1986 Understanding Of Number Concepts In Low

B. DENVIR AND M. BROWN UNDERSTANDING ATTAINING

OF NUMBER

7-9 YEAR

OF DESCRIPTIVE

OLDS:

CONCEPTS

PART

FRAMEWORK

IN LOW

I. D E V E L O P M E N T

AND

DIAGNOSTIC

INSTRUMENT ABSTRACT. Three studies were carried out into the development of number concepts in pupils aged 7 to 9 years who were considered to be 'low attainers' in mathematics. This paper reports the search for a descriptive framework, the development of a diagnostic assessment instrument and a longitudinal study. A subsequent paper (Denvir and Brown, 1986) reports two teaching studies. Support was found for the two main hypotheses namely that: (i) A framework can be identified which describes the orders in which children acquire number concepts (ii) This framework can be used to develop a diagnostic assessment instrument which will provide a description of pupils' understanding of number. The aspects of number which were considered were counting, addition, subtraction and place value. The number skills which children had grasped were inferred from solutions and strategies offered by them to questions posed in a series of interviews. BACKGROUND The rationale for the present study developed as a result of both authors' involvement in the Schools Council project ' L o w Attainers in Mathematics 5-16' (Denvir et al., 1982). Visits to a large number of schools manifested a need for diagnostic assessment linked to prescriptive teaching. This need was subsequently strongly supported by Bennett's findings (Bennett et al., 1984) that teachers were frequently unsuccessful in matching number tasks to the conceptual stages revealed by 6 and 7 year olds. Chiefly in the last decade considerable research has been carried out, notably in the U.S. (e.g., Gelman and Gallistel, 1978; Schaeffer et al., 1974; Steffe and Johnson, 1971; Carpenter and Moser, 1979) but also in France (Comiti, 1981; Descoudres, 1921; Vergnaud, 1982) the U.K. (Hughes, 1981; Matthews, 1983) and I s r a e l (Nesher, 1982) into the development of number understanding in 3-11 year olds, especially in counting, addition and subtraction. Since 1981, when this present study began, a considerable literature has been published and this now incorporates significant work in place value (Brown, 1981; Bednarz and Janvier, 1982; Steffe, 1983; Resnick, 1983) and a clear move towards proposing theories which will explain how understanding develops (Resnick, 1983; Riley et al., 1983). The work described in this paper builds on m a n y of the earlier findings. Carpenter and Moser (1982) describe three main types of strategy for solving simple addition problems; 'Count All', 'Count On' and 'Recall' and the first two of these are regarded by them (Carpenter and Moser, 1983) and other authorities (e.g., Fuson et al., 1982; Steffe et aL, 1983) as characteristic Educational Studies in Mathematics 17 (1986) 15-36. 9 1986 by D. Reidel Publishing Company.

16

B. DENVIR AND M. BROWN

of distinct stages of development of number understanding. Consequently it should be possible to make inferences about a child's understanding of number from observing that child's repertoire of strategies. In particular Steffe et al. (1983) and Fuson et al. (1982) regard the ability to count on as crucial to an understanding of number. Steffe regards the child who is unable to make 'deliberate extensions', i.e., to count on in addition with the intention of keeping a record of the increment, as 'numerically pre-operational'. Riley et al. (1983), building on the work of Steffe et al. (1971), Carpenter and Moser (1979, 1982) and Nesher (1982), explain the development of pupils' ability to solve different semantic categories of addition and subtraction word problems in terms of their 'part-part-whole schema'. Resnick (1983) also uses this schema to explain children's development of place value understanding. The child is thought to progress through three stages in the ability to represent numbers with base ten blocks. At stage one two digit numbers are seen as a whole comprised of two parts, such that one part is all of the tens and the other part is the remaining units. By stage two the child appreciates that there are other non-canonical equivalent representations in which there are more than ten units. At Resnick's stage three the child can map from transactions with blocks to the written algorithm and vice versa, giving semantic meaning to carry and borrow marks. AIMS The general intention of this study was to shed light on the learning of number concepts by children aged seven to nine years who were considered to be low attainers in mathematics in a way that would help teachers to provide effective learning experiences. The aims were to: (i) find a framework for describing low attainers' acquisition of number concepts; (ii) develop a diagnostic instrument for assessing children's understanding of number; and (iii) design, carry out and evaluate a remedial teaching programme. It was proposed that whilst the orders of acquisition of many skills would be independent, the hypothesised framework would contain 'hierarchical strands', within which the acquisition of some 'easy' skills would be a prerequisite condition for the acquisition of more difficult skills. These 'strands' would be the means of ascribing developmental stages or levels of number understanding to pupils.

