In my opinion skill-automation is the key to accessing higher-level problem-solving skills in mathematics (and possibly other fields of work as well). But what do I mean by skill-automation?

I don’t know if skill-automation is an officially recognised term but I will explain what I mean when I use this term; a skill has been automated when it can be used and applied with very little, if any, conscious thought and effort. Some skills can be automated quite easily, others require much more effort to automate and there may be yet other skills that can’t be automated.

Automating skills is essential in mathematics because it means that you are able to focus your mental energies towards the more intricate aspects of a problem without getting bogged down with trivial aspects. For example, addition of fractions is a skill that can be automated quite easily (and I don’t mean that it can be just done on an electronic calculator or similar device) and it might seem like a very small and insignificant skill and yet, a failure to automate this skill means that whenever you need to add fractions together as part of a problem you have to strain to either remember the way that you’ve been shown or to try and figure it out from scratch. Both of these are a huge drain on mental energy – it’s like having to think about every single step that you take while walking or having to figure out how to walk every time you want to take a step.

The lack of automation of certain skills is something that often frustrates my students’ attempts to solve problems. Some skills that are often not automated to a sufficiently high degree are

- Addition, multiplication, subtraction and division of whole numbers and fractions
- Simple algebraic manipulation and re-arranging equations and formulae and solving linear equations
- Expanding brackets, collecting like-terms and simplification of expressions
- Pythagoras’ Theorem and trigonometry in right-angled triangles
- Recognising and solving quadratic equations

Of course there are other skills as well. The point is that once these are automated (which doesn’t take that long) you don’t have to worry about them and you can concentrate on the more challenging aspects of a problem.

So how do you automate a skill? Well it can happen in all kinds of ways – all of which take time to a certain degree – but the main way is through practice and initially, conscious and directed effort. At first the skill will be far from automated and you will have to think very carefully about what you’re doing (think about when a child learns to walk) but through repeated exposure to situations where the skill is required then, if you are paying attention and concentrating, you will find that you start to spot certain patterns – sometimes these patterns may be very difficult to express in words but you acknowledge them all the same. You will start to take advantage of these patterns subconsciously and through necessity as problems become more challenging – the skill is starting to be automated. Continual exposure to challenging problems will cause you to see more patterns which will often be subconsciously incorporated into your problem-solving. Of course there may be a limit to this – a skill may only need to be automated to a certain degree; the law of diminishing returns will kick in so it will make further automation very time-consuming but possibly unnecessary anyway.

The key here is practice – you cannot automate a skill without a great deal of practice and this is often why many of my students have failed to automate these skills – they simply haven’t practiced enough. They complain that they struggle to add fractions together and yet they refuse to practice this skill. Would you expect to learn to play the piano without bothering with the practice? No, it would be absurd to even think that it could be done. Admittedly, adding fractions is not very exciting (and I’m not even going to patronise you by claiming that it is!) but it IS an essential skill to automate if you want to get on to solving more challenging problems; the same applies for other skills. Learning scales is maybe not a very exciting aspect of learning to play the piano – but I would guess that any piano teacher would insist on scales being learned (and automated) otherwise higher-level piano playing becomes a practical impossibility.

So if you want to get on to solving more interesting and challenging problems in mathematics then you’re going to have to get these basic skills sorted first – there’s no way around it I’m afraid!