I think that I’ve done pretty well with Leo in math. Of course, I’m biased, so don’t take my word for it; We recently got back the results of a standardized nation-wide math test that Leo took at the end of last year — that is, at the end of 4th grade — on which he scored …well, without getting into too much detail, let’s just leave it at, pretty darn well, scoring well above his grade level.
Now, granted, being a US Standardized test, the performance expectations are not extremely rigorous. But off the strength of that confirmation that we’re doing something right, we decided to kinda go for a reach goal, and set Leo’s 5th grade personalized math goal is to get 80% correct on the real SATs (possibly including sitting for one at the end of the school year!)
Now, there are a few issues with Leo’s math. One is that although Leo can definitely do a lot of math that’s well beyond his grade level, he’s really bad a explaining himself — what’s sometimes called “showing your work”, or even just explaining it to me (so it’s not just a problem with writing things down in an organized way, although he definitely has that problem as well)! This is a problem, but it’s hard to tell whether this is a math problem; Explanation is in-and-of-itself a huge separate skill and a topic that I’m planning to write about separately.
A related question is whether Leo actually “understands” what he’s doing? That is, even if he can get the right answer, does he “understand” the underlying principles?
Like “explanation”, “understanding” is a huge and complex topic that deserves its own treatment. One way in which explanation and understanding are related is that you pretty much need to understand what you’re doing in order to give a coherent explanation. The opposite is a little more complex: It may be that Leo can’t give good explanations because he doesn’t really understand the math he’s doing, or it could equally well be that he’s not good at giving explanations, which is (as above) in-and-of-itself a complex skill. The fact that he gets the answers right is some evidence to the positive, but the question remains a valid one; he could just be very good at rote procedures.
Now, before going further, there’s another very difficult problem, that is, what does it actually mean to “understand” math? I’m obviously not going to deal with that here; I’m not even competent to deal with that; I’m not sure anyone is. That’s more like a matter for philosophy of math, not my areas of expertise, which are closer to cognitive science.
So, not dealing with either of the hard questions of what understanding or explanation really are, I think I can address the “Does Leo actually understand what he’s doing?” question pretty clearly in practical terms. At the same time, I’m going to explain something about my teaching philosophy, because these are closely connected.
In a previous post I wrote a brief aside about my “educational (or teaching) philosophy”. Here’s what I wrote there: “[O]ne of its pillars [of my teaching philosophy] is this: You only get people’s (esp. children’s!) attention for a couple of minutes at a time, so be sure to do tiny fun things, and build them up over days, weeks, months, and years to reach where you want to go.”
As stated, this is only “one of” the pillars of my teaching philosophy. Actually, a much more important principle is one that I was inspired to by an actor who I happened to hear interviewed on the radio (probably Fresh Air), many years ago. This particular actor’s family moved to France when she was a child. She did not speak any French, and the interviewer (probably Terry Gross) asked if that was hard? In response the actor said something like: “I’ve never thought that I needed to completely understand everything I read the first time through.” (Nb. I’m paraphrasing, of course, as I can’t remember the exact words. This is probably not at all what she said, but it’s the bit that I recall.)
One of the reasons that this idea spoke to me is that I have had exactly the same experience with every math course I’ve ever taken! I floundered the first time through pretty much every advanced math course, but the second or third time I encountered the same topic, I understood more of it, and then even more of it, until I pretty much got it entirely.
From these two thoughts — First, that you shouldn’t expect to understand every concept involved in what you are doing in detail the first few times you encounter them, and Second, that you only get people’s (esp. children’s!) attention for a couple of minutes at a time — comes my entire teaching philosophy: Grab kids attention by working interesting and challenging problems from very early on, and don’t worry too much about whether they understand everything about what’s going on, and repeat a lot. If you can get and keep kids attention as they encounter the same concept 10 then 20 then 30 times in 10 then 20 then 30 different interesting settings, they’ll understand a little more at each level, until they pretty much understand everything there is to understand about it…or at least enough to count as a high school or undergraduate “understanding” of math.
