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Using Reasoning Tasks to Develop Skills Necessary to Learn Independently
A Capstone Project
Submitted in Partial Fulfillment
of the Requirements for the Degree
of Master of Arts in Teaching: Mathematics
Fenecia Lynn Foster
Department of Mathematics and Computer Science
College of Arts and Sciences
Graduate School
Minot State University
Minot, North Dakota
Summer 2012
ii
This capstone project was submitted by
Fenecia Lynn Foster
Graduate Committee:
Dr. Laurie Geller, Chairperson
Title First and Last Name
Title First and Last Name
Dean of Graduate School
Dr. Linda Cresap
Date of defense: Month day, year
iii
Abstract
Type the abstract here. Do not indent. It should be one block paragraph. The
abstract is a summary of your paper.
iv
Acknowledgements
Type your acknowledgements here. Indent each paragraph 0.5 inch. You
can thank whomever you choose.
v
Table of Contents
Page
Abstract .................................................................................................................. iii
Acknowledgements ................................................................................................ iv
List of Tables ....................................................................................................... viii
List of Figures ........................................................................................................ ix
Chapter One: Introduction .......................................................................................1
Motivation for the Project ............................................................................1
Background on the Problem.........................................................................1
Statement of the Problem .............................................................................1
Statement of Purpose ...................................................................................2
Research Questions/Hypotheses ..................................................................2
Definitions....................................................................................................2
Summary ......................................................................................................2
Chapter Two: Review of Literature .........................................................................3
Heading One ................................................................................................3
Heading Two ................................................................................................3
Heading Three ..............................................................................................3
Heading Four ...............................................................................................3
Summary ......................................................................................................4
Chapter Three: Research Design and Method .........................................................5
vi
Setting ..........................................................................................................5
Intervention/Innovation................................................................................5
Design ..........................................................................................................5
Description of Methods................................................................................5
Analysis Strategy .........................................................................................6
Expected Results ..........................................................................................7
Timeline for the Study .................................................................................7
Summary ......................................................................................................7
Chapter Four: Results and Interpretations ...............................................................8
Results of Data Analysis ..............................................................................8
Interpretation of Results ...............................................................................8
Summary ......................................................................................................8
Chapter Five: Conclusions, Action Plan, Reflections, and Recommendations .....10
Conclusions ................................................................................................10
Action Plan.................................................................................................10
Reflections and Recommendations for Teachers .......................................10
Summary ....................................................................................................11
References ..............................................................................................................12
Appendices .............................................................................................................13
Appendix A: Title of Appendix A .............................................................14
Appendix B: Title of Appendix B ..............................................................15
vii
Appendix C: Title of Appendix C ..........................................................................16
viii
List of Tables
Table Page
1. Title of Table 1...........................................................................................xx
2. Title of Table 2...........................................................................................xx
3. Title of Table 3...........................................................................................xx
ix
List of Figures
Figure Page
1. Caption or title of Figure 1.........................................................................xx
2. Caption or title of Figure 2.........................................................................xx
3. Caption or title of Figure 3.........................................................................xx
Chapter One
Introduction
Life rarely takes place as anticipated and planned. People often encounter
unfamiliar and confusing issues, problems, and situations for which they are
challenged to respond, react, and creatively solve. I am beginning this project in
the midst of an unprecedented and devastating flood. This event has challenged
homeowners including myself to draw on logical reasoning skills and the ability
to think analytically during this lengthy period of uncertainty. This event has also
challenged a wide array of public officials to draw on their logical reasoning skills
as they make endless decisions in the best interest of the community and in an
attempt to predict potential issues that have not yet developed.
This flood has reminded me of the importance of being able to apply one’s
reasoning skills in order to navigate through unfamiliar and confusing situations.
Likewise, in the classroom, my students struggle with independently identifying
and applying their critical reasoning skills when required to explore unfamiliar
concepts. Thus, my project will focus on ways to equip my students with the
reasoning habits necessary for success inside and outside of the classroom.
Motivation for the Project
As I reflect on my first three years of teaching, one thing I would like to
improve upon is inspiring my students to become independent learners and
thinkers. I believe an independent learner is one who can patiently and
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confidently reason through a given task. As a teacher, it is my goal to equip my
students with reasoning strategies and habits that will help them approach the
multitude of issues and problems that they will face over the course of their lives.
I am interested in learning about research-based methods that will assist me in
developing these skills in my students. I believe that if my students can leave my
classroom equipped with reasoning habits then they will be able to successfully
approach the multitude of mysterious and confusing situations of which life is
comprised.
Background on the Problem
I teach in a small, rural school and I am the only math teacher for grades 7
through 12. My class sizes range from three students to 20 students per class. I
make a concerted effort to use many differentiated instruction techniques and
appeal to the students’ multiple intelligences. Yet despite my efforts to engage
my students in the learning process, the most common response to an
investigation or a problem set is a blank stare or an immediate raise of the hand.
