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Analogue thinking in maths By David Coulson, 2012
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Two people stand in a park, facing each other across a distance of about 20metres. One person throws a ball towards the other, and the other personcatches it. Question: is this mathematics?
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Before you answer this question, I want you to consider whether you thinkthat maths is necessarily something that goes on in a classroom, andsecondly, whether you think maths necessarily involves numbers. Then Iwant you to consider how much mathematics would be involved if we got arobot to go out and take a turn at catching the ball.
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Whatever the ball-catcher is doing, whether you regard it as mathematics ornot, he is finding the solution to a mathematical problem without referenceto numbers. In approximately one second of observation, the ball-catcherhas judged which way the ball is moving and how fast it is moving, and haspositioned himself to intercept the ball with an outstretched hand that hasto be positioned to the nearest centimetre in order to catch the ballproperly.
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If I was to get a robot to do that, the robot would have to be loaded with alibrary of kinematic formulae for measuring the ball‟s speed and direction,and defining a path for the hand to travel in to intercept the ball.So is the ball-catcher doing mathematics or not? It‟s not the kind ofmathematical thinking kids do at school. This is a kind of mathematicalprocessing that does not use numbers but somehow represents the motionof objects in an analogue fashion, and for that reason I want to call itanalogue thinking.
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Analogue thinking is very fast and goes on in a very primitive part of thebrain. This must be true because dogs can catch frisbees in flight andseagulls can catch breadcrumbs in flight, and I don‟t see many of themdoing physics courses at university.Analogue thinking is automatic and irresistible: we can‟t stop ourselves fromanalysing objects in motion even when we want to, and this is proven simplyby throwing an object at someone‟s head. If he doesn‟t duck in time it‟sbecause he failed to see it coming, not because he failed to analyse itsmotion.
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This may lead us to think that analogue thinking is subconscious. Thiswould be wrong because a person catching a ball in flight is very aware ofwhat he is doing. More so, he‟s concentrating so much on the task it‟sdifficult for him to do anything else at the same time.But this in turn allows us to see that ball-catching is a process of refinedapproximation, and if ball-catching is an example of analogue thinking inaction, then it implies that analogue thinking in general works in the sameway; that is, it works by constantly improving rough estimates.
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Be that as it may, the steps in the process of catching the ball – if there aresteps at all – occur below our ability to recognise and understand them. Weknow that all of our attention is fixed on catching the ball but we don‟t knowwhat it is we are doing with all that attention. All we know is that we have to“keep our eye on the ball”, in order to catch it. Somehow seeing the balltranslates directly into catching the ball.
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To summarise, analogue thinking is quick and imprecise and seems tooperate through very primitive parts of the brain. By contrast, conventionalnumerical and algebraic maths are characterised by the manipulation ofsymbols on a piece of paper. These symbols could be strings of digits orletters of the alphabet or even differential operators. This kind of thinkingcould be called symbolic thinking, and I would like to use that term for therest of this discussion.
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Symbolic thinking is very precise and does not require storage in the senses.An analogy can be made with digital computer technology, which storesinformation in a language that does not depend on the medium in which itis stored: data is data, whether it is stored on hard disk in magnetic form, orstored on CD-ROM, in optical form.By comparison, analogue thinking is like old-fashioned recording tape: datais stored in the medium of the senses, which means that quantities areimagined as lengths or heights or weights or distances or volumes ortemperatures or durations, all of which are as big or small as the quantitiesthey represent.
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Symbolic thinking works very well because (normally) an answer is foundafter a finite sequence of steps, requiring a finite period of time, and theanswer is stated with no loss of precision and requires no furthermodification.But symbolic thinking can be cumbersome with very simple problems. Forexample, suppose I am required to find the difference between these twonumbers: 102 - 67
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If I am to solve this problem purely symbolically then I have to go througha number of steps whereby digits are crossed out and adjusted so that a 2can become a 12. Yet I know already that the answer is going to be a bitbigger than 30 and all I really need to do from there is find out what theextra amount is. 102 - 67
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I can do this instead by imagining the two quantities as floors in a multi-storey building. The extra bit on top is 2 and the extra bit underneath is 3.The problem has been solved utilising avisual representation. The process was 102not entirely analogue, but that is not the 30point: analogue thinking guided me 67quickly towards a symbolic answer. 102 - 67
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I would like to propose that analogue mathematical thinking and symbolicmathematical thinking are compatible, and further that this is what peopledo all the time when they are solving problems outside of the schoolclassroom, where mathematical problems are generally simple and can besolved without resort to pen and paper.Since this appears to be a natural way to handle quantities, I would like tosee its use encouraged in the classroom, not to the expense of traditionalsymbolic thinking but as an adjunct to it.
