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'I' IS for INDUCTION © ATM 2007 ƕ No reproduction except for legitimate academic purposes ƕ [email protected] for permissions 'i' IS FOR INDUCTION Tim Rowland I' ve recently been thinking about different kinds of probably little interest. Another factor, I suggest, is problem-solving activities, how students have whether you feel you have a secure analytical responded to them and what they have learned from overview of the situation presented in the problem. engaging with them. I'll begin by asking you to take By that, I mean whether you feel able to approach it five minutes to consider, if you will, hawyou mightgo with some general case in mind. about tackling each of the following problems. (Feel That's how I described my initial reaction to free to put the rest of my article aside, if any of them Painted cube in MT143 [1]. I feel the same about really interests you). Boundary. I can 'see' a general square. I 'count' the 1 Boundary: An integer-sided square is subdi- non-corner boundary unit squares ('side minus 2',4 vided into a grid of unit squares. H ow many unit times) and add th e 4 corner squares. Anoth er person squares lie on the boundary? might prefer to collect some numerical data on the 2 Diagonal: How many unit squares lie on the boundaries of particular square. Example -a4-by-4 diagonals of the above square grid? square has 12 unit squares on the boundary. Try it 3 Fifteen: Choose a positive whole number and with some children in your class . What do th ey write down all ofits factors, including the chosen choose to do ?Was that what you expected? M ore number and 1. N ow add all the digits of those difficult - how will you constrain their choices(or factors(th at sum would be 5 if you had chosen try to avoid constraining them) by the way you 13). Repeat the whole process on the new present the problem, the materials you offer them, number. Keep going. Try different start ing the recording m ethods you suggest (or do not), or numbers. the prior judgements you make about how they are 4 Stairs: In how many different ways can you likely to approach the problem? ascend a flight of stairs in ones and twos? I have included Polygram for my own benefit. I 5 Partitions: The number 3 can be 'partitioned' invented the problem (without claim to originality) into an ordered sum of (one or more) positive but have not yet worked on it, and so I can think numbers in the following four ways: 3, 2 + 1, aloud as I write. I have no instant sense of how it 1+ 2, 1+ 1+ 1. In how many ways can other might develop. I know that for a regular pentagram, positive numbers be partitioned? the angles add to 180°, but that's all. Well, not quite. 6 Sums ofsquares : In how many different ways I suspect th at the angles of every pentagram will have can a prime number be written as a sum of two the same sum. I know that a hexagram consists of square numbers? What about non-primes? two distinct triangles,and so the angle sum will be 7 Polygrarn: A polygram is constructed by 360°. And so, almost despite myself, I have one joining alternate vertices of a polygon with conjecture about polygrams in general. But, unlike straight lines (the best-known example is a with Boundary, Ihad to examine some data, some pentagram). At each vertex of the polygram, an particular cases, to gain a sense of what might be the internal angle is formed between adjacent edges. case for Qlty polygram. What is the sum of the se angles ofa pol ygram ? For me, Diagonal (which I also devised as I 8 Painted cube: A cube is made up from lots of started to write) lies somewhere between Boundary little cubes, and the surface is painted red . How and Pclygram. I anti cipate that the 'rules' for even- many little cubes have three painted faces? Two and odd-sided squares will be slightly different, fac es? One? None? be cause only the latter have a 'middle' unit square Your response to each problem, your heuristic on both diagonals. Ah ... it wasn't as obvious as for solving it, will depend on a number of factors. Boundary, but Ihave a growing sense of the general One of these will be whether you have encountered case . and worked on the same problem before. If you I doubt wheth er anyone who has not worked on have, it may offer little challenge to you and (or read about) Sums of squares would have a feel for MT167 JUNE 1999 23 © ATM 2007 ƕ No reproduction except for legitimate academic purposes ƕ [email