The Nature and Development of Mathematics: Cross Disciplinary Perspectives on Cognition, Learning and Culture

127 Pages·2017·6.88 MB·English

by  John Adams, Patrick Barmby, Alex Mesoudi (eds.)

The Nature and Development
of Mathematics
From an infant s first grasp of quantity to Einstein s theory of relativity, the
human experience of number has intrigued researchers for centuries. Numer­
acy and mathematics have played fundamental roles in the development of
societies and civilisations, and yet there is an essential mystery to these concepts,
evidenced by the fear many people still feel when confronted by apparently
simple sums.
Including perspectives from anthropology, education and psychology, The
Nature and Development of Mathematics addresses three core questions: Is maths
natural? What is the impact of our culture and environment on mathematical
thinking? And how can we improve our mathematical ability? Examining the
cognitive processes that we use, the origins of these skills and their cultural
context, and how learning and teaching can be supported in the classroom, the
book contextualises each issue within the wider field, arguing that only by tak­
ing a cross-disciplinary perspective can we fully understand what it means to be
numerate, as well as how we become numerate in our modern world.
This is a unique collection including contributions from a range of renowned
international researchers. It will be of interest to students and researchers across
cognitive psychology, cultural anthropology and educational research.
John W. Adams is a Lecturer in Psychology and a Fellow of the Wolfson
Research Institute for Health and Wellbeing, Durham University. His research
interest is in cognitive development and its relation to educational attainment,
which includes the role of working memory in the development of mathemati­
cal skills.
Patrick Barmby is a Senior Lecturer in Mathematics Education at Wits Uni­
versity in Johannesburg. His research interests include use of representations
in classrooms, problem solving and mathematical thinking, attitudes towards
mathematics and eye-tracking research.
Alex Mesoudi is Associate Professor in Cultural Evolution at the University of
Exeter and conducts research into human cultural evolution and social learning
using lab experiments and theoretical models. His work intersects anthropol­
ogy, archaeology, biology and psychology.
The Nature and Development
of Mathematics
Cross Disciplinary Perspectives on
Cognition, Learning and Culture
Edited by
John W. Adams,
Patrick Barmby
and Alex Mesoudi
Routledge
Taylor & Francis Croup
LONDON AND NEW YORK
Contents
First published 2017
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon, 0X14 4RN
and by Routledge
711 Third Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2017 Taylor & Francis
The right of John W. Adams, Patrick Barmby and Alex Mesoudi to be
identified as the authors of the editorial material, and of the authors for
their individual chapters, has been asserted in accordance with sections 77
and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or
utilised in any form or by any electronic, mechanical, or other means, now
Preface vii
known or hereafter invented, including photocopying and recording, or in
any information storage or retrieval system, without permission in writing Outline of the book viii
from the publishers.
Trademark notice: Product or corporate names may be trademarks or
registered trademarks, and are used only for identification and explanation 1 Introduction: will 3 into 1 go? A multi-disciplinary
without intent to infringe. view of emergent maths 1
British Library Cataloguing-in-Publication Data JOHN W. ADAMS, PATRICK BARMBY AND ALEX MESOUDI
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data 2 Nature/genes: is maths natural? 21
A catalog record for this book has been requested
2.1 Nature/nurture and the origin of individual differences in
ISBN: 978-1-138-12441-7 (hbk) mathematics: evidence from infant and behavioural genetics
ISBN: 978-1-315-64816-3 (ebk)
studies 23
Typeset in Bembo
ELENA RUSCONI AND JANET F. MCLEAN
by Apex CoVantage, LLC
2.1.1 Commentary 41
UBIRATAN D’AMBROSIO
AND MANOEL DE CAMPOS ALMEIDA
2.2 An animal’s sense of number 43
GIORGIO VALLORTIGARA
2.2.1 Commentary 66
ELENA RUSCONI AND JANET F. MCLEAN
2.3 Ethnomathematics and the emergence of mathematics 69
UBIRATAN D’AMBROSIO AND MANOEL DE CAMPOS ALMEIDA
2.3.1 Commentary 86
GIORGIO VALLORTIGARA
3 Culture/environment: the impact of society 90
3.1 Early emergence of quantitative knowledge: implications for
educational practice 91
MIX
Papwltam DAVID C. GEARY
FSC ratpoiwiM* aouroM Printed and bound in Great Britain by
wwwJKorp FSC* C013056 TJ International Ltd, Padstow, Cornwall
3.1.1 Commentary 105
SIEGHARD BELLER AND ANDREA BENDER
vi Contents
3.2 Number systems in Oceania: the cultural evolution of
Preface
a cognitive tool 110
ANDREA BENDER AND SIEGHARD BELLER
3.2.1 Commentary 131
STEPHEN LERMAN
3.3 Culture, the environment, and the impact of society:
an educational perspective 133
STEPHEN LERMAN
3.3.1 Commentary 148
DAVID C. GEARY
4 Improving maths attitudes/learning 153 In 2014 the co-editors were awarded funding from Durham University’s Insti­
4.1 Improving working memory to enhance maths performance 154 tute of Advanced Study to award a visiting fellowship to Professor David Geary
and to organise a Multi-disciplinary Mathematics Workshop to bring together
JON! HOLMES AND DARREN L. DUNNING
different perspectives on developing mathematical knowledge to see how they
4.1.1 Commentary 174
can inform instruction and learning, particularly in the classroom.
