Question of style
Aural learners learn by listening. Visual learners learn by looking.
Kinaesthetic learners learn by doing. But just what do they do? The phrase kinaesthetic learning is sometimes taken to mean any activity that involves the use of apparatus. But if the focus of the teaching is primarily on the correct use of the apparatus, rather than on the mathematical understanding that the apparatus is designed to develop, then it may not lead to much learning. Pupils will just follow the instructions to use the equipment, without necessarily relating what they are doing to maths.
Kinaesthetic learning calls for a lot more than a pile of cubes or a pair of scissors and some scrap paper. It involves pupils in using their whole being, engaging all their senses to feel or imagine what is happening.
Visual, aural and kinaesthetic learning are all intertwined: together they can lay down a memory that will be recalled as a total experience, not just as a set of rules for getting right answers. For example, when I factorise the number 12, I can imagine pulling a block of 12 cubes apart to make two sets of six, three sets of four, or two sets of two threes.
Long multiplication by the "area method" conjures up an image of a rectangle whose area must be found. Square numbers are square numbers because they are square, not just because they are whole numbers multiplied by themselves etc. My understanding of these concepts is based on my memory of creating and working with physical models - I am a visual and kinaesthetic learner. But because I have made the models and have understood the maths they represent, I do not need to actually handle them again in order to recall them.
On the other hand, what I recall is most certainly not a rule or a formula: it is more like a moving picture - a sort of waking dream. This, I suggest, is kinaesthetic learning. Kinaesthetic experiences that embody and convey mathematical principles can be very powerful. But they are worthwhile only if pupils think about and understand the maths - and this takes time.
Take that old chestnut, multiplying and dividing by powers of 10. The key to understanding this concept is movement. When 4 is multiplied by 10, so that it becomes 40, it moves. It moves one place to the left.
But the trouble is, it doesn't. Few teachers would ever actually teach their pupils that they should "add a nought' to multiply a number by ten - but that, in reality, is what happens. If written symbols are used to represent 4 multiplied by 10, for example, then the teacher may talk about the 4 moving one place to the left, but to the pupil it is obvious that the 4 stays put. The 0 just takes up its position after it.
Alternatively, a sliding model that demonstrates what happens when a number is multiplied or divided by a power of 10 does show the movement. It can be made from a sheet of A4 paper folded into quarters, with two windows cut in one panel and separated by a decimal point. A one- or two-digit number, followed by some zeros, is written on a strip of paper. This paper feeds through one end of the folded paper to slide past the windows. As the strip slides to the left, the number is multiplied by 10. As it slides to the right, it is divided by 10.
The vital role played by the zero as a "place holder" in, say, the number 570 becomes clear with this sliding model. If we did not have the zero to fill the units column then there would be nothing to distinguish 570 from 57. Even if the paper strip is too short to show them all, we can imagine a string of zeros trailing off to the right, ready to slide up as the number on the strip is multiplied by 10 again and again. A larger model, and a longer strip, allows for numbers with more digits - but a model made from a sheet of A4 paper, with two non-zero digits and three zeros, will get the idea across. It will give pupils a visual and kinaesthetic "picture in the mind" of the movement of the digits across the decimal point that they are far more likely to remember than any number of written exercises on printed sheets.
Images and models like these have one driving purpose. They all offer a model to think with, something that the pupil can use to make sense of the maths. Symbols and rules may mean little in themselves, and are easily forgotten. But once the conventions have meaning, pupils are far more likely to recall and use them effectively.
Tandi Clausen-May is principal research officer, Department of Assessment and Measurement, National Foundation for Educational Research www.nfer.ac.uk
* More ideas can be found in Teaching Maths to Pupils with Different Learning Styles by Tandi Clausen-May (Sage Publications) www.sagepub.co.uk