Since my class sizes are so small I have the time to address each student
individually. My instincts draw me toward the raised hand and the pleading look;
thus, the very small student-to-teacher ratio enables a high-level of immediate
feedback and direction. Although this may appear beneficial to the learning
environment, students seldom experience the wait time necessary to develop their
own reasoning habits. My experience and observation has led me to believe that
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students in my school consistently demonstrate learned helplessness. When
encountering a task or problem, my students default to asking me rather than
determining how they can use their own knowledge, skills, and resources.
I would like to implement a strand of instructional activities for each
standard that will teach my students reasoning skills to help them become
successful independent learners. My goal is for the students to use these
reasoning habits to support their learning and understanding of mathematics and
then to apply these strategies to real life experiences outside of the classroom.
The world is dynamic. Many of my students will be involved in jobs and
careers that have yet to be invented. It is impossible for me to expose my students
to every possible mathematical skill and concept they will encounter during their
lives. But, I can help my students cultivate reasoning habits they can apply to
every situation they will encounter. The math classroom must be a venue where
―students learn to apply strategies in solving multi-part problems, establish
connections between multiple pieces of information, and use reasoning to
determine which tools are applicable and how to use them‖ (Achieve, 2008, pp. 6-
7). The development of logical reasoning skills in all students is imperative in
preparing students for citizenship, for the workplace, and for further study
(National Council of Teachers of Mathematics [NCTM], 2009).
13
Statement of the Problem
In my opinion, rote memorization and a discrete skill set no longer best
serve students. To be successful in the 21st century, students must be critical
thinkers, problem solvers, communicators, collaborators, information and
technology literate, flexible and adaptable, innovative and creative, effective
communicators, and initiators (Wagner, 2008).
Prior to the notion of 21st century skills, George Polya laid out a 4-step
problem solving process in his book, How to Solve It. The process is first to
understand the problem, second to devise a plan, third to carry out the plan, and
fourth to look back (Polya, 1957). My students demonstrate a general
understanding of this process, but they run into a roadblock when required to
independently apply the process. This roadblock prevents my students from being
successful mathematics students and from being able to apply their knowledge to
situations involving mathematics outside of the classroom.
My students struggle to understand a given problem and to identify and
apply appropriate reasoning strategies to solve the problem. If my students are
told which procedure to use on a given task, then they are often able to solve the
problem. In my classroom I must learn how to teach my students how to delve
into a problem from the beginning. I must learn how to cultivate reasoning habits
in my students.
14
Statement of Purpose
Using Polya’s 4-step process as the foundation, I will focus on developing
reasoning habits with my Algebra and Geometry students. The purpose of this
study is to use the NCTM’s (2009) reasoning habits to assist my students in
analyzing a problem, implementing a strategy, seeking and using connections, and
reflecting on a solution. The NCTM recommended the following as methods for
developing reasoning habits:
o Provide tasks that require students to figure things out for themselves.
o Ask students to restate the problem in their own words, including any
assumptions.
o Give students time to analyze a problem intuitively, explore the
problem further by using models, and then proceed to a more formal
approach.
o Resist the urge to tell students how to solve a problem when they
become frustrated; find other ways to support students.
o Ask students questions that will prompt their thinking.
o Provide adequate wait time after a question for students to formulate
their own reasoning.
o Encourage students to ask questions of themselves and one another.
o Expect students to communicate their reasoning orally and in writing.
o Highlight and reflect on exemplary explanations.
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o Establish a classroom climate in which students feel comfortable to
share and critique in a productive manner. (p. 11)
Utilizing the above strategies, my plan for cultivating reasoning habits
includes research and implementation. First, I will research instructional methods
for teaching reasoning habits, facilitating discussions, and asking engaging
questions. I will prepare discussion prompts and questions for each lesson that I
will have available to use instead of providing the students with answers. Second,
I will use the research-based methods to create lessons and tasks that will focus on
reasoning on a daily basis.
Research Questions/Hypotheses
Which methods can I use to cultivate reasoning habits within my students?
The NCTM (2009) recommended a number of strategies, but which methods will
work for my students? What will be the effects of the development of reasoning
habits with Algebra and Geometry students in decreasing the presence of learned
helplessness? My goal is to help my students become successful independent
learners who can analyze problems, implement strategies, make connections, and
reflect on the results.
Summary
The ability to draw on reasoning habits is imperative for success inside
and outside of the classroom. My students currently struggle to understand
problems and to determine how to begin to solve the given problem. Through a
16
focused effort to learn about instructional methods and to apply those methods in
the classroom with my students, my goal is to decrease the presence of learned
helplessness in my classroom and to develop independent learners who have the
ability to reason through any given task. In the next chapter I discuss research
findings regarding students’ abilities to reason and solve problems as well as the
needs of the students and the role of the teacher in cultivating reasoning habits.
Chapter Two
Review of Literature
A veteran teacher stated that learning began to happen when ―I shifted my
focus from trying to manipulate my students to learn to showing them how to
learn and helping them see the value in learning‖ (Jackson, 2009, p. xiii). As a
teacher, my goal is for my students to learn how to learn. I want to determine
whether a focused effort to develop reasoning habits improves a student’s ability
to learn independently. In this chapter I described the current research with
regard to students’ abilities to reason and solve problems. I summarized the needs
of students to be prepared for college and careers and the role of teacher when
developing reasoning habits. Finally, I presented the benefits of teaching
reasoning habits to students.
