Contributed by:

Its purpose was to see whether teachers could affect the quality of student mathematical thinking and solution writing by teaching students five key processes of mathematical thinking we had identified, and by providing students with opportunities to evaluate sample student solutions using traits describing these processes.

1.
University of Nebraska - Lincoln

DigitalCommons@University of Nebraska - Lincoln

Summative Projects for MA Degree Math in the Middle Institute Partnership

Five Processes of Mathematical Thinking

Toni Scusa

Yuma, CO

Follow this and additional works at: https://digitalcommons.unl.edu/mathmidsummative

Part of the Science and Mathematics Education Commons

Scusa, Toni, "Five Processes of Mathematical Thinking" (2008). Summative Projects for MA Degree. 38.

This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at

DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Summative Projects for MA

Degree by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.

DigitalCommons@University of Nebraska - Lincoln

Summative Projects for MA Degree Math in the Middle Institute Partnership

Five Processes of Mathematical Thinking

Toni Scusa

Yuma, CO

Follow this and additional works at: https://digitalcommons.unl.edu/mathmidsummative

Part of the Science and Mathematics Education Commons

Scusa, Toni, "Five Processes of Mathematical Thinking" (2008). Summative Projects for MA Degree. 38.

This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at

DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Summative Projects for MA

Degree by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.

2.
Five Processes of Mathematical Thinking

Toni Scusa

Yuma, CO *

Math in the Middle Institute Partnership

Action Research Project Report

in partial fulfillment of the MA Degree

Department of Teaching, Learning, and Teacher Education

University of Nebraska-Lincoln

July 2008

* I began the program as a fifth grade teacher at Paxton, Nebraska but have since moved

to Colorado

Toni Scusa

Yuma, CO *

Math in the Middle Institute Partnership

Action Research Project Report

in partial fulfillment of the MA Degree

Department of Teaching, Learning, and Teacher Education

University of Nebraska-Lincoln

July 2008

* I began the program as a fifth grade teacher at Paxton, Nebraska but have since moved

to Colorado

3.
Five Processes of Mathematical Thinking

Abstract

My research project was to investigate key processes of mathematical thinking in

my seventh grade mathematics classroom. Its purpose was to see whether I could affect

the quality of student mathematical thinking and solution writing by teaching students

five key processes of mathematical thinking I had identified, and by providing students

with opportunities to evaluate sample student solutions using traits describing these

processes. Every two weeks, students attempted solutions for a given problem and rated

their work according to the specific characteristics identified as key to mathematical

thinking. Every other week I gave the class sample student work at varied proficiency

levels to rate according to a rubric and they discussed or defended their decisions. I found

that student reasoning, whether written or oral, did improve over time as we emphasized

these processes, although the change was slow and in small steps. Student engagement

was also affected by the time we spent working in large or small group activities. The

change, however, did not occur without an investment of substantial effort and time on

my part and the students’. Learning about specific processes to emulate, model and then

use to evaluate another’s work is an in-depth task that does not happen quickly or easily.

Abstract

My research project was to investigate key processes of mathematical thinking in

my seventh grade mathematics classroom. Its purpose was to see whether I could affect

the quality of student mathematical thinking and solution writing by teaching students

five key processes of mathematical thinking I had identified, and by providing students

with opportunities to evaluate sample student solutions using traits describing these

processes. Every two weeks, students attempted solutions for a given problem and rated

their work according to the specific characteristics identified as key to mathematical

thinking. Every other week I gave the class sample student work at varied proficiency

levels to rate according to a rubric and they discussed or defended their decisions. I found

that student reasoning, whether written or oral, did improve over time as we emphasized

these processes, although the change was slow and in small steps. Student engagement

was also affected by the time we spent working in large or small group activities. The

change, however, did not occur without an investment of substantial effort and time on

my part and the students’. Learning about specific processes to emulate, model and then

use to evaluate another’s work is an in-depth task that does not happen quickly or easily.

4.
Traits of Good Mathematical Thinking 2

INTRODUCTION

My problem of practice was to pursue the idea of traits of good mathematical thinking

based on the five process standards. I teach at a school with a 70% Hispanic population with

about half of the population qualifying for free or reduced lunch and a high mobility rate. Last

year there was about a 26% gap between those considered minority or low income who achieved

a proficient or advanced rating on their Colorado State Assessment and those students classified

as white or non poverty level who were proficient or advanced in Math. Since I began

participating in the Math in the Middle Institute, I had been trying weekly problem solving

activities in my classroom chosen to focus on developing good mathematical Habits of the Mind

and had been toying with the idea that I needed to model for students how to approach the

problem solving of a Habits-of-Mind type problem. The students I have in seventh and eighth

grade had poor reasoning skills overall, and needed to develop good problem solving behaviors

and familiarity with different formats of representation. When given a problem solving activity,

most students truly had no idea of where to start.

I have been an elementary teacher for more than 20 years and during that time have had

to become familiar with Six Traits Writing (now called Six Traits Plus One and developed by

Northwest Regional Educational Laboratory). With that program, students are taught the key

traits of good writing: Voice, Word Choice, Fluency, Organization, Ideas, Presentation and

Conventions. Students are taught to look for these traits in other students’ writing and then to

evaluate them in their own writing using a set of pre-made rubrics. I wondered if there might be

a list of key traits for mathematical thinking, similar to those identified for writing. Could the

processes of mathematical thinking be taught? Was this focus missing from my teaching? How

could I model the kind of process that I had seen as a Math in the Middle participant so that my

INTRODUCTION

My problem of practice was to pursue the idea of traits of good mathematical thinking

based on the five process standards. I teach at a school with a 70% Hispanic population with

about half of the population qualifying for free or reduced lunch and a high mobility rate. Last

year there was about a 26% gap between those considered minority or low income who achieved

a proficient or advanced rating on their Colorado State Assessment and those students classified

as white or non poverty level who were proficient or advanced in Math. Since I began

participating in the Math in the Middle Institute, I had been trying weekly problem solving

activities in my classroom chosen to focus on developing good mathematical Habits of the Mind

and had been toying with the idea that I needed to model for students how to approach the

problem solving of a Habits-of-Mind type problem. The students I have in seventh and eighth

grade had poor reasoning skills overall, and needed to develop good problem solving behaviors

and familiarity with different formats of representation. When given a problem solving activity,

most students truly had no idea of where to start.

I have been an elementary teacher for more than 20 years and during that time have had

to become familiar with Six Traits Writing (now called Six Traits Plus One and developed by

Northwest Regional Educational Laboratory). With that program, students are taught the key

traits of good writing: Voice, Word Choice, Fluency, Organization, Ideas, Presentation and

Conventions. Students are taught to look for these traits in other students’ writing and then to

evaluate them in their own writing using a set of pre-made rubrics. I wondered if there might be

a list of key traits for mathematical thinking, similar to those identified for writing. Could the

processes of mathematical thinking be taught? Was this focus missing from my teaching? How

could I model the kind of process that I had seen as a Math in the Middle participant so that my

5.
Traits of Good Mathematical Thinking 3

students could experience the same kind of learning? I wanted my students to be exposed to the

kind of problem solving activities we had in Math in the Middle and to experience the struggle of

figuring out ways to reason, prove and solve as we had. What would be the best way to achieve

this in the middle school classroom?

I decided to develop a rubric of those traits or particulars for mathematical reasoning

similar to what is available presently for teaching and assessing Six Traits Writing. I then used

this list of characteristics and rubrics to work on more difficult problems focused on

mathematical habits of thinking with my class. We used the rubric as a class to discuss and

assess student sample answers and in the evaluation of sample individual work. In the process I

modeled what it takes to make good mathematical thinking. I provided ideas of how a solution

could be changed to improve the attempt.

I had not assigned many harder problems yet in the year that would qualify as a problem

specifically chosen to help students work on mathematical habits of the mind. Frankly the

students I had were not in a place where they could do even the simplest type of word problem.

Their biggest challenge was they had no idea of even where to begin. I felt as if they had

somewhere learned it was the ANSWER that was the most important part, and everything else

was static. I wanted students to see that the struggle has value. I wanted them to be able to work

at a more independent level and not have as their key strategy- - ask Ms. Scusa for help.

I wanted to have a classroom with students of all ability levels who would have the

confidence to try something new or different. They would look forward to challenging

mathematical problems. They would want to learn new and better strategies and would be

anxious to hear from others about alternative strategies. They would “see” the big picture and

understand the value in attempting to solve problems, but I needed to make it achievable. We

students could experience the same kind of learning? I wanted my students to be exposed to the

kind of problem solving activities we had in Math in the Middle and to experience the struggle of

figuring out ways to reason, prove and solve as we had. What would be the best way to achieve

this in the middle school classroom?

I decided to develop a rubric of those traits or particulars for mathematical reasoning

similar to what is available presently for teaching and assessing Six Traits Writing. I then used

this list of characteristics and rubrics to work on more difficult problems focused on

mathematical habits of thinking with my class. We used the rubric as a class to discuss and

assess student sample answers and in the evaluation of sample individual work. In the process I

modeled what it takes to make good mathematical thinking. I provided ideas of how a solution

could be changed to improve the attempt.

I had not assigned many harder problems yet in the year that would qualify as a problem

specifically chosen to help students work on mathematical habits of the mind. Frankly the

students I had were not in a place where they could do even the simplest type of word problem.

Their biggest challenge was they had no idea of even where to begin. I felt as if they had

somewhere learned it was the ANSWER that was the most important part, and everything else

was static. I wanted students to see that the struggle has value. I wanted them to be able to work

at a more independent level and not have as their key strategy- - ask Ms. Scusa for help.

I wanted to have a classroom with students of all ability levels who would have the

confidence to try something new or different. They would look forward to challenging

mathematical problems. They would want to learn new and better strategies and would be

anxious to hear from others about alternative strategies. They would “see” the big picture and

understand the value in attempting to solve problems, but I needed to make it achievable. We

6.
Traits of Good Mathematical Thinking 4

would have a lot of work in groups and lots of discussion time. I would need to use many

examples and be consistent in modeling, since this type of Math work would not be something

they were used to.

I came up with the idea of focusing on specific processes of mathematical thinking- -

practices I had decided my students needed to be successful mathematics students. I decided to

try to teach these practices to students, and I figured out the characteristics that each of these

embodied. They were based on five key areas 1) Representation, 2) Reasoning and Proof, 3)

Communication, 4) Problem Solving, and 5) Connections. If these look familiar, it is because

they are the five process standards from the National Council of Teachers of Mathematics

(NCTM, 2000). It was my thinking that each of these process standards from NCTM had a

specific set of behaviors that one could use to characterize each. My task was to make such a list

for each of these five process standards and develop problem solving activities that afforded

students opportunities to work on each.

My research project focused on teaching the students these specific processes and what

they looked like. We spent time learning about each process and its identifying traits. We spent

time evaluating student work based on this list, discussing the work’s merits, and then worked on

improving our own abilities to achieve good mathematical solutions with these characteristics in

PROBLEM STATEMENT

Problem of Practice

Many teachers in my Math in the Middle sessions had commented on the thinking ability

of their students and the students’ abilities to apply what they knew to a problem solving

situation. Members of my cohort talked about the difference between a product skill, such as

would have a lot of work in groups and lots of discussion time. I would need to use many

examples and be consistent in modeling, since this type of Math work would not be something

they were used to.

I came up with the idea of focusing on specific processes of mathematical thinking- -

practices I had decided my students needed to be successful mathematics students. I decided to

try to teach these practices to students, and I figured out the characteristics that each of these

embodied. They were based on five key areas 1) Representation, 2) Reasoning and Proof, 3)

Communication, 4) Problem Solving, and 5) Connections. If these look familiar, it is because

they are the five process standards from the National Council of Teachers of Mathematics

(NCTM, 2000). It was my thinking that each of these process standards from NCTM had a

specific set of behaviors that one could use to characterize each. My task was to make such a list

for each of these five process standards and develop problem solving activities that afforded

students opportunities to work on each.

My research project focused on teaching the students these specific processes and what

they looked like. We spent time learning about each process and its identifying traits. We spent

time evaluating student work based on this list, discussing the work’s merits, and then worked on

improving our own abilities to achieve good mathematical solutions with these characteristics in

PROBLEM STATEMENT

Problem of Practice

Many teachers in my Math in the Middle sessions had commented on the thinking ability

of their students and the students’ abilities to apply what they knew to a problem solving

situation. Members of my cohort talked about the difference between a product skill, such as

7.
Traits of Good Mathematical Thinking 5

knowing multiplication facts, and a process skill, such as providing reasoning and proof. From

this discussion, we decided that we would like to know more about teaching the second type of

skill in the classroom.

Reading TIMSS (the Third International Mathematics and Science Study) and other

studies comparing U.S. classrooms with classrooms in other countries, told my cohort members

and me what we knew in our hearts already—we wanted to provide in our classrooms something

beyond providing practice on rote skills and memorization. Many of us did not know how to

begin. I wondered if I could come up with a list of specific behaviors for each of the five process

standards and then create some sort of a system or “vehicle” that could be used to teach these

process skills in the mathematics classroom. Could it be done in such a way that it could be

replicated from year to year with consistency? What kind of support would my students need?

How would I go about making it easy to understand and imitate?

I hoped that teaching specific strategies of problem solving to my students would not

only increase student confidence as they learned to work problems and we worked through the

steps of reasoning, but would also require higher level thinking and have real world applications.

I believed students who reasoned and solved problems were much better equipped to function in

today’s society than those who did not have this practice.

I also believed establishing clear cut behaviors of mathematical thinking, modeling the

process and learning to evaluate sample work helped to equate students of differing levels of

ability as ALL learn the steps to better reasoning and problem solving skills. I thought there must

be a better way to improve problem solving and reasoning than by merely providing more

practice doing problem solving and reasoning. I asked myself, “What happens in the real world?”

If I were a coach who wanted better basketball players, for example, I would break basketball

knowing multiplication facts, and a process skill, such as providing reasoning and proof. From

this discussion, we decided that we would like to know more about teaching the second type of

skill in the classroom.

Reading TIMSS (the Third International Mathematics and Science Study) and other

studies comparing U.S. classrooms with classrooms in other countries, told my cohort members

and me what we knew in our hearts already—we wanted to provide in our classrooms something

beyond providing practice on rote skills and memorization. Many of us did not know how to

begin. I wondered if I could come up with a list of specific behaviors for each of the five process

standards and then create some sort of a system or “vehicle” that could be used to teach these

process skills in the mathematics classroom. Could it be done in such a way that it could be

replicated from year to year with consistency? What kind of support would my students need?

How would I go about making it easy to understand and imitate?

I hoped that teaching specific strategies of problem solving to my students would not

only increase student confidence as they learned to work problems and we worked through the

steps of reasoning, but would also require higher level thinking and have real world applications.

I believed students who reasoned and solved problems were much better equipped to function in

today’s society than those who did not have this practice.

I also believed establishing clear cut behaviors of mathematical thinking, modeling the

process and learning to evaluate sample work helped to equate students of differing levels of

ability as ALL learn the steps to better reasoning and problem solving skills. I thought there must

be a better way to improve problem solving and reasoning than by merely providing more

practice doing problem solving and reasoning. I asked myself, “What happens in the real world?”

If I were a coach who wanted better basketball players, for example, I would break basketball

8.
Traits of Good Mathematical Thinking 6

down into a set of skills I wanted my players to learn- - dribbling, shooting, passing, defending

etc… and then we would practice and practice. I would help players by evaluating weaknesses

and strengths myself and help them to assess themselves. I would teach them what to do in

different situations to use those skills. I would not just put out some basketballs and then say to

the group, “Go, get better at basketball.” Just providing the basketballs helps those who are

already skilled by providing the time for them to get better, but it does not help the ones who

need to address specific lack of skills. Not dividing problem solving into a set of skills we could

practice would be the same for math class. More practice without attention to improving skill

would only help those who already were skilled. I did not want a classroom where the strong got

stronger and the rest did not have a clue of how to become better. The gap between those with

mathematical skill and those who did not have this ability would only widen.

