teaching for transfer using geometer’s...

Teaching for Transfer Using Geometer’s SketchpadReflective Teaching Project

Final ReportMSU-MAET 2007

Christopher Lloyd KerlinMay 6, 2007

Statement of General Question

As a middle-school mathematics teacher, I am concerned about my students being able to transfer the knowledge they learn in the Geometer’s Sketchpad environment to problem-solving and application situations out of the environment. The 2005/2006 school year was the first year I used Geometer’s Sketchpad in my classroom. I noticed that my students understood the mathematical concepts when doing the activities in the environment, but when I asked them to solve conceptually related problems back in the regular classroom they were not able to transfer their new knowledge to similar problem solving situations. They were not able to connect the big concept they learned using Geometer’s Sketchpad to skill related questions typically seen in math textbooks and on standardized tests.

I teach seventh grade mathematics at Cairo American College in Cairo, Egypt. My students are twelve and thirteen years old with a fairly even balance of males and females. My students are mostly from the upper ranges in socio-economic status. Our school is comprised of forty-seven percent U.S. citizens, fourteen percent host country nationals, and thirty-nine percent are other nationalities. Their achievement characteristics are quite evenly spread out based on the results I have seen from their IOWA scores with percentile ranks from the tenth percentile to the ninety-ninth percentile in total math.

In this action research project I will be studying how my teaching strategies used with Geometer's Sketchpad can help my students transfer their knowledge to problem-solving situations. I feel this problem is important for many reasons. I feel that the dynamic environment of Geometer’s Sketchpad offers my students an opportunity to truly understand abstract mathematical concepts I am to teach them during the year, and it gives them the opportunity to think like mathematicians. This research will help me by focusing my energies on improving my teaching strategies when using Geometer’s Sketchpad so that my students can transfer what they learn in The Geometer’s Sketchpad environment to other problem-solving situations and real-life applications out of the environment. This will result in my students becoming better real-life problem solvers, and it will give them a strong mathematical foundation to build upon for future studies in advanced mathematics courses.

Reflective Teaching ProposalChris Kerlin 2

Review of Related Research and Information

Geometer’s Sketchpad

As stated in Exploring Geometry (1999), Geometer’s Sketchpad encourages a process of discovery that more closely reflects the way mathematics is invented: a mathematician first visualizes and analyzes a problem, making conjectures before attempting a proof (Key Curriculum Press, 1999). Geometer’s Sketchpad therefore provides an ideal environment for my students to become mathematicians themselves, and to develop the thinking routines associated with being a mathematician. In the past my students have initially struggled with truly understanding the abstract mathematical concepts I am to teach. This in turn inhibits their ability to transfer their initial understandings of the concepts to other situations. By getting them to think and be like mathematicians in The Geometer’s Sketchpad environment they will have a better initial understanding of the mathematical concepts and they will be better equipped to transfer that knowledge to other situations.

There are three important mathematical features Geometer’s Sketchpad has which enhance deep mathematical understanding for students and will help them transfer their understanding to other situations. They are: 1) Continuous motion, 2) Connectivity, and 3) Communication (Sinclair & Crespo, 2006). Continuous motion refers to the software’s ability to actively engage the students in moving and manipulating shapes while looking for relationships throughout the process. Connectivity, for example, refers to the programs capability to enable the students to look at algebraic ideas from a geometric perspective. And communication refers to the fact that the software gives students a meaningful context to use, learn, and understand mathematical vocabulary and notation, which furthers their understanding and transference of the concepts they are using. All three of these components are important factors in establishing a good initial understanding of the mathematics my students are studying which will enable them to transfer what they learned to situations outside of the environment.

Other aspects of Geometer’s Sketchpad that can facilitate transfer for my students are that they are in active control of their learning and they can go at their own pace in constructing their meaning of the mathematical concepts we are studying. As Hannafin (2001) recommended, “affording students limited control, honoring their need to “mess around,” and letting them set the pace are all part of successful active learning situations that encourage transfer” (p.143).

Teaching for Transfer

The main focus of my action research project was teaching for transfer when using Geometer’s Sketchpad in my seventh grade mathematics classroom. The pedagogical approach I implemented was grounded upon the research I did last summer, and the instructional design had several components. The first was to try to frame my Geometer’s Sketchpad lessons into meaningful contexts for my students. After the stage was set my students constructed their own knowledge of the mathematical


concepts by completing the Geometer’s Sketchpad activities individually or with a partner. Upon completion of the activities, when time permitted, I put students together to further refine the ideas they encountered in the activity. The students were then given a homework assignment asking them to transfer what they learned in the Geometer’s Sketchpad activity to the assignment problems, some of which were similar situations and some, which were different. As a class, the next day we would discuss the connections between the two activities and revised the homework. Once a unit of study was completed I gave an assessment, which was identical to the pre-assessment given. Again, they were asked to transfer what they learned throughout the unit to the assessment questions. Some of the questions were similar situations, and some were not similar to what they had done in the Geometer’s Sketchpad activity itself, and the homework assignments they were given.

The first component of my instructional design was to frame my Geometer’s Sketchpad lessons into meaningful contexts for my students. As stated in Jonassen (1999) the learner will have a deeper understanding of new skills and concepts if they are initially and consistently constructed in meaningful contexts to which they can be related. For transference to happen for my students I needed to make sure they initially understand the concepts I would be asking them to apply at a later time. By setting the lessons in a meaningful context, and using The Geometer’s Sketchpad environment itself, I provided my students with a context they could draw upon when I ask them to transfer their new knowledge at a later time to new situations.

During and after the Geometer’s Sketchpad activities my students worked individually, in pairs and in groups of four. Through these groupings, students had many opportunities to discuss and share their ideas and thoughts on the knowledge they constructed. In fostering this type of learning environment I created a classroom atmosphere that encouraged students to articulate and to reflect on their cognitive and motivational processes during learning and problem solving. As noted by The National Research Council, 2000, there is evidence that learners who have a high degree of self-regulation also tend to be highly motivated and competent in using their knowledge productively (as cited in De Corte, 2003, p.144).

When students completed the homework assignments and assessments I was looking at their ability to make two types of transfer called near and far transfer. Near transfer is a type of transfer where students are asked to apply knowledge learned in one context to very similar contexts. Far transfer is a type of transfer where students are asked to apply knowledge learned in one context to very different contexts (Perkins & Salomon, 1992).

In terms of looking at two possible catalysts making transfer happen for my students, I will be using the terms reflexive (low road) transfer and mindful (high road) transfer. Reflexive or low road transfer entails that the learner is drawing upon their automated routines by stimulus conditions that are akin to those in the original learning context. Mindful or high road transfer entails that the learner is making conscious arduous abstraction and a quest for connections (Perkins & Salomon, 1992).


The new instructional strategies I incorporated into my overall unit design fell into two broad categories: “hugging” and “bridging”. These terms are spoken of as being two instructional strategies that are best practice to foster transfer. “Hugging” is the type of instruction where the learner is directly engaged in similar situations compared to the performances desired. “Hugging” seems to be more associated with leading to reflexive (low road) transfer. “Bridging” is the type of instruction where the learner is encouraged to make abstractions and search for possible connections, and it usually leads to mindful (high road) transfer (Perkins and Salomon, 1992). The “hugging” in my instruction will be achieved by giving my students similar homework problems to what they experienced in the Geometer’s Sketchpad activities. I will also be achieving this when those same types of problems appear on the post assessment. The “bridging” in my instruction will be achieved with the help of the constructivist nature of the Geometer’s Sketchpad activities. Because the students are often discovering the initial concept in the Geometer’s Sketchpad environment, this naturally leads them to doing problems in their homework assignments that they have not seen or experienced before. This will force them to wrestle with their initial understanding of the concepts, to challenge themselves to make connections, and to apply their new knowledge to a new context.

In designing my pedagogical approach to include the strategies of “hugging” and “bridging” throughout the Geometer’s Sketchpad component, the homework component, the communication component, and the assessment component, I feel I have included an idea brought up by Evans (1999). He states that to facilitate transfer pedagogical approaches should incorporate “a balance of generality and situational features in teaching the initial task; and providing practice on a range of initial and target tasks” (p.41)

Overall, I feel my instructional interventions were based on sound research on learning theories; research on teaching for transfer; and research on Geometer’s Sketchpad. According to De Corte (2003) using “an instructional intervention based on the current understanding of principles for designing powerful learning environments can yield successful transfer [for students]” (p. 145).

Intervention

Description of Intervention

The purpose of my intervention was to help my students transfer the knowledge they learned in The Geometer’s Sketchpad environment to other problem-solving situations. In the 2005/2006 school year, when I implemented The Geometer’s Sketchpad activities from Key Curriculum Press, I reverted back to a more behaviorist approach when teaching the lessons. This in part was because I was more concerned with the new technology I was using in my class; the new classroom management issues I was facing; and trying to understand my new role as facilitator during the activities, which are constructivist activities in nature. I feel that by using this behaviorist approach where I would simply introduce the lesson objectives; do a short demonstration of the activity;


and provide feedback on what they were to get out of the activity impacted my students’ ability to take true ownership of the task, and to truly understand what they were to get out of the activity. I believe this in turn resulted in my students being unable to apply the concepts learned in The Geometer’s Sketchpad environment to the problems I was asking them to do back in the classroom without the use of the environment. I believe that by implementing a methodology that incorporates aspects of social constructivism with a cognitivist perspective throughout my lessons I was able to facilitate the transfer of knowledge learned in The Geometer’s Sketchpad environment to problem solving situations.

I began my unit planning by evaluating the assessments and the Geometer’s Sketchpad activities that go with them. I was looking for what exactly it was that my students were to be able to do after completing the unit. After breaking those tasks down I began looking for homework assignments that would ask my students to transfer what they had learned in the Geometer’s Sketchpad activity; that would provide practice problems to reinforce the concepts learned; and that would help them perform well on the post-assessment.

For the first unit, Points, Lines and Angles, before any teaching occurred the students did a pre-assessment. I introduced the first unit with a meaningful context set through a discussion with my students about where in the real world do we see examples of points, lines and angles. My students worked with a teacher-selected partner and shared one computer in the computer lab. They were told to rotate time on the computer equally and to share their ideas with each other while they completed the activity. After each activity early finishers were asked to discuss the activity with other early finishers refining their understandings. When most had completed the activity a class discussion summarized what they were to get out of the activity. After each activity a homework assignment was given with problems that were related to the concept they had just learned in the Geometer’s Sketchpad activity. The following day we discussed, and graded the homework. During the discussion of homework I looked for opportunities to connect what they had experienced in the Geometer’s Sketchpad activities to their homework problems. At the culmination of the unit the students completed a post assessment, which was exactly like the pre assessment. The students were not allowed to keep the pre assessment, nor did we go over the pre assessment after they did it. They were only allowed to see the pre assessment when they got their post assessments returned to them. Following the post-assessment the students were asked to complete a reflection answering questions about their interpretation of their ability to transfer the knowledge learned in Geometer’s Sketchpad to their homework assignments.

The content standards addressed in the Points, Lines and Angles unit were:

TechnologySocial, Ethical and Human Issues Standard 5: The student develops positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits and productivity.


5.1 Use content-specific tools, software and simulations (e.g., environmental probes, graphing calculators, exploratory environments, Web tools) to support learning and research.Technology Productivity Tools Standard 6: The student uses technology tools to enhance learning, increase productivity and promote creativity. 6.1 Students use content-specific tools, software and simulations to support learning and research.

