Teaching Middle School Mathematics

Douglas K. Brumbaugh

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781136756214
• Edition: 1
• Topic: Education
• Subtopic: Education General

Middle school teaching and learning has a distinct pedagogy and curriculum that is grounded in the concept of developmentally appropriate education. This text is designed to meet the very specific professional development needs of future teachers of mathematics in middle school environments.Closely aligned with the NCTM Principles and Standards for School Mathematics, the reader-friendly, interactive format encourages readers to begin developing their own teaching style and making informed decisions about how to approach their future teaching career. A variety of examples establish a broad base of ideas intended to stimulate the formative development of concepts and models that can be employed in the classroom. Readers are encouraged and motivated to become teaching professionals who are lifelong learners.The text offers a wealth of technology-related information and activities; reflective, thought-provoking questions; mathematical challenges; student life-based applications; TAG (tricks-activities-games) sections; and group discussion prompts to stimulate each future teacher's thinking. "Your Turn" sections ask readers to work with middle school students directly in field experience settings.This core text for middle school mathematics methods courses is also appropriate for elementary and secondary mathematics methods courses that address teaching in the middle school grades and as an excellent in-service resource for aspiring or practicing teachers of middle school mathematics as they update their knowledge base.
Topics covered in Teaching Middle School Mathematics:
*NCTM Principles for School Mathematics;
*Representation;
*Connections;
*Communication;
*Reasoning and Proof;
*Problem Solving;
*Number and Operations;
*Measurement;
*Data Analysis and Probability;
*Algebra in the Middle School Classroom; and
*Geometry in the Middle School Classroom.

1

Introduction

Focal Points

• Teaching what mathematics is
• Working at teaching mathematics
• Planning your lessons
• Technology
• Role of mathematics in the world
• Expand concepts or definitions
• Analyze a problem in more than one way
• Use an idea in more than one setting
• Conclusion
• Sticky questions
• TAG (tricks, activities, and games)
• References

Many studies have been conducted to identify the qualities that all good teachers possess. The only two that are consistently identified are warmth and a sense of humor. There is agreement on a need to know the mathematics being taught if one is to be an effective teacher of mathematics. Not all agree that there is a necessity to know about how to teach the mathematics known. Some hold the opinion that if one knows mathematics, one can teach it. This book will focus on the idea that knowing the mathematics is not sufficient to teach it. An effective teacher of mathematics must know the mathematics being taught AND how to present that knowledge in dynamic, interesting, understandable, motivational, and appropriate terms.

If you are not competent and confident with the subject matter, you will struggle to create positive mathematical experiences best suited for students. To that end, you must have the desire to learn mathematics, how to educate the students, and how to supply the optimum mathematics environment for each student. As you continue to investigate new mathematical knowledge and effective teaching strategies, you can stimulate students to follow your lead to new knowledge, each at their respective level and pace. A true teacher is always willing to learn new methods and strategies for introducing concepts to students.

As stated in the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), an effective teacher of mathematics will be able to motivate all students to learn mathematics. Similar ideas are raised in NCTM’s Principles and Standards for School Mathematics (NCTM, 2000). Mathematics opens doors to careers, enables informed decisions, and helps us compete as a nation (Mathematical Sciences Education Board [MSEB], 1989). The only constant today’s students will face in their working lives will be change. It is predicted that today’s K-12 student will change careers—not jobs—as many as five times, which implies major reeducation. Students must learn to absorb new ideas, adapt to change, cope with ambiguity, perceive patterns, and solve unconventional problems (MSEB, 1989). Without these abilities, today’s students will have a difficult time in their working future.

A large segment of our society is willing to make statements like “I hated math in school,” “math, yuk,” and “I did not do well in math.” If someone said, “I cannot read,” countless individuals would express sorrow, perhaps pity, and then skirmish to find ways to help that person learn to read. Why is there not such a cry for the one who confesses an inability to do mathematics? It is our commission to set the wheels in motion that will change this attitude.

Teaching what Mathematics Is

What is mathematics? If we are going to be concerned about teaching mathematics to others, we need some idea of what it is we are teaching. Check the definition of mathematics in a dictionary. Are you comfortable explaining the definition to others? An eminent mathematician once said that he would not be “satisfied with his knowledge of mathematical theory until he could explain it to the next man he met on the street” (Newman, 1956, p. 4). As teachers of mathematics, we are responsible for explaining complex concepts that incorporate real-life mathematical activities, while putting things in words students can relate to and understand. The effective teacher must also emphasize student-centered instruction. The learning environment (social, cognitive, affective, and physical) should be conducive to the optimal learning of each individual.

