Non-mathematical thinking in maths By David Coulson, 2012
Quite likely the most logical thing a person will ever do in his or herlifetime is solve a math problem at school. Cold, dispassionate, rational,callous, the rules of mathematics lead us to the answer as efficiently as acomputer programme, ...... Which kind of suggests that to be good at mathematics you have to besome kind of a robot. And yet in my experience as a teacher and as apractitioner of maths, I find that being very unmathematical at times makesme a better mathematician. That is what I want to discuss in this article: thevalue of thinking in thoroughly unmathematical ways when it comes tosolving mathematical problems.
Long before logic was codified, our ancestors on the plains and in thepaddocks were judging distances and weights and other „measurable‟quantities in ways that did not require numbers. How we did this is still amystery. But we can demonstrate this strange magic in a number of wayswhen we pick up a ball and do things with it.Some years ago, as a way of passing time in a park, I asked my son to throwa ball to me from a distance of twenty metres or so while I stood facing theother way. At the instant the ball left his hand, he would shout “Now!” and Iwould turn around and assess the situation and catch the ball, usually withone hand.
How am I doing this? I know the rules of kinematics and can calculatevelocities and distances, but I don‟t have time for this when I‟m positioningmyself to catch the ball. Even if I did, I lack the input variables from whichto calculate an outcome. What‟s the ball‟s initial speed? Which way is itflying? Could anyone judge this to the three decimal places required toensure my hand intercepts the ball‟s path at the exact right instant?Most sportsmen know less about physics than I do and yet they can catch aball in flight too. What we are doing in those few seconds remains largely amystery, but whatever it is, it is totally non-mathematical; it has to bebecause dogs and seagulls can do it too. Don‟t ask me how, they just do it.
Being able to do things such as this of course gave us humans a head start insurvival, and from there a head start in the construction of civilisation. Butthe non-numeric estimation of distances and speeds has somehow beensidelined by the arrival of mathematical logic in the last three hundredyears. Trust in rational thinking somehow got turned into mistrust ofirrational thinking, and by that I mean the determination of numbers bynon-numeric means.
Take the case of estimating percentages. Now 100% 90%we know that ninety percent is smaller than ahundred percent, but only by a little amount.So ninety percent of a number constitutesmost of that number. Knowing this, howwould you determine ninety percent of $400?
I had cause to look at this question just a few days ago while teaching areally nice kid who I can see will never be particularly good at mathsbecause she lives in a very non-mathematical world outside of school hours.Maths is just something she does for half an hour or so during the schoolday, between all the important stuff that goes on in her life. So as much as Ilove this kid, I don‟t really feel inclined to take her through all of the rah-rah-rah of percentage calculations. Quite the reverse, I feel that myguidance will only be useful to her if it produces easy results in a non pen-and-papery sort of way and in as little time as possible.
What I said to this young girl was “Write the numbers on the board withoutthe dollar sign and the percentage squiggle. Turn the word „of‟ into a timessymbol because „of‟ in mathematics always means „times‟.“Now rub out all the zeroes in front and behind the numbers. What do youhave left?”Of course, she says to me, “Nine times four”.“Right!” I say. “So what‟s nine times four?”
In a few seconds she gives me the answer. And then I feel a headachecoming on because at this point I‟m supposed to tell her how the twozeroes in the $400 that we rubbed out make the answer a hundred timesbigger and that the percentage sign means we have to make the answer ahundred times smaller and that the zero behind the 9 that we rubbed outmakes the answer ten times bigger again. Sigh....
That‟s what we have to do when we rely exclusively on numeric logic. Butwe don‟t have to. There are other parts of the brain silently screaming at usfrom the inside saying “It‟s $360, doofus!! The answer‟s right in front ofyou!”What we need to do at this critical point in the lesson – and I‟m pleased tosay that this is what I did with young what‟s-her-name – is to scale the 36 upand down by factors of ten to get one which makes non-numeric sense.I say, “We have a number which starts with a 3 and a 6. It can‟t be $36because it‟s too small. It must be $360 because that‟s about the right size.$360 is smaller than $400, but only by a small amount.”
Oh dear, says my guilty conscience, I‟ve broken the rules. I‟ve abandonedthe sacred path of logic for a grimy shortcut to the answer.I‟ve done what? I‟ve done exactly what people do in the real world, eventhose that don‟t know they‟re doing it, which is to let a non-numeric senseof how big an answer ought to be guide their calculation to the correctanswer. It‟s a Yin and Yang effort by two diverse parts of the brain, workingtogether to produce an answer that would have taken much longer if eachpart had worked alone.