NUMBER CONCEPTS OF LOW ATTAINERS

17

METHOD All the data in the assessment part of the study were gathered from individual interviews with 7-9 year olds in which items were presented orally and in which frequent use was made of practical materials. The work was carried out in three stages: (i) Responses made in the pilot study helped to identify which skills it was appropriate to assess. (ii) The assessed skills were extended and defined more precisely during the main assessment study and the items to assess the skills were developed and refined. Pupils taking part in the main assessment study also participated in a longitudinal study. (iii) Finally the Diagnostic Assessment interview was trialled on a wider sample of pupils. Organisation of the samples is shown in Table I. Throughout the Pilot and Main Assessment Studies predictions were made about what hierarchical dependencies there might be between acquisitions of different number concepts. These were based on observation of pupils' behaviour, reflection o n their responses, logical analyses of the mathematics and evidence from research literature. From the Diagnostic Assessment Interview results with a wider sample of pupils, relationships between performances on every pair of skills was investigated numerically using item-item Loevinger coefficients (Loevinger, 1947). The Diagnostic Assessment Instrument was also used to examine changes in performance of a group of seven pupils over a'period of two years. RESULTS: THE ASSESSMENT INTERVIEW The interviews aimed to elicit each child's repertoire of strategies for dealing with number. The aspects which were considered included: (i) Strategies for adding and subtracting small numbers in 'sums' and word problems. (ii) Commutativity of addition. (iii) Enumerating grouped collections. (iv) Strategies for adding larger numbers: place value. (v) Piagetian tasks: conservation of number and class inclusion. These aspects will be discussed separately below.

Strategies for Adding and Subtracting Small Numbers Similar responses were observed in this study to those found by Carpenter and Moser (1979, 1982, 1983) and they are classified according to Carpenter

* N.F.E.R. (1969-80).

Diagnostic Assessment Interview (DAI) trials

Assessment studies

7 5to 9-3

Main study and longitudinal study 7 5to 9-6

8-3 to 9~)

Pilot study

Ages

TABLE I

Performance o f pupils in four classes assessed using N F E R * Basic M a t h s Test ' A ' for 1st year classes (grade 2) and 'B' for 2nd year classes (grade 3). Initially all pupils with Quotient < 90 were assessed. Later pupils with higher performances selected to add information a b o u t harder skills.

Teachers of four classes asked to select 3 or 4 pupils with lowest attainment in mathematics. 7 o f these selected on basis o f lowest attainment.

Teacher asked to select 10 pupils with lowest attainment in mathematics. 5 of these selected, on basis of lowest attainment, in preliminary interview.

M e t h o d o f selection

Organisation and samples

41

5

Sample size

6 Over 3 m o n t h s

6 Over 3 m o n t h s

N u m b e r of interviews/ teaching sessions

C: D: working class -multiethnic

middle/ working class multiethnic

B;

working class -multiethnic

A;

School

~0 9

>

~7

t;0

OO

NUMBER CONCEPTS OF LOW ATTAINERS

19

and Moser's terminology. On a few occasions, children simply 'knew' the answer to a 'sum' (Recalled Fact). In this example, however, Pal uses 'Derived Fact': BD: Pal: BD: Pal:

Sarah has some sweets. She has 5 toffees and 6 fruit gums. How many sweets has she got altogether? Eleven. Eleven. How did you do that? Easy. Add on - you add on - I added on five. I t o o k . . , like I took off one. Off six, added both fives then put back a one and added it up. It's eleven.

San uses 'Counting on': BD: San: BD: San: BD: San:

Eight add seven? Fifteen. How did you do that? Counted forwards. Counted forwards. What numbers did you say? I said nine, ten, eleven, twelve, thirteen, fourteen, fifteen.

'Counting up' is grouped together with 'Counting On' and 'Counting Back'. In this example Bri 'Counts Up'; the answer is the increment needed to count from the smaller to the larger. BD: Bri:

Alice has seven felt tipped pens, Graham has eleven. How many more has Graham got? (Using fingers) Eight, nine, ten, eleven (looks at fingers) . . . Four.

and Clo counts back: BD:

Clo: BD: Clo:

At a birthday party there are eleven people altogether. Some are children and some are grown ups. There are four grown ups, how many of them are children? E r . . . E l e v e n . . . Four grown ups (pause). There's only seven children. Seven children. How did you do that? O h . . . E r . . . I had eleven. And I had four. I went eleven, ten, nine, eight for the grown ups and I believed seven, six, five, four, three, two, one to be the children.

In 'Counting All' (with models) the child uses physical objects such as fingers or counters to model the problem, e.g.:

20

B. DENVIR AND M. BROWN

BD:

J o h n h a d 25 conkers, he gives eleven to his b r o t h e r , h o w m a n y has he g o t left?

Don:

( C o u n t s 25 stones, c o u n t s eleven a n d r e m o v e s them, recounts r e m a i n i n g g r o u p ) F o u r t e e n . H e ' s got f o u r t e e n left.

TABLE II(a) Frequency of strategies for carrying out additions and subtractions used by each child during Pilot Study

Derived Fact Recalled Fact Counting On/Back Counting All (with models)

Clo

Don

Bri

Pal

Loy

0 4 16 9

0 2 0 16

14 2 5 4

1 5 11 5

0 I 4 9

A further distinction a p p e a r e d in the main s t u d y between ' c o u n t i n g all' a n d c o u n t i n g f r o m one'. In response to '5 + 3'? Ch. typically c o u n t e d all, a s k i n g for cubes a n d c o u n t i n g o u t first five, then three, then c o m b i n i n g the two collections a n d c o u n t i n g f r o m one. O n the o t h e r hand: Ph:

F i v e a d d three? (Stares with fixed gaze straight ahead). 1, 2, 3, 4, 5 . . . 6, 7, 8. Eight.

