J U L I A MATTHEWS

A S U B T R A C T I O N E X P E R I M E N T W I T H S I X A N D S E V E N

Y E A R O L D C H I L D R E N

ABSTRACT. This paper gives an account of an experiment involving 176 children of 6 and 7 years and their difficulties with subtraction problems. By using two screening questions, four levels of attainment were established and two teaching progzammes were then devised to help the children forward to their next level. The results show that it is possible to assess a level of understanding in subtraction quickly and accurately and that, through the teaching prog~ammes, pre-requisite skills needed to establish a sound foundation for sub- traction can be taught. The experiment indicates, however, that an attempt to symbolise too soon is a danger to be avoided.

1. I N T R O D U C T I O N

This paper reports on an experiment on the understanding of two aspects of

subtraction by 6 and 7 year old children, namely 'take away' and 'difference between'. A preliminary experiment had highlighted the difficulties encoun- tered by such children when faced with simple subtraction problems (involving only numbers below 10). Before proceeding to the experiment reported below, it was decided first to see whether the difficulties in understanding resolved themselves with age and maturity. Accordingly, an investigation was made involving four categories of subjects, age range 11 to 27 (Matthews, 1981). This showed that many of the difficulties remained, and so confirmed the value of studying the problems of the 6 and 7 year olds more deeply. The children are at the interface between pre-operational and early concrete operational

thought (see, for example, CoUis in Keats et al., 1978), and it was decided to investigate the possibility o f helping them forward from one stage to the next

by means of teaching programmes in the specific context of subtraction prob- lems.

Previous research on subtraction has concentrated on identifying compara-

tive difficulties o f types of subtraction, e.g., 'take away', 'difference between',

(see, for example, Gibb (1954) and the review of current research given by Carpenter et al. (1982)). Published material for young children, such as work-

books and work-sheets, mainly ignore the children's errors and their causes and, at least implicitly, encourage premature symbolism. The research reported

below not only identifies the difficulties but also offers at least partial remedies.

Educational Studies in Mathematics 14(1983) 139-154. 0013-1954/83/0142-0139 $01.50. Copyright �9 1983 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

140 J U L I A MATTHEWS

2. D E S I G N OF E X P E R I M E N T

It was decided to use about 150 children (half experimental , half control) from

schools in the Greater London area. The experimental group would be taught

by myself on a one-to-one basis on each of my visits.

Two screening questions were devised:

(i) 'Take 9 c u b e s . . , now give me 6. How many have you left? '

(ii) 'Give yourself 9 cubes from this pile. Now give me 3 from the pile

(NOT from his set). 'How many more have you than me? '

According to their answers, the children were categorised at different levels

of abil i ty for their understanding o f subtraction. Those unable to answer either

questions were to be placed at Level 1 ; those answering (i) but not (ii) were to

be placed at Level 2. For these Level 1 and 2 children, a series of activities (Pro-

gramme A) was devised to help them to Level 3, i.e., to be able to answer both

questions correctly.

Programme A was carried out verbally and no written recording was

required. It was carried out on a one-to-one basis, using sets o f small objects

found in any classroom (buttons, counters, etc.) and a toy ladder with two t iny

dolls. The first activity was:

'Put out 5 counters for yourself in a line. Now I'll put out 4 (underneath the child's line). Have you got more or have I?' The children generally agreed they had more and were then asked 'How many more?' A child saying '5' (i.e., his set) instead of '1 more' was corrected by, for example, pointing out that his line had only 1 more than mine and the problem left to the following day.

The second activity involved the ladder and the dolls: "

'My doll has gone up 5 steps but yours has only gone up 3. How many more steps has mine gone up than yours?'. If the child said '2', he was then asked 'How many more steps will your doll have to go up to reach mine?' The 'difference' between the two numbers was then discussed.

