Mathematics for Elementary Teachers via Problem Solving - Softcover

Masingila, Joanna O.; Lester, Frank K.; Raymond, Anne Miller

Softcover

ISBN 10: 0130173452 ISBN 13: 9780130173454

Publisher: Prentice Hall, 2001

Synopsis

This combination activity manual and resource handbook provides a two-part learning system for prospective K-6 teachers. It uses activities to help develop a deep and lasting understanding of the mathematical concepts, procedures, and skills that are essential for teaching mathematics to elementary school children. Chapter topics include getting started in learning mathematics via problem solving; numeration; operations on natural numbers, whole numbers and integers; number theory; data and chance; fraction models and operations; real numbers: rationals and irrationals; patterns and functions; geometry; and measurement. For teachers of mathematics at the elementary school level.

About this title

Excerpt. © Reprinted by permission. All rights reserved.

Give a child a fish and you feed her for a day. Teach a child to fish and you feed her for life.

The ancient Chinese saying above tells us how important it is to have good teachers for our children. Indeed, there are few professions or occupations that are as important to the welfare of our society and culture as teaching. The purpose of the activities in this book is to help you develop a deep and lasting understanding of the mathematical concepts, procedures, and skills that are essential to being able to teach young children, in particular children in the elementary grades. We believe that if you develop such deep and lasting understanding, you will be well prepared to teach mathematics to many children and thereby help to prepare them to lead productive, informed lives once their school days are over.

The Way Mathematics Is Taught Is Changing

Those who would teach mathematics need to learn contemporary mathematics appropriate to the grades they will teach, in a style consistent with the way in which they will be expected to teach.

All students, and especially prospective teachers, should learn mathematics as a process of constructing and interpreting patterns, of discovering strategies for solving problems, and of exploring the beauty and applications of mathematics.
(Everybody Counts: A Report to the Nation on the Future of Mathematics Education, 1989, pp. 64, 66)

These two quotes shown are taken from a report written over 10 years ago for the United States' National Research Council by a group of concerned mathematics teachers. The authors of the report insisted that it was time to change the way that mathematics was taught at all levels, kindergarten through university. Since this report was written, the nature of mathematics instruction has begun to change. In the past, mathematics instruction was viewed by many as an activity in which an "expert"—usually the teacher—attempted to transmit her or his knowledge of mathematics to a group of students who usually sat quietly trying to make sense of what the expert was telling them. This passive transmission view has been replaced by a new view in which mathematics is seen as a cooperative venture among students who are encouraged to explore, make and debate conjectures, build connections among concepts, solve problems growing out of their explorations, and construct personal meaning from all of these experiences.

Principles of the Problem-Based Approach

The activities contained in our books have been created with the new view of mathematics teaching and learning promoted by the American Mathematical Association of Two-Year Colleges (AMATYC) and the National Council of Teachers of Mathematics (NCTM). In particular, we developed the activities with the following documents in mind: AMATYC's publication, Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus, and the NCTM's publications, Curriculum and Evaluation Standards for School Mathematics, Professional Standards for Teaching Mathematics, Assessment Standards for School Mathematics, and Principles and Standards for School Mathematics.

From these five documents we developed a set of principles to guide the development of all activities.

1. All Activities Are Based on the NCTM Standards.

Special emphasis is placed on the five process standards of the NCTM: problem solving, communication, reasoning, connections, and representations. First and foremost, students should be engaged in the solution of thought provoking problems. Not only should students learn to solve problems, but they should also learn mathematics via problem solving. The second major standard is communication. Knowing mathematics is of little value if one cannot communicate mathematical ideas to other people. NCTM's third major standard is reasoning. Among other things, reasoning deals with the ability to think through a problem and to carefully evaluate any solution that has been proposed. The fourth of the major standards involves making connections. To really understand mathematics, one must be able to see connections between various mathematical ideas, and between "school" and "real world" mathematics. Finally, the way in which mathematical ideas are represented is vital to how students can understand and apply those ideas. Representations should be viewed as essential ingredients in supporting the development of deep understanding.

2. Solving Problems Regularly and Often is an Essential Part of Developing a Good Understanding of Mathematics.

In order for you to improve your ability to solve mathematics problems, you must attempt to Solve a variety of types of problems on a regular basis and over a prolonged period of time. We also believe that ability to solve problems goes hand-in-hand with the development of an understanding of mathematical concepts, procedures, and skills. Put another way, as you solve problems you will develop better understanding of the mathematics involved in the problems. And, as you develop better understanding of mathematical ideas, you will become a better problem solver.

3. Problem Solving Involves a Very Complex Set of Processes.

There is a dynamic interaction between mathematical concepts and the processes used to solve problems involving those concepts. That is, heuristics, procedural skills, control processes, awareness of one's cognitive processes, etc. develop concurrently with the development of an understanding of mathematical concepts.

