What math education could (and perhaps should) look like

The opinions of a maths teacher

Dave Coulson, 2013

1: Modularise it

Chop maths into the bits and pieces that characterise it; arithmetic, algebra, trigonometry, geometry, probability, calculus, statistics, matrix methods, ....

The problem with maths now is that it is one long mule track through all of these domains with no particular destination in sight. And so a person has to become proficient in all these things whether or not the student finds it interesting or useful or applicable to the life journey they have in mind after leaving school. That’s not the way it should be.

How about organising these subjects into a tree-like structure so that a student veering towards a career in accounting can take courses that are suitable in that profession, as can a young carpenter, as can an aspiring scientist. A secretary would probably look more favourably upon maths if it were about juggling numbers in a spreadsheet than about passing half of an exam on algebra. Let people pick and choose what kind of maths they learn because not everyone aspires to be a theoretician.

2: Take out the cohort effect

Allow people to opt out of maths at an early age without embarrassment as their interests diverge from those of their age-mates. Similarly, allow people to opt back into the system without embarrassment as their circumstances and aspirations change. Consequently it should not be unusual to see a 20 year-old seated next to a 15 year-old in a course on, say, trigonometry.

This should be a natural development from modularising maths into its natural subtopics. With so many different-aged people coming and going from the courses, there will no longer be a cohort of kids who’ve been shunted along, class by class since thy were five years old. And when that happens, you’ll see that no-one cares anymore if one of them drops out or takes a year off and returns to school a bit richer and wiser. Abandoning maths early in life will no longer mean anything bad. Staying fervently on track won’t make you a nerd either. It just allows you to be who you are without peer pressure.

3: Associate courses with courses.

Maths is meaningless if learned in a vacuum, and while I have no problem with kids learning a branch of maths simply out of curiosity about it, I am also keen to acknowledge that there won’t be many kids who are like that. No, a bigger lot of kids will learn geometry because it is a parallel requirement for carpentry or a pre-requisite for engineering. Kids will learn statistics because it is necessary for accounting and social sciences. Others will just follow their noses and end up in places none of them can yet imagine - just like many university students do today - but they will have arrived at their final destination by combining an interest in maths with an interest in something else.

This is not just about kids choosing a career path and building their education around that. That precludes the possibility of kids changing their minds, which as we know happens all the time. But in the same way that learning a language is enriched by learning the culture and history of a country, so learning maths will be enriched by exploring its ties to science, construction, graphics, aviation, medicine and social research.

4: Make it optional

Think about it. No-one really needs maths beyond the ability to count on their fingers. Lifestyles are enhanced by numerical skills that exceed basic counting, but how much enhancement do you really need? Surely that’s a decision the learner makes, not a bunch of strangers. Let’s get rid of this nonsense of kids learning to multiply matrices so that they can pass half an exam so that they can go off to drama school and never see matrices again until their own kids grow up and go to school.

Think how nice it will be for the teachers to have only the kids who want to be there. Think about how nice it will be for the kids who are keen to learn to be able to learn without the presence of the angry, bored kids who would leave if only they could. Imagine how nice it would be for a frustrated kid to ditch a subject that patently is useless to him/her, let alone boring.

Some grownups fear that if we allow maths to be an option in life, no-one will take it. Not so. You just might be surprised at how fashionable math becomes when it’s a choice. I’ve seen it happen, time and time again.

5: Make exams mean something

If you pass or fail an exam, it should mean something. You train people in a given art so that they became better at it, and you examine them at a given level of study so that you can be sure they will understand what’s going to be taught at the next level up. This is wholly different from deciding a kid is smart enough to be released from captivity into society if he/she understands half of a maths exam. Today’s maths exams function mostly as intelligence tests. Only those people moving on to research careers in science really benefit from the content of these exams, and they are a minority.

Frankly, to progress to level 2 trigonometry (for example), I’d like kids to be able to understand all or nearly all the material taught in level 1 trigonometry, simply because it shows the kids are genuinely ready for the material taught at level 2. That’s kind of like expecting a heart surgeon to be completely competent at heart surgery, not just half competent on a good day.

6: Let kids meet practitioners

Yes it’s good to have professional teachers, but it’s also nice to have people with industry experience. How about a mix of the two, with a balance point left open to debate? We could have aviators talking about windspeeds and attack angles, which are applications of vectors. Of course math teachers can talk about these things too, but imagine the personal stories a working professional can bring to the classroom! We could have bridge builders talking about force diagrams and the trigonometry that comes from that. We could have visits from electrical engineers to talk about wave forms and complex numbers, computer game designers who can talk about matrix methods, market research people talking about correlations and causality, artists talking about perspective drawing. And guess what? If it’s an occasional thing, and if the kids in the class are all kids who want to learn this stuff, the once-a-week visit to the school down the road could be the best part of a professional’s work week.

7: Tell stories

As part of a broader desire to put maths into context, tell stories of where the mathematical material came from. This does a couple of things:

One, it strengthens a student’s understanding of a block of knowledge by letting him/her see how it came about, what its limitations may be, why it is expressed in the notation it is expressed in, what competing methods may have existed at the time and why they fell out of use, and why perhaps it’s time to bring those alternatives back to life (as sometimes happens). After all, the story of mathematics isn’t finished yet.

Two, it appeals to a natural learning style we are born with and which we celebrate in virtually everything else we do in life, from watching TV to trading gossip with our friends. We can’t stop ourselves from being drawn into stories. Tell stories in a classroom and see for yourself the instant change in behaviour.

7: Tell stories

Physics teachers know this. They explain their science wonderfully by talking about the tinkerers and thinkers who changed the world from the confines of their laboratories, sometimes by accident. Hans Christian Orsted was teaching his class that there was no relationship between electricity and magnetism and went so far as to set up a demonstration for them to witness, only to find that there (&$%#@#) WAS a relationship! It’s one of physics’s most celebrated stories.

Why can’t math teachers teach like this too? There’s no shortage of stories.

Trigonometry was once regarded as an inferior kind of maths because it wasn’t precise. Respectable mathematicians wouldn’t touch it. And so for a long time it was just that cooky stuff that astronomers did in order to plot the positions of stars on a star map. No-one was really sure what they hoped to achieve by doing that, but the astronomers seemed to think it was worth doing. And so they developed a tool that was about as useful as the thing they applied the tool to.

7: Tell stories

Matrix algebra was essentially useless until computer programmers grabbed hold of it during world war two. They were trying to design aircraft propellers and none of the customary mathematical techniques of the day worked. Ten years later a fellow wrote a book on the subject and now bewildered high school kids all over the world are wondering why they are learning it. Everyone teaches maths as if it’s been around forever, but my parents are older than some of this stuff. Maths is still growing!

Calculus was invented twice, by two highly eminent scientists who absolutely hated each other. For a long time there was English calculus and European calculus, and neither of the two sides was interested in trading notes.

7: Tell stories

Most of the maths that kids learn at high school wasn’t even invented by mathematicians but by people who needed a new tool to solve a problem they were working on at the time. Many of them didn’t even realise they had done anything significant.

Gauss became an expert on ellipses while measuring the distance between the north pole and Paris, perhaps the most pointless mathematical task in history. And yet he became so good at ellipses he could find a lost planet in the asteroid belt just five years later, and in the process became world famous. No-one cared about ellipses, they just wanted to hear about the planet!

Interested? All maths could be like this.

- David C, 2014

dtcoulson@gmail.com