TABLE II(b) Frequency of strategies for carrying out additions and subtractions used by each child during Main Study

Recalled Fact Counting On/Back/Up Counting from One Count All (direct physical modelling)

Ch

Je

Pe

Ph

Dn

Jy

Th

2 0 0 12

2 0 0 30

1 7 1 24

8 13 2 7

4 7 3 19

4 23 0 6

11 20 0 1

In the main study, no use was made of 'Derived Fact'. The three children ( D o n , Ch, Je) w h o d i d n o t use a c o u n t i n g o n strategy used c o u n t all with m o d e l s a l m o s t exclusively: their responses to discussions o f the c o u n t all s t r a t e g y c o n f i r m e d the evidence t h a t c o u n t i n g on was n o t in their repertoire. I n Je's response to a c o m b i n e p r o b l e m :

NUMBER

BD: Je: Je: BD: Je: Be: Je:

BD: BD: Je: BD: Je: BD: Je: BD: Je:

C O N C E P T S OF LOW A T T A I N E R S

21

Gary has some red marbles and some green marbles. He has 9 red marbles and 6 green marbles. How many marbles has he got? What was it again? (Counting fingers) 1, 2, 3, 4, 5, 6, 7, 8, 9 red a n d . . . How many green? (Repeats question) M m m . . . can I use your fingers? (Spreading fingers for counting) Use which you want. ('Puts away' her nine and counts BD's) 1, 2, 3, 4, 5, 6. (Now 'spreads' her nine again - counting each one) 1, 2, 3, 4, 5, 6, 7, 8, 9, (lets her nine 'go' and points to BD's) 10, 11, 12, 13, 14, 15. Fifteen. And you used my fingers as well as yours. Could you have done it with just your fingers? No. Why not? I ain't got enough. Suppose it was 9 red and 7 green. Could you work that out with your fingers? (Puts up 9 fingers, again counting each one). Nine. And what was it? 9 red and 7 green (Puts away 9 fingers and counts 7 of her own, then looks at them) Can I use yours?

These responses confirm work by other authors (Steffe et al., 1983; Fuson, 1982; Carpenter and Moser, 1982) in which a transition is observed between 'count all' and 'count on' and this is explained theoretically (Steffe et al., 1983). No such clear-cut transition occurs from either of these strategies to the use of recalled facts. Responses given by these low attaining 7-9 year olds were similar to responses given by younger children in the Carpenter and Moser (1982) and Riley et al. (1983). Commutativity of Addition There were three methods of determining whether children perceived that the addition of numbers was commutative. Note was taken of whether, in a series of written sums, they remarked on pairs of sums such as 3 + 6 and 6 + 3; their strategy for solving sums in which the smaller addend came first (e.g., 3 -t- 18) was noted; and the time children took to respond to sums of the form 1 + n and n + 1 was noted. These times are shown in Table III.

22

B. D E N V I R A N D M. B R O W N TABLE III Commutativity of addition for sums of the form n + 1, 1 + n. Time (in seconds) taken to give answers to sums of the type n + 1, 1 + n (0 < n < 10) in the Main Study Je

Pe

Ph

Dn

Jy

Th

2+I 3+1 4+1 5+1 6+1 7+1 8+1 9+1

2 3 8 8 9 10 8 9

1 1 1 1 1 1 1 1

1 1 1 1 1 1 I 1

2 1 2 1 1 1 1 1

2 I 1 1 1 1 1 1

1 I 1 1 l I 1 1

1+2 1+3 1+4 1+5 1+6

6 10 9 8 9 12 5

1 1 1 1 1

3 ! 1 5 1

3 1 1 5 1

1 1 1 1 1

1

1

1

1

1+8

2 3 8 6 11 10 10

1

1

1

1

1 +9

7

14

1

2

2

1

1 + 7

Je solved nearly all the sums by counting all, but Pe, had two distinct strategies forn + l a n d l + n. N.B. These sums were all given in one interview, but not in the tabulated order. BD:

( s h o w s 4 + 1)

Pe:

( L a u g h s ) F i v e ( O n e s e c o n d t a k e n to a n s w e r ) .

BD:

And how did you do that?

Pe:

I t ' s easy. I t was v e r y easy. I j u s t d o n e it.

BD:

Okay. (Shows 8 +

Pe:

N i n e . ( O n e s e c o n d to a n s w e r ) .

1).

BD:

And what about that one?

Pe:

J u s t d o n e it.

BD:

H o w d i d y o u d o it?

Pe:

E i g h t a d d o n e is nine.

BD:

O k a y . ( s h o w s 1 + 7).

Pe:

Seven . .. seven . .. one add seven ...

one add seven ...

I

d u n n o . ( T e n s e c o n d s to " I d u n n o " ) . BD:

H o w c o u l d y o u w o r k it o u t ?

Pe:

I dunno really...

I c o u l d d o it o n m y fingers.

T h i s s u g g e s t s a s t r a t e g y f o r d e t e r m i n i n g w h e t h e r c h i l d r e n w h o use a c o u n t i n g o n s t r a t e g y f o r a d d i t i o n a p p r e c i a t e a n d c a n use t h e f a c t t h a t a d d i t i o n is c o m m u t a t i v e .

NUMBER

CONCEPTS OF LOW ATTAINERS

23

Counting Grouped Collections Children were asked to say 'how many' were in grouped collections under various conditions; groups might contain 2, 5 or 10 individual items; there may or may not be ungrouped single items included in the collection; and individual items in the grouped collection may or may not be visible at the time of the enumeration. In one item four opaque bags, each containing ten sweets were opened in turn and the child encouraged to examine the contents. Having established that each bag, in its turn contained ten sweets, the four bags and three loose sweets were displayed and the child asked 'How many sweets?'. From the seven children in the main study sample, there were six substantially different responses: 1. Count all as one (CA1) Ch and Je both gave the same response: Je:

(Counts, pointing to each bag and each sweet in turn) 1, 2, 3, 4, 5, 6, 7. Seven.

