Interpreting multiple representations

“The good thing about this was, instead of like working out of your textbook, you had to use your brain before you could go anywhere else with it. You had to actually sit down and think about it. And when you did think about it you had someone else to help you along if you couldn't figure it out for yourself, so if they understood it and you didn't they would help you out with it. If you were doing it out of a textbook you wouldn't get that help.”

Mathematical concepts have many representations; words, diagrams, algebraic symbols, tables, graphs and so on. These activities are intended to allow these representations to be shared, interpreted, compared and grouped in ways that allow learners to construct meanings and links between the underlying concepts. In most mathematics teaching and learning, a great deal of time is already spent on the technical skills needed to construct and manipulate representations. These include, for example, adding numbers, drawing graphs and manipulating formulae. While technical skills are necessary and important, this diet of practice must be balanced with activities that offer learners opportunities to reflect on their meaning. These activities provide this balance. Learners focus on interpreting rather than producing representations. Perhaps the most basic and familiar activities in this category are those that require learners to match pairs of mathematical objects if

they have an equivalent meaning. This may be done using domino-like activities. More complex activities may involve matching three or more representations of the same object. 

Typical examples might involve matching:

1. times and measures expressed in various forms (e.g. 24-hour clock times and 12-hour clock times);

2. number operations (e.g. notations for division – see below);

3. numbers and diagrams (e.g. decimals, fractions, number lines, areas);

4.algebraic expressions (e.g. words, symbols, area diagrams – see below);

5.statistical diagrams (e.g. frequency tables, cumulative frequency curves). 

The discussion of misconceptions is also encouraged if carefully designed distractors are also included. The examples below show some possible sets of cards for matching. They show how learners’ attention can be focused on the way notation is interpreted, and common difficulties may be revealed for discussion. 

The activities offer learners a number of mathematical statements or generalizations. Learners are asked to decide whether the statements are ‘always’, ‘sometimes’ or ‘never’ true, and give explanations for their decisions. Explanations usually involve generating examples and counterexamples to support or refute the statements. In addition, learners may be invited to add conditions or otherwise revise the statements so they become ‘always true’. This type of activity develops learners’ capacity to explain, convince and prove. The statements themselves can be couched in ways that force learners to confront common difficulties and misconceptions. Statements might be devised at any level of difficulty. They might concern, for example:

1. the size of numbers (“numbers with more digits are greater in value”);

2. number operations (“multiplying makes numbers bigger”);

3.area and perimeter (“shapes with larger areas have larger perimeters”);

4.algebraic generalisations (“2(n + 3) = 2n + 3”);

5.enlargement (“if you double the lengths of the sides, you double the area”);

6. sequences (“if the sequence of terms tends to zero, the series converges”);

7. calculus (“continuous graphs are differentiable”); . . . and so on.

Throughout this process, the teacher’s role is to:

1.encourage learners to think more deeply, by suggesting that they try further examples (“Is this one still true for decimals or negative numbers?”; “What about when I take a bite out of a sandwich?”; “How does that change the perimeter and area?”);

2. challenge learners to provide more convincing reasons (“I can see a flaw in that argument”; “What happens when . . . ?”); ! play ‘devil’s advocate’ (“I think this is true because . . .”; “Can you convince me I am wrong?”).