MathematicsProblem Solving:Standard 1: The student uses a variety of strategies in the problem solving process.1.4 Apply various problem-solving strategies.1.6 Explain thinking processes both orally and in writing.Geometry:Standard 7: The student understands and applies basic and advanced properties of the concepts of geometry.7.1 Identify corresponding, vertical, opposite, interior, supplementary, and complementary angles.7.2 Apply angle rules and use to compute missing angle values for above angle types. 7.4 Classify types of angles (right, straight, obtuse, acute).7.5 Measure and construct angles.7.6 Identify parallel, intersecting and perpendicular lines.7.7 Estimate sizes of angles.

For the second unit, Triangles, I followed the exact same process as above except my meaningful context setting was a tad weak due to poor planning on my part. Also, after reading their reflections from the first unit I decided to allow each person a computer in the computer lab, but they had to sit next to their teacher-selected partner and work together on the activities. The end of the unit reflection asked them their interpretation of their ability to transfer what they learned in Geometer’s Sketchpad to the assessment, and which was more beneficial to their learning, the homework assignments, the Geometer’s Sketchpad activities, both or neither.

The content standards addressed in the Triangles unit were:

TechnologySocial, Ethical and Human Issues Standard 5: The student develops positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits and productivity. 5.1 Use content-specific tools, software and simulations (e.g., environmental probes, graphing calculators, exploratory environments, Web tools) to support learning and research.Technology Productivity Tools Standard 6: The student uses technology tools to enhance learning, increase productivity and promote creativity.


6.1 Students use content-specific tools, software and simulations to support learning and research.

MathematicsProblem Solving:Standard 1: The student uses a variety of strategies in the problem solving process.1.4 Apply various problem-solving strategies1.6 Explain thinking processes both orally and in writingGeometry:Standard 7: The student understands and applies basic and advanced properties of the concepts of geometry.7.3 Classify triangles by the lengths of their sides7.9 Label the legs and hypotenuse of right triangles7.10 Find the measure of a missing angle in a triangle7.15 Use Pythagorean Theorem to find missing side lengths of right triangles

For the final unit, Quadrilaterals, I followed the same process again but there was no meaningful context setting established due to poor planning on my part, and this time we were in the classroom with teacher-selected pairs and groups of three sharing one laptop. I also used the new Smartboard in my classroom to introduce and summarize the Geometer’s Sketchpad lessons. The end of the unit reflection was poorly constructed by me and gave no meaningful data in regards to my project.

The content standards addressed in the Quadrilaterals unit were:

TechnologySocial, Ethical and Human Issues Standard 5: The student develops positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits and productivity. 5.1 Use content-specific tools, software and simulations (e.g., environmental probes, graphing calculators, exploratory environments, Web tools) to support learning and research.Technology Productivity Tools Standard 6: The student uses technology tools to enhance learning, increase productivity and promote creativity. 6.1 Students use content-specific tools, software and simulations to support learning and research.

MathematicsProblem Solving:Standard 1: The student uses a variety of strategies in the problem solving process.1.4 Apply various problem-solving strategies1.6 Explain thinking processes both orally and in writingProperties & Concepts of Numbers:Standard 2: The student understands and applies basic and advanced properties of the concepts of numbers.


2.3 Construct Venn diagrams given dataGeometry:Standard 7: The student understands and applies basic and advanced properties of the concepts of geometry.7.3 Classify quadrilaterals7.10 Define Polygons

Connection to learning theories

My action research project had elements of constructivism, and social constructivism with a cognitivist perspective. The constructivism component was quite evident in the Geometer’s Sketchpad activities. Students were given a task in the activity. They proceeded to explore the concepts embedded in the activities with their partners. Through the exploration phase they were faced with constructing their own ideas of the concepts, and solidified their understandings by testing and conjecturing their ideas. Through this process they came to understand the mathematical concepts of the activities.

Another constructivist component incorporated into the first unit, Points, Lines and Angles was using a meaningful context in the introduction phase of my lesson - lending credence to the belief that if the learner is motivated by knowing where the concept applies to other problem solving situations, or real-life situations, they will take a more active role in the learning process (De Corte, 2003).

The social constructivist aspect was evident in two areas of my lessons. First, my students worked in pairs during The Geometer’s Sketchpad activities. This fostered the sharing of ideas between the students whereby they could gain validation in their thought processes or get re-directed when their thought processes were leading them to misconceptions – meaning they constructed their new knowledge in a social setting. Second, social constructivism was evident when I instructed the early finishers to get together with another pair of early finishers to share their findings. This enabled the same type of discourse that occurred in their original pairs thus furthering the refining of their conceptual understanding in a social context.

The cognitivist perspective I incorporated into my lessons aligns with the ideas that there are three kinds of learning processes: proceduralization, tuning, and automization (Grenno, Collins, and Resnick, 1996). The proceduralization aspect of what I did was the constructivist Geometer’s Sketchpad activities where my students are building their so-called procedural or conceptual knowledge. The tuning aspect of my proposal occurred in two places of my instructional design. First, in the social constructivist nature of the discourse between the early finishers where my students got feedback from each other if their responses were correct or incorrect, and second, when my students completed the homework assignments refining their understandings learned in Geometer’s Sketchpad activity. And finally, the automization component of my methodology was the assessment piece to see if the transference of the knowledge acquired in The Geometer’s Sketchpad environment was evident. As stated by Grenno,


Collins, and Resnick (1996), “In the cognitive perspective, transfer is assumed to depend on acquiring an abstract mental representation in the form of a schema that designates relations that compose a structure that is invariant across situations” p. 23.

Role of technology

The technology I used was Geometer’s Sketchpad. The Geometer’s Sketchpad environment is a dynamic environment where my students could explore, look for relationships, hypothesize, and test conjectures about the mathematical concepts we were studying. It provided an opportunity for my students to construct their own understanding of the abstract concepts we were studying. The effects of this technology were to make the abstract mathematical concepts more concrete for my students using an engaging, visual, interactive approach, thereby helping them truly understand those abstract mathematical concepts. As stated in Exploring Geometry (1999), Geometer’s Sketchpad encourages a process of discovery that more closely reflects the way mathematics is invented: a mathematician first visualizes and analyzes a problem, making conjectures before attempting a proof (Key Curriculum Press, 1999). Another affordance the technology provided was that it gave my students the ability to save their work, complete it during the next class period, and reflect on it at a later time. The technology could also allow the students the ability to print out their sketches so they could discuss their work with each other out of the computer lab environment increasing the value of the experience. One constraint of the technology was that my students did not have the software at home, and it was not accessible to them during their lunch hours. This meant I had to devote more class time when activities were not completed in the time frame that I had anticipated, and it also meant the amount of time that my students could explore and try different approaches or strategies was limited to the time I allowed during class. Another constraint of the technology was that the activity book I used had activities that were not set in real-life contexts, and some were not set in what I term, meaningful contexts.

Participants

I teach seventh grade mathematics at Cairo American College in Cairo, Egypt. This year was the first year we had totally heterogeneous groupings in the seventh grade. My students are mostly from the upper ranges in socio-economic status. Our school is comprised of forty-seven percent U.S. citizens, fourteen percent host country nationals, and thirty-nine percent are other nationalities. Their achievement characteristics are quite evenly spread out based on the results I saw from their IOWA scores with percentile ranks from the tenth percentile to the ninety-ninth percentile in total math. On the whole, our students tend to want to do well in school and care about their grades, however they sometimes focus more on the grades than on how they are truly understanding the material they are to learn.

There were twenty-eight case study students who I chose to gather data on for my project. I decided after the implementation of my project was complete to use case studies because I felt I needed specific evidence of transference from individual


students to truly answer my research questions. Due to this late decision, I was limited to these twenty-eight students because they were the only students who saved all of their work from the units in their portfolios. I categorized the students according to gender and ability level in mathematics. In determining what ability level they were I referred to their standardized test scores from the previous year, when available, and from their performance in my class throughout the year. I had five high ability males, five high ability females, four medium ability males, seven medium ability females, three low ability males and four low ability females. High ability students had standardized test scores for total math in the 80th percentile or above. Medium ability students were in the 50th to 79th percentile, and low ability were below the 50th percentile. In the high ability males category there were two advanced ESL students, one from Poland and the other from Korea, and in the high ability females there was one beginning ESL student from Korea.

Materials and Resources

1. I used the computer labs, laptops and a Smartboard at our school. Each computer had The Geometer’s Sketchpad software installed on it.

2. I used Holt Pre- Algebra and Holt Math Course 2 textbooks for homework problems.

3. I used Geometry Activities for Middle School Students with The Geometer’s Sketchpad, Unit 2: Points, Lines, and Angles; Unit 3: Triangles; and Unit 4: Quadrilaterals as an introduction to my new teaching methodology.

4. I used the Wrap-Up’s from the book mentioned above as a pre-assessment and post-assessment tool to see if the skills and concepts learned in the Geometer’s Sketchpad activities were present before the lessons, and if transference occurred after the lessons.

5. I used student reflection writing pieces as a qualitative evaluative tool informing me about my students’ perceptions on their ability to transfer their new knowledge to other situations.

Specific Research Questions

For my action research project I am trying to find out how my teaching strategies used with Geometer's Sketchpad can help my students transfer their knowledge to similar problem-solving situations?

1. How did the change in my teaching methodologies when using Geometer’s Sketchpad impact my high, middle, and low achieving students’ ability to obtain near transference? How did it impact far transference?

2. Which mechanism, reflexive (low road) or mindful (high road) transfer, helped my students transfer their knowledge?


Data Collection

In order to adequately evaluate the effects of the intervention, the data collected involved both quantitative and qualitative aspects. The quantitative data includes pre and post assessment scores and homework scores. The qualitative data is the reflection responses from all of my students at the end of the first two units, which I was able to quantify numerically based on the yes, no, sometimes, and multiple choice questions in the reflections.

Quantifiable Data

The first thing I did at the beginning of the year was to gather the IOWA test scores for all of my students. This data was not available on some of the new students entering seventh grade. The purpose of this data was to aid me in my decision-making process for which kids could be considered high, middle or low achievers in my project. When making the decision about my case study students I also took into account what I saw in terms of their mathematical thinking during the year.

Before each unit of study my students took a pre-assessment that was exactly like the Wrap-Up or post-assessment for the end of the unit. The purpose of this data was so I would know what knowledge they already possessed before we began the unit. They were given instructions to do the best they could and it was all right not to know the answer. They were told this would just inform me if anyone already knew the material and if I would have to plan my instruction accordingly. I recorded the scores for case study data and whole class data.

At the end of the Geometer’s Sketchpad lesson/activities I assigned homework problems that were from the Holt series of math textbooks. Some of the assignments contained problems that were very similar, if not exactly like what my students experienced during the Geometer’s Sketchpad lesson/activity. This would be an example of near transfer with most likely reflexive transfer as the mechanism. Other problems were different in nature where the students were going to be asked to apply their new knowledge. This would be an example of far transfer with most likely mindful transfer as the mechanism. The purpose of this data was to provide me with evidence of near or far transfer, and reflexive or mindful transfer. It also gave me evidence of how my instructional strategies of “bridging” and “hugging” may have helped foster this transfer.

I then did an item analysis for each of my case study students to evaluate whether or not the different types of transference occurred. I recorded how many problems they missed on six separate assignments during the three units of study. I then decided to narrow that down to focus on three assignments because these three assignments were good examples of near and far transference. When completing my analysis, I looked at how they performed on certain portions of their assignments that were related to the different types of transference.


