The Nuffield Green ProblemsAuthor(s): Geoffrey MatthewsSource: Mathematics in School, Vol. 2, No. 3 (May, 1973), p. 19Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30210988 .

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The Nuffield Green Problems by Geoffrey Matthews, Centre for Science Education, Chelsea College

Lower secondary teachers have a problem. There is a rapidly growing realization that children are more different than had previously been supposed. It is in fact becoming increasingly more difficult to receive a class and plunge in the first day with the whole lot of them at Chapter 1, whether it be Revision

of, Fractions or An Introduction to Sets. Even when the children are ruthlessly setted, the teacher's conscience will tell him that some children will get there at once, some slowly, some perhaps not during the current year; but the problem is aggravated by the rapidly growing tendency to "mixed ability grouping" in the first two secondary years. It is difficult to deny the social justice of this: children develop in different ways particularly between the ages of 9 and 13, and assignment to "1H" gives a high risk of going on to "2H", "3H" ... regardless. But however socially desirable it may be, the "mixed ability" set-up brings fresh problems to the beleaguered class teacher. And to crown all, the primary schools are such anarchists that there is no guarantee that all the children will have "done" or "not done" a particular topic.

It was with this problem in mind that the Nuffield Mathe- matics Project turned to production of secondary materials, its remit being to produce a "contemporary scheme for children aged 5 to 13". The idea of yet another complete "course" was ludicrous-secondary private armies abound already-but what seemed necessary was a link between the primary assignments and the secondary course-books. Part of this link has taken shape as a series of "modules of work" with cards for the child- ren and notes for the teacher, each designed to take a group say a couple of weeks, with titles some old, some new (Decimals, Number patterns, Topology, Integers, etc.). These can supple- ment text-books and provide the flexibility whose necessity has been indicated above.

But of equal importance has been the production of series of more or less unrelated problems, the so-called Green Set (and later, by demand, the Red and Purple Sets). These problems are set out on individual cards, and in addition a teacher's book contains them all together with copious notes on solutions and possible extensions. The aim behind the cards was two-fold: (i) To provide (rather secretly) a revision course for topics which might be hoped to have been "done" earlier (ii) to provide opportunities for the children not just to "do" a problem but to extend and generalize it. We will deal with these twin objectives in turn.

(i) The "revision" aspect The range of concepts which can be acquired by most primary children has been described elsewhere (e.g. by the "map" dis- played in the Nuffield Checking-Up guides). To take just one example, lower secondary-children will all have acquired some idea of intersection of sets, but some will have done this through specific experiences and others will have got the.idea simply through intuition. The card-writer had to make up a problem which would not be too "obvious" to the first category ("Oh! it's that old problem again") nor too difficult for the second. A resulting card was:

Mrs. Brown breeds dogs. At present she has eleven of which seven are spaniels and eight are puppies. How many spaniel puppies has she?

Make up a similar problem.

As with many real-life problems, there is more than one possible answer: the "intersection set" may contain 4, 5, 6 or 7 spaniel puppies.

This one example has illustrated the "revision" principle and the reader can trace for himself the topics behind other problems in the set.

(ii) The "generalization" aspect The cards are based on a principle which I have called "closed and open". "Closed" problems are theteacher's problems, which have a definite answer and can be dull; "open" ones often are too dreamy or vague for much mathematics to emerge ("investigate the traffic", etc.). The "closed and open" problem has the best of both worlds: something definite to do ensures progression but when the task is accomplished, it should be a tradition that the child doesn't just come up with the answer but tries to make up a better problem himself or generalize the given one. This is really mathematical behaviour. When a mathematician is told that the angle-sum of a triangle is 2 right angles, he immediately asks "Well, what about a quadrilateral, a pentagon, an n-gon? What sort of analogue might there be in 3 dimensions?" etc., etc.

To take another example from the Green Problems;

The lines on the right are part of a very large grid. Starting from A a man may move along the lines in four directions, north N, east E, south S, west W. A

journey of one square east, two squares south and one square west can be recorded as ESSW. Find five different journeys which will bring him back tow his starting point. s

This problem, taken straight, is pretty dull, but it has the advantage that anyway most children could "have a go" at it, and the central point is that encouragement should be given not to the child who goes through the pack, superficially getting an answer, but to the one who will spend as much purposeful time as possible on each problem. For example, with this one, children can discover the identity element by considering NS, EW, NNSS, etc.; one child invented algebra by writing S7 "because I got tired of writing SSSSSSS"; another made matters worse by inserting diagonal lines; another by extending to 3 dimensions; and so on, and so on.

Many of the problems can be used over a wide range of mathematical ability. One, more purely "open" than most, reminds the reader of triangular numbers and then simply asks him to "define trapezial numbers". Even the slowest of 11-year- olds have had fun with this (having discovered what a trapezium was) and at the other end of the scale a Ph.D. from Korea, a formidable woman, left a class for 4 days and returned with a thesis on trapezial numbers, containing the most unlikely and splendid generalizations.

What about having a go now-how would you define trapezial numbers? But it is far more rewarding to add to your problem bank yourself. The final Green card (number 53 of a set of 52) simply says "Make up your own problems", and if the children get that far they have taken the first step towards becoming creative mathematicians.

The Green Problems are of course obtainable, along with the other Nuffield materials, from the publishers at 11 Thistle Street, Edinburgh.

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