Where's the Wonder in Elementary Math?

Encouraging Mathematical Reasoning in the Classroom

Judith McVarish

• Publisher: Routledge
• Year: 2012
• Print ISBN: 9780415957151
• eBook ISBN: 9781135908898
• Topic: Education
• Subtopic: Education General

This book argues that even in today's high-stakes testing environment, 'teaching to the test' need not be teachers' only focus as they introduce young children to mathematics. Judith McVarish demonstrates how building a community of learners and using problem solving to engage students can help teachers encourage students' disposition to creative thinking and reasoning—skills that can otherwise become lost due to the pressure of the many other expectations placed upon both teachers and students. This book offers strategies for infusing mathematics learning and reasoning into elementary school classrooms while meeting curriculum and testing mandates. The teacher researcher component of each chapter provides a vehicle for teachers to bring their own expertise and questions back into the teaching and learning equation.

Chapter 1
Setting the Stage

The 28 fourth graders in Mrs. Thomas’s class are congregated on a rug in front of the windows talking about the large graphic representation on the floor in front of them. The discussion is animated and filled with reasoning and speculative theories. No one is raising a hand to speak or waiting for a nod from the teacher. The dialogue moves from student to student with only the occasional question from the teacher. The students have spent the previous weeks surveying all 1,000 people in their school about the question, “Do you like school?”

Many discussions have already taken place and decisions made regarding how to collect, sort, and represent the data. Should the answer be provided verbally? Or would it be better to require a written response? Students even entertained the ideas of collecting the data in large group settings rather than surveying individuals or of using a collection box to gather the responses. Once the data have been collected, countless hours were spent working on how to categorize the responses. These categories included the students in grades K–6, their classroom teachers, the kitchen staff, the two custodians, the nurse, the principal and vice principal, the librarian, gym teachers, art and music teachers, and the resource room teachers and counselors, plus the three secretaries who work in the office.

Students gave reasons pro and con to support placing the classroom teacher responses with the data from their individual grades or into a category of their own. They also had to decide whether the office workers were considered administration or whether another category had to be created. If classroom teachers were included in the grade-level data, students then had to consider what to do with the data from special subject teachers. The next phase in the process was to determine how to visually display the results. Students experimented with several different graphing styles and concluded that a simple bar graph allowed readers to see the results most clearly. Yet this provided another dilemma of how to represent the data since a one-to-one representation would create bar lengths that were unmanageable. Students grappled for some time before arriving at a mathematical key of 5 people per inch for their graphic representation in order to accommodate the large number of responses in some categories.

Now, as students sat on the carpet in a semicircle, looking at their 8-foot by 4-foot finished graphic representation, discussion focused on the fact that all categories showed more YES responses than NO responses— except the third-grade category.

Susie: Who went in to the third grade to ask them?

Alisha and Brianna: We did.

Susie: Did you say, “WHO likes school?” (Susie demonstrates by speaking with a dreaded intonation) or did you say, “Who likes school?” (Again Susie demonstrates with a lilt at the end of the question as if asking who likes ice cream.)

Alisha and Brianna proceeded to demonstrate the neutrality of their voices during the visit to the third grade.

Susie: I think it is because it’s not cool to like school in the third grade. I would have said “no” if someone asked me last year.

Jorge: Yeah, remember how we were afraid of not being cool?

Mrs. Thomas: Is there a way you might have prevented the cool factor in your questioning?

Tim: Have them write down their answer.

Kim: But they would still think we expected them to say NO. I think we should have given them a secret ballot kind of vote.

Max: Maybe it’s not the third grade they don’t like. I mean, it’s only October, so it could be second grade they didn’t like.

Susie: I don’t think it was second grade though, ’cause little kids like school more.

Brianna: Well we know it’s not the third-grade teachers they didn’t like. They’re awesome.

Alisha: Yeah, it’s not the teachers. I went and asked Mrs. Blake what she thought, and she didn’t think they’ve been in third grade long enough to not like it.

Mrs. Thomas: So, how could we find out for sure?

It is obvious these students have had many opportunities to exchange mathematical ideas because this discourse is free flowing. Students are building on the math thinking of their classmates—questioning each other and responding in a manner that is more collaborative than defensive or competitive. Collaborative dialogue such as this encourages students to think critically. The teacher’s questions were not aimed at mathematics facts, such as “How many fewer third graders like school than fourth graders?” Rather they were targeted at getting students to think and solve problems. Thoughtful teacher questions elicit responses that require more than computational, procedural, lower-level math thinking.