2. Guesses (G) Pe~

BD: Pe: BD: Pe: BD: Pc: BD: Pe:

Twenty. Twenty. How did you decide that? I just guessed. Do you think your guess is right? No . . . dunno really. How could you find out if it was right? Tip out the sweets and count them. Yes, you could tip them out. Could you do it any other way? N o . . . I don't think so. (Proceeds to tip out sweets from each bag, put them together in one collection, then count in ones).

3. Count in ones - Guesses number in each bag C1/G) Dn:

(Gazes at first bag) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (transfers gaze to next) 11, 12, 13, 14, 15, 16, 17, 18, 19 (next) 20, 21, 22, 23, 24, 25, 26, 27, 28 (next) 29, 30, 31, 32, 33, 34, 35, 36, 37 (looks at loose sweets) 38, 39, 40 . . . . forty.

During this time Dn did not appear to use her fingers and did not seem to think, when questioned, that she had used them. It does seem likely,

24

B. D E N V I R

AND

M. BROWN

however, that she was matching her count to some mental image of ten objects. In Fuson's terms Dn was 'counting entities' and in Steffe's she was a 'counter of figural items'. 4. Counts in ones (C1)

Jy:

How many? (She looks at a bag, puts up ten fingers and counts, touching each finger in turn on her lower lip) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (and repeats, transferring her gaze to each bag in turn) 1 1 . . . 20, 2 1 . . . 30, 3 1 . . . 40 (then points to each loose sweet) 41, 42, 43.

. 'Place Value' strategy, incorrectly co-ordinated (PVc) Th: BD: Th:

Dunno. Could you work it out? I could count in tens. (Point to each of the four bags in turn) 10, 20, 30, 40. (Points to each of the loose sweets in turn) 50, 51, 52.

6. 'Place Value' strategy (PV) Ph: BD: Ph:

Forty three. Forty three. How did you decide it was 43? (Points to each bag, then each sweet in turn) 10, 20, 30, 40, 41, 42, 43.

Th's error could easily have been passed off as a 'careless slip'. But evidence from her behaviour as well as from other children interviewed at other stages of the study suggests that children who make this type of error do so frequently. Table IV summarises children's strategies and changes in strategies for enumerating collections grouped in tens and ones during the Main Study. Results support Resnick's (1983) notion that children find it easier to count when only one 'denomination', i.e., only ls or 10s or 100s has to be counted than when 2 or 3 denominations are present. The results suggest that several skills are needed for an 'economical' solution: (i) knowledge of the number word sequence for tens; (ii) appreciation of the structure of the grouped collection; (iii) the ability to stop the 'tens number word sequence' and begin the 'ones number word sequence'; (iv) ability to co-ordinate the end of the groups of ten and the beginning of the individual items with the change in the number word sequence.

25

NUMBER CONCEPTS OF LOW ATTAINERS TABLE IV Strategies for enumerating collections grouped in tens in the Main Study Date

Jan (1) Jan (1)

Objects used

Individual Tens only or Strategiesused by: items visible? tens and ones? Y/N 10s/10s & ls Ch Je Pe Ph Dn Jy

Straws Y Dienesbase Y ten blocks Feb (1) Sweets N March (1) Sweets N March (1) Straws Y March (2) Straws Y March (2) Sweets N

Th

10s & ls 10s & ls

C1 C1 CA1PV

C1PVc C1 PVc C1 C1PVc PVc CA10 Cl

10s & 10s & 10s & 10s 10s &

CA1 C1/G C1 C1 C1

G PV G PV C1PV C1 PV Cl PV

ls ls ls ls

CA1 C1/G C1 C1 C1

C1 CI C1 C1 PV

C1 CA10 CA10 PV PV

PVc CA5 PV PV PV

C1 = counts in ones. CA1/CA5/CA10 = counts each 'package' as 1(5, 10). G = Guesses. C1/G = Counts in ones, guessing how many count words to say for each 'package'. PVc = 'Place Value' strategy incorrectly co-ordinated. PV = 'Place Value' strategy. Thus, children m a y pass t h r o u g h a sequence o f stages in using a place value strategy to enumerate a collection o f grouped items, namely: (1)

C o u n t i n g each 'package' as one, regardless o f numerosity o f packages (--. incorrect answer).

(2)

C o u n t i n g each item as one ( ~ correct answer).

(3)

A t t e m p t at place value strategy by counting grouping n u m b e r for each package and one for each u n p a c k e d item, but lacking co-ordination ( ~ incorrect answer).

(3b)

Successful place value strategy.

Strategy for Adding Larger Numbers: Place Value D u r i n g the pilot, for orally presented addition and subtraction sums with larger numbers (e.g., 18 + 24), children usually resorted to ' c o u n t all', although a few 'counted on' for addition. N o child showed an appreciation o f the place value structure o f numbers, by for example, adding tens and units separately. It is, o f course, likely that children would have responded differently if the 'sums' had been presented in the conventional 'vertical algorithmic' format. In the main study children were instead presented with the m o s t simple oral questions in which a 'place value strategy' for addition might be observed, e.g., '20 + 4'. It was usually very easy to tell, from the

26

B. DENVIR AND M. BROWN

time taken (cf. Resnik, 1983) whether there had been an 'automatic' response '24' or the child had counted on: '21, 22, 23, 24'. Further evidence was provided by the child's description o f the strategy. F o r example for 40 + 9, K a r replied '40 + 9?