ALEX MESOUDI This workshop (also supported by the Wolfson Research Institute) exam­
ined the emergence of early mathematical concepts, looking at the interaction
4.2 What is mathematics? Perspectives inspired
between the cultural and historical, the psychological, and the educational per­
by anthropology 119
spectives. The event provided an opportunity for practitioners and academics
JENS H0YRUP
to come closer together in their understanding of the challenges that maths
4.2.1 Commentary 197 education presents, both to the learner and the educator. The activity reflected
GILAH C. LEDER the need for closer links to be established between theory and practice and the
need for more inter-disciplinary synergies to be explored.
4.3 Attitudes to mathematics, its teaching and learning:
The keynote address was given by Professor Geary, a world expert on the
educational perspectives 201
psychological development of mathematical knowledge in children, whose
GILAH C. LEDER
research also examines this issue from evolutionary and neurobiological per­
4.3.1 Commentary 219 spectives. Other world leading academics were invited to provide a unique
JONI HOLMES AND DARREN L. DUNNING inter-disciplinary program of talks; some of those academics have provided
chapters for this book.
5 Discussion: does it all add up? 224 The workshop aimed not only to bring multiple perspectives together, but
JOHN W. ADAMS, PATRICK BARMBY AND ALEX MESOUDI to bridge the gap between theory and practice.
I really enjoyed the Workshop and found all of the presentations very interesting.
Index 235
As a teacher, it was good that there was a mix of theory and elements that
could have a practical application in the classroom.
(Teacher participant)
Through reading this volume we hope that researchers, teachers and education
professionals will engage with this research in the hope of building partnerships
in the wider community.
Outline of the book 1 Introduction
Will 3 into 1 go?
A multi-disciplinary view
of emergent maths
John W. Adams, Patrick Barmby and Alex Mesoudi
In this introductory chapter, we will attempt to present three different perspec­
The human experience of number has intrigued researchers for centuries. From
tives on the process of learning mathematics. We will begin with a psychologi­
an infant’s first grasp of quantity to Einstein’s theory of relativity we live in a
cal view of learning mathematics and then present an anthropological view of
world where numeracy and mathematics are fundamental. Diverse academic
the development of mathematical understanding. We will then present an edu­
perspectives attempt to explain key issues in our mathematical world, each
cational view, thinking about the perspectives held in mathematics education
bringing a wealth of knowledge and experience. This book aims to present
with a specific focus on understanding. We conclude by relating these three
three such perspectives to the understanding of what it means to be a numerate
perspectives, gaining an indication of the common themes that emerge.
human.
Now why would we want to try and present and relate these different per­
This edited volume brings together world renowned academics from Anthro­
spectives? In trying to combine the psychological and anthropological views
pology, Education and Psychology to address some fundamental questions on
with the educational perspective, ultimately, we want to see whether the con­
the human experience of number. Their contributions are related to three core
nections that we can make provide greater insight and possibly ways of develop­
questions which we posit as the following:
ing our educational perspectives on the learning of mathematics. Quite possibly,
we might find that we struggle to make connections, that the different per­
• Is maths natural?
spectives clash. If this is case, once again, this would raise interesting questions
• What is the impact of our environment on mathematical understanding?
from which all the different perspectives could benefit. As we will highlight
• And how can we improve human mathematical ability?
particularly in the educational part of this chapter, one way that we can view
the notion of‘understanding’ is as a network of connections made between dif­
The chapters provided demonstrate the diverse research on these issues, but we
ferent ‘representations’ of a given concept. Therefore, in attempting to look for
hope to show that these perspectives can learn from each other and enhance
the connections, or the instances where we find it difficult to connect, among
our understanding.
the different disciplinary perspectives, we aim to develop our understanding of
what is involved in learning and understanding mathematics.
