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My grade two teacher demonstrated "venn diagrams" one day, it absolutely blew my mind. She had a bunch of coloured shapes scattered on the floor, then put one h

My grade two teacher demonstrated "venn diagrams" one day, it absolutely blew my mind. She had a bunch of coloured shapes scattered on the floor, then put one hoop over the triangles, another hoop over the red shapes, and then, what to do with the red triangles? My six-year-old brain loved it when she put one hoop overlapping the other one. And yet to this day I refuse to memorize the multiplication table. Rote learning is easier for the teacher: easier to teach, easier to mark. It's not just mathematics, it's all the subjects in school. Many of the "geniuses" merely survive school, nothing more.

soVeryTired 6y ago
I found myself thinking about multiplication tables in the shower today. I think you can get away with knowing the tables for 2, 3, and 7 really well. The rest is just tricks: 4: 2x twice 5: add a zero and divide by 2 6: a 3x followed by a 2x 8: 2x three times 9: 3x twice Beyond that, it’s repetition of those tricks, plus the distributive property of multiplication over addition.
- TeMPOraL 6y ago
  Nice, though I still prefer the trick for 9 that I was taught: 9: 10x, then subtract once.
  - jjk166 6y ago
    Another: for X from 1 to 10, the digits of 9x will add to 9, and the leading digit will be x-1. You can do this visually by putting up all your fingers and putting down the Xth one - this splits your fingers into two groups, the first is the tens place digit while the second is the ones place digit.
    - jmlundberg 6y ago
      They keep adding to nine. 137×9 = 1233 . 34457×9=310113. When it breaks down: 99x9=891 sums to 18, sums to 9.
      - jjk166 6y ago
        99 on its own is also an example where it breaks down, which is why I stopped at 10, but yes all multiples of 9 in base 10 have digits summing to a multiple of 9.
- jmlundberg 6y ago
  Also (x+1)(x-1) = xx -1 helps if you know the squares 1x3 = 2x2-1, 2x4 = 3x3-1, 3x5 = 4x4-1, 4x6 = 5x5-1, 5x7 = 6x6-1, 6x8 = 7x7-1, 7x9 = 8x8-1, ...
mcphage 6y ago
> And yet to this day I refuse to memorize the multiplication table. > Rote learning is easier for the teacher: easier to teach, easier to mark. You're right, but there's still a good reason to memorize the multiplication table: more advanced math builds upon lower level arithmetic, and so the less time you have to spend working on the low level details, the more time you can spend focusing on the higher level understanding. So you memorize the multiplication table so that you don't have to think about it anymore.
- xphos 6y ago
  I'd agree that knowing basic arithmetic is important but the issue is deeper in America. The rules and problem type issue is a direct result of uninspired math teaching. Teaching arithmetic for the sake of arithmetic is boring and uninspired and leaves a person incapable of doing both the arithmetic and math. You don't need arithmetic if you have nothing to apply it too. Arithmetic is a tool but a tool without a problem to apply it to is easily forgotten. Take this from someone who could get through advanced complex analysis but who is also bad at arithmetic. Time plus checking solves all arithmetic issues but arithmetic never solves a realistic problem. Even balancing checks(not that is useful today) if you only know how to computer you'll be at a lost until you know what to compute.
- jjk166 6y ago
  Any math higher level than arithmetic doesn't actually require doing arithmetic. If you want to find the hypotenuse of a triangle with side lengths A and B, the answer is sqrt(A^2 + B^2). It shouldn't matter how long it takes you to multiply A with itself because you shouldn't actually be doing that. Even if one insists on getting a numerical answer for say a word problem, you can use a calculator. If knowing your times tables provides any benefit to learning higher level math, it's merely in mitigating the inefficiency of bad instruction.
  - alan-crowe 6y ago
    The mathematics itself might not require doing arithmetic, but human psychology forces us to approach the abstract via the concrete. A facility with arithmetic helps with building a pile of concrete examples as a springboard for the leap to the abstract. For example, the usual example of a cyclic group is {0,1,2,3,4,5} with addition modulo 6. Maybe we start with 12 rather than 6 because it is familiar to old people used to analogue clocks. We end up with one example for each size. But we could also give {1,2,3,4,5,6} with multiplication modulo 7. Now the abstract idea of an isomorphism has a concrete example. [0->1, 1->3, 2->2, 3->6, 4->4, 5->5]. And we get a new perspective on generators. {0,1,2,3,4,5} has an obvious generator: 1. We are tempted to view 5 as an extra nuisance generator. But we notice that 6x6 = 36 = 5x7+1 means that 6x6 = 1 in the multiplicative group so 6 isn't a generator. And 2x2x2 = 8 = 7+1 == 1 mod 7, so 2 isn't a generator either. Viewed abstractly, product groups look dull and easy. The Klein Four group is a product C2 x C2, and has four elements {(0,0), (0,1), (1,0), (1,1)}. Obviously the only set of generators is {(0,1),(1,0)} and there are two automorphisms, the identity and the map that swaps (0,1) and (1,0). Those with a facility with arithmetic might play with {1,3,5,7} and multiplication modulo 8. Or {1,5,7,11} and multiplication modulo 12. But beware, you are venturing into dangerous territory. Which one of 3,5, and 7 is (0,1) and which is (1,0)? You are going to discover that the previous paragraph is wrong. (1,1) looks different from (0,1) and (1,0). It looks inferior and not a generator. But actually there are three minimal sets of generators {3,5}, {5,7}. and {3,7}. Playing with similar examples gives a second insight into what mathematicians mean by isomorphism. The first insight is that names don't matter: an isomorphism is what a ConLanger would call a relex = relexification. You invent a new language by inventing new words for the same old stuff. The second insight is that we build compound names, with internal structure. We have primitive names, 0 and 1. We build compound names (0,0), (0,1), (1,0), and (1,1). Then the mathematical concept of isomorphism treats our set of compound names as a mere dust of points, discarding the structure. An isomorphism can as easily map (0,1) to (1,1) as to (1,0) because they are just names, like 3,5,7. A loose notion of isomorphism as "same shape" can trip you up, because the structure of the names isn't part of the shape. Knowing your times tables well enough to play happily with simple examples can help learners appreciate the subtleties of advanced math.