These activities were to be repeated at least five times over a period of 10

school days, with the numbers altered (but not above 9). A month after the

conclusion o f the activities, the children would be post-tested on the two screening questions.

The remainder of the children, i.e., those who answered both (i) and (ii) cor-

rectly in the initial screening, were to be placed at Level 3. For these children,

Programme B was devised to see whether they could use symbols.

The first part of Programme B consists o f questions requiring the child to

record, using symbols, the results o f a problem he had already solved with the

help o f concrete aids. (Details of these questions are given below in the results.)

Any child successfuly answering all these questions was to be placed at Level 4

SUBTRACTION WITH 6/7 YEAR OLDS 141

and removed from the experiment. The remainder would proceed to the second part of Programme B (see Appendix), which consists of a number of teaching routines devised to help the children to answer the questions of the first part of the programme. The children would then be post-tested on these questions a month after the completion of about 15 ten-minute teaching ses- sions. The following diagram shows the route taken by the various children, starting with the screening questions (i) and (ii).

,= y ~ ,. No

142 JULIA MATTHEWS

3. EXECUTION 176 children were chosen at random from four different schools, one within the Inner London Education Authority and the other three in Outer London. The children came from varied backgrounds, mainly working class, and were all in the first term of their last school year in the infants' school, so that they were at least 6 years old and a few had just reached 7. In the nine classes involved, all the children were accustomed to 'doing' written addition and sub- traction tasks, often unsuccessfuly.

After an informal meeting with each child individualy (when a game was played and a friendly rapport established), the next meeting started with the two screening questions, so that the child's level of ability, for the purpose of the experiment, could be determined.

The 176 children concerned were processed as follows:

Yes

Yes 47

( Level 41

No P

99

146 , I

44

No 30

(could not be~ ( selecte(:l tor "~ \ matched ) ~

S U B T R A C T I O N WITH 6/7 YEAR OLDS 143

The children were listed within each class according to age and then selected

by me al ternately as experimental or control . Although all the children were

' top ' infants, that is, between 6 and 7 years old, I decided to divide them into

two age groups for closer analysis o f the results.

6+ (children between 6.0 and 6.6)

7 - (children between 6.7 and 7.0).

Table I shows the groupings and levels from each o f the four schools, following

the two screening questions.

TABLEI Asfi~menttoprogrammesbyage groups

School 1 School 2

Programme 6 + 7 " X Programme 6 + 7--

A 12 6 18 A 7 5 12 B 3 7 10 B 3 1 4

15 13 28 10 6 16

School 3 School 4

Programme 6 + 7-- r. Programme 6 + 7--

A 8 3 11 A 8 14 22 B 0 3 3 B 1 4 5

8 6 14 9 18 27

(These numbers refer, of course, to the experimental group and a similar number, equally placed, were in the control group.)

4. R E S U L T S

P r o g r a m m e A

Table II shows success/failure at post-test after Programme A, the four schools

and both age groups being combined.

TABLE II Programme A overall success/failure at post-test

Experimental Control

Success 46 22 68 Failure 17 41 58

63 63 126

144 J U L I A MATTHEWS

The children in the experimental group did significantly better than the

children in the control group applying X 2 test (t7 < 0.001). In Table III, the schools are combined and the children separated into age

groups.

TABLE III All schools, age differences (Programme A)

Experimental Control

Age group 6 + Success 23 11 34. Failure 12 24 36

35 35 70

Age group 7 - Success 23 1 24 Failure 5 27 32

28 28 56

In the 6 + age group, the experimental group did significantly better than the control group, applying the X 2 test (p < 0.10), but the significance was not

so great as for the overall result shown in Table II. In Table IV, the age groups are combined but the schools separated.

