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Is there any other way to visualise multiplication apart from repetitive addition. I've been thinking about it for quite a while now that we're introduced to multiplication as solely repetitive addition in the lower grades. Is that approach correct? Is there any other way to visualise it?

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    $\begingroup$ Area/volumes/hypervolumes of shapes? That's how ancient Greeks used to think of numbers in general $\endgroup$ Apr 6, 2017 at 16:47
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    $\begingroup$ How about areas? Is that repeated addition? Sure you can put your rectangle on a grid and count the boxes, but what is going on in your head? Another mental model is a change in scale. And if you were multiplying by a fraction, do you think of that as repeated addition? $\endgroup$
    – Doug M
    Apr 6, 2017 at 16:49
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    $\begingroup$ Peano defined multiplication as an operation that is associative and distributes over addition. It is never defined as "repeated addition" but due to that distributive property, over the natural numbers, that is exactly what it is. $\endgroup$
    – Doug M
    Apr 6, 2017 at 17:06
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    $\begingroup$ "Somehow that's a bit non intuitive to me. I needed an intuitive way to wrap my head around multiplication." Why? If you need an intuitive idea then what's wrong with repetitive addition? That is how it arose after all. It's defined purely abstractly as any method of combining to objects to get an object that follows certain rules. I could define a knight times a bishop equals a rook if I wanted to (and it followed my rules). As such, repetitive addition, area calculation, combining rearrangements of sets, rotating by angles,etc. all satisfy the rules and can be multimplication. $\endgroup$
    – fleablood
    Apr 6, 2017 at 18:19
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    $\begingroup$ I actually like stretching things longer and scaling things. Not sure why that is unintuitive to you. I like it because it avoids the mistake that you can't multiply 2 apples with 3 oranges. I also like area and work and physics problems because it always that if there were any meaningful way to conceptualize an "apple-orange" (as there are ways to conceptualize foot-pounds and man-hours) then you could multiply 2 apples by 3 oranges. Yes, they are repetitive addition on infintisimals (not really) but ... what's wrong with that. $\endgroup$
    – fleablood
    Apr 6, 2017 at 18:28

2 Answers 2

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I have recently been teaching this concept to my own child (about first grade math). My explanations have revolved around a couple of concrete representations.

  1. Length of rods

    We use Cuisenaire rods to make arithmetic operations more concrete. So, for multiplication if you have five rods of length 2, say the child can see and feel that that is the "same" as one rod of length 10. I'm amazed at how quickly my child has learned the corresponding lengths of these rods and can quickly do arithmetic by saying something to the effect of "a brown, and two reds makes ..."

  2. Area of geometric figures

    This is very similar to 1. This may use the rods again, but the pattern would be arranged in a rectangle. The question I ask in this case is something to the effect, "if we have five rows of green rods, how many white rods (unit length) is that equivalent to?"

  3. Counting groups of fingers

    This has actually worked the best for my child. For $3 \times 2$ for example, I just hold up two hands with three fingers each. We can then count them. Of course, you don't need to use fingers, apples work just as well.

I recognize that many of these have the potential to fall back on the idea of repeated addition but I don't think that is necessarily a bad thing. Mathematics is all about building abstractions. As one concept becomes more concrete (addition in this case), it becomes a block to build upon with further abstraction.

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a geometric way could use Thales' theorem and look at the multiplication as a "stretch" of the original value. This would be useful when you start multiplying for numbers between 0 and 1, since it shows that the result could be less than the number you start with.

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