At the end of each unit my students took a post-test, which was the Wrap-Up that I used as the pre-assessment. The purpose of this data was so I could ascertain which knowledge they were able to obtain near or far transfer, and reflexive or mindful transfer in comparison to the pre-test. I did an item analysis for each of my case study students and recorded their scores breaking them down by the types of transference they related to. I also recorded the whole class data for total scores without the breakdown.

Qualitative Data

At the end of the units I gave my students reflection assignments. I gathered whole class data and case study data on what their perceptions were about their ability to transfer what they were learning. The last reflection I assigned was poorly written so I will not refer to it, as it has no information about transference. On the first reflection they were asked to comment about their ability to transfer what they learned in the Geometer’s Sketchpad activity to their homework assignments. They were also asked if they felt the computer lab activities enhanced their understanding of the concepts, whether they liked the activities or not, and if they had any suggestions. On the second reflection they were asked to comment about their ability to transfer what they learned in the Geometer’s Sketchpad activities to the assessment. They were also asked which of the following: the Geometer’s Sketchpad activities, the homework assignments, both or neither, helped them understand the concepts better. The purpose of this data was to see if there were emerging themes that represented evidence of transference according to the students’ perceptions.

Data Analysis

Quantifiable Data

During the data analysis phase I had to really sit down with my Geometer’s Sketchpad activities, homework assignments, and assessments and categorize what I was asking of my students in terms of types of transference, the mechanism for transference, and which instructional strategy I was employing to achieve them. Starting on the next page are the three tables I constructed to show the links between the Geometer’s Sketchpad activity, the homework assignment and the assessment for each unit. Below the first table for the Points, Lines and Angles unit I have included the rationale for my decisions on why I categorized something as near or far, reflexive or mindful and “hugging” or “bridging”. Copies of the Geometer’s Sketchpad activities, homework assignments and assessments are attached for your reference. (see appendices)


Points, Lines and AnglesGeometer’s

Sketchpad ActivityNear/Far,

Reflexive/Mindful, Hugging/Bridging

Homework Assignments N/F, R/M, H/B Assessment

A. Exploring Angles: 1. Estimate, Measure and classify Acute (A), Right (R) and Obtuse (O) Angles. 2. Given picture name A, R, or O.

1. From A to D near and reflexive transference, and “hugging” examples.

D. Holt 5-1: Classify and Name figures. 1. Name A, R, O, complementary, supplementary, and vertical angles given picture. 2. Given picture find angle measure for vertical angles. 3. T/F supp., comp., O, and find angle measures.

4. From D to T1 and T2 near and reflexive transference, and “hugging” examples.

T1. Given picture tell A, R or O.

T2. Given picture identify as comp., supp., or vertical and find the missing angle measure.

B. Exploring Special Pairs of Angles: 1. Given examples come up with the definitions for complementary, supplementary and vertical angles.

2. From B to D far and mindful transference, and “bridging” examples.

E. Holt 5-2: Parallel lines with transversal. 1. Given picture find angle measures and name congruent angles using concept.

5. From E toT3 near and reflexive transference, and “hugging” examples.

T3. Given picture classify as corresponding, alternate interior or alternate exterior, and give the angle measure.

C. Exploring Angles formed by parallel lines and a transversal: 1. Given picture name corresponding, alternate interior, and alternate exterior angles. 2. Come to conclusion that pairs of corresponding, alternate interior and alternate exterior angles are congruent.

3. From C to E far and mindful transference, and “bridging” examples.

F. Angle Puzzle. (Given after assessment.) 1. Find angle measures with many intersecting lines applying concepts learned.

6. From T2 and T3 to F near and reflexive transference, and “hugging” example.

7.T4 questions are examples of far and mindful transference, and an example of “bridging”.

T4. Sometimes, Always or Never questions on corresponding, vertical, alternate interior, alternate exterior and supplementary angles.

The rationale for classifying #1 as near/reflexive transference and “hugging” is because the students are doing nearly the same thing in their homework assignment as they were doing in the Geometer’s Sketchpad activity.


The rationale for classifying #2 as far/mindful transference and “bridging” is because the students are applying their new understandings of the definitions to problems in a new context. They are actually finding the missing angle measures and naming congruent angles applying their understanding of the definitions.

The rationale for classifying #3 as far/mindful transference and “bridging” is because the students are applying the new knowledge about the angle measures and congruency to a new context. They are actually finding the missing angle measures.

The rationale for classifying #4 as near/reflexive transference and “hugging” is because the students are doing nearly the same thing in a similar context as in the activity.

The rationale for classifying #5 as near/reflexive transference and “hugging” is because when the students are finding the angle measure that is the same thing they did in the activity, but naming them was new.

The rationale for classifying #6 as near/reflexive transference is because the task is similar to other diagrams of intersecting lines they worked with, but this was more complicated. It had more lines; therefore it is a similar situation.

The rationale for classifying #7 as far/mindful transference and “bridging” is because the students are applying all the new knowledge about all the concepts to a new context. Questions are in a new format in which they are asked to think about all the possibilities of a statement and respond sometimes, always or never.

Triangles UnitGeometer’s



Homework Assignments

N/F, R/M, H/B Assessment

A. Exploring Properties of Triangles: 1. Discover all angles in a triangle sum to 180 2. Discover size of angle/size of side relationship. 3. Discover the sum of any 2 sides is greater than the 3rd.

1. From A to E near, reflexive transference, and “hugging” example.2. From A to T1, T2 and T3 far, mindful transference, and “bridging” example.

D. Holt 7-6: Triangles: 1. Given picture classify triangle by its sides and angles. 2. Given picture identify how many of each type are present.

4. From E toT2 near and reflexive transference, and “hugging” examples.

T1. Given side lengths, is the triangle possible? Why or why not?

T2. Given 2 angle measures, find the 3rd and put the sides in order from shortest to longest.

B. Exploring Types of Triangles: 1. Can you have various types? (Equilateral Obtuse, Isosceles Right, etc.) 2. Always, sometimes or never questions. (A right triangle is equilateral.)

2. From B to D far and mindful transference, and “bridging” examples. B to F near, reflexive, and “hugging”.

E. Holt 7-8: Angles in Polygons: 1. Given picture with 2 angle measures find the measure of the 3rd angle.

5. From D to T4 near and reflexive transference, and “hugging” examples.

T3. Given side lengths, put the angles in order from smallest to biggest.


C. Exploring the Pythagorean Theorem: 1. Discover the Pythagorean Theorem for all right triangles and its converse.

3. From C to T5 near, reflexive, and hugging.

F. Holt Pre-Alg 5-3: Triangles: 1. Sketch triangle type if possible, if not – why?

6. From F to T4 near and reflexive transference, and “hugging”.

T4. Sketch triangle type if possible. If not, why is it not possible?

T5. Simply state the Pythagorean Theorem.

Quadrilaterals UnitGeometer’s



Homework Assignments N/F, R/M, H/B Assessment

A. Exploring Quadrilaterals: 1. Given each quadrilateral figure, discover the side length and angle measure properties for them. 2. Always, sometimes or never questions. (A rhombus is a square.)

1. From A to C near, reflexive transference, and “hugging” examples.

C. Holt 7-7: Quadrilaterals: 1. Give all quadrilateral names that apply to a picture. 2. True/False and explain. (All rhombuses are squares.) 3. Name all quadrilaterals given written descriptions of properties.

4. From C to T1 near and reflexive transference, and “hugging” examples.

T1. Given figures of quadrilaterals, and properties, list all names it could be called, and its most specific name.

B. Diagonals in Quadrilaterals: 1. For each quadrilateral, when are the diagonals congruent, perpendicular, and bisect each other.

2. From B to T1 near and reflexive transference, and “hugging” examples.

D. Holt Pre-Alg 5-4: Triangles 1. Sketch quadrilateral given properties, if possible. 2. Venn diagram given in this section.

5. From D toT2 near and reflexive transference, and “hugging” examples.

T2. Draw a Venn diagram for the quadrilateral family.

The next thing I did was to grade and record all of the students’ pre and post assessments for each class I teach. I calculated and recorded the class averages and the total average for each unit. I then did the same process for my case study students. Below are the tables showing the results:

1. Points, Lines and Angles (assessment is out of 28 points).A block (16) B block (15) D block (18) G block (14) H Block (14)

Pre-Test Avg 10.6 / 38% 10.3 / 37% 9.1 / 32% 6.4 / 23% 9.3 / 33%Post-Test Avg 24.4 / 87% 24 / 86% 24.2 / 86% 24 / 86% 24.7 / 88%

All classes: 77 students. Pre-Test Avg.: 9.2/33%; Post-Test Avg.: 24.3/87%.

Case study studentsGender/Ability Male/High(5) Male/Med(4) Male/Low(3)Pre-Test Avg 10.1 / 36% 9 / 32% 7 / 25%Post-Test Avg 25.5 / 91% 25 / 89% 23.5 / 84%

Female/High(5)

Female/Med(7) Female/Low(4)


Pre-Test Avg 12.9 / 46% 7.8 / 28% 7.4 / 26%Post-Test Avg 26.6 / 95% 22.4 / 80% 21.3 / 76%

2. Triangles (assessment is out of 25 points).

A block (16) B block (16) D block (17) G block (16) H Block (16)Pre-Test Avg 3.1 / 13% 4.9 / 20% 2.7 / 11% 3.8 / 15% 8 / 32%Post-Test Avg 19.5 / 78% 18.6 / 74% 21.5 / 86% 21.2 / 85% 20.2 / 81%


Case study studentsGender/Ability Male/High(5) Male/Med(4) Male/Low(3)Pre-Test Avg 3.4 / 14% 6.8 / 27% 0.3 / 1%Post-Test Avg 24.5 / 98% 21.6 / 87% 17.1 / 69%

Female/High(5)


Pre-Test Avg 5.3 / 22% 6.3 / 25% 3.8 / 15%Post-Test Avg 24 / 96% 21 / 84% 18.1 / 73%

3. Quadrilaterals (assessment is out of 30 points).

A block (17) B block (17) D block (17) G block (17) H Block (18)Pre-Test Avg 5.9 / 20% 7.4 / 25% 8.1 / 27% 5 / 17% 7 / 23%Post-Test Avg 26.4 / 88% 24.6 / 82% 27.4 / 91% 26.6 / 89% 26.8 / 89%


Case study studentsGender/Ability Male/High(5) Male/Med(4) Male/Low(3)Pre-Test Avg 9.7 / 32% 7.5 / 25% 1.7 / 6%Post-Test Avg 28.8 / 96% 27.1 / 90% 22.5 / 75%

Female/High(5)


Pre-Test Avg 10.7 / 36% 7.7 / 26% 5.6 / 19%Post-Test Avg 28.5 / 95% 28.2 / 94% 24.9 / 83%

I did not break down the students’ scores for the Quadrilateral assessment, as I did in the other two assessments, because I categorized all types of questions on this assessment and homework assignments as examples of near and reflexive transference with the instructional strategy of “hugging” in place (refer to Quadrilaterals unit table above). Therefore, I think the scores from the Quadrilateral assessment show positive results for how “hugging” all of my instruction to all of the desired outcomes lead to a greater probability of near and reflexive transference occurring for all of my students.

