Ratio and Proportion: Research and Teaching in Mathematics Teachers’ Education (Pre- and In-Service Mathematics Teachers of Elementary and Middle School Classes)

264 Pages·2012·4.488 MB·English

by  David Ben-Chaim, Yaffa Keret, Bat-Sheva Ilany (auth.), David Ben-Chaim, Yaffa Keret, Bat-Sheva Ilany (eds.)

Ratio and Proportion

Ratio and Proportion
Research and Teaching in Mathematics Teachers’ Education (Pre- and
In-Service Mathematics Teachers of Elementary and Middle School Classes)
David Ben-Chaim
The Technion- Israel Institute of Technology, Israel
Yaffa Keret
The Open University of Israel, Israel
Bat-Sheva Ilany
Beit Berl Academic College, Israel
A C.I.P. record for this book is available from the Library of Congress.
ISBN: 978-94-6091-782-0 (paperback)
ISBN: 978-94-6091-783-7 (hardback)
ISBN: 978-94-6091-784-4 (e-book)
Published by: Sense Publishers,
P.O. Box 21858,
3001 AW Rotterdam,
The Netherlands
https://www.sensepublishers.com/
Printed on acid-free paper
All Rights Reserved © 2012 Sense Publishers
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording or otherwise, without written permission from the Publisher, with the
exception of any material supplied specifically for the purpose of being entered and
executed on a computer system, for exclusive use by the purchaser of the work.
TABLE OF CONTENTS
Chapter 1: Introduction ........................................................................................ 1

PART I: STRUCTURE OF BOOK AND TEACHING MODEL 5
Chapter 2: Structure of Book................................................................................ 7
Chapter 3: A Model for Teaching Ratio and Proportion
Using Authentic Investigative Activities .......................................... 13

PART II: THEORETICAL BACKGROUND 21
Chapter 4: A Mathematical Perspective of Ratio and Proportion ...................... 23
Chapter 5: Proportional reasoning—A psychological-didactical view .............. 49
Chapter 6: Research and new approaches in pre- and
in-service mathematics teacher education ........................................ 61

PART III: AUTHENTIC INVESTIGATIVE ACTIVITIES, INCLUDING
DIDACTIC COMMENTS AND EXPLANATIONS 71
Chapter 7: Authentic investigative activities— Introduction ............................. 73
Chapter 8: Group 1: Introductory activities ........................................................ 83
Chapter 9: Group 2: Rate activities .................................................................... 95
Chapter 10: Group 3: Ratio activities ................................................................. 125
Chapter 11: Group 4: Stretching and Shrinking: Scaling Activities ................... 145
Chapter 12: Group 5: Indirect Proportion Activities .......................................... 181
Chapter 13: Group 6: Additional Activities
for Practice and Enrichment (All topics) ........................................ 195

PART IV: ASSESSMENT TOOLS 219
Chapter 14: Introduction to Assessment Tools .................................................. 221
Chapter 15: Questionnaire: Attitude Toward Ratio and Proportion ................... 225
Chapter 16: Diagnostic Questionnaire in Ratio and Proportion ......................... 233
Chapter 17: Assessing Research Reports and
Building a Student Portfolio ........................................................... 245

PART V: ANNOTATED RESOURCES AND BIBLIOGRAPHY 253
Chapter 18: Annotated Resources ...................................................................... 255
Chapter 19: Bibliography ................................................................................... 269

v

CHAPTER 1
INTRODUCTION
The concepts of ratio and proportion are fundamental to mathematics and
important in many other fields of knowledge. Many phenomena can be expressed
as some proportional relationship between specific variables, often leading to some
new, unique entity. Conceptualization and comprehension of these concepts, not to
mention skills and competence in using them, facilitate mathematic awareness.
Even more importantly, these skills foster the ability to use relational reasoning,
otherwise known as proportional reasoning, which is crucial to the development
of analytical mathematical reasoning.
Studies world-wide have shown that most pre- and in-service mathematics
teachers not only have insufficient understanding of basic mathematical content
knowledge, they also lack basic pedagogic-didactic abilities for the mathematical
subjects taught in primary and middle schools. Their comprehension of
mathematics is predominantly technical and schematic, and they have neither
assimilated understanding nor formulated knowledge at such levels that can lead to
sufficient and adequate fluency in the concepts. In other words, the difficulties that
many pre- and in-service mathematics teachers have in understanding the concepts
of ratio and proportion result in the inability to properly teach them.
The intrinsic difficulty that pre- and in-service teachers alike have in
understanding the concepts of ratio and proportion, combined with the general lack
of instruction for teachers, ultimately results in teachers with negative attitudes
towards teaching this subject, especially amongst those teaching mathematics in
elementary school.
Unfortunately, most teaching institutions world-wide do not provide teachers
destined for teaching mathematics in elementary or middle schools any systematic,
in-depth, comprehensive instruction of these mathematical topics from a psycho-
didactic perspective.
This book comes to solve this problem.
While this book is primarily intended for use by instructors in teacher-
training institutions who are preparing pre- and in-service teachers to teach
mathematics in elementary and middle school, it is also suitable for use in
continuing teacher education programs, and for anyone interested in teaching or
studying the subject of ratio and proportion.
This book is an ideal sourcebook for constructing a comprehensive teaching
unit on the subject of ratio and proportion by applying and adapting the unique
teaching model presented. This teaching model combines theoretical instruction
with practical investigative activities, thus offering teachers firsthand experience in
1
CHAPTER 1
solving authentic problems along with the deeper understanding that results from
analyzing published research reports. (These reports offer, among other things,
varied and diverse solutions to problems that may arise when actually teaching the
subject in class.) By giving novice (and experienced) teachers deep pedagogic
mathematical understanding combined with concrete, hands-on experience, a
positive change in attitude and confidence is elicited, improving their ultimate
teaching experience, and benefitting both teacher and student alike.

