Difficulties in the acquisition of counting skills

Richard Cowan

School of Psychology and Human Development

Institute of Education University of London

Talk (with corresponding slide numbers)

1.             I am honoured by the invitation to talk to you today about children's counting. Many years ago on a one hot summer's day there was a knock on my door at the Institute of Education. I opened it to discover a gentleman in shorts, smiling. It was Professor Vicente Bermejo. He was kind enough to present me with a copy of his book ‘El niño y aritmética' and several journal articles. I regret to say my grasp of Spanish has not increased sufficiently since then. So, much of what I shall tell you may have already been more elegantly expressed by him.

2.             The research on young children's number development that I have been involved with has been only possible with the collaboration of many people, listed here, and the support of the Nuffield Foundation.

3.              I going to begin by emphasizing why I think counting is important. Then I shall talk about the components of counting: mastering the number word sequence, using the count sequence to determine numerosity, and understanding counting. I shall then review what we think we know about why children who can count do not count to determine relative number, and how they learn to understand the significance of counts. Finally I shall consider some recent work that suggests the importance of a process, subitizing, which may be vital in supporting the development of understanding of counting, and even more generally in understanding arithmetic.

4.             Counting is important because it is the most reliable method of determining the exact numerosity of a set and the numerical relationship between sets. If you know how to count you have can solve any computational problem in arithmetic with whole numbers.

5.              In this picture there are some red dots and some blue dots. Many people when asked for a judgment of whether there are more red or more blue, think it looks as though there are more blue dots. A few think there are more red dots. Some may think there are the same number. Counting is the most convincing way to settle differences of opinion (there are in fact 16 of each colour).

6.             Any whole number addition, subtraction and multiplication problem can be solved by counting, though of course this may not be the quickest method. Counting can also be used to solve division problems such as in this picture.

7.             The three components of counting I wish to emphasize are the number word sequence, the procedure for using it to determine the numerosity of sets, and understanding why counting works.

8.             Mastering the number word sequence involves learning a sequence of basic number words, which in English and Spanish are the words for the numbers from 0 to 19 and each of the decade words for 20, 30.. and so on up to 90, learning the multidigit words for  100, 1000, and so on, and the compound rules that  tell us how to combine these basic words and basic and multidigit words to generate particular spoken numbers, such as for numbers like 45 and 106

9.             Websites document the astonishing variety of number words in different human languages. Important differences between languages exist in the number of basic words the child has to master. There are far fewer basic number words in Chinese (only 0 to 10). In contrast a child learning Hindi has essentially to learn different number words for each of the numbers from 0 to 100. Another dimension of difference between languages is in the length and number of syllables of individual words: all French number words for numbers from 1 to 13 are single syllables but none of the Italian number words are. Some number words are quicker to say than others. Compounding  rules differ in different languages: in US English the conjunction ‘and' is not used to link different parts of the number, so 45 is said ‘forty-five' and 106 is said ‘one hundred six'.  UK English introduces ‘and' for numbers above 100, e.g.  ‘one hundred and six' , Spanish uses the conjunction ‘y' for numbers between 20 and 100, e.g. ‘cuarenta y cinco' for 45.

10.         It is likely that these differences between languages affect the ease with which children learn the number word sequence. After all, a smaller basic number set means the child has less to learn. One syllable number words make the child's task of telling where one number word ends and the next begins easier (Fuson, Richards, & Briars, 1982). Shorter words are easier to say and to remember, which is one reason why Chinese children can remember longer sequences of numbers than US children (Stigler, Lee, & Stephenson, 1986). Simpler compound rules are easier to grasp and make the decimal system more transparent, e.g. the English equivalent of the Chinese for 45 - ‘si-shi-wu' is ‘four-ten-five'.

