From Streamed Teaching to Mixed Ability Teaching in Secondary School MathematicsAuthor(s): Helen WillmoreSource: Mathematics in School, Vol. 3, No. 4 (Jul., 1974), pp. 4-6Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211225 .

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Figure 4

pattern, but it is not to be compared with the interplay of shapes in the heptomino solution. The relationship of the sides is not very artistic and the number of pieces occurring in the rectangle is too small. (See figure 4.) Anyone who does not wish to embark on the heptomino problem, may try out his powers of spatial combination in this last problem.

I do not know whether octominoes can fill a rectangle. Most likely it is possible to prove that they cannot be so combined since they consist of an even number of squares. The probability that this is so is exactly one in two. One rainy day during my holidays I shall probably discover the proof, if no-one gets there before me.

There are other spatial problems of a similar kind. Enclosed

spaces can be made into cuboids. The question then arises whether an assortment of such enclosed spaces can fill given cuboids or blocks: the problem is one well known to every bricklayer. However, this is leading us too far afield. And, after all, posters come in two dimensions only. Printing would present insoluble problems.

Footnote 1. This leaflet goes with poster, "Symmetrical Asymmetry", a mathematical problem which developed into a piece of modern art. Printed in colour, size 61x90 cm. Using the same shapes and colours, four different effects have been produced by rotating the colours used for each of the four groups of 21 heptominoes. (The centre pages of this issue show one of these posters-Ed.) The four different versions are designated A, B, C and D. They may be purchased separately for 65 fr. each, post and packing included, or for 220 fr. for a set of all four posters, from the publishers:

v.z.w. Hoogland/TEK, Korte Schipstraat 16, B-2800, Mechelen, Belgium.

Footnote 2. The Belgian Minister of Culture has classified the posters as a work of flemish abstract art. They can be seen at the "Masereel Centrum" for graphic art.

Footnote 3. Further polyomino problems and solutions may be found in Chapter 13 of the first Scientific American book of Mathematical Puzzles and Diversions by Martin Gardner, published by Simon & Schuster (New York) in 1959, and available as a Pelican Book in Great Britain.

From Streamed Teaching to

Mixed Ability Teaching in

Secondary School Mathematics by Helen Willmore, Fulford School, York

My first five years of teaching mathematics had been confined to children in streamed situations in a Grammar Technical School. I then decided to explore the realms of mixed-ability teaching. It occurs to me that in the near future, several teachers--whether they choose to or not-may be forced to undergo a similar transition. It is with these teachers in mind that I intend to explain my decision to undertake this venture, to describe the set up in which I have been working and to relate some of my experiences and mistakes, hoping that this will give them some insight into what to expect.

On the whole, I had enjoyed my teaching at the Grammar Technical School. The streamed organisation had lent itself to a great deal of class teaching and the actress streak in me was finding a very satisfactory out- let in the "chalk and talk"! I was fully aware of the fact that these so-called homogenous groups contained a fair range of ability. I used to pitch the level of my instruction in the hopes that the majority of children would understand, attempting at the same time to suggest more complicated follow-up work for the brighter pupils, whilst ensuring that the weaker pupils were at least being entertained--even if their mathe- matical intake was rather low! Occasionally, I would introduce group work, making use of professionally manufactured work cards but this tended to be the exception rather than the rule. As time passed by, it became increasingly evident to me that more teaching than learning was going on and however much I tried to salve my conscience with catering for the extremes,

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I was-deep down inside me-dissatisfied by the lack of catering for the individual child. However, I had to admit that the existence was a "comfortable" one and to make the effort to move to new pastures and explore mixed-ability teaching was not an easy decision. My imagination did not have to work overtime to envisage a class of children all engaged in different topics and all seeking information about what to do next at the same time. Having a mind which normally can only cope with one thing at a time and being extremely partial to order in my classroom, I dreaded the chaotic consequences. However-nothing ventured nothing gained-and in September 1972 I was lucky enough to join the mathematics department at lulford School, York, which was well known to be mathematically exciting.

