Using Reasoning Tasks to Develop Skills Necessary to Learn Independently

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

Spring 2012

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This capstone project was submitted by

Fenecia Lynn Foster

Graduate Committee:

Dr. Laurie Geller, Chairperson

Dr. Narayan Thapa

Dr. Ryan Winburn

Dean of Graduate School

Dr. Linda Cresap

Date of defense: April 26, 2012

Abstract

Despite my efforts to engage my students in the learning process, the most common response to an investigation or a problem set was a blank stare or an immediate raise of the hand. This project focused on using reasoning tasks to cultivate reasoning habits. I implemented a strand of instructional activities for each state mathematics standard that taught my students reasoning skills to help them become successful independent learners. My goal was 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. Students were presented with reasoning tasks regularly, but completed a task via a template one time per week during the course of the study. Although the templates were not necessary at all times, they helped my students engage in the reasoning process and helped me identify specific reasoning habits that the students could apply and specific reasoning habits that needed further attention. The results of the study showed an increase in students being able to identify helpful strategies, a shift in ownership of learning, and a willingness by the students to be more patient when attempting to solve a problem. Also, students made significant growth in their ability to analyze a problem, implement a strategy, seek and make connections, and reflect on a solution as well as their overall ability to reason.

Acknowledgements

First of all, I would like to thank my mom and dad for their love, dedication, and inspiration. They have supported me through all of life’s adventures and have modeled the value of education. They have taught me how to teach and have shown me what it means to truly love what you do.

I would like to thank my husband, Dan. Just after we met he affirmed my love of math and helped me discover that my true passion is in the classroom and I have not looked back since.

I would like to thank my chairperson, Dr. Laurie Geller, for her endless assistance and support at all hours of the day.

Finally, I would like to acknowledge my students. They inspire, challenge, and motivate me to continually improve as a teacher and as a person. They make me laugh, smile, and cry. They have given me the opportunity to know what it feels like to love what I do.

Table of Contents

Page

Abstract iii

Acknowledgements iv

List of Tables viii

Chapter One: Introduction 1

Motivation for the Project 1

Background on the Problem 2

Statement of the Problem 3

Statement of Purpose 4

Research Questions/Hypotheses 5

Summary 5

Chapter Two: Review of Literature 6

Current Research 6

Research on the Needs of the Student 7

Research on the Role of the Teacher 9

Benefits of Reasoning Habits 11

Summary 11

Chapter Three: Research Design and Method 12

Setting 12

Intervention/Innovation 13

Design 13

Description of Methods and Analysis Strategy 14

Expected Results 16

Timeline for the Study 17

Summary 17

Chapter Four: Results and Interpretations 18

Results of Data Analysis 18

Pre- and post-surveys 18

Reasoning tasks and templates 25

Journal 34

Interpretation of Results 34

Summary 40

Chapter Five: Conclusions, Action Plan, Reflections, and Recommendations 41

Conclusions 41

Action Plan 42

Reflections and Recommendations for Teachers 43

Summary 44

References 45

Appendices 48

Appendix A: Institutional Review Board Approval Letter 49

Appendix B: Research Participant Consent Form 50

Appendix C: Student Participant Assent Form 52

Appendix D: School Administrator Consent Form 54

Appendix E: Reasoning Habits Survey and Open-Ended Questions 55

Appendix F: Reasoning Task Template 57

Appendix G: Sample Algebra Reasoning Task 58

Appendix H: Reasoning Task Rubric 59

List of Tables

Table Page

1. Pre-Survey and Post-Survey Results for Statements 1-15 19

2. Pre-Survey and Post-Survey Measures for Statements 1- 23

3. Pre-Survey and Post-Survey Responses of Open-Ended Question # 25

4. Initial Task and Final Task Results by Class 26

5. Initial Pre-Task, Final Task, and Cumulative Increase Results by Student 28

6. Initial Task and Final Task Results by Individual Criteria for

Analyzing a Problem 30

7. Initial Task and Final Task Results by Individual Criteria for

Implementing a Strategy 31

8. Initial Task and Final Task Results by Individual Criteria for

Seeking and Using Connections 31

9. Initial Task and Final Task Results by Individual Criteria for

Reflecting on a Solution 32

10. Sampling of Reasoning Task Mean Scores from Throughout the Study 33

11. One-Sample T-Tests Between Initial Tasks and Final Tasks 33

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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 began this project in the midst of an unprecedented and devastating flood. The event challenged homeowners including myself to draw on logical reasoning skills and the ability to think analytically during a lengthy period of uncertainty. The event also challenged a wide array of public officials to draw on their logical reasoning skills as they made endless decisions in the best interest of the community and in an attempt to predict potential issues that had not yet developed.

The flood 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 struggled with independently identifying and applying their critical reasoning skills when required to explore unfamiliar concepts. Thus, my project focused 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 reflected on my first three years of teaching, one thing I wanted to improve upon is inspiring my students to become independent learners and thinkers. I believed an independent learner is one who can patiently and confidently reason through a given task. As a teacher, it was 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 was interested in learning about research-based methods that will assist me in developing these skills in my students. I believed that if my students left my classroom equipped with reasoning habits then they would 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 ranged from three students to 20 students per class. I made 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 was a blank stare or an immediate raise of the hand.

Since my class sizes were so small I had the time to address each student individually. My instincts drew me toward the raised hand and the pleading look; thus, the very small student-to-teacher ratio enabled a high-level of immediate feedback and direction. Although this may appear beneficial to the learning environment, students seldom experienced the wait time necessary to develop their own reasoning habits. My experience and observation led me to believe that students in my school consistently demonstrated learned helplessness. When encountering a task or problem, my students defaulted to asking me rather than determining how they can use their own knowledge, skills, and resources.

I implemented a strand of instructional activities for each standard that taught my students reasoning skills to help them become successful independent learners. My goal was 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 was 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).

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 was 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 demonstrated a general understanding of this process, but they ran into a roadblock when required to independently apply the process. This roadblock prevented 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 struggled to understand a given problem and to identify and apply appropriate reasoning strategies to solve the problem. If my students were told which procedure to use on a given task, then they were often able to solve the problem. In my classroom I had to learn how to teach my students how to delve into a problem from the beginning. I had to learn how to cultivate reasoning habits in my students.

Statement of Purpose

Using Polya’s 4-step process as the foundation, I focused on developing reasoning habits with my Algebra students. The purpose of this study was 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.

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 included research and implementation. First, I researched instructional methods for teaching reasoning habits, facilitating discussions, and asking engaging questions. I prepared discussion prompts and questions for each lesson that I had available to use instead of providing the students with answers. Second, I used the research-based methods to create lessons and tasks that focused on reasoning on a daily basis.

Research Questions/Hypotheses

Which methods could I use to cultivate reasoning habits within my students? The NCTM (2009) recommended a number of strategies, but which methods would work for my students? What would be the effects of the development of reasoning habits with Algebra students in decreasing the presence of learned helplessness? My goal was 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 struggled to understand problems and to determine how to begin to solve the given problem. Through a focused effort to learn about instructional methods and to apply those methods in the classroom with my students, my goal was 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 discussed 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.

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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 was for my students to learn how to learn. I wanted 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.