Strategies and Errors in Secondary MathematicsAuthor(s): Kathleen HartSource: Mathematics in School, Vol. 16, No. 2 (Mar., 1987), pp. 14-17Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214189 .

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STRATEI AND ERRORS IN SECONDARY MATHEMATICS

by Kathleen Hart "Concepts in Secondary Mathematics and Science" (CSMS) finished in 1979 having formulated hierarchies in 11 topics commonly taught in the secondary school. The project generated a large amount of data some of which concerned the occurrence of specific errors. The sub- sequent research project "Strategies and Errors in

Question 1

Find the area of

5

e 2

13 year olds giving error

'Error Answer' answer Correct

Question 2

Find the perimeter

h h

tin

Question 3

Add 4 on to 3n

5e2, e10, 10e 42 7% e+10

hhhht 27 57% 4ht 5ht

3n4, 7n 45 22%

Fig. 1 Algebra errors illustrated by single items.

14 Mathematics in School, March 1987

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Secondary Mathematics" (SESM) was funded to inves- tigate the cause and cure of some of these errors. There are now three books available, which describe the results of the research in Ratio and Proportion, Algebra and Fractions. This article is to remind readers of that research and to give a brief overview of the results.

SESM used the CSMS results in Algebra, Ratio and Proportion, Graphs, Fractions and Measurement as base data. Some 10 000 children had been tested on one or more (usually two) of the CSMS test papers and a record was available of the incidence not only of the correct answers but also identified wrong answers. Usually these specific wrong answers had been given during interviews using the same items and so the researchers were able to offer tentative descriptions of the reasons that underpinned the stated error. More information was needed however before any ideas concerning ways of helping children overcome their misconceptions could be suggested to teachers. A basic philosophy of the research was that the errors being committed were indicative of the cognitive processes of the children and investigation of the reasons they were made would result in valuable insights of the child's reasoning patterns.

Methodology of SESM Phase One The first task was to select from the CSMS results those erroneous answers and interview statements which might form a fruitful area for investigation. The errors and the proposed problem for research in the topics of Algebra, Fractions and Ratio and Proportion are described below.

CSMS Algebra (n = 2923) In Algebra the problems investigated and highlighted by

these erroneous respones, were essentially the meaning children give to letters when used in generalised arithmetic and their interpretation of even the most elementary laws of algebra.

CSMS Fractions (n = 555) Question 1 3 - 5 (The interpretation of the division sign)

Answers: 3/5 or .6 12 1 r. 2 5/3 or 1

Age 12+ 35 5 18 3%

13+ 31 10 18 9%

Question 2 Equivalence and its use in addition

12 yr 13 yr 14 yr 15 yr

3/8 + 2/8 = 5/16 9 20 14 17%

1/3 + 1/4 = 2/7 18 29 22 20%

Question 3 How many fractions lie between 1/3 and 1/2?

14 yr 15 yr

Answer 1 30 30%

Fig. 2 Fractions Errors.

(Do children think fractions are numbers?) The CSMS survey results in almost every topic had

shown that children have a marked reluctance to use fractions. Kerslake6 hypothesised that a major cause of error was the child's restricted view of the meaning of a fraction.

CSMS Ratio and Proportion (n = 2257) Question 1

Mr. Short

You can see the height of Mr. Short measured with paper clips. Mr. Short has a friend, Mr. Tall. When we measure their heights with match- sticks:

Mr. Short's height is four matchsticks Mr. Tall's height is six matchsticks How many paper clips are needed for Mr. Tall's height?

Age 13+ 14+ 15+

Correct 28.1 30 42

Answer 8 51 51 39% D B

A

'8

E

These two letters are the same shape, one is larger than the other. AC is 8 units. RT is 12 units. The curve AB is 9 units. How long is the curve RS? U

S

R

12

V ,T

Age 13+ 14+ 15+

Correct 14 20 29%

Answer 13 44 40 34

Fig. 3 Error in Ratio and Proportion.

The CSMS interview data suggested that to obtain answers such as these on the Ratio items the children used addition rather than multiplication, i.e. in an enlargement of a length "b" to a corresponding length "a" the pupil assumed an addition of "a - b" had taken place and conse- quently increased all other lengths by adding this amount.

Phase Two A new sample of 13, 14, 15 year olds obtained from "mixed ability" classes or from those sets considered "average" in attainment was obtained from five schools. These children were asked to complete the appropriate CSMS test paper and then the individuals who produced the identified incorrect response were interviewed. Initially they were asked to give their reasoning when obtaining correct an- swers and then to describe how they attempted the specific questions which resulted in "the error", later they were given other questions to test whether their method was consistently used. This phase is illustrated by a discussion of the investigation into the incorrect addition strategy in Ratio and Proportion.