Fulford comprehensive school divides each of its first three years into mixed ability classes in which most of the pupils stay for all their lessons. A total of

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about sixteen children are withdrawn from the first two years and spend all their time in a special remedial group. In spite of the latter, my classes contained a very much wider range of ability than I had previously experienced. The mathematics department selects six or seven different topics to be studied by a year-group each term. Carefully graded work cards on each topic are prepared by the department so that the same topic can be studied at a variety of different levels of ability. The work cards are teaching materials in themselves but they also provide references for further work in many different text books and professionally produced work cards. At the end of each work card-or possibly two or three cards-the child consults the teacher as to which is the next piece of appropriate work. Each child explores a topic as far as his ability allows. In the second and third years, all the children attempt the topics in the same order-an order so planned to give them an insight into the building up of the subject as a whole. Even in this system, it is not long before several different topics are being studied at the same time. In the first year the topics are chosen so that it does not matter in which order they are attempted. The children can choose the order for themselves but again they are encouraged to relate the work they do in the different topics so that they do not think of mathematics as consisting of several separate water-tight compartments. The cards offer a variety of ideas for individual work, group work, work with apparatus and book work. In order to test retention as well as understanding, a written test is set, not immediately after the completion of a topic but after some different work has been attempted. Class lessons are given at appropriate times and serve as good introductions to a new topic, useful reminders of computational skills and occasionally to introduce a new idea not covered in the syllabus. The department is fortunate to have four specialist rooms. Each room has approximately ten of each of the relevant text books-complete sets being unnecessary owing to only a few children requiring them at the same time.

The First Few Critical Days This, then, was the set up into which I landed in September 1972. I admitted to the children that although I was not new to teaching, I was unfamiliar with the mixed ability organisation. This proved to be a good move as the children themselves were used to being in mixed ability classes and seemed quite pre- pared to give me their support over the first few critical days. I shall never forget on my first day challenging a boy who had left his chair and was walk- ing around the room. In reply to my indignant tone, he very politely explained that his work card required him to fetch some scissors and he was looking for them! He smiled so understandingly at my hurried apology and the experience taught me that if children were going to wander around for equipment, work cards, books, etc., then the sooner I organised my room the better. I had a long bench running the whole length of the room with cupboard space and drawers underneath. I counted the text books, labelled them and assigned them a position where the children would know where to find them and where to return them after use. I carefully labelled all the drawers and appointed monitors to check the contents of the equipment drawer at the beginning and end of each lesson. I divided up the work cards into partitioned boxes and arranged the boxes on top of the bench so that the children could help them- selves to their appropriate card on arrival and return the cards to the correct places on departing. It came home to me how lucky I was to have all my lessons in

one room. From now onwards when I have a new form, I shall profitably spend the first few lessons explaining where everything can be found. I suppose this seems rather obvious but I was only used to children who produced their own books and equipment out of satchels and who needed nothing else.

Another mistake committed on the first day-and for several days after-was worrying about the noise. It was in fact-on the whole!-a healthy hum of organised work but the contrast to children who always sat quietly in their desks was so incredible that to me it sounded deafening. My initial reaction was to ask for silence-within seconds I realised how ridiculous this was when the aim of several of the work cards was to promote lively discussions! I hastily apologised for my unreasonable demand and settled for "as much quiet as possible"! There were times when I could have quite cheerfully banned the ringing hand-calculators from the room! However, after some time, I began to actually enjoy the "working noise" and now if there is ever a moment of silence, we all stop what we are doing to find out what is wrong.

My fears of masses of people all wanting different help at the same time were soon realised and I had queues of children on both sides of my desk. To do them justice, they were waiting their turn in a quiet and orderly fashion-largely I suspect because the sport's field is just outside my room! From a mathe- matical point of view, however, queueing time was time wasted and I had to find a solution. One of my colleagues suggested that I might make a set of answer cards so that children who were just requiring routine marking could do it themselves and then I would re-mark the work later. I wrote out some answer cards and placed them in carefully labelled drawers. I had found it essential in the first few weeks to collect the childrens' books in at the end of each lesson. This involved a great deal of hard work in the evenings but it was the only way that I could try to assess the ability of each individual and to get a global picture of how the children were responding to the work cards and to me. It soon became obvious from the books, that most of the children were capable of adequate routine marking. Children still consulted me after they had done their own marking so that I could advise them about the next piece of work but the saving in time was considerable and the size of queues quickly diminished.

My mind was so conditioned to every one doing something different, that it was some time before it occurred to me that perhaps two children in the same queue might be wanting to query the same or a similar problem. It is amazing how often this situation does happen and I now make a point of asking the children why they are waiting so that I don't waste time repeating the same information. Now that the size of my queues had become more manageable-preferably two but never more than four--I was able to integrate them all into the discussion in hand, whether they were at that particular stage or not and there was no more gazing out of the window! Another method I employed was to send a child to another child for help. This experiment has proved worth while if only because in the course of trying to help another, the child may realise that his own understanding is inadequate.

