Course Handout  The Psychology of
Numeracy
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written and published in Wales by Derek J. Smith (Chartered Engineer). It forms
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First published online 14:00 BST 16th June 2003,
Copyright Derek J. Smith (Chartered Engineer). This version [2.0  copyright] 09:00 BST 3rd July
2018.
This material previously appeared in Smith (1998; Chapter 3). It is repeated here with minor amendments and supported with hyperlinks. 
1  Introduction
For
a variety of reasons, interest in the topic of mathematical cognition has been
on the increase in recent years. It is, for example, an area of appreciable
theoretical interest in its own right, being one of several areas of cognition
where research findings are beginning to throw light on the deeper workings of
the mind. As a result, academic journals have been themed on the subject [examples] and universities have
set up special research units [example]. In addition,
the press has mobilised itself in a crusade to
improve the nation's numeracy, and governments have urgently been seeking  and
occasionally even listening to  professional advice. There is, however, a
major barrier facing all these agencies, namely that we do not really
understand the mental representation of anything, let alone numbers; nor do we
really understand the nature of the mental processes which manipulate one set
of ideas to give you another set; nor, therefore, do we really understand what
is actually being constructed during the period of construction. In short, we
do not really understand cognition. Take, for example, our mental
representation of the number "1". What does it mean to think
"1", and how does this differ from "2", or "10",
or "1000", or "10,000", or "10,000,000,000"? Do
we think of that number of beads on a table top, or of that number of points on
an infinitely long straight line (the "number line"), or of that
number of depressions on a keypad, or what? And what does it actually mean when
we ask "How many?"? If we have ten things in sight do we reply
"ten" in absolute terms because we somehow know what ten is (semantic
knowledge), or do we know that if we start counting upwards from 1 we will
quite quickly get to 10 (procedural knowledge). But it cannot be the latter
because there are plenty of numbers we cannot actually count up to, such as
10,000,000, or minus 5, or 1.33452. Equally, it cannot be the former
either, because there is insufficient room in our semantic memories to have an
entry for every possible number, most of which will never ever get used. And
once we have learned to count sweets, say, does it automatically mean we shall
be able to count, say, toys? Once you start looking for them, the problems come
thick and fast, and in the next section we look at how research with
braindamaged adults helps us understand some of them.
2  Evidence from Acquired Dyscalculia
According
to Grewel (1952), it was around the turn of the 19th
century that neurologists first noted specific disorders of calculation
following brain damage. Henschen (1919) carried out
an early survey of the area and named the disorder acalculia
(literally without calculation), although nowadays the preferred name
for this type of disorder is dyscalculia (literally disordered
calculation), because the deficit is rarely as total as the prefix a
would imply. Thus .....
"Acalculia
was observed in occipital, frontal, parietal and temporal lesions [], in the
absence of aphasia. Henschen, therefore, concluded
that the integrity of several areas of the cortex is essential for calculation
and that a separate system, independent of the apparatus for the use of
language and of music, exists in the human brain." (Grewel, 1952, p397.)
Another
early source was Walther Poppelreuter, a German
military surgeon who published his observations of a series of head injured
soldiers (Poppelreuter, 1917/1990). This contains 52 detailed
neurological case reports, each one including the results of a calculation
test. There are repeated instances of calculation impairments with damage to
the rear left head (patients GF, BP, EJ, FG, SA, HJ, and others). In a later
paper (Poppelreuter, 1923, cited in Humphries, Riddoch, and Wallesch, 1996), he
reported on patient Merk, who had gunshot wound
damage to both parietooccipital cortices. As might
be expected, Merk suffered serious visual processing
defects, but was also impaired at calculation, being unable, for example, to
write numbers to dictation.
Not
long after that, Gerstmann (1930) described a
syndrome in which four distinct signs tended to cooccur. These were (a)
"finger agnosia", an inability to tell one finger from another, (b) rightleft
disorientation, (c) writing difficulties, and (d) calculation difficulties.
This syndrome seemed to follow damage centred on the
angular gyrus of the dominant hemisphere. The angular gyrus (Brodmann's Area 39) is situated at the junction of the temporal
and occipital lobes (see Figure 1), and is adjacent to that part of Area 19
which was subsequently identified by Kleist (1934) as
serving counting and number recognition. The syndrome is still recognised today, and is usually referred to as Gerstmann's Syndrome (see Benton, 1987, for a
detailed review, if interested.)