Remember those lessons in psychology, of people who act decisively in amoment of crisis when leadership is needed? In the absence of completeinformation, these people make a snap decision based on what they sense isright, and use information to justify (or undermine) their conclusions asthey go along. Natural leaders, the fly-by-the-seat-of-the-pants variety, arethose who use their gut intelligently to anticipate which way the numbers areflying, and catch the outcomes correctly most of the time. Moreso, they areunencumbered by numerical thinking for most of the process, and canconcentrate on the whole job instead of just one part of it. And of course, iftheir thinking is wrong, they can change course rapidly and recover fromtheir mistakes faster than if they had analysed it methodically.
Here‟s another example. What‟s 70% of $2.10?Strip away the mumbo-jumbo and you have 7x21 = 147.How big should the answer be?Gut felling tells me to look for an answer which is bigger than half of $2.10,so I should choose $14.70. Problem solved.What about 72% of $2000?Sounds mean, but really this is just 72x2 =144.Common sense tells you the rest of the story.
Now does this mean we should chuck away rational procedures when doingmaths and adopt yet another cock-eyed shortcut? No way! Of course thatwould be wrong. But be cognisant of the ability, enthusiasm and futureorientation of your students and design your lesson accordingly.
I work with young scientists all the time (10-12 year olds, most of them) andcan justify to them why it is that we make a number ten times bigger or ahundred times smaller according to the positions of the decimal points hereand there. But I do not want to teach this to people who simply want to getthe answer. Their joy is in getting the answer quickly so that they can seewhat the answer means in the middle of a much bigger and much moresatisfying project. Even the young scientists who are interested in all the rah-rah-rah of percentages for their own sake are keen to add non-numeric,seat-of-the-pants thinking to their toolbox because getting the right answerwithout effort is somehow strangely satisfying.
For those of you that are interested, the way to teach the full rah-rah-rah ofpercentages without getting caught up in the need to divide by a hundred ortimes by a hundred whenever the percent sign (dis)appears, is to seedecimals and percentages as two different languages for the same thing.Percents could be French and Decimals could be Italian. Both languagesare talking about the bits and pieces between the whole numbers. Eachlanguage is strong and weak in different areas, so it‟s helpful to be able totranslate from one language to another to take advantage of the strengths ofeach.
If I want 20% of (say) $2.40, I‟ll start by converting 20% into its decimalequivalent. Now I have 0.2 x 2.4.To know where the decimal point in the answer should go, just look at thenumber of decimal digits in the two input numbers. Altogether there aretwo decimal digits, so there will be two decimal digits in the answer (hence$0.48).
Boom. No crap about putting numbers on top of a hundred and cross-multiplying fractions. The procedure is explained in terms of a translation,which in fact it is.
Consider what an advantage it can be to admit that there are nonnumericparts of the brain that can work in parallel with the numerical parts to guidea person towards the answer. Look what happens in trigonometry, forexample. 1m 2m
We know that the angle in the corner of a triangle can be found as theinverse Tangent of a fraction made up of the vertical and horizontal sides ofthe triangle. But which way up does the fraction go? Is it 1 over 2 or 2 over1? 1m 2m
Look at the picture and see what your instincts tell you. “Use the force,Luke,” I‟ll sometimes say to a student at this point, and wave my armsaround in some mystical sort of way. 1m 2m
The angle will either be a big number or a small number, depending onwhich way up the fraction goes. The bigger the fraction, the bigger theangle. The picture is telling me to look for a small angle, so I should begoing for a small fraction, which means 1 over 2. 1m 2m
The thinking I‟ve just used is non-ritualistic, non-mnemonic, evennonmathematical. Has it slowed me down? Absolutely not. In fact itquickened my pace. Metaphorically, it‟s allowed me to catch the ball inflight and carry on with the game. 1m 2m
A song by Sting contains the words “Emotions are the sail and blind faith isthe mast”, which suggests that two parts of our character acting in unisoncan take a person places that each characteristic alone could not.In the case of mathematics, it‟s not emotions allied with faith that gives usthe results but two regions in the brain which we may not even have namesfor, one rational and logical and the other non-mathematical and logic-less.Putting the two together make for a very fine mathematician.-DC, Dec 2012.