A psychological perspective
The emergence of numeracy skills in children is crucial for their cognitive
development. Like the development of language, number plays an important
role in acquiring the mathematical skills we need to take an active role in soci­
ety. Psychology is interested in how these abilities emerge and what accounts
for individual differences in these abilities. Lev Vygotsky (1896-1934) con­
sidered conscious thought as the progressive build-up of representations and
processes as a result of interactions with the environment, given the basis of
elementary biological ‘givens’. This classic view holds true today as scientists
consider the interplay between nature and nurture. What abilities do we natu­
rally develop and what abilities do we learn?
2 John W. Adams et al. Introduction 3
The idea that mathematicians are born and not bred has had some support from a number of studies which examined adult ability to judge set size. He
over recent years as psychologists have studied babies’ sensitivity to number. noted that adults responded quicker to small set sizes than large sets, but that the
A classic study by Starkey, Spelke and Gelman (1990) found that if you show reaction time (RT) data showed adults ‘count’ small and large sets differently.
babies two things and then another two things they gradually lose interest and Small sets up to five items had a 40-millisecond increment per count item, five
start to look away for longer and longer periods. Then you show them a set and above, a 300-millisecond increment. Thus, for sets above four or five items
with three items, and they become interested again. Following, you show them we appear to start a process of verbal or sub-verbal counting to mentally rep­
more sets with three, and they lose interest, and then you show them a set with resent larger quantities. Thus, subitizing seems to be a perceptual skill different
two items, and they gain interest again. Researchers change what the objects from verbal counting.
are and vary lots of the visual features to control for possible confounds. This The other system proposed by Feigenson, Dehaene and Spelke (2004) is the
method, known as the preferential looking design, arguably demonstrates that Approximate Number System (ANS). This is the ability to make numerical
babies are sensitive to changes in quantity. discriminations using ratio acuity in accord with Weber’s Law (m).That is, the
One suggestion is that infants are sensitive to the ratio between items; how­ ability to detect a change (Just Noticeable Difference) in a stimuli at least 50%
ever, while the infants in this study could discriminate two versus three items, of the time. Libertus and Brannon (2010) found that the ANS of 6-month-old
they performed below chance on four versus six items. While it is plausible infants shared the same ‘signature of ratio-dependent discrimination’ as seen in
from an evolutionary perspective that infants, and indeed all animals should older children, adults and non-human animals. They also noted that individual
be able to discriminate between quantities (number sense), is it likely that they differences in numerical acuity are relatively stable as the children develop.
will also have a rudimentary ability to perform or understand mathematical Hence, individual differences in the acuity of the ANS have been suggested
operations? as having a causal role in an individual’s mathematical attainment (Booth &
The preferential looking design was also used by Karen Wynn (1992) to dem­ Siegler, 2008; Halberda, Mazzocco & Feigenson, 2008). Libertus, Feigenson and
onstrate that 5-month-old infants have some notion of simple addition and sub­ Halberda (2011) found that ANS predicted formal math ability (as measured by
traction. Given simple addition and subtraction events to view using no more the Test of Early Mathematics Ability; Ginsburg & Baroody, 2003) in children
than three objects, the infants looked longer at unexpected results (1 + 1 = 1) 3-5 years old. However, the study did not control for other cognitive abilities
than at expected ones (1 + 1 = 2).Wynn argued that the infants behaviour sug­ such as non-verbal intelligence (NVIQ), working memory (WM) or speed of
gests that they have a true conception of the ‘individual entity’ of items, and of processing, and the amount of variance ANS explained was only 13% of solu­
numerical relationships (p. 750). tion accuracy and less for co (6%) and RT (5-8%) respectively.This suggests that
Butterworth (1999a) proposed that we are born with a number module which other influences are potentially more important in predicting mathematical
‘categorize(s) the world in terms of numerosities’ (p. 7).This ability he argues ability than simply numerical acuity.
can explain individual differences in children’s mathematical ability. Developmen­ While the ANS can account for some early variance in number discrimina­
tal dyscalculia is the term used by some to label individuals with mathematical tion tasks, the extent to which it can predict later more complex mathematical
difficulties that cannot be explained by a lack of schooling, low intelligence or ability is unproven. Clearly, this early system has to integrate with the formal
gross physiological cause. Like its counterpart for literacy, dyslexia, develop­ mathematical systems we encounter later in life. First of these is the number
mental dyscalculia is considered to be partly heritable suggesting a genetic link. system, the verbal representation of quantity. An important inter-disciplinary
However, as mathematics is such a wide-ranging domain that includes geom­ issue is to explore whether the cultural tools (number systems, language, writ­
etry, algebra and calculus, it is difficult to accept that a faulty ‘number module’ ing symbols) adopted by different cultures build upon the emergence of these
(‘a little knot of cells over the left ear’, Butterworth, 1999b) can account for all elementary ‘givens’, or is mathematics socially constructed?
