Tall, David

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University of Warwick institutional repository: http://go.warwick.ac.uk/wrapThis paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please refer to the repository record for this item and our policy information available from the repository home page for further information.  To see the final version of this paper please visit the publisher’s website. Access to the published version may require a subscription. Author(s): David Tall Article Title: Misguided Discovery Year of publication: 1990 Link to published version:  Publisher statement: None  Published in Mathematics Teaching, 132 27–29 (1990).Misguided DiscoveryDavid TallIt seems that the world is changing more now than it ever did before. So we musteducate our young to be flexible in their thinking and to be able to face new problemswith confidence. Part of this trend is seen in the introduction of problem-solving andinvestigations into the mathematics curriculum. The problem with this change is that itneeds a new kind of thinking by teachers as well as pupils, and it is making demandsway beyond those of traditional mathematics. Are we ready to face the new challenge?Is it fair to teachers and to children to introduce new techniques almost overnightwithout adequate research and adequate teacher preparation?I very much want to see our educational system succeed, and passionately believe thatwe must equip our children to face problems that we have never seen. But such adrastic change requires far greater time for preparation, experience and reflection thanour system is prepared to give and, as we learn from our mistakes, it is our childrenwho must accept the consequences.Jamie – as we will call him to protect his identity – is fifteen years old. He is a giftedchild who is expected to get a grade A in his GCSE Mathematics. His teacher is facedwith the new curriculum and presents the class with an extended project. Instead ofteaching children the full intricacies of euclidean geometry, the idea is to encouragethem to discover some of the ideas for themselves and to write them up as work tocontribute to their GCSE assessment. They have been introduced to the idea of makingconstructions using straight edge and compass and have experienced a number oftechniques: bisecting angles, subdividing lengths into a number of equal intervals, andso on. The investigation is to use straight edge and compass to construct a number ofgiven figures and then to go on to see what angles, other than 30˚, 45˚, 60˚, 90˚ can beconstructed.Jamie attacks the problem with relish. He demolishes the given figures and begins onwhat he sees as the main event. He attacks the problem systematically. If only he couldtrisect an angle, he could divide 30˚ into three and get 10˚. He could then bisect it to get5˚, and with a little more effort he could divide this into five equal parts to get 1˚. Hecould then build up an angle of any size (which is a whole number of degrees). So far,so good. His logic is OK. The only trouble is: can he trisect an angle?Of course Jamie hasn’t heard that mathematicians vainly sought a solution of thisproblem for two thousand years, nor that a young French mathematician who showedDavid Tall Misguided Discovery- 2 - to appear in Mathematics Teachingthat it was impossible was ridiculed by professional mathematicians of his day beforebeing tragically killed in a duel. Jamie is young, vigorous and untainted by seriousfailure. He believes it can be done.He has an idea. He knows how to trisect a line, why not use the same method to trisectan angle? So he takes an isosceles triangle with a 30˚ angle between the equal sides andtrisects the opposite side.B ACDivide AC into three equal parts to (attempt to)trisect angle BHe does it as accurately as he can and then measures the resulting trisected angles. Theyall equal 10˚.He bisects one of these angles, then divides the 5˚ angle into five parts using the sametechnique.BSubdivide AD into five equal pieces to (attempt to) construct 1°ADWith very careful drawing, as best as he can measure, each of the tiny angles is onedegree. So he has conjectured that he can do it, he has tested his conjecture empirically,and it works.He goes on to show how he can construct a number of regular polygons by workingout the angles required and constructing them using his theory. Again, he hasconjectured that he can do it, he has tested his conjecture empirically, and it works.Triumphantly he uses a computer word-processor to write his material up neatly, withhis solutions all set out in his best possible draughtsmanship. Typical of many boys hewrites in a terse, pointed style, stating the results with only the briefest of explanations.The previous year he had the experience of getting every answer correct in a test yetobtained only 87% because he did not explain answers whose truth he considered to beself-evident. But this has not daunted him, for  now he believes that his hour has come;David Tall Misguided Discovery- 3 - to appear in Mathematics Teachingthis time he has made a real discovery and he does his best to write it up. It takes manyhours, but it is worth it. At last it is finished and it looks good. He submits it forassessment and waits eagerly for the marks...He gets a grade D. He has failed.Jamie’s teacher knows that it is theoretically impossible to trisect an angle. He knowsthat almost all of Jamie’s work is based on a false premise. So it is wrong. But heseems not know that it is possible for the process in the proof to be correct, eventhough the premise is false. If Jamie could have trisected an angle then divided a 5˚angle into five equal parts then he could have done all the things he claims. Hisargument is faultless. His premise is faulty.At least, his theoretical premise is wrong. In a practical sense he has done well. InThinking Mathematically John Mason and his co-authors tell us that the process ofconjecture and proof consists of three stages:convince yourselfconvince a friendconvince an enemy.Young Jamie did two out of three. He convinced himself, he convinced not one, butseveral  friends in his class that he was empirically correct, in fact they were veryimpressed. But he failed to convince the enemy. The enemy, in this case the teacheracting as assessor, did