As a mathematics teacher, it was important for me to identify and clearly communicate

the expectations I had for the classroom. Creating a list of traits and a rubric helped me state and

communicate my expectations with respect to high quality reasoning for ALL students. My

students had multiple opportunities to discuss what makes good reasoning and were able to view

reasoning through modeling techniques. By incorporating all of these various strategies centered

on the traits of good mathematical reasoning, I believed good mathematical thinkers would

emerge in my classroom.

Research Questions

The purpose of my research then, was to determine if one could teach mathematical

thinking in a systematic manner. I taught my students the five process standards (the “Processes

of a Mathematical Thinker”) and saw what would happen when mathematical thinking was

taught in a structured way that showed students how to evaluate their work and the work of

down into a set of skills I wanted my players to learn- - dribbling, shooting, passing, defending

etc… and then we would practice and practice. I would help players by evaluating weaknesses

and strengths myself and help them to assess themselves. I would teach them what to do in

different situations to use those skills. I would not just put out some basketballs and then say to

the group, “Go, get better at basketball.” Just providing the basketballs helps those who are

already skilled by providing the time for them to get better, but it does not help the ones who

need to address specific lack of skills. Not dividing problem solving into a set of skills we could

practice would be the same for math class. More practice without attention to improving skill

would only help those who already were skilled. I did not want a classroom where the strong got

stronger and the rest did not have a clue of how to become better. The gap between those with

mathematical skill and those who did not have this ability would only widen.

As a mathematics teacher, it was important for me to identify and clearly communicate

the expectations I had for the classroom. Creating a list of traits and a rubric helped me state and

communicate my expectations with respect to high quality reasoning for ALL students. My

students had multiple opportunities to discuss what makes good reasoning and were able to view

reasoning through modeling techniques. By incorporating all of these various strategies centered

on the traits of good mathematical reasoning, I believed good mathematical thinkers would

emerge in my classroom.

Research Questions

The purpose of my research then, was to determine if one could teach mathematical

thinking in a systematic manner. I taught my students the five process standards (the “Processes

of a Mathematical Thinker”) and saw what would happen when mathematical thinking was

taught in a structured way that showed students how to evaluate their work and the work of

9.
Traits of Good Mathematical Thinking 7

others. I examined the use of a rubric to identify the characteristics of mathematical thinking and

whether effective rubric use would influence the quality of student reasoning and engagement in

problem solving situations. I sought to answer the following research questions:

What will happen to the quality of student written reasoning when students use a rubric to

evaluate their work?

What will happen to the level of student engagement in small group discussions when using

the five traits of mathematical thinking to solve problems?

What will happen to the quality of student oral explanations of solutions when using the

traits of a mathematical thinker to guide student solutions?

What will happen to my teaching when I specifically set aside time to teach traits of

mathematical thinking and deliberately spend time on mathematical discussion and

reasoning?

LITERATURE REVIEW

I looked at the literature for current trends and research. What did it say? Could one teach

mathematical thinking? What were the keys to mathematical thinking? I believed that if I

determined the answers to questions like these, I could use problem solving to teach and develop

successful mathematics students.

In the research examined within the United States and other countries, problem solving

was being used as both a means and an end result. In the past ten to twenty years, the trend in

problem solving had been similar. It had been to concentrate on developing mathematic skill and

not just arithmetic skill by developing or emphasizing problem solving. It was a difference I had

heard before amongst colleagues - - teaching math could be divided into two realms- - teaching

others. I examined the use of a rubric to identify the characteristics of mathematical thinking and

whether effective rubric use would influence the quality of student reasoning and engagement in

problem solving situations. I sought to answer the following research questions:

What will happen to the quality of student written reasoning when students use a rubric to

evaluate their work?

What will happen to the level of student engagement in small group discussions when using

the five traits of mathematical thinking to solve problems?

What will happen to the quality of student oral explanations of solutions when using the

traits of a mathematical thinker to guide student solutions?

What will happen to my teaching when I specifically set aside time to teach traits of

mathematical thinking and deliberately spend time on mathematical discussion and

reasoning?

LITERATURE REVIEW

I looked at the literature for current trends and research. What did it say? Could one teach

mathematical thinking? What were the keys to mathematical thinking? I believed that if I

determined the answers to questions like these, I could use problem solving to teach and develop

successful mathematics students.

In the research examined within the United States and other countries, problem solving

was being used as both a means and an end result. In the past ten to twenty years, the trend in

problem solving had been similar. It had been to concentrate on developing mathematic skill and

not just arithmetic skill by developing or emphasizing problem solving. It was a difference I had

heard before amongst colleagues - - teaching math could be divided into two realms- - teaching

10.
Traits of Good Mathematical Thinking 8

students process and product, or in other words, the teaching of arithmetic vs. teaching of

mathematics. I wondered if it was possible to do both.

All articles I researched for the six countries of the United States, China, Singapore,

Australia, Japan, and Portugal mentioned problem solving as a major focus of the country’s

mathematics emphasis with concentration on development of higher level thinking skills for at

least the past ten years, if not longer. The difference from country to country was in the

curriculum and textbooks used, and in the degree of influence the teacher had over the change.

The emphasis on problem solving has meant in some cases a change in teaching

strategies, classroom atmosphere, and/or a change in the role of the student and teacher. In any

case, it has meant using problem solving in the classroom to achieve problem solving success. It

has meant using problem solving to teach mathematics while at the same time helping students

learn how to problem solve. This emphasis has been as much about the process as about the

product. The big question seems to have been—how does one teach the process of problem

I wondered as I read what the research literature would suggest. Was there a need to

create in students certain procedures in order for them to be successful mathematicians? Would I

be teaching students the qualities of good mathematical thinking by using these? Could these

processes be grouped? What key identifiers could be listed under each? Would promoting these

skills also promote higher level thinking and reasoning?

When I identified this list, I planned to focus my research project on teaching these

significant processes and reaching higher level thinking skills by helping students learn about

solving problems and the traits of a mathematical thinker while actively problem solving. By

using what characteristics I could find in common amongst the literature, I thought I could teach

students process and product, or in other words, the teaching of arithmetic vs. teaching of

mathematics. I wondered if it was possible to do both.

All articles I researched for the six countries of the United States, China, Singapore,

Australia, Japan, and Portugal mentioned problem solving as a major focus of the country’s

mathematics emphasis with concentration on development of higher level thinking skills for at

least the past ten years, if not longer. The difference from country to country was in the

curriculum and textbooks used, and in the degree of influence the teacher had over the change.

The emphasis on problem solving has meant in some cases a change in teaching

strategies, classroom atmosphere, and/or a change in the role of the student and teacher. In any

case, it has meant using problem solving in the classroom to achieve problem solving success. It

has meant using problem solving to teach mathematics while at the same time helping students

learn how to problem solve. This emphasis has been as much about the process as about the

product. The big question seems to have been—how does one teach the process of problem

I wondered as I read what the research literature would suggest. Was there a need to

create in students certain procedures in order for them to be successful mathematicians? Would I

be teaching students the qualities of good mathematical thinking by using these? Could these

processes be grouped? What key identifiers could be listed under each? Would promoting these

skills also promote higher level thinking and reasoning?

When I identified this list, I planned to focus my research project on teaching these

significant processes and reaching higher level thinking skills by helping students learn about

solving problems and the traits of a mathematical thinker while actively problem solving. By

using what characteristics I could find in common amongst the literature, I thought I could teach

11.
Traits of Good Mathematical Thinking 9

the keys to mathematical thinking and use these characteristics to create a rubric of the traits to

evaluate example student work and to evaluate our own class work. The key questions I tried to

look for in the research literature were 1) What are the keys to successful mathematical thinking?

2) How does a student become a good mathematician? 3) What are the traits necessary in order

to demonstrate proficiency in mathematics? And 4) Could these traits be lumped together in

some way under specific processes of mathematical thinking?

Problem Solving as a Process

Several researchers noted using problem solving as a process in order to promote higher

level thinking and reasoning. Many mention some common skills in problem solving I wanted to

pursue. According to Segurado (2002) good problem solvers are confident in their abilities.

It is possible to provide students of this school level a mathematical experience of

doing investigations. Students are able to approach the tasks and move in the

direction of becoming confident in their abilities, of enlarging their ability to

solve and formulate problems and of communicating and reasoning

mathematically. (p. 72)

Costa and Kallick (2000) say those good at problem solving are risk takers. Students who

practice what they call responsible risk taking show a willingness to try out new strategies or

techniques and are willing to test new hypotheses with an attitude of “What’s the worst thing that

can happen? We’ll only be wrong?” Costa and Kallick also list persistence among the skills of

those good at the problem solving process. They say in order to be successful problem solvers,

students must not give up when encountering a difficult problem, even if they are not used to

such struggle.

Persistent students have systematic methods of analyzing a problem. They know

how to begin, what steps must be performed, and what data need to be generated

and collected. They also know when their theory or idea must be rejected so they

can try another…If the strategy is not working, they back up and try another. (p.

22)

the keys to mathematical thinking and use these characteristics to create a rubric of the traits to

evaluate example student work and to evaluate our own class work. The key questions I tried to

look for in the research literature were 1) What are the keys to successful mathematical thinking?

2) How does a student become a good mathematician? 3) What are the traits necessary in order

to demonstrate proficiency in mathematics? And 4) Could these traits be lumped together in

some way under specific processes of mathematical thinking?

Problem Solving as a Process

Several researchers noted using problem solving as a process in order to promote higher

level thinking and reasoning. Many mention some common skills in problem solving I wanted to

pursue. According to Segurado (2002) good problem solvers are confident in their abilities.

It is possible to provide students of this school level a mathematical experience of

doing investigations. Students are able to approach the tasks and move in the

direction of becoming confident in their abilities, of enlarging their ability to

solve and formulate problems and of communicating and reasoning

mathematically. (p. 72)

Costa and Kallick (2000) say those good at problem solving are risk takers. Students who

practice what they call responsible risk taking show a willingness to try out new strategies or

techniques and are willing to test new hypotheses with an attitude of “What’s the worst thing that

can happen? We’ll only be wrong?” Costa and Kallick also list persistence among the skills of

those good at the problem solving process. They say in order to be successful problem solvers,

students must not give up when encountering a difficult problem, even if they are not used to

such struggle.

Persistent students have systematic methods of analyzing a problem. They know

how to begin, what steps must be performed, and what data need to be generated

and collected. They also know when their theory or idea must be rejected so they

can try another…If the strategy is not working, they back up and try another. (p.

22)

12.
Traits of Good Mathematical Thinking 10

The development of good problem solving techniques takes time, however. Ponte (2007)

cites some examples in Portugal in which problem solving or mathematical investigations were

used in a school setting. Having no one right answer seemed to generate some insecurity for

students. According to Ponte, as time went on, the activity improved in quality, and, with teacher

support and continuation of the work, student confidence in their abilities grew. The voiced

“unpleasantness” by some that the activities required high personal perseverance lessened.

Allowing students to struggle and develop persistence is not always easy for teachers

either. Ben-Hur (2006) believes that among teachers are two camps of thought when it comes to

allowing students to “struggle.” One camp seeks to take the shortcut of teaching key words,

algorithms and other tricks that work for given types of problems. He believes that this shelters

students from the uncertain nature of problem solving. The other camp of teachers seeks ways to

enhance reflective practices thus provoking students through use of cognitive dissonance. Wood

(2001) says, “In order to create these situations for mathematical learning in classrooms, teachers

must resist their natural inclination to tell students information, make the task simpler, or step in

and do part of the task” (p. 116). Therefore, to develop students who are persistent problem

solvers who take risks, teachers need to exhibit those qualities as well.

Fan and Zhu (2007) talk about a framework for problem solving modified from Polya’s

problem-solving model and published in a syllabus by the Ministry of Education in 1990. Its list

includes developing a plan, carrying out the plan and/or modifying the plan if necessary and

ending with seeking alternative solutions and checking for reasonableness. Students good at

problem solving do all of these things.

Costa and Kallick (2000) say that as students increase in their problem solving ability,

they become more flexible in their thinking. They consider, express or paraphrase other points of

The development of good problem solving techniques takes time, however. Ponte (2007)

cites some examples in Portugal in which problem solving or mathematical investigations were

used in a school setting. Having no one right answer seemed to generate some insecurity for

students. According to Ponte, as time went on, the activity improved in quality, and, with teacher

support and continuation of the work, student confidence in their abilities grew. The voiced

“unpleasantness” by some that the activities required high personal perseverance lessened.

Allowing students to struggle and develop persistence is not always easy for teachers

either. Ben-Hur (2006) believes that among teachers are two camps of thought when it comes to

allowing students to “struggle.” One camp seeks to take the shortcut of teaching key words,

algorithms and other tricks that work for given types of problems. He believes that this shelters

students from the uncertain nature of problem solving. The other camp of teachers seeks ways to

enhance reflective practices thus provoking students through use of cognitive dissonance. Wood

(2001) says, “In order to create these situations for mathematical learning in classrooms, teachers

must resist their natural inclination to tell students information, make the task simpler, or step in

and do part of the task” (p. 116). Therefore, to develop students who are persistent problem

solvers who take risks, teachers need to exhibit those qualities as well.

Fan and Zhu (2007) talk about a framework for problem solving modified from Polya’s

problem-solving model and published in a syllabus by the Ministry of Education in 1990. Its list

includes developing a plan, carrying out the plan and/or modifying the plan if necessary and

ending with seeking alternative solutions and checking for reasonableness. Students good at

problem solving do all of these things.

Costa and Kallick (2000) say that as students increase in their problem solving ability,

they become more flexible in their thinking. They consider, express or paraphrase other points of

13.
Traits of Good Mathematical Thinking 11

view, can state several ways of solving the same problem, and evaluate the merits of more than

one course of action. Students who have this habit of mind in place become systems thinkers.

They analyze and scrutinize parts, but also shift their perspective to the big picture.

The Australian Mathematics Education Program (AMEP), established by the Curriculum

Development Centre (CDC), in its first national statement of basic mathematical skills and

concepts (CDC, 1982) states,

Problem solving is the process of applying previously acquired knowledge in new

and unfamiliar situations. Being able to use mathematics to solve problems is a

major reason for studying mathematics at school. Students should have adequate

practice in developing a variety of problem solving strategies so they have

confidence in their use. (p. 3)

Good problem solvers do just that. When given an unfamiliar problem, they know what to do and

can switch strategies because they have an unofficial list of problem solving strategies to call

Successful problem solvers are agile users of what Schoenfeld (1994) calls the tools and

logic of mathematics. That ability is improved through the solving of “good problems.”

Schoenfeld defines a good problem:

Good problems can introduce students to fundamental ideas and to the

importance of mathematical reasoning and proof. Good problems can serve as

starting points for serious explorations and generalizations. Their solutions can

motivate students to value the processes of mathematical modeling and

abstraction and develop students’ competence with the tools and logic of

mathematics. (p. 60)

So, to be good at problem solving a student must exhibit the following: 1) show

confidence in solving problems; 2) demonstrate persistence when encountering a difficult

problem and refuses to give up; 3) when given an unfamiliar problem, knows what to do and can

switch strategies if one is not working; and 4) has an unofficial list of problem solving strategies

to call upon when solving problems.

view, can state several ways of solving the same problem, and evaluate the merits of more than

one course of action. Students who have this habit of mind in place become systems thinkers.

They analyze and scrutinize parts, but also shift their perspective to the big picture.

The Australian Mathematics Education Program (AMEP), established by the Curriculum

Development Centre (CDC), in its first national statement of basic mathematical skills and

concepts (CDC, 1982) states,

Problem solving is the process of applying previously acquired knowledge in new

and unfamiliar situations. Being able to use mathematics to solve problems is a

major reason for studying mathematics at school. Students should have adequate

practice in developing a variety of problem solving strategies so they have

confidence in their use. (p. 3)

Good problem solvers do just that. When given an unfamiliar problem, they know what to do and

can switch strategies because they have an unofficial list of problem solving strategies to call

Successful problem solvers are agile users of what Schoenfeld (1994) calls the tools and

logic of mathematics. That ability is improved through the solving of “good problems.”