the difficulties experienced by humans solving problems ‘mathematically’. French neuroscientist Stanislas Dehaene has done more than most to map
Despite this reservation, it is important that children in the classroom and out how we humans process numerical information. The Triple Code model
beyond become fluent in everyday numeracy. This means building upon their proposes that specific areas of the brain are responsible for different numeri­
rudimentary number sense and developing skills beyond those displayed by cal processes (Dehaene & Cohen, 1995).The model posits three autonomous
animals. This is where our mathematical culture plays a crucial role. Feigen- interconnected modules located in the brain which together process numerical
son, Dehaene and Spelke (2004) propose that number sense is represented by and mathematical material. The ‘Visual Arabic number form’, situated in the
two distinct systems for dealing with numerical representations that are robust right hemisphere, is responsible for Arabic numeral reading and writing. Tasks
across different modalities of input. such as parity judgments and efficient digit transcoding are linked to this area.
The Precise Number System (PNS) partly relies on a magnitude processing The ‘Auditory verbal word frame’, situated in the left hemisphere, is responsible
skill known as ‘subitizing’, the ability to quickly recognise the size of sets of for auditory and spoken input/output. This includes our counting procedures
objects.This ability was described by Klahr (1973), who summarised evidence and arithmetic facts, which are represented verbally.
4 John W. Adams et al. Introduction 5
The third module is the ‘Analogue magnitude representation’, which is Memory and strategy use need to develop in tandem as children have to learn
bilaterally located in the temporal/parietal area. Responsible for subitizing and to be good strategic learners in order to improve their performance. Metacogni­
estimation, numbers are represented as an analogue locus on an internal num­ tive information increases the probability that children will learn a strategy and
ber line. With links to language/verbal number systems this system provides apply it to a new situation. Pressley, Borkowski and O’Sullivan (1985) argued
humans with two modes of mathematical thinking: the visual (approximate), that good learners possess two important elements of metamemory about strate­
which relies on the sense of number magnitude and estimation, and the verbal gies (MAS): general strategy knowledge (a recognition that strategies require
(exact), which is language dependent. Evidence for this comes from a study of effortful use but ultimately, if applied properly, enhance learning) and specific
bilinguals who were asked to solve problems in both their native and second strategy knowledge (when, where and how to apply the appropriate strategy).
languages (Dehaene et al., 1999). It was found that performance in exact maths Strategy use is highly variable throughout the lifespan (Siegler, 1999).This
was impaired when using their non-native language, but not for approximate variability allows us to select strategies that fit the demands of a particular
maths. problem. In developmental studies of mathematical cognition early childhood
The development of the mental number line appears to develop gradually, is characterised by a shift from using computational strategies to an increas­
with children shifting from a logarithmic to an increasing use of the linear num­ ing use of fact retrieval (Kaye, 1986). This transition is correlated with both an
ber scale. Siegler (1996) proposed a Multiple Representations Model where increase in speed of solving simple arithmetic and a decrease in the number of
both representations are available, with better numerical magnitude compre­ errors. Geary (1993, 2004) highlighted several skill deficiencies demonstrated
hension facilitating the shift to linear representation. Some research suggests by children with Mathematics Disorder (MD). In solving arithmetic problems
that this shift can be encouraged through the use of number board games in the MD children frequently use developmentally immature strategies such
the classroom (Ramani & Siegler, 2008; Whyte & Bull, 2008). This suggests as counting, and in those instances where they use retrieval, they produce a
that education can have a significant effect on these domain-specific cognitive high proportion of errors. Evidence also indicates that MD children are slower
abilities. Indeed, Stanislas Dehaene commented in an article that ‘the impact of counters than typically developing children when using a computational strat­
education is probably much greater than any initial difference’ in our ability to egy to solve basic arithmetic.
represent number (Dehaene, 1999). Mathematical ability is a valued skill in societies and has long been a core
So far, the focus of this review has been on domain-specific cognitive abili­ component of intelligence (IQ) tests. Societal influences can have a profound
ties. However, domain-general cognitive abilities such as language, memory effect on how mathematics is perceived and psychology recognises the signifi­
and problem solving play an important role in mathematical performance. As cant influence that environmental and societal factors have on cognition. For
already noted, verbal areas of the brain are used to store mathematical facts and example, societal norms have implicitly and explicitly suggested that math­
count. The ability to store and process information is vital and another cogni­ ematics is a more suitable subject for boys. Sex differences in visuospatial cogni­
tive ability linked to this is working memory. tion are often cited as a possible cause (see Halpern et al., 2007).