(Puts up 9 fingers) 41, 42, 43, 44, 45, 46, 47, 48, 49'

and had clearly counted on, but C o r must have used a 'place value strategy' when he t o o k under one second to reply '49' and said in describing his strategy: 'Well I had 40. A n d I had 9, and it's 49' Table V shows the items asked at different stages o f the main study and the strategies used by each child. TABLE V Strategies used in Main Assessment Study for adding two digit numbers Date

Jan (1) Feb (1) Feb (1) Feb (1) Feb (1) Feb (2) Feb (2) Feb (2)

Item

38 + 20 + 40 + 10 + 20 + 5+ 6+ 27 +

Strategy for:

7

4 9 6 6, 30 + 6, 40 + 6, 50 + 6 20 30 10

Je

Pe

Ph

Dn

Jy

Th

DM DM DM DM DM DM DM DM

DM CO CO CO CO DM DM DM

DM CO CO CO PV PV PV DM

CO CO CO CO CO CF1 DM DM

CO CO CO CO CO CO CO CO

CO CO CO PV PV PV PV CO

DM = direct modelling with physical objects - count all. CF1 = counts from one (without physical objects). CO = counting on. PV = place value strategy. The results suggest that adding ten to a 2-digit n u m b e r is more difficult than adding units to a decade number. Table V, like Tables II, III, and IV suggest that pupils' responses to these four categories o f questions were highly consistent: on none o f them did any child use quite different strategies each time. F u r t h e r m o r e the changes which did. occur between interviews shown in Tables IV and V are towards a m o r e sophisticated strategy, indicating learning. These results are generally in agreement with Resnick's (1983) model o f n u m b e r development in which she sees an elaboration o f the concept o f cardinality and the part-part-whole schema accounting for children's solution

NUMBER

CONCEPTS

OF LOW

ATTAINERS

27

to word problems and also for the stages through which children progress in their grasp of place value concepts. A new model of children's elaboration of the part-part-whole schema for cardinal number is proposed here and shown in Table VI. This is an extension and modification of Resnick's model. It does not represent a 'natural' progression since it relates to children within an educational system and within a culture that heavily emphasises certain aspects of mathematics. At a general cultural level, the standard number word sequence and canonical form are emphasised; within the educational system th-ese are usually also heavily emphasised, but other aspects may also be emphasised and this is likely to affect not only the rate, but possibly the order of elaboration. It was noted above that children rarely count back for 'take away' but an appreciation of this strategy and an awareness of the inverse nature of addition and subtraction appeared to be important aspects in mental computation, e.g., for 43 - 28 Miw:

43 take 2 8 . . . forty take 2 0 . . . 8 . 2 0 . . . 2 0 . . . 3 from 8 . . . is f i v e . . . 5, 19, 18, 17, 16, 1 5 . . . fifteen.

Similar observations were made by Fuson (Fuson et al., 1982). T A B L E VI

Elaboration of cardinality of number Stage of elaboration of part-part-whole schema

Example of number behaviour

None

Can state how many in a simple collection

1.

(a) Can count on to add

(b) Can solve 'compare and 'missing addend'

Additional concept from earlier stage

The whole is the sum of the parts - neither more nor less

2.

(a) Can count grouped collection (b) Can add tens and units, mentally, when there is no regrouping

The parts may have different sizes (tens and ones)

3.

Can add, mentally with regrouping

The different sizes are related to each other (I0 • I = 1 • 10): all numbers must be in canonical form

4.

Can take away, mentally, with regrouping

Numbers can be expressed in non-canonical form

5.

Mentally can solve sharing problems

Numbers can be expressed as products of other numbers

28

B. DENVIR AND M. BROWN

Piagetian Tasks This consistency of response was not apparent in every type of question. Performance on the Piagetian tasks (number conservation, class inclusion) was much more variable as it was on another item similar to the number conservation task but specifically designed for this study. Children were asked to compare two collections of similar but distinguishable objects which were grouped differently (Figure 1).

collectionof bluestors ~-X ~'~ ~

Fig. 1. Illustration of an item in which pupil is asked to compare two collectionswhich are grouped differently. The child was asked: 'Are there more blue stars, more yellow stars or the same number of blue and yellow?' Several strategies were observed: only the first one is successful: (i) (ii)

(iii)

(iv)

Counting each collection one by one and comparing (C1) Comparing the number of 'single' items in each collection and ignoring the grouped items (CS), e.g., in second item (Table VII) says 4 x 4 + 2 > 3 x 6 + 1 'because 2 there and only 1 there'. Comparing the total number of 'packages' in each collection, irrespective of size, i.e., comparing the number of groups plus the number of singles but ignoring the number of items per group (CNP), e.g., in the first item 3 x 4 + 2 > 2 x 7 + 1 because 5 there and only 3 there'. Comparing the number of items per group but ignoring the number of groups and the number of single items (CGN), e.g., in first item says 2 • 7 + 1 > 3 x 4 + 2 'because 7 is more than 4'.

As Table VII shows only three of the seven children showed a consistent response. Furthermore whilst two assessments are insufficient for drawing many conclusions, there appears to be no consistent trend in the changes of strategy.

