Proof and ReasoningAuthor(s): David WellsSource: Mathematics in School, Vol. 16, No. 5 (Nov., 1987), pp. 47-48Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214408 .

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Letters to the Editor

The National Curriculum for Mathematics

Dear Sir and Madam, I agree with Mr Baker that a review of the school curriculum is both desirable and necessary. I also agree that the national curriculum needs to cater for a wide interest and ability range and that courses should be adapted to meet new needs as they arise. Nevertheless I am concerned that Mr Baker makes no reference to the increasing use of mathematics by the biological and social services within recent years, though their math- ematical needs are very different from those of physics.

Traditionally school mathematics has been taught in three stages which can be described as practical, pure and applied respectively. Practical mathematics involves using mathematics to measure and describe many aspects of the everyday world. Pure mathematics makes no use of applications, and at the third stage, the rele- vant scientific theory is expressed in terms of mathematical formulae which can be applied to specific problems. When the only school science that made much use of mathematics was physics, it was not unreasonable to teach co- ordinate geometry and calculus as pure math- ematics to able sixth-formers, and then to apply them to problems from Newtonian physics. This combination of pure and applied mathematics has served many generations of students very well, but today's students also need to make more use of practical mathematics. Applications to the less exact biological and social sciences can, and should be, used to include a practical introduction to graphical algebra, differential coefficients and descriptive statistics.

I am not convinced that the National cur- riculum should be enforceable by law, or that all pupils should be tested at 7, 11, 14 and 16 years of age. Examinations at 11 and 16, part of the present system, serve a useful purpose for decid- ing the next stage for these pupils: examinations at 7 and 14 are surely unnecessary.

I also question the need for the school syl- labus to be determined by the best "minds", surely the best "teachers" would be a more appropriate choice.

Freda Conway Leicester

Maths View

Dear Sir and Madam, Having been concerned with "training" Primary School teachers for 25 years my comments on the SSNCS in Vol. 16, No. 3 voice my disquiet on the effect on young children.

The Government will be referred to as G. and points in the Statement as (1), (2), etc.

I. G. uses words, particularly "curriculum" in a confusion of senses

In (3) "it draws upon individual talents" but in (4) this is narrowed to "a level of attainment and statement of content" which is suggestive of uniformity of instruction leading to measurable results, and in (9) "a list of subject titles" and in (11) "essential content, skills and processes taught". In (11) it is assumed that a specified progression has been followed. In Infant Schools this would be contrary to the best practice of developing Maths as it arises in experience and harks back to the Twenties, with mechanical Arithmetic of addition followed by subtraction, "tens and units", etc. So in (13) "the testing of what can be tested" would either be different for different classes or at a low level of "sums". How can this be modified externally?

In (11) I cannot imagine what is meant by setting attainment levels before specifying con- tent, skills and processes. Does it mean, "The 7 year old shall understand Mathematics" or some other vague aspiration? If so how can such a phrase be a measurable attainment?

I1I. What is G's Educational Psychology? It does not appear to make allowance in practice for individual differences

(i) in ability (ii) the stage reached at entry whether due to

home ambience, ability (iii) health and attendance at school, often

variable in the first years (iv) comprehension and use of English/

School language (v) that some children at the test age of 7 have

had only 2/3 of teaching time of some others

(vi) continuity of teachers (vii) the catchment area of the schools. Nor has it considered the effect of a formal test, with the work out of the ken of many of those sitting. The only hope of testing what a child understands is the observation of its skilled teacher.

Ill. What confidence has G. in the qualified professionals employed in Education?

(4) pays lip service to the existence of teachers with "skill who can adapt the work to the pupil and develop new approaches", but in (5) does not trust their judgment on content or level of attainment either considering them singly or corporately in school or with the specialists in the L.E.A. (6) and (7) are even more disparag- ing, "hardly anyone knows what the schools seek to teach" and (7) suggests that at the moment the staff do not know where they are and assumes that they are not committed and will not be until they have a good curriculum imposed on them. It is not clear in (13) who decides on the test items nor the nature of the assessment of the "class work" which with young children will be different for each of the 30 + doing it simultaneously and may well have no written work to be kept as evidence. The judgment and recording of this in any appro- priate manner would be highly skilled, especially as it is to be subjected to moderation at some later date. Incidentally, as the generality of teachers are not considered trustworthy pro- fessionally, who will be given the role of external moderators?