Current Research
Problem solving in the mathematics classroom involves engaging in tasks
that promote conceptual understanding, foster the ability to reason and
communicate mathematically, and capture interest and curiosity (Marcus & Fey,
2003). While it may seem practical to teach problem-solving skills and strategies
in isolation, no evidence supports the effectiveness of this practice. In fact, when
a group of teachers in several studies emphasized problem solving over skills,
there was no change in their students’ computational performance (National
Research Council [NRC], 2001). Rather, problem solving and reasoning ought to
18
be embedded in tasks that involve relevant math concepts and skills in an
intriguing, speculative, and challenging manner (Lester & Charles, 2003; Schoen
& Charles, 2003). Effective teaching can take a variety of forms as the teacher,
the student, and the content interact. Problem solving and reasoning are most
effectively taught through a combination of tasks and discourse that emphasize
multiple solution strategies, engaging explorations, giving reasons for solutions,
and making generalizations (Cai & Lester, 2010).
Recent research and publications by the National Research Council’s
(2001) Center for Education, the NCTM (2005), and the Common Core State
Standards Initiative (CCSS) (2010) emphasized the importance of mathematical
thinking. The NRC highlighted five strands for mathematical proficiency that
included conceptual understanding, procedural fluency, strategic competence,
adaptive reasoning, and productive disposition. The NCTM emphasized the five
process standards that included problem solving, reasoning and proof,
communication, connections, and representations. The CCSS combined the work
of both organizations and developed standards for mathematical practice and
content. The standards for mathematical practice described skills and abilities
that students need to possess in order to be successful. Students need to be able to
make sense of problems and persevere in solving them, reason abstractly and
quantitatively, construct viable arguments and critique the reasoning of others,
model with mathematics, use appropriate tools strategically, attend to precision,
19
look for and make use of structure, and look for and express regularity in repeated
reasoning. The stances of these three organizations identify the priority of
cultivating reasoning habits within students.
Research on the Needs of the Student
Educators have been charged with the task of preparing students to be
college and career ready. Thus, a common topic of study and discussion is what it
means to be college and career ready. Educators ask, ―How do we prepare
students for jobs that don’t exist? How do we teach our students knowledge that
we’ve not yet discovered?‖ Many (Erwin, 2004; Wagner, 2008; Zhao, 2009)
replied that students must be taught 21st century skills in order to be successful.
That skill set includes being able to analyze, synthesize, evaluate, compare and
contrast, manipulate, and apply information individually along with the ability to
collaborate and communicate with others (Erwin, 2004). Students must be critical
thinkers, problem solvers, communicators, collaborators, information and
technology literate, flexible and adaptable, innovative and creative, effective
communicators, and initiators (Wagner, 2008).
Students must be able to offer skills that are in demand and adapt to a
changing society. Skills in demand today may become irrelevant in the future
(Zhao, 2009). Since useful knowledge changes as societies change, education
must also reflect the change and move from teaching in the information age to
teaching in the conceptual age. Daniel Pink (2006) emphasized tapping into the
20
right brain abilities to design, tell stories, symphony or put the big picture
together, empathize, play, and pursue meaning.
The NCTM (2005) connected the 21st century skills to mathematics in its
Principles and Standards for School Mathematics, specifically through the
learning principle. Students must be flexible in their learning and possess
conceptual knowledge that can respond to an increasingly technological world.
The mathematics classroom is the ideal venue to develop the skills students need
to be successful. Samuel Otten (2011) echoed this sentiment:
Countless benefits arise from the ability to recognize the crucial features
of a problem, to uncover the latent assumptions at play, to think carefully
and without fallacy, to devise symbols and diagrams that aid such
thinking, and to communicate clearly and precisely, all of which can be
cultivated in the mathematics classroom. (p. 23)
The mathematics classroom is a place where students can encounter the
unfamiliar and use their curiosity to persist in finding a solution.
Jeremy Kilpatrick (1983) stated, ―The quality of the lives our citizens lead
depends on whether they are equipped with mathematical tools for thinking about
problems that confront them‖ (p. 306). A specific skill necessary for success is
the ability to reason through a problem. Reasoning involves analyzing the
problem, implementing a strategy, seeking and using connections, and reflecting
on the solution (NCTM, 2009). Reasoning is more that memorizing; it involves
21
cognitive skills such as problem solving and critical thinking (Zhao, 2009). While
it is also possible to reduce problem solving to a set of strategies, it is imperative
to recognize that reasoning and problem solving are about interpreting,
describing, explaining, and modeling situations and not simply knowing which
strategy to use for a particular situation (DeMatteo, 2010).
Reasoning requires action. Learners must be actively engaged in the
processing of information (Marzano, 2007). ―Fostering depth of students’
mathematical knowledge requires classrooms in which students are actively
involved in solving problems that require them to make connections among
content areas and to develop mathematical reasoning habits‖ (NCTM, 2009, p.
