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> Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work

> Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math. I remember the agony I experienced as a 3rd grader for not being able to memorize "multiplication tables". Instead I took the approach of trying to understand what the mathematical meaning of multiplication is. This was to my extreme academic & mental detriment as we were not graded on understanding, but instead graded on how fast we could answer math questions. I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles. Obviously, my concern over this was totally pointless as I've never been in a career environment where I was told "just memorize the answer, don't bother trying to understand anything".

eli_gottlieb 5y ago
>This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math. Also university-level math for everyone who doesn't get up to taking proof-based analysis.
ajuc 5y ago
> Instead I took the approach of trying to understand what the mathematical meaning of multiplication is. There's not much to think about tho, is there? You take 3 boxes of 4 apples each, how many apples is there? Or were you thinking about groups, rings and fields of integers? I highly doubt it at that age ;) Ultimately you need to know multiplication tables instinctively to be able to do algebra, so they make you learn them. I don't think there's any way to skip it, just like you can't be good at football without doing a lot of cardio. For the record - I also hated memorizing multiplication tables and thought I will be a writer because clearly math isn't for me. Fortunately in the next semester we had word problems and I loved that part of math.
- spywaregorilla 5y ago
  I think there is. For example, multiplication is not visually intuitive on a number line, but it is very intuitive shown as a grid of row x columns. Doing the math out helps build an intuition for things like commutative properties and what not.
  - ajuc 5y ago
    Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times. > Doing the math out helps build an intuition for things like commutative properties and what not. Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway. I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.
    - spywaregorilla 5y ago
      > Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times. Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times. Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.
      - ajuc 5y ago
        > Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times. That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa. I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.
charcircuit 5y ago
>I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles. and this is why kids are taught to memorize these lookup tables. When doing algorithms such as long division or multiplication these lookup tables can make the process much faster.
MikeSchurman 5y ago
I don't know what the answer is, but we're currently going through this with our kids, but the opposite. There is now a push to not memorize the multiplication table, but treat multiplication as repeated addition. So to get 6 * 7 you do 6+6+6+6+6+6+6 or 7+7+7+7+7+7 (I forget which is 'correct' as it's been a few years). That's good because it gives understanding, but when you're trying to multiply 2 4 digit numbers, it's really useful to have your tables memorized! We're now going into a new school and their biggest advice to new students is to memorize their multiplication tables, as they'll have trouble otherwise. Maybe they should do understanding followed by memorization?*
- spywaregorilla 5y ago
  > I forget which is 'correct' as it's been a few years God I hope they're not teaching one of them is correct and one of them is incorrect.
  - ivansavz 5y ago
    Unfortunately it is according to some of the common core grading rubrics (in the US). 5 x 3 can be understood as five groups of three items, or three groups of five items, and students at a particular grade level are "encouraged" to use one interpretation over the other to the point of taking off points when they naturally use the commutative property (and or choose to chunk differently). I forgot what it was exactly but I read about a parent (rightfully) complaining about this. I don't think this is anywhere in the common core standards, but probably the teacher/school was using some pre-packaged rubric and the teacher didn't bother to override using better judgment.
- gavinray 5y ago
  I picked up this trick intuitively as a very young kid, perhaps 7-8 I break multiplication and division up into chunks of 1's, 5's, and 10's so that it's mostly just adding a zero to the end, or doing that and then cutting it in half. I never learned multiplication tables, long division, lattice multiplication or anything else. just did this in my head. So say you want to find 86 * 32 86 * 10 = 860 860 * 3 = (2400 + 180 = 2580) 2580 + (86 * 2) = 2752 It starts to break down for bigger numbers depending on how "uneven" they are but it's been pretty useful in general
ghaff 5y ago
I'm not sure that's a great example. The principles of multiplication are pretty straightforward. However, especially before calculators were commonplace (one can probably debate if there's sufficient value in memorizing multiplication tables today), there was a lot of benefit in doing multiplication and other simple operations quickly and mentally. There are a ton of basic things that we memorize without necessarily knowing how to get there from first principles off the top of our heads. This is true with programming as well. Sure you may look up APIs or specific syntax but if you had to look up every detail every time you put fingers to keyboard, you'd be really slow.