IV. Underlying assumptions which are debatable or even controversial

(a) There are no good L.E.A. curricula. Speaking of Mathematics, the L.E.A. in the West Country spent endless time and corporate efforts on Primary Maths with large volumes covering every aspect and also on various forms of record card/profiles to be kept up throughout a child's Primary years. They met the immense difficulties of specifying levels and assessing the degree of real understanding. Teachers went on courses about the content and to develop their own knowledge. What will a National curriculum in this area achieve that these did not? (b) The age of 7 is a key one for testing. Is this merely because by a historical accident of the English State System children have to move from one building to another? Did Plowden not recommend the change to be at 8 when the early skills in language and reading might be better consolidated and more usable as tools of learn-

ing? Is a test system with ranking not likely to produce the ills surrounding the old 11 + with its emphasis on failure rather than on encouragement, possibly caused by undue parental anxiety. And will not this same pres- sure force the teachers to revert to the old rote methods which cannot contribute to understanding? (c) That parents will be better informed about their child's progress from the result of a written test near the end of the time spent in any school than from the usual discussions with the class

teacher and from seeing the children's actual work throughout the time. (d) That the time and effort spent in this formal testing, recording and moderation is more profit- able than if it were spent on teaching. (e) That all the advice of the H.N.I. on how to improve standards in school will be fulfilled by this plan. (f) That there is an adequate supply of teachers with the requisite knowledge and understanding of Maths to teach the subject at the appropriate level of the compulsory period. There never has been since I entered the profession. (g) That it is possible to improve the quality of teaching by legislation, see (3) and (15), "a curriculum required and enforceable by law", "attainment targets and subject curricula speci- fied in legislation". What body will police the system? What punishments will be given if the targets are not met? (h) That the teachers should be expected to do all the extra work including detailed record keeping to submit for moderation and to face appeals which may be lodged against their decisions, without a much improved staffing rate. (i) That G. knows better than the experts.

Cecily M. Nevill Bath

Proof and Reasoning Dear Sir and Madam, I was interested to read the article Proof in the Middle Years by John Cable (M in S, March 1987) about the problem, "Given that the angle-sum of a quadrilateral is 360 , prove that the angle-sum of a triangle must be 180+", and I would not wish to dispute his conclusion when he writes, "I am reinforced in my belief that proof, of the above kind at least, is too abstract for the majority of 12-year-olds ...".

I am however disturbed by the possible im- plications that might be drawn from the article by readers who do not stop to consider the many questions that are begged by the essen- tial qualification, "of the above kind at least".

In particular, it might be taken as undermin- ing attempts by teachers to reinstate proof in the mathematics classroom after a quarter of a century curing which it has been almost entirely absent.

Certainly it is likely that if a particular proof "of the above kind" is chosen by the teacher for presentation to pupils, then pupils will find it extremely difficult.

This however is to tackle the problem the wrong way round. Rather, it is necessary for teachers to stop imposing proofs on pupils, and essential instead to look for examples of thin- king in children's work which has the charac- teristics of proof, and then to interpret and explicitly label such thinking as proof whenever it occurs.

In particular, teachers must stop thinking of proofs as necessarily being long with many individual steps, and accept that even the short- est chain of reasoning is a proof provided it displays the usual features such as clarity, con- viction and communication.

It is also necessary to stop associating proof with particular parts of mathematics, for example with geometry. A sequence of steps in algebra, for example, which correctly infers that if x2= 5x+ 6 then x= 6 (or - 1 ) is just as valid a proof as any proof of a theorem or rider about triangles.

That it is more mundane, and, perhaps, re- quires no original thought, does not deprive it of the status of a proof. Originality has never been a feature demanded of proofs in general, only a feature that is often found in proofs of original theorems.

Let me give just one simple example: let the problem be, "Are there any squares in this sequence? 3 6 12 24 48 96 192 ...?"

Mathematics in School, November 1987 47

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Let the solution be, "No there are not. Each number is double the previous number, so the factors of every number in the sequence are several 2s and one factor of 3. So none of the numbers can be a square because the factors of a square always come in pairs."

Then I will confidently assert that that so- lution is a proof by any reasonable criterion, which happens to have nothing to do with geometry, and is a proof moreover which might well be produced in this or some equivalent form by a pupil.

I emphasise the reference to geometry because a recent publication, School Math- ematics in the 1990s (G. Howson and B. Wilson, CUP 1986) suggests that if geometry receives any less emphasis in school syllabuses then "proof at school level will be severely restricted", and continues, "This should mean a recasting of courses at university level - not merely complaints that 'students come to us with no understanding of the nature of proof'."

I suggest it is an illusion of, perhaps, univers- ity mathematicians, that pupils cannot and do not prove conclusions in their everyday math- ematical activity, and that there is no reason at all why all pupils - those few who go on to university and the many who do not - should not perfectly well appreciate the nature of mathematical proof through their own problem-solving activity.