NUMBER CONCEPTS OF LOW A T T A I N E R S

29

TABLE VII Strategies for comparing two collections which are grouped differently Date

Item

Ch

Je

Pe

Ph

Dn

Jy

Th

Feb.

Compares2 • 7 + l(blue stars) with 3 x 4 + 2 (yellow stars) individual stars visible

CGN

CGN

CI

C!

CNP

C1

CS

March (1)

Compares 4 x 4 + 2 (blue stars) with 3 x 6 + 1 (yellow stars) individual stars visible

CNP

CNP

CGN

C1

C1

C1

CS

In the final version o f this t a s k the c o m p a r i s o n was between different c o l o u r e d sweets g r o u p e d t o g e t h e r in c a r d b o a r d boxes so that, a l t h o u g h each i n d i v i d u a l sweet h a d previously been e x a m i n e d by the child, at the time t h a t the c o m p a r i s o n was m a d e each i n d i v i d u a l sweet was n o t directly visible ( F i g u r e 2). orange sweets: 5 in each box, 3 'loose' ones

2 in each box

Fig. 2. Final version of item for comparison of two collections grouped differently.

RESULTS: THE D I A G N O S T I C ASSESSMENT I N T E R V I E W In all, 47 skills were assessed for each o f the 41 pupils interviewed in the D . A . I . These skills were s h o w n in T a b l e VIII(a). There were two m a j o r outcomes o f the analysis o f children's performances in the D . A . I . b a c k e d u p b y the q u a n t i t a t i v e d a t a collected in the assessment studies: (a)

Levels o f P e r f o r m a n c e

(b)

T h e Descriptive F r a m e w o r k

Levels of Performances W h e n each o f the skills was o r d e r e d a c c o r d i n g to facility a n d each o f the pupils o r d e r e d b y overall r a w score it was possible to g r o u p the skills into

30

B. D E N V I R A N D M. B R O W N TABLE VIII(a)

Skills assessed in D.A.I. in order of difficulty (hardest first). The categories of word problems is that given by Riley et al. (1983) 3. 6. 47. 4. 45. 7. 5. 2. 20. 33.

15. 46. 12. 8. 24. 25. 34. 9. 10. 1. 17. 26. 13. 16. 19. 21. II. 14. 22. 18. 23. 40. 27. 35. 38. 39. 44. 28. 29. 30. 32. 37. 31. 41. 36. 42.

43.

Mentally carries out two digit 'take away' with regrouping Uses multiplication facts to solve a 'sharing' word problem Perceives 'compare (more) difference unknown' word problem as subtraction Models two digit 'take away' with regrouping using 'base ten' apparatus Appreciates concept of class inclusion, without any hint or help Mentally carries out two digit take away without regrouping Uses multiplication facts to solve a 'lots of' word problem Mentally carries out two digit addition with regrouping Uses counting back/up/down strategy for 'take away' Bundles objects to make a new group of ten in order to facilitate enumeration of a collection which is partly grouped in 10s and ls Uses repeated addition or repeated subtraction for a 'sharing' word problem Partly appreciates concept of class inclusion Uses derived facts for addition Mentally carries out two digit addition without regrouping Can repeat the number sequence for counting in 10s from a non-decade two digit number Can repeat the number sequence for counting backwards in 10s from a non-decade two digit number Makes quantitative comparison between two collections which are grouped differently 'Knows answer' when taking ten away from a 2-digit number 'Knows answer' when adding ten to a 2-digit number Models two digit addition with regrouping using 'base ten' apparatus Knows number bonds (not just the 'doubles') Interpolates between decade numbers on a number line Models two digit 'take away' without regrouping using 'base ten' apparatus Uses repeated addition for a 'lots o f word problem Solves 'compare (more) difference unknown' word problem Counts in 2s and ls to enumerate a collection grouped in 2s and ls 'Knows answer' when adding units to a decade number Models two digit addition without regrouping using 'base ten' apparatus Counts in 5s and ls to enumerate a collection grouped in 5s and ls Solves 'compare (more) compared set unknown' word problem Counts in 10s and ls to enumerate a collection grouped in 10s and ls Knows numbers backwards from 20 Orders a selection of non-sequential two-digit numerals Appreciates structure of grouped collections Solves sharing problems by direct physical modelling Solves 'lots of' problem by direct physical modelling Appreciates conservation of number Appreciates commutativity of addition for sums of the form 1 + n Uses a counting-on strategy for addition Reads a selection of non-sequential two-digit numerals Repeats numbers in correct sequence for counting in 2s, 5s and 10s Uses counting on strategy when provoked Repeats numbers in correct sequence to 99 Knows numbers backwards from 10 Compares collections and states whether equal Can say numbers in correct sequence to 20, can solve addition and take away by direct physical modelling Makes 1 : 1 correspondence

NUMBER CONCEPTS OF LOW ATTAINERS

31

TABLE VIII(b) Performance of wider sample in Diagnostic Assessment Interview classified according to level of performance S k i l l Ho. ylll(s)

TABLg

) 6

,

i.~i!::%.::>..:>>::!:!.i "': ",".~i!~i~ii.!i~i!i~ii

2 33

iI

U

,4

'.'..,

:..::.v,.....

' :'+i".%

',~~

:.)':'..'.;:'.,7:Z.2'.v:7::)v:z,: :r

":-::.)'