In terms of overall interpretation of learning occurring, the pre and post assessment data shows positive growth results during each unit for all of my classes, and every category of my case studies. Of course I was very happy with the results, but what does


this say about how the use of “hugging” or “bridging” in my instructional design impacted the different ability levels of students to obtain near or far transference? When I look at the scores above, I notice that the scores for the Triangles unit are a bit lower than the other two units. This is particularly evident in the scores for my low ability males (69%),low ability females (73%), and my A and B block classes (78% and 74%, respectively). So, I asked myself, “Why did this occur?” The answer I believe has to do with the fact this was the only unit in which there was no homework (“hugging”) provided where my students could practice and deepen their new conceptual understandings learned in the Geometer’s Sketchpad activity. The specificities of what I am talking about are on the first part of the Triangle unit assessment. The first nine out of fifteen questions deal with the concepts of angle size/side size relationship, and whether or not triangles can be made given certain side lengths. These concepts were not practiced in any of the students’ homework assignments, which means I did not “hug” my instruction to the desired outcomes, but I did try to “bridge” my instruction by revisiting the concepts before the assessment through a discussion about their experience in the activity. In other words, the Triangle unit was the only unit in which I asked my students to directly transfer what they learned in a Geometer’s Sketchpad activity to the assessment. As I said, I did go over the concepts with them the day before the assessment, but I believe this was only enough of a reminder for the high ability and medium ability students to make that “bridge”, which is evidenced by their higher scores. As you can see from the data below, and from the post assessment average of 73% for low ability females, it is most evident my low ability females did not make that “bridge” between the Geometer’s Sketchpad activity and the assessment questions. This, I believe, is a good example of what I felt my students struggled with during my first year of using Geometer’s Sketchpad, lacking the ability to make the far/mindful transfer directly from the Geometer’s Sketchpad activity to the new context of the assessment questions. This is evidence to me that it is critical for all ability levels, and especially my low-ability level students to receive the “hugging” component, or homework assignment, so they can further “play” with the concepts that were in the Geometer’s Sketchpad activity. By doing this I will be increasing the likelihood of at least near and reflexive transference occurring for all students.

Triangle Assessment First 9 Question Performance (Out of 12 points)Far/ Mindful Transference with “Bridging” Strategy

Gender/Ability Male/High(5) Male/Med(4) Male/Low(3)Pre-Test Avg 1.4 / 12% 3.25 / 27% 0.3 / 3%Post-Test Avg 12 / 100% 10 / 83% 10 / 83%

Female/High(5)


Pre-Test Avg 2.25 / 19% 2.8 / 23% 1.5 / 13%Post-Test Avg 11 / 92% 10.2 / 85% 7.5 / 63%

For the Points, Lines and Angles unit assessment I recorded how each student scored on certain parts of the assessment based on the type of transference evident in the questions. I did this because this was the only assessment that had a component of far/mindful transference and “bridging” where my students really had to understand the


concepts deeply to perform well. Specifically, it is the last six questions on the Points, Lines and Angles assessment that I am referring to as an example of far/mindful transference with the instructional strategy of “bridging” at play. I think this is a good example of “bridging” my instruction, and of far and mindful transference because these questions were in a new context and not practiced in any homework assignments or Geometer’s Sketchpad activities. I was asking my students to make connections using the concepts to come to a conclusion about a statement. In these questions the students were asked to think of all possibilities concerning the various concepts they learned and apply that knowledge to decide if certain statements were sometimes, always, or never true. For example, “A pair of angles are vertical and acute.” The students would have to realize that you could also have a pair of angles that are vertical and obtuse or vertical and right to come to the conclusion that the answer is sometimes. This is evidence to me that my students had to make some deeper connections with the concepts, and apply their knowledge in a new setting to answer these questions correctly.

As you can see from the data below, the strategy of “bridging” my instruction to the last six questions was average at best. As you go from high ability students to the lower ability students the data follows a pattern of decline for the rate of transference. I believe if I were to “hug” my instruction next time to the desired outcomes by practicing these types of questions, and the thinking routines associated with them, the results would be better. If you look at the averages for the first fourteen questions, it looks like near and reflexive transference occurred for all ability levels. Again, this was achieved with the instructional strategy of “hugging” in place through the two homework assignments and the Geometer’s Sketchpad activities.

Near/Reflexive and Far/Mindful Transference with “Bridging” or “Hugging” Data

Gender/Ability

Pts., Lines and Angles Total Post Assessment%

Questions 1-14. Near/Reflexive.Averages

Questions 15 – 20. Far/Mindful. Averages

Page 224 Homework Scores (out of 24).


Male/High(5) 91% (21.5/22) 98% (4/6) 67% 90% 93%Male/Med(4) 89% (21.5/22) 98% (3.5/6) 58% 85% 88%Male/Low(3) 84% (20.2/22) 92% (3/6) 50% 68% 73%

Female/High(5)

95% (21.3/22) 97% (5.2/6) 87% 95% 88%

Female/Med(7) 80% (19.5/22) 89% (3.5/6) 58% 87% 91%Female/Low(4) 76% (18.5/22) 84% (2.8/6) 47% 84% 88%

The importance of “hugging” my instruction to the desired outcomes on the assessments with the homework assignments and the Geometer’s Sketchpad activities was also evident in the data from the Quadrilaterals unit. All ability levels, except the low ability males, showed improvements from their homework scores to their post assessment scores. I believe this is strong evidence that the instructional strategy of “hugging” leads to near and reflexive transference for the majority of students.


Near/Reflexive Transferencewith “Hugging” Data

Gender/AbilityQuadrilaterals Post

Assessment %Page 380 Homework Scores (out of 18)

Male/High(5) 96% 88%Male/Med(4) 90% 72%Male/Low(3) 75% 72%

Female/High(5) 95% 86%Female/Med(7) 94% 88%Female/Low(4) 83% 67%

Qualitative Data

Analyzing the students’ end of the unit reflections was informative in terms of learning what the students’ perceptions were about their ability to transfer what they learned in the Geometer’s Sketchpad activities and the homework assignments to the post assessment. As I read through each classes responses I categorized and recorded their responses as yes, no and sometimes. Through this process I noticed a few themes emerging, the most significant theme being the students ability to make the connections between the Geometer’s Sketchpad activities, the homework assignments and the post assessments.

The first question on the Points, Lines and Angles reflection gives some insight as to the students’ ability to visualize the Geometer’s Sketchpad activity at a later time. In my interpretation it tells me that the majority (74%) of the students had positive feelings about their ability to transfer what they were learning in Geometer’s Sketchpad to their homework assignments. It also was the first time in which a student responded about the connection between the Geometer’s Sketchpad activity and the homework assignment. This tells me that some of the students were mentally aware of the instructional strategy of “hugging” being present.

The second question gives information about whether or not the students felt like the Geometer’s Sketchpad activities helped them understand the concepts we were studying. The majority (90%) said the Geometer’s Sketchpad activities did help them understand the concepts better because they were active participants in their learning, as opposed to being lectured to about the concepts. Also of note is that again, more students made references to being aware of “hugging” being used in the instructional process, and it being beneficial for their learning.


The first question on the Triangles reflection gives some information on whether or not the students felt they could transfer the knowledge they learned in the Geometer’s Sketchpad activities to the post assessment. The majority (79%) responded affirmatively to the question with an even larger portion of students, compared to the first reflection, mentioning the “hugging” strategy being beneficial to their learning.

The second question was very informative as an even higher number of students mentioned both the Geometer’s Sketchpad activities and the homework assignments helped the students the most in understanding the concepts better. The responses mentioned how the students felt like they learned the big idea or main concept in Geometer’s Sketchpad and got to practice it and master it by doing the homework assignments.

Whole Class Data

The analysis of all the student responses is included below so that the reader can glean more information as it relates to my project, and so that you can understand how I came to the conclusions I mentioned above. There were eighty-two students who responded in the first reflection, and seventy-five students for the second reflection.

Student Reflection September 9, 2006Points, Lines and Angles

1. When doing your homework do you find yourself thinking back to the activities we did in the computer lab (or do you refer to the activity sheets)? Please elaborate. 74% YES, 12% SOMETIMES, 7% NO, AND 6% OTHER. The “yes” responses where about the students remembering back to the Geometer’s Sketchpad activities; having a clear picture in their mind of what they experienced; making visual references of what they experienced; the activities showing up in their mind; and one student shared the connection they made between the homework assignment and the Geometer’s Sketchpad activities. The “no” responses where about one students’ learning style, and another student finding Geometer’s Sketchpad activities confusing. Here is one of the “no” responses: “To be completely honest, no, Geometer’s Sketchpad is fun, but it doesn’t help me understand too well. After you told me that complementary angles are angles that sum up to ninety degrees, that’s how I learned complementary angles. Geometer’s Sketchpad is fun, but it doesn’t really help me learn new things.”

2. Do the activities in the computer lab help you understand the concepts we are studying? Please elaborate. 90% YES, 7% SOMETIMES, AND 2% NO. The “yes” responses fell into two main categories. One category was that “seeing” the concepts visually helped the students, and two, that “doing” it versus just reading, talking or listening about it helped as well. Two more students shared the connection between the homework and the Geometer’s Sketchpad activities. The “no” responses where that the activities didn’t help them study for the post assessment.


3. Do you like the activities using Geometer’s Sketchpad? Why or why not? 91%

YES, 4% SOMETIMES, 4% NO. One of the “yes” responses mentioned she liked knowing why things worked the way they did in math and made her more sure of her answers.

4. What suggestions do you have to make the computer lab activities more

meaningful for you in terms of helping you learn? Many made references to exploring the program at a deeper level by wanting to explore constructions. Also from this question I thought a larger portion wanted their own computer to work on, but after a second look at the responses it was only 10% of the students that expressed that opinion. So, for the next unit I did make a change and I put each student on a computer for more interaction for all.

Student Reflection October 1, 2006Triangles/Geometer’s Sketchpad Reflection

1. When taking the test, are you able to transfer what you learned in the Geometer’s Sketchpad activities to the problems on the test? If so, please explain. If not, which problems on the test were you unable to connect to the Geometer’s Sketchpad activities? 79% YES, 13% SOMETIMES, AND 7% NO. Several references to the concept of “hugging” were made in the “yes” responses. The “no” responses varied from Geometer’s Sketchpad being confusing; learning more from the homework; or connecting more with what we did in the regular classroom than in the computer lab. The “sometimes” responses had to deal with some of the students not making connections with some of the problems on the assessment.

2. Which helped you understand the concepts on the test better? Please explain.a. 32% The Geometer’s Sketchpad activities in the computer lab.b. 13% The homework problems in the packet.c. 53% Both a and b.d. 2% Neither a or b.

The responses to answer “c” say something about the concept of “hugging” in my instructional design and the benefits of using it to help in the transfer of knowledge. I had many great responses about how in Geometer’s Sketchpad they learned the big idea or concept, and how the homework was where they solidified their understanding of the concept by looking at the details of it, and where they got to practice the concept.

Case Study Reflection Data and Sample Responses

Points, Lines and Angles ReflectionFor question one on the Points, Lines and Angles reflection 85% (22/26) of my case study students responded “yes”, they could think back to the Geometer’s Sketchpad activities when doing their homework. While only 15% (4/26) responded “sometimes”.


The “sometimes” responses came from two medium ability males and two medium ability females.

Sample “yes” responses:1. “Yes, I do find myself thinking back because most of the activities for homework

were in the Sketchpad.” Medium ability male. I believe this is an example of a student who recognizes the use of “hugging” in my instruction.

2. “I think of the Geometer’s Sketchpad because there is now a clear picture in my mind from the examples, which were very clear.” High ability male.

3. “When I did my homework, and the words complementary, supplementary and vertical come up, if I try and figure something out I find myself remembering the demonstration in Geometer’s Sketchpad.” High ability female.

4. “Yes, I think back by remembering how the angles change. For example, when we were doing complementary angles I can remember when we moved the dot to move the angles. And it helps me remember it.” Medium ability female.

Sample “sometimes” response:1. “Sometimes I try to remember what we did in the computer lab, but I think using

the worksheets is much easier.” Low ability female.

