The Importance of Order in Mathematics Teaching

I will get onto Maths a bit later in the article – bear with me!

When I moved into International teaching, one of my personal aims was to learn a foreign language. Having always had languages as my weakest subject in school, and never really feeling I got to grips with even how to learn one, it was time to sort that out. Everyone says if you want to pick up a language, just move to the country and learn from the natives – right? Sounds simple, but I’ve come to realise that the circumstances you’re in really make a difference.

So now, I’m living and working in Japan, and I’m pretty sure I’ve managed to pick out one of the trickiest languages to learn, for a native English speaker at least. Although basically all conjugations are regular, the sentence structure, concepts, and of course the complicated writing system with 3 alphabets (4 if you include the roman script we use in English as well), all combine to provide a continual stream of confusion. In addition, conjugations can be added to other conjugations, sometimes creating words that are barely recognizable from the original ‘stem’. Combine all this with the bubble one finds oneself in when working abroad, with work in English and the majority of friends being English speakers, and you have a situation which isn’t as simple as just ‘move to the country and learn from the natives.’

However, I have started to get the hang of how to learn more effectively, given the circumstances I find myself in. I owe a lot to the Japanese learning website Wanikani, which is aimed at helping the user learn how to read over 2000 Japanese characters, in around 18 months to 2 years (The majority of these characters are in their list – take a look!]. Bear in mind this takes native Japanese the majority of their school life to learn. So, how is this possible for a non-native to learn so many in such a short time?

The answer lies in the system used, known as SRS (Spaced Repetition System). The idea being that, when you learn something new, it initially enters only your short-term memory. This means that in a relatively short amount of time, you will forget it, or at the very least take a long while to recall it. When Wanikani teaches a Japanese character, it will ask you to review it again 4 hours later. If you get it wrong, it needs to be reviewed another 4 hours later, if wrong again, another 4 hours later, and so on. If you get it right, it will come up 8 hours later. If then wrong, back to 4, if right, up to 24 hours later. This process continues with each letter, increasing the gap each time in an attempt to gradually move the character into your long-term memory. I have been using the site very successfully and it is clear that the system works, provided on keeps on top of reviews and does them as soon as possible when they come up (sleep permitting). Ultimately if you are able to keep to timings it is possible learn some characters having only seen them 9 times in total (including the initial ‘lesson’).

The image below is a visual of their SRS system. The intervals between reviews are 4 hours, 8 hours, 24 hours, 3 days, 1 week, 2 weeks, 1 month, and 4 months.

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Wait though, this is a Maths website – right? Why am I talking about this? Well, I was thinking about how all schools have a limited time for Maths (and all subjects really!), and how can we use time more effectively, and I think the SRS has the potential to make a big difference. If students see a new concept, they need to have a brief reminder of it very soon after. For example, if I have taught students about using Sine in right-angled triangles that day, I suggest to them that the same evening they should try and do 1-2 questions on it, as this helps it move deeper into their memory. Forcing them to recall prior knowledge before is slips away is the key to helping topics be remembered in the long-term. Whether they choose to do this or not is of course up to them.

However, this all involves a very carefully designed scheme of work, an example of which can be found in the Maths Free Resource Library (check the ‘Resourced Scheme of Work’ folder). As much as possible we have designed our schemes to allow reviews of prior work at regular intervals. For example, in Year 7 we have basic algebra at the start, just how to write calculations algebraically, and simplifying like terms. The next unit has some area and perimeter of basic shapes in, and in here we introduce equations, starting with a review of writing expressions. This allows students to try and remember a topic they covered not too long ago, and then they see how to apply it. All the equations in here only involve positive integers.

The first unit after Christmas has some negative numbers in it, as well as fractions, and students see how these can appear in equations, reviewing both how to write expressions and solve equations at the same time. There is an angles topic later in the year where students need to form and solve equations as well. This is effectively spaced repetition, where students revisit topics throughout the year as much as possible, rather than having them as separate blocks. There are some percentages and probability questions later in the year which have been set up to involve areas and perimeters from before Christmas. This continues into Year 8, with a review lesson at the start of each unit to remind students of work they covered the previous year. Of course, this is not a one-size-fits-all solution, but so far it has proved to be an effective way of using time.

In short, I think that in order to get the most efficient use from the limited time we have, it is vital for teachers to continually look for opportunities for students to recap prior work within the topics they are currently doing. Maths I feel offers lots of opportunities to do this, although it will certainly require a good amount of prior planning. Get the post-it notes out and write the year’s topics on them, as well as topics they can link with. Then arrange them on desks and start discussing where to put everything – that’s what we did!

Check out the Year 7 scheme of work in the Free Resource Library to see what this might look like when completed!