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There's a lot of this in mathematics education. Unfortunately, a lot of math teachers don't realize themselves the importance of the things that they're setting

by dhosek 6y ago

There's a lot of this in mathematics education. Unfortunately, a lot of math teachers don't realize themselves the importance of the things that they're setting up in students' education. Having been through the educational process for getting a secondary math education certification, it felt like too much of this was left implicit. Worse still, there were cases like in abstract algebra where students would do exercises on the group a+b\sqrt(-2) without any indication of the significance that this would play later in the algebra course. Heck, high school algebra, I think students learning that being able to solve for x in ax+b=c equations turns into a goal or transforming more complex problems into that simpler problem.

lmkg 6y ago
My advisor for my math degree told me straight-up "You don't really understand a math course until you take the next class in the series." I'm really glad he told me that, because it sharpened my perception of how well I understood different subjects. It also shaped which classes I took for the rest of college, tending towards depth over breadth. Too many students think that math is just an arbitrary collection of arbitrary steps. It all makes sense, but you have to have at least one teacher contextualize it for you in order for all of it to click.
- riverlong 6y ago
  Hell, I don't think anyone really understands fractions until they've taken Elementary Set Theory and constructed the reals, and I don't think anyone really understands high-school trig until they've taken real analysis, etc. It's a great feeling to come back to the "simple" topics from early in school, and to realize how much rich knowledge they communicate, and the level of detail that really sits underneath them.
jerf 6y ago
There's also cases where we have students spend a long time on truly useless stuff because none of the teachers involved understand that it's essentially useless. The example I'm seeing with my kids right now is "mixed fractions". They bang on and on about them... but they're just a special case that is basically used by cookbooks, and not much else. You don't need to know what 37 8/17s + 98 3/24s is, because the only mixed fractions you're likely to see are ones you'd see in recipes. 3½ is pretty large as mixed fractions come in the real world. I don't think I've ever seen in my entire life a mixed fraction as large as 12 4/5. They're always written some other way; 64/5, 12 + 4/5, 12.8, but never a mixed fraction. In high school, I remember spending endless hours on trig identities. They are useful only in a similarly small situation. (And I actually think they're a fun target to fire Taylor polynomial expansions at, as a good exercise for using them and understanding infinite series, but even then, rather than "memorize this pile of identities" it becomes "prove or disprove this statement:", which is different.) There's a dozen things that time could be better spent on, by purely pragmatic or purely academic standards. But nobody responsible for the curriculum realizes this. I also recall some really abortive graph theory introductions and set theory... I'm not even sure I see the value in the terrible "15 minutes in high school" types of introductions of those topics. Nobody comes away with anything of either (again) academic or practical value, despite the huge value of the topics themselves.
- frandroid 6y ago
  > "mixed fractions" I strongly disagree. Being able to bring two quotients to the same denominator, so that you can compare apples and oranges on the fruit scale, is a essential skill at not getting screwed by The Man. This is a core life numeracy skill.

by dhosek 6y ago

lmkg 6y ago

My advisor for my math degree told me straight-up "You don't really understand a math course until you take the next class in the series." I'm really glad he told me that, because it sharpened my perception of how well I understood different subjects. It also shaped which classes I took for the rest of college, tending towards depth over breadth. Too many students think that math is just an arbitrary collection of arbitrary steps. It all makes sense, but you have to have at least one teacher contextualize it for you in order for all of it to click.

riverlong 6y ago
Hell, I don't think anyone really understands fractions until they've taken Elementary Set Theory and constructed the reals, and I don't think anyone really understands high-school trig until they've taken real analysis, etc. It's a great feeling to come back to the "simple" topics from early in school, and to realize how much rich knowledge they communicate, and the level of detail that really sits underneath them.

jerf 6y ago

There's also cases where we have students spend a long time on truly useless stuff because none of the teachers involved understand that it's essentially useless. The example I'm seeing with my kids right now is "mixed fractions". They bang on and on about them... but they're just a special case that is basically used by cookbooks, and not much else. You don't need to know what 37 8/17s + 98 3/24s is, because the only mixed fractions you're likely to see are ones you'd see in recipes. 3½ is pretty large as mixed fractions come in the real world. I don't think I've ever seen in my entire life a mixed fraction as large as 12 4/5. They're always written some other way; 64/5, 12 + 4/5, 12.8, but never a mixed fraction. In high school, I remember spending endless hours on trig identities. They are useful only in a similarly small situation. (And I actually think they're a fun target to fire Taylor polynomial expansions at, as a good exercise for using them and understanding infinite series, but even then, rather than "memorize this pile of identities" it becomes "prove or disprove this statement:", which is different.) There's a dozen things that time could be better spent on, by purely pragmatic or purely academic standards. But nobody responsible for the curriculum realizes this. I also recall some really abortive graph theory introductions and set theory... I'm not even sure I see the value in the terrible "15 minutes in high school" types of introductions of those topics. Nobody comes away with anything of either (again) academic or practical value, despite the huge value of the topics themselves.

frandroid 6y ago
> "mixed fractions" I strongly disagree. Being able to bring two quotients to the same denominator, so that you can compare apples and oranges on the fruit scale, is a essential skill at not getting screwed by The Man. This is a core life numeracy skill.