Summer 2004

Talking Maths

Getting Real with Maths Teaching

What chance of success is there in mathematics to introduce ‘authentic pedagogy’—content that is meaningful and connected to the student’s own life? PAUL WHITE looks at how two facets of mathematics curriculum provide some answers, but also some questions.

DOES THE common cry in mathematics classrooms—‘when are we ever going to use this?’—make futile the current moves by educational authorities to adopt innovative elements of Fred Newman’s Authentic Pedagogy in mathematics teaching? After all, ‘Authentic’ means authentic or significant for students. So can school mathematics be ‘authentic’? Let’s take a closer look at the nature of mathematics.

The mathematics curriculum cannot be seen as a single entity, but must be seen in two separate stages. In 1990, mathematics education researcher Lesley Booth described these stages as a shift from ‘empirical’ to ‘invented’. Empirical mathematics is that which arises from real-life situations. Invented mathematics is that which depends solely on other mathematics.

Empirical mathematics is the main focus of the early school years, where the relation to concrete experience is hard to ignore. There has been a recent emphasis on numeracy as the mathematics required to participate meaningfully in our current society. Special initiatives like Count Me In Too in the primary years have been very successful in making mathematics authentic to young children. The numeracy focus has also resulted in considerable attention being paid to topics such as financial mathematics, chance and data, and graphs of real-life situations right up until the end of secondary school. Unfortunately, at the secondary level the situation with empirical mathematics is not so rosy.

There is strong evidence that many students learn mathematics without making any connection with situations where the mathematics arises, and that this disconnection explains why they find it difficult and can’t see the point of what they are doing. Consider, for example, learning about ratio. The well-known research of Kath Hart indicates a weak link between the concept of ratio and real-life contexts. A cursory look at some text books shows why— a teaching approach that starts with an abstract definition of ratio as the comparison of two quantities of the same type. Research has shown this abstract approach to teaching empirical mathematics is not effective with most students. Yet, teaching without links to experience seems to be very common. For example, each of 35 Diploma of Education students at a Sydney university recently prepared a lesson on empirical mathematics. Thirty of the students chose a non-linked, abstract approach. When asked to explain their choice, the students referred to how they had been taught themselves, to the resources available to them to use in the lesson (texts, kits, etc), and to their desire ‘not to confuse the students’.

An alternative would be to start with a range of real-world situations (eg sharing, percentage swings in elections, odds in betting, proportions in mixing cement or cakes) which the students know about and then see how they all involve a similar type of comparison between like quantities. The abstract notion of ratio is then the end not the starting point. Would such an approach make the mathematics more authentic?

What about invented mathematics? A common description is that it is very abstract, meaning that it is totally removed from reality. The essence of this claim is that mathematics is self-contained because it uses everyday words; but their meaning is defined precisely in relation to other mathematical terms and not by their everyday meaning. Furthermore, mathematics contains objects that are unique to it. For example, although everyday language occasionally uses symbols like x and p, things like x0 and ÷(-1) are unknown outside mathematics.

Being able to stand alone is a crucial feature of invented mathematics. All the many advances of the last few centuries rely on it. Historically, mathematics has become increasingly independent of experience as more systems and structures have been invented. Mathematicians look for completion—ways to extend current ideas and results to more general ones and so build bigger mathematical structures. For example, powers of 2 and 3 in expressions like 152 and 23 arise in real-world situations involving area and volume. The empirical concept of a power is then applied to expressions for very large and small numbers and to compound interest calculations. To incorporate these ideas into a complete system which stands on its own, however, requires the invention of concepts like zero, negative, rational and irrational powers. At each point in this extension/completion process, it is crucial that the new objects be related to each other and the previous objects in such a way that they can be operated on without any appeal to any external meaning they might have.

If invented mathematics is self-contained and totally removed from reality, how can it be authentic and connected to life?

It would be unfair to suggest that teaching in the higher grades never attempts to apply new mathematical knowledge in some way. Such attempts usually fall into one of two categories.

Artificial exercises. For example, asking about the volume of a cube of volume 64 cm3 shrinking at a rate of 96 cm3 per minute. As one student doing this example (who was obviously well connected) observed, the cube was in its last moments of existence. This type of example has no connection with reality. Instead, it only requires students to strip away a facade of context and uncover the mathematical exercise underneath.

Applications in finance and statistics. These topics are essentially empirical mathematics, which is why realistic applications can be found easily.

It would appear that large sections of the senior mathematics syllabus deal with abstract ideas where it is too difficult to find related realistic contexts. Does this mean that invented mathematics should be treated as a special case where authenticity is not relevant or possible? Maybe not!

Most of the invented mathematics studied in schools is either based on empirical mathematics or on invented mathematics which is based on empirical mathematics. Mathematicians do not invent mathematics out of thin air. So, no matter how advanced a piece of mathematics is, there is at least a tenuous link back to reality which need not be a purely academic affair—it can have a human face. The history of mathematical thought is a rich source of real-life stories and excitement, which can place mathematical results in a human perspective.

There are a number of ways personal meaning beyond the classroom can be achieved with invented mathematics.

1. Students may see success with algebra and calculus as contributing to their overall competence, their self-esteem and future career prospects.
2. The challenge in a problem is another. For example, a dozen undergraduate students all agreed they loved the challenge of a good problem and often found it difficult to drag themselves away from a problem to continue with mainstream work.
3. And yes, the creation of self-contained, consistent and complete systems within the world of mathematics can provide aesthetic satisfaction to some people.

So teaching higher-level mathematics using challenge, purpose and narrative can provide a meaning beyond the classroom, but maybe only for the ‘true believers’. Others may, as one student put it, see it as ‘maths for maths sake’. Can all mathematics be authentic? Empirical mathematics certainly can be. Maybe invented mathematics will never be authentic for everyone, but certainly it can be for others.

Reference

Hart, K M (Ed.) (1982). Children's Understanding of Mathematics: 11–16. London: John Murray.

Paul White is a senior lecturer at Australian Catholic University, Sydney.

The author owns the copyright in this article. For information related to the reuse of this work in any form please contact the publisher denise.quinn@curriculum.edu.au