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In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + c) = ab + ac$].

What's bothered me for a long time is that while combining like terms [$ax + bx = (a + b)x$] is equivalent to distribution, it seems more basic and fundamental. It's used as an addition process (instead of a multiplicative one); it seems commonsensical in terms of unit addition (3 feet + 5 feet = 8 feet), which was is referenced by some as the "great principle of similitude"; and so forth.

So what is the rationale for taking distribution as axiomatic, and proving combination afterward? Why is it not better pedagogy to take combination as fundamental, and then prove distribution from it?

(Edit) I've cross-posted this question on the Mathematics Educators site: https://matheducators.stackexchange.com/questions/9601/why-is-distribution-prioritized-over-combining

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    $\begingroup$ Note that "distribution" and "combination" as you appear to be using them are axiomatically equivalent, i.e., we define them to be two sides of the same process. We give it the name "distribution" or "distributive property", and if we were to try to use the term "combination" we would run into trouble with statistics... $\endgroup$ – abiessu Sep 2 '15 at 2:56
  • $\begingroup$ More specifically, $ax+bx=(a+b)x$ is a two-way definition; you can apply it in either direction across the equal sign. $\endgroup$ – abiessu Sep 2 '15 at 2:58
  • $\begingroup$ Now crossposted to MathOverflow: mathoverflow.net/q/218003/1916 $\endgroup$ – Zev Chonoles Sep 11 '15 at 3:59
  • $\begingroup$ And crossposted to Math Educators SE: matheducators.stackexchange.com/q/9601/5404 $\endgroup$ – Zev Chonoles Sep 11 '15 at 5:35
  • $\begingroup$ And deleted the cross-post on MathOverflow. $\endgroup$ – Daniel R. Collins Sep 12 '15 at 2:03
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(The question was cross-posted to MESE; below, I cross-post my answer, as well.)

This grew a bit long for a comment.

(My first note is similar to Alexander Woo's remark about factoring polynomials; perhaps he intended "polynomials" to subsume the case here, in which we add constant functions...)

Given $413 + 91$, it may not be clear that this can be re-written as $7 \times 59 + 7 \times 13 = 7(59+13)$.

(Plenty of people seem to believe that $91$ is prime; an earlier comment cites J. Conway as observing as much, and I would guess it is related to memorizing the $10\times10$ or even $12\times12$ times tables.)

Meanwhile, $7(59+13)$ can have the $7$ distributed mechanically.

Still, the act of viewing an equation from both perspectives is certainly important, and I think what you observe here is not altogether different from the tendency to write $1 + 1 = 2$ significantly more often than $2 = 1 + 1$, which is known to cause problems with viewing the equal sign, $=$, as an operator, i.e., operationally rather than relationally, cf. MESE 7964 and the nice response of D. Hast.

You did ask for some rationale; perhaps we can look historically to Euclid's Elements. If we are to believe the translation provided here, then we have Euclid remarking in the same order. A similar translation is found, e.g., in the following source:

Drucker, T. (Ed.). (2009). Perspectives on the history of mathematical logic. Springer Science & Business Media.

A note from the aforecited:

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Finally, a separate but related comment: I observe a prioritizing in the presentation of the difference of squares and their factorization, i.e., more often than not I see: $a^2 - b^2 = (a+b)(a-b)$.

This phenomenon seems to prioritize "combining" over "distribution," and is often accompanied by the same for the sum and difference of cubes. Anyhow, there are consequences in this case, as well. Few students consider such a property in computing, e.g., $47 \times 53$.

(Ask students to compute that product in a few ways and see if it even comes up!)

I feel confident that most important will be for students to see that the same mathematical information is presented by an equality regardless of which expression is on which side, and to give students the opportunity to consider both presentations and their various ramifications.

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    $\begingroup$ Best response I've seen so far. I'm not sure it answers the "why this way first" question, but it's very thoughtful and the Drucker/Euclid reference is great. Thanks for that, and taking the time to cross-post. +1. $\endgroup$ – Daniel R. Collins Sep 14 '15 at 2:01
  • $\begingroup$ On the issue of 91, consider the most elementary checks for factors: (2) ends in an even digit, (3) digits sum to multiple of 3, (5) ends in 0 or 5, (11) is a repeated digit, or (other) is a square from the basic times tables. 91 is the only composite number below 100 that fails to be detected by all of these checks. $\endgroup$ – Daniel R. Collins Sep 14 '15 at 7:34
  • $\begingroup$ On the issue of "viewing an equation from both perspectives", one detail I'd point to is that technically this is not absolutely immediate; it also involves 2 applications of the commutative property to transform distribution into the other direction (or vice-versa). So if we're being carefully axiomatic then it's a slightly further step than some people want to recognize, I think. $\endgroup$ – Daniel R. Collins Sep 14 '15 at 7:39
  • $\begingroup$ RE: 91, right. RE: both perspectives, I mean that: $a(b+c) = ab + ac$ and $ab + ac = a(b+c)$ convey equivalent mathematical information. These are the "both" to which I refer; there is no need for commutativity (I'm not comparing, e.g., left and right distribution). $\endgroup$ – Benjamin Dickman Sep 14 '15 at 7:53
  • $\begingroup$ But again my question is in regards to combining like terms of the form $ax + bx = (a+b)x$, which is related to distribution via commutativity. $\endgroup$ – Daniel R. Collins Sep 14 '15 at 8:13

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