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Mathematical Ulterior Motives
- the mother of all ulterior sites -
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- the mother of all ulterior sites - |
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by David Fielker I have been working part-time at a private American international school, which caters mainly for children of business men and women who find themselves in England for two or three years, and whose parents are anxious that they keep up with the American curriculum. About 25% are not American, but most children have previously attended either public schools in the US or other similar international schools. It is a completely different environment from that of the inner-city secondary modern and comprehensive schools I used to teach in, with good facilities, interested students, supportive parents, and a maximum of 16 children in a class! I salve my conscience partly by having to face the problems of the very traditional teaching which was most students’ previous experience, and some parents who prefer that sort of approach; and partly by thinking that I deserve this sort of retirement! Furthermore, my wife is the Head Teacher, so I do not have a lot of choice!! My role is mainly an in-service one, and I work with different classes and their teachers, write notes for the staff, and generally act very much like the ILEA group consultants based at my Centre used to do. One of the characteristics of American mathematical education, at least in this private sector, and in spite of the influence of the ‘Standards’ produced nearly ten years ago by the National Council of Teachers of Mathematics [1], is that the teaching of algorithms tends to follow a rigid pattern. Addition of one-digit numbers is introduced using what some people in old-fashioned days used to call ‘vertical addition’. Then there follows two-digit addition without carrying, two-digit addition with carrying, two-digit subtraction first without and then with ‘regrouping’, and so on until a few years later one reaches long division. Children are generally told what to do, and however well the process might have been supported with structural materials, I find too many children who do not remember what to do, and invariably those who do remember are not able to explain why anything works. Explanation appeared anyway to be a strange
idea in mathematics when the school opened three years ago. The class of 11-year-olds that
I took over were used largely to giving numerical answers. The question "How did you
get that?" was something they had not met before, and even when they possibly knew
how they arrived at an answer, they found it difficult at first to put it into words.
Gradually they and the other classes with whom I worked improved their descriptive skills,
and they and their teachers - became more and more used to discussion as a normal part of
their mathematics lessons, and the ideas of the students began to take precedence over the
instruction of the teacher. Why does it work? One of the may ways in which I can tackle the problem of standard algorithms half-learned is to take a leaf out of my own book [2], and present some unfamiliar algorithms, with an invitation to the students to explain how and why they work. This worked well last year with the Fourth Grade classes, who are 9- and 10-year-olds. I had been spending a lot of time with them and their teachers, and the students were used to contributing their own ideas. The teachers were willing to ‘'revise multiplication’ by presenting a more visual approach to the standard algorithm using an ‘area model’, and they were also ready to tolerate my introduction of some new ideas. The ‘area model’ begins by setting out, say, 123 times 456 on squared paper to scale, and ends up diagrammatically thus:
The 13th century Italian algorithm known as the Gelosia method fitted well into the area model, and the students were able to explain it and use it. (To readers who do not know this an example will be self-explanatory. To multiply 123 by 456 one sets the numbers in the gelosia or grid thus:
The method is the same as for the area model, except that the insertion of the partial products across the diagonals enables the addition of them to ‘slide’ down top right to bottom left, taking care of ‘carrying’ automatically.) We also explored the connections between so-called ‘Russian’ and ‘peasant’ methods. In the area model, the partial products can be written out underneath each other and added, and later the model can be made more sophisticated in order to telescope the partial products, as in the standard algorithm. I discussed with the students how, the number of partial products is normally reduced from, in this case, four:
to two in the standard algorithm:
Why do we stop there? Why can we not manage with just one ‘partial’ product? This is something I can demonstrate. I write the two numbers to be multiplied in the usual way:
and first I concentrate on the ‘ones’ place: 3 times 5 is 15, so I put down the 5 and ‘carry’ the ten:
Now I look at the ‘tens’ place. Tens come from ones times tens or tens times ones: 2 times 5 is 10, plus 4 times 3 is 12, which makes 22, plus the 1 which I carried, making 23: put down the 3 and carry the 2 tens:
Now for the hundreds. These come from tens times tens: 2 times 4 is 8, plus the 2 to carry is 10.
