Mathematics Rebooted: A Fresh Approach to Understanding

287 Pages·2017·10.84 MB·English

by  Lara Alcock

MATHEMATICS REBOOTED
MATHEMATICS REBOOTED
A Fresh Approach to Understanding
LARA ALCOCK
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PREFACE
What do adults choose to learn? Interests are personal, but many learn
languages, or read about science, history, politics, economics, philosophy, or
psychology. And many enjoy both classic and contemporary art, literature, plays,
and films. Not everyone values these things equally, but many are proud to know
at least a little about each. I’ve written this book because I would like to see
mathematics added to the list. Not because mathematics is a standard school
subject, but because it is a cumulative intellectual endeavour with a long history
and a wealth of clever and interesting ideas. I wouldn’t expect a layperson to
know many details or to have any idea about the cutting edge, but I’d like more
people to feel that they could speak confidently about key mathematical
concepts and approaches to reasoning.
I got into a position to write this book by studying mathematics then
mathematics education. Mathematics education as an academic discipline
overlaps with mathematical cognition as a branch of psychology: both study
ways in which people learn and think about mathematics. And research in these
areas has revealed a lot. We know quite a bit about typical errors,
misconceptions, and sources of confusion in children and in adults. We have
good theories about how some of these arise, and we’re testing them with
intervention studies designed to improve teaching and learning. Education is
complicated, of course—anyone who has been in a class of 30 teenagers knows
that intellectual development requires more than good lesson plans. But teachers
and researchers know about numerous stumbling blocks in mathematical
thinking, and they will recognize much of the content in this book.
That said, this is not a book about research—I use my knowledge about both
mathematics and education in a more cavalier way than I would in academic
writing. I explain why some ideas are naturally confusing, but this book is
essentially an account of how I think about the subject. Like every teacher, my
thinking is heavily influenced by my early experiences, and I do not try to hide
that—I point out places in which I suspect that my way of understanding is
idiosyncratic. But I don’t include jokes, puns, or attempts to make mathematics
interesting. In my view, there is no need to make mathematics interesting—it is
fascinating all by itself.
This book has multiple intended audiences, so readers with mathematical
backgrounds will notice that sometimes, when I introduce an idea, I skate over
the subtleties. That’s deliberate: I think it can be important to consolidate a
simple version first. Sometimes the subtleties don’t appear until considerably
later in the book, so I hope that such readers will be patient. In particular, people
who have studied higher level mathematics will have been told to be wary of
intuition based on visual representations. That is sensible advice, and
mathematicians offer it when they want students to question their assumptions
and to justify their ideas within an established theory. Teaching disciplined
reasoning within established theories is a valid aim, but it’s not mine in this
book. My aim here is to communicate with nonspecialists about mathematical
ideas. And I really like pictures, so I use them a lot, albeit discussing their
limitations. Similarly, I start each main chapter with basic ideas, but I mean
basic in an everyday rather than mathematically foundational sense. In my
experience, learners need to work down to foundational ideas just as they need to
work up to advanced ones. So the more foundational discussions appear at the
end of the book, not the beginning.
To conclude this Preface, I would like to thank many friends and colleagues
for their help and feedback. For carrying this book through the practicalities
from proposal to finished product, thank you to Dan Taber of Oxford University
Press. For the usual extraordinarily patient and attentive copyediting and
typesetting, thank you to Charles Lauder Jr., and to Karen Moore and her team.
For extremely valuable feedback on drafts of the content, thank you to the
reviewers of the original proposal, and to Nina Attridge, Sophie Batchelor,
Louisa Butt, Jane Coleman, Lucy Cragg, Jo Eaves, Ant Edwards, Cameron
Howat, Hazel Howat, Matthew Inglis, Jayne Pickering, Artie Prendergast-Smith,
and David Sirl. Their input taught me that educated nonspecialists find algebra
and logic easier than I expected, but diagrams harder. This, together with
numerous detailed comments, improved my writing in ways that I hope will help
all readers.
Finally, I’d like to dedicate this book to my parents, Angela and Eric Alcock,
who always supported me but never pushed. That allowed me to develop a
genuine love for mathematics, and I’m grateful for it.
CONTENTS
Introduction
1 Multiplying
1.1 Famous theorems
1.2 Multiplication made easy
1.3 Properties of multiplication
1.4 ‘Multiplication makes things bigger’
1.5 Squares
1.6 Triangles
1.7 Pythagoras’ theorem
1.8 Pythagorean triples
1.9 Fermat’s Last Theorem
1.10 Review
2 Shapes
2.1 Tessellations
2.2 Regular polygons
2.3 Regular tessellations
2.4 Interior angles
2.5 Mathematical theory building
2.6 Semi-regular tessellations
2.7 More semi-regular tessellations
2.8 Algebra and rounding
2.9 Symmetry: Translations and rotations
2.10 Symmetry: Reflections and groups
2.11 Symmetry in other contexts
2.12 Review
3 Adding up
3.1 Infinite sums
3.2 Fractions
3.3 Adding fractions
3.4 Adding up lots of numbers
3.5 Adding up lots of odd numbers
3.6 Powers of 2
3.7 Adding up powers
3.8 The geometric series
3.9 The harmonic series
3.10 Convergence and divergence
3.11 Review
4 Graphs
4.1 Optimization
4.2 Plotting points
4.3 Plotting graphs
4.4 (or b)
4.5 More or less?
4.6 Intersecting lines
4.7 Areas and perimeters
4.8 Area formulas and graphs
4.9 Circles
4.10 Polar coordinates
4.11 Coordinates in three dimensions
4.12 Review
5 Dividing
5.1 Number systems
5.2 Dividing by 9 in base 10
5.3 If and only if
5.4 Division and decimals
5.5 Decimals and rational numbers
5.6 Lowest terms
5.7 Irrational numbers
5.8 How many rationals and irrationals?
5.9 Number systems
5.10 Review
Conclusion
Why didn’t my teachers explain it like that?
What is it all for?
What do mathematicians do?
What shall I read next?
References
Index