Depending on the ContextAuthor(s): Peter JohnstonSource: Mathematics in School, Vol. 13, No. 2 (Mar., 1984), pp. 35-36Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216211 .

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Book Reviews

Activities for Primary Mathematics by Peter Smith MacMillan Education, 50 pp, paper a1.95 ISBN 0 333 30694 5

This book is a compilation of forty activities suitable for reinforcing a variety of mathematical ideas in the classroom. Some of the activities require apparatus, usually cards, to be made but nothing very time consuming or elaborate is needed.

Concepts included range from number work and place value to domino games, magic squares, co-ordinates and activities to en- courage strategies and logical thinking. The book is a useful source of ideas for encouraging the enjoyment of maths and at the same time it covers aspects of the curriculum which are central to the subject.

The publishers have graded the activities into groups for which they consider them appro- priate, early 5 to 7 years, middle 6 to 8 years, advanced 8 to 10+ and remedial.

This book would be a useful addition to the maths section of any staffroom library.

LESLEY JONES

Confident Mathematics Teaching 5-13 by Edith Biggs NFER, ISBN P7005 0581-4, paper a4.95

This book is the report of a long term in-service project involving teachers and children from 12 first and middle schools. It records the teachers' own attitude to mathematics, how this affects their teaching methods and how the in-service work changed both their approach and the children's responses. The title might mislead teachers into expecting specific advice on how to acquire confidence.

The book provides a frank account of the "ups and downs" of a research worker and makes lively, interesting reading with the author's ebullience and optimism shining through. The results of action research do not lend themselves easily to generalisations and Dr Biggs has con- fined herself to subdued optimism in interpreting her conclusions. In some of the middle schools and one first school the results showed a favourable change in the teachers' attitudes and methods; this change was reflected by the quality of response from the pupils.

The message to all teachers is "Do ... talk ... record and use an encouraging manner with the children ..." but, simple as this message may seem, Dr Biggs concludes that "at least three years are needed for changes in teaching style to be effected and that is in the schools receiving support and input". Teachers and co-ordinators have to be convinced that the provision of activities and the creation of opportunities for discussion about these activities are vital to the successful teaching of mathematics. These conclusions have far reaching implications for the education of teachers and for those re- sponsible for the organisation of school mathematics.

I was disappointed in not finding references to Dr Biggs' actual work with the children except that "it was set out elsewhere". However, the book should prove of great interest to all con- cerned with the initial and in-service training of teachers.

J U LIA MATTHEWS

Understanding Mechanics by A. J. Sadler and D. W. S. Thorning

Oxford University Press, 354 pp and index paper a5.00

After seeing the television caption, "Live from the Moon", life for the teacher of Mechanics can never be quite the same again. The subject has indeed taken to the skies and become "celestial mechanics" when the American astronauts thoughtfully performed Newton's "guinea and feather" experiment on the Moon for all Earth's children to see. "Weightlessness" has become part of our vocabulary. In 1929, in one of its earliest teaching Reports, the Mathematical Association pointed out that mechanics is a subject which, more than most, can give reality to Mathematics by showing how it may be applied - words repeated in the Second Report in 1965. Furthermore, "There is perhaps no branch of mathematical instruction for which a pupil comes prepared with a larger body of intuitional knowledge than he does for mech- anics". For the mechanics teacher, swings and roundabouts really are grist to the mill, whilst statistics, the other field of school applied math- ematics, often seems not to venture beyond coins and dice. But it is also true that "you may watch falling bodies for an eternity yet without mathematics there would be no law of gravita- tion". Experiment has its limitations - "pupils need to be led on to ever smoother surfaces to envisage the abstract limiting conception of a body moving on a perfectly smooth horizontal plane". Theoretical mechanics has held an honoured place in British schools - and it complements the attractive experimental work taking place in Physics lessons. If it is to be taught- and learnt - well, it demands much of the teacher who must guide the beginner to grasp ideas which human thought acquired only slowly and painfully.

The authors of this book are concerned, "to present topics in as simple and direct a manner as possible and to give ample opportunity for practice". Care is taken to give plenty of simple and carefully graded examples and the need for clear diagrams is well illustrated. (Diagrams illustrating problems on colliding spheres con- tain a total of 182 spheres, surely a record for the course?)

There are eighteen chapters covering the usual statics and dynamics of a particle for "single subject" mechanics and at the end of each chapter except the first on "Vectors", there is a group of questions from the 'A' level papers of various Boards. Answers to examples are given. The book is attractively printed and produced.

