: The teacher will foster motivation, create demand, and elicit motivation

in students by providing support through encouraging students to explore the concept

of radians using the technology, providing technological support, presenting activities

as relevant to authentic experience, chunking activities and maintaining an

appropriate pace of exploration and instruction, and using questioning to elicit student

curiosity, as well as helping students identify potential misconceptions and areas of

understanding requiring further development.

• Construction: The teacher will guide students through the activity by offering

assistance using the technology, using questioning to provoke the identification of

relationships, encouraging exploration and analysis, assisting students in developing

links with prior knowledge and new relationships, and providing feedback to guide

exploration and evaluation.

• Refinement: The teacher will provide students with an opportunity for reflection and

encourage critical thinking about the relationships drawn between arc length, radius,

central angles and radians. Teachers will monitor discussion, offering input and

questions to further develop student knowledge and promote reflection. Teachers

will encourage students to share their work and collaborate to engage in discussion.

Additionally, the teacher will present students with a novel application of these

relationships to solidify their knowledge and broaden its application.

Student Role:

• Motivation: Students will critically examine their current knowledge to assess their

current understandings of angular measure. Students will actively complete activities

1 and 2, identifying present knowledge structures that are being challenged or

confirmed as well as being opened to the possibilities of limitations and a need for

new structures. Students will articulate their understandings and use the tools

provided to articulate further development needs.

• Construction: Students will actively participate in the construction of new knowledge

through completion of activities 3 and 4 using GSP to visualize the desired

relationships, combined with previous knowledge to thoroughly examine the data.

Students will complete these activities, ensuring their ability to articulate the process,

observations, relationships, conclusions, and remaining questions in collaboration

with others. Students will identify key aspects of their experience that led to the

creation of relationships.

• Refinement: Students will actively reflect on the learning process and identify the

mathematical relationships they developed. Students will participate in

collaborations with peers and the teacher to engage in discussions, providing relevant

information and receiving input from others to deepen understanding and promote

further thinking. Students will apply their new knowledge to actively try to solve a

novel problem.

Technology Support: Students will use GSP to develop an understanding of radian measure

by analyzing the relationship between the radius of a circle and arc length. Students will use

construction and measuring features of GSP to assist them in the construction of

understanding.

Step of LfU

Framework Activities & Design Strategies

Motivation:

Experiencing

the Need for

New

Knowledge

Fostering Motivation:

• Activity 1: Students will perform research using an Internet browser to

answer the questions: “Why are angles measured in degrees?”, “Why

is a full circle 360°?”, and “What is a degree?”

i. Through this research, students will start to discover that angular

measures in degrees are a rather arbitrary form of measurement

with no concrete mathematical foundation as to why there are 360

of them in a full circle. Students will start to wonder about the

mathematical validity of this unit of measure.

ii. The design of this activity is purposefully shaped to create a doubt

in students’ understanding about their current understandings of

angular measure. This activity will initiate critical thinking about

degrees as a form of angular measure and motivate students to

explore the meaning of angular measure in greater depth.

iii. Technology is used as a research engine to provide students with

information about angular measures in degrees necessary for this

activity. Students are exposed to a wealth of information that may

reveal misconceptions or areas of understanding that require further

development.

Creating Demand:

• Activity 2: Students will be presented with the following problem:

A construction company has recently hired you to build them a pulley

system to lift heavy objects. If the radius of the pulley must be 9 feet

and rotates at a rate of 60°/!"#, is it possible to construct a pulley

that will raise the heavy objects at least 8 feet off the ground in one

minute?

i. This problem presents students with an opportunity to critically

think about the relationship between the distance travelled on the

circumference of the pulley (ie. the arc length), the radius of the

pulley (ie. the radius of a circle), and the angle through which the

pulley rotates (ie. the central angle subtending the arc length).

Students must critically analyze the context of this question to

realize that the arc length will be equal to the distance that the

weight is raised off of the ground. Students will start to

hypothesize between these relationships.

ii. The design of this activity creates a demand for a relationship

between arc length, radius length, and central angle measure.

Students will start to identify that their current knowledge about

angular measures in degrees in insufficient to analyze this

relationship.

Eliciting Curiosity:

i. Activity 2 will elicit curiosity of a relationship between arc length,

radius length, and central angle measure once students determine

that their current knowledge about angular measure in degrees is

not providing sufficient knowledge to solve this problem. Students

will start to wonder about alternative relationships and potentially

alternative forms of angular measure that are possibly less arbitrary

than angular measures in degrees.

Knowledge

Construction:

Building

New

Knowledge

Structures

Fostering Knowledge Construction:

• Activity 3: Students will perform the following constructions and

calculations using GSP:

1. Construct 5 circles, C, of various sizes (use the circle feature on the

left tool bar in GSP). Perform the following steps for each circle.