4. The Teacher's Role in Fostering Healthy Problem-solving Performance Is Vitally Important.

Problem-solving instruction is likely to be most effective when it is provided in a systematically organized manner under the direction of the teacher. Our philosophy is that the role of the teacher changes from that of a "dispenser of knowledge" to a "facilitator of learning." With respect to problem solving and reasoning, this implies that the teacher does very little lecturing on how to solve specific types of problems and much more posing and discussing of a wide variety of non-routine and applied problems. The teacher also focuses on helping you make connections between the mathematics you are learning and its application to the workplace or home.

5. Cooperative, Small-group Work Is Encouraged.

The standard arrangement for working on the activities in the Student Activity Manual is for you to work in small groups. Small group work is especially appropriate for activities involving new content (e.g., new mathematics topics, new problem-solving-strategies) or when the focus of the activity is on the process of solving problems (e.g., planning, decision making, assessing progress) or exploring mathematical ideas.

6. Assessment Practices Are Closely Connected to Instructional Emphases.

We believe that the teacher's instructional plan should include attention to how your performance will be assessed. In order for you to become convinced of the importance of the sort of behaviors that a good problem-solving program promotes, it is necessary to use assessment techniques that reward such behaviors. As a result, we encourage teachers to use various alternative assessment methods such as providing opportunities during tests for you to work with a group of your classmates to solve certain problems on the tests. We also encourage teachers to assess your ability to discuss your understanding of mathematical concepts and procedures in writing and orally.

Features

Hands-on Exploration through Group Work

The activities in this text are designed to engage you in doing real mathematics through small-group exploration. We have two mottoes that should be followed:

• Mathematics is best learned by active, "hands-on" exploration of real problems.
• If "two heads are better than one," then three or four heads are even better!

These mottoes arise from our conviction that the best learning occurs when you are engaged actively in making sense out of problematic situations. Thus, it is your responsibility to make sense out of the activities, rather than wait for the teacher to tell you what is important or how to solve the problems.

Consequently, the activities in this book include almost no explanations with them. It will be your responsibility to work with the students in your group (some students like to refer to their groups as "teams") to solve problems and develop good understanding of the mathematics involved. The teacher's job is to encourage you, to offer gentle assistance without giving too much specific guidance. This style of learning may be a new experience for you and it may even be a bit uncomfortable for you at first. But, be patient! As you gain experience working in a group with others rather than depending on the teacher to tell you everything you should know, you are likely to find that you are becoming more and more independent of the teacher and increasingly in control of your own learning.

Activities Grouped by Chapter

The activities are broken into chapters that conform to ten different mathematical topics. Each chapter begins with

• an overview of the subject addressed in the activities and an outline of the activities
• "The Big Mathematical Ideas" addressed in the activities
• "NCTM Principles and Standards Links"

Each chapter concludes with "Things to Knob" that reviews your understanding of the material and includes lists of

• Words to Know
• Concepts to Know

A set of "Exercises & More Problems" helps test your mastery of the skills learned in the activities and expand your grasp of the concepts. This section is divided into several different types of problem sets including:

• Exercises
• Critical Thinking
• Extending the Activity
• Writing and Discussing

Activities Linked to Student Resource Handbook

As noted previously, the activities in this manual do not include text discussions and explanations of concepts. To provide this information with the activities would be detrimental to helping you become a good problem solver and an independent learner of mathematics. However, we developed the Student Resource Handbook to provide you with useful explanations of the mathematical concepts and procedures that are explored in the activities. The Student Resource Handbook and Student Activity Manual are designed to work hand-in-glove.

• FYI—At the beginning of each activity, an FYI box lists relevant topics in The Student Resource Handbook that correspond to the content of the activity. You may find it helpful to review the referenced topic as a content refresher while working the activity.
• Tools Appendix—A selection of the paper manipulatives used in selected activities are found in the Tools Appendix of the Student Resource Handbook. You may find it helpful to make copies of those tools in preparation for working a given activity. In addition, the Tools Appendix includes lists of useful resources for teachers and a synopsis of the NCTM Standards.

Use of Technology

The use of technology tools ranging from a Graphing Calculator, to a statistical software package, to Geometer's Sketchpad can be valuable in exploring mathematical ideas. Although we do not provide any instruction on how to use various tools, a number of our activities are written with the assumption that some sort of technology tool will be used while working them.

Supplements

Web Site (www.prenhall.com/masingila)
The Web site includes a number of support features for students and faculty including:

• "Projects"–a selection of more complex group activities that take more time than the activities in the text and often require outside research be done to arrive at a solution.
• Links to useful articles and publications that suggest scoring rubrics for exploratory, group based classes, teaching suggestions, etc.
• Contact information for the authors to provide a support for instructors as they embark on this new approach to teaching.
• Bulletin Boards for shared information

Instructor's Resource Manual (ISBN 0-13-018989-8) – a complete set of solutions to the activities in the text is available from the publisher to adopters of the Mathematics for Elementary Teachers via Problem Solving.

Edition details

• Publisher: Prentice Hall
• Publication date: 2001
• ISBN 10: 0130173452
• ISBN 13: 9780130173454
• Binding: Paperback
• Language: English
• Number of pages: 492