Importantly, this isn’t the same theory as “throw ’em in the deep end and they’ll sink or swim”; it’s closer to scaffolding, but a version of scaffolding where you do interesting problems, no matter how hard they are (or almost no matter, anyway), and work them together, and eventually, after doing 10 or 20 or 30 problems that conceptually overlap one another, they’ll have got it all figured.
The key is to keep it fun, and, frankly, early math just isn’t all that much interesting. So instead we mostly work AoPS AMC 10 and 12 contest problems, SAT practice problems, and as many math puzzle books as I can get my hands on. (Yes, the ones from Martin Gardner, of course, but there are so many more, and his are good, but actually aren’t actually the best!)
Returning the question of whether Leo actually “understands” — whatever that means — I want to give an an example that came up just this morning, which I think points this out perfectly. We were working this problem from an SAT practice test:
Which comes from here: https://cdn.kastatic.org/KA-share/sat/2-5LSA08-Practice3.pdf
The SATs, or at least the SAT practice tests, are apparently intentionally confusingly written. The extra “a” factor, and using “c” and “d” to represent the vertex, are just extra confusing pointless noise. I was even having trouble figuring this one out, even though, as you’ll see in a second, it’s actually nearly trivial. (There’s another problem in this same test that’s even more confusing; I showed it to a mathematician and even he was confused by it!)
I tried to talk Leo through turning it into vertex form, and a couple of other over-complex approaches. It was simple enough to approximately plot the parabola, and find the zeros, esp. as they give you the factorized form, so you can just read the zeros off!, But the “a” was throwing us off … how does the “a” play into it? If you had the polynomial form, you’d have an “a” in every element of the polynomial…what role would that play?
Leo actually figured this problem out before I did (although, in my defense, not a LOT before I did! 🙂 If you know the zeros (-4 and 2) then the vertex of a parabola is always half way between them. 🤦🏻♂️!!! From there it’s easy: Set a=1 to get it out of the way, then half way between -4 and +2 is … actually, we got this wrong at first … forgot zero! … it’s -1 (best to use the mean: (-4+2)=-2/2=-1), and then plug x=-1 into the equation, and, voila: -9. The “a” is just a scaling factor, so the answer is A: -9a.
(Just for fun — if you can call it that — we confirmed the zeros by cross-multiplying the factors into the polynomial form (aka. FOIL) and applying the quadratic formula.)
Leo made one other interesting observation, while we were considering this problem: For some reason we started talking about higher degree polynomials; I don’t recall why. I pointed out that when the degree of the polynomial is odd, X values less than zero go to decreasing Ys (because odd powers of negative numbers are negative), whereas Xs greater than zero go upward (even powers of negative numbers are positive!), resulting in the famous twisty form of the plots of odd-degree polynomials, and the famous parabola of the even ones.
Pondering this briefly, Leo said, excitedly: “Oh, I can use that to figure out whether infinity is odd or even, by plotting a polynomial of degree infinity, and if it has the left-down shape, then infinity is odd, but if it goes up, then infinity is even!”
I’ll have to think about this, but not too hard. My sense is that there’s some bug in this thinking; a the very least, I’m not sure you can plot a polynomial of degree infinity (infinity isn’t really a number — that’s one of those concepts that Leo’s about half-way to.) But, still, even if his assertion isn’t completely sensible, at least it’s coherent. It’s sort of like learning French by just moving to France — you get a few words and phrases here and there, and you hook French culture to your own understandings, and build on them over time, and eventually, you’re French!
So, does Leo “understand” all this? Well, he realized how to find the vertex (shortly) before I even did, and is coming up with interesting, if slightly outlandish, hypotheses about polynomials of infinite degree. So, okay, maybe he doesn’t get some aspects of this area of math, but heck, I probably don’t either; I learn something new almost every time we work an advanced problem together!
All in all, I’d say we’re both doing okay so far, having fun with math!