11.         Many parents attempt to teach their children to count. There is considerable variation between children in what they know in the preschool years. Nevertheless, the range of numbers that a child can count up to expands considerably in the first years of schooling. Also flexibility increases - children become capable of counting up or down from any number within their range, and this has implications for their arithmetic. Their grasp of the number word sequence becomes more fluent: they can recite the number word sequence more quickly and accurately. This means that recitation requires less conscious attention and so allows them to devote more attention to other processes - also important for arithmetic.  Another aspect of development is that they gain more insight into the numerical relationships embedded in the number sequence, such as the successor relationship, that ‘n + 1' is the number after ‘n' in the count sequence and that ‘n -1' is the number before (Baroody, 1995), and the composition of numbers, such as recognizing that 45 is composed of four tens and a five.

12.         As the number sequence is verbal, children whose language development is impaired are likely to have more difficulty mastering it. Short term memory functioning is also likely to affect sequence learning. As the verbal number sequence is a cultural invention then children need to be introduced to it and supported in developing their knowledge of it. Children who lack experience and appropriate support either at home or at school are likely to develop more slowly. As children are often left to figure out some of the relationships themselves, differences in intelligence might also be important

13.         We recently studied a group of children who seemed particularly likely to have difficulty mastering the number word sequence. Children with specific language impairment are those who show impaired phonological processing and impaired understanding of language despite average or better nonverbal intelligence. Previous research with such children by Barbara Fazio has found that they show poor development of count sequence knowledge. However as they also have problems with short term memory functioning it was not clear what the contributions of memory difficulties and linguistic difficulties were. Much previous research on children with number difficulties has highlighted short term memory functioning as the crucial variable.

14.         Our study (Cowan, Donlan, Newton, & Lloyd, 2005) compared three groups of children: an SLI group, consisting of 55 children between 7 and 9 years old with recognized language impairment despite average or better nonverbal intelligence, 44 of the children were in Mainstream Schools and 11 were in Special Schools; an Age Control (AC) group of 57 children matching the SLI group in chronological age and nonverbal intelligence, drawn from the same schools as the mainstream SLI children, or from schools with similar catchment areas to the Special Schools; and a Language Control (LC) group of 55 children, matched with the SLI group on oral language comprehension and with similar age-corrected nonverbal intelligence. The LC group were much younger - between 4 and 6 years old

15.         The count sequence tasks involved no objects. In one trial, we just asked them to recite the number sequence starting with 1. We stopped them when they reached 41. Another trial asked them to count back from 25. We started by counting with them 25, 24, 23, and then let them continue by themselves. We also had three counting on tasks to assess how they crossed decade, century, and millennium boundaries. Like the counting back task we started by counting with them and then left them to continue.

16.         We also assessed their short term memory functioning using standardized measures of each component in the Baddeley model (Baddeley, 2003), their understanding of grammatical contrasts using the TROG (Bishop, 1983), their nonverbal reasoning using the Raven's Coloured Progressive Matrices (Raven, Raven, & Court, 1998). We also asked each child's teacher to indicate the extent to which they had covered the curriculum: we were concerned that the children with SLI might have missed out on number work at school because of the extra attention that had been paid to supporting their language development.

17.         In this graph the averages are shown by spots and the variation within groups by the vertical lines. The children with SLI were no different whether they were in mainstream or special schools and neither differed from the much younger LC group. In contrast the children in the AC group were much more successful. Variation within each group was marked.

18.         As well as resembling the LC group in their language comprehension, the SLI groups showed similar short term memory functioning so we attempted to determine the relative importance of these by entering them, as well as instruction, and nonverbal reasoning in a multiple regression. This indicated that differences in language comprehension, nonverbal reasoning, instruction, and central executive functioning were most important in accounting for differences in count sequence knowledge. Incidentally, a separate analysis of the LC group also indicated that language comprehension and central executive functioning independently accounted for variation in count sequence knowledge. 

19.         To determine numerosity by counting requires more than just the ability to recite the count sequence. It also requires co-ordination with marking off items in the set to be counted so that each item is counted once and only once and knowing how this provides the numerosity of the set.