Working memory (WM) is a cognitive system responsible for the temporary This norm, reinforced through peer interactions, parental attitudes and
storage of information required to support ongoing everyday activities such as teacher expectations has been found to negatively affect girls’ mathematical test
mental arithmetic (Adams & Hitch, 1997). Baddeley and Hitch’s (1974) multi- performance and ultimately reinforce the girls’ own stereotypes and behaviours.
component model of WM consists of a central executive, a limited capacity This phenomenon is known as ‘stereotype threat’ (Spencer, Steele & Quinn,
component responsible for the control of attention, supplemented by domain- 1999).
specific verbal and visuospatial stores. Numerous studies (e.g. Gathercole et al., In conclusion, studying mathematical ability is a classic example of the inter­
2004; Holmes & Adams, 2006) have reported that children’s performance on action between nature and nurture. Psychology has adopted various approaches,
working memory tasks that measure the capacity of their WM components biological, cognitive and social, to explore this question and there is still more
are related to attainment on UK national curriculum assessments of English, to learn. To fully capture the complexity of this interaction complementary
mathematics and science. research in psychology, education and anthropology is valuable and the aim of
Sue Gathercole s (e.g. Gathercole, Lamont & Alloway, 2006) hypothesis is this collection of original articles in this book is to highlight each perspective’s
that working memory acts as a ‘bottleneck’ for learning in classroom activities individual contributions and to suggest possible synergies for future research.
for children with poor working memory capacities. Recent attempts to boost
attainment through WM training have shown some promise (Holmes, Gather­
An anthropological perspective
cole & Dunning, 2009; St Clair-Thompson et al., 2010), but further research
is needed to develop a full understanding of the training, transfer in particular. The broad discipline of anthropology explores and links both biological and
Here, education can provide valuable insights in how metacognitive knowledge cultural explanations of human behaviour. Taking a biological anthropologi­
can support this learning process. cal perspective, we can ask whether our numerical abilities have biological
6 John W. Adams et al. Introduction 1
evolutionary roots by examining the presence and distribution of numeri­ that chimpanzees that have been extensively trained on number-related tasks
cal abilities in other species (Cantlon, Platt & Brannon, 2009; Haun et al., can discriminate quantity up to 10 even with the more challenging item-
2010). Studies with various species - mostly apes but also distantly related by-item presentation (Beran & Beran, 2004).
species such as domestic chickens - suggest a common and innate ability to Simple arithmetic has been examined in non-human species by adapting
mentally represent number and perform simple arithmetic across vertebrate looking-time paradigms from developmental psychology to test whether indi­
species. Humans, however, are unique in the extent of our capacity for cul­ viduals look longer at unexpected numerical violations (e.g. 1 + 1 = 1 or
tural, as well as biological, evolution (Mesoudi, 2011a). Beyond any innate 2-1=2) than expected arithmetic operations (e.g. 1 + 1 = 2 or 2 - 1 = 1).
number ability, mathematical knowledge is invented, learned and transmitted Rhesus monkeys (Hauser, MacNeilage & Ware, 1996), tamarins (Uller, Hauser &
over successive generations through observation, teaching and other forms of Carey, 2001) and lemurs (Santos, Barnes & Mahajan, 2005) all show this ability
cultural (i.e. non-genetic) transmission. Hence, anthropologists have studied for these small magnitudes. Interestingly,Vallortigara and colleagues (seeVallor-
the cultural evolution of mathematics throughout history, as well as contem­ tigara, this volume) have shown comparable number discrimination and arith­
porary cultural variation in mathematical knowledge and use (the field of metic in newly hatched domestic chicks using similar paradigms but with balls
ethnomathematics: Powell & Frankenstein, 1997). Cultural history and cul­ rather than food items, where the balls constitute social partners onto which
tural variation are inextricably linked: the cross-cultural variation that we see chicks would ordinarily imprint (Rugani et al., 2009).
