Some Thoughts on Numeracy

Some Thoughts on NumeracyAuthor(s): Janet DuffinSource: Mathematics in School, Vol. 7, No. 5 (Nov., 1978), pp. 26-28Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213424 .

Accessed: 22/04/2014 11:06

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 130.239.116.185 on Tue, 22 Apr 2014 11:06:12 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mathas

http://www.jstor.org/stable/30213424?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Somle ghts on

numeracy

by Janet Duffin Department of Educational Studies, University of Hull

I was in a primary school some weeks ago where they told me the problem they have with the children when they try to introduce the top class (8-9-year-olds) to the normal vertical form of recording division. Oh yes, the children understand all about division, they share from the time they come to us and they can distinguish between the sharing activity (doling out one by one) and how many threes there are in 12 when they take threes from the total and they can record these processes as 12 + 3= 4. It is when they come to 3F63 3 into 6 goes 2, 3 into 3 goes 1 3F72 3 into 7 goes 2 and 1 over, 3 into 12 goes 4 3 T18 3 into 1 won't go, 3 into 18 goes 6 that the problems begin to arise and with three different situa- tions in three simple sums this is perhaps not surprising.

The problem intrigued me. I recalled the confusions we set before children in reading and early arithmetic: reading from left to right, number line from left to right, numbers grow from right to left in a place value system, three operations are worked from right to left, the fourth from left to right'. Look- ing at the vertical recording of the operations one cannot help noticing the way that division stands out like a sore thumb because it is so different:

54+ 54- 27x 27 27 12 12154

Is this method of recording really essential? One remembers that it replaced 3163 some years ago to bring it more into line with the process for long division. Clearly there have been problems but has this really solved them? Don't we need to examine the recording of all the operations to see whether we have really found the most efficient for all of them? Could we not find one for division which would not be quite so different from the other three? Is the only reason we record division this way that we want it for long division late on? Is that a good enough reason? Indeed might we not begin to question its usefulness even for the division of larger numbers when we can speculate the possibility of pocket calculators for all in the not too distant future? Such a suggestion is not to say that we can get rid of division, for an understanding of mathematical operations is perhaps even more necessary with calculators in order to check for human failure in feeding in the data. No, what I am feeling after is a different way of recording division, one that might not cause confusion and one that might also be more useful to us in general numeracy because it can be used

26

in later work as well. I recall that Fletcher2 first records division as 12/3 instead of 12+ 3 and I want to build on that. The primary school would have been delighted to have something to help them but they didn't really want to try anything new and untried themselves, they felt that their expertise lay in teaching the techniques not in changing them.

The following day I was in a middle school where their problem was more that their children were finding it difficult to accept Fletcher's treatment of addition and subtraction together. This too is a problem that interests me and I made a mental note to come back to it later, only murmuring something about inverses on this occasion. I wanted to find their reaction to the division problem. On the whole, by the time the children reach them this is not in fact a particular problem but they were interested in investigating my idea about developing Fletcher's introductory notation for division. We recalled though that Fletcher himself in a later book3 does use the standard form for division but rejects the usual form of words,

17 1165 "17 into 1 won't go, 17 into 16 won't go, 17 into 165", in favour of seeing 1' as one hundred and 7 as seven tens and realising that 17 certainly will go into them. Is this merely to cater for the demand for the long division process or is it seen as a necessary step in the development of numeracy? Let Us look at 12/3. It says: How many threes are there in 12 or divide 12 into three equal parts, so it is a natural for intro- ducing the notation for fractions at a later stage when 1/3 is used to represent the idea of breaking 1 into three equal parts. Continued experience of division recorded in this way so that children find from that experience and from the written practice that

36 24 12 36+09 24+06 12- 3

9 6 3

all lead to the same numerical answer, forms a natural basis for equivalence of fractions, perhaps the most important thing we want children to know about them, certainly in the early stages.

Let us look at some

1 2 3 4 5 2 4 6 8 10

What do we observe about these? The top numbers (use the term numerator if you wish) are all half the bottom numbers, or the bottom numbers are all twice the top ones; indeed here



is another way of recording the two-times-table where the top numbers tell us how many two's.

Or consider 3 6 9 12 15 4 8 12 16 20

What can we say about these? The top numbers are all three- quarters of the bottom numbers. Here the other way round is a little more difficult but the ability to verbalise the conclusion links with understanding of ratios, interest, profit and loss at later stages and for the more able it does no harm to lay foundations for these stages.

You may be feeling that this is off the point of division, but not really, for when we come to division by larger numbers, which is all that long division in fact is, it means that we can in many cases reduce the division to a simpler one for

124 62 31 16 8 4

A real understanding and knowledge of tables would enable that to be seen as 73/4 and that 31=(7x 4)+3 without the stylised recording of 4[ 31.