TABLE IV Schools differences (Programme A)

School N Experimental group Control group

Success Failure Success Failure

1 18 14 4 9 9 (78%) (22%) (50%) (50%)

2 12 9 3 3 9 (75%) (25%) (25%) (75%)

3 11 4 7 0 11 (36%) (64%) (0%) (100%)

4 22 17 5 7 15 (77%) (23%) (32%) (68%)

From these figures it seems intuitively likely that one factor affecting the number of errors made by children is the schoo l which they attended. (School 1, for example, seems 'better ' than School 3.) A deeper investigation into this area, however, would require the use of a larger number of subjects.

SUBTRACTION WITH 6/7 YEAR OLDS 145

Table V shows the percentages of success, after the post-testing of Pro- gramme A, of boys versus girls. The four schools are combined and so are the

age groups.

TABLE V Sex Differences (Programme A)

Boys Girls

x/ 19 27 46 x 9 8 17

28 35 63

Using the • test, this result shows no significant difference between the suc-

cess rates of boys and girls. Of the 126 children screened (experimental and control), at the end of the

experiment, 66 children succeeded in reaching Level 3. However, six were still at Level 1 and 52 at Level 2 and all these must be considered to be at 'mathe- matical risk'. Fifteen of these children, some from each of the four schools, and all from the experimental group, are the subject of a special study to be reported later.

Programme B

The questions used in the pre- and post-tests are in two parts, except for the preliminary question (a). The first parts are oral, with materials available as required; the second parts involve the recording in figures and symbols of the results of the first parts.

Table VI gives the overall results for the pre and post-tests, x /denotes success and x failure. For example: x x means failure at both pre- and post- testing, while: x x /means failure in pre-test but success in post-test. The actual questions are given and discussed below. In addition to the 44 children originally assigned to Programme B, a further 30 children, who had successfully completed Programme A, were included, (15 experimental, 15 control).

Analysis o f Programme B results

Question (a): ' I f you had 9 cubes and gave me 3 of them, how many would you have left?'

The results shown in Table VI are not surprising as this was a deliberately

146 J U L I A MATTHEWS

TABLE VI Success/failure at pre-and post-test, (age groups and schools combined)

Part 1 Part 2

Question XX Xx/ ~/~/ x/X EorC XX Xx/ ~/x/ ~/X

(a) 0 55 32 0 E 1 3 31 2 C

(b) 0 1 35 1 E 5 9 21 2 0 1 34 2 C 6 8 21 2

(c) 1 3 31 2 E 2 8 18 9 1 4 30 2 C 9 3 22 3

(d) 3 5 29 9 E 25 10 2 0 5 3 23 6 C 34 1 0 2

(e) 6 8 23 0 E 30 7 0 0 4 8 20 5 C 36 0 0 1

(f) 5 10 19 3 E 24 7 6 0 8 5 17 7 C 28 4 2 3

'easy' question, put in to give the children confidence. No written work was

involved.

Question (bj Part 1: ' I f you gave me 6 bricks and then took 2 of them back for

yourself, how many would I have left? '

The high success rate, as shown in Table VI, indicates that a simple subtrac-

t ion sum in the ' take away' category, with concrete materials available and not

involving place value, is well within the understanding of most children of 6

and 7 years.

Question (b) Part 2: 'Now write down in numbers and put in the signs to show

what you did so that we can read it like a sum.'

On the Fisher Exact Test, the experimental group did not do significantly

bet ter than the control group, so that Programme B activities did not specific-

ally help in bridging the gap between the 'doing' and the 'writing' of this prob-

lem. However, both groups showed progress during the 3 to 4 months between

the pre- and post-testing. In fact, as shown in Table VII, all the children taken

together did significantly bet ter on the post-test than on the pre-test, taking the • test (p < 0.01).

Question (c) Part 1: ' I f you took 5 bricks and then took 4 more, how many

would you have altogether? '

S U B T R A C T I O N WITH 6/7 YEAR OLDS 147

TABLE VII Success/failure: all children,

Question (b) Part 2.

