How should we teach mathematics? It’s a difficult question, one that people have been discussing and arguing over for centuries. Everyone has their own views, and I have mine. I’m sure I will make several posts on different aspects of this topic.

We can separate the teaching of mathematics into two parts, because solving a problem can often be divided into two parts. Solving a problem is sometimes done by firstly coming up with an equation that represents the problem, and secondly by actually solving the equation. The two parts are different. For example, the first part might result in a quadratic equation, and for the second part one needs to actually solve the quadratic equation. In summary:

- Find the equation.
- Solve the equation.

I would like to consider number 2 here. This is what I call skills, or technique. For example, solving a quadratic equation comes under the heading of “skills.” It involves using a formula. It is rote learning, it is algorithmic. After you have learnt the formula, and solved 100 (say) quadratic equations, you have learnt this skill. You can then solve pretty much any quadratic equation that comes your way. Solving quadratic equations is learned by practice. After enough practice you learn the skill, and then no real thought is required to apply this formula, it is robotic. In fact, it is so robotic that computers can do it.

These algorithmic skills are a part of mathematics, but they are not the same thing as mathematics. One problem that the subject of mathematics has is that many students emerge from school thinking that “skills = mathematics.” They form the false opinion that solving quadratic equations (and things like that) is all there is to mathematics. That’s it. That’s maths. Of course this is not true. There is a lot more to mathematics than skills. There is also part 1 of a problem – find the equation from the given problem. This requires understanding of concepts. Solving problems comes down to (I know this is a simplification)

- Find the equation (concepts)
- Solve the equation (skills).

We can and should test our students on skills alone. In fact, I believe in testing students a lot when it comes to skills, because skills are learnt by practice. Testing skills can be done my computer, by multiple choice. Tests can be taken over and over again by students. We can have testing centres, like the one in this article. At the testing centre there should be staff on hand so that students can ask questions, so they can find out where they went wrong. A testing centre could also offer instructional videos. Students can watch these videos and take the tests in their own time.

Computer software can now solve quadratic equations and perform other mathematical skills calculations better than any human. One can therefore argue that there is no need to teach skills at all! Computers can do it.

Some of our mathematics courses consist entirely of skills, like the one I am currently teaching to engineers. Is it any wonder that so many people think that mathematics = skills? We have only ourselves to blame.

My thoughts on this may be slightly different. If students only learn one part of mathematics, it should be skills. Of course they should also learn concepts, if they are able. But it is an unavoidable fact that only a small fraction of the population (maybe 5-10% of a leaving cert cohort?) have the ability to view a problem at the correct level of abstraction needed to instantiate or express it mathematically. I also think that understanding follows use, and not vice-versa, and that students should therefore have a solid bedrock of skills independently of whether their pocket calculator can solve quadratic equations.

But those who are able should be pushed to develop beyond the rote application of formulae at the earliest opportunity. In this regard some aspects of the project maths syllabus may be a step in the right direction. I was talking to a secondary school teacher who was actually quite pleased that some of her students, who had been expecting A’s in higher level maths, only emerged with B’s. This was because, although they had learned all the course material, they were not up to the conceptual challenge of some of the questions. This teacher’s view was that a student like that fully deserves their B, but no more than a B.

On the point of skills versus concepts, and only a small percentage of the population having the ability to express a problem mathematically:

I am thinking here at a lower level than you. I am thinking of younger children as well, and I’m thinking of the translation between mathematical symbols and words. Somehow, our children don’t think of mathematics as a language, and don’t learn that understanding mathematical concepts requires us to translate. An example I like is

(1) Suppose you know the sum and product of two numbers. How would you find the numbers?

or

the sum of two numbers is 50 and the product is 456. Find the numbers.

This question can also be presented as

(2) Solve the quadratic equation x^2-sx+p=0

or

Solve x^2-50x+456.

Translating (1) into (2) is what I’m talking about.

Or an even simpler example:

(1) x is a number between 3 and 7.

or

(2) |x-5|<2.

Many seem to have a problem with understanding (2) but have no problem understanding (1). Why can nobody translate between (1) and (2) ? This translation requires understanding the concept. It's very simple, but I do think there is a difference between (1) and (2) here.

Understanding follows use, I do agree. This can begin at the youngest age – children don't need to be 18 before starting.

The project maths story is encouraging.