today in mathematical knowledge is a snapshot in time resulting from the These studies demonstrate that quantity discrimination and simple arithme­
past divergence and convergence of historical lineages of thought. Linking tic with quantities up to approximately four in magnitude are common across
these domains - the biological evolution of numerical cognition and the cul­ many species, and likely have deep evolutionary roots. While the majority of
tural evolution and diversification of mathematical knowledge - constitutes research has been conducted with primates,Vallortigara and others’ recent work
the major but much-needed challenge in the study of mathematics that we with chicks demonstrates that such abilities might be common across all ver­
address in this volume. tebrate species, not just humans’ closest relatives, thus pushing the origin of
numerical cognition much further back in time. What this highlights is that the
gap between chicks and chimpanzees - both of which struggle with quantities
The biological roots of numerical cognition
much above four - is miniscule compared to the gap between the mathematical
Evolutionary anthropologists study aspects of behaviour and cognition across achievements of humans and all other species. It is to this apparent qualitative
different species, with the aim of revealing the biological evolutionary roots difference between humans and other species that we turn next.
of those traits (Nunn, 2011). Most research in evolutionary anthropology has
been conducted on primates, and especially great apes, given that such species
The cultural origins of human cognition
are most closely related to ours (Haun et al., 2010). Consequently, many stud­
ies over the last few decades have examined the ability of monkeys and great Recent work in evolutionary or bio-cultural anthropology suggests that humans
apes to discriminate between different quantities of items and perform sim­ are behaviourally distinct from other species not because of our advanced tech­
ple arithmetic with those quantities. Obviously such tests must be non-verbal, nical cognitive abilities, but because of our advanced capacity for culture and
given that non-human species lack language in its full human sense (Hauser, social learning (Boyd, Richerson & Henrich, 2011; Hill, Barton & Hurtado,
Chomsky & Fitch, 2002). 2009; Tomasello & Herrmann, 2010). In other words, rather than individual
Hanus and Call (2007) showed that all four non-human great ape species - humans being smarter than individual non-human animals in solving problems
chimpanzees, orangutans, gorillas and bonobos - have the ability to select the and acquiring knowledge from the environment using dedicated, domain-
larger of two quantities of food pellets. Above-chance selection was found for specific cognition, it is instead our ability to learn from one another and gradu­
all pairs of quantities up to 10 when those quantities were presented all at ally accumulate a pool of knowledge over successive generations that marks
once, although more errors were made when comparing large quantities with out humans from other species. In a neat demonstration of this principle, Her­
small differences in magnitude (e.g. 9 vs. 10) than small quantities with small rmann et al. (2007) tested adult chimpanzees, adult orangutans and 2.5-year-old
magnitude differences (e.g. 2 vs. 3) or large quantities with large magnitude human children on a range of tests of physical cognition, including the quan­
differences (e.g. 5 vs. 10). However, when the pellets were presented one by tity discrimination and arithmetic tests described in the previous section, and
one rather than all at once, the apes performed above chance only for small social cognition, including the ability to imitate another individual’s actions
(<4) quantities. Similarly, free-ranging, untrained rhesus monkeys have diffi­ and non-verbal communication via pointing and eye-gaze direction. No dif­
culty with quantity discrimination via item-by-item presentation for magni­ ferences were found between the species’ average performance on the tests of
tudes greater than three (Hauser, Carey & Hauser, 2000). While these studies physical cognition, whereas the children, even at this young age, significantly
used animals that have little or no explicit training, other studies have found out-performed the adult great apes at the tests of social cognition. Many other
8 John W. Adams et al. Introduction 9
studies have found that human children and adults are unique in the extent The cultural evolution of mathematical knowledge
and scope of their ability to copy and learn from other individuals (often called
From an anthropological perspective, therefore, the history of mathematics is a
‘social learning’), even to the extent of copying irrelevant or opaque actions
paradigmatic example of the cumulative cultural evolution of human knowl­
and knowledge (Horner & Whiten, 2005; Lyons,Young & Keil, 2007; Mesoudi,
edge (Mesoudi et al., 2013b;Wilder, 1968; see also Hoyrup, this volume), made
2011a, Ch. 9), While other species can certainly learn from one another, such as
possible by our capacity to transmit and modify (or ‘evolve’) knowledge over
chimpanzees’ nut-cracking or ant-dipping tool-use behaviours (Whiten et al.,
many generations. In the version of mathematical history that we are most
1999), their social learning is not of high enough fidelity to preserve and accu­
familiar with in the West, we can mark out the beginnings of formal mathemat­
mulate knowledge over successive generations of chimpanzees in the way
ics by the invention of numerical notation and basic arithmetic by Babylonian
human knowledge accumulates over human generations.
scholars in around 2000 BC, followed by the invention of geometry and algebra
Consequently, we can see formal mathematical knowledge as a product of
by Greek and Arab scholars, which then allowed Newton, Liebniz and other
this human capacity to learn from one another and accumulate cultural knowl­
Europeans to invent calculus and mechanics.This is a cumulative process, with
edge over successive generations (Mesoudi, 2011b). People acquire increas­
each advance predicated on the previous advances.
