# How and what shall we teach?

Posted by: Gary Ernest Davis on: January 1, 2010

Today I am posting something written around 1995 by my friend and colleague Nigel Smith, who taught mathematics at Hordle-Walhampton School in Lymington, and at Twyford School in Winchester, England.

Nigel died of brain cancer, aged 44, about 7Â  years after he wrote this piece.

I hope to post very soon an account some of Nigel’s work at Twyford School, highlighting his experimental, thoughtful and creative approach to teaching mathematics, and his clever use of technology.

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by Nigel Smith, Hordle Walhamption School, Great Britain
Most mathematics teaching, I would contend, is fundamentally dishonest. One has only to consider the nature of mathematics to realize that much of what is learned in classrooms is but a pale reflection of mathematical activity. The main problem is that many teachers regard mathematical knowledge as an immutable collection of facts that should be passed on from generation to generation.

This point of view has been under attack ever since Godel first demonstrated that a mathematical statement can be true but not provable. Unfortunately it has only been relatively recently acknowledged that this view has important consequences for the way that mathematics should be taught. Mathematics is a very human activity and so is bound by its historical, social and cultural contexts.

Historically, mathematics has developed and advanced through the medium of problem solving, primarily within a socio-cultural context. Sometimes this problem solving has been collaborative (e.g., Hardy and Ramanujan) and at other times the result of debate and argument (e.g., Frege and Russell). It is through mathematical activity such as problem solving that mathematical concepts begin to acquire their meaning.

As Wittgenstein and others have observed, the meaning of a mathematical concept is defined by its use. From a constructivist standpoint, individuals build meaning for a concept from their interpretation of its use in this context. This “negotiated meaning” would, I suggest, provide an excellent focus for school mathematics. From an early age, children should be made explicitly aware of the nature of mathematics. It is not wrong to tell them that mathematics is fallible, that its meaning is negotiated and that it is bound within its historical context.

By creating an open, questioning approach to learning, mathematics becomes a much more rewarding and uniquely human activity. Recently, many mathematics educators have argued forcibly that a problem-oriented curriculum would help promote understanding of fundamental ideas in mathematics. However, one has to ask whether a problem-oriented approach to learning will by itself provide sufficient motivating force for change to be considered viable. Such a curriculum would have to be “uninhibitedly speculative,” giving children a key role in the solution process. But how is this to be delivered?

### Three Examples

From a simple starting point the process of problem solving can, through multistage development, build up the meaning and applications of many fundamental concepts. For example, the concept of a moving point initially expressed in coordinate form can be built upon in stages to ultimately encompass a large network of interdependent concepts including equations of lines in a plane, parabolas, circles, ellipses etc. It may also serve as the geometric basis for the development of variable and function.

Or considers an example from a Japanese lesson: developing the concept of linear equations with two variables. The initial starting point here is the rolling of two 12-faced dice. The question posed is “In what cases will the sum of two times die A plus B equal 15?” From this initial starting point, can be built the concept of ordered pair solutions of an equation with two variables.

Finally, an example from my own classroom. I begin with the concepts of “odd” and “even” number, introduced both numerically and using squared paper to give a visual interpretation of both concepts. I move on in stages to consider the consequences of simple operations (e.g., odd + odd = even; even + odd = odd), initially in purely numerical terms. This is then extended to multiplication by considering pairs of factors from 1 to 40. From this the class develops evidence of why it is that even numbers have potentially more factor pairs than odd numbers.

These examples demonstrate the potential of multistage problem solving and how it can be used within a socio-cultural context. Indeed, one of the most exciting possibilities is the idea of nested concepts. The odd-and- even number example, for example, leads naturally to many related concepts including implications of the ratio of factor pairs of even numbers to odd numbers being 2:1. This sets the stage for a discussion of infinite sets and the idea of parallel infinities being analogous to parallel universes.

### Solving Problems as a Means to Understanding

Many critics believe that mathematics teachers should return to teaching well-grounded basic concepts using an algorithm-dominated approach. Problem solving has, critics believe, significantly impoverished children’s acquisition of basic number skills. Unfortunately, many seem unable to make the distinction between problem solving per se and problem solving as a technique for establishing a network of interdependent concepts.

This is a crucial distinction if we are to adopt a radical reform schedule. Typical goal-oriented problems can actually hinder learning because they place a high cognitive load on the problem solver. Although this is an entirely separate type of problem-solving activity from that described above, many people believe that this is the only type of problem solving– something to occupy children on a rainy Friday afternoon. They fail to see that problem solving can be a powerful method of introducing and reinforcing essential mathematical ideas.

Some influential industrial groups seem to want in a workforce that can follow mathematical rules without necessarily understanding how they are derived. Consequently they are primarily concerned with the lack of familiarity that many school leavers have with the rules of arithmetic and their applications in the workplace. If only they could see how a problem-solving approach, properly implemented, could actually strengthen the understanding and application of these rules, they might agree with the radical reforms that I am suggesting. From my own experience, I am convinced that algorithms are more readily accepted in a problem-oriented curriculum than in a rule-dominated one because they are seen as a means to an end, not merely as the end in itself.

In my opinion, mathematics teaching is much more than just teaching mathematics. At the school level, it is one of the few subjects that can instill a sense of the worth of mental discipline, the ability to think for oneself, and the validity of always asking “why?” If taught in a way that challenges a child to respond, then surely this must give that child the chance to reflect, not only on what mathematics means, but also on how the process of thinking can help the child negotiate the perilous course that we label adulthood.

### 2 Responses to "How and what shall we teach?" This is argument is very similar to the essence of “A Mathematician’s Lament.” by Paul Lockhart.

I’d like to make this happen when I begin teaching in 2011.

AAC  Best of luck, Alexander, and stay in touch. 