The preceding paragraph describes a constructivist’s approach to teaching mathematics. For the constructivist, students must become active participants in the learning environment. If either the teacher or the student falls short of their responsibility, the net result will be a less-than-satisfactory learning environment. Students need steps of learning that allow for consolidation and success. Students can then practice these steps in groups or alone, with the teacher or other experienced individuals nearby to assist as needed.

No longer can we accept the idea that mathematics is only for the best and brightest. Too many basic mathematical concepts are an integral part of our daily lives. As teachers of mathematics, we must be able to motivate all students to learn. It is our responsibility to create an appropriate atmosphere so that can happen for each and every student. Without such a goal, we, as a nation, risk widening the gap that exists in our society between those who can and cannot do mathematics.

Some say mathematics is a way of thinking. That becomes evident to us as we do proofs, examine patterns, or organize our approaches to new and different problems. Others say mathematics is a language. We can understand that statement as we talk in our special language of Xs and Ys, graphs, and patterns. But, if that language is one spoken only by a select few, what good is it? If we accept that all students are capable of learning mathematics, it is imperative that we include all of them in our inner circle of basic mathematical language. If we reject the premise that all students can learn mathematics, what do we do with those who do not learn the fundamentals of our subject? How do we prepare them to become productive members of our society?

Mathematics is an organized structure of knowledge. If you understand the structure, survival in the world of mathematics is possible. Without the structural understanding, failure, as far as being able to function mathematically, is imminent. Too many students are willing to accept being unable to function mathematically. Did you experience applications of mathematics in real-world settings that were appropriate for your age and interests as you progressed through school? Remember, as you ponder that last question, that you are mathematically oriented. Individuals like those in your class who did not see the applications and understandings are the ones you will be trying to entice into becoming interested in learning mathematics. Far too many students assume there will be no need for mathematics in their future world. How foolish that statement is. Yet, how common it is. If we look through the eyes of students, maybe that statement makes sense. It is our task to get students where they see a need for the power of mathematics in their daily lives. Some instructional strategies to motivate students and prevent math anxiety include the following (partially from Martinez & Martinez, 1996):

Creating an anxiety-free mathematics classroom, which could include different seating arrangements like circles and small groups, where the teacher takes the role of facilitator of learning to help students construct their own knowledge.

Matching instruction to students’ cognitive learning level (concrete, pictorial, and abstract).

Planning instruction that connects mathematics to familiar situations in students’ everyday lives (context relevant activities).

Incorporating games, humor, and puzzles into mathematics instruction.

Teaching mathematics through reading and writing.

Empowering students by using technology and collaborative or cooperative teaching/learning procedures.

Using peer tutoring as a means of enhancing student learning.

Establishing mentors within the classroom environment to help new students, those who need additional motivation, and so on.

Modeling mathematical thinking and learning by teachers and students.

Incorporating the use of manipulatives into instruction that broaches a new topic.

Mathematics abounds with examples of the study of patterns. It seems natural and useful to those of us who are in the field. How do we communicate that value to our students? For example, the Fibonacci (Leonardo of Pisa, 1175–1230) sequence is 1, 1, 2, 3, 5, 8, …, where each new term is found by listing the sum of the two preceding terms; it supposedly grew out of the study of rabbit reproduction starting with two mature adults, one of each gender. This sequence of numbers appears in a variety of settings. For example, if the clockwise and counterclockwise spirals of a sunflower are counted, the results will always be two successive terms in the Fibonacci sequence. Although Fig. 1.1 is only a partial picture of the center of a Sago Palm, the spirals should be visible to you. Can you see the spirals in the pinecone? There are several Internet sites that show pictures of the spirals. One is http://www.pims.math.ca/education/2000/bus00/sunflower.

Fig 1.1. Courtesy of Linda S. Brumbaugh

A Lucas sequence begins with any two numbers, not one and one, and is built like a Fibonacci sequence. A student writes a 10-term sequence for the class, that the teacher cannot see. The class needs to ensure the addition is correct. When the 10 terms are listed, the teacher looks at them and announces the sum of all 10 terms.

	Numeric Example	Algebraic Example
	9	x
	4	y
	13	x + y
	17	x + 2y
	30	2x + 3y
	47	3x + 5y
	77	5x + 8y
	124	8x + 13y
	201	13x + 21y
	325	21x + 34y
Sum	847	55x + 88y

This example can be used in several places in the curriculum. Addition prac...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Contents
• Preface
• About the Authors
• 1. Introduction
• 2. NCTM Principles for School Mathematics
• 3. Representation
• 4. Connections
• 5. Communication
• 6. Reasoning and Proof
• 7. Problem Solving
• 8. Number and Operations
• 9. Measurement
• 10. Data Analysis and Probability
• 11. Algebra in the Middle School Classroom
• 12. Geometry in the Middle School Classroom
• Solution Manual
• Tag Solutions
• Index