Schoenfeld defines a good problem:

Good problems can introduce students to fundamental ideas and to the

importance of mathematical reasoning and proof. Good problems can serve as

starting points for serious explorations and generalizations. Their solutions can

motivate students to value the processes of mathematical modeling and

abstraction and develop students’ competence with the tools and logic of

mathematics. (p. 60)

So, to be good at problem solving a student must exhibit the following: 1) show

confidence in solving problems; 2) demonstrate persistence when encountering a difficult

problem and refuses to give up; 3) when given an unfamiliar problem, knows what to do and can

switch strategies if one is not working; and 4) has an unofficial list of problem solving strategies

to call upon when solving problems.

14.
Traits of Good Mathematical Thinking 12

The Process of Reasoning and Proof

Problem solving requires more than listing or summarizing an answer solution. In order

to help students think mathematically, they must be given opportunities to conjecture, test

these conjectures and prove or reason. This is the process of reasoning and proof. It is what

some other countries call a mathematical investigation that promotes learning mathematics

with understanding. Wood (2001) states,

Learning mathematics with understanding is thought to occur best in

situations in which children are expected to problem solve, reason, and

communicate their ideas and thinking to others. Moreover, it is thought

that situations of confusion and clash of ideas in which students are

allowed to struggle to resolution are precisely the settings that promote

learning with understanding. (p. 116)

Wood sees the heart of reform as a transformation in the ways students learn and teachers teach

mathematics and that the ways of learning and teaching result in students knowing a different

kind of school mathematics. One of its byproducts is a mathematics student who can reason. A

student who is good at reasoning can adequately explain his or her thinking and do more than

just list the procedure or summarize the answer.

A student who possesses good reasoning can use data to make, test, or argue a conjecture.

According to Diezmann, Watters and English (2001), a student with good reasoning is able to

speculate, test ideas and defend or argue them through contextualized problem solving tasks.

Segurado (1998) talks about a study of sixth grade students who had initial difficulties with

investigation activities but notes that the performance of the pupils evolved during the study,

citing improvement in their capacity to observe, conjecture, test and justify, as well as

communicate mathematically.

Ponte (2007) says these mathematical investigations should begin with a question that is

very general or from a set of little structured information from which one seeks to formulate a

The Process of Reasoning and Proof

Problem solving requires more than listing or summarizing an answer solution. In order

to help students think mathematically, they must be given opportunities to conjecture, test

these conjectures and prove or reason. This is the process of reasoning and proof. It is what

some other countries call a mathematical investigation that promotes learning mathematics

with understanding. Wood (2001) states,

Learning mathematics with understanding is thought to occur best in

situations in which children are expected to problem solve, reason, and

communicate their ideas and thinking to others. Moreover, it is thought

that situations of confusion and clash of ideas in which students are

allowed to struggle to resolution are precisely the settings that promote

learning with understanding. (p. 116)

Wood sees the heart of reform as a transformation in the ways students learn and teachers teach

mathematics and that the ways of learning and teaching result in students knowing a different

kind of school mathematics. One of its byproducts is a mathematics student who can reason. A

student who is good at reasoning can adequately explain his or her thinking and do more than

just list the procedure or summarize the answer.

A student who possesses good reasoning can use data to make, test, or argue a conjecture.

According to Diezmann, Watters and English (2001), a student with good reasoning is able to

speculate, test ideas and defend or argue them through contextualized problem solving tasks.

Segurado (1998) talks about a study of sixth grade students who had initial difficulties with

investigation activities but notes that the performance of the pupils evolved during the study,

citing improvement in their capacity to observe, conjecture, test and justify, as well as

communicate mathematically.

Ponte (2007) says these mathematical investigations should begin with a question that is

very general or from a set of little structured information from which one seeks to formulate a

15.
Traits of Good Mathematical Thinking 13

more precise question and produces a number of conjectures along the way. One tests these

conjectures, and in the process forms new questions or validates the first line of thinking. He

says problem solving investigations call for abilities that are beyond computation and

memorization and require higher order abilities related to communication, critical spirit,

modeling, data analysis, logical deduction and metacognition. Such learning of mathematics is

active learning, not passive. Ponte says the student is called to be an active participant in such a

problem. He or she is called on to be a mathematician, think for himself, evaluate decisions and

the work done.

Problem solving is a situation in which the role of the student and teacher might change.

Schoenfeld (2007) calls it a highly productive learning environment where students are

encouraged to take on intellectual problems, students are given authority in addressing such

problems, students are accountable, and students have adequate resources to do all of the above.

Wood (2001) states,

Mathematical reasoning best develops in classes that have highly interactive

situations and in which teachers make possible all students’ active participation in

the interaction and discourse. (p. 112)

Some believe mathematical reasoning requires direct instruction. Students who are unfamiliar

with reasoning and problem solving processes need direct instruction in how to reason. Ben-Hur

(2006) says that students who perform poorly need to learn how to process mathematics and that

they need instruction that targets the problem solving processes they fail to do efficiently and

that this instruction is too often absent.

A student who is good at mathematical reasoning uses a variety of reasoning methods and

proof and listens to others’ mathematical thinking. This is determined, in part, by the classroom

teacher and the classroom atmosphere. Yeo and Zhu (2005) recommend that classroom teachers

more precise question and produces a number of conjectures along the way. One tests these

conjectures, and in the process forms new questions or validates the first line of thinking. He

says problem solving investigations call for abilities that are beyond computation and

memorization and require higher order abilities related to communication, critical spirit,

modeling, data analysis, logical deduction and metacognition. Such learning of mathematics is

active learning, not passive. Ponte says the student is called to be an active participant in such a

problem. He or she is called on to be a mathematician, think for himself, evaluate decisions and

the work done.

Problem solving is a situation in which the role of the student and teacher might change.

Schoenfeld (2007) calls it a highly productive learning environment where students are

encouraged to take on intellectual problems, students are given authority in addressing such

problems, students are accountable, and students have adequate resources to do all of the above.

Wood (2001) states,

Mathematical reasoning best develops in classes that have highly interactive

situations and in which teachers make possible all students’ active participation in

the interaction and discourse. (p. 112)

Some believe mathematical reasoning requires direct instruction. Students who are unfamiliar

with reasoning and problem solving processes need direct instruction in how to reason. Ben-Hur

(2006) says that students who perform poorly need to learn how to process mathematics and that

they need instruction that targets the problem solving processes they fail to do efficiently and

that this instruction is too often absent.

A student who is good at mathematical reasoning uses a variety of reasoning methods and

proof and listens to others’ mathematical thinking. This is determined, in part, by the classroom

teacher and the classroom atmosphere. Yeo and Zhu (2005) recommend that classroom teachers

16.
Traits of Good Mathematical Thinking 14

try to establish a communicating environment for interaction that encourages students to verify,

question, criticize, and assess others’ arguments.

Students in tune with the characteristics of good reasoning ask good questions. Costa and

Kallick (2000) say these students link a sequence of questions to test hypotheses, guide data

searches, clarify outcomes or illuminate poor reasoning. They see the significance and power of

good questioning and that it can lead to better understanding.

In summary, those students successful at mathematical reasoning and proof can: 1) use

data to make, test, or argue a conjecture; 2) adequately explain the reasons behind his or her

mathematical thinking and can do more than just explain the procedure or summarize the answer;

3) use a variety of reasoning methods and proof; and 4) listen to others’ mathematical thinking.

The Communication Process

Problem solving and good mathematical reasoning are probably two of the most

important characteristics of a successful mathematical thinker. Another that is probably equally

important is mathematical communication. What makes a student a good communicator

mathematically? After 23 years in the classroom, I knew what it did not entail. A student who is

poor at communicating cannot explain his or her thinking. He or she does not have the ability to

justify with examples and does not see feedback as important.

Students who are successful at mathematical communication, however, seek clarification.

It happens as part of that communicating environment that Yeo and Zhu (2005) alluded to, that

allows for interaction and enables students to question, criticize, and clarify. It is part of a

community of learners Engle and Conant (2002) call sense-making communities- -highly

productive learning environments that can either support or inhibit the sense-making inclinations

in students. Ponte (2007) says it is in this struggle for explanation that clarification happens. The

try to establish a communicating environment for interaction that encourages students to verify,

question, criticize, and assess others’ arguments.

Students in tune with the characteristics of good reasoning ask good questions. Costa and

Kallick (2000) say these students link a sequence of questions to test hypotheses, guide data

searches, clarify outcomes or illuminate poor reasoning. They see the significance and power of

good questioning and that it can lead to better understanding.

In summary, those students successful at mathematical reasoning and proof can: 1) use

data to make, test, or argue a conjecture; 2) adequately explain the reasons behind his or her

mathematical thinking and can do more than just explain the procedure or summarize the answer;

3) use a variety of reasoning methods and proof; and 4) listen to others’ mathematical thinking.

The Communication Process

Problem solving and good mathematical reasoning are probably two of the most

important characteristics of a successful mathematical thinker. Another that is probably equally

important is mathematical communication. What makes a student a good communicator

mathematically? After 23 years in the classroom, I knew what it did not entail. A student who is

poor at communicating cannot explain his or her thinking. He or she does not have the ability to

justify with examples and does not see feedback as important.

Students who are successful at mathematical communication, however, seek clarification.

It happens as part of that communicating environment that Yeo and Zhu (2005) alluded to, that

allows for interaction and enables students to question, criticize, and clarify. It is part of a

community of learners Engle and Conant (2002) call sense-making communities- -highly

productive learning environments that can either support or inhibit the sense-making inclinations

in students. Ponte (2007) says it is in this struggle for explanation that clarification happens. The

17.
Traits of Good Mathematical Thinking 15

more that students are asked to do these kind of tasks, the more their approximation of what

makes a good mathematical thinker (and therefore what makes a good communicator) will

Costa and Kallick (2000) state that those who are successful at mathematical

communication understand that it is okay to struggle and to let others know when one is

struggling. They also mention that when others come up with new ways to solve a problem, good

communicators ask for an explanation or try to figure why that makes sense. They hear beyond

the words said to the mathematical meaning and can consider other ways to solve. They explain

They demonstrate their understanding and empathy for another person’s idea by

paraphrasing it accurately, building upon it, clarifying it, or giving an example of

it. We know students are listening to and internalizing others’ ideas and feelings .

. . After paraphrasing another person’s idea, a student may probe, clarify, or pose

questions that extend the idea further: ‘I’m not sure I understand. Can you explain

what you mean by . . .’(p.23-24)

The ability to explain what one is thinking mathematically and clarify one’s thinking and

the thinking of others will result in not only in an increase in understanding, but in the ability to

take risks. This however, depends on the classroom atmosphere. Wood (2001) says the

classroom needs to be an atmosphere of acceptance for all views that is not threatening and yet is

challenging to the students allowing them to struggle when appropriate.

A student who is successful at math communication 1)is able to explain his/her thinking

clearly and concisely; 2) seeks clarification; 3) realizes it is okay to struggle in math and make

mistakes; and 4) when others come up with new ideas, asks them to explain or tries to figure why

that makes sense.

The Process of Representation

Being able to get a clear idea of what a student is thinking is often difficult unless a good

explanation and representation of the solution is provided. Clarke, Goos and Morony (2007) say

more that students are asked to do these kind of tasks, the more their approximation of what

makes a good mathematical thinker (and therefore what makes a good communicator) will

Costa and Kallick (2000) state that those who are successful at mathematical

communication understand that it is okay to struggle and to let others know when one is

struggling. They also mention that when others come up with new ways to solve a problem, good

communicators ask for an explanation or try to figure why that makes sense. They hear beyond

the words said to the mathematical meaning and can consider other ways to solve. They explain

They demonstrate their understanding and empathy for another person’s idea by

paraphrasing it accurately, building upon it, clarifying it, or giving an example of

it. We know students are listening to and internalizing others’ ideas and feelings .

. . After paraphrasing another person’s idea, a student may probe, clarify, or pose

questions that extend the idea further: ‘I’m not sure I understand. Can you explain

what you mean by . . .’(p.23-24)

The ability to explain what one is thinking mathematically and clarify one’s thinking and

the thinking of others will result in not only in an increase in understanding, but in the ability to

take risks. This however, depends on the classroom atmosphere. Wood (2001) says the

classroom needs to be an atmosphere of acceptance for all views that is not threatening and yet is

challenging to the students allowing them to struggle when appropriate.

A student who is successful at math communication 1)is able to explain his/her thinking

clearly and concisely; 2) seeks clarification; 3) realizes it is okay to struggle in math and make

mistakes; and 4) when others come up with new ideas, asks them to explain or tries to figure why

that makes sense.

The Process of Representation

Being able to get a clear idea of what a student is thinking is often difficult unless a good

explanation and representation of the solution is provided. Clarke, Goos and Morony (2007) say

18.
Traits of Good Mathematical Thinking 16

that developing an appropriate visual representation of the information in a problem is crucial to

successful problem solving. This is another identifying characteristic of successful mathematical

thinking. Students need practice, however, in presenting and defending their answers and

repeated chances to show what they are thinking and how the problem was solved, if they are to

improve at this skill.

A successful math thinker has a variety of representation strategies in his/her repertoire

that he/she can call upon when needed. The Agenda for Action (NCTM, 1980) made as one of its

eight recommendations that problem solving should be expanded to include “a broad range of

strategies, processes, and modes of presentation that encompass the full potential of

mathematical applications” (p. 2). Those good at the process of representation have an unofficial

list of ways to present the problem and its solution that expresses thinking in a variety of ways

for example: words, drawings or pictures, charts or graphs, as well as written explanations. Costa

and Kallick (2000) say these kinds of thinkers use representation to help show exactly what he or

she was thinking when figuring out a problem and arriving at a solution. When confronted with a

problem, students who are good at representation suggest strategies for gathering data or for

solving the problem that may incorporate more than one method. Students who have found this

success can list the steps needed to solve a problem and can tell where they are in the sequence.

When asked to explain their solution, they can give their conclusion and describe the reasoning

process that brought them there. They can move easily from one kind of representation to

another and know the right or appropriate representation to use and when to use it.

A successful math student good at the process of representation: 1) has an unofficial list

of ways to represent a problem and its solution; 2) uses a range of representation in expressing

my thinking, (for instance- - words, drawings or pictures, charts or other graphs); 3) uses

that developing an appropriate visual representation of the information in a problem is crucial to

successful problem solving. This is another identifying characteristic of successful mathematical

thinking. Students need practice, however, in presenting and defending their answers and

repeated chances to show what they are thinking and how the problem was solved, if they are to

improve at this skill.

A successful math thinker has a variety of representation strategies in his/her repertoire

that he/she can call upon when needed. The Agenda for Action (NCTM, 1980) made as one of its

eight recommendations that problem solving should be expanded to include “a broad range of

strategies, processes, and modes of presentation that encompass the full potential of

mathematical applications” (p. 2). Those good at the process of representation have an unofficial

list of ways to present the problem and its solution that expresses thinking in a variety of ways

for example: words, drawings or pictures, charts or graphs, as well as written explanations. Costa

and Kallick (2000) say these kinds of thinkers use representation to help show exactly what he or

she was thinking when figuring out a problem and arriving at a solution. When confronted with a

problem, students who are good at representation suggest strategies for gathering data or for

solving the problem that may incorporate more than one method. Students who have found this

success can list the steps needed to solve a problem and can tell where they are in the sequence.

When asked to explain their solution, they can give their conclusion and describe the reasoning

process that brought them there. They can move easily from one kind of representation to

another and know the right or appropriate representation to use and when to use it.

A successful math student good at the process of representation: 1) has an unofficial list

of ways to represent a problem and its solution; 2) uses a range of representation in expressing

my thinking, (for instance- - words, drawings or pictures, charts or other graphs); 3) uses

19.
Traits of Good Mathematical Thinking 17

representation(s) to help others know exactly what he or she was thinking, how he or she figured

it out, and how the problem was solved; and 4) can move easily from one kind of representation

to another and knows the right or appropriate representation to use and when to use it.

