Exploring Euclidean Geometry

Blame the Textbook

First published April 20, 2020.

One of the most important life skills is to be able to adjust and adapt any aspects of your plans when they are failing to achieve a goal. The rigidity of school prevents this.

Broken Toilet

Let's say one day you tried to flush a toilet and nothing happened. Suspecting that something might be wrong, you press the lever again and nothing happens. You press the lever a couple more times and still it refuses to flush. At this point a reasonable person would conclude, "The toilet is not working" and call over a plumber or fix it themselves if they know how. Now consider a student who cannot solve probability problems about the Binomial Theorem. Consider some random school textbook that has the following explanation

Binary? The possible outcomes of each trial can be classified as "success" or "failure."

Independent? Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial.

Number? The number of trials n of the chance process must be fixed in advance.

Success? On each trial, the probability p of success must be the same.

and presents problems like these ad nauseum.

A coin is tossed 10 times.  What is the probability that it shows exactly 3 heads?

Boring. I would only present something like this twice (once for the worked example and another for the check-up). At this point we get it, and want to actually do something interesting with a concept for once.

The materials are scattered, the textbook presents a total of zero worked examples, and the student doesn’t know where to actually find good materials. He tries to read the textbook but can’t understand it, and tries to do some problems but doesn't know how. So what does the average student conclude? Instead of concluding that "This textbook is not working and my teacher’s lecture makes no sense, so let’s fix the problem of having bad material," they instead conclude, "This is too hard for me. I’m not good at math. I’m just not one of the smart ones." And society completely reinforces this message. [1]

Slow Change

Let's say you tell a technician your computer isn't working because you got a virus installed on your computer. They would probably get it fixed in a timescale of days. Now let's say you tell your teacher that the textbook isn't working for you. It's boring, stupid, whatever. It would probably get fixed on a timescale of never. In fact, even if there are many legitimate problems, there will not be a substantial change. "But Common Core works fine for us" is not a valid defense either - nobody should have the power to decide how other people teach when they are so far removed. [2]

But let's say customer support sucks and your computer doesn't get fixed. What then? The wrong thing to do is to throw a fit at the world and complain about how unfair it is or how stupid it is that these people cannot do their job. You've got to fix the computer by yourself. It is absolutely unfair that the state of education is what it is and it is worth fighting to fix it. But you've got to fix the computer in the first place, and there are plenty of nice people willing to help you do this.

The smart ones

I think a big misconception people have about "smart kids" is that the reason they can skip so far ahead in school math or take 6 APs without caring or whatever is because they know more stuff. It's because they're used to a specific set of well-defined tasks. [3]

The difference is more like asking someone who has never really fixed anything to fix your toaster versus someone who works in an auto repair shop. Even though neither of them has fixed a toaster, one of them will be done much before the either. [4]

Shelve it for now

But even the broken toilet cannot fully capture the sheer absurdity of this situation. Unlike a clogged toilet, which may present some problems unless dealt with in a timely manner, not learning a certain concept should not usually cause many problems and can be shelved for later.

There are plenty of reasons that people might want to shelf an assignment for later (and not even that much later, mind you) if given the freedom to do so. In fact, there are plenty of people who choose to do this despite ominous warnings that "you will not learn the material and are going to fail this class if you take one day off" or the more silly "but your grade will drop." I do not think the institution is in a place to judge these reasons, and the best reason to take a break for the day is because you need it, not because the day is 6 or 7 when taken mod 7. [5]

Of course you should learn useful stuff eventually. But a sufficiently self-motivated person knows for themselves when eventually is, and people who aren't self-motivated really don't care what the establishment thinks "eventually" is. [6]

Silver Lining

Fortunately some teachers do not care for this approach of "you must do exactly as I say exactly when I say." Society [7] seems to think that there is something wrong with teachers giving students a little bit of leeway when they occasionally turn things in late with little to no explanation. But I think it is a sign of respect when teachers believe their students when they need a break. There are plenty of reasons why people can get out of "the zone" - the best thing to do is to help them get back in the zone, rather than pass judgement on them for getting out of the zone in the first place.

After all, I'd be very upset if every time I called a plumber over, I got blamed for the toilet not working.


[1] Side note: I had no idea this was how school taught Binomial Theorem until now. I was an ardent believer of having schools actually teach combinatorics, especially as it is the topic with the least theory, but seeing what it has done to middle school competition math has changed my mind. I'm not saying that having clear criteria is always bad. But if you're introducing people with the mathematical maturity of a 6th or 7th grader in the Math Club to counting, you're much better off saying "this is what counting problems feel like, go do some," especially when counting is the subject about intuition.

[2] In fact, I think it is explicitly harmful for people who can prompt change through an uninformed opinion to be expressing them. This is probably one of the more altruistic reasons celebrities do not make a big deal of political opinions.

[3] It does help that much of the stuff taught is repeated in 3 or 4 different classes, but I maintain that this is not the primary reason.

[4] This is not implying that getting good grades becomes useful in life later. I just couldn't think of anything genuinely useless for this analogy. See Paul Graham for why learning to get good grades is actively harmful.

[5] There is nothing special about certain periods of 24 hours in spans of 168 hours. There is very much important about the period of time when you decide "I absolutely cannot do this anymore and am burnt out, please let me stop."

[6] This is not to say good students want to care when school thinks "eventually" is - it's just easier for the establishment to force them to care.

[7] "Society" refers to everyone except for the people involved (students, teachers). Administration is included in "society."