My interest in mental arithmetic has been on and off and I’m definitely not adept at the craft, but I thought I’d share my opinion.
I have a few books that cover the subject. They are full of tricks and techniques that are supposed to help in becoming proficient at the skill of mental arithmetic. Formulas, tables, shortcuts, and lots and lots of exercises.
I’ve come to think that the last item in that list is the most important. In fact, the rest feels utterly useless for me.
I tried Trachtenberg, Vedic mathematics, memorizing tables etc. (by rote, as I’m not a big fan of Loci). None of that really gave me a sense of progress or enlightenment.
I think the two most important skills pertaining to mental calculation are number sense and short-term memory. I also kind of think that memory is akin to sensing, so I feel tempted to lump those two together and call it “feeling connected with numbers.”
Techniques like the Method of Loci are undoubtedly immensely useful for committing stuff like multiplication tables to long-term memory, but I suspect you don’t need them to perform mental calculation at a level that would blow your own mind. In fact, I’ve found in my own endeavors that often amassing information may be a detriment to deeper internalization and systemic understanding of a subject, i.e. it can detach and prevent you from “feeling” it. This is definitely true for me, YMMV.
Were I to start seriously practicing again, I would invest all effort into developing a closer connection with numbers. It would be the key factor in adopting and utilizing all the tricks and techniques, whereas - in my opinion - the latter is relatively useless in teaching the former. When you know (and feel) single-digit numbers and their relations inside-out and around the block, it’s a firm ground on top of which to build the rest.
If you’re a beginner, tricks (i.e. shortcuts) may do more harm than good, although they do have the benefit of giving experiences of accomplishment and excitement early on to keep you motivated. I just think that most of the tricks are naturally occurring phenomena that reveal themselves to you organically through the progressive deepening of that “direct” connection with numbers. The ones you’d be less likely to discover on your own are much more useful after your mind is ready to fully utilize them. If you just keep crunching numbers you may one day find that you’ve accidentally memorized the multiplication table you never got around to memorizing as if a prerequisite for learning to calculate.
I would even advise against refactoring before the most basic and essential number sense has been developed to a comfortable level. If you feel tempted to simplify the problem, it should be indication that that’s the one thing you should avoid. So if you hate working with digits closer to 10 and carrying a lot, that’s exactly what you need to be doing more, not less.
My advice of course only pertains to learning, not competing. You don’t want to be fast - you want to be thorough, involved and comfortable. Increased processing speed will be a side-product of learning to think fluently. If you only push for speed, you may be neglecting something more essential to actually building up that speed later. Habits may be difficult to break and as far as I understand learning involves both reinforcing and pruning of neural connections, so I would put thought into what kind of practice routine might give my brain the most functional and extensible programming (instead of an island-skill or a gimmick).
If you have trouble visualizing the calculations (as do I), memory techniques will definitely help there, and you don’t even need a massive memory palace (or your preferred mental realm) to retain steps of up to (or beyond) 10x10 digit calculation in memory (assuming you would be using a sensible set of PAO for example).
When I was doing multiplication I tended to work from sides toward the center, i.e. if I were to calculate 59x89, I would proceed like so:
Left: 5x8 = 40
Right: 9x9 = 81
I try to stick these in memory and give them “identity” of residing at the edges. Next I get the number in between:
5x9 + 9x8 = 45 + 72 = 117
Then, in case of three-digit numbers, I would reduce them down to two by moving the hundreds to the tens of the number to the left:
40 becomes 50
117 becomes 17
Then I would visualize the edge pairs 5081 and stack the center pair on top of it:
Then I start reading:
1 0, 1
7 8, 15
5 2 5 1
Of course, this is a terrible way of doing it and you may well be ahead by miles, but it’s something that felt natural to me at the time. It’s a trick for managing the process and a poor one at that. The best I have managed was 36^6 using cubing and squaring in sequence but the speed wasn’t spectacular (I think it took me over 40 minutes!). I kind of lost the motivation after struggling too much with visualization while not having the interest to learn to utilize more robust memory techniques. I also forced myself to memorize squares of numbers up to approx. 60. After all the effort I felt my number sense was as abysmal as ever, and I found myself compensating for it by using tricks. Next time I know better and will concentrate fully on addition and subtraction (complements and carry).
I don’t totally downplay the ingenious tricks taught in some of the literature I’ve read on the subject, but for me they mostly felt too context-specific and gimmicky. I believe that mental arithmetic is more a simple, acquired skill and not an intellectual endeavor. Focusing too much on theory and techniques and too little on repetitive, monotonous drills ended in disappointment for me. Now, I’d leave all those books on the shelf and start grinding column addition instead!