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What‟s 90 percent of $400? 100% 90%The classic symbolic approach is to strip awaythe percent sign and the dollar sign and all thezeroes behind the numbers and reduce theproblem to multiplying 9 by 4 to get 36.We then add two zeroes from the 400 andone more zero from the 90 to get 36000 andthen divide everything by a hundred becausethe 90 is actually a percentage instead of anormal number.
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That‟s very circuitous processing for an 100% 90%answer that I can see right up front is going tobe $360. How do I know this? Verysimply, the answer has to be slightly smallerthan $400 because 90 percent of a numberconstitutes most of that number.This is analogue thinking in action. It did notobscure my ability to process the problemsymbolically. Rather, it facilitated it, allowingme to jump to an answer that looked right assoon as I had a range of alternatives.
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Another nice example of analogue thinking and symbolic thinking workingtogether can be found in a paper by Guershon Harel and Larry Sowder(http://dx.doi.org/10.1207/s15327833mtl0701_3).A swimming pool is to be filled up from water delivered to it by two pipes. 30 20
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The pipe on the left can fill the pool in 30 hours if used by itself. The otherpipe takes 20 hours. The question is, how long will it take to fill the poolwhen both pipes are used? 30 20
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The pipe on the left can fill the pool in 30 hours if used by itself. The otherpipe takes 20 hours. The question is, how long will it take to fill the poolwhen both pipes are used?Got the answer yet? 30 20
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I looked at this problem for about a minute, considering a number of lazyoptions that might work, and then applied the technique that produced theonly sensible answer. None of the approaches I considered had anyresemblance to the formal processes I learned at school. I did not havetime, I did not have pen and paper, and if the truth is to be told, I was justtoo damned lazy. 30 20
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Experience tells us that the numbers 30 and 20 must play a part in thesolution. So consider the water from the pipes flowing into the pool asrectangles that approach from the left and right.Eventually they will meet in the middle, having filled the tank in the ratio of2:3. 30 20
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How do I know this? The truth is that I don‟t, at least not at this stage. Butmy experience of objects in the real world – analogue thinking – suggeststhat this ought to be true, so I‟m going to run with the concept and see if itworks. 30 20 2 3
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The pool is going to be filled in a fraction of the time it takes either pipe todo the job alone. That should be 2/5 of something or 3/5 of something.Logic would tell me which but I don‟t have the patience right now for logic.I simply want the answer. 30 20 2 3
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Which one is it? 2 2 x 30 ? x 20 ? 5 5 3 3 x 30 ? x 20 ? 5 5 30 20 2 3
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Which one is it? 2 2 x 30 ? x 20 ? 5 5 3 3 x 30 ? x 20 ? 5 5 At this point another kind of thinking kicks in: some kind of common sense tells me that the answer should not depend on which pipe I choose. Therefore one answer on the left should match an answer on the right.
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2 2 x 30 ? x 20 ?5 53 3 x 30 ? x 20 ?5 5 This is the one I want.
