# Exam-based orientations

STATISTICS. By Fiona McGill, Stewart McLennan, Jane Migliorini

MECHANICS. By Martin Adams, June Haighton, Jeff Trim

Stanley Thornes pound;18 each

INTRODUCING MECHANICS. By Brian Jefferson and Tony Beadsworth. Oxford University Press 18.50.

ADVANCING MATHEMATICS FOR AQA: DISCRETE MATHEMATICS 1. By Victor Bryant.

STATISTICS 1. By Roger Williamson, Gill Buque and, Jim Miller, Chris Worth

MECHANICS 1. By Ted Graham Heinemann pound;8.99 each.

HEINEMANN MODULAR MATHEMATICS FOR EDEXCEL AS AND A-level (New Editions): MECHANICS 1. By John Hebborn, Jean Littlewood, Fred Norton

STATISTICS 1. By Greg Attwood, Gill Dyer,, Gordon Skipworth

PURE MATHEMATICS 1. By Geoff Mannall and Michael Kenwood. Heinemann pound;8.25 each

CAMBRIDGE ADVANCED LEVEL MATHEMATICS (OCR): PURE MATHEMATICS 1 amp; 2. By Hugh Neill and Douglas Quadling pound;9.95

PURE MATHEMATICS 3. By Hugh Neill and Douglas Quadling pound;7.50.

MECHANICS 1. By Douglas Quadling pound;7.50

STATISTICS 1. By Steve Dobbs and Jane Miller pound;7.50

DISCRETE MATHEMATICS 1. By Stan Dolan pound;9.95. Cambridge University Press

I imagine publishing houses cracked open a few bottles of champagne when Curriculum 2000 was announced. With most syllabuses suffering major changes, a whole range of new textbooks was required. Many have arrangements with exam boards to produce more or less "official" texts to accompany the various courses.

Of the books reviewed here, only those from Oxford and Stanley Thornes lack an exam-board tie-in. Heinemann has grabbed a major slice of the market with a new scheme for AQA Syllabus B and a revision of its existing Edexcel scheme. Cambridge also has two schemes: the OCR series reviewed here and the AQA Syllabus A scheme that replaces SMP 16-19. MEI, the one other syllabus, also has its own range of books.

The "exam-board" texts, which are mostly written by senior examiners, will undoubtedly sell strongly. The implied guarantee is that they will closely match both the syllabus and the style of the examination. However, I am not sure that this is entirely to be welcomed when the new modular pattern A-level maths is already in danger of degenerating into five terms of exam cramming, rather than a satisfactory maths education.

In this comparative review of the main features of the books it will be useful to bear in mind what is known about effective teaching. In the King's College report Effective Teachers of Numeracy (1997), Mike Askew and colleagues described three orientations towards teaching maths: connectionist, transmission or discovery oriented.

Though they only looked at primary teachers, there is every reason to believe that their findings apply at all levels. The research demonstrated that the most effective teachers were connectionists. These teachers see maths as a vast network of ideas. They believe that students "learn by being challenged and struggling to overcome difficulties". In contrast, transmission-oriented teachers typically teach topics in isolation, believing students "learn through being introduced to one problem at a time and remembering it", while discovery teachers believe that students' own strategies are most important.

The Stanley Thornes series is predominantly transmission-oriented with many worked examples. On the positive side, weaker students will find comfort in the careful descriptions of what to do. However, there is little wider discussion. This is most noticeable in Statistics, where we find examples masquerading as definitions, and pseudo-definitions such as "a random variable is a numerical variate whose alue depends on chance", which obscure the true nature of the concept. Pure Mathematics is better, but still contains unsatisfactory features. For example, any definition of a tangent has to be preceded by a satisfactory definition of gradient at a point on a curve. This subtle point is the real reason for defining gradient via limiting slopes of chords, not the spurious ones given.

In Mechanics, assumptions appear to be made quite arbitrarily: when air resistance is ignored in a projectile problem there is no attempt to consider in which direction the solution might be affected.

OUP's Introducing Mechanics is also mainly transmission-oriented, but with good introductory discussions to each chapter. It engenders an image of mechanics as a coherent whole, unlike some of its modular rivals. The coverage is comprehensive, with a useful chapter on dimensional analysis. It also has more on the modelling process and more careful definitions than the book from Thornes. On the other hand, it lacks the summary boxes that are a feature of its rival.

The remaining books are all based on specific exam modules. They include either learning objectives (at the start of a chapter) or key point summaries (at the end). The AQA books have both. Of the Mechanics 1 books, Quadling's (OCR) book is easily the best. He finds room for "connectionist" discussion and substantial sets of exercises. A major difference in style is that Quadling discusses the modelling aspects as he goes, while all his rivals push most of the modelling into a preliminary chapter.

The AQA and Edexcel books have some introductory remarks for each chapter but worked examples dominate the presentation.

The Statistics 1 books all make a reasonable stab at explaining concepts and motivating the methods employed to solve problems. Again the Cambridge text is the best.

The two Discrete Mathematics Books each have encouraging features. Bryant includes historical background and pointers to more advanced ideas. Also, the text is not so dependent on worked examples as the others in the AQA series. The OCR book has some nice sections for more able students that provide justifications of some standard algorithms.

The Cambridge Pure Mathematics books make a careful distinction between rigour and motivational discussion. For example, where the Thornes book stumbles over tangents, Quadling and Neill are sure-footed, revealing the key issue: that one only knows the co-ordinates of a single point on the tangent line where two are needed to establish a gradient. Again, this is a connectionist book: students are guided through exploratory exercises to determine some results for themselves.

The Edexcel Pure Mathematics book from Heinemann also contains more exploratory work than its applied stablemates. It has a slightly unsatisfactory definition of tangent as a line that just touches a curve, but this is better than the Thornes book, which does not explain the idea at all. The Pure books are thicker than those covering applications: perhaps some modules are more equal than others.

Though all three of the "exam-board" series have similar-sized books for comparable modules, the denser print of the Cambridge series makes it a much more substantial work. More importantly, the Cambridge books are mathematically superior and provide better links between topics. An experienced "connectionist" teacher could fill in the gaps in the more transmission-oriented books, but the Cambridge books are the best bet for a teacher who is still developing this vital skill.

Steve Abbott is president of the Mathematical Association, editor of 'Mathematical Gazette' and deputy head of Claydon High School, Suffolk

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