The First YearAuthor(s): Derek BallSource: Mathematics in School, Vol. 9, No. 3 (May, 1980), pp. 18-19Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213554 .

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The first year

by Derek Ball, University of Leicester, formerly at the College of St Mark and St John, Plymouth

"You won't miss Plymouth when you move to Leicester. The view from the top of the high rise flats will remind you of the view from the top of the cliffs at Wembury." We were at a party following the last session of the first year of our Diploma in Mathematical Education course. The 19 course members, the two tutors, the external moderator and a few spouses were eating, drinking and talking and my wife was being teased about our move to Leicester by three men on the course. Earlier in the evening we had been waving our arms about in the air with our eyes shut and catching hold of spare hands - an activity which resulted in people experimentally stepping over other people's folded arms and someone burning off bits of string with a cigarette lighter, all in the cause of proving that two opposite knots in a loop don't cancel each other out - or do they?

We began the course on a Thursday evening last September, we being two tutors - Richard Harvey and I - and 19 "students" from infant, junior, secondary and special schools. We had interviewed the "students" well before the course started and felt they would all have a great deal to contribute. In addition we felt we had a lot to learn from one another, so we were determined to team teach the whole course. In practice this meant that each of us has been responsible for half of the evening each week and the other tutor has been present and has contributed as he saw fit. Before the course started we felt that the diversity of professional background amongst the course members would mean that the group would need to split for some of its discussions, but so far no one seems to have felt this to be necessary.

When we began the course we did not really know what was going to happen - we were very conscious of the fact that this was our first time on a completely new course. We had written the syllabus and submitted the course description to the Mathematical Association and they had validated it. But actually having to teach it was another thing. For the first session we arranged the chairs in groups around tables. This facilitated small group discussion as well as practical work and has been the arrangement which every course member seems to have regarded as natural (and essential) since. The other contribution to small group discussion on this first occasion, as subsequently, was our trip to the bar at half-time. It very quickly became apparent that the mathematical content aspect of the course was going to be valued not so much for the information imparted as for the quality and quantity of mathe- matical activity which took place. The first term's mathematics was about number and we spent some time using some less familiar number apparatus (such as the Papy minicomputer) at "our own level", as they say. In other words we just used it.

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We also became more proficient at mental arithmetic amongst other things. What we frequently discovered as a result of our discussions and heated arguments was that some of the sessions seemed to leave us all with more rather than fewer problems at the end. But no one seemed to mind - except one week when Richard suggested finishing off three different lines of enquiry for "homework" and one teacher complained that the children in his class would have to work in three separate groups for maths over the next few days. When the second term's look at sets and relations confirmed that we were all valuing activity more than content and when we remembered that the topic for the third term was to be geometry of all things, it became apparent that it was not going to be easy to set a traditional examination paper in mathematics at the end of the first year. So instead we contrived a paper which was given to the students four weeks in advance and which they were free to discuss and work on as they wanted in preparation for their ordeal. The paper is appended to this article. What we hoped was thlat the paper would give students the opportunity of demonstrating that they had really learnt some mathematics and we feel in retrospect that it did give this opportunity. The only real trauma associated with the examination was marking it. Our external moderator, who had given us support and encourage- ment in setting this sort of paper was equally helpful in ensuring that it was appropriately marked.

The mathematics education part of the course has perhaps given us more headaches. Our syllabus suggests a rather formal approach, which in practice seems to have been considerably modified, especially as the year has gone on. We find that setting essays which really help teachers come to terms with some of their real problems in their classrooms is not nearly as easy as it sounds (or perhaps it doesn't even sound easy!). We are still battling with this problem. Knowing about Piaget's ideas seems a long way away from knowing how (if at all) Piaget's ideas can help me with my class next week. The valuable bits of the mathematics education work seem to have been the personal bits: one teacher having tried some experiments with sub- traction with his children felt he had been seriously under- estimating their abilities; two teachers from the same school were involved in an ongoing argument about the merits of mixed ability teaching; one teacher either has a dreadful fear of calculators or else feels she can do all a calculator can do quicker anyway (I'm still not really sure which); another teacher viewed the Leapfrog television programmes one week and asserted that they would be useless for his children, only to return the following week and remonstrate with the rest of the group for rejecting the programmes without trying them.