not check the accuracy of Jamie’s empirical evidence, for if hehad done, he might have been a little more sympathetic. In Jamie’s putative trisection of30˚, a little intricate trigonometry shows that the actual angles are 9.9˚, 10.2˚ and 9.9˚ –so close to an exact trisection that it is unlikely to be seen as different in the drawing ofa fifteen year old – even in the drawing of a professional draftsman for that matter!Jamie’s class have done a lot with calculations to make them as accurate as possible.Jamie tried to get the most accurate possible trisection of an angle. And it was close!But not close enough in the eyes of the teacher.I sympathise with Jamie. When I was sixteen I was told that it was not possible totrisect an angle and I promptly spent the next fortnight trying to do it. I rememberdistinctly noting that the “angle at the centre was twice the angle at the circle” andsomehow tried to get the angle at the circle redrawn abutting the angle at the centre sothat the sum of the angles gave a new angle divided into two parts, one twice the other.I never managed the construction; I was unsuccessful but not defeated. Years later atuniversity I was shown that constructions using straight edge and compass essentiallyinvolve drawing straight lines and circles, and finding where they meet is equivalent toDavid Tall Misguided Discovery- 4 - to appear in Mathematics Teachingsolving linear and quadratic equations. More complicated constructions lead tocombinations of quadratic equations which turn out to be quartic equations, equationsof degree eight, then sixteen, and so on. There is no way that one can solve a cubicequation by such a construction and it is easy (for a more mature mathematician) toshow that x=cos10˚ is the root of such a cubic equation. Hence trisecting the angle 30˚is theoretically not possible, but Jamie got it close as a gnat’s whisker.In carrying out his investigation, Jamie had done (almost) all the right things: he had aplan, he attacked the problem systematically and, where he made conjectures he testedthem in the best ways he knew how. He is a problem solver. His only failure is that helacks the knowledge and sophistication to see a very subtle error, which amounts to thedistinction between what can be done theoretically with a theoretical straight edge andcompass, and what he can achieve practically with his own instruments.Perhaps if he had tried his trisection idea on larger angles, he might have seen that themiddle of the three angles is actually a bit bigger than the two on either sides. But hedidn’t. That is a level of sophistication far beyond what might be expected of a noviceproblem-solver.Yet it is well-known in mathematics that some of the greatest advances come throughmaking such mistakes. A favourite quote of mine comes from Poincaré, a great andvisionary mathematician who was himself not above publishing an incorrect proof of atheorem:It is impossible to study the works of the great mathematicians, or even thoseof the lesser, without noticing and distinguishing two opposite tendencies, orrather two entirely different kinds of minds. The one sort are above allpreoccupied with logic; to read their works, one is tempted to believe theyhave advanced only step by step, after the manner of a Vauban1 who pusheson his trenches against the place besieged, leaving nothing to chance. Theother sort are guided by intuition and at the first stroke make quick butsometimes precarious conquests, like bold cavalrymen of the advancedguard.Jamie is such a bold cavalryman, cut down in his prime. Yet the assessment of hisprogress failed to reward him for attempting to make such a visionary leap into thedark. What is needed is the realization that the child should be rewarded forsuccessfully completed a number of phases of the problem-solving process and that heneeds careful guidance to encourage him to review his work more carefully. To denythe existence of mistakes in problem-solving is to deny the process itself, for fromerroneous conjectures, can come improvements, refinements and true progress.                                    1 Sebastien de Vauban (1633-1707) was a French military engineer who revolutionized the art of siege craftand defensive fortifications.David Tall Misguided Discovery- 5 - to appear in Mathematics TeachingI believe that the role of the teacher is as a mentor, who must use the wisdom ofSolomon to know when to stand apart and let the child learn through making mistakesand when to intervene to guide the child into a fruitful and powerful path. The childshould be encouraged to discover, and also to reflect on those discoveries to learn tobecome an independent thinker. But when the child lacks specific tools to do a job andmoves into an area where serious errors may be committed, there comes a time whenthe mentor should intervene and help to guide the discovery. It is a difficult equilibriumfor the teacher to maintain and beyond my own ability to consistently make the rightchoice.The achilles heel of our new system is that we have introduced new (and relativelyuntried) techniques of teaching and assessing problem-solving with insufficient teacherpreparation to carry them out and with little attempt to understand the learning processesinvolved. When the teacher adopts the complementary role of assessor to judge what itis that the child learns, it seems that the role of teacher must fall temporarily inabeyance. When the teacher has little experience of children operating in such anunfamiliar context and even less experience of assessing what it is that the child isdoing, then, without appropriate guidance, the child may suffer.Jamie has learnt from his experience. What he has learnt is that he should not attempt todo anything imaginative which may rock the boat. To succeed in what remains of hisGCSE and recover the failure in his first investigation, he told me, with some feeling,that he must “keep his nose clean” and only attempt “fairly standard things that theteacher can understand”. I do not think that this what we want children to learn fromtheir investigations. But Jamie has discovered it, and it is a misguided discovery.

Misguided discovery

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Misguided discovery

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