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Timothy Johnson's avatar

I really like this pattern! The main challenge I see with applying it is to break down a concept enough for the target audience to follow along. For example, the article that was shared about Squirrel AI says they've broken down the math curriculum into 10,000 separate steps. That's pretty intimidating for the average teacher to try to reproduce.

My current project is to try to help my two-year-old son learn to count. He loves to read his counting books, and we've read some of them literally hundreds of times. But so far he still seems to be missing the point. He thinks that counting means saying the words, "1, 2, 3", but he hasn't figured out that he's supposed to match the numbers with the objects that he's counting. (He also stops at 3 - anything larger than 3 is "so many").

After reading this post and thinking some more, I think maybe the concept of one-to-one correspondence is actually a prerequisite for counting. I have some ideas to try on how to make that more concrete for him, but I'd love to hear if anyone else has suggestions.

Lastly, I read the examples of concept ladders that you shared. Unfortunately, it's not quite true that sqrt(x^2) = x. That works for positive numbers, and maybe you're assuming the students haven't learned negative numbers yet? But in general, sqrt(x^2) actually equals the absolute value of x.

The mathematician's explanation would be that you can only invert a function that's one-to-one, but just typing that sentence makes me start to feel like this teacher:

https://www.smbc-comics.com/?id=3565#comic

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Hazel Woods's avatar

Separate comment to address a general "how could this fail?" point that applies to a lot of things related to math. There's a general principle that students should learn not just the "how" but the "why", that they should understand rather than just learn a set of boring old rules. This is at least 95% true, but it has to be done in a way that avoids a trap.

Imagine you teach someone that 1 + 2 is the same as 2 + 1, and so on, so you can do a sum in any order you like (they're probably too young to need to know the difference between "associative" and "commutative"). Maybe later on they learn that 1 * 2 is the same as 2 * 1, and there's a general pattern here. The trap to avoid is that halfway through a longer problem, they encounter 1 - 2 and remember "you can swap things around in arithmetic" and turn it into 2 - 1.

Or you teach them how to cancel products in fractions (2*3/2*5 = 3/5) and next thing you know, they cancel (2+3)/(2+5) = 3/5 too. We've all been there. Lockhart's lament, which I unfortunately couldn't join the reading group for, is big on building mathematical intuition; there's some book-length arguments on similar topics by Jo Boaler that I'm familiar with too. The trap is that if student A is encouraged to use her intuition, and generally gets the right answer that way, that's good; if student B uses their intuition and ends up cancelling sums, that's not good. Similarly with Lockhart's "area of a triangle with height h and base b" example, if student B thinks about it and comes up with "h + b" as a result, that's wrong, even if they have a mathematical-ish justification for it: "well if you increase the height then the area gets bigger, and if you increase the base then the area also gets bigger, and both of those principles apply if you add the two". There's a danger that student B learns the meta-rule "they tell me all the time to THINK, but every time I do that I'm told I've got the wrong answer. Maybe it's just a lie they tell children." The basic risk to all attempts at building mathematical intuition - which is not just good, but necessary too - is that until it's fine-tuned enough in an individual student, it can produce random results and lower grades than just learning a formula. That doesn't mean one shouldn't do it, but that one actively has to watch out and intervene to help the "student B" types along the way.

I've privately tutored a few students in the past who were struggling with school math taught in a "learn to think" style, and sometimes they were so confused by the problems that they fell back on heuristics like "find all the numbers in the word problem and multiply them" because at least that looks like they're doing something to the teacher. They were a lot happier, and their grades improved, when for some categories of problem that made up a good part of their tests we went through explicit formulas or procedures that they could just reel off in the test without "thinking". Classic example: the math rule of 3 for proportions (2 apples cost $8, how much are 5 apples?). After seeing my students struggle with the "just think about it" approach, I taught them the old-fashioned method: make a 2x2 table, "scenarios" go in rows, label the columns ("amount", "price") then multiply the two things diagonally opposite and divide by the one in the corner. That completely hides why it works, but it got some of my students from a failing to a passing grade.

A related problem is that in math especially, I find it's often much easier to get an intuition for some rule in general, than it is to properly sort out the edge cases when it does or doesn't apply. This is as true for "higher" math like calculus (famous example: de l'Hopital's rule - a minefield of edge cases if there ever was one), and the example you presented here (the intuition looks like it should extend to 0^0, which I'm sure is covered in the full version of your notes), and for example it's easy to reason from 2x=2y giving x=y and 3x=3y giving x=y and so on that ax=ay always reduces to x=y. That doesn't mean you can't build this up as a concept ladder (in fact it's a great idea), as long as you make sure the student doesn't think they've understood before they get to the warning about if a could be 0.

I think my ideas here are best seen as a complement to, rather than a rebuttal of, concept ladders - which I really like as an idea by the way! Or perhaps being aware of this failure mode is a way to make much more effective concept ladders and intuition-building exercises.

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