20.         Two claims have been made about the relation between children's grasp of the requirements of counting and their skill in counting. One proposal is that some understanding is in place even before children develop much skill in counting. Gelman and Gallistel (1978) claims that this understanding is implicit. In contrast, Briars and Siegler (1984) suggested that understanding develops from observation and experience of counting.

21.         Testing whether children possess implicit knowledge is challenging because one cannot just ask children, or indeed adults, to explain why counting works and use their explanations as a criterion. Nor is observation satisfactory. A child counting correctly and using counts to determine numerosity may understand what they are doing but they may just be producing a performance without understanding. Incorrect counts may be due to a lack of understanding but they may also result from other causes. Even adults miscount sometimes but we do not take this as evidence that they do not understand what they are doing.

22.         Developmental psychologists have shown considerable ingenuity in devising methods for assessing children's implicit knowledge of counting. One method is error-detection. Children are asked to take on the role of critics and comment on another's counting. If they can spot flaws in another's counting this seems to show knowledge of how counting should be. Another method is to use unconventional counts that the child is unlikely to have seen. If they accept unusual but legitimate counts while rejecting error counts this supports the view that their understanding of counting does not derive from their experience of being taught to count. Asking children to predict the results of recounts in different conditions is another method.

23.         Gelman & Meck (1983) found most 3-year-olds correctly rejected puppet miscounts of numerosities that were larger than those they could reliably count themselves. These miscounts included trials where the puppet said the number words in a different order, where the puppet omitted to count an object or counted the same object twice, and where the puppet said something other than the last number word reached when asked how many were in the set.

24.         Subsequent studies have usually found the same success in error detection. Questions have been raised about what error -detection shows. In particular does failure to detect errors prove a lack of understanding - or might it be due to a failure to pay attention? Does success show understanding of counting or might it result from the child determining the numerosity of the set in another way? There is also something strange about an adult manipulating a puppet and pretending that the puppet is counting. Some children may be more familiar with such pretend play than others. Might lack of familiarity with games make some children hesitant?

25.         Error-detection on its own is inconclusive about the basis of children's success. After all, even 3-year-olds can count to some extent so they may have learnt about what counting requires from their instruction in counting. Also detection of deviation from conventional counting is not necessarily a sign of understanding. Gelman & Meck (1983) found successful error detection combined with acceptance of unconventional counts. Other studies have reported less success with unconventional counts. The explanation of this discrepancy remains uncertain.

26.         Professor Bermejo has long been studying children's understanding of counting. He distinguishes six levels of understanding of the relation between counting and numerosity (Bermejo, 1996). In the first level, children do not understand questions about numerosity. In the second level they answer with a number-word sequence but do not refer to each item of the set. In the third level they answer by counting the set again. In the fourth level they repeat the last number word of the count, even when the set has been counted backwards. In the fifth level they show some awareness of when the last number word said is not the numerosity but it is only in the final level that children make accurate cardinality responses.

27.         In a recent study, Bermejo, Morales, & deOsuna (2004) conducted an intervention with children in Level 4. They found that exposing children to conflict between last number words from conventional and unconventional counts resulted in substantial progress.

28.         The order-irrelevance of counting refers to the fact that as long as each item is counted, the order in which they are counted - left to right, right to left, or starting in the middle- does not matter. Gelman & Gallistel (1978) found young children were willing to count sets in different orders. Children in Gelman and Meck's (1983) study accepted counts in unusual orders but those in Briars & Siegler's (1984) study were less tolerant.

29.         Art Baroody (1984) claimed acceptance of counts was insufficient to show understanding or order-irrelevance. He argued that the crucial test was whether children believed that counts in different orders should yield the same numerosity. His method of testing order-irrelevance was to ask a child to count a set of items and then ask them to predict the result of counting the set in a different order. He reminded them of the results of their first count to prevent difficulties due to forgetting. Despite this, over half his sample of 5-year-olds did not repeat the number they had obtained from their first count. This was despite their willingness to count the set in a different order.

