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Pro's and cons of number line model vs color counter model

When teaching multiplication to elementary schoolers, the "number line model" and "color counter model" are both widely used techniques. Can somebody help me to understand some of the pro's and cons for either model? Thank you

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    $\begingroup$ I am not familiar with the "color counter model". Could you explain this or link to a resource? $\endgroup$
    – Nick C
    Commented Dec 6, 2021 at 18:49
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    $\begingroup$ I also have never heard of the color counter model, but it seems to be this method: youtube.com/watch?v=Yhoz1g35alw. It seems overly complicated (introducing 0 as sum and imposing an un-commutativity of factors). $\endgroup$
    – Jasper
    Commented Dec 6, 2021 at 20:36
  • $\begingroup$ I suspect that students will be most comfortable with whatever structure was used to model addition and subtraction of integers. Integer multiplication is hardly the right time to unveil a new paradigm for contextualizing integer quantities, especially if you want the students to leave the unit thinking that the rules are unified and elegant. $\endgroup$
    – Matthew Daly
    Commented Dec 6, 2021 at 23:02
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    $\begingroup$ Some educators are bent on particular explanations or approximations or tricks they like to call "strategies" or "methods". Number line and counters are just two ideas of many to help kids learn numeracy, and the more, um, "strategies" (why are they not "tactics"?) they know, the better. That is, I would not pick one over another, I would use as many as possible, each one in an appropriate context. $\endgroup$
    – Rusty Core
    Commented Dec 7, 2021 at 2:35

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The color counter model is as follows: model an integer as a collection of positive and negative particles. I will use an ordered pair $(P,N)$ to denote the number of positive and negative chips present.

The interesting thing here is that different pairs can represent the same integer. $(3,0) = (4,1) = (5,2)$, and $(0,2) = (1,3) = (2,4)$ for instance. $(a,b)$ is equivalent to $(c,d)$ if and only if $a+d = b+c$.

This is actually the official definition of the integers when you build them up from scratch. As such it is logically prior to the number line from a rigorous development. It also rhymes with the definition of equivalent fraction: in the foundations of mathematics, we actually define the rational numbers as equivalence classes of pairs of integers (ordinate not being equal to zero) with $(a,b) ~ (c,d)$ if and only if $a*d=c*d$.

This is all far too abstract for school children, but they can learn to "cancel" pos/neg pairs and "create" pos/neg pairs when it suits them. This is valuable work, and can support learning the rules for symbolic manipulation of signed numbers.

I think that both models are valuable. Negative numbers are strange beasts, and it takes a lot of work to understand their internal consistency, how their behavior is the "only sensible" way to make the properties of arithmetic work out, how they can sensibly model real world phenomena (temperature, debt, etc).

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  • $\begingroup$ Upvoted for being an excellent explanation of why primary school math teachers should be familiar with the color counter model. But the number line model is equally grounded in formalist arithmetic, since the recursive definition of multiplication in $\mathbb N$ is "repeated addition". The difference between the models is that the number line model is visualizing multiplication in $\mathbb Z$ by extending multiplication in $\mathbb N$ and the color counter method by extending addition in $\mathbb Z$. $\endgroup$
    – Matthew Daly
    Commented Dec 8, 2021 at 17:50
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As someone who never heard of the color counter model, I find it overly complicated.

I assume that the question is asked in the context of signs when multiplying integers. While both methods arrive at the correct results, I can't see any advantages of the color counter approach.

I'd rather fear that this method hides the fact - and more importantly, the reason - why products of integers with equal sign are positive.

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    $\begingroup$ is there something wrong with my first sentence? I don't quite get what you're saying. $\endgroup$
    – Jasper
    Commented Dec 7, 2021 at 14:03
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    $\begingroup$ ^_^ "[S]omeone who never heard of the color counter model" is intended to describe you but there is no reference to you in the remainder of the sentence. I believe that is what the young folks call a dangling participle. For penance, you must listen to Weird Al's "Word Crimes". $\endgroup$
    – Matthew Daly
    Commented Dec 7, 2021 at 15:18
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    $\begingroup$ Better now? Weird Al is epic! $\endgroup$
    – Jasper
    Commented Dec 7, 2021 at 20:53
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    $\begingroup$ "Better now?" - No. It should be something like, "As someone who never heard of the color counter model, I find it overly complicated" or "To someone like me, who never heard of the color counter model, it seems overly complicated". $\endgroup$
    – Rusty Core
    Commented Dec 7, 2021 at 22:20
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    $\begingroup$ When building the number systems up from scratch, we literally define the integers as equivalence classes of pairs of natural numbers under the equivalence relation $(a,b) \sim (c,d)$ iff $a+d = b+c$. This is the "color counter model". Important things for students to grapple with in an age appropriate manner. $\endgroup$
    – Steven Gubkin
    Commented Dec 7, 2021 at 23:39

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