2. Mark center O (Click to highlight O, the click “Display” !

“Label”).

3. Construct line OP (Click to highlight O and P, then click

“Construct” ! “Segment”).

4. Measure the radius OP in centimeters and label OP (Click to

highlight O and P, then click “Measure” ! “Length”).

5. Starting at P, create an arc length of length OP on the

circumference of the circle. Label this point Q (Click to highlight

P, then click “Construct” ! “Arc on circle”).

6. Construct line OQ (Click to highlight O and Q, then click

“Construct” ! “Segment”).

7. Measure and label arc length PQ and length OQ in centimeters

(Click to highlight O and Q, then click “Measure” ! “Length”,

then click to highlight P and Q, then click “Measure” ! “Arc

Length”)).

8. Calculate the circumference of each circle (! = 2!!) (Click to

highlight C, then click “Measure” ! “Circumference”).

9. Answer the following question in relation to each circle:

a. Approximately how many arc lengths PQ would you need to

go around the circumference of each circle?

10. Measure and label the central angles (an angle whose vertex is at

the centre of a circle and the arms are radii of the circle), ∠!"#,

for each circle (Click to highlight P, O, and Q, then click

“Measure” ! “Angle”).

11. Answer the following question in relation to each circle:

a. Compare the central angles, ∠!"#, for each circle. What do

you notice about all five?

b. If central angle ∠!"# = !"#!!"#$"%, write a definition for a

radian in relation to the arc length and radius length of a circle.

Following the exploration, students should have developed the

following conclusion:

• If central angle ∠!"# = !"#!!"#$"%:

The radian measure of an angle is a ratio that compares the

length of an arc of a circle to the radius of the circle. Ie.

!"#$%&"!!"!!"#$%!!"!!!"#!$% = !"#!!"#$%!!"#$%& .

Figure 1 is an example of one constructed circle :

Figure 1

" This activity has been purposefully designed to provide

students with an opportunity for the construction of knowledge

using a technological tool for assistance. This provides a

visual representation of a challenging mathematical concept

and allows students to have a tangible manipulative to explore.

" This activity will ensure that students develop an

understanding of a radian and form their own explanation.

• Activity 4: Students will use their knowledge about degrees and the

knowledge constructed in Activity 3 to complete the following,

developing a relationship between degrees and radians:

" Given a circle with radius r:

• One full circle (or one complete rotation) is _________° The arc length for this rotation is _________ or the

_________________ of the circle with equation

_________________

Therefore, !"#!!"#$%!!"#$%& = !!!!!!!!!!!!!!!!!! .

Therefore, 360° = _______!"#$"%&

• Half of a circle (or half of one complete rotation) is

_________° The arc length for this rotation is _________

Therefore, !"#!!"#$%!!"#$%& = !!!!!!!!!!!!!!!!!! .

Therefore, 180° = _______!"#$"%&

Therefore, 1!!"#$"% = _________!"#$!!"

Describe a strategy to convert from radians to

degrees:

Also, 1!!"#$"" = _________!"#$"%&

Describe a strategy to convert from degrees to

radians:

" This activity provides an opportunity for students to connect their

prior knowledge about degrees with their newly acquired

knowledge about radians, creating valuable links in their

understanding. Students must analyze their findings from Activity

3 in conjunction with prior knowledge about degrees and

circumference of a circle to develop conversion formulas in a way

that is connected to their understandings.

Observation: GSP as used in Activity 3 provides students with an

opportunity to use investigative tools to identify relationships between arc

length, radius length, and central angle measure through the exploration of

data. This provides students with an opportunity for the visualization of a

radian and allows students to use investigations to develop, use, and

interpret the data. Using knowledge from Activity 3 and prior knowledge

about degrees, students further develop their understanding of the

relationship between radians and degrees in Activity 4, connecting prior

knowledge to the formation of new knowledge. This meaningful

connection helps form connections in student understanding.

Communication: Students must complete this activity with a partner,

communicating about steps taken throughout the process. Students are

asked to directly communicate their findings in steps #8 and #10 in

Activity 3 and at the conclusion of Activity 4 to develop conversion

methods. Students will receive direct communication from the teacher as

well through guided questioning and guided assistance if necessary.

Students will receive indirect communication from the results of Activity 3

integrated with Activity 4; if students successfully complete the activity,

they should be able to draw generalized conclusion, but if not, they will

need to re-evaluate their findings and make adjustments.