For question two 92% (24/26) of the case study students responded “yes”, the computer lab activities help me understand the concepts better, while 8% (2/26) responded “sometimes”. The “sometimes” responses came from one high ability male and one low ability female.

Sample “yes” responses:1. “Yes, they do because all of the time the homework is the same as the activities

we do in the computer lab.” Medium ability male. Reference to “hugging”.2. “Yes. They help me see the reasons why it’s like that. Like why complementary

are complementary.” High ability female.3. “The activities in the computer lab helped me remember things better than if we

just talked about it with a few examples. I remember it better if I do it.” High ability female.

Sample “sometimes” response:1. “No, not really but sometimes they do.” Low ability female.

For question three about whether they liked the Geometer’s Sketchpad activities or not, 92% (24/26) responded “yes”, while 8% (2/26) responded “no”.

Sample “yes” responses:1. “Yes, I do because the questions make us sure of our answers and why.”

Medium ability female.


2. “Yes, I do like Geometer’s Sketchpad because it helps me understand what we were learning. And I like physically doing it instead of verbally.” Medium ability female.

3. “Yes, I like them because when you want to change an angle and see what it equals it show you. Like supplementary angles both angles change.” Low ability male.

Sample “no” responses:1. “No, because I forget it soon.” Low ability female.2. “Not really since we aren’t getting to make shapes unlike in animation (class).”

Medium ability male.

For question four about suggestions, two responses from low ability students struck me as interesting. One said, “I think we should always work as a big class because it helps me understand more.” I feel this shows how a low ability student found the constructivist nature of the activities difficult. And the other one said, “There should be more time to finish everything and a bit more activities.” Time in the computer lab was one of the constraints I mentioned earlier in my report. Usually, that type of constraint impacts the lower ability students more than other students due to the pace at which low ability students tend to work.

Triangles ReflectionFor question one on the Triangles reflection 88% (23/26) of my case study students responded “yes”, I can transfer what I learned using Geometer’s Sketchpad to the assessment problems, 4% (1/26) responded “no” and 8% (2/26) responded “sometimes”. The “no” response was from a low ability female and the “sometimes” responses were from a low and medium ability females.

Sample “yes” responses:1. “Yes, I was able to connect the activities in Geometer’s Sketchpad to the test. I

would think about what was on the worksheet and remember what I did on the screen and that would help me answer a couple of questions.” High ability female.

2. “Yes, I was able to connect Geometer’s Sketchpad to the questions on the test. For example, I remember the sheet I did that shows how to be able to make a triangle given the side lengths. (#1-3)” High ability female. She is referring to her ability for far and mindful transference.

3. “Yes, I could connect the questions on the test to the Geometer’s Sketchpad activities because all of the questions were based on what we learned in Geometer’s Sketchpad.” High ability male. He recognizes the instructional strategies of “hugging” and “bridging”.

4. “Yes, many things, for example, the equilateral triangle, I remember all the angles were sixty degrees (acute) and the sides were the same, and I couldn’t move the sides in any way. So, when the test asked for right and obtuse equilateral triangles, I knew that you can only have acute equilateral triangles.” Medium ability male. This is an example of a student who has achieved near and reflexive transfer with “hugging” in place.


Sample “no” response:1. “No, I am unable to connect the Geometer’s Sketchpad. It is easier for me to

remember what’s on the worksheet like on the test when we had to explain why it’s possible. It is easier for me to remember the results I got on the worksheet.” Low ability female.

Sample “sometimes” response:1. “Sometimes I can but sometimes I can’t. It’s kind of weird especially since I can’t

explain. Sorry!” Low ability female.

For question two on the Triangles reflection 31% (8/26) of my case study students responded that the Geometer’s Sketchpad activities helped them the most, 11% (3/26) responded that the homework assignments helped the most, and 58% (15/26) responded that both helped them the most.

Sample “Geometer’s Sketchpad” responses:1. “The Geometer’s Sketchpad activities in the computer lab helped me more to

understand the concepts of the test better. This is because we are actually doing it, experiencing it, and figuring out the concept for ourselves. For the Pythagorean Theorem, we had to figure it out by ourselves and it stuck better in my mind that way.” High ability female.

2. “A, because the homework didn’t teach me really and Geometer’s Sketchpad explained stuff to me instead of asking me questions without telling me about what it’s about.” Medium ability male.

Sample “Homework assignments” responses:1. “B, because the homework packets were not involved with computers and you

could do them anywhere. The Sketchpad needed computer help.” Medium ability male.

2. “Mostly B. It was easier to concentrate rather than exploring Geometer’s Sketchpad, I spent most of the lesson trying to figure out where the buttons are than actually working on the questions.” High ability female.

3. “Mostly B because it helped me more, had more problems and I was alone so I did it by myself instead of someone helping me a lot, (that was and example)!” Medium ability female.

Sample “Both the Geometer’s Sketchpad activities and homework assignments” responses:

1. “C = Both A and B but mostly A, because I was able to apply the knowledge I learned in A (Geometer’s Sketchpad) to my work in B (homework assignments).” High ability female.

2. “The both A and B. The reason is the main idea I got was from the Geometer’s Sketchpad but with the homework I can get the exact idea and reviewed the big idea into smaller parts.” High ability male.

3. “Both helped me because Geometer’s Sketchpad I got to draw it and on the worksheets I actually get to try and figure it out with numbers. After what I’ve


learned in Geometer’s Sketchpad I get to apply it on the worksheets.” Medium ability female.

4. “Both B and A because Geometer’s Sketchpad helped me understand it by trying it out and the homework helped me practice and review.” Medium ability female.

Reporting Findings (What happened? What did you see?)

Points, Lines and Angles UnitBecause our middle school is in the early stages of implementing the Connected Math curriculum, my seventh graders had no instruction at all on Geometry concepts in sixth grade. This meant that my baseline data was fairly low as evidenced by the low pre assessment scores, especially on the Triangles unit and the Quadrilaterals unit. This made it somewhat easy to deal with the preexisting knowledge my students came to me with because it was very low. Therefore, I really did not focus on the pre assessment scores when looking at whether near or far transfer occurred, except for the first six questions of this first assessment.

For the Points, Lines and Angles pre assessment there were only ten students out of seventy-seven that did not get at least five out of the first six questions correct concerning acute, right and obtuse angles. Most students had learned these concepts in fifth grade. Therefore, on the post assessment all seventy-seven students scored either a five or six out of six on that topic. This means that there is no evidence that my instructional strategies impacted the students’ ability to transfer their knowledge about acute, obtuse and right angles because they already possessed it. This is the only occurrence of an overabundance of preexisting knowledge through the three units.

For this unit the students had a teacher selected partner and one computer to share. There were three Geometer’s Sketchpad activities that the students completed for this unit. The first was Exploring Angles where they got a review of right, acute and obtuse angles. It was a good first activity because the content was easy for them, which made it easier for them to get comfortable with the Geometer’s Sketchpad environment. This was followed by the activity Exploring Special Pairs of Angles where the students derived the definitions of complementary, supplementary and vertical angles. For some students this seemed to be a review as well, but for others it did not because quite a few struggled with completing the questions that asked them, “What do you notice about the sum of the angles?”

After completing these two activities I assigned page 224 for homework to see if the students could transfer what they learned to other problems. All of the questions on the homework assignment that dealt with acute, obtuse or right angles I categorized as near and reflexive transfer using the instructional strategy of “hugging”. I did this because all of the problems on the homework assignment were just like what the students did in Geometer’s sketchpad. As for all of the questions on the homework assignment that dealt with complementary, supplementary and vertical angles I am categorizing those as far and mindful transfer with the instructional strategy of “bridging”. This is because


the students only came up with the definitions for the vocabulary in the Geometer’s Sketchpad activity. They did not apply those definitions to problems until the homework assignment. Therefore, this first assignment is evidence of both near/reflexive and far/mindful transference with both instructional strategies in place.

As you can see from the data below, with the exception of my low ability males, all ability levels performed quite well on the homework assignment. When looking at the low ability male scores, one male missed four questions, another missed six and a half, and the last one missed thirteen. With such a small sample that one score really brought the whole average down. When I looked closely at this one males assignment I noticed that he got all of the questions about right, obtuse, and acute angles correct, but any questions involving complementary, supplementary or vertical angles he got those incorrect. Which leads me to believe he had difficulties taking the definitions he learned in Geometer’s Sketchpad and applying them to new questions in a new context.

Therefore, I believe that with the instructional strategy of “hugging” and “bridging” in place all ability levels, with the exception of the low ability males were able to achieve near and reflexive transference, as well as far and mindful transference. The low ability males did achieve near and reflexive transference as indicated by getting all of the questions correct that dealt with acute, obtuse and right angles. The good results for far and mindful transference most likely were affected by the fact that for most students this content seemed like a review, so this is not the strongest evidence for that type of transference.

Near/Reflexive and Far/Mindful Transfer from Geometer’s Sketchpad to Homework.

Gender/Ability


Male/High(5) 90%Male/Med(4) 85%Male/Low(3) 68%

Female/High(5) 95%Female/Med(6) 87%Female/Low(4) 84%

The third Geometer’s Sketchpad activity was Exploring Angles Formed by Parallel Lines and a Transversal. In this activity the students explored the angle relationships when two parallel lines are cut by a transversal. They were introduced to corresponding angles, alternate interior angles and alternate exterior angles. By moving lines, making the angles different sizes, and by observing the angle measurements the students came to the conclusion that corresponding angles are congruent, alternate interior angles are congruent and alternate exterior angles are congruent. The students liked this activity and the majority of them were successful in terms of learning about those relationships in the Geometer’s Sketchpad activity.


Following the activity, I assigned page 230 for homework to see if the students could transfer what they learned to other problems. On the homework assignment the students were asked to apply what they learned to find angle measurements, and name congruent angles given several different diagrams of parallel lines cut by a transversal. Because this is a new and different context compared with what they did in the Geometer’s Sketchpad activity, I am categorizing this as an example of far and mindful transference with “bridging” in place.

Again, all ability levels, with the exception of the low males, performed quite well on their homework. This time one male missed zero, one missed one problem, and the other male, which is a different male than the one mentioned above, missed thirteen and a half. Since there were only eighteen problems on this assignment this student definitely was not able to make the “bridge” from the Geometer’s Sketchpad activity to the homework assignment. It is encouraging though, that the other two males did quite well and were able to “bridge” their understanding to the homework.

In this example we have more positive evidence for far and mindful transference with the instructional strategy of “bridging” occurring for all ability levels, except for one low male. I believe this is stronger evidence for far and mindful transfer than the previous example because this was new material for all the students.

Far/Mindful Transfer from Geometer’s Sketchpad to Homework.

Gender/Ability


Male/High(5) 93%Male/Med(4) 88%Male/Low(3) 73%

Female/High(5) 88%Female/Med(6) 91%Female/Low(4) 88%

As I mentioned in my data analysis, the post assessment results for this unit can be broken down into two parts. The near and reflexive transfer part, and the far and mindful transfer part. I described earlier how I came to those conclusions, and as you can see below the results for all ability levels are quite positive on the part with “hugging”, and near and reflexive transfer. The far and mindful transference, on the other hand, may have occurred for the high ability students, but it is debatable. And for the medium and low ability students it looks like far transfer did not occur. I believe these results happened because as we went over the assessment as a class, it became evident the majority of my students did not think about drawing pictures to test out their ideas. Those that did draw pictures were more likely to think about the different scenarios that could happen in the problem, and were more likely to answer the questions correctly.


