How to Study Maths

Below is a poster with Eight Tips for Learning Maths. A PDF version can be found here.

When learning new ideas it is natural to go through a process of gradually deepening understanding.

Stage 0 - I don't understand...yet!

The initial stage of understanding something new is not understanding it! 

Stage 1 - I can follow an explanation

The first stage of understanding something new is being able to follow an explanation or example given by the teacher or a peer.

Stage 2 - I can do a similar example

After following an example demonstrated, the second stage of understanding is being able to answer a similar problem independently. This is often called a 'Your Turn' problem.

Stage 3 - I can solve problems now

The third stage of understanding is being able to apply your new knowledge to solving unfamiliar problems that use the knowledge in different ways. Doing it immediately after learning it, students are usually cued to the fact they need to use their new learning to solve the problem.

Stage 4 - I can solve problems at a later date

The fourth stage of developing understanding is to be able to solve unfamiliar problems after some time (a month or so) has passed. It is common for students to be able to solve problems in the lesson when it is taught, but to forget a few weeks later. This is completely normal. But to create durable understanding, students need to attempt to solve problems again after a break in time. Teachers sometimes use starter activities or homework to try to emulate this.

Stage 5 - I can use flexibly out of context

The ultimate level of understanding is being able to use knowledge in a variety of different scenarios, when there is no hint as to what knowledge is required explicitly given. This is what experts in a topic can do. This could be solving a quadratic equation that appears in a worded problem or knowing you need to add fractions to solve a particular geometric problem.

The barrier to this level of understanding is usually spotting when you need to use a particular piece of knowledge. Novices usually get stuck on the surface details of a problem (e.g. the context surrounding the problem), whereas experts can easily look past the surface elements and see the deep structure below. This allows them to easily identify what mathematics they need to use to solve the problem.

The word 'test' makes many people anxious. However, here we are not talking about a test given to students by their teacher to assess them. We are talking more broadly about testing your own knowledge.

The "Testing Effect" is one of the most solidly researched findings in cognitive science. Essentially it states that if you test yourself on what you want to learn, then you will learn it better than if you 'study' it (read it, highlight it, etc). The basic principle behind this is that to learn something, you need to think hard about it, and testing yourself makes you think hard.

Test-Enhanced Learning (Henry L. Roediger III and Jeffrey D. Karpicke) discusses two experiments that showed this (though there are many more examples). In the graph below from the article, we can clearly see that the group who Studied four times (SSSS) performed significantly worse than the group who Studied once and then Tested themselves three times (STTT) after a weeks delay.

This means that when studying Mathematics, you should be testing yourself as much as possible. But what does this mean? Actually answer questions! And do it without reading your notes. 

It is very easy to trick yourself into believing you can do something because you understand the notes or the example, but following an example is very different to being able to apply knoweldge to solving an unfamiliar problem (see stages of understanding above). If you watch a video, or read your notes, you are not doing the thinking, and so you are not doing the learning.

The power of testing is all about you (or your teacher, friends, parents) testing yourself to see if you can do it. There is even a paper called the Ten Benefits of Testing, which sumarises how beneficial this approach can be (a summary is provided below).