Making Connections As A Process Skill

Problem Solving, Reasoning and Proof, Communication, and Representation all lead to

making better connections between mathematical problems and/or concepts. Schoenfeld (2007)

calls it sense-making and says that what is reflected in the current standards based curricula is an

understanding that a successful mathematical thinker can develop conceptual understanding in

the context of solving problems. According to Ben-Hur (2006),

Meaningless action can only reproduce, copy, or imitate other actions. It does not

result in transfer to other than identical situations. The meaningless repetition,

copying and imitation that are typical in mindless practice (and lack of thinking)

render students unable to know what to do with standardized test items that fall

outside those drills practiced. Meaningful learning results in conceptualization. (p.

32)

Successful mathematical thinking means noticing how ideas are related. Costa and

Kallick (2000) say it is making higher level connections that allows the student to draw forth a

mathematical event and apply it to a new context in a way that connects familiar ideas with new

concepts or skills. Ben-Hur (2006) states,

When it appears that students have grasped a new concept, the teacher must direct

them to apply the new concept consistently to new situations. New applications

shape and reinforce the new concepts. Adding variations to the concept helps the

learner to reach a greater generalization of the concept and to embrace a wider set

of possible applications. (p. 35)

In China, this is done by teaching with variation in which a series of related problems are

presented to students. Cai and Nie (2007) say the use of variations is not only an instructional

approach, but also an effective way to solve mathematical problems.

representation(s) to help others know exactly what he or she was thinking, how he or she figured

it out, and how the problem was solved; and 4) can move easily from one kind of representation

to another and knows the right or appropriate representation to use and when to use it.

Making Connections As A Process Skill

Problem Solving, Reasoning and Proof, Communication, and Representation all lead to

making better connections between mathematical problems and/or concepts. Schoenfeld (2007)

calls it sense-making and says that what is reflected in the current standards based curricula is an

understanding that a successful mathematical thinker can develop conceptual understanding in

the context of solving problems. According to Ben-Hur (2006),

Meaningless action can only reproduce, copy, or imitate other actions. It does not

result in transfer to other than identical situations. The meaningless repetition,

copying and imitation that are typical in mindless practice (and lack of thinking)

render students unable to know what to do with standardized test items that fall

outside those drills practiced. Meaningful learning results in conceptualization. (p.

32)

Successful mathematical thinking means noticing how ideas are related. Costa and

Kallick (2000) say it is making higher level connections that allows the student to draw forth a

mathematical event and apply it to a new context in a way that connects familiar ideas with new

concepts or skills. Ben-Hur (2006) states,

When it appears that students have grasped a new concept, the teacher must direct

them to apply the new concept consistently to new situations. New applications

shape and reinforce the new concepts. Adding variations to the concept helps the

learner to reach a greater generalization of the concept and to embrace a wider set

of possible applications. (p. 35)

In China, this is done by teaching with variation in which a series of related problems are

presented to students. Cai and Nie (2007) say the use of variations is not only an instructional

approach, but also an effective way to solve mathematical problems.

20.
Traits of Good Mathematical Thinking 18

Making good connections means seeing how mathematical concepts are connected to

others and to the real world. Abrantes et al. (1999) cite an initiative from Portugal called Project

Mathematics For All developed in 1990. They say that investigation activities in the curriculum

stimulate a holistic way of thinking that goes beyond application of knowledge or procedures in

isolation and implies the connection of ideas from different areas of mathematics. When asked

to make these higher level connections between concepts, however, students can struggle. Ponte

(2007) warns that these opportunities for students to consolidate their knowledge and undertake

new learning may highlight weak points in their thinking that may need to be addressed.

Costa and Kallick (2000) describe students good at making mathematical connections as

students who like to know when others think of a solution strategy in a different way. They say

these students are able to build upon, and consider the merits of another’s ideas. They reflect the

desire to understand how others are thinking and to keep making sense out of the problem or

Therefore, a student who is successful at making mathematical connections: 1) likes to see

how mathematical ideas are related; 2) connects new problems to old by asking, “Where have I

seen a problem like this before?”; 3) likes to see how mathematical ideas or concepts are

connected to other subjects and the real world; 4) can easily connect familiar ideas to new

concepts or skills; and 5)likes to know when others think of a solution strategy in a different way.

What did this mean for my classroom? There is a difference between having a list of

mathematical strategies to choose from and knowing when to use one or the other and having the

decision made for you. Making these connections is about seeing relationships and increasing the

level of learning but it takes time.

Making good connections means seeing how mathematical concepts are connected to

others and to the real world. Abrantes et al. (1999) cite an initiative from Portugal called Project

Mathematics For All developed in 1990. They say that investigation activities in the curriculum

stimulate a holistic way of thinking that goes beyond application of knowledge or procedures in

isolation and implies the connection of ideas from different areas of mathematics. When asked

to make these higher level connections between concepts, however, students can struggle. Ponte

(2007) warns that these opportunities for students to consolidate their knowledge and undertake

new learning may highlight weak points in their thinking that may need to be addressed.

Costa and Kallick (2000) describe students good at making mathematical connections as

students who like to know when others think of a solution strategy in a different way. They say

these students are able to build upon, and consider the merits of another’s ideas. They reflect the

desire to understand how others are thinking and to keep making sense out of the problem or

Therefore, a student who is successful at making mathematical connections: 1) likes to see

how mathematical ideas are related; 2) connects new problems to old by asking, “Where have I

seen a problem like this before?”; 3) likes to see how mathematical ideas or concepts are

connected to other subjects and the real world; 4) can easily connect familiar ideas to new

concepts or skills; and 5)likes to know when others think of a solution strategy in a different way.

What did this mean for my classroom? There is a difference between having a list of

mathematical strategies to choose from and knowing when to use one or the other and having the

decision made for you. Making these connections is about seeing relationships and increasing the

level of learning but it takes time.

21.
Traits of Good Mathematical Thinking 19

Ponte (2007) warns that a change to a curriculum that asks students to make conjectures,

and then postulate about them, defending and/or debating is very different from simple recall of

facts, figures and procedures. He notes that this change means students need time to understand.

Ben-Hur (2006) says that there needs to be varied and balanced attention of instructional time

spent on exercising and drilling procedural skills and time spent on discussion of concepts and

that these concepts cannot simply be passed from one person to another by talk. “Teachers must

not assume that meaning is transported from a speaker to a listener as if the language is fixed

somewhere outside its users.” (p. 34). He says that it is necessary to guide students’ reasoning

toward the accepted view through carefully thought out/guided questions, and by engaging

student in self-evaluation, and reflection.

Ponte (2007) attributes some of the difficulty to an initial conception by the students of

their role and the teacher’s role, the belief that there is always only one right answer and that it is

the teacher who establishes the validity. This began to change as time went on, but it changed

slowly. It is realized by researchers that “developing students’ ability in a higher level in solving

challenging mathematics problems could take a longer time than expected” (Fan & Zhu, 2007, p.

The research says it is important to allow time to discuss what students are learning and

to think about thinking, thereby making mathematical connections. This metacognition is seen by

Lester (1994) as the driving force behind problem solving and its influence on cognitive behavior

as well as student beliefs and attitudes. Lester is quick to caution the degree of influence of

metacognition, however, is not known for sure. It is generally accepted though, that “teaching

students to be more aware of their cognitions and better monitors of their problem-solving

Ponte (2007) warns that a change to a curriculum that asks students to make conjectures,

and then postulate about them, defending and/or debating is very different from simple recall of

facts, figures and procedures. He notes that this change means students need time to understand.

Ben-Hur (2006) says that there needs to be varied and balanced attention of instructional time

spent on exercising and drilling procedural skills and time spent on discussion of concepts and

that these concepts cannot simply be passed from one person to another by talk. “Teachers must

not assume that meaning is transported from a speaker to a listener as if the language is fixed

somewhere outside its users.” (p. 34). He says that it is necessary to guide students’ reasoning

toward the accepted view through carefully thought out/guided questions, and by engaging

student in self-evaluation, and reflection.

Ponte (2007) attributes some of the difficulty to an initial conception by the students of

their role and the teacher’s role, the belief that there is always only one right answer and that it is

the teacher who establishes the validity. This began to change as time went on, but it changed

slowly. It is realized by researchers that “developing students’ ability in a higher level in solving

challenging mathematics problems could take a longer time than expected” (Fan & Zhu, 2007, p.

The research says it is important to allow time to discuss what students are learning and

to think about thinking, thereby making mathematical connections. This metacognition is seen by

Lester (1994) as the driving force behind problem solving and its influence on cognitive behavior

as well as student beliefs and attitudes. Lester is quick to caution the degree of influence of

metacognition, however, is not known for sure. It is generally accepted though, that “teaching

students to be more aware of their cognitions and better monitors of their problem-solving

22.
Traits of Good Mathematical Thinking 20

actions should take place in the context of learning specific mathematics concepts and

techniques . . .” (Lester, 1994, p. 667).

In order for students to improve in problem solving, they need to learn what it is that

makes for good problem solving, or in other words what makes for good mathematical thinking.

Clarke, Goos and Morony (2007) call this working mathematically and refer to the

metacognition as cognitive engagement. Ben-Hur (2006) calls it concept-rich instruction, which

he says is founded on two key principles of 1) learning new concepts reflects a cognitive process

and 2) process involves reflective thinking which is greatly facilitated through mediated learning.

So, what did this mean for my research project? It became the reason I wanted to

investigate using problem solving to practice good mathematical thinking. I wanted to see if time

spent practicing, discussing and evaluating sample work could be used to promote a deeper,

higher level of thinking. Lovitt and Clarke (1988) promoted using problem solving as the most

effective way to teach. It was seen as a teaching methodology that involves teaching through

applications and modeling through which students learn by grappling with real world problems.

That is what I hoped to do—use problem solving to solve non-routine problems, develop good

problem solving habits and representation, learn more about problem solving strategies at the

same time, and think about as well as discuss, these experiences thereby promoting

communication and mathematical connections as well.

I would use the lists I had noted for each of the 5 process standards of 1) Problem

Solving, 2) Reasoning and Proof, 3) Communication, 4) Representation, and 5) Connections to

develop rubrics I could use with my students to teach for and develop the characteristics of each

of these processes.

actions should take place in the context of learning specific mathematics concepts and

techniques . . .” (Lester, 1994, p. 667).

In order for students to improve in problem solving, they need to learn what it is that

makes for good problem solving, or in other words what makes for good mathematical thinking.

Clarke, Goos and Morony (2007) call this working mathematically and refer to the

metacognition as cognitive engagement. Ben-Hur (2006) calls it concept-rich instruction, which

he says is founded on two key principles of 1) learning new concepts reflects a cognitive process

and 2) process involves reflective thinking which is greatly facilitated through mediated learning.

So, what did this mean for my research project? It became the reason I wanted to

investigate using problem solving to practice good mathematical thinking. I wanted to see if time

spent practicing, discussing and evaluating sample work could be used to promote a deeper,

higher level of thinking. Lovitt and Clarke (1988) promoted using problem solving as the most

effective way to teach. It was seen as a teaching methodology that involves teaching through

applications and modeling through which students learn by grappling with real world problems.

That is what I hoped to do—use problem solving to solve non-routine problems, develop good

problem solving habits and representation, learn more about problem solving strategies at the

same time, and think about as well as discuss, these experiences thereby promoting

communication and mathematical connections as well.

I would use the lists I had noted for each of the 5 process standards of 1) Problem

Solving, 2) Reasoning and Proof, 3) Communication, 4) Representation, and 5) Connections to

develop rubrics I could use with my students to teach for and develop the characteristics of each

of these processes.

23.
Traits of Good Mathematical Thinking 21

I wanted to focus on real-life problem solving situations that would ask students to apply

mathematic skill and yet also have real meaning for them. I knew that this would be a difficult

task for some because it wasn’t the kind of mathematics they were used to and it might not be

apparent to students why the struggle was necessary. Would they be able to trust me in that

PURPOSE STATEMENT

My study was to determine if focusing on the key traits of a mathematical thinker

namely: Communication, Representation, Reasoning and Proof, Problem Solving and

Connections and learning about the characteristics I had come up with for each of these five

processes would improve my students’ mathematical thinking. Based on the literature review,

the following master list of processes would be used to create rubrics for student use (see Figure

I investigated whether these items could be taught to students using a systematic,

organized approach with carefully selected problems and by providing students with specific

support structures to help them learn how to model their mathematical thinking. I provided

rubrics or checklists for students to use, used group work, spent time modeling the thinking or

metacognition involved, and specifically chose examples of student work that exemplified the

good, fair and poor aspects of a solution and the reasons why.

I wanted to understand whether and how mathematical thinking could be taught in ways

similar to how teachers try to use Six Traits Plus One to teach better writing.

I wanted to focus on real-life problem solving situations that would ask students to apply

mathematic skill and yet also have real meaning for them. I knew that this would be a difficult

task for some because it wasn’t the kind of mathematics they were used to and it might not be

apparent to students why the struggle was necessary. Would they be able to trust me in that

PURPOSE STATEMENT

My study was to determine if focusing on the key traits of a mathematical thinker

namely: Communication, Representation, Reasoning and Proof, Problem Solving and

Connections and learning about the characteristics I had come up with for each of these five

processes would improve my students’ mathematical thinking. Based on the literature review,

the following master list of processes would be used to create rubrics for student use (see Figure

I investigated whether these items could be taught to students using a systematic,

organized approach with carefully selected problems and by providing students with specific

support structures to help them learn how to model their mathematical thinking. I provided

rubrics or checklists for students to use, used group work, spent time modeling the thinking or

metacognition involved, and specifically chose examples of student work that exemplified the

good, fair and poor aspects of a solution and the reasons why.

I wanted to understand whether and how mathematical thinking could be taught in ways

similar to how teachers try to use Six Traits Plus One to teach better writing.

24.
Traits of Good Mathematical Thinking 22

Characteristics of the Five Processes of a Mathematical Thinker

Process 1 Connections- A student who is successful at making mathematical connections - -

o likes to see how mathematical ideas are related.

o connects new problems to old by asking, “Where have I seen a problem like this

before?”

o likes to see how mathematical ideas or concepts are connected to other subjects and the

real world.

o can easily connect familiar ideas to new concepts or skills.

o likes to know when others think of a solution strategy in a different way.

Process 2 Representation—A student who is successful at representation - -

o has an unofficial list of ways to represent a problem and its solution.

o uses a range of representation in expressing my thinking, (words, drawings or pictures,

charts or other graphs . . . )

o uses representation(s) to help others know exactly what he/she was thinking, how

he/she figured it out, and how the problem was solved.

o can move easily from one kind of representation to another and knows the right or

appropriate representation to use and when to use it.

Process 3 Communication- - A student who is successful at communicating mathematically- -

o is able to explain his/her thinking clearly and concisely.

o seeks clarification.

o realizes it is okay to struggle in math and make mistakes.

o when others come up with new ideas, asks them to explain or tries to figure why that

makes sense

Process 4 Reasoning and Proof—A student who is successful at reasoning and proof- -

o Can use data to make, test, or argue a conjecture.

o Can adequately explain the reasons behind his/her mathematical thinking and can do

o more than just explain the procedure or summarize the answer.

o Uses a variety of reasoning methods and proof.

o Listens to others mathematical thinking.

Process 5 Problem Solving- - A student who is a successful problem solver - -

o shows confidence is solving problems.

o demonstrates persistence when encountering a difficult problem and does not give up.

o when given an unfamiliar problem, knows what to do and can switch strategies if one

is not working.

o has an unofficial list of problem solving strategies to call upon when solving problems.

