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> You'll be crippled trying to learn things like decimals, fractions, exponents or god forbid algebra or calculus if whenever you have to add you have to be "making ten" in your head. It needs to be memorized.

/joke: they should just start with category theory -- it will produce a fine new generation of HN readers ;-)

> Thus we have convoluted methods of teaching addition like this that in some way do make sense, but are mostly just another layer of confusion.

I think everyone conceptualizes and encodes basic arithmetic stuff in their own way. Some memorize it symbolically, some visualize, some think of number lines, graphs and so on.

Trying to "help" by providing tables, coins and all kind of "aides" just confuses some students who started to encode it differently.

I think everyone conceptualizes and encodes basic arithmetic stuff in their own way. Some memorize it symbolically, some visualize, some think of number lines, graphs and so on.

I've always had a tetris thing going on in my head.

Ha. Real mathematicians start from axiomatic set theory.

If you can't grasp conceptually why 8+3=11 then you should just memorize it.

You should memorize it either way.

Imagine if every time you looked at some basic arithmetic, you had to make the choice, am I going to take the time to figure this out and lose my train of thought, or am I going to gloss it over and move on? Imagine having to gloss over 8+3 as "some number in the lowish tens." You might know how to add 8 and 3 conceptually, but to actually work it out would take some extra effort.

Thus, memorization. It's highly underrated.

I find it amusing that you say "imagine it", since it's like this for me. People think differently :)

I see 8+3 as a number in the lowish tens. If I want the exact result I quickly increment "9-10-11" in my head. Some special cases (0,1,9 and 5+5) are easier.

I guess ability in arithmetic isn't really correlated with ability in calculus, etc., since math has always been my forte.

I'm studying at a japanese university right now and I see the effect of raw memorization daily. It's not pretty. I don't think it's highly underrated.

  > ... raw memorization daily. It's not pretty.
  > I don't think it's highly underrated.

Raw memorization is dreadfully damaging. Equally, having to add 5 and 8 by counting up is a complete block to other tasks. What's needed is a range of techniques and skills, each assisting the others, and each helping students to get that insight that stops math being a collection of tricks and unmotivated rules, and starts it being a coherent set of principles, facts, tools and techniques.

I think they should start with memorization, and let each individual develop they own heuristics and shortcuts. When those abstract shortcuts are enforced and don't match with the ones a person is already using, it can get confusing.

I very much agree. Unfortunately, memorization is out of vogue in the teaching community. You don't get a PhD in teaching for saying what we already do works so you have to come up with newfangled ways of teaching that, sadly, often miss the mark. They are then, of course, quickly copied by all the textbook companies.

> You don't get a PhD in teaching for saying what we already do works

But it doesn't; at least not for everyone.

I had to quit maths at high school because it was just too frustrating memorizing apparently useless and meaningless formulas and definitions. My teacher wouldn't explain (she probably didn't even understand) what any of it was.

Later, at the university, the beauty of calculus suddenly made sense of everything, and I got a renewed interest in maths. I now hold a master's degree in algorithms and take university math in my spare time for fun.

I curse my high school math teacher's name to this day. I hope she retired early.

You mentioned Kumon, and I felt a cold chill go up my spine from a long lost, forgotten childhood trauma. <nod/>

I was good at math, but my mother ran a Kumon center as her small business, and I had to do Kumon so that she could use me as an example. I felt sorry for those kids. In the end, I think the repetition method helped a lot of the students. But the style is so different from the American educational system that, from the perspective of someone who did fine in public school, the Kumon method just felt.. dirty. Like corporal punishment.

My sister worked at Kumon for a while.

For those folks here who don't know, they're a Japanese juku (cram school) which expanded to America mostly to service the children of Japanese parents abroad. (A common fear is that a student who spends a few years in an American public school prior to returning to Japan will be academically crippled.)

I have only two things to say about mathematics education in Japan: it works very well for students in the academic track, and it isn't considered strange to have girls say they like it. So the next time you hear someone saying how we need to address gender balance, there is your answer: soul crushing drills!

I think culture might be a confounding variable...

After repeating enough times, eventually it will click and you'll understand it.

I fully agree. My parents like to tell this story about how I sucked at math and one day, at Disney World, there was a ride with cars going by. The cars had numbers on them and somehow seeing that made addition click in my head. I spent the rest of the day annoying by finding some sum for every number I saw in the park.