N= 74 x/ ~<

pre-test 46 28 post-test 59 15

This addition question was deliberately included in order to make sure that

the children were able to cope with addit ion, even though they had previously

associated me always with questions concerned with ' take away ' or 'difference

between' . Table VI gives reassurance on this point , showing that over 80% of

the children in each group were successful in the pre-test and over 90% of each

group were successful in the post-test.

Question (c) Part 2: 'Now write down in numbers and put in the signs to show

what you did so that we can read it like a sum.'

Table VIII shows the success rate o f all the children together on this ques- t ion.

TABLE VIII Total children, success/failure

pre- and post-test, Question (c) Part 2

N = 74 x/ X

pre-test 52 22 post-test 51 23

The success rate is not so high as for Part 1. In fact, in spite of the apparent

simplicity o f the problem, less than 70% of each group succeeded in the pre- or

post-test. Overall, the children made no progress from pre- to post-test. This

indicates that even under almost ' ideal ' teaching conditions of a one-to-one

daily contact , I did not succeed in ' teaching' with understanding the use of the

plus symbol in the relatively short period of three to four weeks to those child-

ren who were not ' ready ' for it . It is difficult to see how the programme could

be radically improved unless the child was ' ready' . This state of readiness is

dependent upon a wide range of experiences over a period which cannot be

specified as each child's needs are different and children have different rates of progression.

Question (el) Part 1: 'There is a pile o f bricks there and we will have some each.

You take 9 and I'U take 6. How many more have you than me?'

148 JULIA MATTHEWS

The majority of the children were able to answer this question correctly at the pre-test but the success rate is not quite as great as Question (a) and Part 1 of Question (b). This reflects the difficulty experienced by children in compar- ing two sets of objects, as against operating on one set only.

Question (d) Part 2: 'Now write down in numbers and put in the signs to show what you did to get the answer so that we can read it like a sum.'

This question involves not only 'difference' but also use of the minus sym- bol and this combination defeated the majority of the children. At the post- test, 68% of the experimental group and 97% of the control failed to write the 'sum' correctly (see Table IX). Nevertheless, the improvement in performance of the experimental group was significantly better than that of the control

group (Fisher Exact Test, p = 0.002).

TABLE IX Success/failure at post-test experimental/

control, Question (d) Part 2

, / x z

E 12 25 37 C 1 36 37

13 61 74

Question (e) Part 1: 'From this pile of bricks, if you gave me 7 and gave your- self 9, how many more would you have than me?'

The success rate at the pre-test of both experimental and control groups is less than that of Question (d) Part 1, (see Table VI). The post-test results show that the experimental group did not do significantly better than the control

group (Fisher Exact Test).

Question (e) Part 2: 'Now write down in numbers and put in the signs to show what you did to get the answer so that we can read it like a sum.'

This question, involving 'difference between' and the use of the minus sym- bol appeared similar to Question (d) Part 2. However, only one child from the total of 74 recognised this as a subtraction problem at the pre-test, almost cer- tainly because the smaller number was mentioned first in the question. At the post-test the improvement in performance of the experimental group was sig- nificantly better than that of the control group (Fisher Exact Test, p = 0.006), see Table X.

S U B T R A C T I O N WITH 6/7 YEAR OLDS 149

Success/failure TABLE X

at post-test experimental/ control, Question (e) Part 2

~/ x z

E 7 30 37 C 0 37 37

7 67 74

Question (f) Part 1: ' I had some sweets and gave my friend 3 and now I've got 6 left. How many did I start with?'

This question was posed as a mental problem. Although a box of counting aids was on the table in front o f the child, these were not referred to as specifically as they had been in the previous questions. Very few children attempted to use the aids as substitutes for the sweets: the problem itself appealed to them.

Table VI shows that 62% of all the children were successful at the pre-test.

The improvement in performance from pre- to post-test of the experimental

group was significantly better than that of the control group (Fisher Exact Test, p = 0.043).