Figure 1 Master List of Processes of a Mathematical Thinker

METHOD

I started by choosing five problems from the notebook of sample problems my principal

had given me. It was a book of Exemplar Problems my school had purchased to use in the

classroom. Mine was problems for grades 5-8 with concepts students were expected to learn

sometime that year. These exemplars came with sample student answers at four proficiency

levels. I wanted to choose problems that would have some application to things the students

Characteristics of the Five Processes of a Mathematical Thinker

Process 1 Connections- A student who is successful at making mathematical connections - -

o likes to see how mathematical ideas are related.

o connects new problems to old by asking, “Where have I seen a problem like this

before?”

o likes to see how mathematical ideas or concepts are connected to other subjects and the

real world.

o can easily connect familiar ideas to new concepts or skills.

o likes to know when others think of a solution strategy in a different way.

Process 2 Representation—A student who is successful at representation - -

o has an unofficial list of ways to represent a problem and its solution.

o uses a range of representation in expressing my thinking, (words, drawings or pictures,

charts or other graphs . . . )

o uses representation(s) to help others know exactly what he/she was thinking, how

he/she figured it out, and how the problem was solved.

o can move easily from one kind of representation to another and knows the right or

appropriate representation to use and when to use it.

Process 3 Communication- - A student who is successful at communicating mathematically- -

o is able to explain his/her thinking clearly and concisely.

o seeks clarification.

o realizes it is okay to struggle in math and make mistakes.

o when others come up with new ideas, asks them to explain or tries to figure why that

makes sense

Process 4 Reasoning and Proof—A student who is successful at reasoning and proof- -

o Can use data to make, test, or argue a conjecture.

o Can adequately explain the reasons behind his/her mathematical thinking and can do

o more than just explain the procedure or summarize the answer.

o Uses a variety of reasoning methods and proof.

o Listens to others mathematical thinking.

Process 5 Problem Solving- - A student who is a successful problem solver - -

o shows confidence is solving problems.

o demonstrates persistence when encountering a difficult problem and does not give up.

o when given an unfamiliar problem, knows what to do and can switch strategies if one

is not working.

o has an unofficial list of problem solving strategies to call upon when solving problems.

Figure 1 Master List of Processes of a Mathematical Thinker

METHOD

I started by choosing five problems from the notebook of sample problems my principal

had given me. It was a book of Exemplar Problems my school had purchased to use in the

classroom. Mine was problems for grades 5-8 with concepts students were expected to learn

sometime that year. These exemplars came with sample student answers at four proficiency

levels. I wanted to choose problems that would have some application to things the students

25.
Traits of Good Mathematical Thinking 23

would learn in seventh grade and also interest them. Next I planned how to collect data and

lastly, how to present the problems to the students and gather information.

I gave individual folders to the students to hold their work. Each folder had a pocket for

the students to put their problem and any work they did and another for parents that would hold

any information I sent home. I created a personal letter to parents explaining the procedure, a

calendar, and a sample evaluation. I divided the research period into five 2-week sessions. Each

problem was allotted two weeks from start to finish. The first week students were given the

problem and asked to come up with individual ideas pertaining to the solution. We handed out

the problems on Monday and discussed on Friday. After Friday’s discussion, students revised

their first drafts. Parents knew these were handed out on a Monday and that students had to have

an initial guess at the solution by Friday. On Fridays, I asked for volunteers to give us their ideas.

I also asked for any questions the students had or clarified any concerns warranted. Problem 1

was the Lawn Mower Problem. It related to area and perimeter.

The second week of Problem One I taught a specific process. I decided to teach Problem

Solving each week along with one of the four other processes. I thought Problem Solving was

the hardest to evaluate. Most of what happens with this process needs to be observed when it

happens or is internal and difficult to identify, discuss and assess and so I wanted students to

have as much experience with Problem Solving as possible. For example in Problem One, I

taught students about Problem Solving and Representation. We spent time looking at the

characteristics of these two traits and during that second week, we also looked at sample student

work and rated their solutions according to the rubrics I created for that purpose. Time was spent

discussing what made a good solution, what the problem solving process looked like, and what

would learn in seventh grade and also interest them. Next I planned how to collect data and

lastly, how to present the problems to the students and gather information.

I gave individual folders to the students to hold their work. Each folder had a pocket for

the students to put their problem and any work they did and another for parents that would hold

any information I sent home. I created a personal letter to parents explaining the procedure, a

calendar, and a sample evaluation. I divided the research period into five 2-week sessions. Each

problem was allotted two weeks from start to finish. The first week students were given the

problem and asked to come up with individual ideas pertaining to the solution. We handed out

the problems on Monday and discussed on Friday. After Friday’s discussion, students revised

their first drafts. Parents knew these were handed out on a Monday and that students had to have

an initial guess at the solution by Friday. On Fridays, I asked for volunteers to give us their ideas.

I also asked for any questions the students had or clarified any concerns warranted. Problem 1

was the Lawn Mower Problem. It related to area and perimeter.

The second week of Problem One I taught a specific process. I decided to teach Problem

Solving each week along with one of the four other processes. I thought Problem Solving was

the hardest to evaluate. Most of what happens with this process needs to be observed when it

happens or is internal and difficult to identify, discuss and assess and so I wanted students to

have as much experience with Problem Solving as possible. For example in Problem One, I

taught students about Problem Solving and Representation. We spent time looking at the

characteristics of these two traits and during that second week, we also looked at sample student

work and rated their solutions according to the rubrics I created for that purpose. Time was spent

discussing what made a good solution, what the problem solving process looked like, and what

26.
Traits of Good Mathematical Thinking 24

made for good representation. Students were invited to revise their drafts of a solution and

resubmit by the following Monday.

The following week, we started over again with Problem Two. It was called Fair Game

and involved probability. Students were given the problem on Monday (when I collected their

final solutions from Problem One) and had until Friday to come up with an idea for a solution.

Problem Two’s second week I revisited the process of Problem Solving and taught about a new

one—Reasoning and Proof (See Appendix B-1 and 2 for the sample list and rubric used in class).

Each week I used the posters and checklists I had made to teach about the processes of

good mathematical thinking. Students also used these when evaluating the student work

examples for each of the problems. At the end of the second week of each problem, the students

filled out a learning log over that particular problem. The problems, the order in which they were

covered and the processes taught follows (See Figure 2).

Problem Mathematical Topic Processes Taught

#1 – Lawn Mower Problem Perimeter and Area Problem Solving &

Representation

#2 – Fair Game? Probability Problem Solving &

Reasoning / Proof

# 3 – Cake Decorating Problem Solving &

Pascal’s Triangle, Patterns

Dilemma Communication

Problem Solving &

#4 – Babbling Brook Patterns & Formulas

Connections

Pascal’s Triangle, Patterns &

#5 – House of Cards All 5

Formulas

See Appendix A-1 and A-2 for the five problems and corresponding learning logs

Figure 2 Problem and Topic Timeline

I had planned to focus on a new process/list of characteristics each time we started a new

problem so that by the end of the ten weeks, students would be familiar with all five methods.

During that time, I thought that we could evaluate each other’s work and our own work for the

made for good representation. Students were invited to revise their drafts of a solution and

resubmit by the following Monday.

The following week, we started over again with Problem Two. It was called Fair Game

and involved probability. Students were given the problem on Monday (when I collected their

final solutions from Problem One) and had until Friday to come up with an idea for a solution.

Problem Two’s second week I revisited the process of Problem Solving and taught about a new

one—Reasoning and Proof (See Appendix B-1 and 2 for the sample list and rubric used in class).

Each week I used the posters and checklists I had made to teach about the processes of

good mathematical thinking. Students also used these when evaluating the student work

examples for each of the problems. At the end of the second week of each problem, the students

filled out a learning log over that particular problem. The problems, the order in which they were

covered and the processes taught follows (See Figure 2).

Problem Mathematical Topic Processes Taught

#1 – Lawn Mower Problem Perimeter and Area Problem Solving &

Representation

#2 – Fair Game? Probability Problem Solving &

Reasoning / Proof

# 3 – Cake Decorating Problem Solving &

Pascal’s Triangle, Patterns

Dilemma Communication

Problem Solving &

#4 – Babbling Brook Patterns & Formulas

Connections

Pascal’s Triangle, Patterns &

#5 – House of Cards All 5

Formulas

See Appendix A-1 and A-2 for the five problems and corresponding learning logs

Figure 2 Problem and Topic Timeline

I had planned to focus on a new process/list of characteristics each time we started a new

problem so that by the end of the ten weeks, students would be familiar with all five methods.

During that time, I thought that we could evaluate each other’s work and our own work for the

27.
Traits of Good Mathematical Thinking 25

traits we were learning about but that was asking students to assimilate too much information in

too short of a time frame. It was asking a lot to require students to learn about a trait, internalize

that information and then apply it so quickly.

The first time we tried to rate each other’s oral observations I could tell I was asking way

too much and way too quickly. The students were having difficulty remembering what we had

discussed and were either acting very confused or going down the rubric choosing the highest

score without any thought whatsoever. Some also kept coming to me to have me define what

some of the words on the rubric meant. I knew then that the vocabulary was not student friendly

and needed to be modified and I needed the time to modify them. I wanted my students to not

only have the skills internalized to make correct decisions but also to take the time and thought

in evaluating oral explanations for it to have any real meaning. I made the executive decision to

delay evaluating our own work using the rubrics and would revisit after Problem 3.

The second time we tried oral observations was during the second week of Problem 3. I

split the students into four color coded teams. I chose a team leader for each group based on their

overall work ethic and level of cooperation. We discussed particular jobs for each group member

and concentrated on two traits only – Representation and Communication—two I felt that

students could rate easily according to the rubric because these two processes are particularly

easy to observe or identify according to our list of characteristics. Each group presented and

when finished, rated the other groups as a team. I rated each group as well and tallied the results.

Each group got an evaluation sheet from me with my comments and the scores from the other

groups for the two traits of Communication and Representation. The whole process went much

more smoothly this time.

traits we were learning about but that was asking students to assimilate too much information in

too short of a time frame. It was asking a lot to require students to learn about a trait, internalize

that information and then apply it so quickly.

The first time we tried to rate each other’s oral observations I could tell I was asking way

too much and way too quickly. The students were having difficulty remembering what we had

discussed and were either acting very confused or going down the rubric choosing the highest

score without any thought whatsoever. Some also kept coming to me to have me define what

some of the words on the rubric meant. I knew then that the vocabulary was not student friendly

and needed to be modified and I needed the time to modify them. I wanted my students to not

only have the skills internalized to make correct decisions but also to take the time and thought

in evaluating oral explanations for it to have any real meaning. I made the executive decision to

delay evaluating our own work using the rubrics and would revisit after Problem 3.

The second time we tried oral observations was during the second week of Problem 3. I

split the students into four color coded teams. I chose a team leader for each group based on their

overall work ethic and level of cooperation. We discussed particular jobs for each group member

and concentrated on two traits only – Representation and Communication—two I felt that

students could rate easily according to the rubric because these two processes are particularly

easy to observe or identify according to our list of characteristics. Each group presented and

when finished, rated the other groups as a team. I rated each group as well and tallied the results.

Each group got an evaluation sheet from me with my comments and the scores from the other

groups for the two traits of Communication and Representation. The whole process went much

more smoothly this time.

28.
Traits of Good Mathematical Thinking 26

The next time we worked on oral reasoning evaluation, we split into new color coded

groups with new team leaders for Problem 5/Week 10. After a brief discussion on what made a

good team leader, the old leaders picked new group leaders and then new groups were chosen.

This time after presentations of solutions, teams rated themselves. I also rated each group and

tallied the scores. We rated groups on four traits: Representation, Communication, Reasoning

and Proof, & Problem Solving. We worked with Connections the least, so we did not rate this

trait. I felt the students were really getting the hang of it! By the end of the project, we were

learning to evaluate when in small groups but were a long way from doing so individually.

These were my research questions and the data collection instruments I used for each:

(Figure 3)

What will happen to the quality of student written reasoning when students use a

rubric to evaluate their work?

Administration of a pre-, mid- Student Learning Log for Student work and scores of

and post- problem set. (See each of these problems.(See work for Problems 1, 3, and

Appendix A-1 for sample problems and C-1 Appendix A-2 for Learning Log example 5. (See Appen-dix C for scores &

for scores) and C-5 for scores)

Appendix D for sample student work)

What will happen to the level of student engagement in small group discussions when

using the five traits of mathematical thinking to solve problems?

Individual Interviews (See Pre and Post Survey (See Small Group Interviews (See

Appendix A-3 for sample interviews) Appendix A-5 for sample survey) Appendix A-4 for sample interview)

What will happen to the quality of student oral explanations of solutions when using

the traits of a mathematical thinker to guide student solutions?

Journal and/or anecdotal Students scored oral Peer evaluations of oral

records of class worked explanations according to explanations (See Appendix C-7

problems (See Appendix A-6 for rubric (See AppendixA-7 for sample oral for sample scores and rubric)

sample teacher journal) rubric & scores)

What will happen to my teaching when I specifically set aside time to teach traits of

mathematical thinking and deliberately spend time on mathematical discussion and

reasoning.

Journal and/or anecdotal records. (See Appendix A-6 for sample teacher journal)

Figure 3 Question and Instrument Table

The next time we worked on oral reasoning evaluation, we split into new color coded

groups with new team leaders for Problem 5/Week 10. After a brief discussion on what made a

good team leader, the old leaders picked new group leaders and then new groups were chosen.

This time after presentations of solutions, teams rated themselves. I also rated each group and

tallied the scores. We rated groups on four traits: Representation, Communication, Reasoning

and Proof, & Problem Solving. We worked with Connections the least, so we did not rate this

trait. I felt the students were really getting the hang of it! By the end of the project, we were

learning to evaluate when in small groups but were a long way from doing so individually.

These were my research questions and the data collection instruments I used for each:

(Figure 3)

What will happen to the quality of student written reasoning when students use a

rubric to evaluate their work?

Administration of a pre-, mid- Student Learning Log for Student work and scores of

and post- problem set. (See each of these problems.(See work for Problems 1, 3, and

Appendix A-1 for sample problems and C-1 Appendix A-2 for Learning Log example 5. (See Appen-dix C for scores &

for scores) and C-5 for scores)

Appendix D for sample student work)

What will happen to the level of student engagement in small group discussions when

using the five traits of mathematical thinking to solve problems?

Individual Interviews (See Pre and Post Survey (See Small Group Interviews (See

Appendix A-3 for sample interviews) Appendix A-5 for sample survey) Appendix A-4 for sample interview)

What will happen to the quality of student oral explanations of solutions when using

the traits of a mathematical thinker to guide student solutions?

Journal and/or anecdotal Students scored oral Peer evaluations of oral

records of class worked explanations according to explanations (See Appendix C-7

problems (See Appendix A-6 for rubric (See AppendixA-7 for sample oral for sample scores and rubric)

sample teacher journal) rubric & scores)

What will happen to my teaching when I specifically set aside time to teach traits of

mathematical thinking and deliberately spend time on mathematical discussion and

reasoning.

Journal and/or anecdotal records. (See Appendix A-6 for sample teacher journal)

Figure 3 Question and Instrument Table

29.
Traits of Good Mathematical Thinking 27

FINDINGS

What will happen to my teaching when I specifically set aside time to teach traits of

mathematical thinking and deliberately spend time on mathematical discussion and

reasoning?

Just as it took time for students to develop familiarity with the traits and how to use, it

took time to develop skill in teaching process skills. The more that I concentrated on this, the

more adaptations and changes I made in the process and the more honed my teaching skills

became. The change, however, was happening slowly.

My teacher journals showed that the time devoted to discussing and modeling a process

skill was not something I was used to doing. In my journals, I noted that even though I thought I

had a well laid out plan and purpose and structure and felt well prepared to teach what I had

planned, it was frustrating to put myself out there on a limb so to speak and try something new

(something I had not been taught before Math in the Middle classes). In my journal I wrote,

It is hard for me as a veteran teacher to do something so new that makes me feel like a

beginning teacher all over again. No one has ever taught me HOW to teach

mathematical thinking. Although I think this is the approach for MIM, it is hard for me

to teach it to students. I feel like I am constantly unprepared. I am also always stressed

about the time issue. I have limited amount of time to spend on this. I have limited

amount of time to spend teaching my PELS (Power Essential Learnings). I have my

other Math class and papers to check. This taking the time to journal and data collect is

also difficult. (Teacher Journal, Week of January 21, 2008)

My journals showed that what I had originally planned needed modification. I decided

almost right away that the rubrics I planned to use were not going to work. It was very slow

FINDINGS

What will happen to my teaching when I specifically set aside time to teach traits of

mathematical thinking and deliberately spend time on mathematical discussion and

reasoning?

