From Objects and Images to Mathematical Ideas

From my research in Hungary, one of the key things that has struck me is the way in which multiple representations and images are used to support the development of very abstract and complex mathematical ideas. This month's website will seek to illustrate this and offer some problems and resources that help students to develop notions of abstract mathematical ideas through exposure to a range of representations of them. This all sounds rather obscure and abstract in itself so let me illustrate this with two examples one from a class of 7 year old students and the other from a class of 15 year olds.
For the younger students the concept under consideration was the idea of the number six. During the course of one lesson focused on this number, the children were offered a range of iconic and symbolic representations of six and asked to identify collections of six objects. This range comprised:
Pattern on a die, finger pattern, collections of objects, collections of actions, the Cuisenaire rods, the 'number picture', (you can find them here doc, pdf  ), dominoes with six spots, Roman numerals, the symbol 6, number line, 6 o'clock on an analogue clock face, coins.

I would argue that this rich range of representations of 6 enabled the children to abstract a notion of the 'sixness' of six that transcended the different representations. All the different representations have their value and potential applications: some stress the notion of a number representing a collection and so build from counting such as a collection of objects of actions; others stress aspects of the structure of six such as the finger pattern which draws attention to six being one more than five; others emphasise the wholeness of six as an entity that supports students away from a counting notion of number such as the symbol 6, the 6 Cuisenaire rod or even potentially the die pattern that can be recognized without counting the spots; some stress the place of six in the sequence of counting or Natural numbers such as the number line.
For the 15 year olds, the lesson that I observed was focused on the solution of simultaneous equations involving trigonometric functions. In this lesson the students were able to identify solutions to complicated pairs of equations through their knowledge of the meanings of the functions that were being considered. They were able to sketch the relevant graphs of the functions, consider their ranges and domains and use these ideas to produce solutions or to identify when solutions could not be found.
This seemed to me to be linked to my observations of lessons with the younger children. In both cases the students had access to a range of representations, images and mathematical models and were able to bring appropriate images to the problems with which they were presented.
In some of our problems this month we offer a range of representations of mathematical ideas to work with, in others we offer a problem that lends itself to solution with one representation in mind. In all of the problems we are seeking to explore the power of a variety of representations, images and models with a view to supporting students in enriching their understanding of various abstract mathematical concepts through exposure to this variety. The aim of the exercise is to deepen students' understanding of the abstract mathematical concepts involved in the process of generalizing from a variety of models and representations.

Stage 1

Matching Numbers is an interactive game in which the task is to match different representations of numbers in pairs. In fact the set of cards has four possible representations of each of the numbers so children can discuss how each of those representations shows the number. When they have played the game themselves, they can make their own sets of number cards showing different representations of numbers using this blank set.

How do you see it? is an activity with a difference in that as the children work on these they will have the opportunity to exhibit their own individual ways of thinking about simple calculations. The article referred to in the teachers' notes will enable you to explore some of the findings about those calculations that children find more difficult because of the order of the information.

Let's divide up offers a story scenario in which three different conceptions of division are presented. We hope that children will be able to explore the different conceptions to deepen theri understanding of the mathematical operation of division. They may be encouraged to make up their own stories about that involve division conceived of as sharing, grouping, successive subtraction or the inverse of multiplication.

Stage 2

Matching Fractions is another interactive game of memory but this time there are four representations of a number of fractions to match in pairs. We have a tendency to use pizzas as our main representation of fractions for young children and this can cause them problems with developing their conceptual understanding of fractions so this is a useful activity to tackle this as it offers a range of different representations including fractions of quantities bigger than one. Once again children can create their own fraction representations and their own game using the blank card set.

Stage 3

In What Numbers Can We Make?, students are invited to work with numbers chosen from a linear sequence. In order to explain the patterns they find, they need to explore ideas from modular arithmetic, which can be represented geometrically, numerically, and algebraically. The same representations can be adapted in the follow-up problems Take Three From Five, and What Numbers Can We Make Now?. An understanding of algebraic and graphical representations is also required when playing Diamond Collector.

Stage 4

Factorising with Multilink and Pair Products both make the links between number and algebra through geometric representations. Students are encouraged to generalise by going beyond simple pattern-spotting and engaging instead with the underlying structure.

Stage 5

Polar Bearings brings attention to the relationship between cartesian and polar coordinates. By striving to represent certain curve shapes using different systems, students will realise that choice of coordinates is actually arbitrary and can lead to useful algebraic simplification (or unnecessary complication!). The problem Trig Reps looks at different representations of 'trigonometry' and encourages students to derive many familiar properties of sine and cosine using each representation. Through engagement with this activity students will hopefully gain a deeper understanding of trigonometry and realise that certain representations are more useful for solving certain types of problem. Both problems reinforce the important mathematical notion that the 'underlying' mathematics can be dressed up in different ways as required. It reminds me of the popular saying that 'There is no such thing as bad weather, only the wrong choice of clothes....'