Pupils' misunderstandings of mathematical terms give us a detailed insight into how language and mathematics can interact and create barriers to learning. Take the response given by 10-year-old Andrew, Year 6, to the following (written) question: Q Why is 17 a prime number?
A Because at that age you're still young.
Literal responses like this are common; but while they may amuse researchers, they can feel like a nightmare for class teachers. Confident teachers usually ask three questions. Why does it happen? How serious is it? What can I do about it?
Pupils benefit from lucid communication. Mathematics poses particular challenges to this. Many mathematical terms - like mean and volume - have very different meanings when used in ordinary English.
Other terms, such as area and difference, have a more precise use in mathematics. Some terms have meanings that change according to the aspect ofmathematics being covered. Take, for example, "base" as used in "number" (as in "base two" or "base 10") and then in "shape" (as in "base of a triangle").
And the flow of mathematical script isn't always presented in a way where clues can be gleaned from other parts of the text - unlike reading a book. All the ingredients, then, for creating a bewildering confusion for the inexperienced learner.
Indeed, once you identify the full extent of possible misunderstanding, then the successes and progress that pupils make in mathematics seem all the more worthy.
Knee-jerk solutions to language ambiguities are unlikely to have any long-term benefits. Recently I've been working closely with groups of Year 5 and 6 children on creating "home-grown" mathematical dictionaries.
The task is simple. Together we collect words used in their textbooks and then they create definitions andor illustrations for each term. The emphasis is on openness and honesty - without them you'll probably end upwith a large number of blank returns. The results have been illuminating, and somewhatstaggering, as these few examples indicate: Place value - The cost of a house
Volume Sound - control on the TV
Trapezium - A circus act
Square - Like a rectangle but with all sides the same
Scale - A drawing of a fish
How important are these findings? Do they suggest weaknesses likely to affect a pupil's attainment in future work? In trying to interpret these responses, I'm reminded of a lecture comment made by the late Richard Skemp, president of the International Group for the Psychology of Mathematics Education: A concept without a word is like a suitcase without a handle - difficult to grab hold of. A word without a concept is like a handle without a suitcase. Absolutely useless!
To come to some conclusions for my own work, discussion between and with pupils is used to discover the nature of their misunderstandings. In Skemp's terms, are we talking "handles" or "suitcases"? Do pupils have a developing understanding of the underlying concepts behind these terms?
An early analysis suggests that pupils have a clear understanding of place value with whole numbers (though not with decimal fractions), but their knowledge of some geometric shapes and their properties are superficial: many have difficulties in recognising shapes in different orientations, or with irregular sides - a condition I now term "logic-blockitis".
This situation hasn't arisen because the the language of mathematics has been neglected. All recently-published schemes emphasise a carefully-structured introduction to mathematical terminology, though some seem over-indulgent in what they hope pupils will acquire. Difficulties created by language persist because printed text is an inadequate vehicle for communicating meaning.
In a content-burdened curriculum where time is limited, there is a temptation to teach towards "handles" in the hope that "suitcases" will be forthcoming as a natural consequence. Superficial analysis of pupils' understandings can lead to treating mathematical language as a discrete subject, taken out of context. This results in terminology being taught in the belief that it helps in the development of concepts. It doesn't. Pupils remember most and learn best when they are engaged in an activity they find meaningful and interesting.
An approach that seems to give positive results relies on a very simple model. By keeping instructions, organisation and activity simple, teachers can focus on discussion - this is the key to developing pupils' understanding of terminology: open-ended activities supported by a balance between open and closed questions. For an example, see the diagram on the right.
The confidence of teachers to sustain discussion of this sort with a whole class seems to have waned. It is a confidence we need to rediscover, and it needs to attract management approval and supp-ort. And activities like this need to be supported by mathematical language games so pupils can maintain mastery of newly-acquired terms.
So what happens to Andrew and his "prime numbers"? He now gives a clear definition, but feels that history's choice of vocabulary makes learning unnecessarily difficult. (Why didn't they choose a fresh word? It wouldn't have been difficult to make one up!) He is currently interviewing family, friends and those in school who happen to have a "prime" age, for an English project (his favourite subject).
I suspect that all of us are open to making literal interpretations when confronted with specialised vocabulary in an area that is still new to us. It is through regular use with appropriate feedback that we progress to fluency. For my part - and in my quest to be more adept with the computer - I have become skilled in an act called "dragon drop" using the computer's mouse to move around text. I genuinely believed I could see the flashing cursor turn into a dragon shape before my eyes. It was not until I saw this process written down that I realised that the term was known as "drag and drop". The dragon has now disappeared. I had the suitcase but not the precise handle.
Discussion in mathematics will have a similar effect of minim-ising misunderstandings in the classroom; sensitive interaction will eliminate any embarrassment created by ambiguities. Let's help pupils slay a few dragons of their own.