© 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] . bounds to th e top of the stairs at once [. . .]. T he teacher's invitation to scepticism about pattern s [3, p114] seems like rather an empty gesture. M ost of the H ere deduction is portrayed in terms of descent, time, a few special cases do point the way toeternity.' just as a syllogismis presented on the written page - T he point of the teacher's proof-provoking question me thodical, steady, safe, descending. By contrast, is not to achievecertainty - which is already assured induction is framed as daring, creative, ascending. - but to en courag the quest for insight . As H ersh W hewell disc usses the symbiotic relationsh ip says, the primary pur pose of proof in the teaching between induction and deduction . They must be cont ext is to explain, to illuminate why some thing is

24 MT167 JUNE 1999 © ATM 2007 ƕ No reproduction except for legitimate academic purposes ƕ [email protected] for permissions th e case rath er th an to b e assured th at it is the case . The 1997 Ofsted video Teachers count features There is a genuine problem h ere, esp ecially but one tea cher, K ate, with a class of 10- and l l -year- by no m ean s exclusively with younger pupils. So olds. In the middle phase of th e 'Num eracy hour' often, explan ation is of a quite different ord er of lesson, th echildren investigate the Jailer problem in difficulty from inductive conviction. N umber th eory, sm all gro ups, before being brou ght toget her by Kate in parti cular, is so amenable toconjecture yet so (p resuma bly for some directteaching).T he solution resistant to proof. For example (see S lims ofsquares) , tu rns out to hinge on th e fact that every sq uare Fe rmat's proof by descent that prim es of the form number has an odd number of factors. In fact, Kate 4k+ 1 can be uniquely exp ressed as the sum of two expl ains th is to th e class byreference to (what we squares pre sents a ch allengeeven to mathematics rec ognise as) a generic example. She po intsout that undergrad uates, every factor of 36 has a distinct co-factor, with th e At least th e theorem (about expressing primes as exception of 6, an d so itmustfollow that 36 has an sums of squares) is of major sign ificance. This is odd number offactors. She th en generalises, 'One of more than can be said of th e (admittedly cu rious) the factors of a square number is a number times outcome of Fifteen - the conjectu re th at every suc h itself (sic); th at's why it'sasquare number, isn't it?' seq uence arrives and remains fixed at 15 soo ne r or H er choice of 36 is in teresting - small enough to be later. In a recent issue of Equals, Fifteen is accessible with mental arithmetic but with sufficient comm ended as a good Ke y Stage 2who le-class factors to be n on-trivial. N ot surprisingly, no starter [8J. I would agree that it offers amot ivating referen ce is made in thecommentary to this aspect contextfor work on divisorsand gives rise to a nice of her teach ing and proof strategy. tree of sequences. Beyond that, I fecivery uncom- fortable with it, because th ere seems to be little Empirical and structural prospect of eitherpupil or teache r being ab le to 4 exp lain why each sequence should incl ude 15 • T here generalisation is a plethora ofsuch'cha in'investigations, such as Inductive inference has to start with examples, but if H appy and sad num bers, which are to do with investigatio n finishes with 'spo tting a patt ern' (or summing th e sq uare s of th e digits ofan integer, and even stating a for mula) it rem ains at the level of iterating on that sum.' Fo rme, Fifteen is uncomfort- naive empiricism .As my students said, '\XTe ll, it has ably re m inisce nt of th e n otorious T hw aites' up to no w, so it seem s rea sonable tosuppose th at it C onjecture,' a dead-end for the classroom if ever always will' . Liz Bills and Ihave tried to distingu ish there was one. between two kinds of generalisation .' I am appealing for teachers, in choosing problem • Empiri cal- that which 'merely' gene ralises fro m starters, to have a sense of wheth er inductive conj ec- tabulated numerical data ture isalmostcertain to be a terminus for th e •Structural- deriving from an overview of the investigation. Is that wh at th ey want? Every time? situation from which the data arises. What message will th at give about the nature of ...We emphasise that on e form of generalisation mathematics, apart (hope fully) from being good is achi eved by considering th eform of resu lts, fun? W hat price the 'Ah a!' of insight, of rational , whilst theother is mad e by looki ng at th e und er- connected mathematical knowledge? lying m eanings, stru ctures or pro cedures [12]. Ide ally, we m ight hop e for pupils to generate Empirical gene ralisation may in tim e become th eirown expla nat ions. M ore often, we might structural; if kn owing that becomes knowing why, attempt to point th em towardsways of perceiving when an exp lan ation for wh at is obse rved becomes th e probl em situa tions th at have the potential to available. Empirical generalisation is authentic, prompt explan atory insight. I b elieve that the important but in complete, mathematic s. M y 'generic exam ple' canplaya cr ucial role in th is latter opening problem set was chosen to exemplify some res pect. T h e generic examp le is a confirming situations wh ich seem to require da ta to be collected instance of aproposition, carefully presented so as to (and typically tabulated) b efore any progress provideinsi ght as to why th e prop osition holds true towards empirical gen eralisatio n is possible, as well for that single instance. as others where structural generalisation is an T he generi c example involves m aking expli cit immediate pr ospect .The response (em pirical or th e reasons for the truth of an assertion by structural) is in part a fun ction of th estu dent an d me an s of op erati ons ortran sformations on an th e cla ssroom enviro nment determined by the ob ject tha t is not th ere in its own right,but as teacher, not solely of the problem in isolation. a characteristic representative of the class Again, I suggest that it is important for teachers [10,p219]. to have a sens e of how different individ ualswill I have argued elsewhere [Ill for th e unrealised respond to agiven starting point - empirically or pedagogic potential of gene ric examples, an dwill structurally. At different times, both are desirable. If settle for one topical illustrationhere. we want stude nts to experienc e and reflect up on