The children get quite excited by this, and ask me to multiply three digits by three, and then larger numbers! Apart from the excitement, what they get out of watching me do this varies a great deal. Mainly, there are obviously some very important manifestations of place value involved. Not everyone understands everything immediately. Why should they? One of the things about differentiation of the syllabus is that we should be able to arrange activities from which every member of the class can get something appropriate. For some children this will just be pure excitement, and that is no small thing for students who in their previous schools have often learned to hate, or be bored by, mathematics. Most will achieve some greater insight into place value. The brighter students will tackle the algorithm for themselves.
There has been some research into the ability of children to create their own algorithms. This semi-retired person no longer has easy access to it all, but I remember from various reports, read and heard, that the CAN Project, an off-shoot of PRIME, organised a large number of primary schools all over the country into a scheme in which no standard algorithms were ever taught, and children were encouraged always to invent their own, with a free use of calculators, structural apparatus, pencil and paper and mental methods. Hearing or reading about such activities is never the same as trying them out yourself. During my years in in-service education I taught one-off lessons with children as a regular part of all courses, and I worked with groups of children in schools. I did a lotof work on calculator algorithms [3], but somehow I had never got around to algorithms for the four rules of number. Occasionally an opportunity arises spontaneously. I was once asking the class how one did a subtraction,
and why you ‘could not take 5 from 3’. Matt said you could, and you got ‘negative 2’. So we wrote that down, and underneath wrote the 30 that was the result of subtracting 40 from 70,
and 30 minus 2 as 28. A new algorithm was born! (No, it is not new to many of us, because other children have produced it many times before, but it was new, to this class.) Usually it is too late to ask our students to create a new algorithm from scratch so to speak. Even if the teachers have not done so first, most students at the appropriate stage have already been taught standard algorithms in their previous schools. However, it transpired that no-one in the Fourth Grades had learned the standard algorithm for long divisions When it ,as due to come up on the syllabus I asked the teachers if I might try to see what the children themselves could do. I came in to each class in turn and presented a problem. The only restriction was that they were not allowed to use calculators. (In one class I was asked why not! I explained that I realised that using a calculator was the most sensible way to tackle the problem, but that I was interested in seeing what they could do without one. One of the things I have learned while working with American children is that I often have to be fairly explicit about my reasons for working in a particular way!) A firm has 10,000 bottles of lemonade to be delivered to supermarkets in crates, each of which has 36 bottles. How many carets should be ordered? In one class someone immediately said, "That's divisions" and someone else said, "We haven't done division yet!" I think that a couple of years previously this situation would have been met with consternation and no-one would have known what to do. Not now. Each class immediately set to work, in groups of their own choosing.
Others were finding their own ways of speeding up the subtraction. Some subtracted 72s, some 108s, some 360s. Nicole and Jessica added 360s, and found that 14 lots of 360s were 5040, and twice this was 10,080, which was close and represented 280 crates.
Nada and Sam drew some nice diagrams of crates with 36 bottles, grouping them in threes to make 108 bottles, and eventually deciding that writing the ‘36’ in the crate as more efficient than drawing all the bottles! They then began to add the 108s together.
Cory multiplied 500 by 36 and got 10,800. This was 800 too many, so he began to subtract 36s from it to see how many crates less than 800 would be needed, but soon realised that subtracting 108s was quicker.
Tim and Joe used their recently learned multiplication algorithms to calculate various multiples of 36, finding among other things that 36 x 36 = 1296. Unfortunately they misread the final digit as a zero, so they calculated that 8 x 1290 was 10,320, and this represented 8 x 36 = 288 crates, so they could then make the necessary adjustment. Hannah, having given up her subtractions, was now multiplying different numbers by 36 to try to get near 10,000. 20 gave 720; 40 gave 1440. Because of the way she set out her work on the paper it wa not easy to tell in which order she chose her numbers of crates, but soon she arrived at 280 giving 10,080, 279 giving 10,044, 277 giving 9972. She wrote "278 crates and 8 bottle spaces left". Brian and Chris worked through a similar sequence which gave them (not always correctly!):
Both they and Erika and Michael recorded the problem as
and the latter used the ‘area’ multiplication algorithm for some successive approximations.