As
to the relevance of clinical observations to mainstream cognitive theory, McClosky (1992/3) provides a good example of what can be
achieved. He takes a cognitive neuropsychological approach to
mathematical cognition, that is to say, he tries to deduce how the normal brain
processes numbers from what happens in individual cases of brain damage. The
generic name for defects in mathematical ability arising from such problems as
head injury, stroke, or tumour is acquired
dyscalculia (namely disordered calculation, absent before the
insult, and acquired as a direct result of it), and, for epidemiological
reasons, acquired dyscalculias usually affect only
adults. The neuropsychological literature contains occasional reports of
acquired dyscalculics, including .....
(a) Warrington
(1982): Patient
DRC, originally reported by Warrington (1982), had brain damage in his left
parietaloccipital cortex. He could read and write numbers, could judge which
of two numbers was larger, and could give reasonable rough estimates of
magnitude for variables like height. Thus, when asked to multiply 3 by 4 he
replied "13, roughly". However, even for simple formal problems such
as 5 + 7 his performance was slow and inaccurate. He commented that he often
knew the rough answer to a problem but could not come up with the exact answer.
He was also able to do what we might call "counting onwards":
thus 7 onwards from 5 would go 6  7  8  9  10  11  12! So the semantic
concepts of adding and equalling, as well as the
number concepts of 5, 7, and 12, were all individually intact, but the mental
process which in normals returns a precise sum from
two given numbers was somehow faulty. His arithmetical knowledge had
been damaged, in other words, whilst his number knowledge remained
intact.
(b) Dehaene and Cohen (1991): Dehaene and Cohen (1991) report on patient
NAU, a 41yearold executive salesman with a large left temporoparietooccipital lesion resulting from a head trauma, who
could successfully detect 2 + 2 = 9 as being false, but not 2 + 2 = 5. As with
Warrington's case (above), this implies that the ability to estimate has
survived, as has the relevant number knowledge, but that the arithmetic skills
have gone. Which implies, in turn, that normal mathematical
cognition involves both processes simultaneously, each providing some sort of
mental plausibility checking for the other.
(c) Benson
and Denckla (1969): Benson and Denckla
(1969) report on a 58yearold man with a left hemisphere lesion. The ability
to respond correctly to written or spoken arithmetic problems was retained
provided the answer could be pointed to. Spoken or written answers, however,
were often inaccurate. Thus for the problem 4 + 5 he replied "eight",
wrote "5", and pointed (correctly) to "9". As with the
nonfluent aphasias [glossary],
this implies that the semantic system can lose the power to control its output
processes, and this implies, in turn, that those processes were physically
separate in the first place.
(d) Ferro
and Botelho (1980): Ferro and Botelho
(1980) report on patient AL, a 40yearold righthanded woman with a lesion at
the left temporaloccipital junction resulting from a posttraumatic cerebral haematoma. Upon investigation, AL could process the numbers
in arithmetic problems and could carry out arithmetic to oral instruction.
However, if the problems were presented in writing, she was easily confused as
to the operation required on them. Thus 721 + 36 was mistakenly multiplied
instead of added, albeit the correct product (that is to say, 25,956)
was obtained. The same authors also observed a similar defect in patient MA, a
52yearold male, and called this defect asymbolic
acalculia  "a variety of acalculia characterised by a
failure to differentiate the arithmetical symbols". They explain it thus:
"The arithmetical signs
are the symbols of a semiotic system different from written language and
numbers. They are like an ideographic notation because each sign has a strict
and universal value and does not combine in more complex symbols. [] Our
patients behaved as if these signs were stripped of their names and of the
corresponding computational rules." (Ferro and Botelho, 1980, p179.)
(e) Cipolotti (1995): Cipolotti (1995)
reports on patient SF, a 52yearold righthanded male with probable
Alzheimer's disease, who was able to read numbers out loud if written in word
form (eg. "twenty
seven" was read out correctly), but unable to do so if written in arabic form (eg. "27"
was read as "two hundred and seven"). Reading of normal text,
however, was flawless.