101). Reasoning is rooted in context. The CCSS (2010) called for students to be
able to reason abstractly and quantitatively, meaning that students need to be able
to decontextualize and contextualize as they work with a given problem.
Reasoning is also messy. In order to learn how to reason, students need to
experience messy, meaningful mathematical situations and be challenged to
construct their own understandings (DeMatteo, 2010). Students must be
challenged to struggle.
Research on the Role of the Teacher
An effective teacher holds high expectations of her students, motivates her
students to value learning, and is versatile in her methods (NRC, 2001). These
three facets of effective teaching relate to the NCTM’s (2009) recommended
22
strategies for developing reasoning habits. Many of those strategies involve the
role of questions for teachers and students. Teachers must resist the urge to give
answers when a student becomes frustrated. A teacher should be a ―coach,
mentor, and facilitator, not a purveyor of information‖ (Collins, 2010, p. 43). It is
also necessary to provide wait time following a question to give students time to
formulate their own reasoning as ―instant answers rob students of the gift of
pondering‖ (Hirsch, 2010, p. 62).
Types of questions and questioning techniques are powerful teaching
strategies. Marzano (2007) stated, ―When used effectively, questioning
techniques can be one of the most flexible and adaptive tools in a teacher’s
arsenal‖ (p. 108). Questions ought to confront students with an authentic problem
that will stimulate their curiosity. Various types of open questions give students
the opportunity to elaborate, clarify, explain, identify relationships, and engage in
metacognition (Barell, 2003). A good question is one that has no immediate
answer and requires thinking, feeling, and application of previous knowledge.
Good questions ―engage our minds in complex processes of analysis – posing
problems and resolving them, uncovering unstated assumptions, and searching for
evidence that will lead us to logical, reasonable conclusions‖ (Barell, 2003, p. 80).
Student questions are opportunities to identify what students do and do not
understand (Cavey & Mahavier, 2010). A teacher’s response is often just as
critical as the questions since a response has the power to stifle or stimulate a
23
student’s reasoning (Cavey & Mahvier, 2010). Students must be encouraged to
ask probing questions (NCTM, 2009). In addition to asking, ―What information
is given? What am I trying to find? What else do I need to know?‖ students ought
to ask inferential questions of themselves such as ―Why would that be true?‖
(Marzano, 2007).
In order to develop reasoning habits, students must be given tasks and
opportunities to figure out things for themselves. Tasks are central to students’
learning as they shape the students’ opportunities to learn and their views of the
subject matter (NRC, 2001). Reasoning tasks need to ―promote sound and
significant math contents, reflect students’ understanding, interests, and
experiences, support the range of ways that diverse students learn math, engage
students’ intellect by requiring reasoning and problem solving, help students build
connections, promote communication‖ (NCTM, 2009, p. 102). The role of the
teacher is to develop, revise, and select tasks that promote the development of
reasoning habits and are mathematically meaningful.
Benefits of Reasoning Habits
The NCTM (2005) stated that when students are challenged with
reasoning tasks, they develop into autonomous learners who ―become confident in
their ability to tackle difficult problems, eager to figure things out on their own,
flexible in exploring mathematical ideas and trying alternative solution paths, and
willing to persevere‖ (p. 21).
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Summary
Twenty-first century skills such as the ability to think critically, solve
problems, communicate, collaborate, be innovate, and be create are imperative for
success. The mathematics classroom is the ideal environment to cultivate 21st
century skills, specifically the ability to reason. Recent literature recommends
developing reasoning skills through the use of questions and reasoning tasks. In
the next chapter I discuss the methodology I will use to implement the action
research project.
Chapter Three
Research Design and Method
This study will determine whether a focus on the cultivation of reasoning
habits results in the development of independent learners. During the study, I will
implement a strand of research-based instructional activities for each standard in
the Algebra and Geometry classes that focuses on reasoning. In this chapter, I
discuss my plans for cultivating reasoning habits and then describe the methods I
will use to analyze my results.
Setting
I am in my fourth year of teaching. I taught for one year on a reservation,
and I am in my third year of teaching at my current school. Both schools are
small K-12 schools that have approximately 60 students in grades 7 to 12. I have
always been the only math teacher for grades 7 to 12. Despite the challenge of
preparing for and teaching seven different classes on a daily basis, I enjoy the
small school setting because I have the opportunity to develop powerful
relationships with my students and I am able to try pedagogical approaches that I
believe will benefit my students.
This study will focus on one high school algebra class comprised of 14
students and one high school geometry class comprised of 20 students. The
students vary in grade level from 8th
grade to 11th
grade. The school is located in
26
a small, rural North Dakota town with 56 students enrolled in 7th
through 12th
grades.
Circumstances that will affect the study include student attendance,
student motivation, and student participation. Since I have so few students in
each class, attendance issues will affect the data that I am able to collect. Student
motivation and meaningful participation are necessary components of the study as
the students will only be able to develop reasoning habits if they are open to
learning and growing as students.