Question (fj Part 2: 'Now write down in numbers and put in the signs to show what you did to get the answer so that we can read it like a sum.'

The success rate at the pre-test o f all children was very low: only 11 from

the 74 children (15%) were able to answer correctly. At the post-test, three

from the original successful 11 children changed their minds and gave an in-

correct written answer. The experimental group did not do significantly better than the control group at the post-test, (Fisher Exact Test).

The problem, however, was tackled by some children, quite correctly, as an

addition sum and they wrote either:

3 + 6 = 9 or 6 + 3 = 9.

When this happened, the child was shown the following list and asked to choose the correct sum:

(i) 6 - - 3 = 9 (ii) 3 - - 9 = 6 (iii) 3 - - 6 = 9 (iv) 9 - - 3 = 6

(v) 6 = 3 -- 9. The choice o f answer was interesting: most of the children who had written

150 JULIA MATTHEWS

6 + 3 = 9, chose answer (i) and those who wrote 3 + 6 = 9, chose answer (iii), presumably keeping to their original figures and ignoring the minus sign. A small number chose the correct answer (iv).

Table XI shows the levels of all the children who took part in the experi- ment.

TABLE XI Levels at beginning and

end of experiment

Level Beginning End

1 30 6 2 96 52 3 44 109 4 0 3

5. CONCLUSIONS

Teaching Programme A affords perhaps the most important and immediate application to the classroom. In the experimental group, 73% of the children who started at Levels 1 and 2 benefitted from the programme, in that it took them to Level 3.

The two screening questions proved invaluable diagnostically and could be of immediate value to teachers. They were, in fact, given to a far larger sample country wide (N = 1133). Sixty-four percent of the children concerned (all 6 and 7 year olds) were unable to answer both questions and so could profitably be given Programme A.

The dramatic results of Programme A can partly be explained by the advant- age of the one-to-one contact, but they also reflect how children can progress when given tasks suited to them, rather than the impersonal randomness of what happens to be on the next page of a work book.

Programme B must be viewed more circumspectly. It was aimed at helping children from the concrete mode to the symbolism of problems concerned with 'take away' and 'difference between'. This programme did not have such a dramatic success rate as Programme A simply because many 6 and 7 year old children are not ready for the necessary symbolism. In fact, only three of the 74 children were totally successful (See Table XI). There were, however, a number of partial successes, notably in the Part 1 of the questions, where no written work was required.

Regarding the second parts, Question (b), involving one set of objects and the minus sign, was comparable in difficulty to Question (c), involving two sets

SUBTRACTION WITH 6/7 YEAR OLDS 151

and the plus sign. The figures in Table XII indicate that these questions reflect appropriate levels for activities for some of the children of this age range. It would be difficult, however, to argue for the inclusion in the curriculum of questions like (d) and (e) for all 6 and 7 year olds. These questions deal with both 'difference be tween ' and symbolism. The children were able to deal with some symbolism with a considerable success rate, as shown by the results of Questions (b) and (c). They also showed their ability to deal with "difference

between' , but only orally, in Part 1 of Questions (d) and (e). It is undoubtedly the combination of the two which causes the difficulty. Nevertheless, the experimental group did significantly better than the control group on Part 2 of Questions (d) and (e) (see Tables IX and X), so that the programme benefit- ted a minority of children.

Table XII shows the overall percentage of successes at the post-testing of all children, thus indicating the suitability of the questions concerned. The success rates of questions of comparable difficulty are grouped together and averaged.

TABLE XlI Overall average percentage success at post-

test of all children (Programme B)

Questions Part 1 Part 2

(b) and (c) 94 74 (d) and (e) 87 14

The results in general indicate that, for a minority of children, it is possible to help them in a comparatively short time with an appropriate programme, to make vital steps forwards towards an understanding of 'difference between', involving written symbolism.