protected] for permissions what it's about with out generating some data, such 'processes of th e same mind' . Without induction as 53=2 2 +72 (uni que ap art from order). T he there is nothing tojustify by deduction; but it is the number 59 cannot be so expressed, wh ereas business of ded uction, writes Whewell, to 'establish 2 2 2 2 65= 1 + 8 or 4 + 7 • th e solidity ofher companion's footing'. ' Why should we want to draw attent ion to Induction induction by naming it in the classroom? Because to celebrate induction is to highlight the hu manity of Some, but not all, of the problems listed above seem mathematics and the character of mathematical to invite the sear ch for regularity among examples, in invention. I call two witnesses in support of this order to make predictions about other particular claim. cases (such as the sum of th e angles of a l O-g ra rn) and to arrive at conjectur es of a more general kind. Ana lysis and natu ral ph ilosophy owe their T he word 'conjecture' captures the idea that mo st important discoveries to this fruitful knowledge of the general case, or any case beyond means, which is called induction. N ewton th edata in hand, is for the moment provisional, was indebted to it for his theoremsof the tentative. T he process of arriving at such a con jec- binomial and the principle of universal tur e from a finite data-set is induction (or inductive gravity [4, pI76] . inference), with asmall 'i'.Unfortun ately, mathe- The purpose of rigour is to legitimate the maticians are conditioned to associate 'induction' conquests of the intuition. (H adamard, with Math ematical Induction (with a capital'!'), qu oted by Burn [5, p l j ). which is a schema for a particular kind of proof about propositions to do with all natural nu mbers. On a personal note, I would add th at some of the I r 'call, at school, summing cubes of integers to work that hasgiven me the greatest pleasure to write observe that each partial sumwas thesquare of a (and thanks frequent ly to MT) to publish, has triangul ar nu m ber as a prelude to Proof by consisted of account s of the indu ctive muse at work M athematical Induction. in myself.' T he consequencehas been a powerful desire tooffer opport unities to thestudents I have 3 3 3 2 P + 2 = 3" 1' +2 + 3 = 6\ 1'+2' + 3'+4'=10 • •• taught in school and at university, to experience th e T he proof-process (Induction) was carefully same delight and sense of mathematical 'one-ness' . named, but not th e process (indu ction) of arriving at the statement to be proved. I am sure little has Proof and naive empiricism changed. Given the current concern for 'proper' m ath ematical vocabulary [2], perhaps thetime has Lest I be misun derstood, it is import ant to remark come to introduce 'inducti on ' into the language that there is m ore to mathematics than induction . (and practice!) of the mathematics classroom. Believing is not the same as knowing. Indeed, Iamspeaking of induction here as a scientist induction carries th e danger of premature conviction. wo uld, in relation to discovery or invention . Recently, I worked on Partitions with some Inductive reasoning takesthe thinkerbeyond the primary PGCE students. Thestudents accumulated evidence, by somehow discovering(by generalisa- data about partitions of 2, 3 and 4, and observed a tion) someadditional knowledge inside themselves. do ubling pattern . Some chec ked that it extended to T hemechanism which enable s an individu al to partitions of 5. They were finished! Those who arrive at plausible, if un cer tain, belief about a whole formulated th is as p en) = 2n-1 were more than population , an infinite set, from actual knowledge of finished, a nd well satisfied. "But how do you know", afew items from the set, is mysterious. T he nine- I asked, "that it will continue to do uble eoery tim e?" teenth-centuryscientist William Whewell captures T he typical responsewas along the line, "Well, it has the wonder of it all: up to now, so it seems reasonable to sup pose that it Induction mo ves upward, and deduction always will ", T his is naive em piricism at work. As downwards, on the same stair [ . .]. Reuben H ersh has rema rked, Deduction descends steadilyand methodi- In the classroom, convincing is no problem. cally, step bystep: Induction mounts bya Studentsare all too easily conv inced. Two leap which is out of the reach ofmeth od. She special cases will do it [7, p396] .
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