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By comparison, pure symbolic thinking would have requireda process rather like this: Let f1 be the flow rate from the left, N litres and f2 be the flow rate from the right. f1 30 hours Let N be the amount of water in the pool. N litres f2 20 hours
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By comparison, pure symbolic thinking would have requireda process rather like this: Let f1 be the flow rate from the left, N litres and f2 be the flow rate from the right. f1 30 hours Let N be the amount of water in the pool. N litres f2 20 hours T is the time taken to fill the pool N litres with both pipes open. f1 f2 T hours
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By comparison, pure symbolic thinking would have requireda process rather like this: N litres N N N f1 f1 f2 30 hours 30 20 T N litres f2 20 hours T is the time taken to fill the pool N litres with both pipes open. f1 f2 T hours
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By comparison, pure symbolic thinking would have requireda process rather like this: N N N f1 f2 30 20 T 1 1 1 30 20 T
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By comparison, pure symbolic thinking would have requireda process rather like this: N N N f1 f2 30 20 T 1 1 1 20 30 1 30 20 T 600 600 T T 12 hours
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Now you may be thinking by this stage that I‟m trying to put an end toformal thinking in algebra and that I want to let loose on the world ageneration of kids who rely on intuition for their answers. No! That is not atall the case. Of course I want kids to learn traditional mathematicalmethods. I‟m a maths teacher. Some of these kids we are teaching todaywill go on to confront some of the toughest mathematical problems inexistence, and they will need all the preparation they can get. Does it hurtfor us to add another tool to their toolbox?
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Analogue thinking at the expense of symbolic thinking would be wrong andnaive. But analogue thinking and symbolic thinking working in unison are apowerful combination and can make a good mathematician and ingeniousone. Stories about Albert Einstein and Stephen Hawking come to mind.Some of the best mathematicians in history were famous for their leaps ofintuition and their visual approach to mathematics; perhaps these aremetaphors for analogue thinking.
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The question to ask now is, Do we value analogue thinking when we see itin the classroom? Do we even recognise it? If we could, would we teach it?Or do we mark kids down for jumping to an answer instead of following theformal route? If so, does analogue thinking go into hibernation during theschool years and re-emerge only later in life, when it is safe to do so?
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We can get a sense of what kids think of maths when we ask them thequestion “How long is a metre?”Most kids will answer “100 centimetres”. Some kids will answer “100millimetres” and a few will answer, hesitantly, “Ten centimetres? Tenmillimetres? Something like that.”In my experience no-one answers the question by putting their hands out infront and saying “THIS long.” In fact, even when you ask the kids to showyou the length of a metre using their hands, about half of them still can‟t doit because they don‟t know how long a metre ought to be.
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This suggests to me two possibilities: Either the kids have never learnt howlong a metre is – which is pretty sad if the kids have been doing maths atschool for several years – or the kids just don‟t think that putting theirhands out in front of them is a respectable way of answering a mathquestion. They have learned that maths is about numbers, not aboutquantities.
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It seems to me that if kids are to utilise their worldly experience ofquantities in the middle of a mathematical problem, we should first of allmake sure that they actually have real-world experience to draw on.This can be achieved by letting kids experience for themselves what a metrelooks like, what a centimetre looks like and what a millimetre looks like. Ofcourse, good teachers do this all the time, but perhaps the message is notgetting through. Perhaps it would be more memorable if the kids were tospend time building things that required repetitious measurements ofmetres, centimetres and millimetres. Or maybe there are other ways wecould make the learning more memorable.
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Some years ago I was working with a kid who had no idea what a kilogramwas, except that it was something to do with weights. Looking around forsomething that ought to weigh a kilogram or so, I eventually passed her ahammer and asked her to swing it back and forth a few times to get a senseof its weight. So there she was for half a minute, walking around in mystudio, swinging a hammer back and forth. Whatever else she learned fromme that day, I expect the most memorable part of the lesson was swingingthat hammer around, simply because it was an odd thing to do, andappealed to the senses more than numbers on a page would.
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Generally speaking, when I ask a kid to work on a mathematical problem,I‟ll ask him or her (as we‟re approaching the conclusion) how big or smallthe answer ought to be. Preparation like this often protects a student fromaccepting preposterous calculation errors, but also will allow a student tofind a shortcut when one exists.
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Here‟s a problem that I think illustrates the point quite nicely: Cos(x) Sin(x) 0.9 What‟s x?
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Here‟s a problem that I think illustrates the point quite nicely: Cos(x) Sin(x) 0.9 What‟s x?If we tackle this the traditional way, then this is a very sticky problem.This is symbolic thinking. We regard the objects in the equation assymbols to be manipulated, and therefore it makes no difference whetherthe number on the right-hand side is big or small.But if you look at the problem analogously, then the size of the numberon the right-hand side becomes very significant.