More Time for Each Child Having solved the queueing problem, I was actually able to start teaching. It had worried me very much until then that most of my time had been spent keeping order in the class room and marking books. Now that I no longer felt a sense of panic each time a child came

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out to my desk, I was able to spend more time going over the work with the child. I also had time to wander round the classroom or to call out children who needed extra attention. This in itself created new problems-it was impossible to see every child in each lesson and I found selection difficult. At first I must admit that I erred o:1 the side of spending time with the less able as I fully realised that if neglected, there could have been some disciplinary problems here. I had also discovered, by looking at their books, one or two children who had virtually done no work in class but because they had kept very quiet, had remained un- noticed by me. These were also candidates for atten- tion. However, as I became more organised and more confident, I somehow found time for attempting to stretch the imaginations of the brighter children. I feel very strongly that the latter should not be neg- lected in a mixed ability situation.

One of my most difficult problems was to decide which card a child should tackle next. Although I didn't appreciate this at first, this task was probably the most important aspect of my teaching. I had to select which set of learning experiences were suitable for that particular child at that moment. If I chose correctly then the work card should take over the necessary instruction and the child should learn and develop an understanding through carrying out the assigned tasks. Obviously my choice improved in accuracy as time went by and I became more familiar with each child and the material available. It was some time, however, before I realised what effect my sugges- tions were having on the children. They were returning to their tables and comparing notes and several were feeling "slighted" that I had suggested work which was at an easier level than perhaps their friends were tackling. To overcome this, I now use a different technique. When a child comes to me for advice, I outline several different tasks, explaining their degrees of difficulty and then leave the choice to the child. This works very well and on the whole the children choose sensibly. In fact, their choice nearly always co-incides with mine.

Frequently the children are encouraged to work in groups which they form with their friends. Interaction of various opinions and differing ideas lead to healthy discussions and often stimulating results. If I feel that their discoveries are worth sharing with the rest of the class, I occasionally allow a group to take a class lesson. This has proved to be a great success and I have been impressed by the way the groups have demanded and received complete class participation in a very sensible and organised way. There is always the danger of a work card dominated system seeming rather monoto- nous to children, and the greater the variety of experiences offered to them the better.

Up to now, I have given each child a grade for each piece of written work completed. I would be the first to admit that these grades are rather meaningless-- particularly if the work has been done by a group which is bound to carry its passengers. However, the children delight in this reinforcement and it serves just as well as a tick in my mark book to provide a record of work each child has attempted. The marks from the tests are the only useful indication of attainment. At first I found these tests rather difficult to organise as I was not used to children doing tests at the same time as others carrying on with their normal routine. I found that I just hadn't got enough eyes both to ensure that those doing tests were working under test conditions and to supervise the rest of the class. I restricted the number of those doing tests in each lesson to four and suggested that they might sit together at a quiet table 6

away from the rest of the other activities. If they were attempting the same tests, I would split them up onto different tables or perhaps use the side bench but I kept them all in the same direction from me so that I could see them all at a glance.

Another item which I found difficult to organise-- and still do-is to set homework twice a week, when the children have been engaged in very different tasks in class and vary so much in ability. One solution is to suggest to the children that they continue with what- ever they have been doing in class. However, text books and equipment are confined to the class room and if children are using cards which need these, alternative work has to be set. I often make use of computation and grade the examples in such a way that everybody can do something. What I prefer to do is to select a topic which is new to all and to set work in the form of an investigation which can be tackled at different levels according to ability. However, it is not easy to find an ample supply of such suitable topics.

Hard Work It is certainly true that mixed ability teaching involves a great deal of hard work for the teacher in the even- ings. I still find it necessary to take in books for mark- ing twice a week. Preparation of the work cards is also very time consuming as a great deal of thought has to be given to the make up of these important teaching materials. Actually, I rather enjoy marking and planning new work. Unless one is prepared to do a fair amount of "follow up" work, the whole scheme of mixed ability teaching falls down.

Finally, in case it is not evident from what I have written so far, I must say how much I have enjoyed my experiences with mixed ability teaching. I would also like to thank both the mathematics staff and the children at Fulford School for helping me over all initial hurdles. Obviously I have still a great deal to learn but this will always be the case throughout a teaching career. I must say that never before have I experienced so much success in a classroom. Because all the children are attempting work within their own range of ability, they are all regularly achieving success in some form or other and with a lot of happily work- ing children what more could a teacher want?

New Mathematics Building for Harrow School

Work is about to start on a new Mathematics School which will consist of a new floor above and beyond the Physics School, completed in 1970 and shown on the right of the artist's impression. The accommodation includes 10 classrooms, with ancilliary accommodation, and was designed by Group Architects/Engineers DRG of Bristol. The expected cost is s298,000.

)Photograph Henk Snoek Photography & Associates

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