The results, however, also confirm the view of the Cockcroft Report that " . . . a premature start on formal written arithmetic is likely to delay progress rather than hasten i t" (D.E.S., 1982). This view was mooted 10 years earlier by the Nuffield Mathematics Project (Nuffield, 1972, p. 62) and later by Lunzer et al. (1976, p. 59). An even earlier warning against premature symbolism was made by Renwick (1963):

A child of seven should not be supplied with a symbol system which will be appropriate to his stage of development at nine - he is not helped but hampered in his efforts to bring order to his child's world.., a pattern of symbols is the projection of a pattern of ideas'.

The research reported above does not, however, simply reflect these previous findings, but suggests positive action to help children already involved in the system.

All the children in the experiment were doing written sums using addition

152 JULIA MATTHEWS

and subtraction symbols, most without understanding and with random suc- cess. In fact, the majority of the children could not cope with Programme A (involving no written work), let alone Programme B. Sadly, this reflects the pic-

ture in many parts of the country, with 6 and 7 year olds 'doing sums' with

very little real understanding. The work reported in this paper gives a strong

indication that teachers of young children couM alter this unsatisfactory situ-

ation if they would:

(a) resist the pressures applied by parents, colleagues and others to plunge

the children into premature symbolism, and

(b) identify at an early stage the levels of attainment of their pupils (as indicated by the screening questions in this paper), and then ensure that they were given mathematical tasks attuned to their ability and level of attainment.

APPENDIX: PROGRAMME B (PART 2) ACTIVITIES

BI: 'Take 7 blue cubes and 2 yellow ones and put them in a line. Now I will take 7 blue ones and put them in a line under your line. How many more have you than me?' (Tell if necessary.) 'How many fewer have I than you?' (Tell if

necessary.) 'So the difference between 9 and 7 is 2 and we write it like this: 9 - 7 = 2, that is nine minus seven equals two.' Let child write 9 - 7 = 2. 'Now just let's look at your 9 cubes . . , take away 7 . . . how many are left?' (Child will probably say'2', if not tell him.) 'So, 9 take away 7 leaves 2 and we write it like this: 9 -- 7 = 2.' Let child write and repeat '9 -- 7 = 2'.

B2: 'You have 5 cubes, and we'll pretend they are sweets. If you took 2 away and gave them to me how many would you have left?' (Tell if necessary.)

'So 5 take away 2 leaves 3 and we write it like this: 5 -- 2 = 3. Now suppose you still had your 5 sweets and I bought 2 from the shop, how many more

have you got than me?' (Tell if necessary.) 'How many fewer have I got than

you?' (Tell if necessary.) 'How many more would I have to buy so that we both had the same?' (Tell if necessary.) 'What is the difference between the

number you have which is 5 and the number I have which is 27' (Tell if neces- sary.) 'So the difference between 5 and 2 is 3 and we write it like this: 5 -- 2 = 3 just like you wrote it for 5 take away 2 leaves 3.' (Let child write and repeat the equation.)

B3: [Picture of birds provided, 9 in top row and 7 (initially concealed) in lower row].

'Here is a picture of some yellow birds - count t h e m . . , yes 9 of them. Now

S U B T R A C T I O N WITH 6/7 YEAR OLDS 153

some black birds come along.' (Show lower half o f picture.) Count the black b i r d s . . , how many more yellow birds are there than black birds?' (Tell if

necessary.) 'So the difference between the number of yellow birds which is 9

and the number of black birds which is 7, i s . . . 2, and we write that difference like this: 9 - 7 = 2. Now let's pretend that the black birds fly away, each

taking a yellow bird for a partner.' (Put a pencil between the 7 pairs of

'at tached' birds and the 2 'unattached' ones.) 'How many birds are l e f t ? . . . that 's r i g h t . . . 2'. (Now conceal bot tom line o f birds.) 'So there were 9 yellow

birds and 7 flew away, there are 2 left and again we write it like this: 9 -- 7 = 2? (Let child write and read aloud '9 minus 7 equals 2'.)