Motion in a vertical circle is not discussed and only three examination questions on the topic appear. This is uncharacteristic of the book. There are twenty-five pages on resultant and relative velocity and twenty-seven pages on centres of gravity. The existence of a centre of parallel forces is not discussed in the section on the centre of gravity of a rigid body and there are points in the book where the teacher will wish to supplement a rather brisk treatment in the text. It is worth five minutes of anyone's time to point out that the Polygon of Forces is true in three dimensions. The treatment of vectors is a case in point. If vector notation is to be introduced, then at this level more use should be made of it. The book does not go beyond i and j, addition and subtraction in two dimensions. Scalar product is not introduced and the extension to k showing vectors at work in three dimensions is not made. On p. 292, the derivative of a vector is merely asserted and the thoughtful pupil who raises his head above the parapet to enquire how this exciting idea of applying the calculus to vector quantities might be justified, is left unanswered. Why not establish the idea of the derivative of a vector and put it to work to obtain the acceler- ation components in circular motion?

There is a dangerous moment on p. 213 when

solving projectile problems by substituting in standard formulae seems to be implicitly en- couraged but the teacher will avert this and typically the book is plentifully supplied with worked examples solved from first principles, very supportive for beginners.

Beginnings, the way a thing is first done, are, as every teacher knows, of major importance. The human race had to do some unlearning of wrong conceptions in mechanics and it is notorious that too long an apprenticeship with uniform acceleration formulae leads to attempts to solve SHM problems using v= u + at. Can we not improve on the trite introduction to Newton's 2nd Law as F= ma? To express it as rate of change of momentum being proportional

d to the impressed force - (m v) oc F does at least

dt allow for the possibility that m can be variable as in the case of the rocket - or even the falling raindrop. Then there is nothing to unlearn later.

The authors reach variable acceleration and variable force in chapters 16 and 17 of this eighteen-chapter book, but there is no reason why pupils who have done calculus cannot tackle them earlier - and have longer to get used to the handling of simple harmonic motion. A question on the rise and fall of the tide just gets into the SHM section so the Moon, whose motion Newton complained gave him a head- ache, just appears, albeit anonymously.

It is not only elastic strings which "go slack under compression" (p. 318) - so do pupils learning mechanics. This book sets out to avoid compression, to maintain a gentle but deliberate pace and to achieve understanding and con- fidence by building up a picture of the wood going from tree to tree. In the hands of a perceptive teacher who can use the book as a stimulus to curiosity, it will help another gener- ation of pupils to understand ideas about matter and motion, which as curricular fashions come and go, remain in the nature of things. BRIAN COOPER

Letters to the Editors

Shaking a Six Dear Sirs, When running a tuck shop I found it wise to put the "booze" I was running low on at the left- hand end of the row of samples, and the variety I had most of in third place.

Is this relevant to the interpretation of Table 1? G. R. Addington Hall Packenham Bury St Edmunds

Depending on the Context Dear Sirs, In the children's comic Whoopee there is a "Calculator Corner" where a puzzle is set. Often this takes the following form: a verbal question is set in which the child is asked to do an arithme- tic calculation on his/her calculator, and then find the verbal answer by turning the calculator upside-down and reading the display as a word (or words). For example: "Which group once had hits with "Tragedy" and "Massachusetts"? Work out the sum 2672 x 999+ 341 x 4- 2000=

If the child has the cheaper type of calculator ("arithmetic" in the terminology of David Johnson's article "Let's use more appropriate descriptors", Mathematics in School, Vol. 9, No. 4), which again executes operations in the sequence in which they are entered, then he obtains the intended answer of 5339.338 and hence BEE GEES. However, he will not obtain it

Mathematics in School, March 1984 35

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if he has the more expensive "algebraic" type (again using Johnson's terminology). In this case the above sequence of key-strokes will produce instead the value

(341 x4\ (2672 x 999)+ (

2

4

2000 / The use of such an example from a comic with

the two types of calculator can help enliven a class discussion of the use of brackets in ex- pressions such as the above.

Also, further to the examples I give in "Depending on the context" (Mathematics in School, Vol. 12, No. 2), the above provides an- other instance of a situation where a child experiences, outside the mathematics class- room, the use of mathematical notation in a potentially ambiguous way. In this case, when taken in the context of using a calculator, the ambiguity is assumed to be removed; but, of course, as pointed out above, this can lead to an interpretation other than that intended.

