About the Book
Secret codes are usually employed when someone wants to make things less clear. The intention in this book is to use codes to make the mathematics clearer. The book includes photocopiable activities, plus hints, prompts, and suggestions for further activities. Possible problem-solving strategies are outlined and indicate some of the ways in which the decoding tasks could be followed with other mathematical work. Suitable for KS 2, 3 & 4
Review :
Once in a while something really excellent appears by way of mathematical problems for pupils, and when one meets something like this publication one wonders why the majority of texts and topic books, with their routine and mundane examples, ever get published. The Introduction carefully indicates that the normal curriculum for KS2 or KS3 provides any mathematical knowledge required to solve the problems in this book, but that some of them need a high level of skill in problem solving. This somewhat underplayed statement perhaps sums up what the problems represent: a philosophy that maintains that - with the almost forgotten Cockcroft Report of 1982 - "The ability to solve problems is at the heart of mathematics". One feels that this message is sometimes lost in the current concern for skills and techniques. The skills and techniques are important, but instead of routine practice and application they are here applied to genuine thinking situations that will enhance and deepen understanding of them. Take one of the simpler 'cross numbers', a numerical version of crosswords that is often used merely as practice for computation (see Fig 1). Across: 1 Square number 3 Square number Down: 1 Square number 2 Prime number Here one really has to know something about properties of squares and primes, as well as have the ability to plan a way through the problem. Or, consider finding a Fibonacci-type sequence in which you know that the first four numbers each have one digit, the next four have two digits, and the following two have three digits. The ideas of coding and decoding are taken broadly. There are a couple of substitution codes, but the second has no spaces between the words, though it does include the punctuation. Otherwise, letters stand for digits and boxes stand for digits or letters in a wide variety of topics in which one has to complete sequences, identify jumbled multiplication tables or sets of multiples, solve some unusual equations, or supply missing digits in computations with fractions. 'Hidden text' problems supply statements about time, solid shapes, areas and volumes, quadrilaterals and divisibility tests in which most or all of the words are missing and the only clues are that one knows the lengths of them. The SLIMWAM2 ATM program Counter is adapted to identifying cyclic sequences of units digits in which one only knows whether each term is higher or lower than the preceding! Equiangular and equilateral polygons are coded according to the cycle of sides or angles respectively, and one is invited to identify possibilities. Finally, the removal of squares from rectangles is coded according to the number of 'horizontal' or 'vertical' cuts needed before a square is left; this idea is extended to rectangles where the ratio of the sides is irrational and an infinite cycle of cuts is needed: can you identify the ratio represented by VHVHVH...? A refreshing idea is that practically every page ends with an invitation to make up some similar problems and give them to someone else, with an occasional warning about thinking carefully so as not to make them too difficult, or to decide whether or not there will be a unique solution - since in fact several pages warn that some problems may have more than one solution, but of course do not tell you which ones. There are tacit assumptions of principle here. One is that creating problems is a worthwhile learning process. Another is that pupils should become accustomed to problems which have several solutions, and where this is so there is the additional problem of ensuring that one has them all - a feature recognised but hidden in 'Hints and prompts with suggestions for further activities'. This last section replaces the conventional answers, which are not given because "you know when you have cracked the code" - another refreshing idea! It is addressed to the teacher rather than to the pupil (the problem pages are allowed to be photocopied for them), and presumably supplies the hints so that the teacher can make decisions about how, or whether, to pass them on. The further suggestions contain yet more ideas, just in case one uses up the initial rich source of material. David Fielker is retired. This book contains many different activities using codes to give students mathematically challenging activities. The book begins with some multiplication tables that have been coded using a simple substitution code - i.e. each digit has been coded with a unique letter. The tables are not written in order. The students need to work out which letters stand for which digit. This activity would represent an interesting challenge across many different year groups. I expect that a bright Y6 student would cope well, whilst a Y10 student is probably not so familiar with their times tables and could easily find it too hard! The book continues with some deceptively simple looking cross numbers (with clues such as 'prime number' and 'multiple of 4'). There are some more recognisable coding activities - pieces of text that have been coded and the reader is invited to decode. There were a few pages with the title of hidden text where the reader needs to find a missing letter for each box on the page (to decode sentences such as there are 60 seconds in 1 minute). I am not quite so convinced of the merit of this section. The activities owe more to literacy than mathematics, but perhaps this is an area we ought to be making more of? The final few problems relate to the ATM program Counter which I have not seen, but the instructions are clear enough and the problems are sufficiently interesting without the program. Finally the book concludes with some hints for each page of problems. These hints will be sufficient for any student or teacher to be able to solve the problem. All in all this is an excellent publication and I cannot recommend it strongly enough. I am waiting for the right moment to use it this year - whether with my lower set in Y10 to help us through a long double lesson, or with a Y6 master class session next term. Peter Hall, Head of Mathematics, Tonbridge Grammar School, Kent So who's the author here? No mention of a name on the front. On the inside page? By Mcycb Xrii - never heard of him. Some impenetrable Hungarian mathematician? But no...the decoding here starts on the cover. Mcycb Xrii unscrambles to ATM's very own Derek Ball. (Decoding and Derek begin the same way...) Derek adopts a wide vision of codes, noting that his task here is the reverse of their usual use, which is to introduce a lack of clarity - he is attempting to employ codes to make the mathematics they code clearer to students. And of course, both cracking a code and composing a code can be great fun. So, to real students with this material. I tried this example very happily with a new AS class: _ _ 2 - _ _ 2 = 280 However often I go over the difference of two squares, there are always some students who never quite take it on board. This exercise really does develop an understanding of the factorisation involved, and it brings simultaneous equations in too. Excellent. Then to a carousel of these activities for my retake Intermediate GCSE class. Not the easiest of groups to win over, yet for the most part they were hooked. A multiplication table with coded digits drew out some fine logical thinking, as did the arithmetic sequences with missing digits. The cross-number puzzles really testing their understanding of vocabulary like prime, multiple and cube, providing a powerful motivation to learn these. Some comments were revealing: "R times D is MH, so radius times diameter is..." The substitution code problems here are perhaps better approached with Simon Singh's code-breaking CD - with this the students can watch letters being substituted on the computer immediately, without to much drudgery on their part. I should say that the hidden text puzzles in Decoding Mathematics were the least popular: if you got stuck, there was no real logic that you could employ to get any further. And throughout, if a trio of code came close to spelling something rude, then that inevitably rather overtook the maths involved! There is far more in Derek's book than I have covered here: geometry, fractions, even music. It contains an exciting mix of problems that I shall use again and again. A winner. Jonny Griffiths, Paston College, North Walsham, Norfolk.