30.         Gelman, Meck, & Merkin (1986) suggested that Baroody's procedure may have inadvertently challenged children's confidence in their first counts. Instead of being a helpful reminder, some children may have understood it as a signal that there was something wrong with their first count. Using a different form of questioning led to a more positive estimate of understanding of order-irrelevance.

31.         In our sample of preschool children we found similar effects of type of questioning   (Cowan, Dowker, Christakis, & Bailey, 1996). We also found that several children who predicted there would be the same number if the set was counted in a different number also predicted the same number if the set was recounted after one item had been removed. This was particularly common amongst children who were not so good at counting. Such a pattern makes one doubt whether it is safe to infer understanding of order irrelevance just on the basis of predictions of counts in a different order. In general, children who differentiated between subtraction and different order counts were more likely to be accurate counters but there were some children who succeeded despite limited counting skill.

32.         Subsequent studies have shown that detection of error counts and acceptance of reverse order counts approaches ceiling levels by Grade 2, even in children identified as at risk for maths difficulties (Geary, Hoard, & Hamson, 1999; Geary, Hamson, & Hoard, 2000; Geary, Hoard, Byrd-Craven, & DeSoto, 2004). However some other kinds of unconventional order counts are not accepted as is also shown in a large recent study of Canadian children (LeFevre et al., in press). The explanation is unclear. It might be that it is revealing something of their understanding of counting. Another explanation is that it reflects the educational environment in which a particular way of counting is stressed. Another possibility is that it is due to the difficulty of following some unconventional order accounts sufficiently to judge that all the items have been counted.

33.         I turn now to consider children's counting to compare sets. Many have observed children who can count to determine the numerosity of single sets but do not count when asked to generate sets of a specific numerosity (Wynn, 1990) or to compare sets. Does this lack of counting show a lack of understanding of counting?

34.         There are several possible explanations why children might not count to compare sets. It may be because of the information processing demands of counting, because they do not know how to adapt counting to meet the demands of comparison, because they lack confidence in counting, or because they do not know the limitations of other bases of comparison

35.         The information processing demands should not be ignored. Even counting a single set makes considerable information processing demands as set size increases and when the count list is less familiar. Counting to compare sets requires counting the first set, storing the result, counting the second set, and comparing the results with those for the first set. This is tough without external representation.

36.         When asked to compare sets, some children count the first set and just continue counting to count the second set (Saxe, 1977). Many 3- and 4-year-olds do not appreciate this is less appropriate for comparing sets than counting each set separately (Sophian, 1988).

37.         Lack of confidence might affect likelihood of counting to compare sets. Children learning to count will make mistakes when using counting, whether by making single set errors or by continuing counting. If they receive negative reinforcement without appropriate guidance, they may be discouraged from counting.

38.         Another factor may be ignorance of the limitations of other bases for comparison. Global perceptual bases for comparison are quick and easy but only imperfectly correlated with number. Children may have to discover this before they adopt the more laborious procedure of counting. Rapid enumeration processes such as subitizing develop prior to counting but current accounts do not suggest that we are aware of the process of subitizing - we are just aware of its results. Children may have to learn to differentiate accurate subitizing, limited to small numbers, from estimation that is more approximate.

39.         There is much work that has examined young children's number development by using conflict displays (e.g. Brainerd, 1973; Bryant, 1972; Cowan, 1984, 1987a). These are displays which present a conflict between the relative number judgment suggested by global features and the correct judgment that can be discovered by counting or perceptual correspondence. There are three main types of conflict displays: Lengths Equal, Numbers Unequal (LENU); Lengths Unequal, Numbers Equal (LUNE);  and Lengths Unequal, Shorter Row More (LUSR)

40.          Here is a LENU display. Children frequently claim there are the same numbers of blue and red dots. In fact there are not

41.         Here is a LUNE display. Few children will judge there to be the same numbers of blue and red dots unless they count them or see guidelines linking each blue dot with one and only one red dot.