Intervention/Innovation
I believe that reasoning habits are essential for success in college and
career. Currently, I use multiple pedagogical approaches in my lessons that
include investigations, independent practice, games, lectures, and projects. While
I promote reasoning habits through these approaches, it tends to be a secondary
focus rather than the primary focus. The intent of my study is to determine if a
primary focus on reasoning habits on a regular basis will help my students
develop into flexible and patient independent learners and problem-solvers.
During the study, the students will work on a reasoning task at some point
during the class period three times per week. This task will be given to them on a
template that they will complete. The template will have prompts to guide the
students in analyzing the problem, implementing a strategy, making connections,
and reflecting on the solution. I will create a list of questions and probing
27
statements to accompany each task that I can use to engage the students in the
reasoning process. The tasks will be standards-based and relate to the concepts
that the students are learning. Discussions will take place throughout the task and
time will be given to debriefing the specific reasoning habits. Students will work
collaboratively on half of the tasks and individually on the remaining half of the
tasks.
Design
My action research project is qualitative in design. The results will not be
generalizable to other classes and will only be used to inform my teaching and my
students’ learning. Students will be given pre- and post-surveys to determine their
understanding, use, and opinion of reasoning habits. They will also be given pre-
and post-reasoning tasks that will be assessed based on a rubric to determine if the
focus on reasoning habits has impacted their ability to do things necessary to learn
independently such as analyze a problem, implement a strategy, make
connections, and reflect on the solution. A journal will be kept to document
changes in student responses when presented a task and to document my
responses to students when they ask for help. I will use the results to determine
how to effectively teach reasoning habits and to determine how students learn
reasoning habits.
28
Description of Methods and Analysis Strategy
Prior to the study, my research study, all consent letters, and instruments
will be reviewed and approved by Minot State University’s Institutional Review
board (see approval letter in Appendix A). All participants will be informed of the
study and a letter preapproved by the Institution Review Board will be sent home
to each participant’s parents/guardians asking permission for their student to take
part in the study (see Appendix B). The student participants will also sign a
student assent letter, similar to that of which the parents/guardians will sign (see
Appendix C). Written consent will also be requested from school officials to
allow the study to take place in my classroom. This consent form can be found in
Appendix D. All participants and information about them will remain
confidential.
Once all consent letters have been collected, students will take a pre-study
survey (see Appendix E) to gain an understanding of their current reasoning
habits. This survey will ask students to rate each statement on a strongly agree to
strongly disagree continuum, similar to a five-point Likert scale. Students will
have the opportunity to comment on each of the statements. Students will also
respond to two open-response questions that inquire about their reasoning habits.
The students will then begin the first unit of study for the third quarter.
The results of the pre-survey will be summarized in a table that will show
the percentage of types of responses for each question. The survey ratings will be
29
converted to numbers and the mean, median, mode, and standard deviation will be
calculated for each question. An inductive analysis will be conducted on the
open-response questions to identify common themes and trends.
My students will be given a reasoning task at some point during the class
period three times per week. They will be provided a template (Appendix F) that
they will need to complete as they work through the task. The template will guide
the students in analyzing the problem, implementing a strategy, making
connections, and reflection on their solution. I will prepare a list of questions and
discussion prompts that correspond to each task (Appendix G). I will use that list
to engage students in the reasoning process and to respond to their requests for
assistance. The reasoning tasks will be assessed based on a reasoning rubric
(Appendix H).
At the end of the quarter, students will take a post-study survey. This
survey will be the same as the pre-study survey and will be used to measure the
students’ perception of their reasoning habits at the end of the study compared to
the beginning of the study. Students will also be given a post-study reasoning
task that will be assessed based on the rubric. The post-study task will be
compared to the pre-study task to determine the extent of growth that took place
during the study.
The scores obtained through the reasoning rubric will be analyzed for
individual growth and class-wide growth. Sub-scores will also be calculated and
30
analyzed to determine the growth of a specific reasoning habit and its
corresponding sub-categories. The reasoning task scores will be summarized in a
table that will show the percentage growth by student and the mean, median,
mode, and standard deviation of the class. When comparing the reasoning task
rubric scores of the pre-study task and the post-study task I will use a 95%
confidence level (or 0.05 level of significance) and a t-test of independent
samples to determine if the focus on reasoning habits has significantly increases
my students’ abilities to understand and solve problems. My null hypothesis is
there is no difference in the reasoning habits of my students from the beginning of
the study to the end of the study. My alternative hypothesis is that the reasoning
abilities of my students will significantly increase from the beginning of the study
to the end of the study. This method of analyzing student performance is
appropriate because this is the standard statistical method to determine if there is a
significant difference in two sets of data.
The responses of the post-survey will be compared to the responses of the
pre-survey. I will summarize the results of the pre- and post-surveys in a table
and in narrative form. The survey ratings will be converted to numbers and the
mean, median, mode, and standard deviation will be calculated for each question
as well as the percentage of growth from the pre-survey to the post-survey. An
inductive analysis will be conducted on the open-response questions to identify
common themes and trends.