soVeryTired 6y ago

I found myself thinking about multiplication tables in the shower today. I think you can get away with knowing the tables for 2, 3, and 7 really well. The rest is just tricks: 4: 2x twice 5: add a zero and divide by 2 6: a 3x followed by a 2x 8: 2x three times 9: 3x twice Beyond that, it’s repetition of those tricks, plus the distributive property of multiplication over addition.

TeMPOraL 6y ago
Nice, though I still prefer the trick for 9 that I was taught: 9: 10x, then subtract once.
- jjk166 6y ago
  Another: for X from 1 to 10, the digits of 9x will add to 9, and the leading digit will be x-1. You can do this visually by putting up all your fingers and putting down the Xth one - this splits your fingers into two groups, the first is the tens place digit while the second is the ones place digit.
  - jmlundberg 6y ago
    They keep adding to nine. 137×9 = 1233 . 34457×9=310113. When it breaks down: 99x9=891 sums to 18, sums to 9.
    - jjk166 6y ago
      99 on its own is also an example where it breaks down, which is why I stopped at 10, but yes all multiples of 9 in base 10 have digits summing to a multiple of 9.
jmlundberg 6y ago
Also (x+1)(x-1) = xx -1 helps if you know the squares 1x3 = 2x2-1, 2x4 = 3x3-1, 3x5 = 4x4-1, 4x6 = 5x5-1, 5x7 = 6x6-1, 6x8 = 7x7-1, 7x9 = 8x8-1, ...

