# Sound proof

The revised national curriculum includes more ideas about proof, starting in key stage 1, than previous versions.

The mathematics national curriculum for England states that pupils in KS1 must "explain their methods and reasoning when solving problems". In KS2 pupils should "understand and investigate general statements" and "develop logical thinking and explain their reasoning". They should know that the sum of the angles of a triangle is 180x.

In KS3 pupils have to "investigate whether particular cases can be generalised further and understand the importance of counter-example", "make conjectures" and "distinguish between practical demonstration, proof, conventions, facts, definitions and derived properties". They should "understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices". They should "explain why" the angle sum of any quadrilateral is 360 LESS THAN , and "understand, recall and use Pythagoras's Theorem".

Pupils in KS4 have to show "step-by-step deductions" and "derive proofs using short chains of deductive reasoning". Higher KS4 pupils should be taught "understanding and formulating proofs in algebra and geometry".

The current curriculum is radically different from O-levels in the Sixties. Then, rote learning of proofs was the norm. But that gave way to work on transformation geometry with little reference to proof. The modern curriculum, with using and applying mathematics (and particularly the reasoning strand) now integrated into the programmes of study, teaches pupils to develop their thinking skills in proof from KS1 to KS4.

The new curriculum has a vocabulary associated with learning about proof and includes these words and phrases: know, generalisation, particular case, conjecture, counter-example, exceptional case, justify, assumption, pratical demonstration, proof, convention, definition, derived property, and inferences.

In KS1 pupils have to "know" that the angle sum of a triangle is 180 LESS THAN . How will teachers help them "know"? Will they rip off the three corners of triangles and show that they fit to make a straight line? Is this a proof? No. Teachers will be saying that this demonstration seems to show 180 LESS THAN but does not prove it because the sum might be 179 LESS THAN or because we cannot check all triangles by this method. Pupils will be learning a little about the difference between a demonstration and proof.

Teachers know that pupils must be given opportunities to develop thinking skills in maths. The curriculum provides opportunities for a steady growth in skills concerned with logical thinking and proof.

Two recently published books will assist teachers in providing these opportunities. For teachers in KS1-4, Can You Prove It? by Sue Waring discusses this development by looking at learning about proof in three stages - convince a friend, convince a penfriend and working towards formal proof. There are 86 examples of proofs and "not-proofs" ready for classroom use, with lesson plans and lesson outlines for some of the proofs.

Are You Sure? by Doug French and Charlie Stripp is a book of ideas for upper secondary school teachers. It again shows how to develop ideas of proof and has examples of different types of proof: geometry, number, algebra and calculus. "Try this" boxes are included for students.

Peter Bailey is chair of the publications committee, Mathematical Association.The Mathematics National Curriculum Key Stages 1-4 (1999) DFEE and QCA.Can You Prove It? by Sue Waring , pound;9 + pound;1.50 pamp;p and Are You Sure? by Doug French and Charlie Stripp, pound;7.50 +pound;1.50 pamp;p can be ordered from the MA, 259 London Road, Leicester, LE2 3BE. Tel: 0116 221 0013. Members' discounts are available

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