(f) Cohen
and Dehaene (1991): Cohen and Dehaene (1991)
report on patient YM, a 58yearold male who suffered progressive functional
loss over a period of two years before dying of a malignant glioma. Midway
through this period (mid1987) he underwent a left temporal lobectomy,
following which he was thoroughly assessed for both verbal and arithmetical
abilities. His spontaneous speech remained fluent and grammatical, but with
"pervasive" word finding difficulties, and there were occasional
semantic and phonemic paraphasias. There was reduced
comprehension of complex utterances. He was tested on a total of nearly 2000
textually presented one to eightdigit numbers, and his spoken responses
carefully analysed. Most errors were "of the
same magnitude" as the target, that is to say, a threedigit stimulus would
generate a threedigit incorrect response.
The
brain areas implicated by the preceding studies are shown in Figure 1.
Figure 1 
Brain Areas for Mathematical Cognition: The shaded areas on these diagrams show cortical activation during a simple
mental arithmetic task, as revealed using the rCBF
neural imaging technique. The darker the shading, the greater the increase in
cerebral blood flow from resting levels. Note the heavy involvement of
bilateral frontal cortex. The pointers show other areas implicated by the
literature. [Redrawn from Roland and Friberg (1985,
cited in Dudai, 1989, p163).] If this diagram fails to load
automatically, it may be accessed separately at 
Enhanced from a blackandwhite original in Smith (1998; Figure 3.2), after Roland and Friberg (1985). This version Copyright © 2003, Derek J. Smith. 
3  Evidence from Normal Development
We
begin by refreshing our memories as to the Piagetian view of mathematical
ability. Isaacs (1960) summarises the key assertion
of the Piagetian approach thus: Intellectual growth  both generally, and in
the special area of mathematical ability  is a function of what we have
learned to do with our hands, not our heads. In other words, before we can think
about something we need to have acted it out many times, meaning that
thought is always internalised action. Thus .....
"From the beginning it is patterns of
active behaviour that govern [an infant's] life.
Through these [s/he] takes in ever new experiences which become worked into
[his/her] actionpatterns and continually help to expand their range and scope.
It is through actively turning to look or listen, through following and
repeating, through exploring by touch and handling and manipulating, through
striving to walk and talk, through dramatic play and the mastery of every sort
of new activity and skill, that [s/he] goes on all the time both enlarging
[his/her] world and organising it." (Isaacs, 1960, p5.)
There
are then obvious implications for the development of mathematical skills. If,
as a primary schoolchild, we want to think "one plus one",
then we need to have touched and moved and lifted and squeezed and sucked and
tasted (etc, etc)
"ones" and "twos" and "threes" and "one plus
ones" and "two plus ones" and "three minus ones", and
so on, and so on, and so on, literally thousands of times beforehand. If we
want to think numbers we need to have acted numbers. Indeed, even being able to
count does not mean we necessarily have an idea of number. In Isaacs' view,
counting is just an "enjoyable minor skill" (p11) which can be developed
at a far younger age than even the simplest of the supporting abstract number
concepts. The full Piagetian sequence of events  and it takes some 15 years of
human development to get through it, remember  is as follows .....
actions and action schemas
(birth  2 years)
ß
mental representations
and preoperational thought (2  7.5 years)
ß
concrete concepts and
concrete operational thought (7.5  11 years)
ß
abstract concepts and
formal operational thought (11  15 years)
It
is against this backdrop of general cognitive development that mathematical
development takes place, and, to cut a long story short, Lauren Resnick, of the
University of Pittsburgh's Learning Research and Development Centre, argues
that the real message is the constructive nature of mathematical
knowledge. Mathematical knowledge, she says, "is
not directly absorbed but is constructed by each individual." (Resnick, 1989, p162). To take a constructivist
position on numeracy, therefore, is to accept that mathematics is a complex set
of mental skills, each relying on each other, and taking as a whole many
years of practice to put in place.
Rochel Gelman of the
University of Pennsylvania is a leading theorist in this area, having been
researching mathematical cognition in children since the early 1970s. In one
early study (Gelman, 1972, cited in Gelman, 1980), she addressed the child's concept of number.
