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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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.
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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
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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
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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
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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.
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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,
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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
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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.
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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
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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
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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
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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
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).
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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
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
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
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
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
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.
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
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