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By representing the sine and cosine as graphs, and the difference betweenthe two values as the distance between the two graphs, we can see that thevalue 0.9 forces us well over to the left edge of the number line, wherethe angles are small. 0.9 0.1
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Already I can guess the approximate value of the angle. But even if Idon‟t want to guess, I can replace the original problem with a problemthat is much simpler: Sin(x) 0.1 0.9 x 5.7O ( Correct value 5.48O ) 0.1
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Is the answer good enough for the purpose it serves? That depends onthe purpose it serves. We can still apply the formal symbolic process ifwe need to. The marvellous thing, though, is that we know very quicklyhow many answers to expect and what their sizes ought to be. This canguide our thinking and stop us from going off course.
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Is it possible to be too analogous in one‟s mathematical thinking?Unfortunately yes. I met a young fellow many years ago who seemed toregard every mathematical problem as an exercise in estimation. Asked tosolve any kind of mathematical problem, his answer would always startwith the word “About...” followed by a long pause, and then a numberthat he „felt‟ was right. This fellow was quite difficult to teach(exasperating actually) because he seemed to have no idea that there wereprocesses such as multiplication and addition and so on that could givehim the answer precisely with only a little bit of pen-pushing. Notsurprisingly, his math skill was very poor.
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To a much lesser extent, I‟ve found that kids who are very deeply intosports also tend to have difficulty picking up arithmetic and especiallyalgebra. It‟s not that they aren‟t nice kids (they are nice kids) and it‟s notthat they aren‟t intelligent (they are intelligent), nor is it that they lack adesire to learn. It seems to me that sports requires a highly developedability to judge distances and speeds at a sensory level and that this hasbeen at the cost of developing skill at manipulating abstract symbols.
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At the other extreme, musicians are conspicuously good at maths,particularly algebra. Musicians are used to seeing symbols on a page andunderstanding what these symbols represent. But I wonder how goodmusicians would be at anticipating answers at a sensory level; in otherwords, offering an answer that begins with the word “About...”? When allis said and done, that is a useful skill to have too.
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It seems to me that there must be some optimal way of combininganalogue thinking with symbolic thinking that makes for a bettermathematician. It‟s not about finding a compromise between the two butrather about developing both styles of thinking to a high level andallowing the two styles to benefit from each other; a coupled system, toborrow a term from complexity thinking.
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Let kids play with objects while they are young. And while they‟relearning maths, do not let go too soon the habit of holding up objects inthe air and asking kids to solve problems related to those objects. Let 2+3literally be the addition of two objects that they can see, and later in life let2+3 be the addition of two weights or volumes or temperatures or time-durations so that kids can see what these quantities „look‟ like too.
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In Christchurch, we have become masters at the art of measuring the sizeof earthquakes by the seat of our pants. Two years after the big shake thatdestroyed parts of our city, having experienced dozens of aftershocks ofvarious sizes and depths since then, we can now sit calmly through anaftershock and say “That felt like a 4.5, probably quite deep because itwas a rolling motion rather than a sudden shake.” If we‟re wrong, it won‟tbe by more than ten percent. And when we hear of an earthquake of agiven magnitude somewhere else in the world, we can all too easilyimagine what that would look and feel like.
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In my work I often take kids outside of the classroom and let them seereal objects that must be counted or assessed in some way. I say “Howmany palings in that fence?”, “How far is it from here to there?”, “Howtall is that tree?” All the time I am asking them, “How would you workthat out?” In doing so I am appealing to their sense of scale, and hopethat by doing so I can literally put them inside a mathematical problem,solving it as much by footstep and hand-length as by juggling numbers.
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To conclude, I want to leave you with another kind of mathematicalproblem and ask you to pay particular attention to the way in which youchoose to solve it: How many independent fifteens can you make from three tens and two fives?
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If you solve this by imagining the numbers to be written on cards, and seeyourself moving those cards around on a table so that you can best „see‟the interactions as lines joining the numbers, then you are using analoguethinking of a tactile as much as visual nature, and applying it to themanipulation of symbols; a perfect fusion of analogue and symbolicmethods for solving a mathematical problem. 10 10 5 5 10
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