The middle school was enthusiastic. We must work on that and we are going to - we intend to try to make a logical sequence of work cards developing the idea and to try it on the children. It will come in, too, when we get to decimals I said as I was leaving and sure enough it did a week later when I was in a secondary school watching two lessons on basic numeracy for 11- and 13-year-olds. The first was doing fractions (cancelling and all that), the second changing fractions to decimals. The first was using the stylised process,

1 ? 3

? "How many threes in 6, 2 ones are 2",

3 6

without making the sort of observation that enables children to use their knowledge of tables more effectively: 1 ?

3 12 "If the fraction is one-third what must the unknown number be? What number is one-third of 12? Or more basically, what number goes into 12 three times?

2 ? 3 12

If the fraction is two-thirds, what must the top number be? What number is two-thirds of 12? Could we have got it directly from one-third.

These questions do not provide a method or trick which will always produce the required answer, but in discussing them understanding of equivalent fractions can emerge. Many people seem to want to provide the universal trick or method, partly because they think it is more efficient and partly because they think that the ability to learn from such questions and the verbalisation of observed pattern are beyond the reach of the less able. This is certainly a debatable point and provides a clue to the lack of communication between teachers who are on opposite sides in the argument, but my own feeling is that a stylised trick fails if it is forgotten, while the ability to observe, to verbalise and then to practise what you have seen makes for a more efficient skill in the long run besides being more inter- esting. I find it somewhat strange that those who claim that the less able are "better at practical things" tend to be the very people who want to see these children drilled in stylised sums, which are meaningless to such children unless related to real things and real experiences.

And what about the decimals lesson? How are we going to change one-quarter to a decimal was the question asked. 0.25 was the answer. But how did you do it? The boy began to say, "Well 4 into a hundred goes 25", showing he remembered about equivalent fractions but the method demonstrated was 4 1.00, a legitimate use of the usual recording of division you

1 25 might say. But what was the boy saying? Simply that

1__ 25

4 100 and then he used the place value property that to divide by 10 you simply move the figures one place to the right, by 100 two places to the right.

Two important maths courses from Oxford

Mathematics for Life A course in social mathematics for slow learners Norman Moore and Alec Williams Mathematics for Life is an illustrated social mathematics course for slow learners between the ages of 10 and 16. It contains the basic survival mathematics that children must know if they are to cope adequately with the adolescent and adult world.

Series A (ages 10- 12) 0 19 8347154 a1.75 Series B (ages 12-14) 0198347162 a1.75 Series C (ages 14- 16) 0 19 834717 0 a1.75 Teacher's Book 0 19 834718 9 a2.50 Each series of pupils' books contains three 32-page books

Oxford Comprehensive Mathematics A new course for secondary schools Now complete Oxford Comprehensive Mathematics is a complete, structured course in mathematics for secondary schools covering all the modern syllabuses for C.S.E. and O level. Pupils are encouraged by the lively and colourful approach to think of mathematics as enjoyable and worthwhile. Atthe same time, emphasis is placed throughout the course on traditional skills.

Books 1 and 2 are designed for mixed ability classes, while books 3, 4 and 5 provide two alternative parallel courses leading to C.S.E. or O level. Books 1 and 2 each a1.50 Books 3, 4 and 5 C.S.E. and G.C.E. each a1.75 There are teacher's manuals for each pupils' book and workbooks to accompany books 1, 2, and 3 C.S.E. and 3 G.C.E.

a Oxford University Press 1978 Education Department

Please send me inspection copies of: Oxford Comprehensive Mathematics

Book 1 -] Book 2 O-

CSE Book3 _

CSE Book4 -] CSE Book 5 [

GCEBook3 n GCEBook4 [ GCEBook5 [-

Workbook - Teacher's Manual -I

Mathematics for Life A I- B O CO

Name

School Address

Oxford University Press Education Department (EBL 7), Walton Street, Oxford

27



This method works for many other fractions.

3 6 3 75 - = 0.6 --= 0.75 5 10 4 100

(3 1+1

or use

But what about 1/7 or 1/13? Yes, that is more difficult but what we want is to know how many thirteens there are in 100. We don't know our 13 times-table and it isn't in the syllabus. But if we are going to look more flexibly at numbers and tables what is wrong with knowing a few facts about larger numbers:

13 26 39 52 a pack of cards 14 28 42 @ not much more than 50 so a hint for 100 15 30 45 60 a nice pattern 17 34 51 68 just about halfa hundred 18 36 4 72 a pity yards are going out!

