I woke up this morning thinking about sets.
I recently bought the (Canadian) French, (Mexican) Spanish, and German editions of “Dog Man”. This gave me a set of books in languages I can at least make a real effort to read, and the child already has a set of “Dog Man” books in English.
Yesterday I was thinking about sets in mathematics, and came up with this definition: A set is a collection of distinct objects.
Earlier this year, before Elon Musk killed Twitter, I learned that the set theory that I had learned in my youth was an educational blip, brought into public education by New Math in the 1960s and excised by the 1970s.
So my question this morning was: If the mathematical notion of “set” is basically the same as the non-mathematical notion, why did it fail so miserably? “Set Theory” remains the stain on New Math, the thing that causes titters in the back room from those educators who are old enough to remember it.
Relatedly, I was recently talking to my child (14, now) about his Geometry class, and he complained that there was a new symbol for congruence. Why? Why have one symbol that says that two objects are identical and a different symbol that says that two numbers are identical?
This was not a struggle I had, but this is a struggle I’ve seen other students of Geometry have, and I think it’s tied to the foundational belief of New Math that Set Theory is crucial to understanding mathematics at the K-12 level. But, again, why did it fail so miserably? Comparing a text book from the early 1970s and its late 1970s edition, “Set Theory” isn’t merely downplayed: It’s removed entirely. It goes from front-and-center to missing in the index, even while the rest of the book is effectively identical.
One major reason is that the New Math pedagogical presentation of “Set” was fundamentally flawed: It didn’t correctly articulate the concept of “distinct”, and provided examples of collections that arguably contained non-distinct objects: Stars in the sky, for instance. Yes, every star is distinct, but on a macro level, to a casual observer on Earth, they’re just dots in the sky.
But I think the piece I’ve been struggling with, driven home by the child’s complaint about “congruent” vs “equivalent”, is this concept of “object”. The true subject of mathematics is “objects”, not “numbers”, but there are times that numbers act like objects and times when they act like measurements. Most students seem to see them entirely as measurements, and so they struggle in higher mathematics (even as early as Algebra).
For instance, consider the coordinates (5, 6). This is the address of a location on the Cartesian plain that is five squares to the right and six squres up from the origin. The address involves measurements, so from that standpoint the numbers themselves are measurements. The pairing of the numbers also represents a distance (about 7.8 units) from the origin, although there are plenty of other addresses that are also that distance from the origin (the complete collection of addresses that are that distance from the origin, or from any other random address, is called a circle). So, as an “address” to a specific object (a point in space), (5, 6) represents an object, while it’s also a measurement.
This binary nature is crucial to grokking mathematics, but it’s something that is either entirely missing or at least far under the surface of modern mathematical education. New Math failed to incorporate it, and I think the reason is that they introduced it far too early, in a confusing and not quite correct way. They also then tainted the idea of Set Theory, enough that older educators who still remember New Math but never quite understood what it was trying to get at sneer at the greater picture, in the same way that the Soviet Union’s demise tainted the ideas of socialism and communism, even in the smaller systems for which they work better.
So how to move forward? Develop a more accurate, more student-friendly presentation of sets. Determine what really is the best time to introduce the concept (spoiler alert: it’s not chapter one of a first grade text).