42.         Here is a LUSR display. Many children will judge there to be more in the longer row.

43.         Even small number versions - with only 3 or 4 dots in each row - are often misjudged by 3-year-olds (Michie, 1984). Rarely do children spontaneously count to compare.

44.         Michie (1984) tried several interventions to encourage children to count the items to find out whether each row had the same number. Emphasizing the importance of being right was not effective.  Providing feedback about the correctness of judgments based on separate counts did make children more likely to count in her study, and we found this too (Cowan, Foster, & Al-Zubaidi, 1993). We also found that showing children the consistency of counting based judgments with perceptual correspondence worked. Perceptual correspondence was shown by pairing each item in one row with an item in the other row. When the sets were unequal the unpaired item was at the end of the row. However, although preschoolers would count to compare, they still did not judge consistently with their counting. In this they differed from older children. Both groups were selected for not spontaneously counting. Both showed much increased counting after the intervention. The older children subsequently judged consistently with their counting.

45.         To investigate when children trust count information to judge conflict displays I (Cowan, 1987b) studied three age groups: preschoolers, 5 year olds - in their first year at school, and 6 year olds. In each age group accurate counters were selected.  I used both small number versions of the displays and large number versions. To see whether confidence in counting was relevant the counting was done either by the children or by me (I told them I was very good at counting). It made no difference.  Few preschoolers were reliably correct on all the displays. In contrast almost all 5 -year-olds judged every small number display correctly and the 6-year-olds judged every large number display correctly too. Overall the LUNE displays were judged better than the LUSR displays.

46.          The explanation of the difference between small and large number displays is likely to be the availability of other methods for comparing apart from counting. These include subitizing and perceptual matching. Whereas these can help children judge the small number displays they are not available for larger number displays. The difference between LUNE and LUSR displays is likely to be due to insecure knowledge of relative magnitude. When counts of the two sets yield the same numbers, no knowledge of magnitude is required to make the correct judgment. In contrast when the counts yield different numbers, one needs insight into the count list to determine the more numerous set, especially when the display suggests the opposite judgment.

47.         Subitizing in adults is a rapid and accurate process of enumerating small numerosities (up to 7). It is not just overlearnt counting.  Starkey & Cooper (1995) showed in young children its range is more limited than adults' but appears prior to verbal counting. They also compared different configurations - rows or two dimensional patterns and found children could subitize them. So it is not just pattern recognition. They suggest it may underlie babies' discrimination between small numerosities.

48.         Some believe subitizing is the basis for mapping counting and number words onto representations of numerosity and that subitizing underlies learning what affects numerosity (Klahr & Wallace, 1973). If so, people with defective subitizing at risk for difficulties with number.

49.         Charles, a psychology graduate, had always had problems with mathematics. Butterworth (1999) describes him as lacking the ability to subitize – he had to count even arrays of two dots. Butterworth attributes Charles's difficulties to a defective number module in the brain. 

50.         Another function of this module is magnitude comparison. Most adults know the relative magnitude of numbers between 1 and 9. The speed with which they judge relative magnitude is related to numerical distance. They are faster judging that 9 is more than 5 than that 6 is more than 5. The idea is that in development symbols for numbers become connected with a representation of number, a kind of mental number line.  Charles is much slower and shows the opposite pattern, also attributed to neural abnormality.

51.         Butterworth's (2003) Dyscalculia Screener assesses the two measures – enumeration of small numbers and magnitude comparison – that he believes are subserved by the number module. He does not claim that all children with number difficulties have defective number modules. But children whose number modules, and so subitizing, are compromised should have difficulties understanding counting. As yet there is no method for testing this interesting idea in very young children.

52.         In conclusion, verbal counting is a powerful cultural tool that incorporates many important ideas about number. Children can take a long time to discover them. Successful brief interventions suggest that this is not inevitable. Nonverbal numerosity processes may contribute to learning about counting and number but how remains disputed (Carey, 2004; Gallistel & Gelman, 2005; Rips, Asmuth, & Bloomfield, in press). There is still much to learn about children's counting.

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