[gamegoblin]

In every elementary school I have encountered in the US, one learns the multiplication tables up to 10 in 3rd grade (age ~8). Some schools go up to 12 (presumably because imperial units are somewhat base 12?).

Edit: On a lark, I just tested this on my wife who is not mathy whatsoever, and she was able to remember them up to n=12 (with a slight hesitation for 7^2 == 49).

[wrycoder]

For another lark, go to Kahn Academy and check out how he teaches arithmetic up to third grade.

You have to look really hard to find addition and multiplication tables. They’re there, with the rather lame comment that they’re good to know.

But, they are not in his main stream, which depends on gimmicks to figure out the answer, instead of just rote memorization of the tables (which is good for the brain, as well).

As an example, I know my multiplication table just fine through 12. But, something went wrong in the second grade, and my addition was terrible. I depended for years on tricks: to add three, count up very quickly on the three points of the numeral 3. Same for four. For five, count one point twice.

I was in trouble at adding seven, because I couldn’t do something like add three twice!

I was shocked that Kahn uses tricks like that.

[knaik94]

It's not worth feeling shocked over. When you are approaching fundamentals, it doesn't matter what method you use. The education system failed you. From my experience teaching kids with math anxiety, I learned it's impossible for people to accurately judge where their weakness at the core. But a pattern I found was that they were scared of doing basic things the "wrong" way. Short-sighted teachers worry about the "wrong" way because they feel like it will slow kids. The proper approach is to let kids use whatever way is fastest for them, and they'll learn other methods on their own as they continue practicing problems. Teachers who force a specific method for arithmetic make the problem worse because the kids end up more worried about doing it the right way rather than worrying about getting the answer right.

We live in an age with calculators, no one cares if you use your fingers to count. I have ADHD and feel limited by my working memory often, using fingers or repeating a number I want to remember over and over feels like having extra RAM. Even the way kids are taught to count is different depending on where you live. Studies show that kids who use fingers are stronger in quantitative reasoning. But growing up, I knew teachers who made fun of students for using fingers to count.

Imagining numbers as dots and counting or breaking a number into smaller numbers to add is not a "trick" it's an algorithm that is as valid as any other. It's counterproductive to associate the word "trick" with "wrong".

For a while I wrote my own system of dots to correspond with numbers, 1, 2, 3 I focus on the end points, 4 (I wrote it open) makes a square with four corners if you ignore the extensions, 5 I count when I change directions and the end points, 6 I imagine dots of a domino tile, 7 is basically two layers a four and then the end points of the character, 8 is similar to six but I count the two circle, and 9 is similar to six but I count circle and then both sides of the bottom curve (a 3x3) grid.

Even if my brain gets tired or distracted, I know I can still add by dots because it's so procedural and I don't need to "think", I just remember the starting digit and then count up as I follow the dots. I use saying the word out like as a form of RAM to this day. Repeating a word, to me, uses a completely different part of my mind, so I free up 100% of working memory and cognition. I have "forgotten" numbers while doing mental math and have reminded myself from hearing myself say it. Describing these techniques, I recognize I sound like a literal computer and almost not human, but it's struggle I learned to work past. It works, I can do relatively more advanced mental arithmetic compared to peer even.

For multiplication, I would recommend Anki. This kind of memorizing is what that entire system excels in.

https://www.theguardian.com/science/blog/2012/jun/26/count-f...

[opsunit]

I do the dots thing too. I've never heard anybody else describe it and, to be honest, it's quite comforting to hear I'm not the only one. In fact I don't just imagine dots, I imagine die faces. This obviously gets problematic after six. I too feel limited by my working memory: it makes mental arithmetic of numbers with two digits very very hard without an external store like fingers.

[wrycoder]

So, our methods for addition are nearly the same. I like your seven. Eight is just add ten and subtract two, and nine is subtract one.

But the thing is, doing addition and subtraction should be as fluid as multiplication. Your dots and my points are slower than having the table memorized, where the answer just appears in your head instantly. And so I made an effort to learn the table later in life. But, my memory isn’t great for that stuff anymore, so it wasn’t easy.

The tricks don’t work very well when you are asked to count backwards from one hundred by sevens. (Which used to be on a standard cognitive test!)

I don’t think the system failed me - it was a good school - so, I think it was a joint effort. I didn’t learn the table properly for an unknown reason, and the school didn’t realize that, because my methods worked and were never tested in competition.

I don’t think the tricks are “wrong”, just suboptimal. That’s why I think Kahn should teach tables, and only fall back on other methods when necessary.