When I walk into a classroom and see and hear children actively engaged in questioning each other and talking breathlessly about why they think something is so or not so, I ask myself, “How did this teacher create this kind of mathematics environment? Is this happenstance or is there something behind the scene that is important to understand?” Being involved in teaching and learning that foster discussion, debate, and the construction of mathematical ideas is appealing for students and teachers alike. It taps into the reason most of us went into teaching in the first place. Have you ever met a teacher who claimed his or her career choice was motivated by something other than making a difference, helping children to learn deeply in engrossing and exciting ways, and wanting students to engage in learning of any kind with joy? What happens to dismantle this vision? This book is about exploring ways to foster creativity and critical thinking in the classroom and how to make mathematics learning a joyful journey for all stakeholders. It is intended as a mathematics resource text for pre-service and in-service teachers of elementary school children who want to foster critical thinking and mathematics reasoning in their classrooms.

Young children begin their schooling journey filled with expectation, questions, vim, vigor, openness, and innocence. Where along this learning journey do students develop the stance of “playing school,” the attitude of “doing what the teacher wants,” or “how many minutes until recess?” Why has it become “cool” for students to say that they don’t like school as they progress through the grades? Margaret Donaldson (1978) raises this question in Children’s Minds when she speaks about large numbers of students leaving school feeling defeated and lacking any excitement in the pursuit of intelligence:

In the first few years at school all appears to go very well. The children seem eager, lively, and happy. There is commonly an atmosphere of spontaneity in which they are encouraged to explore and discover and create. However, when we consider what has happened by the time children reach adolescence, we are forced to recognize that the promise of the early years frequently remains unfulfilled. (pp. 13–14)

Throughout this book I will investigate these questions and others and discover ways to fill the classroom culture with mathematics thinking opportunities for both students and teachers. These opportunities will be embedded in the questions that are asked, in the transitions between the activities of the day, and in the homework that is assigned. When the culture of the mathematics classroom shifts from one of compliance and right answers to an interest in the how and why of that which surrounds us, the energy level is raised and the engagement of students and teachers alike becomes heightened.

LEARNING AND TEACHING MATHEMATICS: A HISTORICAL PERSPECTIVE

Historically, mathematics was considered a discipline of right answers, formulas, and procedural rules. It was a male-dominated field of study and considered a discipline where only mathematically gifted students went on to higher levels of mathematical study. The mathematics content offered in elementary school remained primarily in the realm of number operations and functions with little attention given to geometry, data analysis, probability, and reasoning. The teacher stood at the blackboard and demonstrated for students the proper procedure to reach a correct solution. This was typically followed with practice problems for students to complete at their seats, without discourse with fellow classmates.

In the last 20 years, however, there has been a national standards movement in mathematics education that has important implications for both teachers and the students they teach. We are in the midst of a revolution in teaching and learning mathematics that was originally fueled by the publication Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM, 1989). This council is the largest international professional organization in the field of mathematics education. The document suggested a new direction for mathematics teaching and learning and provided the pedagogy to deliver the suggested content and evaluation practices. Elementary school mathematics was no longer limited to the arithmetic of days gone by where students only learned to add, subtract, multiply, and divide. The newly published Standards called for expansion of the elementary mathematics curriculum to include data analysis, probability, patterns and functions, algebra, reasoning, geometry, and communicating mathematical ideas, both verbally and in writing. Although computation skills were addressed in the Standards document, a decrease in attention to rote memorization of facts and algorithms was advocated. Instead of rote computation and memorization, conceptual understanding, connections, and application were to be emphasized. This benchmark publication articulates a vision for mathematics teaching and learning that includes having all students value and become confident in their ability to do mathematics. An important goal is for all students to become mathematical problem solvers who learn to communicate and reason mathematically. The Standards offered guidelines designed to prepare students to be informed citizens in the world into which they would graduate. These guidelines identified exploration, questioning, debate, reasoning, and communication as critical and necessary skills for all students.