The children in the CSMS "Ratio and Proportion" sample who had three out of four times on particular questions given answers like those shown in Figure 3, were designated "Adders". Table 1 shows their general perfor- mance and it can be seen that they were by no means the lowest attainment group.

Mathematics in School, March 1987 15

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Table 1 Performance of Adders, CSMS Ratio Data (1976)

Age Number Levels (Year Group) of Adders 0 1 2 3 4*

13(2) 246 6 138 78 24 14(3) 220 5 138 57 20 15(4) 156 2 83 59 12

*Level 4 items are those on which the addition strategy occurs.

The SESM Ratio sample when asked to describe how they solved items successfully such as that shown in Figure 4, tended to give answers which did not employ multipli- cation. So in part (i) they tended to reason "B's food is C's food and C's food again" and in part (ii) "Find C's food, then add B's and C's food to give A's food". None of the children ever multiplied by a fraction.

Question 2a There are three eels A, B and C in the tank at the Zoo.

15 cm long A

10 cm long B

5 cm long C

The eels are fed sprats, the number depending on their length.

(i) If C is fed two sprats, how many sprats should B and A be fed to match? B A

(ii) If B eats 12 sprats, how many sprats should A be fed to match? A

(iii) If A gets 9 sprats, how many sprats should B get to match? B

Fig. 4 Example of an easy Ratio question (CSMS Ratio Data).

Generally when successful on Ratio and Proportion items they replaced multiplication by repeated addition or worked with small sections of the answer, adding eventually to obtain the total result. In items where these methods would not suffice because a ratio of other than n:2 or n:1 was involved, then the adders still opted for addition but of a constant amount. When faced with the problem of enlarge- ment the adders could recognise that using a - b rather than

a/b when enlarging a figure, provided a diagram which looked incorrect, e.g. they said "It's too long", "Looks more tilted", "Need bit more of a slope". This recognition was of itself insufficient to either (i) stop them repeating the error; or (ii) prompt them to a correct method. From the 45 interviews with "adders" it was possible to formulate those aspects of enlargement problem-solving that they seemed to need for success. These aspects formed the basis of the lesson plans which were then used to teach the "adders". Adders appeared to:

(i) interpret the operation needed to effect an enlarge- ment as addition and not multiplication;

(ii) refuse to multipy by a fraction (since they saw no use for multiplication this is not surprising);

(iii) be unable to find a scale factor. (But why find a multiplying factor if you never multiply?)

Phase Three

For each topic the researchers were able to draw up a list of significant features which distinguished those children giv- ing the identified wrong answers. Teaching ideas designed to address these misconceptions were tried initially on groups of children known to commit the errors. In the case of "Enlargement" the aim of the lessons was that the

"adders" would successfully find the scale factor, given a figure and the enlargement of one side, apply the scale factor successfully and obtain the correctly enlarged figure. Four groups of 14-, 15- and 16-year-olds (n = 4, 5, 3, 6) were chosen on their performance as "adders". All had been taught the topic before in their normal classes but in the research lessons, a particular error was addressed rather than a general overview of the topic being undertaken. Recognising that the adders had shown their preference for "naive methods", a teaching philosophy was adopted which attempted to present a more formal (or general) method only when there was an expressed desire on the part of the group for it. For this to come about, when each obstacle seemed to have been successfully conquered, a new chal- lenge was presented. For example if the group was able to multiply 4 by 1a by saying "one four and half of four" then an example of enlargement requiring "1- x 21" was presen- ted since the method quoted is inadequate for this. At this point each group asked for an explanation of the method of multiplication.

The first teaching module included the teaching of the algorithm for multiplying fractions but little else in the revision of fractions. The effectiveness of each stage of the teaching was tested and then an immediate and delayed post test (five weeks later) administered. Although the continu- ing evaluation and immediate post test showed that the groups had progressed, by the delayed test only partial success could be recorded. Since the reasons for this failure depended heavily on the knowledge of fractions, another module was written in which computation of fractions was replaced by the use of the calculator. Although this required other introductory activities the module overall when tried with "adders" had more success and yet the children did not become dependent on the calculator.

Phase Four Having taught small groups for whom the work was particularly relevant, the researchers taught the same material to intact classes in which there were children who produced the identified responses as well as those who had never considered its use. The intention was to provide classroom teachers with suitable material for their classes which would contain successful pupils, "adders" and others who had different misconceptions. In the three topics described here worksheets or work-cards were used to provide a sequence of about six lessons (a total of approxi- mately four hours).

In Enlargement the ideas for teaching in order to take into account the appearance of the incorrect addition strategy are briefly described in Table 2.

