Colloquial MathematicsAuthor(s): Rosemary SimmonsSource: Mathematics in School, Vol. 7, No. 5 (Nov., 1978), pp. 6-7Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213411 .

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sentence and by detailed editing, employing research criteria for readability. SMP 7-13 is probably still unique in curriculum development terms in having the very closest liaison between the writers, artists and designers. Regular weekly meetings discuss each card's design and format in detail to ensure the fullest implementation of educational ideas. Finally our thorough and widespread testing programme has been essential to discover a classroom organisation structure that really works. The details - individual record cards, assessment tests, practical teachers handbook, separate answer book, and so on, may seem obvious now but they emerged only from long-term classroom testing with more than 20 000 children. I want to stress that the above major points, and many smaller objectives achieved, are listed not in any spirit of self-congratulation so much as genuine satisfaction and happiness that the course will be suitable for children and teachers. In fact breakthroughs are not achieved through good ideas alone, as Edison put it; for every 5% inspiration we need 95% perspiration. Detailed, dedicated and directed hard work is essential to achieve any major advance in curriculum development.

My greatest personal satisfaction, however, is something that has largely motivated my own commitment to SMP 7-13. It is the thought that we can communicate mathematics to individual pupils of all abilities and from all social backgrounds. I want them to find SMP 7-13 (or indeed any other course) readable, enjoyable, and mathematically satisfying, and hence to get more out of education. It has been a source of great happiness to hear almost unanimous reports of children actually enjoying mathematics and being turned on by relevant, interesting and motivating work in- SMP 7-13. The ultimate and fundamental

criterion is simple: it is the success or failure with individual children in the classroom, it is there that SMP 7-13 will prove itself in the future.

Postscript Some unsolicited children's comments on SMP 7-13:

I like SMP because I can work by myself.

It gives me ideas.

I like SMP because it gives you a lot of interesting things to do.

I like SMP because you can work with your friends. And there's lots of things to do. And you find things out like how much things weigh.

I like SMP Maths because you can do all kinds of maths and you can make things.

I like SMP because it is something you can do without having to stay in the classroom all the time.

I like SMP because it gives you lots of things to do. Also you find words which you do not know what they are and you find them out.

You can measure. It has lots of different sums in it. You can count people. It is very interesting.

I like SMP because it isn't just one thing. And because it is more interesting than ordinary maths.

It goes at the speed I want it to. I like measuring and lots of other things.

I think on some cards there are too many sums like 26. I just about like 15.

It's helped me think about how to do maths.

Colloquial Mathematics

by Rosemary Simmons, Southlands School, Biggleswade

In order to be able to suggest what aspects of mathematics should be taught to children to enable them to be mathematically competent and confident in everyday life as adults, I analysed my own use of mathematics over a period of two weeks.

Summarised, the results indicated the basic needs to be:

Familiarity with a wide range of unit values Ability to estimate and compare Oral competence in basic operations with small numbers

(under four figures) Competence in analysis of complex problems, selection of

appropriate units and methods of operation Confident understanding, speed and accuracy

I suggest that there is nothing controversial or extraordinary here. In fact, the list is similar to that which might be found included in the stated aims of any scheme for mathematical education. Yet evidence of non-transference of mathematical skill from school to everyday life is not hard to find. Examples of inadequacy are commonplace - examples like that of the young girl in charge of the supermarket cash register who had to be shown the receipt before she was convinced that a1.92

6

could not be the correct charge for five articles at 48p each, and that the machine had only "rung up" four.

The need to relate school mathematics to "real" situations has long been recognised, and for the past 20 years new mathe- matics schemes have emphasised the importance of developing understanding through practical experience (eg Mathematics for the Majority, The Nuffield Primary Mathematics Project).

If the content of mathematical education is appropriate to the needs of everyday life, why are so many unable to apply the mathematical skill acquired at school to everyday situations?

In this paper I suggest what I consider may be one possible answer to this question.

My hypothesis is that a key reason for non-transference of school learning in mathematics to the reality of life situations is that the processes of mathematical thinking which are needed in everyday life are not the same as those taught in school.

Consider, for example, the petrol pump attendant who, in calculating the cost of four gallons of petrol at 79p, said: "Four eights are thirty-two that will be a3.20 less 4p, please." His method of calculation bears no resemblance to the written form that is commonly taught in school, ie "Four nines are

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thirty-six, carry three. Four sevens are twenty-eight, and three is thirty-one". Conventional computation in the written form is essential for the clear, concise communication of mathematical processes and manipulation. The orderliness and economy of standardised procedure facilitates extrapolation and abstraction. But in ordinary, everyday life these considerations are largely immaterial. The problems to be solved are immediate and specific, and what is required is not the formal application of a generalised "rule of thumb" but an ability to think with mathematical commonsense.

It is generally accepted that there are two styles of expression in language - the formally correct style which is used in writing and the freer, colloquial style which is used in conver- sation. Most people, I suggest, are more articulate in the second. Fluency in both styles is regarded as necessary for adult life and is encouraged in school. Originality of expression, within the conventional bounds of grammar, syntax and punctuation, is applauded.

