Boxed into a corner

19th January 2001 at 00:00
Teaching definitions is a tricky business, warns Keith Jones

Definitions are central to mathematics, and definitions given in maths dictionaries are especially important. While the National Numeracy Strategy restricts itself to proscribing vocabulary checklists, the Qualifications and Curriculum Authority has embarked on defining such terms (see the QCA's Mathematics Vocabulary for teachers in key stages 1 to 4). Apparently, the longer-term plan is that the definitions "will lie behind the programmes of study for mathematics".

Definitions are intimately linked to mathematical classification, in turn linked to notions of proof and proving. In the Strategy, the terms square and rectangle are to be used from reception. The terms for no other quadrilaterals are introduced until Year 4, when oblong appears, while in Year 6, rhombus, parallelogram, trapezium and kite surface.

The Strategy includes classifying shapes as an important objective in Year 4 and, as an outcome, Year 4 pupils are expected to be able to name, classify and describe a rectangle, an oblong, and a square. By Year 6, pupils are expected to "continue to name and describe shapes, extending to: parallelogram, rhombus, kite, trapezium...". This extends to the Year 7 framework, where the requirement is for pupils to be able to "use, read and write, spelling correctly: quadrilateral, square, rectangle, oblong, parallelogram, rhombus, trapezium, kite, arrowhead...".

But what is an oblong? It is not a mathematical term. The QCA defines it as a "rectangle where adjacent sides are unequal" and says "a square is a special case of a rectangle but it is not an oblong because adjacent sides are equal".

This definition raises a hos of questions, but let us focus on one: how should we define something mathematically? The answer is (more or less) universally agreed on by mathematicians: we should employ inclusive definitions, ones that include other things as special cases. First, because they make a number of things simpler, particularly the statement of theorems. So for example, any theorem true of rhombi is also true of squares, as a square is a special case of a rhombus. Second, if you do not employ inclusive definitions you may easily go wrong.

There seems to be no case for using the term "oblong" for quadrilaterals. Defining some rectangles as oblongs makes it harder for pupils to appreciate that squares are rectangles, and undermines the key idea of understanding generalisation in maths when "weaker" concepts are used to generalise towards "stronger" concepts (rather than exclude them).

The Strategy declares that "mathematical language is crucial to children's development of thinking" and that "a structured approach to the teaching and learning of vocabulary is essential if children are to move on and begin using the correct mathematical terminology as soon as possible". The key thing is that it has to be correct terminology, used in a way that enables pupils to explore definitions and appreciate their importance. The national curriculum does not yet contain the term oblong. The danger is that the Strategy does, and the curriculum may well do so in the near future.

See Resources, page 18

Keith Jones works at the Centre for Research in Mathematics Education at the University of Southampton and is a member of the Royal SocietyJoint Mathematical Council inquiry into teaching and learning geometry

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