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 noone 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
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
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