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The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach multiplication when we got to that point (which we did about a year later). I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.) I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her. With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller. It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?

hinkley 6y ago
The new new math isn't just multiplication is addition, it's addition and substraction. 39 x 40 = 40 x 40 - 40
- EvanAnderson 6y ago
  That seems like a reasonable way to do that computation, from a purely practical perspective. Simple "tricks" to arrive at the correct answer quickly don't at all seem like a bad idea. When we were doing addition of two digit integers and working with place value there was a lot of adding the ones separately from the tens, then adding those together. Later, when addition with carry / regrouping came up we used both strategies to solve the problems and demonstrated how they really were the same thing, just following a different "recipe" to get the result.
matheist 6y ago
Math PhD here. You're a fantastic dedicated parent and you did no harm. First, as to the merits of your approach --- I think of multiplication as scaling myself. That's totally valid. Second, whether it's even possible to do harm here --- worst case scenario, the metaphor doesn't make sense to your kid and she doesn't use it in her own thinking about mathematics. Everyone has to develop their own intuitions --- like you had your epiphany --- and she'll develop hers even if you set her on a track that doesn't work, just as part of her learning and doing mathematics.
- EvanAnderson 6y ago
  I appreciate the reply. I didn't feel too terribly off-base, but seeing the discourse here gave me cause to question.
maxerickson 6y ago
I suspect that the problem isn't introducing a given model, rather the problem is then not moving on to additional models. Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.
- EvanAnderson 6y ago
  I will say what we did work thru some multiplication of integers as repeated addition and scaling, just to show that the answers "come out the same". I used the scaling by a fraction to show that the repeated addition method doesn't work on all classes of numbers. (We've talked some about the difference between integers and real numbers.) The gist of my statement was something like: Adding repeatedly is a fine method to solve this kind of problem but it doesn't for all classes of numbers. We can't say, like we can w/ addition and subtraction, that multiplication implies a specific direction of movement (towards or away from zero) on the number line.
  - maxerickson 6y ago
    I wasn't really trying to analyze your approach, more speculating that getting in front of the teacher probably doesn't matter. It's the insight that multiplication isn't strictly equivalent to addition that matters, and illustrating that they are often equivalent probably doesn't block that insight.

hinkley 6y ago

The new new math isn't just multiplication is addition, it's addition and substraction. 39 x 40 = 40 x 40 - 40

EvanAnderson 6y ago
That seems like a reasonable way to do that computation, from a purely practical perspective. Simple "tricks" to arrive at the correct answer quickly don't at all seem like a bad idea. When we were doing addition of two digit integers and working with place value there was a lot of adding the ones separately from the tens, then adding those together. Later, when addition with carry / regrouping came up we used both strategies to solve the problems and demonstrated how they really were the same thing, just following a different "recipe" to get the result.

matheist 6y ago

Math PhD here. You're a fantastic dedicated parent and you did no harm. First, as to the merits of your approach --- I think of multiplication as scaling myself. That's totally valid. Second, whether it's even possible to do harm here --- worst case scenario, the metaphor doesn't make sense to your kid and she doesn't use it in her own thinking about mathematics. Everyone has to develop their own intuitions --- like you had your epiphany --- and she'll develop hers even if you set her on a track that doesn't work, just as part of her learning and doing mathematics.

EvanAnderson 6y ago
I appreciate the reply. I didn't feel too terribly off-base, but seeing the discourse here gave me cause to question.

maxerickson 6y ago

I suspect that the problem isn't introducing a given model, rather the problem is then not moving on to additional models. Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.

EvanAnderson 6y ago
I will say what we did work thru some multiplication of integers as repeated addition and scaling, just to show that the answers "come out the same". I used the scaling by a fraction to show that the repeated addition method doesn't work on all classes of numbers. (We've talked some about the difference between integers and real numbers.) The gist of my statement was something like: Adding repeatedly is a fine method to solve this kind of problem but it doesn't for all classes of numbers. We can't say, like we can w/ addition and subtraction, that multiplication implies a specific direction of movement (towards or away from zero) on the number line.
- maxerickson 6y ago
  I wasn't really trying to analyze your approach, more speculating that getting in front of the teacher probably doesn't matter. It's the insight that multiplication isn't strictly equivalent to addition that matters, and illustrating that they are often equivalent probably doesn't block that insight.