Near/Reflexive and Far/Mindful Transferencewith “Bridging” or “Hugging” Data

After the post assessment I had my students complete a reflection, which I spoke of earlier in my data analysis. From the reflection I learned that many of my students were mentally thinking back to their experiences with Geometer’s Sketchpad. To me this shows “active self-monitoring”, which according to Perkins and Solomon (1992) is one of the conditions that can make transfer more likely to occur. I believe it was through this metacognitive reflection that my students were thinking about their thinking, and in turn deepening their understanding of the math concepts. This, I believe, led them to achieving near and reflexive transfer in almost all instances, and achieving far and mindful transference in a few instances.

One other thing I would like to mention from my analysis of the reflection responses is that while the majority of responses from all ability levels were positive about everything we were doing, it seems the few negative ones that I did get were all from the low ability females. I do not think it had anything to do with their gender, but more with their ability level. I believe some of the negative feedback is due to their feeling about their poor results on the post assessment. I also believe these females didn’t like Geometer’s Sketchpad because of the constructivist nature of the activities. Both females are concrete thinkers, and they are not very confident in their mathematical ability. They also do not like to take risks. This I believe impacted their initial understandings of the concepts making far and mindful transference less likely to occur as evidenced by their score of 47% shown above.

Triangles UnitThis unit I decided to have each student at a computer but sitting next to their partner whom they were to work with as they progressed through the activity. There were three Geometer’s Sketchpad activities that the students completed for this unit. The first was

Gender/Ability

Questions 1-14. Post Assess. Near/Reflexive.Averages

Questions 15 – 20. Far/Mindful. Averages

Male/High(5) (21.5/22) 98% (4/6) 67%Male/Med(4) (21.5/22) 98% (3.5/6) 58%Male/Low(3) (20.2/22) 92% (3/6) 50%

Female/High(5)

(21.3/22) 97% (5.2/6) 87%

Female/Med(7) (19.5/22) 89% (3.5/6) 58%Female/Low(4) (18.5/22) 84% (2.8/6) 47%


Exploring Properties of Triangles where the students discovered that the sum of the angles in a triangle is 180 degrees. They also discovered that to have a triangle the sum of any two sides must be greater than the third, and that there is a relationship between the size of the angles and the size of the sides in all triangles, and vice versa. The second activity was Exploring Types of Triangles where the students were given a type of triangle, like equilateral, and they had to discover if various other types were possibleor not, like equilateral obtuse (no) or isosceles acute (yes). The activity finished with questions where the students could use all the diagrams of the various triangles and respond sometimes, always or never to given statements. For example, “An acute triangle is isosceles” (sometimes). The third activity was Exploring the Pythagorean Theorem where they discovered why the Pythagorean theorem is the sum of the legs squared is equal to the hypotenuse squared in all right triangles. They also learned that the converse was true.

I chose not to analyze the students’ homework performance for this unit because of time issues, and because I feel I have enough data to answer my research questions already.

For this unit I do have the whole class performance data, which I highlighted and discussed earlier in my data analysis section. And I have the data from the post assessment for my case study students. I broke my case study post assessment data into two parts based on the type of transference involved in the questions. The data below shows the positive impacts of “hugging” and “bridging” as an instructional strategy, and there positive impacts on both types of transfer, near/reflexive and far/mindful. As you can see there is good evidence of transfer occurring for all ability levels except for the low ability males (55%) on the near and reflexive transfer, and the low ability females (63%) on the far and mindful transfer.

The post assessment scores for the low ability females were negatively impacted by two of the four females studied. One female mistakenly applied the Pythagorean Theorem to the first three problems, and the other female ended up giving no reasons for her first three answers, and guessing on the last six problems scoring a 33% (4/12). The other two females scored 83% which was nice to see, but I must say one of these females, who did poorly on the Points, Lines and Angles unit, began tutoring with me after school before this assessment, which apparently helped her perform better.

The post assessment scores for the low ability males was impacted by one male who scored 31% (4/13). By looking at his assessment he clearly did not obtain near transfer at all, but he did grasp the concept about the largest angle being opposite the longest side. The other two males scored 65% (8.5/13) and 77% (10/13). I would also like to point out that when I looked back at the homework performance for the assignment related to these questions, I noticed that the low ability males missed an average of five problems on the assignment, which in comparison to the other ability levels is higher than three of the other five ability levels and tied with another. This indicates to me that near and reflexive transference with “hugging” did not occur for the low ability males.


Triangle Assessment First 9 Question Performance (Out of 12 points)Far/ Mindful Transference with “Bridging” Strategy

After the post assessment I had my students complete another reflection, which I spoke of earlier in my data analysis. From this reflection I learned that the majority of my students (79%) felt like they were able to transfer what they learned in Geometer’s Sketchpad activities to the assessment questions. Again, I feel this is evidence of their ability to “actively self-monitor” and think about their thought processes.

The other nice surprise was reading the common responses from the majority of the students about making the connections between the Geometer’s Sketchpad activities, the homework assignments and the assessment questions. On question number two on the reflection 58% (15/26) of my case study students, and 53% (40/75) of all the students I teach, mentioned how beneficial it was to learn the big idea in Geometer’s Sketchpad, and to be able to practice and “play” with the idea on your own during the homework assignments. This makes it quite apparent to me that the majority of my students enjoyed and benefited from the instructional strategy of “hugging”. As one medium ability male stated, “Both A (Geometer’s Sketchpad) and B (homework assignments) helped me understand the concepts because seeing that I knew them on the homework gave me the confidence to know that I can do these types of problems (on the assessment). Also recalling the pictures helped me remember what Mr. Kerlin said in the lab about things.”

Another interesting piece of information from this reflection data was that 100% of my low ability females responded that they thought both the Geometer’s Sketchpad activities, and the homework assignments helped them understand the concepts that were on the assessment. This reaffirms to me that regardless of ability level, it is important to follow up the Geometer’s Sketchpad activity with practice problems so

Gender/Ability

Questions 1-9. Post Assess. (12 pts.) Far/Mindful.Averages

Questions 10-16. Post Assess. (13pts.) Near/Reflexive. Averages

Male/High(5) (12/12) 100% (12.5/13) 96%Male/Med(4) (10/12) 83% (11.6/13) 89%Male/Low(3) (10/12) 83% (7.2/13) 55%

Female/High(5)

(11/12) 92% (12.7/13) 98%

Female/Med(7) (10.2/12) 85% (10.6/13) 81%Female/Low(4) (7.5/12) 63% (10.6/13) 82%


students can continue to refine their understandings, and gain confidence with their new knowledge. Therefore, the instructional strategy of “hugging” is even more important for the lower ability students who usually lack the confidence in themselves to succeed.

Quadrilaterals UnitBy the time I taught this unit my new Smartboard had been installed, so I decided to use the laptops in our school and do the Geometer’s Sketchpad activities in my classroom. The students were in self-chosen groups of two or three sharing one laptop between them. There were two Geometer’s Sketchpad activities that the students experienced in this unit. The first activity was Exploring Quadrilaterals where the students were given a drawing of each type of quadrilateral. The drawing was constructed so that when the students moved and changed the length of the sides or angles the drawing would retain the properties of that type of quadrilateral. They were asked a few questions about the figure, which lead them to a conclusion about which angle and side length properties were unique to which quadrilaterals. It was like they were generating a definition for each type of quadrilateral. Woven into the questions about each type of quadrilateral where statements about the angle and side length properties in which the students had to respond sometimes, always or never. This further refined the students understanding of the various quadrilateral definitions.

The second activity was Diagonals in Quadrilaterals. This activity would have been laborious and tedious, and very time consuming for my students because it required a lot of measuring of side lengths and angle measures. Therefore, I decided to do the measuring for each quadrilateral diagram before the activity, and to do the activity as a whole class using the Smartboard. In this activity the students discovered which quadrilaterals have diagonals that are always congruent; have diagonals that always bisect each other; and which ones have diagonals that are always perpendicular.

In this unit I categorized all of the connections between the Geometer’s Sketchpad activities, the homework assignments and the assessment as near and reflexive transference using “hugging” as the instructional idea. This is because all of the assessment questions were extremely similar to what the students had done before in Geometer’s Sketchpad or their homework.

Overall, compared to the other units this unit had the highest-class average with an 88%. When I look more closely at the scores and compare them to the other units, I notice that for most every class and every ability level the post assessment scores are higher for this unit. The only category that appears a little low is the low ability males at 75%. After looking at their assessments it appears to me that they did not know the meaning of the word perpendicular. They all responded incorrectly to the same question where they answered by naming quadrilaterals that don’t have perpendicular sides. Also, two of the three missed another problem that mentioned perpendicular diagonals. This discovery made me curious as to whether or not the word was used in their homework assignments or in the Geometer’s Sketchpad activity. I looked at the


homework assignment again and noticed that the word perpendicular does not show up. The only time they came across the word was when we discussed it during the Geometer’s Sketchpad activity. This explains why they were unable to make the near and reflexive transference at a higher level.

As you can see below, the homework scores for the low ability students, and medium ability males were quite low. But when it comes to the post assessment scores, the medium ability males and low ability females were able to perform quite well, showing good evidence of near and reflexive transference. Of course, the somewhat poor performance of the low ability males was mostly due to their not knowing what perpendicular meant. Therefore, I feel that this data is a good example of when “hugging” my instruction throughout the learning activities to the desired outcomes on the assessment the likelihood of near and reflexive transference increased for all ability levels.

Near/Reflexive Transferencewith “Hugging” Data

Gender/AbilityQuadrilaterals Post

Assessment %Page 380 Homework Scores (out of 18)

Male/High(5) 96% 88%Male/Med(4) 90% 72%Male/Low(3) 75% 72%

Female/High(5) 95% 86%Female/Med(7) 94% 88%Female/Low(4) 83% 67%

Making Sense of it all: What did you learn?

Conditions for transferAccording to Perkins and Solomon (1992) transfer tends to appear when five conditions are met. The first condition is “thorough and diverse practice” which needs to be given in a variety of contexts. The second is “explicit abstraction” where the learners have abstracted the important elements of a situation. Third is “active self-monitoring” where the learner is reflecting on their own thought processes. Fourth is “arousing mindfulness” where the learner is in a state of alertness towards the activity they are completing. And fifth is “using a metaphor or analogy”. p. 6-7.

I feel I provided “thorough and diverse practice” throughout the three units with my homework assignments supplementing the Geometer’s Sketchpad activities. The one time I did not provide a related homework assignment the level of transference declined, especially with my low ability students.

I feel that my students were able to abstract the important concepts in the activities because of the constructivist nature of the Geometer’s Sketchpad environment. And because the activities are structured in such a manner that leads the students to a discovery or conclusion, their initial understanding of the concept was stronger. This leads to a greater propensity for either type of transference occurring later. This


condition was difficult to meet with my low ability students as they tended to struggle with the Geometer’s Sketchpad activities. This in turn lead to a weaker initial understanding of the concept, which then lowered the probability of transference happening later in the unit.

“Active self-monitoring” was achieved through the students reflecting at the end of the unit and sharing their thought processes in their informative responses. It also was achieved through some of the discussions we had as a class when going over the homework. I would try to concretely remind them of what they experienced in the Geometer’s Sketchpad activity when questions arose about homework problems. By doing this we were sharing our thinking routines when faced with a problem, and increasing the likelihood of being able to transfer these thinking routines to similar and different situations at a later time.

“Arousing mindfulness” was enhanced by the Geometer’s Sketchpad environment itself. Many students expressed how fun and engaging it was to learn math this way. This can only lead to a better initial understanding of the concept and allowing a greater probability of transference happening later. Again, my low ability students may not have met this condition because they struggled with the Geometer’s Sketchpad activity leading to a weak initial understanding and a lower chance of transferring their weak understanding.

