Tricks or sticks and stones

Alan Wigley

Is teaching algebra at odds with trial and improvement ? Alan Wigley urges the case for learning through exploration. Some university mathematicians have made ill-judged remarks recently about mathematics teaching in schools. The national curriculum is blamed for a decline in standards before it has been in place long enough to have influenced the university intake!

If the chronology is wrong, can we at least expect the content to be fairly represented? Apparently not. Take this example from an article "For the love of maths" (TES September 8), reporting on the views of Tony Gardiner from Birmingham University. Maths teachers will have been amazed to read that in key stage 3, simultaneous equations are to be solved by trial-and-improvement methods. Only by such methods? In fact, the national curriculum is very explicit in stating that pupils should be taught "a variety of methods", and to "select the most appropriate for any given problem". Given this level of comment, how can we get a rational debate about syllabus content?

But with the curriculum content settled for five years, it is time to focus on how the content is taught. This is in the hands of classroom teachers and is most critical to pupils' learning. Although critics are right to say that there is a problem with mathematics teaching, some are mistaken, both about the newness of the problem and their proposed remedy. There is a better way forward.

More than 30 years ago, university staff were complaining that many students arrived having been taught by rote, learned a few "tricks" sufficient to get them through exams, but with little understanding of mathematical principles and how to apply them. (Actually, concerns about weak mathematics teaching go back at least 100 years.) Today, we are being told that teachers are so pre-occupied with problem-solving that they are neglecting drill exercises. Supposedly, instead of being taught standard methods, pupils are left to explore their own ways.

The critics of today are wrong on two counts. First, they are wrong to presume that teachers have abandoned wholesale the methods of the past. Although investigative work is found in many classrooms, unfortunately it is not usually linked to the teaching of syllabus content, where a majority of teachers still cling to fairly traditional ways. I would go further, and say that it is because teaching methods have not moved on sufficiently, that standards are still not as high as they could be. The critics are also wrong in prescribing, presumably out of frustration and ignorance, a reinforcement of methods which have failed. To teach mathematics just by setting out rules for pupils to follow is like attempting to teach a language purely by giving rules of grammar - taken on its own, that is just not how we learn. The bad consequences of using mainly rote methods are well-documented - pupils struggle to memorise an increasing burden of disconnected rules, rigidly applied. On the other hand, classroom-based research and development in recent decades has increased our understanding of learning and yielded new approaches to teaching mathematics - approaches which are not yet sufficiently widely known.

Traditional teaching might be characterised thus: the teacher explains necessary mathematical concepts and skills, which the pupils then practise and, hopefully, learn to apply. A more enlightened model works as follows. The teacher carefully selects a problem or situation related to a syllabus topic and known to be challenging or otherwise rich in interest. Pupils begin exploring the situation, bringing their ideas to it and sharing with others. After a suitable time (varying from a few minutes to several lessons), the teacher intervenes to discuss with pupils what has been done, draw threads together and give appropriate direct instruction. Pupils return to the situation better equipped with methods to explore in depth, practise and apply skills and knowledge to related problems.

This approach is supported by the "using and applying" section of the national curriculum: pupils themselves must contribute to the processes of deriving and explaining general principles and procedures if they are to learn mathematics effectively. They can have an investment of interest in the situation. Most of all, they will have some experience to which the teacher's explanations can be related - recognising the need for some new mathematics and being able to make sense of it. Understanding and skills are more secure, improving the ability to adapt known methods to new applications.

To take an example, algebraic skills are regarded as an essential foundation for higher mathematics and its application. Familiar from number work in the primary school, Cuisenaire rods are comprised of a set of coloured sticks of 10 different lengthscolours: l cm (white) to 10 cm (orange). If, for example, 2 red (2 cm) rods are placed end-to-end with a white rod they will match with a yellow (5 cm) rod.

This can be represented by the equation: 2r + w = y.

Pupils have no difficulty with the notation, implicit in this physical analogue, of numbers being the representation of lengths by the initial letter of the colours.

Another way of looking at the arrangement makes it clear that the difference between a yellow rod and two reds is the white rod: y2r = w. Other points of view are represented by yw = 2r, yr = r + w, etc. Doubling up the rods in each row leads to 4r + 2w = 2y, and so on.

Having briefly introduced the idea and the notation, the teacher challenges pupils to find as many different variations as they can from one arrangement of the rods. Later, pupils start with their own examples.

Eventually, they begin to write down variations just from the symbolic arrangements on paper, referring to the rods only to check. In this process, they are learning for themselves the rules for transforming equations. After lots of exercises, the teacher invites pupils to make their rules explicit. But none of this needs to come directly from the teacher. One might pause here to reflect on how involved pupils will feel when invited to engage in this kind of exploration of algebra, rather than just being given rules by the teacher at the outset.

For a second illustration, here is an example of a fairly well-known type of number puzzle: I have three piles of stones. The first pile has seven more stones then the second pile and the third pile has three times as many as the first. There are 28 stones altogether. How many are there in each pile?

This problem is very amenable to trial and improvement methods (choose a number, see if it fits, adjust the number . . .). At an appropriate stage the teacher asks "Suppose the number in pile 1 is x, what can we write down about the other piles?" pile 1 pile 2 pile 3 x x-7 3x So: x + (x7) + 3x = 28.

When solving equations in this context, pupils use their insight into the structure of the problem and their knowledge of number operations to group terms - they do not just apply teacher-given rules. Thus, having reached 5x7 + 28, they might be encouraged to reason "something take away 7 leaves 28, so that something must be 35, so 5x = 35, etc". Pupils might also offer alternative formulations of the problem, for example: (y+7) + y + 3(y+7) = 28. Such exploration facilitates consideration of different expressions and cross-checking of work.

In generating and solving "piles of stones" problems, pupils widen the range of types and gain all the practice needed to develop fluency. The skills of manipulating expressions eventually become automated, but with the vital difference that pupils have a purpose for, understand and are in control of what they are doing. The activity fosters problem-solving skills and leaves pupils better able to recall the methods and extend them to new or more difficult examples.

University colleagues who would criticise should take the trouble to find out and publicise what good teachers are doing. The ablest students may survive indifferent teaching, but we must widen access for a far greater number.

Alan Wigley is mathematics adviser for Wakefield LEA.

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