mcphage 6y ago

> And yet to this day I refuse to memorize the multiplication table. > Rote learning is easier for the teacher: easier to teach, easier to mark. You're right, but there's still a good reason to memorize the multiplication table: more advanced math builds upon lower level arithmetic, and so the less time you have to spend working on the low level details, the more time you can spend focusing on the higher level understanding. So you memorize the multiplication table so that you don't have to think about it anymore.

xphos 6y ago
I'd agree that knowing basic arithmetic is important but the issue is deeper in America. The rules and problem type issue is a direct result of uninspired math teaching. Teaching arithmetic for the sake of arithmetic is boring and uninspired and leaves a person incapable of doing both the arithmetic and math. You don't need arithmetic if you have nothing to apply it too. Arithmetic is a tool but a tool without a problem to apply it to is easily forgotten. Take this from someone who could get through advanced complex analysis but who is also bad at arithmetic. Time plus checking solves all arithmetic issues but arithmetic never solves a realistic problem. Even balancing checks(not that is useful today) if you only know how to computer you'll be at a lost until you know what to compute.
jjk166 6y ago
Any math higher level than arithmetic doesn't actually require doing arithmetic. If you want to find the hypotenuse of a triangle with side lengths A and B, the answer is sqrt(A^2 + B^2). It shouldn't matter how long it takes you to multiply A with itself because you shouldn't actually be doing that. Even if one insists on getting a numerical answer for say a word problem, you can use a calculator. If knowing your times tables provides any benefit to learning higher level math, it's merely in mitigating the inefficiency of bad instruction.
- alan-crowe 6y ago
  The mathematics itself might not require doing arithmetic, but human psychology forces us to approach the abstract via the concrete. A facility with arithmetic helps with building a pile of concrete examples as a springboard for the leap to the abstract. For example, the usual example of a cyclic group is {0,1,2,3,4,5} with addition modulo 6. Maybe we start with 12 rather than 6 because it is familiar to old people used to analogue clocks. We end up with one example for each size. But we could also give {1,2,3,4,5,6} with multiplication modulo 7. Now the abstract idea of an isomorphism has a concrete example. [0->1, 1->3, 2->2, 3->6, 4->4, 5->5]. And we get a new perspective on generators. {0,1,2,3,4,5} has an obvious generator: 1. We are tempted to view 5 as an extra nuisance generator. But we notice that 6x6 = 36 = 5x7+1 means that 6x6 = 1 in the multiplicative group so 6 isn't a generator. And 2x2x2 = 8 = 7+1 == 1 mod 7, so 2 isn't a generator either. Viewed abstractly, product groups look dull and easy. The Klein Four group is a product C2 x C2, and has four elements {(0,0), (0,1), (1,0), (1,1)}. Obviously the only set of generators is {(0,1),(1,0)} and there are two automorphisms, the identity and the map that swaps (0,1) and (1,0). Those with a facility with arithmetic might play with {1,3,5,7} and multiplication modulo 8. Or {1,5,7,11} and multiplication modulo 12. But beware, you are venturing into dangerous territory. Which one of 3,5, and 7 is (0,1) and which is (1,0)? You are going to discover that the previous paragraph is wrong. (1,1) looks different from (0,1) and (1,0). It looks inferior and not a generator. But actually there are three minimal sets of generators {3,5}, {5,7}. and {3,7}. Playing with similar examples gives a second insight into what mathematicians mean by isomorphism. The first insight is that names don't matter: an isomorphism is what a ConLanger would call a relex = relexification. You invent a new language by inventing new words for the same old stuff. The second insight is that we build compound names, with internal structure. We have primitive names, 0 and 1. We build compound names (0,0), (0,1), (1,0), and (1,1). Then the mathematical concept of isomorphism treats our set of compound names as a mere dust of points, discarding the structure. An isomorphism can as easily map (0,1) to (1,1) as to (1,0) because they are just names, like 3,5,7. A loose notion of isomorphism as "same shape" can trip you up, because the structure of the names isn't part of the shape. Knowing your times tables well enough to play happily with simple examples can help learners appreciate the subtleties of advanced math.