Briefly, this book comprises the following sections.

Part One: Structure of Book and Teaching Model, describes this book and
the teaching model developed following five years of research preparing and
giving courses to pre- and in-service teachers in several Israeli teacher colleges and
universities' departments of education. During those years, the teaching concept
was developed, authentic investigative activities were prepared and tested, and a
pioneering project was carried out to test the necessity of such a book. The
premises put forth were presented at international conferences and validated
through discussions with colleagues worldwide.
Part Two: Theoretical Background presents three chapters dealing with the
mathematical, psychological-didactic, and teacher-training aspects of ratio and
proportion.
The mathematical perspective explains how the study of ratio and proportion
progresses through the three sources of knowledge: intuitive (identification of the
existence of some proportional relationship in various comparisons), formal
(comprehension of the different types of proportional problems and the strategies
needed for solving them), and procedural (the perception and understanding of the
mathematical-numerical solutions required to solve any of a wide range of
proportional problems). The psychological-didactic aspect of the topic explains
the results of research concerning the development of proportional reasoning, the
strategies used for solving proportional problems, and sources of difficulties in
understanding the subject. The chapter on training teachers reviews various studies
and approaches in order to round out and give deeper insight to the instruction of
ratio and proportion.
Part Three: Authentic Investigative Activities, is the major content of this
book, and presents a wide, comprehensive range of authentic investigative
activities in ratio and proportion. The activities are divided into five types, from
easiest to most difficult: introductory activities, rate activities, ratio activities,
scaling activities, and activities dealing with inverse proportion. Within each
type, various levels of difficulty are offered, ranging from problems that presume
no familiarity whatsoever with the subject, and culminating with complicated,
difficult ones. Some of the activities also include a second stage, which presents an
even greater level of complication, and which incorporate additional, higher-level
mathematical concepts, such as percentages, functions, and more. Some activities
suggest additional assignments that can be used for further practice and
enrichment. In addition, many of the activities show how problems of ratio and
2
INTRODUCTION
proportion are linked to other mathematical concepts and to other fields of
knowledge.
While the activities in this book are specifically meant for training teachers,
they are all easily adapted for use in elementary and middle schools classes. This
book, therefore, especially this section (Part Three), can serve as a valuable source
for primary-school mathematics teachers.
Part Four: Assessment Tools, provides questionnaires to test the abilities and
attitudes of the participants for evaluation and diagnostic purposes. By conducting
the evaluation tests both at the beginning (pre-testing) and end (post-testing) of the
course, an accurate measure of progress can be made.
Also included in Part Four are explanations and instructions on analyzing
research reports, plus detailed information on the purpose, benefits, and required
content of a student portfolio.
Part Five: A List of Annotated Sources, offers articles from the professional
literature and other resources concerned with the topics of ratio and proportion.
The course instructor can integrate these resources into the teaching process in a
number of ways: by assigning appropriate, timely articles from this section for
analysis; by incorporating these additional resources as part of assigned
homework; or by suggesting these additional sources for in-service teachers to
adopt and present in their classes at school.
A comprehensive bibliography follows.
To the best of our knowledge, there is no book currently available that not only
comprehensively covers the subject of ratio and proportion, but that also presents a
unique teaching model that uses an extensive variety of authentic investigative
activities. As it goes without saying that firsthand, direct experience leads to the
broadest, deepest comprehension of any subject, this book on ratio and proportion
should prove to be a valuable addition to the curriculum of any pre- and in-service
teacher training or professional development program in mathematics education.

3