B4: [Materials required: 16 cubes to represent apples, a number line and a

picture o f two trees.]

'Pretend the cubes are apples and put 9 apples on the first tree and 7 on the other tree. How many more apples are there on the first tree than on the second o n e ? . . , how many fewer are there on the second tree?' (Tell if neces- sary.) 'So the difference between 9 and 7 is 2 and we write it like this: 9 -- 7 = 2, that is nine minus seven equals two. ' (Let child write and repeat this.) 'Now you remember the first tree had 9 apples, well the farmer picked 7 o f them. Use the number line to show how many are left. ' (Demonstrate if necessary.)

'So the difference between 9 and 7 is 2 (point to the 9 and 7 on the number l ine) . . , and 9 take away 7 is 2.' (Point to the first tree with the cubes on.) 'We can write about both o f our discoveries like this: 9 -- 7 = 2. ' (Let child write and read equation.)

BS: ' I f you blew up 7 balloons for a party and your friend blew up 4, show

me on the number line how many more you blew up. ' (Tell and show if neces-

sary and point out the pattern.)

0 1 2 3 4 5 6 7 8 9

'So the difference between 7 and 4 is 3 and we write it like this: 7 - - 4 = 3 ' .

(Let child write and repeat this.) ' I f I had 7 balloons and you took away 3, use

the number line to show me how many would be left. ' (Tell, show and point out the pattern if necessary.) 'So 7 take away 4 leaves 3 and we write it just the same as we write the difference between 7 and 4, that is 7 - 4 = 3. '

B6: ' I f you had 8 sweets and I had 5 fewer than you, show me on the num- ber line how many I would have' (show if necessary). 'So 5 fewer than 8 is 3,

154 J U L I A MATTHEWS

that is the difference between 8 and 5 is 3. We write this: 8 - - 5 = 3. ' (Let child

write and read this.) 'Now suppose I had no sweets at all and you had 8 and

then I t ook away 5 o f yours, how many would you have left - show me on the

number line. ' (Help i f necessary and point out the pattern.) 'We write this the

same as we write the difference between 8 and 5, 8 - - 5 = 3, that is eight minus

five equals three. ' (Let child write and read this.)

B7: 'Here is a sum: 9 -- 5 = 4. Let's read it. Now finish these two stories

about this sum:

Story 1. I had 9 sweets and gave you 5 a n d . . .

Story 2. You had 9 sweets and I had 5 a n d . . .

(Possible finish to story 1 . . . 'I had 4 left ' and to story 2 . . . ' the difference

is 4 ' . I f child finds difficulty in finishing the stories - suggest endings and try

him later with other examples.)

These activities should be spread over about 15 consecutive school days and be

repeated with different numbers (not exceeding 9) on a separate occasion i f the

child falters.

University o f Technology, Loughborough

R E F E R E N C E S

Carpenter, T., Moser, J., and Romberg, T. (eds): 1982, Addition and Subtraction: A Cog- nitive Perspective, Lawrence Erlbaum Associates, New York.

Gibb, E. G.: 1954, 'Take away is not enough', Arithmetic Teacher 1, 7-10. D.E.S. (Department of Education and Science): 1982, Mathematics Counts, Her Majesty's

Stationery Office, London. Keats, J. A., Collins, K. F., and Halford, G. S. (eds): 1978, Cognitive Development, Wiley,

New York. Lunzer, E., Bell, A., and Shiu, C.: 1976, Numbers and the World of Things: A Develop-

mental Study, University of Nottingham. Matthews, J.: 1981, 'An investigation into subtraction', Educational Studies in Mathe-

matics 12, 327-338. Nuffield Foundation: 1972, Nuffield Mathematics Project, Teachers' Guides, J. Murray

and W. R. Chambers, London. Renwick, E. M.: 1963, (7nildren Learning Mathematics, A. StockweU, London.