Just as it took time for students to develop familiarity with the traits and how to use, it

took time to develop skill in teaching process skills. The more that I concentrated on this, the

more adaptations and changes I made in the process and the more honed my teaching skills

became. The change, however, was happening slowly.

My teacher journals showed that the time devoted to discussing and modeling a process

skill was not something I was used to doing. In my journals, I noted that even though I thought I

had a well laid out plan and purpose and structure and felt well prepared to teach what I had

planned, it was frustrating to put myself out there on a limb so to speak and try something new

(something I had not been taught before Math in the Middle classes). In my journal I wrote,

It is hard for me as a veteran teacher to do something so new that makes me feel like a

beginning teacher all over again. No one has ever taught me HOW to teach

mathematical thinking. Although I think this is the approach for MIM, it is hard for me

to teach it to students. I feel like I am constantly unprepared. I am also always stressed

about the time issue. I have limited amount of time to spend on this. I have limited

amount of time to spend teaching my PELS (Power Essential Learnings). I have my

other Math class and papers to check. This taking the time to journal and data collect is

also difficult. (Teacher Journal, Week of January 21, 2008)

My journals showed that what I had originally planned needed modification. I decided

almost right away that the rubrics I planned to use were not going to work. It was very slow

30.
Traits of Good Mathematical Thinking 28

going the first time we looked at them. I was forever explaining what the words meant. I thought

they could be made to be more student friendly. I state in a journal entry,

The time spent on the rubrics already tells me that I am going to have to change the

vocabulary. I spent a lot of time explaining and re-explaining the same words over and

over. Next time we try to use, we will do as a class. I will read, translate, and then they

can mark as we go along. Didn’t realize I was so far off on the wording. . . When the

vocabulary is too difficult, (think of it as the vehicle I am using) the journey is going to

be long and difficult and slow going. (Teacher Journal, Week of January 21, 2008)

In another I stated,

I was going to try to have the students rate each other using the rubrics for the traits

we’ve discussed and rate each other’s presentations (moving to peer and self evaluation

eventually) but they are still so ‘new’ to the process of presenting, I am going to hold off

on this for a bit although I am writing down notes about the quality of their explanations

myself. Also the rubrics I intended to use are still way off in vocabulary and right now

would only frustrate them more (Teacher Journal, Week of January 28, 2008).

I needed to learn to be patient and allow students time to internalize it all.

Facilitating this process and allowing students to experience and arrive at a place of

knowing is difficult and depends on time constraints. I noted that the first time we tried to

evaluate oral observations of each other, it just did not work for a variety of reasons. The

following are journal entries over this time period. I decided students needed something to help

them because applying what they were learning to their own work was proving difficult.

Transferring process skills. Hmmmm. What have I learned about transferring? I think the

biggest thing I’ve learned is that it is not as easy even for my high ability kids to do this

going the first time we looked at them. I was forever explaining what the words meant. I thought

they could be made to be more student friendly. I state in a journal entry,

The time spent on the rubrics already tells me that I am going to have to change the

vocabulary. I spent a lot of time explaining and re-explaining the same words over and

over. Next time we try to use, we will do as a class. I will read, translate, and then they

can mark as we go along. Didn’t realize I was so far off on the wording. . . When the

vocabulary is too difficult, (think of it as the vehicle I am using) the journey is going to

be long and difficult and slow going. (Teacher Journal, Week of January 21, 2008)

In another I stated,

I was going to try to have the students rate each other using the rubrics for the traits

we’ve discussed and rate each other’s presentations (moving to peer and self evaluation

eventually) but they are still so ‘new’ to the process of presenting, I am going to hold off

on this for a bit although I am writing down notes about the quality of their explanations

myself. Also the rubrics I intended to use are still way off in vocabulary and right now

would only frustrate them more (Teacher Journal, Week of January 28, 2008).

I needed to learn to be patient and allow students time to internalize it all.

Facilitating this process and allowing students to experience and arrive at a place of

knowing is difficult and depends on time constraints. I noted that the first time we tried to

evaluate oral observations of each other, it just did not work for a variety of reasons. The

following are journal entries over this time period. I decided students needed something to help

them because applying what they were learning to their own work was proving difficult.

Transferring process skills. Hmmmm. What have I learned about transferring? I think the

biggest thing I’ve learned is that it is not as easy even for my high ability kids to do this

31.
Traits of Good Mathematical Thinking 29

with unfamiliar concepts and the traits are new to them. (Teacher Journal, Week of

February 4, 2008)

I was asking students to digest a lot of information in a short amount of time.

We have not tried again to evaluate oral explanations of each other or self. I plan to get

back to, but it seems a good idea to wait since everything else is happening so slowly.

Also what with the vocabulary issues, I really need to stop and think about how to

change. Do the students need a checklist of some sort when they are working on trait

work? (Teacher Journal, Week of February 4, 2008)

I had originally planned to spend two weeks on each of the five processes I had identified

–solve the problem, discuss possible solutions and eventually rate our oral explanations and a

second week to look at sample student answers and rate according to the rubrics for the processes

learned so far. Two weeks for each of the five identified meant a total of ten weeks to learn, use

the trait and learn to evaluate in other’s work and our own. I found, however, that it was asking

the students to move too quickly. By the end of the research project, we had really only covered

three traits in any great detail. (See Appendix A-7 for a sample of the oral rubric used and A-8

for teacher page used for evaluation).

? What will happen to the quality of student written reasoning when students use a rubric

to evaluate their work?

It took time for students to move through the process of knowing about mathematical

traits, understanding what they were and then applying what they knew to evaluate their own

work and the work of others. Although this change was slow, it DID happen.

with unfamiliar concepts and the traits are new to them. (Teacher Journal, Week of

February 4, 2008)

I was asking students to digest a lot of information in a short amount of time.

We have not tried again to evaluate oral explanations of each other or self. I plan to get

back to, but it seems a good idea to wait since everything else is happening so slowly.

Also what with the vocabulary issues, I really need to stop and think about how to

change. Do the students need a checklist of some sort when they are working on trait

work? (Teacher Journal, Week of February 4, 2008)

I had originally planned to spend two weeks on each of the five processes I had identified

–solve the problem, discuss possible solutions and eventually rate our oral explanations and a

second week to look at sample student answers and rate according to the rubrics for the processes

learned so far. Two weeks for each of the five identified meant a total of ten weeks to learn, use

the trait and learn to evaluate in other’s work and our own. I found, however, that it was asking

the students to move too quickly. By the end of the research project, we had really only covered

three traits in any great detail. (See Appendix A-7 for a sample of the oral rubric used and A-8

for teacher page used for evaluation).

? What will happen to the quality of student written reasoning when students use a rubric

to evaluate their work?

It took time for students to move through the process of knowing about mathematical

traits, understanding what they were and then applying what they knew to evaluate their own

work and the work of others. Although this change was slow, it DID happen.

32.
Traits of Good Mathematical Thinking 30

Problem Name

Lawn Fair Cake Babbling House of

Mower Game Decorating Brook Cards

Total Processes Problem 1 Problem 2 Problem 3 Problem 4 Problem 5

Identified

More than 2 2 3 6 7 10

processes w/

acceptable

explanation

At least 1 or 2 with 11 10 9 10 9

an acceptable

explanation

No explanation 6 1 3 2 2

given or no

processes identified

Figure 4 Key Processes Identification

Change in what students knew about the traits was evident according to their learning log

answers. Students were asked on their learning logs for each problem to identify the traits

focused on for that particular problem and to list reasons why they thought so. In an analysis of

student learning log responses, I looked for the number of processes correctly identified that also

included an adequate explanation and split into three key groups—those who could identify and

explain more than two mathematical processes correctly, those who could correctly identify and

justify at least one or two processes, and those who did not have any process identified or could

not explain correctly. A comparison of processes identified and adequate reasons given is listed

in Figure 4.

Although change happened, it was slow and only for those items students had become

familiar with and were comfortable using. In looking over the learning logs for the first problem

(the Lawn Mower problem) and comparing it to the answers for each successive problem,

students were more likely to correctly identify the traits used and to give adequate reasons for

their thinking. By the third problem, three times as many students were able to correctly identify

three to five traits and provide acceptable explanations than did the first week we tried it. Also,

Problem Name

Lawn Fair Cake Babbling House of

Mower Game Decorating Brook Cards

Total Processes Problem 1 Problem 2 Problem 3 Problem 4 Problem 5

Identified

More than 2 2 3 6 7 10

processes w/

acceptable

explanation

At least 1 or 2 with 11 10 9 10 9

an acceptable

explanation

No explanation 6 1 3 2 2

given or no

processes identified

Figure 4 Key Processes Identification

Change in what students knew about the traits was evident according to their learning log

answers. Students were asked on their learning logs for each problem to identify the traits

focused on for that particular problem and to list reasons why they thought so. In an analysis of

student learning log responses, I looked for the number of processes correctly identified that also

included an adequate explanation and split into three key groups—those who could identify and

explain more than two mathematical processes correctly, those who could correctly identify and

justify at least one or two processes, and those who did not have any process identified or could

not explain correctly. A comparison of processes identified and adequate reasons given is listed

in Figure 4.

Although change happened, it was slow and only for those items students had become

familiar with and were comfortable using. In looking over the learning logs for the first problem

(the Lawn Mower problem) and comparing it to the answers for each successive problem,

students were more likely to correctly identify the traits used and to give adequate reasons for

their thinking. By the third problem, three times as many students were able to correctly identify

three to five traits and provide acceptable explanations than did the first week we tried it. Also,

33.
Traits of Good Mathematical Thinking 31

student answers showed that the traits they knew and could adequately explain were the ones

we’ve been discussing specifically Representation and Communication and Reasoning/Proof.

(See Appendix C-5 for Learning Log Scores by student)

One of the data collection instruments was to compare pre-, mid- and post- problem

scores. The scores were considered to be of either high, medium or low quality. Using a rubric to

evaluate the quality of problem solutions, I rated student work and split into three groups—high

scores of three, medium scores of two and low scores of one--according to their overall

performance on the mathematical processes we had learned about by that time (see Appendix A-

9 for rubric used for teacher evaluation).

Student Solution Scores

Problem #1 Problem #3 Problem #5

Lawn Mower Challenge Cake Decorating Dilemma House of Cards

Hi Medium Low Hi Medium Low Hi Medium Low

4 3 8 7 6 5 7 6 2

Figure 5 Solution Scores

Scores for the problems we worked showed some improvement. Seven out of 15 students

received a high or medium score on the first problem we did. On the third problem scored, 13

received a high or medium score and five a low score. By the fifth problem, 13 received a high

or medium score versus two who had a low score (See Appendix C-1 for all five problem scores

listed by student). Student solutions showed an application of what was learned and an increase

in familiarity with the processes each time we tried a new problem. Although not evident for all

students, scores showed that some students improved and could apply some of the new learning

to their work.

Comparison of the quality of solution answers for pre-, mid- and post problems indicated

some growth. Although small, this was easier to see when comparing the first problem attempted

student answers showed that the traits they knew and could adequately explain were the ones

we’ve been discussing specifically Representation and Communication and Reasoning/Proof.

(See Appendix C-5 for Learning Log Scores by student)

One of the data collection instruments was to compare pre-, mid- and post- problem

scores. The scores were considered to be of either high, medium or low quality. Using a rubric to

evaluate the quality of problem solutions, I rated student work and split into three groups—high

scores of three, medium scores of two and low scores of one--according to their overall

performance on the mathematical processes we had learned about by that time (see Appendix A-

9 for rubric used for teacher evaluation).

Student Solution Scores

Problem #1 Problem #3 Problem #5

Lawn Mower Challenge Cake Decorating Dilemma House of Cards

Hi Medium Low Hi Medium Low Hi Medium Low

4 3 8 7 6 5 7 6 2

Figure 5 Solution Scores

Scores for the problems we worked showed some improvement. Seven out of 15 students

received a high or medium score on the first problem we did. On the third problem scored, 13

received a high or medium score and five a low score. By the fifth problem, 13 received a high

or medium score versus two who had a low score (See Appendix C-1 for all five problem scores

listed by student). Student solutions showed an application of what was learned and an increase

in familiarity with the processes each time we tried a new problem. Although not evident for all

students, scores showed that some students improved and could apply some of the new learning

to their work.

Comparison of the quality of solution answers for pre-, mid- and post problems indicated

some growth. Although small, this was easier to see when comparing the first problem attempted

34.
Traits of Good Mathematical Thinking 32

and the last problem discussed. While not apparent for all students, more students came “on

board” each time we started a new two-week session. These changes included more detail in the

response, clear organization and/or structure, better or more complete representation, and

increased precision. Seven of the problems turned in on the Problem Two (Fair Game Dilemma)

had a clear format or structure to them including an introduction or restatement of the problem, a

diagram or drawing and explanation of how the answer was arrived at, a conclusion or answer to

the problem and possible summary. By the fifth problem (House of Cards) ten student solutions

had a specific format or organization to them and five were partially specific. These solutions

included clearly labeled sections: a title, an introduction or restatement of the problem, and a

solution section with diagram or chart and explanation. Five of these also included a summary or

reflection at the end. Many student answers improved the more we practiced. Several students

continued to include much more detail and were more precise in their explanations. Ellie’s 1

answer for Fair Game (Problem 2/Week 3):

Is it a fair game or not? To figure out the answer to this question, I made a simple table.

On the table, I drew dice that numbered one through six. I drew two sets of dice. One set

of dice was verticle (sic) and the other set was horizontal. Then, I added up the numbers

on the dice. For example, one and one was two, one and two was three . . . Since the

problem had to do with odd and even numbers, I decided to find out how many odd and

even numbers there were. First, I found out how many numbers there were all together.

There was 36. So that meant that there was 6 numbers . . I found out that there were 18

even numbers and 18 odd numbers. So the answer to the question is that it is a fair game.

This was part of the same person’s answer on Problem 1/Week 1:

All names are pseudonyms.

and the last problem discussed. While not apparent for all students, more students came “on

board” each time we started a new two-week session. These changes included more detail in the

response, clear organization and/or structure, better or more complete representation, and

increased precision. Seven of the problems turned in on the Problem Two (Fair Game Dilemma)

had a clear format or structure to them including an introduction or restatement of the problem, a

diagram or drawing and explanation of how the answer was arrived at, a conclusion or answer to

the problem and possible summary. By the fifth problem (House of Cards) ten student solutions

had a specific format or organization to them and five were partially specific. These solutions

included clearly labeled sections: a title, an introduction or restatement of the problem, and a

solution section with diagram or chart and explanation. Five of these also included a summary or

reflection at the end. Many student answers improved the more we practiced. Several students

continued to include much more detail and were more precise in their explanations. Ellie’s 1

answer for Fair Game (Problem 2/Week 3):

Is it a fair game or not? To figure out the answer to this question, I made a simple table.

On the table, I drew dice that numbered one through six. I drew two sets of dice. One set

of dice was verticle (sic) and the other set was horizontal. Then, I added up the numbers

on the dice. For example, one and one was two, one and two was three . . . Since the

problem had to do with odd and even numbers, I decided to find out how many odd and

even numbers there were. First, I found out how many numbers there were all together.

There was 36. So that meant that there was 6 numbers . . I found out that there were 18

even numbers and 18 odd numbers. So the answer to the question is that it is a fair game.

This was part of the same person’s answer on Problem 1/Week 1:

All names are pseudonyms.