Over the summer the students will be starting their investi- gations. Next term they will be getting down to their special studies of some aspect of children learning mathematics. There will also be more mathematics and more mathematics education. Will we profit by some of the lessons we have learnt? I say we, but I am off to Leicester, so I shall be leaving others to get on with it. I am sad to be leaving the course; I feel I have learnt something significant from every member of it. Most of all I

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have enjoyed being involved with a course where so many have grappled with mathematics on so many occasions and have even apparently begun to relish the thought that mathe- matics is hard - and that's what makes it fun.

It would be interesting to hear from anyone who has grappled with the difficulties we have encountered or who has any views on our methods of dealing with the mathematical content or on the style of examination which we adopted.

Appendix The following paper was given to students four weeks before the day of the examination. They were free to discuss it with us or with whom they chose.

Diploma in Mathematical Education - June 1979 Answer FOUR questions Time: 3 hours

1. Explain clearly how the Papy minicomputer can be used to add, subtract, multiply and divide numbers. You should be prepared to discuss specific examples provided at the time of the examination. 2. Just as pentominoes are made by placing five squares edge-to-edge so hexiamonds are made by placing six equilateral triangles edge-to-edge. Draw the 12 possible hexiamonds.

(a) Suppose one hexiamond is secretly chosen. What is the smallest possible number of yes/no questions which can be asked to be sure of discovering which of the 12 hexiamonds has been chosen? Explain your answer.

(b) Give such a set of suitable questions for discovering the chosen hexiamond.

You should also be prepared to answer (a) and (b) for another set of objects described at the time of the examination. 3. Describe and explain a variety of methods of multiplying whole numbers mentally. Apply these methods to find some of the products of whole numbers less than 30. It will be assumed that you "know your tables"!

You should be prepared to demonstrate your methods in connection with products of numbers given at the examination. 4. (a) Give five different nets of a cube. If a ribbon is tied round a cube half-way between two opposite faces and parallel to them draw the position of the ribbon on each of the nets.

(b) Say of each of the pictures in the accompanying diagram whether it is the net of a solid. If it is, describe the solid (either by naming it or in some other way). You should be prepared to answer (b) in connection with other nets given at the examination.

I (i) (ii) (iii) (iv)

(c) Describe the symmetries of the regular tetrahedron. 5. Describe clearly a number of different methods of subtraction. You should be able to demonstrate your methods on numbers provided at the examination. Offer brief but clear explanations of why each method works (NOT how you would teach it to children). 6. Write about classifying triangles and quadrilaterals where the criteria used are equality of lengths of sides and the number of right-angles.

You will be given drawings of triangles and

quadrilaterals at the examination and asked where they fit into your classification. 7. Write about a cube. 8. Identical dice are used to make trains one die thick as shown. The total number of spots on the top, bottom and four sides is counted for each train. If this total is to be a minimum, investigate the sequence which relates the length of the train to the total number of spots counted. Is this a function? Repeat this for trains which are always two dice thick, as shown.

In the examination you will also be asked to write about a slight variant of this situation.

N.B. You may bring to the examination one sheet of paper containing brief notes for any of questions 1, 3, 5, 7 or 8. The notes should not include diagrams or pictures. This sheet of notes is to be handed in with your script.

Supplementary Information in Connection with Mathematics Paper (available at the time of the examination) 1. Discuss these examples in connection with the Papy minicomputer:

55+27, 68+44, 63-35, 4x 23, 75+6.

2. Draw the five different tetrominoes (made by placing four squares edge-to-edge). Answer (a) and (b) also for your set of tetrominoes. 3. Amongst the products you discuss should be these:

8 x 16, 132, 19 x 21, 252, 17 x 29.

4. Answer (b) also in conection with the following nets:

(vi (vi)

(vii)

5. Demonstrate methods on 73-27, 438-165, 4321-1267. 6. Explain where these fit into your classification:

(I) () (ii)

8. Investigate a similar situation about trains of dice two dice thick, where the total number of spots counted are just those spots visible when the trains are made on a flat table top.

Readers teaching the Diploma may like to comment on this paper. We should like to hear of any other interesting forms of assessment that have been used and of any particularly good investigations that have been carried out by Diploma Students - Editors.

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