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In a lot of teaching (particularly math teaching) it seems that when a student doesn't understand the concept, the immediate conclusion is that there must be so

In a lot of teaching (particularly math teaching) it seems that when a student doesn't understand the concept, the immediate conclusion is that there must be something wrong with the way it's being explained. 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. The proper way, I believe, is to just have the student learn the concept by rote. If you can't grasp conceptually why 8+3=11 then you should just memorize it. After repeating enough times, eventually it will click and you'll understand it. Additionally, there's no point in trying to teach the concepts of arithmetic. 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. I believe this also holds true up to even much more complicated concepts than basic arithmetic. For example, I find if I don't understand the proof of some theorem in one of my math classes, if I just copy it out of the book by rote, (translating the notation line by line to something I'm more comfortable with) then I sometimes will suddenly understand it. -- Addendum: I believe Kumon has a good method for teaching math. They put a very strong focus on a rock solid understanding of basic concepts. If things haven't changed, they give kids timed problem sets of basic problems to solve (memorize) with the expectation of 100% accuracy.

thornkin 17y ago
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.
- Deestan 17y ago
  > 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.
rdtsc 17y ago
> 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.
- RevRal 17y ago
  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.
- xiaoma 17y ago
  Ha. Real mathematicians start from axiomatic set theory.
pradocchia 17y ago
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.
- ramchip 17y ago
  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.
  - RiderOfGiraffes 17y ago
    > ... 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.
    - rdtsc 17y ago
      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.
hallmark 17y ago
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.
- patio11 17y ago
  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!
  - silentOpen 17y ago
    I think culture might be a confounding variable...
roundsquare 17y ago
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.

thornkin 17y ago

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.

Deestan 17y ago
> 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.

rdtsc 17y ago

> 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.

RevRal 17y ago
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.
xiaoma 17y ago
Ha. Real mathematicians start from axiomatic set theory.

pradocchia 17y ago

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.

ramchip 17y ago
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.
- RiderOfGiraffes 17y ago
  > ... 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.
  - rdtsc 17y ago
    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.

hallmark 17y ago

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.

patio11 17y ago
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!
- silentOpen 17y ago
  I think culture might be a confounding variable...

roundsquare 17y ago

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.