She did this by independently varying both the number of items in an array and
the distance between them. She took 96 mainly middleclass Philadelphia nursery
schoolchildren (32 each at ages 3, 4, and 5), and preinstructed them that a
row of three green mice was a "winner", whilst a row of two green
mice was a "loser". The children were then randomly assigned to one
of two experimental groups, namely a subtraction group or a displacement
group. Both groups were then reexposed to the stimulus array, save that
with the subtraction group one of the mice had now been surreptitiously
removed and with the displacement group the spacing between the items had
been altered. The children's "surprise" responses and subsequent behaviours were carefully recorded and scored. Subtraction
group children showed much more surprise than displacement group
children. All 48 children noticed the change and 42/48 looked around for the
missing item. On the other hand, only 25/48 of the displacement group
children showed that they noticed the change, and none of them bothered
searching. When interrogated as to what had happened, 45/48 subtraction
group children assumed that someone had taken the missing item. In the displacement
group, 21/25 correctly stated that the change  whilst unexpected 
actually made no difference. This was an important finding because it indicated
that array number had been perceived as well as array size, and was quite
rightly being treated as the more important of the two variables.
Following a decade of similar experimentation (eg. Gelman and Gallistel,
1978), Gelman (1980) was able to argue that
successful counting involves the coordination of no less than five separate
principles, namely:
The OneOne Principle: This is the
principle that each item in an array can be given only one tag.
The StableOrder Principle: This is the
principle that the tags used must come from a stably ordered sequence.
The Cardinal Principle: This is the
principle that the last tag used during the count represents the cardinal
number of the array as a whole, that is to say, how many items are in it.
The Abstraction Principle: This is the
principle that any set of items can be collected together for counting.
The OrderIrrelevance
Principle: This is the principle that the order in which the items are tagged makes
no difference to the end result.
The
cardinal principle has been further investigated by Wynn (1990). She observed
that children between 2:7 and 3:0 prefer to answer the "How many?"
question by repeating the entire count rather than stating the last tag used.
Only at mean age 3:6 do children reliably use the last tag on
its own. The younger children were also weaker at a giveanumber
task. They were asked to give a puppet between one and six items from a pile,
but typically only handed over one or two items. They seemed unable to work out
for themselves that they should continue counting out the items until they
reached the target number. At age 3:6, however, the correct number
of items were handed over, showing a sudden grasp of the cardinal
principle at some time between 3:0 and 3:6 years.
Resnick
(1989) also describes how "change" stories can be used to
determine other developmental dates. The example she gives is: "Ana went
shopping. She spent $3.50 and then counted her money when she got home. She had
$2.35 left. How much did Ana have when she started out?" (Resnick, 1989, p165.) Mastery of such problems typically
comes at 89 years, and the reason it takes this long to achieve is that there
is no "direct mapping" between the story and the equation. Solving
the arithmetical element of the solution, namely 3.50 + 2.35 =
?, is easy once the equation itself has been constructed by properly analysing the meaning and intention of the text.
Mathematical skills of this sort are therefore intimately associated with the
skills of linguistic analysis and problem solving. Indeed one definition of
mathematics might be that it is problem solving with arithmetic. It is
certainly more than just numbers  it is knowing what
to do with them.
4  Evidence from Developmental Dyscalculia
Developmental
dyscalculia is slowness to acquire numerical abilities in otherwise normal
children. It is thus a parallel disorder to developmental dyslexia, and
is often loosely referred to as "number blindness". O'Leary (1995)
lists a typical cluster of symptoms appearing  or at least making themselves
felt for the first time  at around age 8, including early hearing difficulty,
leftright confusion, inability to spot number patterns such as odds and evens,
difficulty learning tables, printing letters and/or numbers in the wrong order,
confusing numbers, poor spelling, slow work, poor mental arithmetic, and poor
number estimation skills. As a result, "number bonds that are second
nature to pupils of similar intelligence never develop, tables refuse to stick,
and basic arithmetical processes remain a mystery" (p18). It has yet to be
decided, however, whether dyscalculia is dyslexia in disguise, or vice versa,
or whether both are manifestations of a single deeper disorder, or even whether
the underlying deficit is with the children or the system which is supposed to
be educating them. It is also far from clear what the normal stages of
mathematical development are, and how much flexibility there is for different
individuals to pass through them in different sequences.