Knowledge

Refinement:

Organizing

and

Connecting

Knowledge

Structures

Reflection: Activity 5: Students will reflect on the knowledge that they

discovered in Activities 1-4 to answer: “Why might radian angular

measure be useful?”. Students will use their knowledge that ! = !" to

solve the initial problem presented to them in Activity 2.

Activity 6: Students will collaborate with peers to compare their findings

and conclusions from activities 1-5. Students will engage in discussion

about process and the formation of their understandings as well.

• These activities provide students with an opportunity to verify and

solidify their understandings. Students will be able to identify with

processes taken by other students as well as learn potentially

different strategies and views that lead to the same conclusions.

Application: Activity 7: Students will be provided with the following

application problem requiring their newly formed knowledge involving

radian measure:

You have been hired to create a system of two gears to move a

conveyer belt. One gear must have a radius of 40 cm and the other

gear, a radius of 70 cm. If the smaller gear rotates through an angle of

300° in one minute, is it possible to construct this conveyer belt to

move at a rate of 2 m/min if the larger gear rotates at a rate of

1!!/150°? o Using GSP, create a labeled diagram representing the

above scenario to help you solve the problem.

" After completing the problem, use GSP to

label/calculate the desired radii, angles, and arc

lengths to verify your calculations.

• This activity provides students with a novel opportunity to apply

their knowledge and test their understandings. Students are

assisted in their learning and understandings through the visual

representation on GSP and the ability to verify their calculations.

This activity will help students solidify their understanding of

radians and its applications.

References

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Appendix

Rationale for

the Use of LfU

Framework:

The development of initial motivation is key to the LfU framework and can be achieved

through a variety of methods and activities centered on a larger goal within the context of an

inquiry-based task requiring the acquisition of specific knowledge. “The LfU approach

recognizes that for robust learning to occur, the learner must be motivated to learn specific

content or skills based on a recognition of the usefulness of that content beyond the learning

environment” (Edelson, 2001). The framework of this PD opportunity allows teachers to self-

identify an area of the curriculum, relevant to their current teaching assignment and of interest

to them, that they feel like they could deliver more effectively to promote deeper

understanding in their students. By placing educators in the role of a learner, the educators

empathize with the feelings, processes required to obtain specific knowledge, and desired

outcomes that frame the learning experience. This process deepens that educators’

understanding of the content as well as the process of delivery in implementing an inquiry-

based activity by “ground[ing] abstract understanding in concrete experience” (Edelcon,

2001). Participation in this LfU framed PD opportunity will help educators develop their

TPCK surrounding the development of their own inquiry-based activities for students designed

around the LfU framework.

The LfU framework promotes the use of technology to support discovery, stating that

“[s]cientific practice increasingly relies on computers to assist with data collection and

analysis, modeling, and prediction” (Edelson, 2001). Therefore, it is important for PD

opportunities to present educators with novel technologies, instigating ideas and the

development of technological implementation as well as to provide educators with an

opportunity to further develop skills using familiar technology to advance their skills and

broaden usage. The variety of presentation formats permitted by technology provides students

with effective forms of communication to further develop individual knowledge structures.

The requirement for further reflection and refinement post-knowledge construction gives

learners an opportunity to extend their knowledge structures past the context in which material

was learned, forming sufficient structures for future recollection, making it applicable to a

broader range of contexts. This helps address the challenge of “changing individual

conceptual frameworks… [by] challeng[ing] the overall conception help by many students and

teachers” by forcing them to consider the acceptability of their knowledge (Confrey, 1990).

Students are also forced to verify the legitimacy and rationality of their findings, exposing

potential falsities and misconceptions as well as solidifying the legitimate understandings and

connections to previous knowledge (Edelson, 2001).

One of the open issues with the LfU framework acknowledge by Edelson (2001) regards an

understanding around “how to initiate and maintain the changes in teachers’ and students’

practices that the LfU model and other constructivist learning approaches entail”. By

providing educators with a PD opportunity framed by the LfU framework as well as focused

on the development of content framed by the LfU framework, educators are placed in the role

of the learner within the framework as well as the role of the educator in the development of

tasks, developing a deep understanding of the requirements for implementation, motivations

behind the practice, and the design of activities to promote desired understanding.

Rationale for

the Use

Geometer’s

Sketchpad:

Geometer’s Sketchpad offers learners a variety of cognitive and social affordances such as:

1. The visualization of geometry though dynamic software such as GSP enables learners

to conceptualize mathematics through the use of technology to provide “access to new

understandings of relations, processes, and purposes” (Olive et al, 2010). Through

concrete experiences, students engage in the development of generalized

understandings in connection with their current understandings through “interactions

with ideas, relations, processes, structures and patterns viewed in new ways” (Olive et

al, 2010). This interaction and visualization provide a tangible quality to mathematics

not always present, providing a context in which mathematics learning takes place in

effective ways to build knowledge and understanding through work with a “model of

mathematics” (Sinclair & Jaciw, 2010). “Dynamic software packages can facilitate

visualization, connecting informal and formal mathematics, and develop perceptions of

mathematics as an instrument rather than an object” (Olive et al, 2010).