31
I will keep a journal throughout the study. This journal will help monitor
any changes in attitudes or habits that I notice with students and myself. I will
also note specific questions, topics, and comments that are made regarding the
reasoning tasks. I will note differences in behavior that students exhibit while
engaging in reasoning tasks compared to other learning activities.
At the completion of the study, I will interpret the surveys, reasoning task
rubric results, and my journal as I determine possible conclusions that can be
made. These sources of data will be triangulated to determine if the behaviors
and comments of my students are consistent regardless of the type of data
representing them.
Expected Results
I hypothesize that the focus on cultivating reasoning habits will
significantly increase my students’ abilities to demonstrate the skills necessary to
learn independently. I anticipate that this focus will benefit my students who
approach the tasks with an open mind. I foresee that this study will be frustrating
for my students who are focused on finishing assignments quickly and rushing to
the answer rather than meddling in the process. I must be prepared and persistent
in order to get accurate results.
Timeline for the Study
This study will take place over the course of a nine-week period during the
third quarter of the academic year. This corresponds to the beginning of January
32
through the middle of March. The consent and assent letters will be sent home
and collected in the middle of December. The pre-survey and the pre-study
reasoning task will be given to the students the first week on January. Reasoning
tasks will be given to the students regularly the following eight weeks. The post-
survey and the post-study reasoning task will be given the first full week of
March. Following the study, the data will be analyzed, summarized, and
presented.
Summary
This action research study will be completed with my algebra and geometry
students. Students will engage in reasoning tasks and use a template to guide them
in the development of reasoning habits. I will use student survey results,
reasoning tasks assessed via a rubric, and my personal journal to analyze the
impact of reasoning habits on the demonstration of independent learning. The
results of my study are discussed in the next chapter.
Chapter Four
Results and Interpretations
Use an introductory paragraph to remind the reader of your purpose and to
give them a brief description of what is included in this chapter.
Results of Data Analysis
Address each data collection method separately and provide its results
(e.g., chapter test, survey, interview, etc.). Be sure to do the following:
Display numerical or statistical results in tables or figures.
Summarize the results of surveys or other instruments.
Theme and summarize narrative data, including representative quotes
when appropriate.
Note: You may need to remind the reader of what you did to analyze the
data while you are presenting the results.
Interpretation of Results
Revisit each research question and present the data that answer that
question. Include the following:
Did you successfully answer your question?
Did you get the results you expected?
Discuss significance and rigor (i.e., quality, validity, accuracy, credibility,
trustworthiness) as needed.
Discuss unusual circumstances as needed
34
Summary
Briefly summarize what you wrote in Chapter Four, highlighting the key
findings, and transition the reader to the next chapter.
Chapter Five
Conclusions, Action Plan, Reflections, and Recommendations
Add an introduction here. Otherwise the two levels of headings, the one at
the top that names Chapter Five and the Conclusions heading beneath it, are too
close and look weird. You can reiterate your purpose and/or tell the reader what to
expect in this chapter.
Conclusions
Draw conclusions about your research questions based on your results.
Someone reading only this section should get a sense of your research purpose
and findings.
Action Plan
Present a plan of action. What will you do now? Will you continue,
modify, or throw out your innovation? Why? Speculate on your ―next steps‖ in
the action research cycle.
Reflections and Recommendations for Teachers
This section is all for you—your opinions, impressions, frustrations, and
celebrations.
What would you do differently?
What were the highlights of your project?
Advice to teachers about your intervention.
Advice to teachers about action research.
36
Summary
This is the last paragraph of the paper. Briefly summarize what you wrote
in Chapter Five and give any last comments that will help wrap up the paper.
37
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Appendices
43
Appendix A
Institutional Review Board Approval Letter
44
Appendix B
Research Participant Consent Form
Using Reasoning Tasks to Develop Skills Necessary to Learn Independently
Fenecia Foster
Purpose of the Research
I am currently completing work towards my Masters of Arts of Teaching:
Mathematics degree through Minot State University. For my final degree
requirement, I am conducting an action research project during quarter three,
January 3rd
through March 9th
, to determine if a focus on reasoning habits in
algebra and geometry will help students develop the skills necessary to learn
independently. Those skills include analyzing a problem, implementing a
strategy, making connections, and reflecting on the solution.
Specific Procedures
Students in my algebra and geometry classes will cover the normal curriculum
while completing reasoning tasks three times per week in class. At the beginning
of the study, your student will complete a survey and a task to assess his/her
current reasoning habits. Throughout the quarter, students will complete
standards-based reasoning tasks. At the end of the quarter, students will complete
the survey and the task again to assess his/her current reasoning habits. Survey
responses, reasoning tasks, and my observations will be analyzed to determine
whether a focus on reasoning habits improves students’ ability to analyze a
problem, implement a strategy, make connections, and reflect on a solution. My
results will be summarized and included in my research paper. None of the
students in my class will be identified in my results. Mr. Schaefer, principal and
superintendent of Sawyer School, has approved this research study.
Duration of Participation
Your student will participate in reasoning tasks during quarter three of the
academic school year. They will be expected to complete two surveys and
multiple reasoning tasks during the duration of the unit.