Temple
(1991) describes two fairly typical cases of dyscalculia. The first of these,
SW, a 17yearold boy who had suffered fits since childhood, had normal ability
to read numbers and judge magnitude, and could verbally describe the principles
of the operations of addition, subtraction, multiplication, and division. Yet
whilst his additions were dealt with adequately (24/24 on one test), his
subtractions were weak (7/15). More specifically, whilst HTU  U subtractions
were adequate (4/4), HTU  TU and HTU  HTU subtractions were either not
attempted at all or else done incorrectly. Temple's second case, HM, was a
19yearold girl with a history of dyslexia but a verbal IQ of 105 and no known
neurological abnormality. She, too, could read numbers and judge magnitude, and
was only slightly below average on tests of addition (with and without
carrying) and subtraction. On multiplication problems, however, she was about
two standard deviations below the norm for her age, and frequently made "bond
errors"  errors where the answer was wrong but from the right
multiplication table (eg. 3 x 4 = 6)  as well as "shift
errors"  errors where one digit of a twodigit answer was one or two
digits removed from its correct value (eg 4 x 9 = 46,
rather than 36).
5  Models of Numeracy
So
what do all these streams of data tell us? Well fortunately the experimental
evidence and the clinical evidence tend to suggest the same general conclusion,
namely that there are at least two fundamentally separate types of mathematical
knowledge  number knowledge and arithmetic knowledge (an
assertion which strongly echoes the increasingly popular distinction elsewhere
in cognitive neuroscience between propositional knowledge and procedural
knowledge). Dehaene (1992/3) goes further,
arguing for modularity of cognition. He clusters the isolated abilities
under three main headings, giving a triplecode model. Here are the headings he
suggests .....
(a) Verbal Number Abilities: These are the
abilities of dealing with numbers as words, and include the basic act of
counting and most rotelearned tables.
(b) Positional Notation: These are the
abilities of dealing with digits, digit positions, and the rules
governing their use.
(c) Comparison and
Approximation: These are the abilities of dealing with numbers as approximate
quantities.
Dehaene summarises his
position thus .....
"Adult human numerical
cognition can therefore be viewed as a layered modular architecture, the
preverbal representation of approximate numerical magnitudes supporting the
progressive emergence of languagedependent abilities such as verbal counting,
number transcoding, and symbolic calculation." (Dehaene, 1993, p35.)
As
is common practice amongst cognitive neuropsychologists, McClosky
(1992/3) then attempts to explain what is going on by using a boxandarrow
model of the mental processing involved. This is all reproduced in Figure 2 .....
Figure 2 
McCloskey's (1992/3) Number Processing Architecture: The resulting number processing architecture
draws heavily upon earlier models in McCloskey, Caramazza,
and Basili (1985), and identifies (a) two number
comprehension systems, one for spoken numbers and one for written
numbers, (b) two number production systems, one for spoken numbers and
one for written numbers, (c) a set of abstract internal representations,
where what we might call the "meaning" of each number is held, and
(d) a set of calculation procedures, which carry out arithmetic on
what has been activated within (c). If this diagram fails to load
automatically, it may be accessed separately at 
Redrawn from a whiteand black original in Smith (1998, p34; Figure 3.3), after McClosky (1993, p113). This graphic Copyright © 2004, Derek J. Smith. 
6  Recent Research
Armed
with models such as McCloskey's, it becomes much easier to understand phenomena
such as those exhibited by the young Brazilian street children studied by
Carraher, Schliemann, and Carraher (1988). They relate the following story of
one such child  PS, a skilled market trader when not at school  when asked to
solve the problem 200  35:
"P.S.: That's easy, one hundred and sixty five (does
not write it down).
E: How did you do it so quickly?
P.S.: Two hundred, minus thirty, one seventy. Minus five, one sixtyfive.
E: Can you do it on paper?
P.S.: OK, I've learned it. I used to know this. (Writes down
200, the minus sign, 35 properly aligned underneath, and underlines.) Zero
minus five, carry the one. (Writes down 5 as the result for units.)
Carry the one (writing down 7, apparently calculating 10  3). Carry the
one. Two minus one. One. (Writes down 1; the obtained result was 175.)"
In
terms of the McCloskey model, when the spoken number input and output routes
are in use, the abstract representations and the calculation procedures
function perfectly (which is hardly surprising, given that the child is orally
highly adept at cash transactions). However, when the calculation needs to be
done on paper, a different set of calculation procedures are needed  specifically,
those of HTU alignment and number carrying  and in these circumstances,
despite the fact that the HTU alignment rules were followed correctly, the
carrying operation was not.