When appropriate reasonings are recognised as proofs by teachers, it will not be necessary to introduce or re-introduce proof into the math- ematics classroom; provided the pupils are mathematically active, proof will be discovered to be already present in their work, requiring only to be dignified with a title, given its due significance and developed as an essential as- pect of doing mathematics.

This can only be to the great benefit of pupils, since it is in proof, in the achievement of personal conviction, clarity of understanding, and active possession of the ideas involved, that so much of the "buzz" of mathematics, so much of its beauty and elegance and satisfaction, resides. David Wells London

A Step in the Right Direction

Dear Sir and Madam, I read with interest your editorial in the March 1987 edition of Mathematics in School.

As pioneers in the field of inter-active video for upgrading of Black teachers in Southern Africa, we have experienced at first hand the problems faced by inadequately trained teachers in poorly equipped classrooms. The following paragraphs illustrate the salient points of our first two years' experience.

Soshanguve is a small township 10 km north of Pretoria. Its most notable feature is an in-service training college for Black teachers from all over Southern Africa. The centre is run by the Depart- ment of Education and Training, which controls 80% of all Black education in South Africa.

In-service training is given in all school sub- jects by yearly 3-week courses in the teacher's subject. These courses take the form of 5 hours of lecturing per day. In January 1986 an interac- tive computer-based video system was installed to do all Maths teacher training at the centre. The system operates on laser disc technology and each student works at an individualised work station. The courseware, secondary level mathematics, has all been written locally and the system's unique feature is its video component, which allows it to actually teach new material.

Initially, the teachers were to be given a brief course in the Maths content of the standard they were presently teaching, this to be followed by a course in the didactics of the subject. It was found, though, that the content level of the incoming teachers was at an average of 34% in standards they were presently teaching - the didactics portion was therefore shelved and the three-week training devoted solely to improve- ment of content.

The success of the courseware and system has been nothing less than astonishing. Teachers in the course improve by an average of 20% indiv- idually - for some this represents more than a 100% gain. The most notable to date has been from 6% on the pre-course test to 76% on the post-course test! What has been equally remar- kable is the subjects' change in attitude over the 21 days. Teachers entering the in-service college are drawn from all over South Africa and are not always amenable to being away from home for 3 weeks and not receiving any payment or recog- nition for course attendance. During the course they develop a strong affinity with their work station, their courseware discs, and are reluctant to take even tea or lunch breaks!

The courseware has been designed as Criterion Referenced Instruction, with the student's performance at the work station being continuously evaluated. The better the teacher's performance, the more difficult the examples that follow.

The most enlightening aspect of Soshanguve has not been the turnaround in the teachers' performance and attitudes; it has been the possi- bilities arising from computer-based education when applied to Black education. In the South- ern African context, several generations of Black students have now missed the last years of secondary school. The possibility of a computer-based self-paced education system that can accommodate these disparate back- grounds and differing abilities provides a so- lution that conventional teaching is not capable of.

Courseware is now being written above and below secondary school level and systems are planned for suburban learning centres.

It may be that in the not too distant future, Black Maths teachers may be afforded the opportunity of returning to their roles as educators rather than high pressure dissemi- nators of facts.

Lindsay C. Vieira Education Division, Learning Technologies South Africa

Mathematics

inschool

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48 Mathematics in School, November 1987

Secretary of State's National Curriculum Statement Dear Sir and Madam, I am responding to Maths View, Mathematics in School, 16 (3).

+4 "The level of attainment to be aimed at and the content of what is taught should reflect the best practice of our good schools." If this is taken literally, a noble "aim" may result in many school/pupils falling short of the unattainable (for them).

+7 The structure of this paragraph is appar- ently logical, but must be recognised as idealistic. The logical chain could easily be broken at several of the links. Desirable though these ideals are they may be harder to achieve than this paragraph envisages.

+11 What is the point of setting "clear and challenging attainment targets ... allowing for variations in ability" if no sanctions are applied to the pupils/schools when these are not attained? And if there are to be sanctions, what good will it be to say that "they should not result in an unduly narrow approach to teaching and leaving"?

+13 Although it may involve an additional bur- den for teachers it is desirable that assess- ment of work done in class at the key ages should be added to tests of attainment related to the targets.

+15 This paragraph is bold and decisive, but it does not follow that the desirable edu- cational aims concerned can be secured by legislation, even following thorough pro- fessional discussion.

A. R. Nicholson Queens University, Belfast

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