Jason and Jakob began by looking at 2 crates which they converted into 200 crates by multiplying 72 by 100, but then followed the same sort of route as Brian and Chris. Eventually everyone could see that 277
crates would hold 9972 bottles, and 278 would hold 10,008, so we needed 278 crates, but
the last crate contained only 28 bottles. Learning algorithms is bad for your health! I could not resist trying this problem on the Fifth Grade classes, some of whom had already learned how to do long division in other schools. Some of these students could do the division successfully, but not may of them could interpret the answer correctly. Some said the answer was ‘277 remainder 28’, but could not say what the 277 or the 28 represented. (One pair could even check the answer by multiplying 277 by 36 and adding the 28 to get 10,000, but could not say how many crates were needed.) Some expressed the answer as 277 7/9 crates, but again could not say how many crates there should be. There was usually some confusion over whether last crate had 8 or 28 bottles, or 8 or 28 spaces! Some knew they had ‘done’ long division but had forgotten how. At first they tended to spend time trying to remember it, rather than finding an alternative method. Others had not learned the algorithm. They
tended at first to be somewhat deterred by the presence of those who had, but
generally they tackled the problem in the same way as the fourth graders had. Pacifying the parents I prepared some notes on the lessons just described, partly for the teachers, but also for the parents of the Fourth and Fifth Grade students. I told them that one of the things I had found, particularly with this age group, was that students coming from other schools had been taught very formally, learning methods of calculation with ,varying degrees of success. In general:
I pointed out that after concentrated work with these students and their teachers, mathematics had become something different from what they were formerly used to in various ways. Among other things:
The transformation had taken a long to achieve, but it was now paying off in several ways:
This way of working involved setting problems and situations to the students which embraced a whole range of mathematical ideas. It was quite different from the traditional approach, in which one set of techniques is taught, practised and applied, before going on to the next. In these particular lessons, students were
Towards- a standard algorithm I also explained for the benefit particularly of traditionally-minded parents, that with all classes I was able to use what they had done to work towards developing a way of doing division on paper. If we are going to subtract multiples of 36 from 10,000, then it is easier to use those multiples which are easiest to calculate, like 360, 3,600, 36,000, etc. So we can begin with the largest possible, in this case 3,600:
(Note how on the right we record how many 36s we have subtracted each time.) We cannot subtract 3,600 from the remaining 2,800, so now we subtract 360s:
When we get down to 280 we can only subtract 36s:
It is easy to see how this method can be shortened, perhaps by writing out the multiples of 36 first, then instead of subtracting two lots of 3,600 at the beginning, subtracting 7,200 and recording it as 200 lots of 36.
And so on. Gradually it becomes more like the standard method which
most people recognise, not many remember, and few understand! This approach, however,
means that the algorithm is developed from what the students themselves have done. It is
therefore something of which they are part, it is based on their ideas, and it is understandable.
Even if they forget it, they can reconstruct it, or use a similar method. The ‘airy-fairy’ approach to the curriculum When I first began teaching at the school, one of the parents was particularly anxious about my methods. Early on she complained to the Head that I had "an extraordinary way of teaching mathematics". My wife agreed, and told the lady that that as why I had been employed! When I met the mother, she said, critically, "You don't use any textbooks!" I told her that the last time I used textbooks was about 1962, and that I had had better than expected successes in getting the students through their examinations, all the way up to A-level. Sometimes it helps to have grey hair! Later on she complained that what I was doing was "all so airy-fairy". I could see what she meant. To someone who was used to a traditional, textbook-driven approach, in which one clearly identifiable topic followed another, what I was doing was indeed ‘airy-fairy’. It was difficult for an outsider, whose evidence about what was happening was only some home-produced questions for homework and her son's reports of what they did in school today, to see how any progress was being made. So, I ended my notes to the parents on this occasion (alas, too late for this mother, who by then had been obliged by her husband's business transfer to take her son back to Indiana), by trying to explain how this apparent jumble of ideas was part of an overall plan.
David Fielker is a former Editor of Mathematics Teaching, and apart from the part-time teaching described in this article he devotes part of a busy retirement to occasional writing and lecturing. References:
Other work by David Fielker: Online articles at Times Educational Supplement. Go to the TES Archive at http://www.tes.co.uk:8484/tp/9000000/PRN/network/library/libraryarc.html and search for David Fielker. (Or just click the titles below.)
Books avaliable at www.amazon.co.uk.
Articles in journals and occasional publications. Acknowledgement:
Reactions: This space is for your reactions to the thoughts above. Write to Mum.
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