Demon King RepliesAuthor(s): Geoffrey MatthewsSource: Mathematics in School, Vol. 24, No. 5 (Nov., 1995), pp. 14-15Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215215 .

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Replies Replies Replies

by Professor Geoffrey Matthews

Judging from Ian Thompson's article "Pre-Number Activities" and the Early Years Number Curriculum, (Thompson, 1995), I appear to be responsible for any calamities that have befallen mathematics education over the past 30 years.

I started the Nuffield Mathematics Teaching Project (age-range 5 to 13) with misgivings as I had "little or no experience of working with primary age children". I had my arm twisted by HMI who told me they had spent a morning discussing who should be recommended for the job and my name was the only one which had emerged. At least I managed to get the title changed, as it was proposed by the Trustees to be age-range 8-13.

My team included Barbara Mogford, a college lecturer with infant and nursery experience; Benita Jackson, an infant teacher; Jim Boucher, a junior class teacher, George Corston, a primary head and Harold Fletcher, an LEA inspector. I also had the great good fortune to meet the head of an outstanding nursery and infant school, who became Dr. Julia Matthews.

My policy not to produce pupil material was, according to Thompson, "probably a wrong decision". I stand by it. For one thing, the state of primary mathematics in 1964 was appalling, consisting almost entirely of "sums" drill and unrealistic "problems". We couldn't expect to get things perfect. The strategy was to produce simply guides for teachers, initially in 90 pilot areas, who would develop

their own curriculum. To be chosen as pilots, LEAs had to agree to set up Teachers' Centres. The shelf-life of the project was aimed to be 10 years, by which time a new generation of researchers and developers would have taken over. This secret army has unfortunately consisted only of a dozen stalwarts in the mould of Ian Thomp- son. Nothing at primary level has been planned on the scale of the Concepts in Secondary Mathematics and Science programme (CSMS) which has become required reading. Instead, the wet blanket of the National Curriculum has descended to discourage individual initiative.

The NC has at least given a frame-work and is to be broadly welcomed. It has cut down the number of ill- considered and random lessons which in the past inspectors have had to endure. All the same, it remains imperative to mobilise an army of researchers to start now and be ready for the next revision in 5 years' time.

Inevitably a posse of publishers jumped on the band- wagon, and if the Nuffield project had produced pupil materials, it would have just become one of many, instead of a catalyst for reform world-wide. (This is borne out by the successor project, "Nuffield 5 to 11", which catered less controversially for teachers not having the benefit of a nearby Centre).

Thompson is on safer if less relevant ground when he takes Nuffield to task for suggesting that subtraction should

14 Mathematics in School, November 1995

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wait until introduction of the integers. We knew this was pie-in-the-sky, and indeed made suggestions if teachers couldn't wait. Nevertheless, the premature use of symbols, notably the minus sign, has been the cause of untold confusion. I still reckon it is bad to introduce + and - more or less at once, a practice encouraged by the SATs where inevitably additions and subtractions proliferate on the same page. It is a misconception that primary children understand inverse operations and so can never be allowed to write 5+3=8 without an auto-pilot 8-3=5, etc. I would refer to Julia Matthews' doctoral thesis, An investi- gation into numerical errors of six and seven year old children [Matthews, 1981].

It seemed worth giving pilot (or otherwise dedicated) teachers a glimpse of the complexities endemic in subtrac- tion, but of course there is no trace of such a utopia in the follow-up "5 to 11". The same goes for fractions and the rationals.

The main thrust of Thompson's paper is that children should be encouraged to count instead of fooling about with "pre-number activities". He has a point. The commer- cial schemes produce acres of expendable work-sheets exemplified by the diagrams at the start of his article, with time-filling instructions such as "colour the squares green". Thompson quotes approvingly (on p. 38) five principles which guide children's early counting experiences. These can be related to matching, ordering, sorting by cardinality, comparisons, the abstract nature of number and its invariance (conservation). This is precisely the menu for the so-called pre-number experiences, which certainly should not be done to distraction.

Thompson makes the strange assertion that "one import- ant attribute of a set is that its elements must have some definable property in common".. He then quotes from "some teachers" texts that Addition can be defined as the

union of disjoint sets. This is nonsense. Addition and union are not synonyms. The truth is clumsier, Addition can be defined in terms of the cardinality of disjoint sets. It cannot be defined in terms of intersecting sets, so the example of the total number of letters in the words boys and girls is ludicrous.

The most potentially worrying part of Thompson's paper is his apparent approval of seven and eight year olds who "appear not to abandon counting even when they have learned some addition, subtraction or multiplication bonds". It is all very well for 5 + 3 to start with 1, 2, 3, 4, 5 and then go onto 6, 7, 8 but this gets laborious for say 57 + 3 or 57 + 32. Counting on is the name of the game, but I am sure that has been in Thompson's mind.

In short, I believe we are on the same side. The only substantial research available to us 30 years ago was that of Piaget, who certainly produced much evidence in favour of the pre-number experiences. We tried not to be over influenced by him, and had we been starting now, we would have had more to go on. However, there is still a vast amount of research to be done.

In any case, I am a bit too old to play the Demon King in this year's pantomime. I shall audition instead for Buttons, past his sell-by date, but cheerful and co-operative. The part of Cinderella will be played as usual by the statemented children. But who will take the part of the Fairy Queen, who will wave her magic wand and produce the perfect curriculum for the whole cast of Villagers at Key Stages 1 and 2?

Anyone for an audition? IE

Reference Matthews, J. (1981) An investigation into Subtraction, Educ. Studies in

Maths, 12, 327-338.

NCEI.- ENATIONAL COUNCIL FOR EDUCATIONAL TECHNOLOGY

Geometry Project The main aim of the GEST geometry project is to provide support materials for teachers wishing to use dynamic geometry software in their classrooms. There are currently three major packages available which all run on the PC or the Apple Mac:

Cabri-geometre, Geometer's Sketchpad and Geometry Inventor. However, things are changing very rapidly and it is expected that Archimedes software will also be available soon.

The two main features of a dynamic geometry package are the construction of points and lines and the measurement of lengths, angles and areas. A constructed point or line always remains dependent on those other points or lines which were used to construct it. For example, if one of the two points which were used to construct their perpendicular bisector is moved (using the mouse), then the perpendicular bisector will also move so that it remains in the correct location; and if the angle between this perpendicular bisector and the line joining the two points is measured, it will be observed to be a constant 90'.

The project started last term with a meeting for the eight teachers working in the four project schools in Huddersfield, Leeds and London. The time was taken up exploring the software, making decisions about which children or which classes would be involved, and exploring appropriate activities to trial. The classroom work and the writing will happen during this term and further meetings will be arranged to discuss the teachers' feedback. The final product will probably be a fairly short and exploratory booklet containing a number of classroom activities, with comments and lesson accounts. There may also be some writing about the curricular implications of dynamic geometry software. The booklet will be printed and distributed before March 1996.

If you are working with dynamic geometry software in your school, or you think that you may move in this direction, please do send the name and address of your school to Ronnie Goldstein at NCET. There may be too many replies for personal contact to be made but, if you enclose an addressed A4 envelope, you will at least be sent copies of the activities being tried.

Milburn Hill Road Science Park Coventry CV4 7JJ

Telephone: 01203-416994 Facsimile: 01203-411418

Dialcom: 84:MTT005

Mathematics in School, November 1995 15

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