2 Key issue 2: What does it mean to act and be mathematical?

What it means to act and be mathematical is an important consideration for teachers. What do you want to pass on to your students? Is being numerate enough? Or are the fundamentals of mathematics and mathematical thinking more than simple arithmetic?

Articulating a view that is informed and coherent, and that you actually believe in, can be difficult. But once you can do that, you will be in a good position to use this as a basis for thinking about teaching mathematics. It is possible to see mathematics simply as a bag of tools – a set of unconnected skills that can be used whenever necessary. Thus, the skills of ‘finding a percentage of something’, ‘finding a fractional part of something’ and ‘finding proportions of something’ are common topics in textbooks. These are often taught as quite separate techniques, and students are expected to learn them separately. However, these ideas are essentially all the same. There is a ‘sameness’ about percentages, fractions, proportions – and helping learners connect ideas and know that they represent different ways of, essentially, seeing the same thing is important to building fluency with ideas in mathematics and not overloading memory.

The idea of connectedness is important not only because it is a powerful way of representing mathematics knowledge, but also because once learners are aware of these connections, they have access to more mathematics when tackling unfamiliar problems and challenges. It is also true that, if teachers stress those connections, students’ learning is improved. If students’ attention is drawn to the notion of ‘square’ as it appears in geometry, in number and in algebraic expressions such as (2a +b)², they are likely to develop a more integrated understanding of the mathematics and may also avoid typical difficulties associated with what 3² or πr² means. A problem in trigonometry might be solved more easily if students are aware of the connections between concepts of numerical ratio and geometrical similarity. This is about making time to explore and think about the generality of ideas and to select from that generality a strategy appropriate for tackling a particular problem.

Figure 4 Thinking mathematically

Show description|Hide description

A graphic of a head with cogs in place of a brain.

Figure 4 Thinking mathematically

In their book Thinking Mathematically, Mason et al. (1982) identified four essentials in mathematical thinking:

• specialising – trying special cases, looking at examples
• generalising – looking for patterns and relationships
• conjecturing – predicting relationships and results
• convincing – finding and communicating reasons why something is true.

Mathematical thinking consists of conjecturing, testing those conjectures, convincing yourself and others of the truth, and relating those truths to others – for example, by asking the following:

• ‘Is this always/sometimes/never true?’
• ‘Are there special cases or exceptions to the rule?’
• ‘Does this result apply to a wider set of problems?’

Such questions and prompts test the limits of students’ understanding and provoke them to think about generalities and connections. By asking such questions a teacher asks for deep learning rather than surface remembering, and by listening to the students’ tentative answers they can uncover misconceptions and act to facilitate a surer understanding.

Dylan Wiliam (1998, p. 6) states the following: