Course Handout - The Psychology of Numeracy
Copyright Notice: This material was written and published in Wales by Derek J. Smith (Chartered Engineer). It forms part of a multifile e-learning resource, and subject only to acknowledging Derek J. Smith's rights under international copyright law to be identified as author may be freely downloaded and printed off in single complete copies solely for the purposes of private study and/or review. Commercial exploitation rights are reserved. The remote hyperlinks have been selected for the academic appropriacy of their contents; they were free of offensive and litigious content when selected, and will be periodically checked to have remained so. Copyright © 2003-2018, Derek J. Smith.
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 brain-damaged 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 parieto-occipital 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 co-occur. These were (a) "finger agnosia", an inability to tell one finger from another, (b) right-left 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 parietal-occipital 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 41-year-old executive salesman with a large left temporo-parieto-occipital 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 58-year-old 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 non-fluent 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 40-year-old right-handed woman with a lesion at the left temporal-occipital junction resulting from a post-traumatic 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 52-year-old 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 52-year-old right-handed 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 58-year-old male who suffered progressive functional loss over a period of two years before dying of a malignant glioma. Midway through this period (mid-1987) 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 eight-digit numbers, and his spoken responses carefully analysed. Most errors were "of the same magnitude" as the target, that is to say, a three-digit stimulus would generate a three-digit 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 black-and-white 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] action-patterns 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)
mental representations and preoperational thought (2 - 7.5 years)
concrete concepts and concrete operational thought (7.5 - 11 years)
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 middle-class Philadelphia nursery schoolchildren (32 each at ages 3, 4, and 5), and pre-instructed 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 re-exposed 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 One-One Principle: This is the principle that each item in an array can be given only one tag.
The Stable-Order 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 Order-Irrelevance 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 give-a-number 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 8-9 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, left-right 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 17-year-old 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 19-year-old 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 two-digit 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 triple-code 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 rote-learned 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 language-dependent 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 box-and-arrow 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 white-and- 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 sixty-five.
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 right-aligned above the other). In the horizontal test, present each question in single-line 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), pp23-24.
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 one-to-one cross-mapping. 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 "hot-spot". And even the known mathematical hot-spots do other things as well, with all the areas shown in Figure 1 being involved in a host of non-mathematical 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