2. Technology affords students with an opportunity for operational understanding by

emphasizing the practice and applications of mathematics (Olive et al, 2010). GSP

offers students an opportunity to visualize, manipulate, and model mathematics in a

dynamic way to improve their understanding, especially in complex situations.

Students are presented with a visual representation of mathematical ideas, such as

graphs that they can be analyzed for relationships and begin to understand by

approximation not requiring the mastery of a specific skill. Although the mastery of

skills such as factoring are essential for successful advances in mathematics, skills such

as this can seem arbitrary to students when presented through a rote approach.

Alternative forms presented through the assistance of technology can help students

understand the concept and see the significance in relation to their prior knowledge and

the potential for building knowledge. “[T]his requires the student to understand and

make decisions about what mathematics might be useful and how it might be used”

(Olive et al, 2010).

3. GSP fosters meaningful interactions between task, teacher support, technological

environment, classroom and social culture, and mathematics by offing opportunities for

discussion regarding findings and process (Olive et al., 2010). Students have an

opportunity to share with and learn from other students’ prior knowledge to solve a

problem. Inquiry-based activities can be encouraged through hands-on activities

requiring active participation and knowledge creation rather than knowledge absorption

through rote memorization. Peers, the teacher, the technology, and the classroom

become resources to aid in the process.

4. GSP strengthens relationships between prior knowledge and new knowledge through

the construction of the basics as well as the novel (Olive et al, 2010). For example, a

process can start out by creating an isosceles triangle by its components, ie. by

recognizing that two side lengths and two angles must be equal, to use in the

construction of new knowledge such as the ambiguous case of the sine law. This helps

draw attention to links to prior knowledge and allows the learner to have a gradual

learning process with opportunities for small successes to encourage perseverance.

Additionally, GSP can help address misconceptions by preventing the formation of

impossible objects, ie. a triangle where the hypotenuse is shorter than another side of a

right triangle. The interaction of “Evocative Computational Objects characterized by

both their own computational nature and the evocative power caused by their

relationship with geometrical knowledge” promotes the understanding of geometric

figures and their construction to develop concrete referents (Mariotti, 2000).

5. The use of technologies such as GSP “encourage a closer relationship between

mathematical knowledge and mathematical practice, providing learners with

opportunities to experiment, visualize, and test emerging mathematical

understandings” (Olive et al, 2010). Students are becoming more educated about real-

life applications of math and future math-focused career options available.

Additionally, students are better prepared for these careers as a result of experimenting

with raw data or novel processes that require them to analyze and interpret the results

to make meaningful connections, much as they would need to do in an authentic

environment.

The promotion of technology in mathematics education benefits the learner through such

software as GSP providing opportunities for exploration and interpretation. The integration of

technology however, provides several affordances to educators as well, requiring the analysis

of effective practices (Olive et al, 2010); some considerations and affordances include:

1. Educators are encouraged to deepen their understandings of the applications of

mathematics, rather than a traditional focus on the teaching of strictly algebra-based

mathematical processes.

2. Educators develop a focus on the relationships between mathematical practice and

opportunities for students to experiment, visualize, test understandings, and extrapolate

and interpret findings (Olive et al, 2010).

3. Educators have an opportunity to experiment through the technology tools as well,

being placed in the role of a learner to better understand the processes in creating

knowledge and effective strategies for activity design.

4. Educators promote learning through encouraging students to take control and make

decisions about their learning and must develop effective strategies for guiding

explorations and encouraging drive (Sinclair & Jackiw, 2010).

5. Educators must effectively use their own feedback as well as the feedback provided by

technology to supplement student understanding and promote further inquiry (Olive et

al, 2010).

“The role of the teacher is fundamental, in order to direct the goal of the discussion and to

guide the evolution of personal senses towards the geometrical meaning of a construction

problem, and more generally to the theoretical perspective” (Mariotti, 2000). The teacher

must be adequately trained in the use of a technology to effectively implement it into an

inquiry-based activity to help students develop internal meanings and personal understandings

of mathematical content. Therefore, it is essential that PD opportunities allow educators to be

placed in the role of a student to better understand the learning process. While a tool such as

GSP can provide students with an opportunity for deeper understanding and offer visualization

not offered through conventional methods of mathematical process, this is only possible when

the activity is designed and administered in a way that fosters exploration and support.