7 ms·

I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is re

by D13Fd 6y ago

I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition. For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases and all of the more advanced concepts the author (and the related articles) discuss can be taught later.

prionassembly 6y ago
You can teach (some intuition for) multiplication without numbers by multiplying lengths to get areas. Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.
- magicalhippo 6y ago
  > Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?
  - prionassembly 6y ago
    Using a calculator. Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.
    - magicalhippo 6y ago
      So teaching multiplication ends up as "whatever the calculator does when I press this button"?
whizzter 6y ago
Having been teaching programming I can see an immediate problem with it, whilst it works great as a starter tying it too hard in the mind will require more un-doing when it comes to more advanced concepts where it isn't an appropriate level. On the other hand knowing the relation is a good hint at how to explain exponentiation. In the end teachers should be aware of what will be needed at higher grades to use tools such as it as a starting point but not ingrain it too much and try to push students onto thinking of multiplication as it's own functional primitive once they get the hang of basic tables and start moving on to long multiplication.
tgv 6y ago
You start by teaching rings, duh. But seriously, her first objection is > To define multiplication as repeated addition is to make multiplication a sub-species of addition. To me, that means the author is wrong from the get-go. Not all cases of multiplication might be easily seen as repeated addition, but to get hung up on a deliberate linguistic misunderstanding? And to add insult to injury, she finishes by proposing to see multiplication as the answer to "how many or how much OF the unit", and shows a diagram with _repeating_ units. Oh, never mind, the answer is at the end: she's got books to sell, and probably tries to improve her sales by appealing to parents with children that can't multiply 83 by 17 when they're 8.
fifilura 6y ago
Lego is a great tool for this. I also taught prime numbers by handing out (round) 1x1 pieces and told him to figure out which ones could be made into squares. The teacher was surprised when - in the middle of a more convoluted explanation for primes - he just said "primes are numbers that can't be arranged as rectangles".
- eigenket 6y ago
  *rectangles (hopefully)
  - deleted 6y ago
    [deleted]
pdonis 6y ago
> I'm at a loss how you would teach it without explaining that it is repeated addition. How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition? Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't always, it only is for whole numbers. Others might be better served by telling them up front that multiplication is a separate operation, but can be "emulated" for whole numbers by repeated addition. (As a child I was in the latter group; I hated it when teachers told me one thing in one grade, and then said "well, what you were taught before isn't actually correct..." in a later grade. How many more times would they change the rules?)
- kmill 6y ago
  Maybe I've been doing too much computer-assisted proofs recently, but I'm not understanding what all the fuss is. At some point you need a `Nat.mul` that's concretely defined, and probably the only reasonable definition is recursively using repeated addition (and if your natural numbers are in unary, which is theoretically the simplest, then even `Nat.add` is going to be defined recursively by repeated incrementing! like taking a bead from a pile one at a time and putting it into another). Then you can define an interface with abstract operations like addition and subtraction, then any given number system can try to implement this interface. Naturals, integers, rationals, reals, polynomials, rational functions, complex numbers, and many others can implement at least the semiring interface. But what's important, I think, is that there's no universal multiplication operation -- it's just a word that fills in for whatever is the right multiplication operation at the moment. It's some ad-hoc polymorphism. Unless you're accepting the real line as being axiomatic (maybe from geometry, where you can define all the basic operations using ruler-and-compass constructions), all the definitions of multiplication for all the above number systems are going to be, eventually, based on the one for the naturals.
  - pdonis 6y ago
    > At some point you need a `Nat.mul` that's concretely defined And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation. But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever. The latter claim is the claim I'm arguing against. > Unless you're accepting the real line as being axiomatic Your computer can't manipulate real numbers. It can only implement finite operations on numbers with a finite bit size, which, as above, is a set of numbers isomorphic to the integers. So if you want to work with the real numbers themselves, abstractly, then you can't expect what the computer does to carry over.
deleted 6y ago
[deleted]
Double_Cast 6y ago
I tend to think of multiplication as a transformation. "repeated additional" almost makes it sounds like we're merely telescoping an interval within a single dimension. Whereas "transformation" (to me), evokes more of a Cartesian Product or Quadrature sort of mental image.

