Research in Mathematics Education
Vol. 15, No. 2, July 2013, xx-xx
BOOK REVIEWS
Space, Time and Number in the Brain: searching for the foundations of
mathematical thought, edited by Stanislas Dehaene and Elizabeth Brannon, London,
Academic Press, 2011, 374 pp., £60.99 (hardback), ISBN: 0123859484
This book is a review of research looking into the cognitive basis of spatial orientation
and magnitudes of quantity, time, and distance. There is emphasis on neuropsychology
and neurophysiology, but the chapters also draw from studies of animal behaviour and
cognitive psychology, as well as cross-cultural studies. The book presents a good
overview of well-established research findings and plenty of new evidence from the
authors’ research. Altogether, the book challenged my previous view of how mathematics
is founded in the human brain.
The book is framed not only as a review of research but also as an outline for future
research agenda, and a look forward is well in place in a field that is developing fast.
Most of the content deals with a ‘general magnitude system’ that several of the authors
claim to be the foundation of the concepts of time, number and space in the human mind.
In addition, the book discusses the two foundational systems for geometry: one for
navigating in three dimensional (3D) layouts and another for recognising two
dimensional (2D) forms. It is not a small task that the authors have accomplished.
However, as the title of the book indicates, they are searching for the foundations but
they do not claim to have found them yet. Further research is needed before anything
conclusive can be said, and I feel that mathematics education researchers are needed for
Book reviews
this attempt to bear fruit. In the present volume, mathematics educators are not present,
although the last six chapters of the book were given a unifying title “Representational
change and education”. Their contribution is needed as there is more to the foundations
of mathematical thinking, especially for more advanced mathematics, than is covered in
this book.
The title of the book reminded me of Lakoff and Núñez’ (2001) “Where mathematics
comes from: how the embodied mind brings mathematics into being”. However, the two
books are working on different levels. Lakoff and Núñez explored how the human mind
can invent and comprehend abstract and complex mathematical ideas, such as negative
numbers, Boolean logic, infinity and limit. They claimed the human mind uses
metaphorical thinking to understand abstract ideas in terms of more familiar domains
often originating in sensory or motor experiences. The present book edited by Dehaene
and Brannon goes deeper to analyse the neural foundations of our basic understanding of
magnitude and location. Moreover, this book explores in depth the presence of such
foundations in more primitive animals such as chicks and rats.
Although I found the Lakoff and Núñez book wonderfully illuminating, this book
provides a more thorough review for the embodied foundations of the number concept,
and calls for some revisions to previous work in this area. For example, Lourenco and
Longo make explicit reference to conceptual metaphors and claim that both number and
time are conceptualised in spatial terms. This is not completely aligned with Lakoff and
Núñez’s basic metaphors of arithmetic, where the most primitive metaphor for number
seems to be a collection of objects, while a spatial metaphor for numbers (point-locations
on a path) is more advanced. The book suggests a minor revolution in our perception of
Research in Mathematics Education
early learning and the emphasis on subitizing and learning of number word sequences as
the origin for learning numbers. Butterworth and Piazza provide additional detail to the
picture, challenging some established views. Piazza discusses two cognitive systems that
are able to identify between one and four elements prior to formal learning of numbers:
the object tracking system (OTS); and, the approximate number system (ANS). There is
evidence in favour of the approximate number system being more important for the
learning of numbers. For example, dyscalculic individuals have impaired ANS, while
their OTS seems to be normal.
The foundations of number and geometry are fairly well covered in this book, but it
includes very little work towards exploring how these findings might inform the early
learning of mathematics and its instruction in schools. The overall message of this book
would suggest utilising the number line more extensively in early mathematics education,
as it would coincide with a pre-existing general magnitude system. Moreover, practising
estimations of time, distance, and numerosity would support improving the accuracy of
the magnitude system, which some authors in this volume suggest is necessary for the
learning of the number concept.