ingly complex mathematical knowledge during childhood and early adulthood
However, in the previous section we suggested the hypothesis that the only
through formal and informal teaching. This knowledge is occasionally modi­
innate component of mathematical knowledge is the basic understanding of
fied and beneficial modifications are selectively preserved and passed on to
quantity and simple arithmetic with numbers no greater than 4 or 5, as seen in
future generations in a process that can be described as cultural evolution due
other species, and all other mathematical knowledge is the product of cultural
to its parallels with genetic/biological evolution (Mesoudi, 2011a; Richerson &
evolution rather than genetically evolved cognitive mechanisms dedicated to
Boyd, 2005).This cumulative cultural evolutionary process was well-captured
mathematics. If this is the case, then this raises the possibility that other non-
by Isaac Newton when he described his own discoveries as only having been
Western societies may have followed very different historical pathways. Indeed,
possible because he could ‘stand on the shoulders of giants’; that is, acquire and
this hypothesis is consistent with, and indeed helps to explain, the curious
incrementally modify the mathematical knowledge that had already been accu­
observation that mathematical knowledge is surprisingly culturally variable. For
mulated by generations before him.
example, Pica et al. (2004) found that Munduruku speakers living in a remote
This perspective helps to resolve a puzzling evolutionary aspect of math­
Amazonian region of Brazil lack number terms for exact quantities above five.
ematical knowledge: its apparent lack of biological fitness benefits. In other
Various tests of numerical cognition demonstrated that Munduruku speakers
species, numerical cognition appears to be specifically tied to adaptively rel­
had comparable arithmetical skills to French speakers for quantities up to three
evant stimuli such as food or social partners (e.g. parents, offspring or mates).
or four, but performed considerably worse than French speakers for apprehen­
Such stimuli have direct fitness consequences: individuals that could keep track
sion and manipulation of exact numbers greater than four. Similarly, the Piraha,
of the number of fruit, offspring or mates that are present in their environ­
another Amazonian tribe, lack exact number terms for quantities above one,
ments would probably have been more likely to survive and reproduce than
and consequently fail to perform exact calculations on any numbers where
individuals lacking such numerical ability, allowing such abilities to evolve
quantities vary across space, time or modality (Frank et al., 2008). As Frank et al.
genetically. However, it is difficult to see how the invention and acquisition
(2008) say, their results suggest that ‘numbers may be better thought of as an
of calculus, quantum mechanics or set theory enhanced their bearers’ survival
invention: A cognitive technology for representing, storing, and manipulating
or reproductive prospects (while Isaac Newton had a huge cultural influence
the exact cardinalities of sets’ (p. 823).
with his mathematical influence, he almost certainly had zero genetic influ­
ence on future generations).This puzzle is resolved by recognising that human In other cases, as Bender and Beller (this volume) note, societies may have
cognition has not evolved to specifically discover, understand and learn such evolved different but equally effective mathematical systems, such as the binary-
advanced mathematical concepts, any more than it has evolved to understand decimal system of the Mangarevan islanders in French Polynesia (Bender &
the rules of soccer, say, or the plots of sci-fi novels. Instead, human cognition Beller, 2014).This system evolved independently from the purely decimal sys­
has evolved to faithfully acquire knowledge from others, wherever that knowl­ tem that emerged in the West and eventually spread across the world (at least
edge leads to. More often than not cultural knowledge is biologically adap­ until the purely binary system used in computing was invented in the West
tive, given that it is our knowledge of medicinal plants, food extraction and centuries after the Mangarevan system), and by all accounts this binary-decimal
processing, tool construction, seafaring navigation and so on that has allowed system was just as effective at performing calculations with large exact numbers
humans to successfully colonise every terrestrial environment on the planet. as the purely decimal system. The field of ethnomathematics has extensively
However, cumulative culture is crucially open-ended and necessarily divorced explored cultural variation in mathematical knowledge beyond Western prac­
from genetic control (Mesoudi et al., 2013a), allowing mathematics to evolve tices (see d’Ambrosio, this volume), not only across non-literate societies but also
culturally with no direct fitness benefits. non-Western literate and industrialised societies (Powell & Frankenstein, 1997).
10 John W. Adams et al. Introduction 11
Summary A child may see a mathematical or scientific idea in quite a different way
than it is viewed by an adult who is expert or experienced in working with
Recent approaches within anthropology exploring the cultural origins ofhuman
the idea. These differences are not simply reducible to missing pieces or
cognition (Tomasello, 1999) and the cultural evolution ofhuman knowledge
absent techniques or methods; children’s ideas also possess a different form
and practices (Mesoudi, 2011a; Richerson & Boyd, 2005) offer great potential
of argument, are built from different materials, and are based on different
to move beyond simplistic ‘nature-nurture’ or ‘biology-culture’ dichotomies.
experiences.