Table 2 Teaching Module for Enlargement

1 Recognition of similar and non-similar figures 2 Outcome of using addition strategy shown and discussed 3 Large diagrams in which the enlargement is 3-> 33, 5-+ 45 to

encourage multiplication 4 Enlargement by 2, 11 5 1/3, 1/5 of figures, to provide examples of diminution 6 Calculator practice 7 Enlargement by scale factors using calculator 8 3 -, 5, 7--+ 3 diagrams to suggest scale factor 9 Finding scale factor using equation

10 More examples

Phase Five Volunteer teachers were recruited to test out the material in their own secondary school classes. The children were tested on the topic being investigated before and after the teaching and again after two or three months (sometimes the long summer vacation). The results for Enlargement, Fractions and Algebra are described below.

16 Mathematics in School, March 1987

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Enlargement Six teachers volunteered to teach the module on Enlarge- ment but only four completed the trial. Figure 5 shows the results obtained by these teachers when using the work- sheets. The graph represents the results of those children who attempted 50 per cent or more of the worksheets.

Percentage Percentage 100- 100-

/II

80- 2 80 -

/ a. 60 -, 60 -

o o 0 , C

(D

40- O 40- - o E

(1) E

a o 20- E 2 20-

E a>

Correct answers Addition strategy answers Correct answers Addition strategy answers

School 1 School 3 School 5 School 6

Number of answers given The percentage is obtained: 100 x Number of answers possible

Fig. 5 A comparison between performances of schools.

Note that each class had almost entirely abandoned the incorrect addition strategy immediately after the lessons and after 11 weeks it reappeared (at a lower level) in only one school.

The materials were designed specifically to combat the incorrect addition strategy and this appears to have been successful. Of the 23 children who would have been regarded as "adders" on the original criterion (three answers of the "addition type" when faced with the four CSMS questions, Fig. 3) only one would have been so classified at the delayed test. From the CSMS longitudinal survey we know that in their own class setting when they received the normal mathematics lessons 500% of those defined as adders at age 13+ were still adders at 15 years of age.

Fractions In the Fractions investigation Kerslake provided activities and worksheets which sought to redress three perceived problems:

(1) that many children conceive of numbers as whole numbers only and do not accept fractions as an extension to the set of natural numbers;

(2) that the "part of a whole" diagram with which most children associate fractions is not a helpful model for operations on fractions and a greater variety of models is required; and

(3) that the idea of equivalence needs to be more firmly based if it is to be successfully applied to the solving of problems in fractions.

Her results demonstrate that these three aspects of Fractions understanding are of different degrees of dif- ficulty. Of particular note is the resistance by many children

to the idea that it is possible for division to result in a fraction. Table 3 shows that although the success rate increased for each of the aspects of Fractions under con- sideration many of the children were still unsuccessful.

Table 3

Results of Fractions Teaching Module

Comparison of mean scores for Division, Equivalence and Number between Class Trials and Main Experiment

Pre test Post test Delayed Post test

Division Class Trials 31 60 54% Main Experiment 29 65 52

Equivalence Class Trials 39 52 54 Main Experiment 40 60 61

Number Class Trials 31 50 55 Main Experiment 35 58 62

Algebra The results suggested that the teaching programme devised on the basis of information gleaned from interviews with children who committed certain errors, improved the gen- eral level of understanding in elementary algebra. The gains were of the order of three to seven items correct out of a total of 21 items. The main improvement for all groups appeared to have been in the correct use of algebraic notation and the acceptance of expressions such as 3n + 4 (see Fig. 1). The use of brackets was not addressed at length by the teaching module and the complexities of the problem resulted in less improvement in performance than in the notation items. How children interpret a letter formed part of the investigation and it was found that with certain classes the programme was highly effective.

Conclusion The results of the investigation point to the conclusion that specific intervention at a particular time seems to have been effective with most children as far as the abandonment of the incorrect strategy is concerned, and with many children when the correct responses were required. The difference between the work attempted in SESM and the usual class "correction" was that the reasons why the children were failing were investigated thoroughly and the remediation took into account their naive and child-methods besides considering the transition to more formal (and gen- eralisable) methods.

In nearly all the classes investigated it was apparent that certain errors were more resistant to change than others during the short remedial sequences designed for use by classroom teachers. Some children did not improve their performance even though using the materials but the approach would seem worthwhile for a substantial number of those committing the errors.

References 1. Hart, K. (1980) Secondary School Children's Understanding of Math-

ematics, research Monograph, Chelsea College, London. 2. Hart, K. (1981) Children's Understanding of Mathematics 11-16: Mur-

rays, London. 3. CSMS team (1985) Chelsea Diagnostic Tests: NFER-Nelson,

Windsor. 4. Hart, K. (1984) Ratio: Children's Strategies and Errors: NFER-Nelson,

Windsor. 5. Boothe, L. R. (1984) Algebra: Children's Strategies and Errors: NFER-

Nelson, Windsor. 6. Kerslake, D. (1986) Fractions: Children's Strategies and Errors: NFER-

Nelson, Windsor.

Mathematics in School, March 1987 17

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