by D13Fd 6y ago

prionassembly 6y ago

You can teach (some intuition for) multiplication without numbers by multiplying lengths to get areas. Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.

magicalhippo 6y ago
> Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?
- prionassembly 6y ago
  Using a calculator. Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.
  - magicalhippo 6y ago
    So teaching multiplication ends up as "whatever the calculator does when I press this button"?

whizzter 6y ago

Having been teaching programming I can see an immediate problem with it, whilst it works great as a starter tying it too hard in the mind will require more un-doing when it comes to more advanced concepts where it isn't an appropriate level. On the other hand knowing the relation is a good hint at how to explain exponentiation. In the end teachers should be aware of what will be needed at higher grades to use tools such as it as a starting point but not ingrain it too much and try to push students onto thinking of multiplication as it's own functional primitive once they get the hang of basic tables and start moving on to long multiplication.

tgv 6y ago

You start by teaching rings, duh. But seriously, her first objection is > To define multiplication as repeated addition is to make multiplication a sub-species of addition. To me, that means the author is wrong from the get-go. Not all cases of multiplication might be easily seen as repeated addition, but to get hung up on a deliberate linguistic misunderstanding? And to add insult to injury, she finishes by proposing to see multiplication as the answer to "how many or how much OF the unit", and shows a diagram with _repeating_ units. Oh, never mind, the answer is at the end: she's got books to sell, and probably tries to improve her sales by appealing to parents with children that can't multiply 83 by 17 when they're 8.

fifilura 6y ago

Lego is a great tool for this. I also taught prime numbers by handing out (round) 1x1 pieces and told him to figure out which ones could be made into squares. The teacher was surprised when - in the middle of a more convoluted explanation for primes - he just said "primes are numbers that can't be arranged as rectangles".

eigenket 6y ago
*rectangles (hopefully)
- deleted 6y ago
  [deleted]

pdonis 6y ago

> I'm at a loss how you would teach it without explaining that it is repeated addition. How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition? Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't always, it only is for whole numbers. Others might be better served by telling them up front that multiplication is a separate operation, but can be "emulated" for whole numbers by repeated addition. (As a child I was in the latter group; I hated it when teachers told me one thing in one grade, and then said "well, what you were taught before isn't actually correct..." in a later grade. How many more times would they change the rules?)

kmill 6y ago
Maybe I've been doing too much computer-assisted proofs recently, but I'm not understanding what all the fuss is. At some point you need a `Nat.mul` that's concretely defined, and probably the only reasonable definition is recursively using repeated addition (and if your natural numbers are in unary, which is theoretically the simplest, then even `Nat.add` is going to be defined recursively by repeated incrementing! like taking a bead from a pile one at a time and putting it into another). Then you can define an interface with abstract operations like addition and subtraction, then any given number system can try to implement this interface. Naturals, integers, rationals, reals, polynomials, rational functions, complex numbers, and many others can implement at least the semiring interface. But what's important, I think, is that there's no universal multiplication operation -- it's just a word that fills in for whatever is the right multiplication operation at the moment. It's some ad-hoc polymorphism. Unless you're accepting the real line as being axiomatic (maybe from geometry, where you can define all the basic operations using ruler-and-compass constructions), all the definitions of multiplication for all the above number systems are going to be, eventually, based on the one for the naturals.
- pdonis 6y ago
  > At some point you need a `Nat.mul` that's concretely defined And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation. But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever. The latter claim is the claim I'm arguing against. > Unless you're accepting the real line as being axiomatic Your computer can't manipulate real numbers. It can only implement finite operations on numbers with a finite bit size, which, as above, is a set of numbers isomorphic to the integers. So if you want to work with the real numbers themselves, abstractly, then you can't expect what the computer does to carry over.

deleted 6y ago

[deleted]

Double_Cast 6y ago

I tend to think of multiplication as a transformation. "repeated additional" almost makes it sounds like we're merely telescoping an interval within a single dimension. Whereas "transformation" (to me), evokes more of a Cartesian Product or Quadrature sort of mental image.