In fact, humans appear to have biological adaptations to acquire and transmit
culture (Boyd, Richerson & Henrich, 2011; Chudek & Henrich, 2011), in this
Furthermore, Neisser, cited in Noddings (1990, p. 8), highlighted that this
case mathematical knowledge which builds on fairly minimal innate numeri­
view of the construction of knowledge blurs the line between perception and
cal cognition that corresponds roughly to what we see in apes, monkeys and
cognition, that the act of perception is constructive in itself. This leads to the
even chickens. Different societies have followed different cultural evolution­
personal nature of meaning of any new mathematical idea (Bishop, 1985). Con­
ary trajectories, some leading to the Western mathematics that we are famil­
frey (1990) also highlighted that this constructive process is subject to social
iar with, and others such as the Piraha failing to reach much further beyond
influences; examples from the classroom being the problems provided by the
our species’ innate capacities, while yet others culturally evolving different but
teacher, the language used to describe the mathematics or the sociomathemati-
equally effective systems such as the Mangarevan decimal-binary system.This is
cal norms within the classroom (Cobb &Yackel, 1996).
not to say that the cultural evolution of mathematics is entirely unconstrained
by cognition - as Bender and Beller (2014) note, the decimal-binary system
• emerged to fit the constraints of the human working memory system, just as
External influences on the construction of knowledge
our decimal system did (a fully binary system in contrast, is simply incompat­
ible with human working memory - although it is entirely compatible with From a Piagetian perspective, external forms of knowledge can provide a sup­
computer memory). But rather, the precise course of the cultural evolution of porting role in the internalised construction of knowledge (Marti, 1996). From
mathematics is more culturally contingent than often assumed. Understanding a Vygotskian perspective, this internalisation of knowledge is more closely
the implications of this hypothesis for mathematics education, both in Western dependent on the use of external sign systems:
and non-Western contexts, is a challenge we address throughout this book.
For Vygotsky, the importance of culturally constructed sign systems (lan­
guage, mathematical systems, iconic systems, mnemotechnic systems) in
An educational perspective
the constitution of the individual’s mental functioning means that the latter
In beginning to consider how we can bring together psychological, anthro­ is closely dependent on the former.
pological and educational perspectives, in particular how the two former per­ (P-71)
spectives inform the latter, we will examine possible processes involved in the
construction of knowledge from an educational perspective. We will begin by Therefore, in considering the way in which pupils have constructed their math­
adopting a social constructivist view (Ernest, 1991), which is arguably the most ematical ideas, we must acknowledge not only the personal nature of their
prominent theoretical perspective on learning mathematics. Lerman (1989, knowledge construction, but also the socio-cultural basis for this knowledge.
p. 211) describes constructivism as consisting of two hypotheses: (i) that knowl­ Focussing more specifically on this issue, let us explore how mathematical
edge is actively constructed by the ‘cognizing’ subject, not passively received ideas are represented in the classroom. When we speak of representations, we
from the environment, and (ii) that coming to know is an adaptive process on have to be clear about the distinctions between internal and external mani­
the subject’s experiential world, and not a discovery of an ‘independent, pre­ festations of representations (Pape & Tchoshanov, 2001). Kaput (1991) refers
existing world outside the mind of the knower’. Developing the first hypoth­ to these as ‘mental structures’ and ‘notation systems’ respectively. Davis (1984,
esis, Fosnot (2005) and Steffe and Kieren (1994) emphasise the continual p. 203) defined mental or internal representations as follows:‘Any mathematical
construction of knowledge through actions experienced by the subject and concept, or technique, or strategy - or anything else mathematical that involves
the reflection on these actions. Developing Lerman s second hypothesis as well, either information or some means of processing information - if it is to be
Tall and Vinner (1981) use the term concept image to describe the total cognitive present in the mind at all, must be represented in some way’. External repre­
structure that develops for a concept, built up through experience of all kinds, sentations on the other hand are ‘materially instantiated’ entities (Kaput, 1991,
and changing as the individual meets new stimuli and experiences. Confrey p. 56), with examples being physical apparatus, marks on paper, pictures, sym­
(1990, pp. 108—109) further highlights the individual nature of the construction bols, sounds, spoken words and computerised objects.These ‘serve to denote or
of knowledge: to exemplify’ mathematical concepts (Perkins & Unger, 1994, p. 2).