Benefits to the Individual
There are no direct benefits in participating in this study, but participation will
give your student additional tools to help him/her become successful in college
and career. The study may show the benefits of reasoning tasks to help students
develop the skills necessary to learn independently.
45
Risks to the Individual
The risks to your student are no more than he/she would encounter in a regular
classroom setting.
Confidentiality All data will be treated confidentially by the researcher. Names of participants
and their data sets will be kept in a locked filing cabinet in the researcher’s
classroom or on a password-protected computer and will be destroyed once the
paper has been defended and approved. The researcher agrees to maintain strict
confidentiality, which means that your student’s name will not be discussed or
divulged with anyone outside of this research project. The researcher will also
make sure confidential information will not be discussed in an area that can be
overheard that would allow an unauthorized person to associate or identify the
student with such information.
Voluntary Nature of Participation During this study, the survey responses and reasoning tasks from your student do
not have to be included. If you decide to allow your student to participate, you
are free to withdraw your consent at any time. If you do not grant consent or
withdraw your consent, your student’s data will not be included in my results and
your student will not complete the surveys. However, your student will still be
asked to complete the reasoning tasks since these are a regular part of my course.
Human Subject Statement
The Institutional Review Board of Minot State University has given me
permission to conduct this research. If you have questions regarding the right of
research subjects, please contact the Chairperson of the MSU Institutional Review
Board (IRB), Dr. Vicki Michels at 701-858-3594 or
Offer to Answer Questions
If you have any questions or concerns now or during the study, feel free to contact
me at 701-624-5167 or email me at [email protected] or Mr.
Luke Schaefer at 701-624-5167. Thank you for your consideration.
Consent Statement
You are voluntarily making a decision whether or not to participate in this study.
With your signature below, you are indicating that upon reading and
understanding the above information, you agree to allow your student’s survey
46
and reasoning tasks to be used in this study. You will be given a copy of the
consent form to keep.
____________________________________
Participant (Please print student’s name)
____________________________________ __________________
Signature of Parent or Guardian Date
____________________________________ __________________
Signature of Researcher Date
47
Appendix C
Student Participant Assent Form
Using Reasoning Tasks to Develop Skills Necessary in Learn Independently
Fenecia Foster
Invitation to Participate
You are invited to participate in a study on the use of reasoning tasks in the
mathematics classroom. This study will examine the impact of focusing on
reasoning habits, specifically on how the focus impacts your ability to analyze a
problem, implement a strategy, make connections, and reflection on your solution.
This study is being conducted by Mrs. Foster, mathematics instructor at Sawyer
School and graduate student at Minot State University.
Basis for Selection
You have been selected because you are in Mrs. Foster’s Algebra or Geometry
class. Your class was chosen because the class size and age level are appropriate
for the study.
Specific Procedures
If you decide to participate, you will be asked to complete a pre-study survey and
reasoning task, participate in standards-based reasoning tasks three times per
week that relate to the current unit of study, and complete a post-study survey and
reasoning task. All of the research will be done in the classroom and the results
will be kept confidential. This study will take place during the third quarter of the
2011-2012 academic year. Mr. Schaefer, principal and superintendent of Sawyer
School, has approved this research study.
Benefits to the Individual
There are no direct benefits in participating in this study, but participation will
give you additional tools to be successful in college and career. The study may
show the benefits of reasoning tasks to help you develop the skills necessary to
learn independently.
Alternatives to Participation
If you decide not to participate, you will still complete the reasoning tasks during
class, but will not be required to take the two surveys. The data from your tasks
will not be used in my results. Your participation or lack there of will not impact
48
your grade in the course. If you do participate, you may withdraw your assent at
any time.
Confidentiality All data will be treated confidentially by the researcher. Names of participants
and their data sets will be kept in a locked filing cabinet or on a password-
protected computer in the researcher’s classroom and will be destroyed once the
paper has been defended and approved. The researcher agrees to maintain strict
confidentiality, which means that your name will not be discussed or divulged
with anyone outside of this research project. The researcher will also make sure
confidential information will not be discussed in an area that can be overheard
that would allow an unauthorized person to associate or identify you with such
information.
Human Subject Statement
The Institutional Review Board of Minot State University has given me
permission to conduct this research. If you have questions regarding the right of
research subjects, please contact the Chairperson of the MSU Institutional Review
Board (IRB), Dr. Vicki Michels at 701-858-3594 or
Offer to Answer Questions
If you have any questions or concerns now or during the study, feel free to contact
me at 701-624-5167 or email me at [email protected] or Mr.
Luke Schaefer at 701-624-5167. Thank you for your consideration.
Consent Statement
You are voluntarily making a decision whether or not to participate in this study.
With your signature below, you are indicating that upon reading and
understanding the above information, you agree to allow your survey and
reasoning tasks to be used in this study. You will be given a copy of the consent
form to keep.
