The topics on this forum have helped me a lot to step up my mental math skills. Currently Im practicing two by two digit multiplication and three by three digit multiplication.
I currently use a training app where I see the numbers. If I would only hear the numbers ( i.e. if someone would ask me a question) it would take me a lot longer to perform the calculation because I also have to memorize the problem and cannot only focus on solving it. Are there any steategies I can pursue that deal with remembering the problem that needs to he solved?
If I understand, you want to multiply a pair of either two-digit numbers or three-digit numbers. When someone tells you those numbers, it gets difficult to remember the numbers long enough to start doing the calculations in your head.
If that is the case, it sounds like you need to put those numbers into your short term memory so you can remember them long enough to start doing the calculation.
Do you have some kind of encoding system for two-digit or three-digit numbers? i.e. for any given number, do generate some kind of mental image for it so it will be more memorable?
If I am wrong about what you are having trouble with, maybe you can break down more where your attempt is failing. If you can focus on one thing that is a barrier to your success, people here will have ideas for how to deal with that.
So I do this kind of calculation occasionally, and I do have some tricks up my sleeve, without using memory systems…
One trick is to visualize/imagine the numbers in a position suitable for (left-to-right) criss cross mutliplication.
That means you would visualize the numbers in your head, and it would look something like this when imagining it:
532
451
If you cannot visualize the numbers well, practice is the key to get better at visualizing.
The key point to this is that you have to see all the numbers at once (in your head). If you only focus on one part, you will have problems seeing the calculations needed for the other parts and it will take you a bunch of time to re-visualize it all.
I would recommend doing an exercise, in which you imagine the two numbers without doing any calculation, but just “looking” at the groups you would calculate with, in order…
The visualization exercise
So let’s say you wanted to do (left-to-right) criss cross multiplication on 73 and 61… you would imagine the numbers:
73
61
and you will first look at the 7 and 6 as if you were to calculate with it. Then you look at the cross - the 3 & 6 and the 7 & 1. then the 3 and 1. Note: do not perform calculations in this exercise - you only train your ability to see the number groups here.
Another trick is to find patterns throughout the digits of the numbers. You can see repetitions or arithmetical relations between digits.
Take 147, for example… it starts with a 1 and every next digit is 3 bigger than the last.
Or take 369… this one is pretty obvious, having a very similar pattern. But it also is the first three multiples of the number 3.
Or 428 - 4 times 2 is 8. or you could think of it more complexly and think 4 plus 2, 2 times is 8 → 4+2+2=8.
Fun fact: The more ways you represent the number, the more likely you are to remember it. That’s because you have more cues to catch onto when you forget the others.
Another tip from me would be to say the intermediate results quietly to yourself… This is helpful if you are quick enough (or have a good enough auditory memory) to get the next result before you forget this auditory memory.
Auditory memory can be trained too, though.
I might later recall some other things I utilize, if there are any more… I will write another reply then.
I am engaged in a similar project and like you I work from sight of the problem, on screen or printed out. I haven’t done much without sight of the problem but when I do, I find that, with effort, I can visualize the problem and work from that image. There’s a tendency to get confused when doing the cross products but with practice and effort that comes too.
After all, even if you work from sight of the problem, you must hold partial products in your head, perform carries etc, this is not much different.
Another thing I do to remember the numbers is something like spaced repetition, just short term.
Whenever I calculate with a number without seeing it, it would be spoken out loud.
At first, I rely on the auditory memory, but before that gets forgotten, I remind myself of (or imagine) the number, then wait like 2 seconds, remind again, then wait like 5 more seconds and keep increasing it in a kind of exponential manner, for the time during which you will use the number.
Let’s do an example where I use previously mentioned tricks… 517,913 (= 517 thousand 913)
517 and 913… 17 and 13 add up to a nice number of 30. 30 is a multiple of ten - that’s the reason for it being nice.
5 and 9… those are 4 away from each other. There are 4 sides to a square, which could serve as some kind of mnemonic - imagining a square.
These kinds of mnemonics help me remember the original number for when I switch to visualizing intermediate results.
Can’t believe I didn’t say anything about short term spaced repetition.