The only condition I did not meet was “using a metaphor or analogy” condition. I feel that this is a tough condition to meet in a mathematics classroom. And actually, I did not even try to come up with an analogy or metaphor for what my students were learning.

Overall, I met four out of the five conditions for when transfer appears. The positive results for near and far transference for nearly all ability levels throughout my project reinforces the importance of meeting as many of the five conditions as you can, and the importance of using the instructional strategies of “hugging” and “bridging” throughout your instructional design.

Answering my research questions.Changing my teaching methodologies when using Geometer’s Sketchpad to include the ideas of “bridging” and “hugging” with elements of constructivism, and social constructivism with a cognitivist perspective proved to be a change for the better for all ability levels of students, but to varying degrees.

High ability females and males consistently demonstrated at a high level their ability to reach near and far transference throughout the units, which is not surprising since these are the students that like math and are interested in it. The only poor score was for the high ability males’ rate of far transference on the Points, Lines and Angles assessment. While the percentage score of 67% looks low, when you take into account that the raw score was 4 out of 6 a case could be made that they were able to achieve far transfer, but not to the level of the females at 5.2 out of 6. As for this being attributed to my new teaching methodology that is debatable. This is because these kids would do well in


most circumstances no matter what methodology a teacher implemented. I do feel that their thinking routines about mathematics were stretched in a new direction though, and that was due to the constructivist nature of the Geometer’s Sketchpad activities where they were asked to think like mathematicians.

Medium ability females and males consistently demonstrated their propensity to achieve both types of transference as well, but not at the performance level of the high ability students. Again, I do not feel this is surprising because these are also bright math students, but when compared to the high ability students these students tend to be a little more concrete in their thought processes. This group also struggled with far transference on the Points, Lines and Angles assessment with a lower average of 3.5 out of 6. I feel these students benefited from the change in my teaching methodology because the homework assignments reinforced the Geometer’s Sketchpad activities, and gave them the extra time they needed to “play” with the concepts we were studying.Low ability females and males had scores that showed they struggled consistently to achieve both types of transference. Again, this is not surprising as these students are the students who lack confidence in themselves and their ability for various reasons, and are still very concrete thinkers. While the results for these students were up and down for both types of transference throughout the three units, I do feel they benefited the most from the change in my teaching methodologies. This is because for the most part the Geometer’s Sketchpad activities really did help them make that “bridge” from abstract concept to concrete thought. It also helped that for the most of them they found the Geometer’s Sketchpad activities more engaging than traditional classroom instruction. And they benefited from the discussions with others about their mathematical ideas, and the immediate reinforcement of the concepts provided in their homework assignments.

Because near transfer was achieved more often, and at a higher performance level for all ability levels, I feel that reflexive transfer was the mechanism responsible for this. This is because the homework assignments were chosen to “trigger” the students’ experiences and understandings gained in Geometer’s Sketchpad. This would give them a chance to get more practice with the thinking associated with the concepts. Then when the students took the assessment they had two experiences in which to draw upon to help them demonstrate their understanding of the concepts. Mindful transference was evident and achieved as well, but not to the level or frequency of reflexive transference. This is mostly due to the difficulty of being able to make connections and abstractions when asked to apply the new knowledge. What would I do differently and why?I should have collected observational data on my case study students when they were completing the Geometer’s Sketchpad activities. I would have noted the students who were struggling with the concepts and activities, and the meaningful comments I heard. This would have been valuable information as to the level of initial understanding of the concepts my case study students possessed going into the homework assignments. This would have resulted in a more accurate assessment on their ability to achieve near or far transfer from the Geometer’s Sketchpad activity to their homework assignment.


Another change I would make has to do with the homework packets I assigned. The first two packets consisted of problems only from the textbooks. The third packet for the Quadrilaterals unit included problems, and the text associated with them that had explanations and examples for the students to refer to at a later time. When I included this information on the last packet, I received many positive comments about the usefulness of the information. They liked having yet another explanation of the concepts and the sample problems.

The next change I would make has to do with two Geometer’s Sketchpad lessons and my closure of them at the end of the activity. The two activities are Exploring Types of Triangles and Exploring Quadrilaterals. Both of these activities involve the students developing definitions for the various types of triangles and quadrilaterals. After reflecting on the lessons, I realize that I should have made the definitions more apparent to the students at the end of the activity. I should have had them summarize their findings to add clarity to their understandings and to check for understanding, and when introducing the activity, I should have mentioned to the students the purpose of the activity is to build definitions through their explorations of the various types of triangles and quadrilaterals.

Along those same lines, I would change another part of my lesson closure by requiring students to print out their Geometer Sketchpad drawings. Many students expressed difficulties when working on their homework assignments and preparing for the assessments because when they looked back at the activity sheets from the computer lab there were no drawings to refer to. Therefore, those students who did not get a good mental picture of the activity or concepts found it difficult to transfer their knowledge from the Geometer’s Sketchpad activities to their homework assignment to the assessment.

Another change I would make involves improving the meaningful contexts I used when beginning the unit and lessons. I was only able to do this during the first unit. By improving this aspect of my instructional design I would increase the usefulness of the concepts for my students. This in turn would lead to a more engaged learner during the activities and homework because they would understand how the concepts are important in other contexts and in real life.

The last change I would make has to do with my low ability students and how to address their needs during the Geometer’s Sketchpad activities. The first thing would be to make sure they are partnered with specific students during the activity. Students that are patient, easy to work with, good at math and can communicate their thinking clearly. This partner student would then be a good role model for what good mathematical thinking is like. The other thing I would do would be to keep a better eye out for these students when they are starting the Geometer’s Sketchpad activity. By making sure they get started on the right track at the beginning they will not end up being so far behind towards the end of the activity. This leads to them not gaining a strong initial understanding of the concepts from the whole experience. By making these two


changes I feel I would give the lower ability students a better initial understanding of the concepts which would lead to a higher rate of transference.

What would other teachers learn about from my project? From my project other teachers would learn about an instructional strategy, which when used with Geometer’s Sketchpad can yield positive results for a high percentage of all students obtaining near or far transference. They would learn the importance of following up the Geometer’s Sketchpad activities with meaningful and purposeful homework assignments that link back to the activities, and they would learn more about the idea of teaching for transfer, and hopefully investigate the topic further.

Final thoughtsWhile the results for all ability levels were positive throughout my project I do feel the results are debatable in terms of truly measuring for near transference or far transference. I categorized the links between the components of my instructional design as near and far transference with “hugging” or “bridging” to the best of my ability and my understandings of the concepts. It was difficult at times to discern what was near transference and what was far transference, and was I “hugging” my instruction or was I “bridging” my instruction. I am sure upon further review by others there might be some tasks within my project that could be looked at as near transference by one person and looked at as far transference by someone else.

Yet despite that debate, I am confident that I have created a good instructional model that benefits all ability levels when using Geometer’s Sketchpad. I feel I have made a positive case for the importance of using an instructional design that incorporates the strategies of “hugging” and “bridging” with elements of constructivism, and social constructivism with a cognitivist perspective.

List of References

Bennett, D. (1999). Exploring geometry with the geometer’s sketchpad. Emeryville, California: Key Curriculum Press.

De Corte, E. (2003). Transfer as the productive use of acquired knowledge, skills, and motivations. Current Directions In Psychological Science. 12, 142-146.

Evans, J. (1999). Building bridges: Reflections on the problem of transfer of learning in mathematics. Educational Studies in Mathematics. 39, 23-44.

Greeno, J. G., Collins, A. M., & Resnick, L. B. (1996). Cognition and learning. In D. Berliner & R. Calfee (Eds.), Handbook of educational psychology (pp. 15-46). New York: Macmillan.


Hannafin, R.D., Burruss, J.D., & Little, C. (2001). Learning with dynamic geometry programs: Perspectives of teachers and learners. The Journal of Educational Research. 94, 132-144.

Jonassen D.H., Peck K.L., & Wilson B.G. ((c)1999). Learning with technology: A Constructivist Perspective. Chapter One: Learning with technology: Technologies for making meaning. Upper Saddle River, N.J.: Merrill.

Perkins, D.N., & Salomon, G. (1992). Transfer of Learning. International Encyclopedia of Education, Second Edition, Retrieved July 15, 2006, from http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm.

Sinclair, N., & Crespo, S. (2006). Learning Mathematics in Dynamic Computer Environments. Teaching Children Mathematics. May, 436-444.

Appendices

Description PageAppendix A Annotated Bibliography 37Appendix B Ethics/Permission/Notification Plan 42Appendix C Permission document 43Appendix D Geometer’s Sketchpad activities, Homework assignments and

assessments45

Appendix A

Annotated Bibliography

De Corte, E. (2003). Transfer as the productive use of acquired knowledge, skills, and motivations. Current Directions In Psychological Science. 12, 142-146.

This article describes how the behaviorist perspective and the constructivist perspective differ when it comes to looking at how to best enable your students to transfer what they know to new problem-solving situations, and what it means to truly transfer knowledge – referred to as a reconceptualization of transfer. The major difference being that within the constructivist perspective the students are in active control of the transfer situation.

The article then describes five characteristics teachers should use to design powerful teaching-learning environments where the likelihood of transference of knowledge can be increased for their students. First, the environment should support constructive learning for all students. Second, the environment should encourage students’ cognitive and motivational self-regulation to develop. Third, there needs to be a social context used in the learning environment. Fourth, the use of situated learning should be used so that students can make personal meanings for themselves, and lastly, the environment

http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm


should provide opportunities for the students to articulate and reflect on their cognitive and motivational processes during learning and problem solving.

This article has pointed out to me that I need to focus on stimulating my students’ articulation and reflection on their cognitive and motivational processes during the situated learning activities they are asked to solve. I also need to sequence those situated learning activities so that my students can become more self-regulated so they internalize what they are doing, and why, during problem-solving situations. Finally, from this article I have an idea for assessing my students’ transference. For example, I would like to ask them questions after the performance assessments focusing on: what activities in class they feel prepared them for the assessment; how those activities prepared them for it; and if they connected this task to an experience they had outside of class and if so, what was it that they connected to.

Evans, J. (1999). Building bridges: Reflections on the problem of transfer of learning in mathematics. Educational Studies in Mathematics. 39, 23-44.

In this article the author challenges the behavioral approach and the situated cognition approach on the transfer of mathematical learning, as most educators think of it, though he lends more credence to the situated cognition approach. He also mentions there is a social dimension to transferability to consider when teaching mathematics. His focus is not on transferring mathematics to other subject areas in school, but applying the mathematics in work contexts, and how one applies out-of-school experiences to the learning of mathematics.

In my view his approach has two main ideas for educators to take into account. First, educators need to analyze the mathematics they are teaching for connections to the workplace. Second, that there is an affective side we need to pay attention to through discourses with our students. By doing this we can impact our students motivation towards the problem-solving experience we are putting in front of them, and we can see how the students are transferring what they have experienced in the past to the new problem-solving experiences they are having in our classes.

This article has pointed out that for transference to be occurring more often for my students I need to find connections between the mathematics I am teaching to the everyday activities they have experienced in the past and the present. It has also made me realize that I need to have more conversations or journal exercises where my students tell me about what they are connecting their thought processes to when they are solving problems so that I can build on that for future lessons. It has also furnished me with many other resources to check on at a later date.

Hannafin, R.D., Burruss, J.D., & Little, C. (2001). Learning with dynamic geometry programs: Perspectives of teachers and learners. The Journal of Educational Research. 94, 132-144.