Now that many children experience the use of algebraic notation in computer programming, but with interpretations sometimes differing from those traditionally used in the mathematics classroom (see e.g. David Tall's article: "Introducing Algebra on the Computer: Today and Tomorrow", Mathematics in School, Vol. 12, No. 5), this issue of dependence on the context is likely to become more significant, and is one of which we should perhaps be more aware. Peter Johnston Department of Mathematics University of Essex

Mathematical Association of Tanzania

Dear Sirs, I would like to mention that several people in Tanzania have shown interest in reading the Mathematics in School journal. Some articles have proved quite interesting among mathemat- ical educators.

On reading Vol. 11 No. 5, I noticed the announcement of the Mathematics Association about the mathematics posters and puzzle cards. I made arrangements to get them ordered through the British Council. The British Council generously presented them to me. After looking at them and admiring their construction I dis- played them to delegates attending the 1983 annual general meeting of the Mathematics Association of Tanzania. The puzzles appeared very interesting and quite a number of the delegates tried to solve some of the puzzle cards. At the end of the meeting several members asked for the address of the Mathematics Association so that they could order some more puzzle cards.

I would like to thank the Mathematics Associ- ation for their effort.

B. R. Seka Senior Curriculum Developer Dar-es-Salaam

Quantitative and Formal Methods

Dear Sirs, I refer to the article by Dietmar Kuchemann in Number 5 of Volume 12. Slight unease with the emphasis of the article was crystallised when I came to the sentence "If children are to be taught a formal method, it is important to show that their own methods are inadequate." This seems to me to imply a misunderstanding of the relationship between what he calls quantitative and formal methods. I would say that the chil- dren need to be conversant with "their own methods" to the point at which they recognise a formal statement for what it is, a way of record- ing part of the thinking process concisely.

Formal methods are no more than a shorthand statement of the obvious; the problem with teaching formal methods arises when the pro- cess is not obvious to the people concerned.

This is essentially the same problem as is met when attempting to use decimal notation before place value is properly understood, e.g. speaking formally of "borrowing" (when subtracting) to someone who has not personally seen the need for and meaning of "borrowing".

Less drastically, I take issue with the author in his assertion that item 7 cannot be dealt with quantitatively without "co-ordination of a great deal of information". If we have a situation in which the participants are able to cope with 3e + 5 = 14, say, then "What do we add to 5 to obtain 14?" is presumably an acceptable ques- tion, in the sense that it will lead to solution rather than confusion. It should not be too difficult to lead on, then, to 3e+5=e+13, where an appropriate question is "What do we add to 5+e to obtain 13 + e?" The difficulty is certainly greater, but not significantly so - unless, indeed, the participants have notreached the pre-requisite stage; and there, surely, is the rub?

Bruce Andrews Ipswich School

Dear Sirs, Last November, travelling back from a margin- ally useful Conference in London, I resolved to produce something in the three hour rail journey. The resulting lesson goes like this:-

The pupils are instructed to head their pages like this:

Q A

a. -

b. =

c.-

d.

e.

I then ask: "What do 'Q' and 'A' stand for?" This is a non-mathematical question and is very easy. It gives me the chance to be complimentary right from the start.

"Now, under the 'A', please write the four Answers: 4, 5, 6 and 7".

"Now tick them, because I know they are right - put your score at the bottom - 4 out of 4."

"Now, you fill in the questions!" There is now scope for the teacher, who knows his class, to dish out praise, very few will need actual help.

Then: "I will now read out the next four Answers: they are, 4, 5, 6 and 7! but this time you may not use the same numbers twice in the set of four questions." There is considerable scope for variety. There may be an opportunity to discuss with some the difference between "numbers" and "figures". "Please Sir, may I use fractions?" "Have I used the 2 twice in 24?"

Game 3 - (I don't call it a game at the start. I wish to establish that we are doing serious mathematics).

"Same four answers, but you may not use the same sign twice."

Game 4- "This time you get bonus points for using more than one sign to a question."

Game 5 - "Now you make up the rules", etc.

Perhaps restricting the domain to.. Integers 0-9, .. only even numbers, .. odd numbers .. prime numbers etc. .. with or without the rules of games 2 to 4.

I like my new game, not only does it permit that rare thing in school mathematics - indi- vidual variety, but it also allows quite ordinary children to glimpse at a fundamental activity in mathematics - playing with the "rules".

A. G. Foot, Education Inspector (Mathematics) Lincolnshire

MATHEMATICS

IN SCHOOL

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36 Mathematics in School, March 1984

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