*Note: This is not a polished edit, just some somewhat disorganized thoughts. Hopefully, I’ll write something more organized later.*

For a long time, I thought I understood set theory. Then, a few years ago, I realized I had somehow messed up what is a fairly rudimentary concept: That sets, by standard definition, do not have repeated elements. That is, {1, 1, 3, 5} is not a proper “set” because it has two “1”s in it.

I registered this, resisted it for a while because it went against what I’d thought, and then finally accepted it. Obviously, there is a useful concept of a collection of things that may have repeated elements, that’s just not typically called a “set”; it’s called a multiset, a collection, a bag, a list… there are several names.

It was only recently that I got a clue as to why I would have messed this up, when Matt Salomone (@matthematician on Twitter) posted a screenshot of a quote from a 1967 book on mathematics education (*Contemporary Teaching of Secondary School Mathematics* by Stephen S. Willoughby), p. 52, on teaching set theory to first graders.

I was confused because, having attended elementary school in the 1970s, I thought that set theory was just a thing that had always been taught in early grades (in a rudimentary fashion), and that somewhere we just decided to stop doing it.

Down the rabbit hole!

I happened to already own a copy of *Holt School Mathematics Levels 51-63* (Eugene D. Nichols et al., 1974). I don’t know if I used that specific book, but I certainly used the series. It was published when I was in second grade. This particular volume appears to be intended for fifth grade.

Sure enough, it starts with “Sets”. I have since tracked down four of the other books in the series (apparently 1^{st}, 3^{rd}, and 4^{th} grades), and they also start with “Sets”.

Looking further, I found other books from the era about set theory for early grades, including 1972’s *The Easy Book of Sets* by David C. Whitney.

At this point, though, I’d already noticed a problem: These books were confusing sets and collections. The cover of *The Easy Book of Sets* includes a cartoon of four seahorses. Under what circumstances would we need to distinguish between, say, individual seahorses? Certainly there might be some, but if our only need is to add groups of seahorses, we wouldn’t need to distinguish them first.

Whitney’s book does treat sets correctly: Each member of a set is given a distinct letter. However, early in the book (p. 15), he discusses sets containing different types of things, giving them a letter based on their name:

“We can say: This set of pets has a cat, a dog, and a parrot. Or we can write: Set P = {c, d, p}.”

Page 18 even refers to a “Correct answer” of labelling the members of a family, as if there is only one way of doing so.

However, on page 24, he writes: “This is a set of stars in the sky It has so many members that we cannot count them all. Any set that has an endless number of members is called an INFINITE set. After listing a few of the members, you write three dots. The three dots show that you could continue writing additional members to the set. Set S = {a, b, c, …}.”

Having established that every letter representing a member of the set should be meaningfully tied to that member, he now just arbitrarily names the members of the set. On which basis is a particular star Star a? Star b?

This is not a specific criticism of this book: The same pattern is evident in the Holt books. And comparing the 1974 to 1978 editions of the Holt books leads to a Twilight Zone moment: Sets, ever-present in the 1974 books, don’t even appear in the index of the 1978 books I could locate.

Yesterday, I wrote some notes by hand, thinking about the problem with teaching addition using sets:

*Adding using sets doesn’t fully make sense, since we normally add like things, and proper sets don’t contain like things (as duplicates).*

*The idea may have been to add magnitudes of sets, but that only works if our sets contain different elements.*

*For instance: If Abe has two apples and Beth has three apples, how many total apples do they have?*

*On one level, Abe has two distinct apples, so he does have a “set” of apples. But how do we distinguish them in a way that makes sense to small children, and doesn’t set them up for confusion later on?*

*That is to say, the physical apples are distinct, but they both belong to the class “apple”. Mathematically, we want to be able to treat them as both “the same” and “distinct”.*

*This is a high concept, something children are likely to struggle with.*

*How would “adults” do it?*

*We could distinguish the apples with subscripts. Abe has the set A = {a _{a1}, a_{a2}}, while Beth has the set B = {a_{b1}, a_{b2}, a_{b3}}.*

*We can see that |A|=2 while |B|=3. Also, |AUB|=5 because 2+3=5.*

*That is, the communal set AUB={a _{a1}, a_{a2}, a_{b1}, a_{b2}, a_{b3}}.*

I think this was the basis of the original idea of teaching “sets” before addition, but I also question that it was properly thought out at the time.

Regardless, though, I certainly failed to notice the difference, and came away from the experience with the belief that sets could indeed contain duplicates. I think this is also perhaps one of the more glaring examples of the disconnect between K-12 education, where collections are easier to understand than sets, and higher mathematics, where sets are important enough to have a communally agreed upon term and an entire theory, while collections don’t even have a communally agreed upon term.

This is not a trivial disappointment. The Fundamental Theorem of Arithmetic can be expressed in terms of collections: For each positive integer k, there is a unique collection of prime numbers such that k = the product of the members of the collection. More formally, where P is a collection of prime numbers: \[\forall k\in N_1 \exists !P: k=\prod_{p\in P} p\]

We can express this as tuples and sets: For each positive integer k, there is a unique set of tuples (p, n), where p is a prime number, n is a non-negative integer, each p is distinct, and k is the product of each p^n. But that’s less elegant.

Note that, use set theory language, we don’t need to restrict k to numbers larger than one: 1 follows from the empty set.

Compare this to a standard definition: “The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979, pp. 2-3).” (Weisstein, Eric W. “Fundamental Theorem of Arithmetic.” From *MathWorld*–A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html)

Ultimately, what we call a “set” is a matter of language, which is a matter of community agreement. And I do wonder if there is in fact any benefit in discussing collections (not sets) in early grades: Was it a truly failed experiment, or was it a misfire that could be implemented correctly? I don’t know.