____________________________________ __________________
Signature of Participant Date
____________________________________ __________________
Signature of Researcher Date
49
Appendix D
School Administrator Consent Form
Dear Mr. Schaefer:
I am completing work toward the Master of Arts in Teaching: Mathematics
degree through Minot State University. As a degree requirement, I am to conduct
a research project in my classroom during the third quarter this year. I am
planning to implement reasoning tasks to determine whether these tasks improve
our students’ ability to analyze a problem, implement a strategy, make
connections, and reflect on solutions. To accomplish this, I would like to work
with the students in my algebra and geometry classes.
During this time, students will take pre- and post-study surveys regarding their
beliefs about reasoning habits. They will complete reasoning tasks three times per
week. I will also be taking notes on my own observations. At the completion of
the study, I will analyze the data from the surveys and my personal journal of
student comments and progress to determine the results. Classroom and student
confidentiality will be observed regarding all data collected and no individual will
be identified by name.
Before the study begins, I will send home consent forms for parents/guardians to
notify them of this project and to request their permission allowing their student
to participate in the research study. I will also give each student an assent form to
notify him or her of this project and to request his or her assent to participate.
Copies of these letters are attached for your inspection. I am requesting that you permit me to carry out this research in my classroom. Please contact me if you have any questions. Thank you for your consideration.
Sincerely,
Fenecia Foster
50
______ I grant permission for Fenecia Foster to conduct the above-mentioned
research in her classroom.
_______ I do not grant permission for Fenecia Foster to conduct the above-
mentioned research in her classroom.
____________________________________________________________
Signature of Mr. Luke Schaefer, Superintendent of Sawyer Public School
________________________
Date
51
Appendix E
Reasoning Habits Survey and Open-ended Questions
Check the one box that most closely describes your approach to and attitudes
toward mathematical reasoning and problem solving. You are welcome to
comment on your response in the margin.
Item Always Usually Sometimes Rarely Never
1. I try to restate a new
math problem in my own
words.
2. If I am given a problem
that is not exactly like the
examples, I can figure it
out myself.
3. Reading a problem
more than once is a waste
of time.
4. When I get the answer
to a problem, I look back
at the problem to see if my
answer makes sense.
5. Solving a math
problem involves finding
a rule or formula that
applies.
6. I have trouble getting
started on a problem that
is new to me.
7. I enjoy exploring how
the math concept that I am
currently learning
connects to a math
concept that I have
previously learned.
8. I enjoy solving
problems that require me
to figure out my own
individual approach.
52
Item Always Usually Sometimes Rarely Never
9. I learn math best when
someone shows me
exactly how to do the
problem and I can practice
the technique.
10. For the math
problems I have
encountered, there are
obvious ways to solve
them.
11. I can think of at least
one way to begin to work
on a math problem that I
have never seen before.
12. If I can’t solve a
problem in five minutes, I
will give up.
13. I enjoy doing many
easy math problems.
14. If I recognize how to
solve part of a problem, I
can figure out the rest of
the problem.
15. I take time to predict
the answer to a problem
before actually doing the
problem.
Open-ended Questions:
Please answer thoroughly.
1. Describe your typical approach to solving a math problem or task. What
strategies do you use? How long will you spend on a problem before stopping or
asking for help? How do you seek help? (ask the teacher, ask a classmate, look in
the book, look on the internet, etc.) What do you expect the teacher to do when
you raise your hand?
2. What does reasoning mean to you? Can you name any reasoning habits? If
so, please list.
53
Appendix F
Reasoning Task Template
Task:
Analyze the Problem:
Implement a Strategy:
Make Connections:
Reflect on the Solution:
54
Appendix G
Sample Algebra Reasoning Task
Task: Rico drives from Boston to Chicago at an average speed of 50 mph and
returns at an average speed of 60 mph. Rico is on the road for 36 hours. What is
the driving distance from Boston to Chicago? (Graham, Cuoco, & Zimmerman,
2010)
Analyze the Problem:
Implement a Strategy:
Make Connections:
Reflect on the Solution:
55
Appendix H
Reasoning Task Rubric
Category Sub-Category 4 3 2 1
Analyzing a
problem
o Identifies relevant math concepts,
procedures, or representations that reveal
important information about the problem
and contribute to its solution.
o Defines relevant variables and
conditions carefully
o Seeks patterns and relationships
o Looks for hidden structure
o Considers special cases or simpler
analogs
o Applies previously learned concepts
o Makes preliminary deductions and
conjectures
Implementing
a Strategy
o Makes purposeful use of procedures
o Organizes the solution (calculations,
data displays)
o Makes logical deductions based on
current progress, verifies conjectures and
extends initial findings
o Monitors progress towards a solution
Seeking and
Using
Connections
o Seeks and uses connections across
different mathematical domains,
different contexts, and different
representations
Reflecting on
a solution
o Interprets a solution
o Considers the reasonableness of a
solution
56
Category Sub-Category 4 3 2 1
o Revisits initial assumptions
o Justifies and validates a solution
o Reconciles different approaches
o Refines arguments so they can be
effectively communicated
o Generalizes a solution to a broader class
of problems and looks for connections
with other problems
Comments:
Points Earned/Points Possible:
