By Harry Red.  Harry is a developer and math fanatic who helps students at exammastery.com

Deep study is a rare commodity. It’s what happens when you’re studying at your peak ability. Mathematics definitely requires lots of deep study. But it doesn’t really come naturally to us. With some practice though, you can train yourself to fall into deep study mode more easily.

1. Make a dent

Ever spent a whole day avoiding the actual hard work you wanted to do? Ended up doing busywork instead?

Everyone’s done that.

Fortunately there’s a way out of it: ignore things that are trivial but urgent (chores etc.) for a while, and make a dent in that difficult, non-urgent thing you’re somehow avoiding. Make some small amount of progress on it.

It doesn’t have to be big. It may take several attempts. It may be just a vague, uncertain feeling of having understood the subject a tiny bit better. But it can release a lot of anxiety – conscious or otherwise – and give you some energy and insight so that you can later nail the thing completely.

In math, small insights can make a really big difference towards understanding some larger concept. A single, apt diagram, a carefully remembered assumption, or an unwritten, abstract and seemingly trivial equation or mathematical fact can be enough to set your mind on a marathon to finally connect all those little dents that you made when you last studied the subject.

2. Jump around when working on questions

At least two bad things happen when you’re neck-deep in math: tunnel vision and myopia. Not seeing other parts of a question, and focusing too much on the minutiae of just one aspect of it.

Both waste time and increase your risk of failure.

So next time you find yourself stuck in tunnel vision or math myopia, stop yourself. Feel free to jump around. Go back to the question and pick out key words or concepts. See if you can quickly verify given results. (Trig identities are usually a good target for this.) Recall definitions – maybe write them down.

The point is to engage with the material at varying levels of breadth and intensity. Actively avoid the thing you were working on: it brought you into tunnel vision, so you want to get as far away from it as possible.

This can save you both study- and exam-time. Both are extremely valuable.

3. Change your views on proofs using the ‘scaffold’ method

Writing a mathematical proof can be much like constructing a building.

First, you need a construction plan, or at least a rough idea of where to begin. Then comes some scaffolding and preparation. Then the actual construction – putting things in place so they’ll stay there when the scaffolding is removed.

Suppose you’re asked to prove something in an exam. You understand clearly what you’re asked to prove, you’ve broken down the problem into smaller parts as best you can, and you’ve got a few ideas of how to get from one thing to another and then to something that is at least close to what you’re asked to show. However, something is still stopping you from actually writing down the proof, because you’re not sure you’ve got the details figured out correctly.

Cases like these are where the scaffold method is most useful. The idea is this: write the scaffolds in, then do the actual construction work.

That is, don’t worry too much about the intermediate steps and their mathematical rigour for the moment. Instead, write down what you are able to write down; for instance, any intermediate steps, leaving some free lines between them. This will set the problem out more clearly in your mind, and may even trigger memories of similar proofs that you’ve worked through during revision.

4. Don’t neglect boring stuff

As strange as it may sound, you need to be aware of how bored you are while you’re studying.

When you’re in the ‘bored’ mode, you’re just trying to get by. It’s likely that you’ll miss very important things if you just coast through a part of your subject without conjuring up actual interest for it.

You need to prevent this. Monitor your boredom levels (only you can do that accurately), and take a break whenever they spike. (When I say ‘monitor’ here, I don’t mean make lists and charts. Just use your subjective judgement.)

Taking a break in such situations is effective because everyone’s level of enthusiasm for something drops once they’ve spent a long time expending mental energy on it. Enthusiasm is replenished with rest and time spent doing more enjoyable or relaxing activities.

So take that break. You will lose a bit of study time, but as long as your time-management is not too far off you won’t lose anything important. Try not to think of this as lost study time. A difficult, less interesting subject is best studied with a fresh mind, not an exhausted, interest-starved one that will make all the wrong connections at the worst possible time.

Sound obvious? Maybe so. You should be taking breaks anyway. But monitoring your boredom levels is a good way of measuring a lot of things in one go. Boredom is complicated. Taking that extra break because your boredom indicator goes critical will help you avoid missing important bits of your subject.

5. Watch out for behaviour deemed ‘interesting’

Sometimes you’ll get questions like this:

a) Investigate the function f(x), particularly its properties with regard to |some topic on the course|. (4 marks)

b) Draw suitable conclusions from this. (2 marks)

I call this a professorial question – it’s asking you something in a roundabout way. These questions might explicitly ask you for ‘interesting’ behaviour, or they might just instruct you to ‘draw suitable conclusions’ from some mathematical argument. In a lot of cases, what they’re really asking you to do is to link the question to ‘interesting’ behaviour that you’ve previously studied in your course. Physics exams are especially prone to this.

It’s like you almost need to know what to do in advance.

The trick is to identify ‘interesting’ special behaviour before the exam starts.

Whenever your lecturer, teacher or textbook goes on about so-called interesting behaviour, pay extra attention. You may be looking at something that is more exam-relevant than you think.

However, it’s not always easy to spot when this is happening. Sometimes it’s obvious. Sometimes you can smell interesting behaviour when one part of a course links in to a much larger field.

For instance, in a course on algebra you may be told that there’s no general algebraic solution for any polynomial of fifth degree or greater, and to get usable numbers out of such equations, we usually need to develop numerical techniques for solving these, which in turn is a huge topic in mathematics.

6. Start with a long attention span

Mathematics contains a lot of information. Simply reading it will take up much of your mental bandwidth.

You’ll need to have a big enough attention span to process everything and avoid missing important bits. The tiniest detail can make all the difference.

But nowadays, it’s hard to start anything with a long attention span. So you’ll likely need to spend (read: waste) time trying to adjust your attention span.

Here’s what you can do. Before you start a difficult exercise, go read a slightly heady but non-math text for 15 minutes. Then start that exercise.

You’ll find that your attention span has expanded a bit.

The reason why I’d recommend reading a non-math book is that you don’t want to exhaust the math faculty of your brain. See what kind of books will work best for you.

Best part? Over time, you can read entire books like this. I’ve read The Book of Five Rings, The Art of War and most of Carl Sagan’s Cosmos using only this method.

7. Return to neutral + leave study hooks for tomorrow

It’s hard to get motivated to do a long, hard math study session.

Here’s what you do. Keep an eye out for interesting topics or problems that you’ve just encountered near the end of a study session. Then, don’t study them. Restrain yourself and leave those topics or exercises for the next session.

That gives you an incentive to get back into the material even if you are distracted and don’t start your next study session when you intended to.

There’s a passive variation on this – call it returning to neutral. That is, don’t leave a tangled mess of study notes on your desk after a long session, because when you come back to it in the morning you’ll lose at least some motivation in cleaning that mess up.

8. Use hard exercises

Pick a few hard exercises on your course. Make sure they’re likely to be relevant for your exam. Struggle with them.

When you’ve solved them, memorize some of the key insights that led you to solve the problem. That’s right: memorize them.

Why spend time on even more memorization? The reason is that math exams often reuse the same kinds of problems. So if you already to know how to solve them at a basic level, you’ll have a real advantage in the exam.

But even if those hard problems don’t turn up in the exam, they provide some of the best teaching you’ll ever get. By definition, they’re going to be weird in some way. And understanding weird things improves your mind.

More math study tips on www.exammastery.com

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