~:?i:2"~!:i':i:i.::i:i:):i:i~:i.~'!i.!:i:?.!:i~r :i:i':):

:i':Z:'i:

...::::: ,;~:,:.~::. " >.:.?:.::?:::ii:U.::: L..::::;i,:;~.::,!::i?::::::?.:.:,i:. ~.."? ::::::::::::::::::::::::::i'.'...'..:..ii!:::.:.< ::::::::::::::,.-?..:i :::::::::::::::

,:

:!:::iT;ii:i:::i!i ili:711: ::::7;i

. ======".:.========:.".:: ===:::::::::::::: =====::::::::::::::::::::::::::::::::::::::::::::::::::: ==================::::::::::::::::::::::::::::::::::::::: ==============::::i::::::::::::: ========.":::!:::"==================== ::~ i :.:i~i[~:i}:~}i:~iii:2:ii~5i:i:~igii!!:~:?:i:i:!i)!ii!i:!::i:i:Xii}}i:ii:!:i:!i?i:}::{ii:iil:~:~::i:: ::":::f.H::::[

b,Ni?ii:!iii!ili:!ki::i~i::!?ii!i!?I~!iii~i!:!i!i:i!iiiii:!iliiii:::i:!ii ~ii~i:i~!ili::::::ili?ii:ili!~i[i?ii ~ u p l l Hame HFER A or B

quotienL

* Absent when NFER test administered. 'levels' defined by a particular range o f facility so that every pupil w h o had succeeded in 2/3 o f the skills at any level had succeeded in 2/3 o f the skills at every preceding level. This is shown in Table VIII(b). Thus it was possible to describe each child's performance in one o f two ways, firstly by detailing

32

B. DENVIR AND M. BROWN

performance on each of the 47 skills and secondly by ascribing one of 7 levels of performance to each pupil. As Table VIII(b) indicates, the first (listing performance on every skill) produced a unique description for each child. Oneskill in Table VIII(b) appears misplaced; skill 20: uses counting back for 'take away', though low in facility is passed by pupils with an overall low score. Carpenter and Moser (1982) found few children using counting back strategies: our results support these findings. However both quantitative and qualitative data suggest that children do reach a stage when they grasp that 'counting backwards will solve a take away'. Whilst, in practice, they may find this difficult, preferring either 'direct modelling' or 'counting on', it seems likely that this is an important notion for children to have grasped. We conclude that, in this case, the assessment should have focussed more closely on the child's perception rather than the preferred strategy.

The Descriptive Framework There are two criteria for portraying a pair of skills in the framework as if one were prerequisite for the other: (i) There should be a logical reason (which can be defended by the researchers) why one skill depends on the other. (ii) All or nearly all of the children interviewed who succeeded on the harder skill must have succeeded on the easier one. The descriptive framework is shown in Figure 3. In addition to relationships between skills in which one skill is 'prerequisite' for the other (indicated by solid lines in Figure 3), some skills appeared to be strongly connected by a logical relationship (e.g., skill 8: mentally carries out 2-digit addition without regrouping and skill 1: models 2-digit addition with regrouping using 'base ten' apparatus) and also by empirical evidence, but some pupils acquired the skills in the reversed order so that neither could be regarded as being prerequisite for the other. These are indicated by broken lines. As hypothesised, whilst the acquisition of some skills forms part of a 'hierarchy' of skill acquisition, others are quite independent. The strongest hierarchical strand appears to be a 'place value strand': counting on (29) ~ counting collections grouped in tens (23) --* modelling addition and subtraction with blocks (14, 13) ~ mentally adding units to decade numbers (1) ~ mentally adding tens to 2-digit numbers (10) ~ mentally carrying out 2-digit take away without regrouping (7). The method of assessment consisted of attempting to discover what strategies each child had for dealing with numbers. On the whole the items

NUMBER CONCEPTS OF LOW ATTAINERS

~

33

'

( | Fig. 3-Descriptive framework,n indicates skill n in Table VIII(a), ~ indicates skitl m pre-requisite for skill n, a n d ( ~ Q indicates strong connection between skill i and skillj. used and the techniques adopted appeared to yield a coherent and sensible description of children's understanding, but nevertheless the method has some important limitations. The child's understanding is inferred from the strategies which that child is observed to use. in fact the child may have the understanding but simply choose to use other strategies all the time. Furthermore there is a motivation aspect. It is likely that some children only produce their most sophisticated reasoning when they are highly motivated to succeed. Although both these limitations were recognised from the outset and interviewing strategies designed as far as possible to overcome the difficulties, nevertheless in some cases the results may provide an underestimate of pupils' abilities.

The Longitudinal Study The D.A.I. was used to examine changes in performance of seven pupils interviewed approximately six monthly over a period of two years. The

34

B. DENVIR AND M. BROWN LEVEL OF PERFORMANCE WITH TIME Level Of Performoncein Assessment Interview

,.w. ph.f" f

3

!h_

8

13

20 Time in months

(a)

RAW SCORE WITH TIME RowScorein Assessment Interview

~xPh

~

Dn

C

h

Ch

0

3

8

13

ZO Time in months

(b) Fig. 4. Changein performance with time for pupils in the longitudinalstudy. timing of these interviews is shown in Table IX. Figures 4(a) and 4(b) show their progress in terms of changes in raw score (i.e., total number of skills successfully performed) and in level of performance, respectively. Several points emerged from an examination of these results: (i) All the children made progress at nearly every stage of the study, but this progress was, in most cases, very slow; so slow, in fact, that a less sensitive instrument might not have registered any change for some of the pupils.

