The transmission approach to teaching mathematics (Askew, 1997) is that in which a series of ‘rules and truths’ are conveyed to learners by means of ‘chalk and talk’ followed by individual practice until fluency is attained. The teacher pre-digests and organises material, gives clearly prescribed instructions, teaches everyone at once in a predetermined manner and emphasises practice for fluency over discussion for meaning. Everything is ‘covered’ from first principles. Teaching is seen as the transmission of definitions and methods to be practised. Learners are generally not encouraged to work collaboratively, show creativity or make decisions about what they learn. (Swan, 2006)
If the teaching is mainly determined by a transmission view, learners are likely to:
- listen while the teacher explains
- copy down the method from the board or textbook
- only do questions that they are told to do
- work on their own
- try to follow all the steps of a lesson
- do easy problems first to increase their confidence
- copy out questions before doing them
- practise the same method repeatedly on many questions.
‘For these learners, mathematics is something that is ‘done to them’, rather than being a creative, stimulating subject to explore. It has become a collection of isolated procedures and techniques to learn by rote, rather than an interconnected network of interesting and powerful ideas to actively explore, discuss, debate and gradually come to understand.’ (Swan, 2005)
Transmission approaches can appear to be effective in the very short-term, but they do not lead to long-term understanding.
Learners whose experiences of ‘doing mathematics’ are mainly as described above, are not likely to learn that mathematics is fundamentally about becoming aware of and expressing generality. Rote memorising of disconnected rules, which are often misapplied and quickly forgotten, does not allow learners to make or recognise generalisations, which bring understanding, and that can be used, applied and explored in other situations. For example, in order to know when to use a particular technique learners need to understand how that technique has been derived, as a generality, from a range of examples. Learners who don’t generalise for themselves, but continue to expect the teacher to remind them of rules, become dependent as learners, and are not equipped to learn efficiently and effectively in the future. They also fail to be excited by mathematics because they feel they are not contributing anything of themselves.
Generalisations, such as ‘each exterior angle of a triangle equals the sum of the opposite interior angles’ are not facts to be remembered blindly; they come about as a result of noticing something and trying to account for it. The act of forming a rule from noticing something from a set of examples is an intensely mathematical thing to do. The following of somebody else’s rule isn’t.
However teachers can encourage learners to use questions in a text-book critically to help them understand some mathematics; they can work on a list of questions rather than just work through it. For example:
- learners can ask themselves how many of the questions they need to do before they understand the idea behind them
- when a learner successfully solves a problem, they can think about what changes they could make to the problem and still their solution method would work
- learners can try to make up, and try to answer, their own questions about the idea that they have deduced is behind those in the text-book
- learners can explain, to each other, to the whole class, or to the teacher, how to do this type of question.
All learners come to school with natural powers, such as the ability to:
- imagine, and express what they are imagining
- specialise and generalise
- conjecture and convince
- organise and classify.
These powers can be developed for making sense of mathematics. When a textbook does the expressing, specialising and organising for learners, learners are discouraged from using their own powers, and they are likely to conclude that in mathematics their powers are not needed or wanted. But tasks that invite learners to use their own powers will attract learners to engage in mathematics, giving a sense of satisfaction. When learners’ conjectures and suggestions are respected, they feel that they have a role to play in, and are contributing to the progress of, a lesson.
Learners should find out for themselves that the source of the authority for truth and correctness in mathematics is not the authors of text books or the teacher, but is reasoning and logic; the authority lies in the structure of mathematics itself. If it is the text-book that asks the questions, learners see their task as trying to match their answer to that given by the text book. If it is the teacher asking the questions then their task is to find the answer that the teacher knows. So learners think that they need the text-book or teacher to validate their answers. But, of all school subjects, mathematics is the one where most clearly you can know that you are right. Achieving the depth of understanding that enables you to know that you are right is an essential part of learning mathematics.