There
has also been considerable recent debate as to whether British children should
be allowed to use calculators or not. Indeed, there are those who blame the use
of calculators for the national decline in mathematical ability. At the British
Psychological Society's 1995 London Conference, Tony Ward, of Luton University,
reported that today's students were worse at mental arithmetic than their
parents' generation (cited in Milton, 1995). And David Burghes, of Exeter
University's Centre for Innovation in Mathematics Teaching, agrees: "All
children must be numerate as a first priority, which will be achieved by
restricted calculator use" (Burghes, 1995, p4.21).
But
it is probably not just the calculator to blame. The rot seems to be deeply
rooted in the methods, timing, and resourcing of the teaching, and even in the
quality of the textbooks used. As far as the methods are concerned, there is
even uncertainty over whether children should be taught to do sums vertically
or horizontally .....
Exercise 1 
Vertical vs Horizontal Adding? 1
Prepare three sets of addition and subtraction problems. A "simple"
set should consist of number pairs between 1 and 9, a "medium" set
should consist of number pairs between 10 and 99, and a "difficult"
set should consist of number pairs between 100 and 999. Half the test items
should involve a "carry one" operation in the U (units) column. 2
Randomly allocate 10 of each type of problem to a vertical test and a horizontal
test. In the vertical test, present each question in columnar format (ie.
each number rightaligned above the other). In the horizontal test, present
each question in singleline format. 3
Compare and contrast accuracy and speed of completion of the two types of
test at the three levels of difficulty. Is there a pattern to your results? This
procedure loosely based on Bierhof (1996), pp2324. Extended Exercise 4
If you had to teach children to add HTU + HTU, which of the following
preparatory sessions would, theoretically, be of more use? (a) 10 practice HT0 + HT0 sums (eg. 350 + 470 = ?) (b) 10 practice HTU + U sums (eg. 357 + 9 = ?) 
As
far as the quality of textbooks is concerned, Helvia Bierhoff, of the National
Institute of Economic and Social Research, has compared the quality of primary
school textbooks and teaching practices in Britain, Germany, and Switzerland
(Bierhoff, 1996). Her findings include .....
As
we have seen, the study of mathematical cognition goes back about a quarter of
a century, and many valuable findings have been made. Nevertheless, the experts
still find it hard to agree on what practical improvements are needed. Drawing
on the American experience, for example, Resnick (1989) puts it this way .....
"Although there is
substantial agreement among researchers and mathematics educators about what
the problem is, debate continues over how to solve it. Some believe that
arithmetic practice should be sharply reduced or abandoned in favour of more conceptually
oriented teaching that focuses on mathematical principles. Others claim that
only a firm foundation in basic number facts and relationships will allow
children to move ahead." (Resnick, 1989, p167.)
The
fact is that mathematics stretches the mind of all involved  both teacher and
learner. If we take the McClosky model as typical, no less than six mental
subsystems have to be trained up and integrated, some of which are arithmetical
whilst others are conceptual. It is hardly surprising, therefore, (a) that
things go wrong, (b) that it is difficult to specify which of the six
subsystems is "underperforming", and (c) that it is difficult to know
what best to do to remedy the matter. Nor has there ever been one universally
accepted way of teaching mathematics. One mathematics teacher voiced his
concerns this way: "Even [after 20 years] I am still learning new ways of
approaching the material to make it accessible. [] We would be more
successful at teaching maths if we had a more realistic view of the teaching
process." (Medcalf, 1995, p4.19; bold emphasis added.)
But
perhaps the greatest barrier to progress is the fact that functional analyses
of cognition like McClosky's are extremely difficult to integrate with
anatomical analyses like Gerstmann's. The diagrams in Figures 1 and 2 simply do
not have a onetoone crossmapping. The brain does not seem to work by
dedicating particular gyri or lobes to particular mental functions. Instead, it
seems to function as a vast distributed processing network, dotted here and
there with the occasional "hotspot". And even the known mathematical
hotspots do other things as well, with all the areas shown in Figure 1 being
involved in a host of nonmathematical functions such as perception, reasoning,
communication, and memory. In short, there is at present no definitive
neuropsychology of mathematics, nor will there be until our ability to analyse
cognitive function improves.
7  References
See the
Master References List
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