N U M B E R C O N C E P T S OF LOW A T T A I N E R S

35

(ii) Whilst no two pupils acquire skills in the same order, the match between each pupil's performance at each interview and the hierarchical framework was extremely good - only 3 skills were at any time acquired 'out of order'. (iii) Each pupils' performance in relation to the framework was used as the basis for two remedial teaching studies which are described in a forthcoming paper (Denvir and Brown, 1986).

SUMMARY

If one assumes that there is a development aspect to children's learning of number, useful prescriptive teaching arising from diagnostic assessment needs to take account of three different aspects of learning: (i) the orders in which children learn, i.e., a framework describing acquisition; (ii) where each individual child is within the framework; (iii) how the individual progresses from one skill to another, i.e., how individuals learn. This paper has described, within a limited range of performance for a small aspect of the mathematics curriculum how the first two aspects have been considered. The third aspect was also dealt with during the research study and will be reported in a subsequent paper (Denvir and Brown, 1986).

REFERENCES Bednarz, N. and Janvier, B.: 1982, 'The understanding of numeration', Educational Studies in Mathematics 13, 33 57. Bennett, N., Desforges, C., Cockburn, A., and Wilkinson, B.: 1984, The Quality of Pupil Learning Experiences, Erlbaum, New Jersey. Brown, M.: 198 I, 'Levels of understanding of number operation, place value and decimals in secondary school children', unpublished Doctoral Dissertation, University of London, Chelsea College. Carpenter, T. P. and Moser, J. M.: 1979, 'An investigation of the learning of addition and subtraction', Theoretical Paper No. 79, Madison, Wisconsin Research and Development Center for Individual Schooling. Carpenter, T. P. and Moser, J. M.: 1982, 'The development of addition and subtraction problem solving skills', in T. P. Carpenter, J. M. Moser and T. A. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 9-24. Carpenter, T. P. and Moser, J. M.: 1983, 'Acquisition of addition and subtraction concepts', in R. Lesh and M. Landau (eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York, pp. 7-44. Comiti, C.: 1981, ~ premieres acquisitions de la notion de nombre par l'enfant', Educational Studies in Mathematics 11, 301-318.

36

B. D E N V I R A N D M. B R O W N

Denvir, B. and Brown, M.: 1986, 'Understanding of number concepts in low attaining 7-9 year olds, part II: The teaching studies', Educational Studies in Mathematics 17 (2) (in press). Denvir, B., Stolz, C., and Brown, M.: 1982, 'Low attainers in mathematics 5-16: Policies and practices in schools', Schools Council Working Paper 72, Methuen Educational, London. Descoudres, A.: 1921, 'Le development de l'enfant de deux a sept ans', Delachaux et Niestle, C.A., Paris. Fuson, K. C.: 1982, 'Analysis of the counting on procedure', in T. P. Carpenter, J. M. Moser and T. A. Romberg (ed.), Addition and Subtraction: A Cognitive Perspective, Elbaum, New Jersey, pp. 67-81. Fuson, K. C., Richards, J. and Briars, D. J.: 1982, 'The acquisition and elaboration of the number word sequence', in C. J. Brainerd (ed.), Children's Logical and Mathematical Cognition, Springer Verlag, New York, pp. 33-91. Gelman, R. and Gallistel, C. R.: 1978, The Child's Understanding of Number, Harvard University Press, Cambridge, Mass. Hughes, M.: 1981, 'Can preschool children add and subtract?, Educational Psychology 1, 207-219. Loevinger, J.: 1947, 'A systematic approach to the construction and evaluation of tests of ability', Psychological Monographs 61, No. 4. American Psychological Association. Matthews, J.: 1983, 'A subtraction experiment with six and seven year old children', Educational Studies in Mathematics 14, 139-154. Nesher, P.: 1982, 'Levels of description in the analysis of addition and subtraction word problems', in T. P. Carpenter, J. M. Moser and T. A. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 25-38. N.F.E.R.: 1969-80, "Basic Mathematics Tests', National Foundation for Educational Research. NFER-Nelson, London. Schaeffer, B., Eggleston, V. H., and Scott, J. L.: 1974, 'Number development in young children', Cognitive Psychology 6, 357-379. Steffe, L. P.: 1983, 'Children's algorithms as schemes', Educational Studies in Mathematics 14, 109-125. Steffe, L. P., von Glaserfeld, E., Richards J., and Cobb, P.: 1983, Children's Counting Types: Philosophy, Theory and Application, Praeger, New York. Steffe, L. P. and Johnson, D. C.: 1971, 'Problem solving performance of first-grade children', Journal for Research in Mathematics Education 2, 50-64. Resnick, L. B.: 1983, 'A development theory of number understanding', in H. P. Ginsberg (ed.), The Development of Mathematical Thinking, Academic Press, New York, pp. 10% 151. Riley, M. S., Greeno, J. G., and Heller, J. I.: 1983, 'Development of children's ability in arithmetic', in H. P. Ginsburg (ed.), The Development of Mathematical Thinking, Academic Press, New York, pp. 153-196. Vergnaud, G.: 1982, 'A classification of cognitive tasks and operations of thought involved in addition and subtraction problems', in T. P. Carpenter, J. M. Moser, and T. A. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 35-59.

Shell M a t h e m a t i c s Education Unit, Centre f o r Educational Studies, Kings College ( K . Q . C . ) , University o f London, 552 Kings Road, Chelsea, London S W I O OUA, England