35.
Traits of Good Mathematical Thinking 33

To try and find out how many trips Randy took, I subtracted four from 80 and 40, then

multiplied. I got an answer of 2,736. I subtracted four again from76 and 36, then

multiplied. I got 2,304. Then I subtracted four from 72 and 32 and multiplied again…I

then got 1600. Next I counted the trips he had made and I got 3 ¾ trips.

Ellie’s explanation was much clearer in the second attempt than in the first. She seemed

to do a better job of taking the reader through the process of what she was thinking at the time

she solved the problem on the second try. Her first problem attempt seems to be a list of

computational steps without an explanation of why she did the things she did and there does not

seem to be any overall goals in mind (See Appendix D-1 for all of Ellie’s work).

In looking at the students whose solutions included a conclusion or reflection on the fifth

and last problem and comparing to their first attempt, I could see more depth and detail in their

last problem’s answers. Although Fred’s work on the first problem was fairly high, his answer

on the last problem showed improvement in that he demonstrated the answer solution in three

ways. The questions asked for the number of cards in a house of cards that is five levels high, ten

levels high and n levels high. He used a picture to explain how to figure for three and five levels,

then made a t chart to show how to figure the answer for up to ten and then generalized to the

formula to show for n levels. He went on to prove this formula by working through it for the

number ten to justify his work on the chart. By showing part of his solution in three different

forms, Fred showed that he not only understood how to solve in more than one way, (and could

generalize for n levels) but also that he had an idea of what makes for good representation (See

Appendix D-1 for Fred’s work).

Fifia’s work on the first problem/week 1 showed a drawing and explanation of her work.

The problem asked if two people are mowing a lawn that is 40 by 80 feet and each wanted to

To try and find out how many trips Randy took, I subtracted four from 80 and 40, then

multiplied. I got an answer of 2,736. I subtracted four again from76 and 36, then

multiplied. I got 2,304. Then I subtracted four from 72 and 32 and multiplied again…I

then got 1600. Next I counted the trips he had made and I got 3 ¾ trips.

Ellie’s explanation was much clearer in the second attempt than in the first. She seemed

to do a better job of taking the reader through the process of what she was thinking at the time

she solved the problem on the second try. Her first problem attempt seems to be a list of

computational steps without an explanation of why she did the things she did and there does not

seem to be any overall goals in mind (See Appendix D-1 for all of Ellie’s work).

In looking at the students whose solutions included a conclusion or reflection on the fifth

and last problem and comparing to their first attempt, I could see more depth and detail in their

last problem’s answers. Although Fred’s work on the first problem was fairly high, his answer

on the last problem showed improvement in that he demonstrated the answer solution in three

ways. The questions asked for the number of cards in a house of cards that is five levels high, ten

levels high and n levels high. He used a picture to explain how to figure for three and five levels,

then made a t chart to show how to figure the answer for up to ten and then generalized to the

formula to show for n levels. He went on to prove this formula by working through it for the

number ten to justify his work on the chart. By showing part of his solution in three different

forms, Fred showed that he not only understood how to solve in more than one way, (and could

generalize for n levels) but also that he had an idea of what makes for good representation (See

Appendix D-1 for Fred’s work).

Fifia’s work on the first problem/week 1 showed a drawing and explanation of her work.

The problem asked if two people are mowing a lawn that is 40 by 80 feet and each wanted to

36.
Traits of Good Mathematical Thinking 34

mow half (and the lawn mower mowed a path that is two feet wide), how many trips would each

person take? Her explanation started out:

First on grid paper, I made my lawn 80 ft. by 40 ft. I labeled it and now I am going to get

to 1600 feet. Randy is going to make his first trip. So I went around and found how many

feet, and then I timesed [multiplied] it by 2 because the mower mows 2 feet at a time.

Then I had to add all the numbers together…

Fifia’s last problem/week 9 started with her own title, “Castle of Cards” and had Roman

Numeral Sections - - an introduction, a solution, an answer, a reflection and a second page with

her charts and diagram. Her answer included:

I used a chart for some of the easier levels (1-5) My chart showed how one side went up

by 1 while the other went up by 3. You just had to add up all the numbers before it to get

the total amount because the chart only shows the number of cards on that level. . . I used

my formula when we went to bigger numbers so I didn’t have to add so much. I also

drew a drawing…

Fifia’s second example shows a much better understanding of solution organization. Her

explanation also included more of the process she used to solve and WHY she chose to proceed

as she did (See Appendix D-1 for Fifia’s work).

What will happen to the level of student engagement in small group discussions when using

the five traits of mathematical thinking to solve problems?

Students liked the time in class spent working on the problem solving packets especially

when we worked in groups. This time together discussing and critiquing sample work increased

their interest level and participation. The answers to the small group interview questions showed

that students liked working with others because they got exposed to other viewpoints. During

group interviews when asked what they liked about working on the problems as a group. Sevie

mow half (and the lawn mower mowed a path that is two feet wide), how many trips would each

person take? Her explanation started out:

First on grid paper, I made my lawn 80 ft. by 40 ft. I labeled it and now I am going to get

to 1600 feet. Randy is going to make his first trip. So I went around and found how many

feet, and then I timesed [multiplied] it by 2 because the mower mows 2 feet at a time.

Then I had to add all the numbers together…

Fifia’s last problem/week 9 started with her own title, “Castle of Cards” and had Roman

Numeral Sections - - an introduction, a solution, an answer, a reflection and a second page with

her charts and diagram. Her answer included:

I used a chart for some of the easier levels (1-5) My chart showed how one side went up

by 1 while the other went up by 3. You just had to add up all the numbers before it to get

the total amount because the chart only shows the number of cards on that level. . . I used

my formula when we went to bigger numbers so I didn’t have to add so much. I also

drew a drawing…

Fifia’s second example shows a much better understanding of solution organization. Her

explanation also included more of the process she used to solve and WHY she chose to proceed

as she did (See Appendix D-1 for Fifia’s work).

What will happen to the level of student engagement in small group discussions when using

the five traits of mathematical thinking to solve problems?

Students liked the time in class spent working on the problem solving packets especially

when we worked in groups. This time together discussing and critiquing sample work increased

their interest level and participation. The answers to the small group interview questions showed

that students liked working with others because they got exposed to other viewpoints. During

group interviews when asked what they liked about working on the problems as a group. Sevie

37.
Traits of Good Mathematical Thinking 35

responded, “You have many ideas to choose from.” Trey agreed. “Yes. It’s not just one. You can

get different ideas of how to do not just how you did.” Eithia answered, “I like having to hear

other people’s opinions on what the answer is.” This was from a student with many learning

problems who is on medication and does not work well with others most of the time. Ted replied,

“You get more explanation from a group” (See Appendix D-2 for interview questions and lists of

Responses to individual interview questions indicated that working together was a way

for students to help each other “arrive” at solution answers that were reasonable and thorough. In

the individual interviews, 17 of the students interviewed said working on the problem solving

packets together as a group helped them get involved in learning and that they preferred it at

least part of the time to working individually. That was evenly represented by students who were

high ability as well as average and low. Fifia said, “You can tell them your ideas and they can

tell you theirs.” Forman replied, “Yes because if I do not know something I can get help.” When

asked what he liked about working on the packets as a group versus as an individual, Fred

answered, “I like working as a group to get the main idea and as an individual or small groups

you can do your own work or get into it farther.” Xavier said “Yes, because they can help me

and I can understand it more.” Fortran replied, “Yes, because we’re talking and discussing the

problems.” Sithe answered, “Working together- that makes me do the work with (a) group” (See

Appendix D-3 for questions and answers).

On the pre and post survey, students indicated they liked working on the problem solving

packets together in class and that they thought it important to listen to one another’s thinking.

For the pre survey students took during the first week of research, 13 out of 19 answered strongly

responded, “You have many ideas to choose from.” Trey agreed. “Yes. It’s not just one. You can

get different ideas of how to do not just how you did.” Eithia answered, “I like having to hear

other people’s opinions on what the answer is.” This was from a student with many learning

problems who is on medication and does not work well with others most of the time. Ted replied,

“You get more explanation from a group” (See Appendix D-2 for interview questions and lists of

Responses to individual interview questions indicated that working together was a way

for students to help each other “arrive” at solution answers that were reasonable and thorough. In

the individual interviews, 17 of the students interviewed said working on the problem solving

packets together as a group helped them get involved in learning and that they preferred it at

least part of the time to working individually. That was evenly represented by students who were

high ability as well as average and low. Fifia said, “You can tell them your ideas and they can

tell you theirs.” Forman replied, “Yes because if I do not know something I can get help.” When

asked what he liked about working on the packets as a group versus as an individual, Fred

answered, “I like working as a group to get the main idea and as an individual or small groups

you can do your own work or get into it farther.” Xavier said “Yes, because they can help me

and I can understand it more.” Fortran replied, “Yes, because we’re talking and discussing the

problems.” Sithe answered, “Working together- that makes me do the work with (a) group” (See

Appendix D-3 for questions and answers).

On the pre and post survey, students indicated they liked working on the problem solving

packets together in class and that they thought it important to listen to one another’s thinking.

For the pre survey students took during the first week of research, 13 out of 19 answered strongly

38.
Traits of Good Mathematical Thinking 36

agree or agree to the question, “I like to work on problem solving in math class.” Fourteen out of

21 agreed or strongly agreed on the post survey (see Fig 6).

15 Question 1 Post

14 Pre

13

I like to work 12

Number of Students

on problem 11

10

solving in 9

math class. 8

7

6 Post

5 Pre

4

3

2

1

0

Strongly Agreed or Strongly Disagreed Strongly Agreed or Strongly Disagreed

Agree or Disagreed Agree or Disagreed

Figure 6 Ques One

Eighteen students rated listening to others’ mathematical thinking as important to them

(strongly agree = 10; agree = 8; Total 19 students answered) on the pre survey. Eighteen still felt

the same way on the post survey.

19 Pre Question 6 Post

18

Listening to 17

others’ 16

15

Number of Students

mathematical 14

13

thinking is 12

important. 11

10

9

8

7

6

5

4

3

2

1 Pre Post

0

Strongly Agreed or Strongly Disagreed Strongly Agreed or Strongly Disagreed

Agree or Disagreed Agree or Disagreed

Figure 7 Ques 6 Answers

agree or agree to the question, “I like to work on problem solving in math class.” Fourteen out of

21 agreed or strongly agreed on the post survey (see Fig 6).

15 Question 1 Post

14 Pre

13

I like to work 12

Number of Students

on problem 11

10

solving in 9

math class. 8

7

6 Post

5 Pre

4

3

2

1

0

Strongly Agreed or Strongly Disagreed Strongly Agreed or Strongly Disagreed

Agree or Disagreed Agree or Disagreed

Figure 6 Ques One

Eighteen students rated listening to others’ mathematical thinking as important to them

(strongly agree = 10; agree = 8; Total 19 students answered) on the pre survey. Eighteen still felt

the same way on the post survey.

19 Pre Question 6 Post

18

Listening to 17

others’ 16

15

Number of Students

mathematical 14

13

thinking is 12

important. 11

10

9

8

7

6

5

4

3

2

1 Pre Post

0

Strongly Agreed or Strongly Disagreed Strongly Agreed or Strongly Disagreed

Agree or Disagreed Agree or Disagreed

Figure 7 Ques 6 Answers

39.
Traits of Good Mathematical Thinking 37

Fifteen students said they like to know when others think of a solution strategy in a different

way. (15 answered strongly agree or agree, 2 = neither agree or disagree). Eighteen answered

strongly agree or agree to this question on the post survey. (See Appendix C-2 for tallies of

results of Pre and Post Survey Questions and and C-3 for a comparison of answers).

19

I like to know

Question 11 Post

18 when others

17

16 Pre

think of a

15 solution

Number of Students

14

13 strategy in a

12 different way.

11

10

9

8

7

6

5

4

3 Pre

2 Post

1

0

Strongly Agreed or Strongly Disagreed Strongly Agreed or Strongly Disagreed

Agree or Disagreed Agree or Disagreed

Figure 8 Ques 11 Answers

More students indicated problem solving work as the time they were most involved on the post

survey than on the pre survey. Three had identified problem solving work in the classroom as the

time they were most involved on the pre survey. Ten said problem solving time on the post

survey. On the post survey six of the ten students who had said homework was the time they

were most involved on the pre survey, had changed their answers to during problem solving (See

Appendix C-2).

Fifteen students said they like to know when others think of a solution strategy in a different

way. (15 answered strongly agree or agree, 2 = neither agree or disagree). Eighteen answered

strongly agree or agree to this question on the post survey. (See Appendix C-2 for tallies of

results of Pre and Post Survey Questions and and C-3 for a comparison of answers).

19

I like to know

Question 11 Post

18 when others

17

16 Pre

think of a

15 solution

Number of Students

14

13 strategy in a

12 different way.

11

10

9

8

7

6

5

4

3 Pre

2 Post

1

0

Strongly Agreed or Strongly Disagreed Strongly Agreed or Strongly Disagreed

Agree or Disagreed Agree or Disagreed

Figure 8 Ques 11 Answers

More students indicated problem solving work as the time they were most involved on the post

survey than on the pre survey. Three had identified problem solving work in the classroom as the

time they were most involved on the pre survey. Ten said problem solving time on the post

survey. On the post survey six of the ten students who had said homework was the time they

were most involved on the pre survey, had changed their answers to during problem solving (See

Appendix C-2).

40.
Traits of Good Mathematical Thinking 38

During which

part of math

12 Pre/Post Survey Changes

class do you Post

11

feel the most Pre Post

involved? 10

9

Number of Students

8

7

6

5

4

Pre

3

2

1

0

Problem Solving Homework Problem Solving Homework

Figure 9 Student Engagement

What will happen to the quality of student oral explanations of solutions when using the traits

of a mathematical thinker to guide student solutions?

Students need more time to work on developing familiarity with the traits of a

mathematical thinker and feel more comfortable with using the information, before they can

apply this knowledge to rate each other’s oral explanations. My journals showed that knowing

about a mathematical trait and applying it by evaluating someone else’s work (or one’s own

work) is a higher level thinking skill that required time for the student to become familiar with

the trait’s characteristics and lots of practice using in order to use for evaluation purposes. I

Students seem to understand we will eventually get to different approaches (and the right

answers) and what high quality work looks like. (Teena says “mine’s not like that, is that

okay?”) They realize others tried the same wrong path first like putting candies in the

middle or trying to use color when counting the # of lines. (on the Cake Decorating

During which

part of math

12 Pre/Post Survey Changes

class do you Post

11

feel the most Pre Post

involved? 10

9

Number of Students

8

7

6

5

4

Pre

3

2

1

0

Problem Solving Homework Problem Solving Homework

Figure 9 Student Engagement

What will happen to the quality of student oral explanations of solutions when using the traits

of a mathematical thinker to guide student solutions?

Students need more time to work on developing familiarity with the traits of a

mathematical thinker and feel more comfortable with using the information, before they can

apply this knowledge to rate each other’s oral explanations. My journals showed that knowing

about a mathematical trait and applying it by evaluating someone else’s work (or one’s own

work) is a higher level thinking skill that required time for the student to become familiar with

the trait’s characteristics and lots of practice using in order to use for evaluation purposes. I

Students seem to understand we will eventually get to different approaches (and the right

answers) and what high quality work looks like. (Teena says “mine’s not like that, is that

okay?”) They realize others tried the same wrong path first like putting candies in the

middle or trying to use color when counting the # of lines. (on the Cake Decorating

41.
Traits of Good Mathematical Thinking 39

problem) I’m still concerned that some are parroting and not understanding.. (Teacher

Journal, Week of February 18, 2008)

And in another journal entry, I noted the same kind of thing,

I think the transference of this trait work and its application is going to involve a lot

of time and emphasis and may not happen as easily as I first thought. Silly me. I

thought I would teach them about the 5 traits, show them the rubrics, give them some

examples, that we’d discuss and then presto! Chango! It would magically appear in

their work. I am constantly reminded of Bloom and his taxonomy. These kids may

have the knowledge now or at least more of it but comprehending it and applying it

are stages of learning that need to be moved through and each student is going to

have to move through this (some slower than others) at his or her own pace. (Teacher

Journal, Week of January 28, 2008)

In another entry I wrote, “I chose this problem because we have been working on perimeter and

area and I thought it would make more sense. I am realizing that taking a computational skill and

transferring into a PROCESS skill is difficult” (Teacher Journal, Week of January 21, 2008).