Why cannot a colloquial form of mathematics, the mathe- matics illustrated by the petrol pump attendant, be actively encouraged for oral use?

That children can be mathematically articulate and capable of originality when uninhibited by conventional methods of manipulation is shown by the following experiment.

I asked a class of 10-year-old children of mixed ability how they would do the sum 49x 3V1/2p. Although most of the children were familiar with the process "long multiplication", because the problem was given orally it did not occur to any child to suggest using the form in which it would have been written

3V1/2p x49

in a textbook or on a workcard as a method of approach. If it had been presented in written form, conditioned by the rule that it is not possible to multiply by an amount of money, it would not have occurred to the children to apply the commuta- tive law

49

x 3V2p

nor would they have known how to multiply by 1/2 in this form. They were insufficiently conversant with the manipula- tion of vulgar fractions to have been able to attempt 49 x 7/2p.

But they did suggest the following solutions, which I trans- lated into equation form on the blackboard exactly as they dictated, using the simple symbols with which all were familiar.

(a) 3V1/2p + 31/2p + 31/2p + . .. (49 times) (b) 10 x 31/2p x 4+ 5 x 3V1/2p + 2 x 31/2p x 2 (c) 10 x 3V1/2p x 5- 3V1/2p (d) 100 x 31/2p + 2- 31/2p (e) 3 x 49p + 49p + 2 (f) 49 x 3p + 49 x V2p (g) 20 x 31/2p x 2 + 10 x 3V1/2p

- 31/2p (h) 4 x 31/2p x 10 + 9 x 3p + 9 x 1/2p (i) 49x4p-49x 1/2p (j) 50 x 4p - 50 x V2p - 31/2p (k) 3p x 50 + 50 + 2 - 31/2p

Several observations of significance can be made from this exercise:

1. Although the calculations appear cumbersome in the written form, for oral use they are more logical than the method which would have been used to calculate the sum on paper.

2. Many more children were able to work out the answer than would have been able to tackle the problem if it had been presented in written form.

3. The considerable variety in methods of approach contrasts with the single method of approach to which children are often restricted in formal computation.

4. In no instance was the 1/2p multiplied first, as it would have been if the conventional method of written multiplication had been followed.

5. Because the problem was presented orally, the children res- ponded orally. No one asked if they could use pencil and paper.

6. Without the restraint of a visible "p" the commutative law was used without hesitation (equation "e").

The children were then asked to work through the list, solving each equation to make sure they agreed that each was logically sound. In this way, they not only had a great deal of practice in basic computation, but they gained a little insight into the dif- ferent ways in which other minds worked. (It also produced an extremely valuable side effect. The written forms produced ambiguities which were not apparent to the children until they arrived at different answers. After argument and discussion, it was they who "invented" brackets to mark off which sections should be worked first - evidence that the understanding of formal mathematics can be strengthened by encouraging the use of the children's own mathematical thinking or logic.)

Some adults who were not considered successful in mathe- matics at school have, nevertheless, highly developed colloquial computation skills for everyday use, and are able to solve, with remarkable speed and accuracy, problems which they would find perplexing in their written form. But there are many others in whom this skill has not developed, who are mentally chained by the computation formulae they were taught to apply to "sums" in school, and so are unable, like the girl in the super- market, to deal intelligently with even the simplest situations met with every day.

In many schools today, recognition of differences in the rates of children's development has resulted in the provision of indi- vidualised programmes, in the form of cards or workbooks, to cater for children working at different levels and speeds within the same class. The individual attention which such provision demands limits the time available for discourse between the teacher and each pupil, who therefore spends a great deal of time in practice which, despite attempts to relate content to real situations, is unreal in that the mathematical problems of every- day life are rarely presented in written form on a printed card, or solved with pen and paper. Carefully structured schemes may help children to understand and become skilled in the formal discipline of written mathematics, but they offer little oppor- tunity for the development of colloquial mathematics. Pro- gressive stages ultimately lead to the page of practice examples of the appropriate computation. The child is led gently towards the successful mastery of the conventional manipulation, which he is expected to apply, even to those examples which, by the use of colloquial mathematics, he could readily solve in his head.

For the skills of colloquial mathematics to be developed, there must be opportunities for oral work. I am not advocating more of the mental arithmetic which is a familiar feature of the mathematics lesson when children are class taught, for this allows for no individual stages or rates of development. Nor is it normally concerned with thought processes. In my experi- ence, it serves to quicken minds and generate alertness at the beginning of a lesson, or to test the class's assimilation of work, and is concerned only with correct answers. Admittedly, some children in this situation will develop, as a means of survival, personal techniques of computation skill, but unless the processes are examined, analysed and discussed, their acquisition is fortuitous, and limited to the few.

The problem is therefore that of providing both for increased opportunities for the oral situation, which is in reality a simu- lation of everyday life, and the individualised programme.

How opportunities to develop and practice colloquial mathe- matics on an individual basis in the classroom situation can be given I have no space here to suggest, but I believe that atten- tion to this aspect of mathematical skill could result in more children being able to deal competently and confidently with the mathematical problems met in everyday life.

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