MT167 JUNE 1999 25 © ATM 2007 ƕ No reproduction except for legitimate academic purposes ƕ [email protected] for permissions

induction, it might be bett er not to offer them with students is that th e sequenc e (Fibon acci) is Boundary - which, in turn, might be ideal forstru c- easily generated recursively, but the formula for the tural gen eralisation, and for discussion of different nth term refuses to yield to difference tables and is but equivalent formulation s of such a gen eralisa- virt ually inaccessible (th ough students still want to tion." L ast year,we be gan aworkshop on know it!) mathemati cal processes with Primary PGCE students by invit ing them to work on Pairuedcube. Conclusion Within m inutes, man y (by no means all)of them T here are tensions here, dida ctic dilemmas. The had writt en things like 6 (n- 2)2 and were asking national trend is towards pedagogic uni formity, a 'What sha ll I do now?' T hey were not 'being lesson structure and teaching styles that 'work' for difficult'; they were experien cing and dem onstrating all teachers and almost all learners. It is far from str uctur al gen eralisation . This year, we be gan with clear, on the evid ence to date, whe th er or how this Stairs. Asastim ulus for experiencing and reflecting will embrace inductive approaches to learning. In on induction and (eventually) explan ation,it was anycase, even the one-time prop onentsof investiga- much m ore effective. tion seem to be increasingly disillusioned . Are we, therefore, to con sign investigational work to th e Train spotting and 'the formula' dustbin of late twentieth centuryschoo l-rnaths I began this article with a set of 'investigation history, along with multibase arithmetic?That woul d starters '. H ow subversive I fcIt in th e 1970s, working beneith er my conclusion nor my wish , notwith- with such material in the classroom! T ime has standing the distorting conseque nces of GCSE revealed the reason for my sens e of unease: assessment over th e last decade. Indeed, I would fan Cockcroft has legitimised our activ ity, and said that the flickering flame of Paragraph 243 [16], adding everybod y should be doing it. Som e of th e tea chers four additional bullet points: who enthusiastically promoted th e place of investi- Investigat ional work at all levels should in clu de gational work in th e curriculum have come to regret opportunities for : the institutionalisat ion of investigation, not least • induction within G C SE cou rsework. T he unexpected regular- • explanatio n ity, the creativity and frisson of mathematical • empirical generalisation discovery, has been forced intoan algorithmic • structural generalisation mould: data-pattern-generalisation (and for mula for So long as we hold onto belief in pupils' entitle- extra marks). Assessment ha s hardened the ment to experience and learn mathematics, I would paradigm. In her recent book, Cand ia M organ has want to emphasise andpromote the centrality of addresse d generalisation in the schoolmathematicscur ricu- ...some of the tension s and contradictions lum, and 'investigation'asavaluable and authentic implicit in the official and practical discourse means whereby students might encounter and expe- [of 'investigation ']. In particular, the idea ls of rience it. openness and creativity, once operationalised through the pr ovision of examples, advice Notes and assessment schemes, become predictable Wh ewell pe rsonifies Induction and Deduction (like and even develop into prescribed ways of the characters of Bunyan 's Pilgrim 's progress) as posing questions or 'extending' probl em s and tho ugh th ey were two characters inhabiting the mind rig id algorithms for'do ing invest igation s' [13, of thescientis t. It is gra tifying to note , moreover, that \XThewell do es no t conform to the stereotype and ma ke p73]. Induction female (illogical, intuitive, uncertain, ap t to D ave Hewitt [14] ha s famously complained th at lead , to seduce her companion, capable of er ror) and in children's driven determin at ion to tabulate D edu ction ma le (logical, secure, the steadying numerical data, ' their attention is with th e numbers influence on his partner). In \XTh ewell's text, bo th cha r- and is thus taken away from the origina l situation'. I acters are portrayed as fema le. personally experience a sense of irritation when 2 If I had to choose just one of my own artic les to take to stu dents insist on an algebraic 'formula' - in th e a des ert islan d, it would be [6J. form of a function fix) - for everyth ing. Thesense of 3 In the case of Farotions, naive empiricism and confi- de nce in do ubli ng can be challenged by working on a closure on ce they have itis palp able.Presumably, different pro blem. Point s are marked un evenly on the they are th e victims of GCSE coursework indoctri- boundary of a circle, and each point is joined to every nation. M y irritation stems, in part, from awarene ss other point . How many regions areformed? that a few stu de nts will be scared off by th e formula, 4 It ma kes mu ch more sense simply to investigate sums an d sense failure because they didn't find it, or feel of divisors - try 2, 5and then 10. anxiety bec ause their undoubted rationality is frozen 5 By 'happy' coin cidence, an articl e on the topic [9J by th e sight of it. One reason I like towork on Stairs appeared in the Mathem atical Gazette in the same