Initial work on rating of oral reasoning skill (on Problem One), was asking students to

apply a newly learned skill too quickly. When students are not ready, this does not work well.

The first time we tried to rate each other’s work was on the first problem, the Lawn Mower

Problem. It kind of turned out like the answers on the pre-survey. Students thought their work

was good enough and did not need revision. I mentioned at that time that I wondered if it was

because students thought revising would mean more work. Also, even considering the difficulty

we had working through the rubrics because of the vocabulary, students took as little time (and

as little thought) as possible to complete and most everyone gave most everyone else perfect

problem) I’m still concerned that some are parroting and not understanding.. (Teacher

Journal, Week of February 18, 2008)

And in another journal entry, I noted the same kind of thing,

I think the transference of this trait work and its application is going to involve a lot

of time and emphasis and may not happen as easily as I first thought. Silly me. I

thought I would teach them about the 5 traits, show them the rubrics, give them some

examples, that we’d discuss and then presto! Chango! It would magically appear in

their work. I am constantly reminded of Bloom and his taxonomy. These kids may

have the knowledge now or at least more of it but comprehending it and applying it

are stages of learning that need to be moved through and each student is going to

have to move through this (some slower than others) at his or her own pace. (Teacher

Journal, Week of January 28, 2008)

In another entry I wrote, “I chose this problem because we have been working on perimeter and

area and I thought it would make more sense. I am realizing that taking a computational skill and

transferring into a PROCESS skill is difficult” (Teacher Journal, Week of January 21, 2008).

Initial work on rating of oral reasoning skill (on Problem One), was asking students to

apply a newly learned skill too quickly. When students are not ready, this does not work well.

The first time we tried to rate each other’s work was on the first problem, the Lawn Mower

Problem. It kind of turned out like the answers on the pre-survey. Students thought their work

was good enough and did not need revision. I mentioned at that time that I wondered if it was

because students thought revising would mean more work. Also, even considering the difficulty

we had working through the rubrics because of the vocabulary, students took as little time (and

as little thought) as possible to complete and most everyone gave most everyone else perfect

42.
Traits of Good Mathematical Thinking 40

scores of five on Representation and Problem Solving. I made the executive decision to toss

these and revise the rubrics. They simply were not ready and needed more experience before we

could try again.

For the third problem, we revisited the evaluation of oral explanations. I divided students

into color coded groups. Each had a team leader and they were given the revised (more student

friendly) rubrics for Representation, Communication and the newest one we have been working

on - - Reasoning and Proof. I observed each group and scored on Representation and

Communication and students scored each other. This has been the first time we have tried it

again since that first week. We spent a whole period preparing the presentations and a whole

another day (we have block schedule) presenting and scoring each other. Three of the four

groups got a total of 200 points or above on Representation (1 had 192) out of 225 total points

possible. Communication scores were not as good but better. Out of 175 total points, their scores

ranged from 131 to 154. Almost everyone on the teams presented or played a part in the

preparations except for a few new students and students who were absent the previous day (See

Appendix C-7 for oral reasoning averages for color coded groups on the Cake Decorating

Teams’ Representation strategies showed some attempts at a variety of methods to solve

and/or represent the answer including the use of a t chart, a drawing or diagram, a written

explanation of the process and the solution as well as the broader application to a formula

generalization. This variety of representation was not present for all groups, however.

As far as Communication was concerned, the work showed a need for ALL members to

participate equally and learn how to function as a group. Overall, communication seemed

segmented and lacking. Some group members made an attempt to explain or adequately

scores of five on Representation and Problem Solving. I made the executive decision to toss

these and revise the rubrics. They simply were not ready and needed more experience before we

could try again.

For the third problem, we revisited the evaluation of oral explanations. I divided students

into color coded groups. Each had a team leader and they were given the revised (more student

friendly) rubrics for Representation, Communication and the newest one we have been working

on - - Reasoning and Proof. I observed each group and scored on Representation and

Communication and students scored each other. This has been the first time we have tried it

again since that first week. We spent a whole period preparing the presentations and a whole

another day (we have block schedule) presenting and scoring each other. Three of the four

groups got a total of 200 points or above on Representation (1 had 192) out of 225 total points

possible. Communication scores were not as good but better. Out of 175 total points, their scores

ranged from 131 to 154. Almost everyone on the teams presented or played a part in the

preparations except for a few new students and students who were absent the previous day (See

Appendix C-7 for oral reasoning averages for color coded groups on the Cake Decorating

Teams’ Representation strategies showed some attempts at a variety of methods to solve

and/or represent the answer including the use of a t chart, a drawing or diagram, a written

explanation of the process and the solution as well as the broader application to a formula

generalization. This variety of representation was not present for all groups, however.

As far as Communication was concerned, the work showed a need for ALL members to

participate equally and learn how to function as a group. Overall, communication seemed

segmented and lacking. Some group members made an attempt to explain or adequately

43.
Traits of Good Mathematical Thinking 41

represent their solution but it was not cohesively presented in such a way to demonstrate the trait

of good mathematical communication.

We tried again in different color coded groups for the fifth and last problem. Again each

team had a leader. Students were given the rubrics for Representation, Communication, Problem

Solving, and Reasoning and Proof. I observed each group and rated them. Students also were

asked to evaluate themselves. This was the second time we have tried evaluating oral

observations. We spent a period preparing the presentations and another day presenting and

scoring. I compared the scores of Representation, Communication because that is what we had

done on the previous oral scoring. This time all of the four groups got a total of 200 points or

above on Representation (the lowest was 202) out of 225 total points possible. Three of the four

groups scored 160 or above on Communication. Out of 175 total points, their scores ranged from

141 to 166. Of the 21 students present, only one person did not play an active part in the

Presentations were much more organized and well rehearsed this time. The second round

had more overall participation from group members, and members seemed to do a better job of

not just telling what the solution was but representing how it was they got to the solution. All

groups represented their solution strategies in a variety of ways including t charts, drawings

and/or diagrams, written explanations of the process, the solution and a generalization to a

formula. The team work also showed more willingness to “go beyond” what was expected to

make their team approach different or unique and showed much more thought. Some groups

included a restatement of the problem at the beginning, a summary at the end and a reflection of

what other problems compared or an example of a harder version (See Appendix C-7 for oral

represent their solution but it was not cohesively presented in such a way to demonstrate the trait

of good mathematical communication.

We tried again in different color coded groups for the fifth and last problem. Again each

team had a leader. Students were given the rubrics for Representation, Communication, Problem

Solving, and Reasoning and Proof. I observed each group and rated them. Students also were

asked to evaluate themselves. This was the second time we have tried evaluating oral

observations. We spent a period preparing the presentations and another day presenting and

scoring. I compared the scores of Representation, Communication because that is what we had

done on the previous oral scoring. This time all of the four groups got a total of 200 points or

above on Representation (the lowest was 202) out of 225 total points possible. Three of the four

groups scored 160 or above on Communication. Out of 175 total points, their scores ranged from

141 to 166. Of the 21 students present, only one person did not play an active part in the

Presentations were much more organized and well rehearsed this time. The second round

had more overall participation from group members, and members seemed to do a better job of

not just telling what the solution was but representing how it was they got to the solution. All

groups represented their solution strategies in a variety of ways including t charts, drawings

and/or diagrams, written explanations of the process, the solution and a generalization to a

formula. The team work also showed more willingness to “go beyond” what was expected to

make their team approach different or unique and showed much more thought. Some groups

included a restatement of the problem at the beginning, a summary at the end and a reflection of

what other problems compared or an example of a harder version (See Appendix C-7 for oral

44.
Traits of Good Mathematical Thinking 42

reasoning averages for color coded groups on the House of Cards Problem and D-1 for sample

team work).

CONCLUSIONS

Mathematical reasoning was a complicated skill. It took lots of practice to become

familiar with the concepts. Before one could apply it to his or her work or evaluate it in someone

else’s work, time was essential to be able to walk through the process and not only learn about

reasoning, but understand it. Mathematical reasoning was harder for those less proficient in the

arithmetic part of mathematics and took longer to develop. It is as if they were concentrating so

hard on the individual parts, that they could not look up and see the big picture. I imagined it as a

new dance step I had taught them and now as they practiced, their head was down, and they were

looking at the footprints and were busy putting one foot in front of the other. For some students,

the why of the mathematical work we were doing, and the answer produced are just disconnected

steps in a process they had long given up understanding. One could also see that in their initial

learning log entries. It was as if they had decided, “If you tell me to add, I’ll add, but if you tell

me I need to subtract, then I’ll do that.”

Meaning was so important and so clearly tied to mathematical understanding. Written

and oral explanation was difficult for seventh graders to put into words. Sometimes the meaning

behind the mathematical operations was unclear. There was not always agreement between what

we had discussed and done (and what they had put down) and what they wrote. Even my

advanced students found it hard to explain why they got what they got. Again, this transfer or

internalizing of what we learned and then applying it was a complicated and time consuming

reasoning averages for color coded groups on the House of Cards Problem and D-1 for sample

team work).

CONCLUSIONS

Mathematical reasoning was a complicated skill. It took lots of practice to become

familiar with the concepts. Before one could apply it to his or her work or evaluate it in someone

else’s work, time was essential to be able to walk through the process and not only learn about

reasoning, but understand it. Mathematical reasoning was harder for those less proficient in the

arithmetic part of mathematics and took longer to develop. It is as if they were concentrating so

hard on the individual parts, that they could not look up and see the big picture. I imagined it as a

new dance step I had taught them and now as they practiced, their head was down, and they were

looking at the footprints and were busy putting one foot in front of the other. For some students,

the why of the mathematical work we were doing, and the answer produced are just disconnected

steps in a process they had long given up understanding. One could also see that in their initial

learning log entries. It was as if they had decided, “If you tell me to add, I’ll add, but if you tell

me I need to subtract, then I’ll do that.”

Meaning was so important and so clearly tied to mathematical understanding. Written

and oral explanation was difficult for seventh graders to put into words. Sometimes the meaning

behind the mathematical operations was unclear. There was not always agreement between what

we had discussed and done (and what they had put down) and what they wrote. Even my

advanced students found it hard to explain why they got what they got. Again, this transfer or

internalizing of what we learned and then applying it was a complicated and time consuming

45.
Traits of Good Mathematical Thinking 43

It was very different to get students to think about math THE PROCESS and not math

THE PRODUCT. Along the way, I had many “So, what’s the answer?” and “Am I right?” This

change in thinking took work and did not change overnight. It was especially surprising to me

however, because it was from some of my brightest students that some of the questions came. I

continued to model as much of the thinking and the process of problem solving as possible to

give students an insight into what was involved and asked them to do the same. I highly valued

time to discuss and learn from each other in the classroom and tried to use that time by asking

higher level questions of my students.

I believed that these students were demonstrating an overemphasis on the answer and an

under-emphasis on the “how” and the “why.” This led them into concentrating on writing down

the answers without supplying the work, or copying from someone else’s paper. By going over

the solutions the second week and concentrating on what made a good solution, I think I steered

my students around that issue.

I also think I saw students who did not see the need to be “involved” in the process. They

viewed grades and homework assignments as something that was done to them, instead of

something they needed to do for themselves. Getting them more involved by incorporating more

small group or whole group activities helped tremendously.

I came to believe the following to be true: 1) I needed to have faith in teaching process

skills and the resolve to spend the time on it, although difficult and time consuming; 2) modeling

the process and the thinking was key to help students learn how to do the process; 3) students

need “visual reminders” or checklists to help light their journey; and 4) although slow and

sometimes hard to see, growth and, therefore, positive change happened.

It was very different to get students to think about math THE PROCESS and not math

THE PRODUCT. Along the way, I had many “So, what’s the answer?” and “Am I right?” This

change in thinking took work and did not change overnight. It was especially surprising to me

however, because it was from some of my brightest students that some of the questions came. I

continued to model as much of the thinking and the process of problem solving as possible to

give students an insight into what was involved and asked them to do the same. I highly valued

time to discuss and learn from each other in the classroom and tried to use that time by asking

higher level questions of my students.

I believed that these students were demonstrating an overemphasis on the answer and an

under-emphasis on the “how” and the “why.” This led them into concentrating on writing down

the answers without supplying the work, or copying from someone else’s paper. By going over

the solutions the second week and concentrating on what made a good solution, I think I steered

my students around that issue.

I also think I saw students who did not see the need to be “involved” in the process. They

viewed grades and homework assignments as something that was done to them, instead of

something they needed to do for themselves. Getting them more involved by incorporating more

small group or whole group activities helped tremendously.

I came to believe the following to be true: 1) I needed to have faith in teaching process

skills and the resolve to spend the time on it, although difficult and time consuming; 2) modeling

the process and the thinking was key to help students learn how to do the process; 3) students

need “visual reminders” or checklists to help light their journey; and 4) although slow and

sometimes hard to see, growth and, therefore, positive change happened.

46.
Traits of Good Mathematical Thinking 44

IMPLICATIONS

So, what does this mean for me? I believe that I have established a case for more in-depth

study of problem solving within my mathematics classroom. The research discusses using

problem solving as a most effective way to teach. It was seen as a methodology that involves

teaching through modeling and applications through which students learn while trying to figure

out real world problems. That is what I hope to continue to do—use problem solving to solve

non-routine problems, develop good problem solving habits and representation, learn more about

problem solving strategies in the process, and think about as well as discuss these experiences

thereby promoting communication and mathematical connections as well.

Through careful and considerate continued research, it is my hope that I can continue to

explore the process of problem solving and provide some answers to what makes a good

mathematical thinker. The quotation may be 62 years old, but it is as true today as it was when

Polya (1945) said it,

Thus a teacher of mathematics has a great opportunity. If he fills his allotted time

with drilling his students in routine operations he kills their interest, hampers their

intellectual development, and misuses his opportunity. But if he challenges the

curiosity of his students by setting them problems proportionate to their

knowledge, and helps them to solve their problems with stimulating questions, he

may give them a taste for, and some means of, independent thinking. (p. V)

According to all that I have read and done, there are several major issues I would like to

keep in mind in the future. First I need to use criteria when choosing problems. I believe I need

to have some kind of idea of what makes a good problem and need to keep that in mind when

choosing the problems for my class to work on. Second, I need to use a rubric to “teach” problem

IMPLICATIONS

So, what does this mean for me? I believe that I have established a case for more in-depth

study of problem solving within my mathematics classroom. The research discusses using

problem solving as a most effective way to teach. It was seen as a methodology that involves

teaching through modeling and applications through which students learn while trying to figure

out real world problems. That is what I hope to continue to do—use problem solving to solve

non-routine problems, develop good problem solving habits and representation, learn more about

problem solving strategies in the process, and think about as well as discuss these experiences

thereby promoting communication and mathematical connections as well.

Through careful and considerate continued research, it is my hope that I can continue to

explore the process of problem solving and provide some answers to what makes a good

mathematical thinker. The quotation may be 62 years old, but it is as true today as it was when

Polya (1945) said it,

Thus a teacher of mathematics has a great opportunity. If he fills his allotted time

with drilling his students in routine operations he kills their interest, hampers their

intellectual development, and misuses his opportunity. But if he challenges the

curiosity of his students by setting them problems proportionate to their

knowledge, and helps them to solve their problems with stimulating questions, he

may give them a taste for, and some means of, independent thinking. (p. V)

According to all that I have read and done, there are several major issues I would like to

keep in mind in the future. First I need to use criteria when choosing problems. I believe I need

to have some kind of idea of what makes a good problem and need to keep that in mind when

choosing the problems for my class to work on. Second, I need to use a rubric to “teach” problem