Stop calling maths mathsAnother rant from David Coulson, about math education, 2012
For crying out loud people, before you venture to change the way math istaught in the western world, before you bask in the statistical glory ofhaving raised class averages for maths by making kids do more hours orby reducing class sizes or by firing bad teachers or by assessing kids moreor even assessing kids less, COULD YOU PLEASE stop viewing maths assome homogenous lump that can be sliced and packaged like a block ofcheese! Maths is NOT the same at both ends! Two units of maths doesnot mean twice as much maths. Twice the amount of time does notmean twice the quality. Twice the score does not mean twice as smart.
Deep at the heart of the math problem is that we are treating maths as ifit were an ‘it’. No. Maths is a multitude of skills that serve a multitude ofpurposes and are therefore only of interest (let alone use) to particulargroups of people.The deepest, most conspicuous fissure in the mathematical cortex is theone that separates algebra from arithmetic. For the uninitiated,arithmetic essentially means computation. If it involves numbers only, it’sarithmetic. If it involves other symbols, it’s algebra. Both regions are hugeand both have texture within them. Arithmetic can be regionalized intowhole number processes such as multiplication and addition, andfractional processes such as, well, fractions (decimals and percentages fitin there too). Algebra can be landscaped into calculus, complex numbers,simultaneous equations and so on.
I’m teasing apart the different clumps that make up maths because weneed to see that our attitude towards teaching it and more particularlyour attitude towards assessing it is fundamentally wrong.An extra year of maths does not make a student any better suited tohis/her chosen career unless the material taught in that extra year isrelevant to that career. And if it is, we would hope that the materialtaught in the preceding year was a stepping stone up to that material.Think of an architect who studies trigonometry for a year after havinglearned geometry for a year, after having learned basic number skills for ayear.Without seeing the texture that makes up the different parts of maths,we will continue to have secretaries who can find the determinant of amatrix but can’t balance the company’s accounts. We will have chefs whocan solve a quadratic but can’t convert Fahrenheit to Celsius.
Think about this: There are people all over the western world who, for abrief time in their childhood, were able to find the angle between twosides of a triangle. Hundreds of millions of people alive today oncemastered that skill and then forgot it, and are struggling now toremember it because their own kids are learning it at school. Thesehundreds of millions of people who once knew trigonometry probablygot their jobs based on a school qualification that in small part derivedcredit from a math test that included possibly one or two questions ontrigonometry that measured how much these people knew abouttrigonometry for six months prior to the examination. Arguably,trigonometry got them the job, in some cases, but only because someonea long time ago in a galaxy far, far away decided it should be so. Thatperson, that place and that time are no longer around, so we needn’t bedoing this anymore. There are better ways to prepare people for life.
Now we all know that physical strength and agility peaks somewhere inyouth and fade pretty quickly thereafter. And because of that, we don’texpect fifty-year-old men to be as good on the sports field as their fullygrown children. Furthermore, we would be reckless to think that achampionship swimmer should be able to perform well in a cricketmatch. BUT THAT’S WHAT WE DO when we look at math qualifications!We see that Johnny got a B+ in maths and expect him to be good withnumbers when in fact he was tested on calculus. And do you think Johnnyis still going to be good at calculus thirty years later?Maths is not just one thing. Do you get it!? Therefore making kids comein on Saturdays to do more of ‘it’ misses the point. And pattingyourselves on the back for making kids score better on tests that measure‘it’ is worse than missing the point; on occasion it’s actually damagingpeople.
An hour of maths with one teacher and an hour of maths with anotherteacher on another topic, these are not interchangeable units. It’s not likeour kids are aircraft pilots who get better by logging more hours in thecockpit.If we are teaching kids geometry, we are teaching them how to measurethings and (more subtly) teaching them how to use formulae. We areleading them into the outdoor world of objects and masses andmaterials. We might well be giving them the first baby steps towards acareer in research, too, but it’s still research to do with the outdoorworld.But if we are teaching bookkeeping and accounting, we are teachingthem something wholly different, and therefore I would like to think thatkids could choose to do one or the other, or both or neither, according towhich way they have chosen their lives to go.
If we are teaching statistics, we are teaching them, subtly, how to viewthe diversity in the world. Statistics, believe it or not, is a veryphilosophical branch of maths and is capable of broadening a kid’sunderstanding of other people, particularly those we don’t get alongwith. But it has a dark side too. Too many people in powerful places usestatistics as a way of reducing understanding to a handful ofrepresentative numbers such as averages. Such thinking regards kids in aclass as an amorphous substance that has to be taught. It regards kids asidentical learning bots with no more say in their progress through lifethan the molecules that make up a block of cheese. I could have a biglong rant on what statistics does wrong, but one rant at a time issufficient.
The philosophy that underscores the good side of statistics - call it Jedistatistics if you will – is that it represents a group of people as a bellcurve. We see that the average of one group is higher than the average ofthe other, but that the two groups overlap, which means that thedifferent averages might be trivial (maybe not). We see also that there isa correlation between where a kid sits in a classroom and how we hedoes at classwork, but we can see that the correlation is imperfect sothat there must be other factors involved. And though we can see thecorrelation, we can’t immediately say what the causative factors are untilwe do more investigation. Simply changing the seating arrangementmisses the point. That’s succumbing to the dark side of statistics: anabuse of correlations and averages, based on an unspoken desire to viewa class as a homogenous lump rather than a collection of individuals. Andright now there are so-called reformers out there in the schools who areusing this kind of thinking to make the statistics better and think that bysome miracle they are making the kids better too. There, I had my rantanyway.
I would like to see most kids learn arithmetic, at least long enough forthem to be able to manage their daily lives once they leave school andleave Mum and Dad. There are areas of arithmetic I would be less keen toforce on kids, such as fractions, but I would be keen to see them offeredas options for kids who are interested as well as teachers who areinterested in teaching these things.
I could not say the same about algebra and the universe of subjectstaught under that label. In my experience with kids – and I have met a lotof kids in my time – there are those who take to algebra like a duck towater and those who take to algebra like a brick in water. You can see inthe first hour that some kids are never going to get it, and struggling withthem is only going to make participants miserable.Between the two extremes of gifted and hopeless lie a huge swath of kidswho can master bits of algebra with a fair bit of effort. But should they?We should be debating with these kids and their parents for a whilewhether their effort will lead them to something desirable.As much as possible, we should be making an algebra class a nice place tobe for those that are keen to learn it. They should not be afraid to admitthat they like algebra, nor should they be distracted by the kids who don’tneed to be there. In a class like this even an average teacher could nothelp but be buoyed by the students and become enthusiastic. That’s awin for everybody, surely.
In speaking about the diversity within maths, we need to be aware too ofthe diversity within math teachers.No I’m not talking about good teachers and bad teachers. (That’s yetanother rant). I’m talking about teachers who are good at manipulatingnumbers in their heads; the Scott Flansburgs of society. I’m also talkingabout the teachers who’ve seen the incredible things that come out ofcalculus such as astronomy and quantum physics and who can teachcalculus enthusiastically, even though they need a dumb-ass calculator totell them the times table.Teachers are sometimes good at one end of maths and lousy at the other.Let math teachers be as diverse as the kids who learn it. In short, havecourses in algebra and courses in geometry and courses in statistics andcourses in basic number-juggling, and stop calling the stuff MATHS![END]