eli_gottlieb 5y ago

>This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math. Also university-level math for everyone who doesn't get up to taking proof-based analysis.

ajuc 5y ago

> Instead I took the approach of trying to understand what the mathematical meaning of multiplication is. There's not much to think about tho, is there? You take 3 boxes of 4 apples each, how many apples is there? Or were you thinking about groups, rings and fields of integers? I highly doubt it at that age ;) Ultimately you need to know multiplication tables instinctively to be able to do algebra, so they make you learn them. I don't think there's any way to skip it, just like you can't be good at football without doing a lot of cardio. For the record - I also hated memorizing multiplication tables and thought I will be a writer because clearly math isn't for me. Fortunately in the next semester we had word problems and I loved that part of math.

spywaregorilla 5y ago
I think there is. For example, multiplication is not visually intuitive on a number line, but it is very intuitive shown as a grid of row x columns. Doing the math out helps build an intuition for things like commutative properties and what not.
- ajuc 5y ago
  Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times. > Doing the math out helps build an intuition for things like commutative properties and what not. Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway. I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.
  - spywaregorilla 5y ago
    > Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times. Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times. Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.
    - ajuc 5y ago
      > Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times. That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa. I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.

charcircuit 5y ago

>I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles. and this is why kids are taught to memorize these lookup tables. When doing algorithms such as long division or multiplication these lookup tables can make the process much faster.

MikeSchurman 5y ago

I don't know what the answer is, but we're currently going through this with our kids, but the opposite. There is now a push to not memorize the multiplication table, but treat multiplication as repeated addition. So to get 6 * 7 you do 6+6+6+6+6+6+6 or 7+7+7+7+7+7 (I forget which is 'correct' as it's been a few years). That's good because it gives understanding, but when you're trying to multiply 2 4 digit numbers, it's really useful to have your tables memorized! We're now going into a new school and their biggest advice to new students is to memorize their multiplication tables, as they'll have trouble otherwise. Maybe they should do understanding followed by memorization?*

spywaregorilla 5y ago
> I forget which is 'correct' as it's been a few years God I hope they're not teaching one of them is correct and one of them is incorrect.
- ivansavz 5y ago
  Unfortunately it is according to some of the common core grading rubrics (in the US). 5 x 3 can be understood as five groups of three items, or three groups of five items, and students at a particular grade level are "encouraged" to use one interpretation over the other to the point of taking off points when they naturally use the commutative property (and or choose to chunk differently). I forgot what it was exactly but I read about a parent (rightfully) complaining about this. I don't think this is anywhere in the common core standards, but probably the teacher/school was using some pre-packaged rubric and the teacher didn't bother to override using better judgment.
gavinray 5y ago
I picked up this trick intuitively as a very young kid, perhaps 7-8 I break multiplication and division up into chunks of 1's, 5's, and 10's so that it's mostly just adding a zero to the end, or doing that and then cutting it in half. I never learned multiplication tables, long division, lattice multiplication or anything else. just did this in my head. So say you want to find 86 * 32 86 * 10 = 860 860 * 3 = (2400 + 180 = 2580) 2580 + (86 * 2) = 2752 It starts to break down for bigger numbers depending on how "uneven" they are but it's been pretty useful in general

ghaff 5y ago

I'm not sure that's a great example. The principles of multiplication are pretty straightforward. However, especially before calculators were commonplace (one can probably debate if there's sufficient value in memorizing multiplication tables today), there was a lot of benefit in doing multiplication and other simple operations quickly and mentally. There are a ton of basic things that we memorize without necessarily knowing how to get there from first principles off the top of our heads. This is true with programming as well. Sure you may look up APIs or specific syntax but if you had to look up every detail every time you put fingers to keyboard, you'd be really slow.