The curiosity of secondary school mathematics education

Many folks will be quick to say that "Mathematics is an exact science."

Is it? In statistics, a random variable is not exactly known, indeed it is explicitly not known. One does not know the number of dots facing up upon the roll of a standard six-sided die. But one does know that it is is not going to be more than 6 or less than 1. One might also say that if the die was fair the probability of any given number on any given throw would be 1/6.

In this case, the exactness refers to that we know exact bounds, and conditional exactness on the likelihood of an event.

You might say that mathematics is the study of what is exact and if something is exact - through the medium of how is it exact?

The mathematics conflict

You will struggle to find a definition of mathematics without including the word "and". Its definition has come to be the conjunction of subdefinitions, as a definition is otherwise either too vague or incomplete.

Here's a cute table of some potential interpretations of mathematics, by certain different people:

Who they are	What they think mathematics is
Me, 6	Numbers
Me, 8	12 x 12 = 144
Me, 10	Infinity plus one
Me, 12	x(x + 2) = x2 + 2x
Me, 14	That thing I'm good at
Me, 16	That thing I'm no longer good at
Me, 18	Hard work
Me, 20	?!?!?
Friend's mum	"Oh I was never good at maths, ha ha"
Politicians	We teach this to kids so they can work and pay taxes
Big Shaq	2 plus 2 is 4 minus 1 thats 3 quick maths

Most mathematics, particularly the pure stuff, is dry. Really, really dry. Almost the science of dryness.

To those who love it, often the love comes from the obsession over the limit of abstractness. For others, it comes from being "good at it". Often these people have a career in it.

Career mathematicians owe it to the masses to recognise that for the majority of people, the mathematics they will learn will never feed either of these urges.

Teaching mathematics

So in that case, how do we present it to children?

We want to maximise the education for the few that will end up as career mathematicians, and maximise the education for the many that won't.

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First, a word of warning. There are some popular misinformed opinions on this.

One amongst some university students and graduates in the hard sciences is that it is worth the inefficiencies. For example, official figures in the UK, show that 49% of people under 30 go into higher education, and 46.5% of those go into sciences. This means that only 22.8% do end up building upon the knowledge gained from their education, and so the other 77.2% are stuck doing trigonometry. Even in my mathematics degree, I could count on one hand the number of times I needed to know anything about trigonometry. At the very least, learning to manage finances would be a better use of time.

Another is the perception that mathematics needs to be taught to everyone as it is the clever subject. A measuring stick of sorts. Some suggesting this might have gone through it already, and confuse it being good because they did it and it actually being good.

It is true though, that many of the smartest students will blindly gravitate to the perceived biggest challenge, and any solution must take this into account.

A better mathematics education

Mathematics needs a split.

When we are first learning mathematics, it is almost completely applicable to all. Arithmetic is needed to pay bills and file taxes.

So it seems logical that they should split at a certain age. There is precedent with this in the way that science splits into biology, chemistry and physics. In the UK, these subjects are split up when the children are aged 13 to 14.

It would make sense, in the UK at least, to split Mathematics into two at this age.

The names for each subject are important to get right, to ensure a good balance between each. For the sake of this article, I shall call them "Pure" and "Applications".

• Pure
  • from ages 13 - 16 (required)
    • Algebra
    • Basic number theory
    • Infinity
    • Equations
    • Formal logic
  • from ages 16 - 18 (not required for 16 - 18 Applications)
    • Proofs
    • Calculus
    • Trigonometry
• Applications
  • from ages 13 - 16 (required)
    • Finances
    • Unit conversions
    • Statistics (probability, normal distribution, z-tables)
    • Geometry (taught in sync with mechanics from Physics)
    • Algorithms
  • from ages 16 - 18
    • Hypothesis testing
    • Algorithms
    • Coding

Additionally:

• statistics and algorithms would be required, but smaller and more refined at 13 - 16
• mechanics would be in physics
• this may be missing some key components

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This exercise got me thinking... there are many subjects where separating into pure and practical parts would make more sense, like:

• physics and engineering
• organic and inorganic chemistry
• physical and human geography

I wonder what academics in those fields would think of an analogous argument for their area?