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?
What do you think it means to act and be mathematical? Try and set out your thoughts in two or three sentences.
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.
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:
If we are to develop in young people the ability to move towards capability as mathematicians, then we should spend less time on projecting our ideas about what it means to ‘be mathematical’, and more time ‘being there’ in the mathematical situation – mathematical be-ing.
Activity 3 Think mathematically
- Choose any two-digit number. In this example, 87 is used
- Add together the digits. For example, 8 + 7 = 15.
- Take this number away from the original. 87 – 15 = 72.
- Specialise by trying another number. If you decide to try 86, 85 or 84, the final answer would also be 72.
Based on the number used, you might make a conjecture that all numbers will yield 72 after completing the steps above. However, a little more specialising will reveal that is not the case. So you might make another conjecture, and specialise, and try to generalise in order to convince yourself and communicate your ideas to others.
Working in this way is important. For example, if a student does not understand what a question is asking, they can learn to try an example (specialise) to see what happens, and if they learn to construct convincing arguments, then they can learn reasons rather than rules.
Apply this way of thinking mathematically to a real-life context that your student might encounter. For example: how much should be charged for a ticket to a school play?