Mathematical Imagining: A Routine for Secondary Classrooms

257 Pages·2020·3.589 MB·English

by  Christof Weber (author)

M
Imagine a plastic cup lying on the floor.
A
Give the cup a nudge so that it begins to roll. What does the path it takes look like?
T
H
E
M
that Christof Inside you’ll find the following:
So begins the journey
Weber takes you on in Mathematical Imagining.
• an introduction to the routine including
A
Along the way, he makes the case that the ability the rationale behind it, facilitation
guidance, and classroom examples
to imagine, manipulate, and explain mathemati- T
cal images and situations is fundamental to all • modifications to implement the routine I
mathematics and particularly important to higher in your classroom within varying time C
constraints
level study. Most importantly, drawing on years of
• 37 exercises broken into four categories: A Imagining
experiments in his own classroom, Weber shows
constructions, problem-solving,
that mathematical imagining is a skill that can be reasoning, and paradoxes L
taught efficiently and effectively.
• discussions of the mathematics involved
in each exercise, including follow-up I A ROUTINE FOR
Mathematical Imagining describes an questions
original routine that gives students space and • instructions on how to create your own m SECONDARY CLASSROOMS
time to imagine a mathematical situation and exercises beyond the book
a
then revise, discuss, and act upon the mental
This one-of-a-kind resource is for secondary g
images they create. You can use this creative
teachers looking to inspire student creativity i
routine in your secondary classroom to glimpse n
and curiosity, deepen their own subject matter
into your students’ thinking and discover i
knowledge and pedagogical content knowledge, n
teaching opportunities, while empowering them
g
and invite all students to access the power of their
to create their own mathematics.
own mathematical imaginations.
C
h
r Christof Weber
i
s
Foreword by John Mason
t
o
is an associate professor in the School of Education at the University
Christof Weber f
of Applied Sciences Northwestern Switzerland. He taught mathematics for twenty-five
W
years at a “Gymnasium” in Switzerland, a post-compulsory public high school preparing
students for tertiary education.
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b
e
r
#MathematicalImagining
#StenhouseMath
.
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Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
.
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
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Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
.
PORTSMOUTH, NEW HAMPSHIRE
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
.
Stenhouse Publishers
www.stenhouse.com
Copyright © 2020 by Christof Weber
All rights reserved. This e-book is intended for individual use only. You can print a copy for your own
personal use and you can access this e-book on multiple personal devices (i.e., computer, e-reader,
smartphone). You may not reproduce digital copies to share with others, post a digital copy on a
server or a website, make photocopies for others, or transmit in any form by any other means, elec-
tronic or mechanical, without permission from the publisher. Every effort has been made to contact
copyright holders and students for permission to reproduce borrowed material. We regret any over-
sights that may have occMurartehde amnadt iwscihlle b Veo prlsetaelsleudn gtosü rbeucntigfye nth iemm U innt seurrbiscehqtuent reprints of the work.
Originally published as by Christof Weber, copy-
right © 2010 by Friedrich Verlag GmbH. Friedrich is not responsible for the quality of the translation.
Library of Congress Cataloging-in-Publication Data
Names: Weber, Christof, author.
Title: Mathematical imagining : a routine for secondary classrooms /
 Christof Weber.
Identifiers: LCCN 2019036366 (print) | LCCN 2019036367 (ebook) | ISBN
 9781625312778 (paperback) | ISBN 9781625312785 (ebook)
Subjects: LCSH: Mathematics—Study and teaching (Secondary)
Classification: LCC QA11.2 .W43 2020 (print) | LCC QA11.2 (ebook) | DDC
 510.071/2—dc23
LC record available at https://lccn.loc.gov/2019036366
LC ebook record available at https://lccn.loc.gov/2019036367
Figure 6.4 © 2019, ProLitteris, Zurich. Foto: Peter Lauri.
Figure 6.5a–6.5d Courtesy Art Affairs, Amsterdam. Foto: Studio Brandwijk, Badehoevedorp.
Figures 7.3 and 7.4 Copyright John M. Sullivan, Technische Universität Berlin, Germany.
Cover design by Cindy Butler
Interior design and typesetting by Shawn Girsberger
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
.
CONTENTS
Foreword vi
Preface to the English Edition ix
Acknowledgments xi
An Invitation into My Classroom xii
Introduction xvi
PART 1 THE MATHEMATICAL IMAGINING ROUTINE: BACKGROUND AND
GUIDELINES 1
CHAPTER 1 What Is the Mathematical Imagining Routine? 2
CHAPTER 2 Implementing the Mathematical Imagining Routine 15
CHAPTER 3 Developing Your Own Exercises in Mathematical Imagining 40
PART 2 EXERCISES IN MATHEMATICAL IMAGINING: A COLLECTION OF
EXAMPLE TASKS 51
CHAPTER 4 About Using the Example Tasks 52
CHAPTER 5 Construction Exercises in Mathematical Imagining 59
CHAPTER 6 Problem-Solving Exercises in Mathematical Imagining 123
CHAPTER 7 Reasoning Exercises in Mathematical Imagining 161
CHAPTER 8 Paradox Exercises in Mathematical Imagining 203
References 224
Index 226
v
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
.
FOREWORD
T
he power and importance of mental imagery has been recognized as long as
people have sat around campfires and told stories. It is through stories evok-
ing images that we experience vicarious emotions and the consequences of being
ruled by them, and it is through forming mental images that human beings plan
or prepare for future action. In the Upanishadic metaphor for human psyche as a
horse-drawn chariot, mental imagery corresponds to the reins, the means by which
the intellect communicates with and directs the horses, the emotional energy made
available by the senses.
The case can be made that all mathematicians exploit imagery at least implicitly,
and in some cases, there is evidence of exploiting it explicitly: for example, Begehr
andS Loemnze t(i1m9e9s8 h, 5e 0c)l ossaeyd a bthoeu tc Juarktaoibn Ss tienin oerrd, tehre t gor edaatr ekiegnh tteheen ctlha-scsernotoumry s goe othmaet ter:
students could better follow and “see” his geometric constructions which he
was describing using words and sometimes his hands.
Philosophers and educationalists such as Alfred North Whitehead and Caleb Gatte-
gno, among many others, have recognized that education is largely about support-
ing learners in bringing to the surface, developing, and using their natural powers.
Of the many powers we all possess, the power to imagine—and to express what
we imagine in words, diagrams, pictures; in manipulating material objects; in ges-
tures, movements, and sound—is at once the most basic and possibly the most far
reaching of them all.
Mathematics exploits and depends on this power to imagine. It is much
more than “glorified arithmetic.” It is about discerning details and recognizing
vi
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
. Foreword vii
relationships between those details and then extracting general properties that
can be perceived as being instantiated in those relationships. Put another way, the
power to imagine beyond the confines of the material world is both essential to
mathematics and what gives mathematics its evident and undoubted power.
do
Even though some people appear to be unable to “see” pictures in their mind,
imagery
they do something in response to the invitation to imagine. So it may be helpful
to allow the word to refer to any and all forms of virtual-sensory experi-
ence. Furthermore, repeated exposure to invitations to imagine can develop the
strength and power of imagery. In my own case, I was much stronger at detecting
errors in strings of symbols, a form of algebraic sensitivity, than explicitly imag-
ining mathematical objects, until I joined a group of people working on and with
mental imagery. Over time, I found that I had developed a geometrical sense of
configurations, which were the subject of my research, even though they were mul-
tidimensional and did not belong in or fit into Euclidean space. I still do not know
whether I literally “see” mentally or whether I have a multifaceted “sense” of what
not
I am thinking.
not
What is perfectly clear is that making use of learners’ powers to imagine
and supporting them in developing that power is like trying to teach someone
to throw a ball with their hands tied behind their back. It makes no sense at all.
In this book, you will find not only some exercises through which to develop
your power to imagine and to express what you imagine but also some very specific
suggestions about how to do this in an effective way pedagogically, making use of
the structure of human psyche. When learners express what they are imagining,
completely unexpected aspects may emerge, and the book gives advice on how to
exploit what students come up with. I myself have looked for ways to avoid or side-
line mathematically inappropriate or unproductive images that emerge, but here
you will find ways to encourage learners to modify or restructure those images
so that they become productive. You will also find explicit proposals for how to
develop your own imagery tasks to use with learners.
There is a common expression in English used in response to what someone is
saying: “I see what you are saying.” Some people advocate also using the phrase “I
hear what you are saying,” especially when the speaker is more audile than visile
in their propensities. But if you really want learners to experience these, to be able
to say, “I see what you are saying” and “I hear what you are saying,” it is neces-
sary to become expert at “saying what you are seeing” and “saying what you are
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher
viii Foreword .
hearing,” in other words, to be articulate in expressing what you are imagining. To
do this requires ongoing and repeated practice at imagining, at becoming aware
that you are imagining, and at being aware of what it is that you are imagining. This
awareness can then feed into becoming aware of how you express what you are
imagining to others. The discipline offered in this book is likely to provide a firm
foundation for this ongoing study.
from
To my mind, the greatest contribution this book can make is to bring the reader
about
to awareness of their own imagery and awareness of how to speak , not simply
, what they are imagining. Through this awareness, the reader may be able to
make a significant educational contribution to the life of learners; for though these
may or may not continue through life—imagining mathematical objects and rela-
tionships—their lives will undoubtedly be enhanced and enriched by becoming
articulate in expressing, in whatever medium suits them, what they are imagining.
Of course, to be maximally effective, this has to take place in a caring atmosphere,
in which the teacher evidently cares for both learners and mathematics. Effective
pedagogy resides in a balance between these foci of concern.
John Mason
Professor Emeritus, Open University
Honorary Research Fellow, University of Oxford
Mathematical Imagining by Christof Weber. © 2020 Stenhouse Publishers. No reproduction without written permission from the publisher