26 MT167 JUNE1999 © ATM 2007 ƕ No reproduction except for legitimate academic purposes ƕ [email protected] for permissions

week that I sub mitt ed the first draft of this article to 8 M. Marsh: '\\?hole class investigations in the primary M T. Alan Beardon's ingenious resolut ion of conjec- scho ol' , Equals, Vol. 4, No. 3, 1998, pp 4-5. tures abo ut cycles and fixed points confirmed wha tI 9 i\. Beardon: 'Sumsof squ ares of digits', ha d suspected: that this is fascinating but non-elemen- Mathematical gazette,Vol 82, .0.495,1998, tary mathematics. pp 379-388.

6 If j is even, halve it; otherwise multiply by 3and add 10 j • Balacheff:'Aspects of proof inpupils' practice in I . BrianThwaites' priz e of £ 1000 to anyo ne who can school m athem atics', in D. Pirnm (ed) Mathematics, prove (or refute) the conjecture - that every such teac hers and child ren , London: H odder and sequence is ultim ately attracted to 1- remai ns Stoughton, 1998, pp .2 16-235 . un claimed . 11 T. Rowland: 'Conviction, explanation an d generic 7 Liz first introduced the distinction and the term s in examples', in A.Olivierand K. Newstead (eds) her thesis [18]. Proceedings of the 22 n d conf eren ce of the inte r- 8 See, for example, Bob Vertes' handling ofPic tur e national group for th e p sychology of Frames (aka Boundaryi with aYear 7 class, on the video m athem atics ed ucation , Vol 4, pp. 65-72: which accompanied Open U niversity course EM235, Stellenbosch SA: University of Stellenbosch, 1998. Develop ing mathem atical thinking. 12 E. Bills: 'Shifting Sands:students'understanding of the roles of variables in hoe! math em atics', unpublished PhD References thesis. Mi lton Keynes: Open U niversity, 1996 . 13 C. Morgan, Writing m athem atica lly ; th ediscou rse TRowland Hyperpainting: mind gamesfor night riders, MT143, 1993, pp .28-29. ofinoes tiga tio n , London: Falmer Press, 1988. 14 D. Hewitt, Trainspottersparadise, MT 140, 1992, pp6-8. 2 N ational Numeracy Project: M athematical vocabu- lary,London:BEAM, 1997. 15 E. Bills and T. Rowland: (in press) 'Examples, G eneralisation and Proof', in L.B rown (ed) 3 WI.Whewell: Nouurn organon reuonaturn, London: John W Parker, 1858. Pro ceedingsof the 1996 Meetings of th e Britis h 4 P.S. Laplace:A philosophical essay onprobabili- Society fo r Research into Learning Mathematics. ties, New York:Truscott and Emory, 1901. York: QED Publications. 5 R.I'. Burn: Apathw ay into numbel- th eory, 16 D.E.S: M athematic s co u nts, London: H er M ajesty's Cambridge: CambridgeUniversity Press, 1982. Stationery Office, 1982. 6 T. Rowland: Realfunctionswhichgeneratethe dihedral

