You may well ask
There is a pendulum swing towards testing which threatens to extinguish the ways of developing mathematical thinking that have been developed in schools in the past two decades. "Teaching to the test" and concentrating on practising techniques will result in over-dependence on textbooks and a move away from exploratory and investigative work.
With John Mason of the Open University's Centre for Mathematics Education, I have been looking for ways to give pupils a sound preparation for mathematics assessments while preserving intrigue and methods of exploration which are at the heart of maths. We have collected questions based on mathematical structures and classifications that can be used to trigger pupils to think more deeply about maths, to revise, reconstruct, and reformulate their knowledge. Their learning can be enhanced by questions which force them to construct an overview of a topic rather than just technical proficiency.
Behind teachers' specific questions lie general forms which express mathematical thinking. Once identified, these general forms can be applied to a wide range of topics to create a rich environment for learning.
Here are some examples of questions with the general formats they are employing: Q Is it always, sometimes or never true that if a is less than b, then a-squared is less than b-squared?
(Is it always, sometimes or never true that...?) Q In the context of quadratic equations, if the answer is x = 5 or - 1, what could the question be?
(If this is the answer, what is the question?) Getting pupils to think hard about underlying meanings and links within the subject enhances their appreciation and flexibility. For instance, Pythagoras's theorem focuses attention on squares of sides of right-angled triangles. Through using the question "Is it always, sometimes or never true?" pupils can be led to wonder why the right-angle is important. A follow-up question such as: "When might it not be true?" can trigger exploration of what is true for squares of sides of triangles without right-angles.
The questions we ask our pupils to answer give them a picture of what maths is like. The kinds of questions asked in many textbooks are usually limited in scope, requiring one correct answer. This can give pupils a view of maths as a set of techniques, rather than a world of structures to explore. But exploration is never finished. By prompting pupils to search beneath the surface features of maths, to construct and express meanings and to raise questions, we have found that a lasting interest can be stimulated.
Questions and prompts to try What is the same and what is different?
Q What is the same and what is different about two bead necklaces made up of coloured beads following rules such as red, blue, red, blue: green, yellow, green, yellow; and red red, blue, red, red, blue. (Key stages 1 and 2) Q What is the same and what is different about regular and symmetric polygons? (KS3) Q What is the same and what is different about the angles 60o, 240o, and 420o, in the context of trigonometry? (GCSE) Q What is the same and what is different about 0.999999... and 1? (A-level and tertiary).
This general question encourages pupils to focus on mathematical properties, classifications, definitions, and accurate use of language. In the first example, the pupils can go on to try to devise a way of recording what is the same and what is different, which is a pre-algebraic experience. In the second, pupils can try to construct polygons which are symmetric but not regular. In the third, calculators can answer the question, and drawing diagrams and investigating trigonometric ratios as functions can help explain similarities and differences. In the last case, pupils will make a step toward understanding why modern analysis has such precise definitions.
How can... be changed to...?
Q How can a rectangle be changed into a square? (KS2) Q How can a fraction be changed to a percentage? (KS2 and 3) Q How can the equation of a line be changed so as to produce a line at right-angles to the original? (GCSE and A-level).
This general question encourages pupils to find out how similar things in maths are related to each other, what has to be changed and what has to stay the same, and to create their own descriptions of processes and techniques. In the first case, the important things about a square will be identified; in the second, pupils will devise their own ways to describe a process; in the last, pupils using a graph plotter can learn more about gradient. Time spent finding out for themselves, even including going down unhelpful routes for a while, contributes to a deeper personal understanding of the underlying structures.
Give a definition which...
Q Give a definition of a rectangle which only mentions its diagonals. (KS2 and 3) Q Give a definition of (symbol for pi) which only mentions areas of circles. (KS3 and GCSE) Q Give a definition of number which includes rationals but excludes irrationals. (GCSE, A-level and tertiary).
Creating definitions helps students look at topics from new directions, and helps them see how applications and outcomes of mathematics relate to the more usual ways of defining concepts. Searching for definitions involves investigating the range of possibilities; it takes time, and attempts at definition can lead to fruitful classroom discussion. In the last case, pupils will be replicating part of the history of numbers.
Anne Watson works in the department of educational studies, University of Oxford . 'Questions and Prompts for Mathematical Thinking' by Anne Watson and John Mason is published by the Association of Teachers of Mathematics, 7 Shaftesbury Street, Derby DE23 8YB