I used to simply visualize the computation. Since my handwriting wasn’t the greatest I quickly got comfortable doing this, and it actually improved my ability quite a lot. For a more proof orientated exam, I would advise doing math more verbally.
Are there any benefits to doing maths in your head? Improving working memory? Nowadays you probably need less mental arithmetic to get a job done, as there are calculators etc… However, if this does help other cognitive functions, I am intrigued.
Are you looking to be able to do mental arithmetic, such as 34 ÷ 56, without writing it down? Or mathematics such as calculus? Or the abstract thinking required for inventing proofs about abstract concepts?
I have ideas for each of these—especially the mental calculation since I do coaching and workshops in this—but my recommendations would be different for each.
For most people, doing basic mental arithmetic is useful as it removes a layer of friction from many activities. Reaching for a calculator might only take 10 seconds, but this is a barrier for handling numbers when it’s part of a larger process, e.g. deciding whether a deal is good enough, remaining engaged in a conversation, etc.
Advanced mental calculation like cube roots is interesting, but not directly helpful.
You’re correct that working memory is super important for mental calculation, but there is minimal evidence that it can be improved. Some drills such as N-back have some evidence and I’m going to investigate them further. But the biggest improvement in mental math comes from using the working memory you already have, but in a more efficient way, for example using the cross-multiplication method already referenced by @flou
Your advice has helped me and I have thought of visualising myself computing the Maths in parallel in 3 different views which help me see the whole picture and I am now able to get insights within seconds by using this method,
Honestly for doing better with a Grade 10 exam you are better of just writing out your working in a way that is already comfortable. Sometimes the mark scheme will even give you marks for showing your method.
A right-angled triangle has shortest side 7 cm and smallest angle 30°. What is the length of the longest side?
If you put just 14 cm then you might not be given marks for your method. But writing the following might be the minimum to get full marks:
x sin 30° = 7
sin 30° = 0.5
x = 14 cm
But if you want to get better at doing these questions without writing answers, I’d suggest the following:
Make sure you’re actually familiar with the method. If you can’t do the mathematics, you can’t do it mentally either.
Start with solving questions that are very similar. E.g. the trigonometry question above but with an angle of 45° rather than 30°, or a length of 2 metres rather than 7 cm. This will show you what the essential steps are.
Expect it to take longer at first than writing your working in full. This is because you’ll have to redo parts of your working when you lose it from your working memory. Over time you’ll be faster.
You can also start with skipping some steps. I’ll give an example below.
Solve x^2 + 6x + 3 = 0 (by completing the square)
(x + 3)^2 – 3^2 + 3 = 0
(x + 3)^2 = 6
x + 3 = ±√6
x = –3 ±√6
You might try to solve these only writing down steps 2 and 4 to start off with, and then after some practice, go straight to the final answer (step 4).
Yes, This is true and I have found that using the method of visualizing the computation of Math’s in parallel from a Question Bank with all types of questions and then practicing them on paper also improves performance and this method is very fast(Maybe only for a person who is beginning Mathematics),
Yeah, I try to use my working memory a lot during maths. I am currently in 10th grade, and I try to do geometry, algebra, quadratic as well as multiplication and division in my head. I tested my working memory to be in the top 8% (8.5 digits in digit span). I am trying to improve this though, so I can do things like calculus which I am yet to learn in my head.
I haven’t used any techniques aside from long multiplication. I am currently struggling with 3x3 digits though. I will try the cross over method, because I don’t think my capacity extends that far doing normal long multiplication.
There is a technique in which you can visualize the computation of Maths in parallel of a book or a video and then try to recall the things which you had visualized after every 10 minutes(or a different but recent time),
But I want to know whether this method wil work,
Credit to @Nagime for the idea of visualising the computation of Maths,