So how many thirteens in 100? Eight of them clearly make 104 so seven of them will be 91 (a useful bit of subtraction there). Moreover for the more able

I I 1 1 I - < < so < < 0.1 100 13 10 0.01 13

so 1/13 starts with 0.07 with other digits to follow. Some children would carry on an complete the exercise this way and it forms a basis on which to build the stylised way as an efficient method of recording what is happening. Moreover I suspect that discussion of the approximate values in this way would certainly please the science teachers and that would be a bonus!

To help the children to gain this sort of facility what about looking at exercises like making up 100 as follows (another idea originating with Fletcher):

(4 x 26)- 4(7 x 13)+ 9

100

Think of one for yourself (9 x 11)+ 1

When you begin to play with numbers like this you scarcely need long division and if you did you could learn it as a way you are driven to when all else fails; in this case it is useful and efficient. Of course once you have met it if you really like it you can always use it and it will check all the other methods for you. And if you have a calculator you can use that, and your work with approximations and what makes up 100 will enable you to check the calculator too - that is to check that you fed it the right instructions.

And think what a spin-off there would be in other decimal work:

27.6 276 92 184 1.55 = 18.4 1.5 15 5 10

Or 17 35 17x 35 1.7 x 3.5 = 1x --= ? 10 10 100

This kind of understanding of tables helps in modulo arithmetic too, for the fact that 100=7(13)+9 means that in arithmetic(mod 13) 100 would be denoted as 9.

I am reminded of a user of mathematics who complained that children nowadays cannot change things like nine-sixteenths into a decimal. Using equivalence of fractions and observing the pattern it is easy (and intase sceptics do not believe me I have done it with bright 8-year-olds and so-called 11 + failures).

1 5 1 5 =-0.5 2 10

1 I = 0.25 4

1 = 0.125 8 1

= 0.0625 16

28

9 8 1 -8_+ 0.5625

I enjoy visiting schools but their obvious preoccupation with, and anxieties about, what society sees as the basic skills, and I see as stylised recording, do worry me. When we talk about basic skills, do we mean being able to carry out recorded arith- metical processes and, if we do mean this, are we right to mean it? Should we not be concerned more with number sense rather than recording techniques and ought we not to consider revising our methods of recording when these appear not to reflect what people actually do with numbers in real life.

By chance the evening of the day I visited the secondary school I went to an adult literacy centre because the probation officer who runs it said one man wanted to do some mathematics as well as learn to read. The man told me he had problems with subtraction and indeed he had. I gave him a sum written down on paper and this is what he did:

51- 27 36

I asked him what he had done. He said; "1 from 7 you can't, 2 from 5 is 3". After a little more ineffective groping on his part I suggested we put away the piece of paper. "What is 27 from 51?" I asked him. After a short pause he gave the answer 24. We discussed what he had done in his mind and at first he tried to pretend he had done what he had failed to do on paper but eventually his method of complementary addition emerged and we did several, all of which he managed without difficulty. On subsequent evenings I returned to the stylised method of recording subtraction but built on his own mental method. He became successful at these but not only that, he eventually returned to his equal additions method of recording and incor- porated it into his new found confidence in his ability to subtract - so sad because he could subtract before I came, he just had no confidence in his ability to do sums. It turned out also that he played darts and he therefore knew the beginnings of all the tables up to twenty and was then intrigued to discover that he could use addition to take these further. Of course I don't flatter myself that he will remember all this without the kind of constant practice that he gets at darts but at least he is beginning to get some number sense and actually to get some pleasure out of playing with numbers. I fear however, that it will be a long time before he overcomes his fear of sums.

SI am enjoying helping this man and I think he is enjoying his new ability to relate arithmetic to his life and the mental facility he has built up in it but I must confess to some anxiety about whether I will really be able to equip him to answer the kind of written test that a future employer might set him. I fear that placed once again in the testing situation in which he failed throughout his schooldays, all his new found interest and confidence will fly out of the window and he will again be paralysed into failure. In that case I should have gained from this experience but I ask myself, "would he?".

These things bother me for I fear that we are not always using the abilities that are within people but are instead trying to force on them techniques that we think they should have. I am not even sure that we are even right in thinking that these techniques are the ones people really need. And if they are not are we right to submit to pressures to make people conform when handled differently, they might reveal considerable inner numeracy that they simply have difficulty in recording as we want it done.

The middle school and I are going to try to see if we can develop division without the stylised technique and we'll let you know what happens when we do.

References 1. Duffin, J. Left-Right-Left - A source of confusion, Mathematical Education

for Teaching, Nov. 1977. 2. Fletcher, H. Mathematics for Schools, Teacher's Resource Book, Level I. 3. Fletcher, H. Mathematics for Schools, Teacher's Resource Book, Level II.



Documents

Some Thoughts on Numeracy