groups, MT 69, 1974, pp 40-47. Tim Rowland works at the Institute of Education, University of 7 R. H ersh : 'Proving is convincing and explaining', London. Educatio n al studies in m athem atics,24, 1993, pp 389-399.

Mathematics Teaching + Micromath Vacancy for Editors During 1999,ATM w ill bechoo sing new editors for Mathematics Teach ing and also for M icromath. The new editors will be responsible forissues from the beginning of 20 01 . The appoin tments wil l be for three years, with an option to extend the controct to six years. Mathematics Teach ing is aquarterly journal and M icromath is issued three timesa year. A fee is pa id for editorship: this is£1600 per issue. It is expected that most ap plications w ill come from a team of two ormore people for each journal. Anyone interested in undertaking this role shouldcontact the ATM office for further information. Applications are expected in writing to ATM by 30th September 1999 and interviews will beheld at the ATM office in Derby during November 1999.

MT167 JUNE 1999 27

The attached document has been downloaded or otherwise acquired from the website of the Association of Teachers of Mathematics (ATM) at www.atm.org.uk Legitimate uses of this document include printing of one copy for personal use, reasonable duplication for academic and educational purposes. It may not be used for any other purpose in any way that may be deleterious to the work, aims, principles or ends of ATM. Neither the original electronic or digital version nor this paper version, no matter by whom or in what form it is reproduced, may be re-published, transmitted electronically or digitally, projected or otherwise used outside the above standard copyright permissions. The electronic or digital version may not be uploaded to a website or other server. In addition to the evident watermark the files are digitally watermarked such that they can be found on the Internet wherever they may be posted. Any copies of this document MUST be accompanied by a copy of this page in its entirety. If you want to reproduce this document beyond the restricted permissions here, then application MUST be made for EXPRESS permission to [email protected]

The work that went into the research, production and preparation of this document has to be supported somehow. ATM receives its financing from only two principle sources: membership subscriptions and sales of books, software and other resources.

Membership of the ATM will help you through

x Six issues per year of a professional journal, which focus on the learning and teaching of maths. Ideas for the classroom, personal experiences and shared thoughts about developing learners’ understanding. x Professional development courses tailored to your needs. Agree the content with us and we do the rest. x Easter conference, which brings together teachers interested in learning and teaching mathematics, with excellent speakers and workshops and seminars led by experienced facilitators. x Regular e-newsletters keeping you up to date with developments in the learning and teaching of mathematics. x Generous discounts on a wide range of publications and software. x A network of mathematics educators around the United Kingdom to share good practice or ask advice. x Active campaigning. The ATM campaigns at all levels towards: encouraging increased understanding and enjoyment of mathematics; encouraging increased understanding of how people learn mathematics; encouraging the sharing and evaluation of teaching and learning strategies and practices; promoting the exploration of new ideas and possibilities and initiating and contributing to discussion of and developments in mathematics education at all levels. x Representation on national bodies helping to formulate policy in mathematics education. x Software demonstrations by arrangement.

Personal members get the following additional benefits: x Access to a members only part of the popular ATM website giving you access to sample materials and up to date information. x Advice on resources, curriculum development and current research relating to mathematics education. x Optional membership of a working group being inspired by working with other colleagues on a specific project. x Special rates at the annual conference x Information about current legislation relating to your job. x Tax deductible personal subscription, making it even better value

Additional benefits

The ATM is constantly looking to improve the benefits for members. Please visit www.atm.org.uk regularly for new details.

LINK: www.atm.org.uk/join/index.html