My biggest expectations for the book concerned research on the foundations of
geometrical thought. Early learning of geometry has generally received less attention than
early learning of number, and I expected to learn more about this area. I was only slightly
disappointed in this respect. The foundations of geometrical thinking received far less
attention in the volume than the approximate number system. Yet, the chapters that
focused on spatial orientation review interesting recent research that ranges from the
different spatial maps in the brain to the navigation performance of children, adults of
Book reviews
different cultures, and adults suffering from impaired geometric orientation. Spelke
concludes that there are two different founding systems for geometry, both having their
shortcomings. Our navigation of 3D layouts is based on innate perception of distance and
direction, but not angle. On the other hand, small manipulable objects (and 2D displays)
are interpreted based on distances and angles, but not on direction. If direction is not
involved in our earliest interpretations of 2D forms, this might explain why children often
draw mirror images when they try to copy numbers or other symbols. Integration of these
two systems is the basis of geometrical understanding.
Much of the research reviewed in the book is done with animals rather than with humans.
Hence, the focus is strongly on the evolutionary foundation of human mathematical
thinking. The power of the book is in summarising what research has found out about the
innate capabilities that underlie the early learning of mathematics. However, the book
also reviews some interesting studies about learning certain mathematical ideas in
different cultures. For example, Izard, Pica, Dehaene, Hinchey and Spelke review, in
their chapter, Dehaene’s and Izard and Spelke’s studies of the Amazonian Mundurucu
people, who have no geometry education and whose language lacks many Euclidian
expressions such as ‘right angle’ and ‘parallel’. A test for geometric intuitions of
Mundurucu and U.S. children and adults provides insight into the significance of explicit
instruction for learning geometrical ideas. Surprisingly, both populations showed highly
correlating results: the difficulty of different tasks and the age-related development was
similar in both populations. Hence it seems that, for example, the centre of a circle or
parallelism are geometrical ideas that will be learned without explicit instruction or
supporting discourse. However, Mundurucu people seemed to be insensitive to ‘sense’
Research in Mathematics Education
(i.e., recognising the difference between an image and its mirror image).
The structure of argumentation in the book is challenging for the reader. The strongest
claims and boldest hypotheses were made in the foreword and the first chapter. I found
myself very suspicious at that phase of reading, because the claims were not supported by
sufficient evidence. As the book continued, the handling was better balanced, making
more explicit distinctions between conclusions based on evidence and hypotheses
suggested for future research. Towards the end, I was more willing to accept the
plausibility of claims made in the beginning, although the evidence in their favour was
not conclusive.
Technically, the book is of good quality. The language is clear, and the index helps the
reader to find information on specific concepts. The illustrations are informative and
pretty to look at, but some text in them is too small to be read easily. Several chapters
also have a glossary of key terms used and the main issues are discussed in separate text
boxes. The book is pleasant to read from cover to cover, and it works well as a handbook.
I found much of the research reported in the book simply fascinating. It is magical how a
spatial map in a rat’s brain fires certain neurones depending on its location in an open
environment, and how this map changes when the rat is in a labyrinth of equal
dimensions. Equally mesmerising are findings that Australian aboriginals who speak
Kuuk Thaayorre organise time-sequenced photographs (e.g., a person aging) from east to
west.
Some information provided by the book could be useful for research design in specific
domains. For research on estimations, the different chapters review several known biases,
Book reviews
such as the influence of regular arrangements. In their article, Cavanagh and He
challenge the established view that eye movements would be necessary for explicit
counting. However, because our attentional resolution in peripheral vision is roughly ten
times worse than visual resolution, we can count objects in our peripheral visual field
only if they are sufficiently far from each other. In another chapter, Kadosh and Gertner
suggest that approximately 20% of the population would be synesthetics with respect to
time, number and space, having vivid and fixed visual perception of numbers and/or time
in specific spatial locations. These synesthetics have been found to be faster and more
accurate in basic processing of numbers, but at the cost of being less flexible.
I see the research on the neural basis of mathematics to be useful for any mathematics
educator, and the book provides a good presentation of research done in this field. Those
who are short of time can read at least Spelke’s chapter “Natural number and natural
geometry”, which summarises nicely the overall picture. For those into research of the
early learning of mathematics, I would consider this book essential reading.
References
Lakoff, G., and Núñez, R. 2001. Where mathematics comes from: How the embodied
mind brings mathematics into being. New York: Basic Books.
Markku S. Hannula
University of Helsinki, Finland
Email: markku.hannula@helsinki.fi
Research in Mathematics Education
© 2013, Markku Hannula