This article describes open-ended learning environments (OELE’s), and the theoretical frameworks used in some of them. Geometer’s Sketchpad is not an OELE but is a tool that teachers can use to make a student-centered learning environment employing the instructivist and constructivist methodologies.

The article is a study of how students and teachers reacted to, and the roles they took on when they used Geometer’s Sketchpad to study geometric content. The themes that resulted were based on power and learning. Power in the sense of who was in control of the learning, and learning in the sense of geometric learning and individual learning the students experienced about their abilities.

This article has pointed out that I need to pay attention to the manner in which I scaffold my questioning sequences – really focusing on the art of teaching and acting as a facilitator during my Geometer’s Sketchpad lessons. It has also provided me with some ideas on how to set up my own study, how to collect my data, and how to possibly analyze my data.

Jonassen D.H., Peck K.L., & Wilson B.G. ((c)1999). Learning with technology: A Constructivist Perspective. Chapter One: Learning with technology: Technologies for making meaning. Upper Saddle River, N.J.: Merrill.

This chapter describes the 5 attributes of meaningful learning that are interdependent: 1) Active (Manipulative/Observant), 2) Constructive (Articulative/Reflective), 3) Cooperative (Collaborative/Conversational), 4) Authentic (Complex/Contextualized), and 5) Intentional (Reflective/Regulatory). It describes how technologies can foster learning by acting as a motivator and guide through thinking and knowledge construction. It also states that the appropriate use of technology is to have students use it to discover new knowledge, converse about it, and reflect upon it to truly construct new meanings for themselves.

One piece of information I will use from this reading is to incorporate more reflection opportunities into my lessons for my students to help solidify their understanding of the concepts we are studying. The reading also reminds me that I need to set my Geometer’s Sketchpad lessons into a meaningful context so that my students see the need for the learning activity they are about to experience.

Monaghan, J. (2004). Measuring learning in situations which attempt to link school mathematics to out-of-school mathematical activities. British Society for Research into Learning Mathematics. 24, 75-81.

This article describes three ways a teacher may measure learning situations that try to link real-world mathematics to school-based mathematics. The author is beginning his research on the topic, and is in the proposal stages of his ideas. His three areas of focus are: 1) concentrating on students’ motives, 2) the mathematics that is being used, and 3) the material resources used – and currently he is focusing on the tools used in mathematics.


At the moment, I feel I may be able to use the first two measures he discusses. For the first one I could collect quantitative or qualitative data comparing how the students’ motives compare to out-of-school practitioners’ motives. This could be evidence that the students would be able to transfer the mathematics they are using in school to real-life mathematics as the practitioners use in the real world. I could use his second measure by collecting written explanations of my students’ strategies and thought processes and comparing them to the strategies and thought processes of general practitioners in the real world.

Perkins, D.N., & Salomon, G. (1992). Transfer of Learning. International Encyclopedia of Education, Second Edition, Retrieved July 15, 2006, from http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm.

This article defines and categorizes different types of transfer; looks at the conditions and mechanisms for transfer; and gives two instructional strategies for fostering transfer. Perkins and Salomon (1992) describe two types of transfer: near transfer and far transfer. Near transfer happens when the new situations the learner is in are closely related to the original situations, and far transfer happens when the new situations are quite different than the original situations.

Perkins and Salomon (1992) also describe two mechanisms for transfer. Reflexive or low road transfer is when the learner’s mind recalls the well-practiced routines of the original situation and applies them to a new situation. Whereas mindful or high road transfer is when the learner makes deliberate abstractions and connections from the original situation to the new situation.

They then go on to describe the five conditions of transfer. The first condition is the use of thorough and diverse practice of the concepts and skills by the learner. The second condition of transfer is the explicit abstraction of the critical attributes of the situations by the learner. The third condition is the active self-monitoring of the learners own thinking processes. The fourth condition is the arousing of mindfulness by the learner in terms of what activities they are engaged in, and their surroundings. And lastly, is the use of metaphors or analogies connecting past experiences with new ones.

In conclusion the authors propose two instructional strategies to foster transfer. They use the terms hugging and bridging (Perkins and Salomon, 1988). Hugging refers to enhancing reflexive or low road transfer by having instruction that directly engages the learners in similar situations as the desired situation – meaning their experience “hugs” the desired outcome. Bridging refers to enhancing mindful or high road transfer by having instruction that focuses the learner on actually making the abstract analysis and connections on what they are doing to what they will be doing in a different situation.

This article has made me realize I will be trying to help my students with near transfer. I want them to be able to take what they have learned in Geometer’s Sketchpad and apply it to similar problems outside of the environment. It has also given me a

http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm


theoretical approach to accomplishing this goal by focusing my attention on using the concepts of hugging and bridging in my instructional design. I will be using the hugging concept by providing my students practice problems after their experience using Geometer’s Sketchpad where they have constructed their initial meanings of the concepts we are studying. I will be using the bridging concept by assigning my students reflection questions after their homework problems and their unit assessments asking what connections they made between the two experiences. We will then follow this up with a class discussion making the abstractions of the connections more apparent to all.

Sinclair, N., & Crespo, S. (2006). Learning Mathematics in Dynamic Computer Environments. Teaching Children Mathematics. May, 436-444.

This article explains what Geometer’s Sketchpad is and how it works, and explains how the dynamic environment has three features that help mathematics teachers teach abstract concepts which can be challenging for students. The three features Geometer’s Sketchpad, and other computer learning environments like it, has are: 1) Continuous motion – where students can move shapes and look for relationships when doing so, 2) Connectivity – which, for example, enables students to look at algebraic ideas from a geometric perspective, and 3) Communication – where students mathematical vocabulary in engaged through the menus and commands.

This article reminds me that I should expect my students to use the language of mathematics in their conversations, written explanations and journaling exercises. By using the terminology more often during the activities and exercises they will have an experience to draw upon so that they can transfer their learning of the vocabulary to their future mathematics classes and focus more on the concepts in those classes. It also reminds me of Paul Goldenberg’s “Fluent Tool Use Principle” where through continued use of the technology students will begin to use it as a problem-solving tool and not just something they use once to learn a certain concept. Our high school teachers use Geometer’s Sketchpad so the skills I teach them in 7th grade using Geometer’s Sketchpad should transfer to the later grades where they will be more likely to employ the software as a tool for solving problems.

Taylor-Powell, E., & Renner, M. (2003). Analyzing Qualitative Data. Program Development & Evaluation. 1-12.

This article is a resource for anyone who will be analyzing and interpreting qualitative data. It describes the different types of narrative data one might collect during research. It then takes you through a five-step process for analyzing the qualitative data. First, you should get to know your data by rereading it several times. Second, you should focus your analysis whether it is by question, topic, time period or event, or by individual or group cases. Third, you should categorize the data by categories or themes, which you can do by using emerging themes or preset themes. Fourth, you should look for patterns within and between categories. And lastly, you should interpret it all paying attention to your themes and original questions you were trying to answer.


The article then goes on to explain how to collect and keep track of all the different types of qualitative data; how to improve the process; and pitfalls the researcher should look to avoid through the whole process.

This article made me realize that I should focus my case study analysis by both question/topic and by group (high, middle, and low achieving students). It has also made me realize I should use a small set of preset categories to look for and record any emerging categories I notice as I progress through the process. Another point the article has brought to my attention is the importance of using a coding scheme and the importance of coming up with a systematic way to track the themes I see through the analysis process. It has also furnished me with some great ideas on making my analysis better by getting someone else involved, and has brought to my attention the pitfall of avoiding generalizations.

Appendix B

Ethics/Permission/Notification Plan

Title of Project: Teaching for Transfer Using Geometer’s SketchpadTeacher Name: Chris Kerlin

1. School/District PoliciesDescribe any policies your school or district has for conducting research in your school. Does your school routinely get permission from parents for students to participate in various activities and research?

There is no policy at Cairo American College, but permission is routinely gathered for these types of activities and research. After discussing my project with my principal, he suggested I send home an informational/waiver form to the parents.

2. Risks and BenefitsAre there any risks for students participating in your project? Describe the potential risks and benefits for students or others participating in your project.

The main risk is that the students end up not understanding the geometric concepts we are studying. The benefits are that my students might have a deeper understanding of the geometrical concepts we are studying, and they are able to apply their new knowledge to other situations.

3. Informing participantsDescribe procedures for informing students, parents, or others about your project. What do parents or others need to know? (This is important for getting ‘buy-in’ from students and parents for the new instructional activities you are using.) (Attach copy or letter or other communications, if any)

I explained and informed parents about my project at our Back to School Night function. I also sent home information about my project in the permission slip form. See form/letter attached.

4. Consent/PermissionDo you need to get consent from students or their parents for their participation in your project? (This would be true only if you are conducting more formal research Describe procedures for obtaining consent from parents, students, or others. (Attach copy or letter or other communications, if any)


My principal asked me to send a letter home asking for permission. See form/letter attached.

5. Participant Privacy: How will you protect the privacy of your students or others participating in your project? What information will you share publicly and how will you ensure confidentiality?

I will remove their names from their written work and put fake initials on it. I will refer to them using their fake initials during audio interviews.

Appendix C

Permission documentSeptember 18, 2006

Dear Parents:

For those of you who were at Back to School Night, you might remember that I spoke

about the fact I am completing my Master’s Degree in Educational Technology this summer.

One component of my degree is to implement and evaluate an Action Research Project. My

Action Research Project involves investigating how my students are able to transfer the

knowledge they acquire when using Geometer’s Sketchpad to problem situations outside of that

environment. As you probably know, we have already begun to use Geometer’s Sketchpad to

learn about the geometric concepts in our curriculum.

As you can imagine, an integral part of my Action Research Project is to collect and

analyze data, looking for evidence of knowledge transfer. The data I am collecting are pre and

post assessment scores, journal reflections, classroom observations, and audio taping interviews

of a few chosen students, based on their genders and ability levels in mathematics.

Your child’s work, and, if chosen, his/her responses to interview questions will be used

only for my Action Research Project and a presentation I must do this summer to report my

conclusions. The work and interview responses will not be reproduced or distributed in any way,

and I will not use any students’ names throughout my project. The purpose of the presentation is


to share my findings with other graduate students in the same program. If you care to see the

final results of my Action Research Project, I will be happy to share them with you.

You may grant me permission to use your child’s work or interview responses by signing

this letter and having your child return it to me. Please feel free to contact me if you have further

questions. Thank you in advance for your support.

Sincerely,

Chris Kerlin

7th Grade Mathematics Teacher

Cairo American College

E-mail: [email protected]

Direct: 202-755-5327

You may use my child’s work, and, if chosen, his/her responses to interview

questions for your Action Research Project.

You may not use my child’s work nor their responses to interview questions for your

Action Research Project.

Name of Student: ___________________________

Name of Parent: ___________________________

Signature: ___________________________ Date: ________________

mailto:[email protected]


Appendix DGeometer’s Sketchpad Activities, Homework Assignments and AssessmentsDescription PageExploring Angles 46Exploring Special Pairs of Angles 48Page 224 Homework 50Exploring Angles Formed by Parallel Lines and a Transversal 52Page 230 Homework 54Points, Lines and Angles Assessment 55Exploring Properties of Triangles 57Page 384 Homework 59Exploring Types of Triangles 60Page 376 Homework 62Page 237 Homework 63Exploring the Pythagorean Theorem 64Triangles Assessment 67Exploring Quadrilaterals 68Page 380 Homework 71Page 240 Venn diagram 72Diagonals in Quadrilaterals 73Page 243 Homework 75Quadrilaterals Assessment 76

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