In the same year, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (NRC, 1989) was published. The National Research Council (NRC) was organized in 1916 to function as the operating agency of both the National Academy of Sciences and the National Academy of Engineering. The NRC is responsible for research in the field of mathematical sciences. Everybody Counts strongly advocated a need to alter the mathematics being taught in the schools, outlining how transitioning from an industrial society to an information society and a technology explosion have dramatically affected business, government, industry, and the home. The authors stated that mastery of mathematical facts and rules alone would no longer suffice to equip students to understand and participate in this ever-changing, technological world, where computation skills need to be coupled with conceptual understanding. The mathematics needed to be successful in the 21st-century workplace would require more mental analysis and communication of ideas and less perfunctory maneuvering of numbers. Technological expertise would be necessary for all (NRC, 1989).

Both the NCTM and the NRC provided the rationales for such a teaching and learning change. The protection of the environment, medical advances, space exploration, nuclear energy, inflation, and national debt were all given as examples of complex issues requiring an informed electorate who could interpret the world and make crucial, reasoned decisions and critically analyze data based on logic and mathematical reasoning (NRC, 1989). There are economic and societal issues that drive this reform, as well as educational demands. The demographics of our society are changing rapidly. One in every three American students is a member of a minority, and in the next decade minority populations will become the majority. Poverty, the need for instruction in English as a second language for our growing and diverse immigrant population, and stresses on family structures raise the stakes for all and underscore the need for true equity in all aspects of society. Mathematics teaching, which has traditionally discriminated against individuals on the basis of gender, class, and ethnicity, now must create “math equity” for all students. Advanced mathematics can no longer be reserved for white males, who now make up the overwhelming majority of those currently dominating the field of mathematics. Women and minorities must be given every opportunity to learn in order to enter fields involving mathematics and technology and to function effectively in the new global environment if we are to eliminate the elitist filter that currently exists and compromises students’ potential to learn mathematics (NRC, 1989).

Again, in 1990, another seminal document by the NRC, Reshaping School Mathematics: A Philosophy and Framework for Curriculum, was published by the Mathematical Sciences Education Board (MSEB), which is governed by the National Research Council. The MSEB, yet another national organization, demanded a change from the instructional practices of 20, 50, or even 100 years ago in mathematics curricula and pedagogy. This document proposed a framework for national school mathematics reform that included improving the education of mathematics teachers through continuous professional development offerings to improve teacher efficacy and raise teachers’ confidence in teaching mathematics, while encouraging alternative teaching methods. The thrust of this report was a drastic reform call for new materials and assessment methods to accompany suggested new content and pedagogy. The report also advocated that money be appropriated for new textbooks that embraced standards-based teaching and learning, as well as for materials such as mathematics manipulatives, computers, and calculators.

In April 2000, after 10 years of reflection and discussion among all constituents concerned with mathematics education, the NCTM published Principles and Standards for School Mathematics: An Overview. It revised, clarified, and refined the original document, while still maintaining the original vision set forth in 1989. The focus of mathematics learning in recent years has been on problem solving, including every aspect of mathematics education from creating problem-based curricula to including open-ended problems on standardized tests to posting problem-solving steps in classrooms.

Based on more than 30 years’ experience as a teacher and teacher educator, I have been influenced by these documents, as well as by the works of Dewey (1938/1963) on constructivism and experiential learning, Vygotsky (1986) on the collaborative construction of understanding and his model of the zone of proximal development, Duckworth (1987) on “the having of wonderful ideas,” Mezirow (1990, 1998) on self-regulation and transformative learning, and Freire (1970, 1973, 1996) on learning as liberation, social justice, and problem posing. The combination of these influences has convinced me that educators best help students understand their worlds by developing students’ habits of wonder and wanting to know.

The Current Scene

School classrooms are situated at the epicenter of a series of expectations and obligations, each with its own set of controls and mandates. These expectations are manifested nationally on a large scale in such mandates as No Child Left Behind and NAEP testing. State controls are often financial or are directed at teacher accountability. Districtwide mandates may be curriculum based, un...

Table of contents

• Cover
• Title
• Copyright
• Dedication
• Contents
• Acknowledgments
• Series Editor’s Foreword
• CHAPTER 1 Setting the Stage
• CHAPTER 2 What Does the Room Teach?
• CHAPTER 3 Who Asks the Questions? Who Answers Them?
• CHAPTER 4 How Can I Tune Transitions to a New Key?
• CHAPTER 5 What Is Real about Homework?
• CHAPTER 6 How Do I De-Fang the Test?
• CHAPTER 7 How Can We Take Critical Thinking Beyond the Classroom?
• CHAPTER 8 What Do Parents Know?
• CHAPTER 9 Is Thinking about Thinking Just a Play on Words?
• References
• Index