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> Common Core teaches the concept of 'proportionality' [over a period of six years] It sounds like you're poo-pooing the centrality and depth of notions of pro

> Common Core teaches the concept of 'proportionality' [over a period of six years] It sounds like you're poo-pooing the centrality and depth of notions of proportionality in Mathematics. Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. > Maybe Common Core can teach Proportionality if you sit down with a kid and give them all six years of it over two days Hell no it can't. How in god's name are you going to teach about fractions and the Riemann integral in two days? To be clear, proportionality is extremely important in both cases -- I'm not sure what it would even mean to successfully teach either of those things without your students noticing something essential about proportionality. Obviously, a competent teacher will notice that some analogies and connections could be useful to point out and elucidate, while some are still a decade away from being useful to the student. > Teach the algorithm. Rote practice. Then teach the concept. This isn't at all contradictory with Common Core... what makes you think it is? For me, "State and motivate the concept. Teach the algorithm. Rote practice. Then teach the concept. Rote practice with the concept as opposed to the algorithm." is far more effective, but anyways, the practice and algorithm is still there. > What happens in reality, is that my kids (a) don't understand the confusing ways its presented Oh right, shitty execution. I'd be unsurprised if the teacher is as confused as everyone else. You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. I know why I'm not a teacher... > and (b) don't make the connection between the different expressions of the same underlying concept... Its actually immensely frustrating for everyone. Well, yeah, learning hard stuff usually is.

johncolanduoni 10y ago
> Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. First off, I'm not getting the proportionality => measure theory link. Proportionality figures in much more prominently in other areas (e.g. projective spaces, geometry in general). Where's the time you allot for studying PU(n)? Second, you forgot to demand that high school science teach kids how to derive hyperfine structure so they can really understand what a hydrogen atom is. You should probably give up on teaching them about any of the other elements at all. > You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. Except what you'll get is as many opinions on what to spend the 6 years on as there are mathematicians. And any discussion is likely to be dominated by group theorists thinking group theory is the most important thing mankind has come up with, analysts thinking the same about calculus, etc. Doesn't sound helpful to me at all.
- throwaway729 10y ago
  So I really had two points: 1. Some ideas (eg proportionality) recur throughout a course of study in mathematics. Pointing out the relationships between these ideas is absolutely a useful thing to do, and a student's understanding of these "simple" concepts should mature throughout their study. That's not some profound debatable point. It's just a really stupid simple observation. CC is full of these stupid simple observations that somehow become controversial because of shitty execution. 2. Shitty execution of CC is an indict of our inability to make education an attractive career choice, far more than it is an indict of CC. Having students memorize relationships between angles and points on a unit circle without making sure they can derive those facts themselves is exactly as useless as showing the derivation without giving them any practice at calculating things themselves, for example. But at least CC gets one thing right -- ideas matter. And yes, you also need practice. But rolling back CC (or adopting it) isn't going to solve the fundamental problem with math education in the US, which is that most people who are really good at math don't become math teachers. For obvious reasons. > I'm not getting the proportionality => measure theory link Do you get the link between proportionality and Lebesgue integration? Then the connection back to measure should be obvious. (edit: better example) My point is that proportionality isn't some "stupid simple" concept that CC has somehow over-complicated. It actually does recur as an idea of central importance throughout mathematics, and drawing out those connections is a good use of instructional time. > Second, you forgot to demand... That's a stupid demand. Teaching measure theory in high school as part of a standard curriculum is also stupid. But that doesn't mean that idea of proportionality don't recur, at varying levels of sophistication, throughout a standard age-appropriate course of study. > Except what you'll get is as many opinions on what to spend the 6 years on Right, well, that's kind of the whole point of having a standardized curriculum... Also, empirically, I've literally never seen this happen. Only in the context of after-school or extra-curricular enrichment stuff do people focus on topics they're passionate about, and in those cases it's entirely appropriate.

johncolanduoni 10y ago

> Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. First off, I'm not getting the proportionality => measure theory link. Proportionality figures in much more prominently in other areas (e.g. projective spaces, geometry in general). Where's the time you allot for studying PU(n)? Second, you forgot to demand that high school science teach kids how to derive hyperfine structure so they can really understand what a hydrogen atom is. You should probably give up on teaching them about any of the other elements at all. > You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. Except what you'll get is as many opinions on what to spend the 6 years on as there are mathematicians. And any discussion is likely to be dominated by group theorists thinking group theory is the most important thing mankind has come up with, analysts thinking the same about calculus, etc. Doesn't sound helpful to me at all.

throwaway729 10y ago
So I really had two points: 1. Some ideas (eg proportionality) recur throughout a course of study in mathematics. Pointing out the relationships between these ideas is absolutely a useful thing to do, and a student's understanding of these "simple" concepts should mature throughout their study. That's not some profound debatable point. It's just a really stupid simple observation. CC is full of these stupid simple observations that somehow become controversial because of shitty execution. 2. Shitty execution of CC is an indict of our inability to make education an attractive career choice, far more than it is an indict of CC. Having students memorize relationships between angles and points on a unit circle without making sure they can derive those facts themselves is exactly as useless as showing the derivation without giving them any practice at calculating things themselves, for example. But at least CC gets one thing right -- ideas matter. And yes, you also need practice. But rolling back CC (or adopting it) isn't going to solve the fundamental problem with math education in the US, which is that most people who are really good at math don't become math teachers. For obvious reasons. > I'm not getting the proportionality => measure theory link Do you get the link between proportionality and Lebesgue integration? Then the connection back to measure should be obvious. (edit: better example) My point is that proportionality isn't some "stupid simple" concept that CC has somehow over-complicated. It actually does recur as an idea of central importance throughout mathematics, and drawing out those connections is a good use of instructional time. > Second, you forgot to demand... That's a stupid demand. Teaching measure theory in high school as part of a standard curriculum is also stupid. But that doesn't mean that idea of proportionality don't recur, at varying levels of sophistication, throughout a standard age-appropriate course of study. > Except what you'll get is as many opinions on what to spend the 6 years on Right, well, that's kind of the whole point of having a standardized curriculum... Also, empirically, I've literally never seen this happen. Only in the context of after-school or extra-curricular enrichment stuff do people focus on topics they're passionate about, and in those cases it's entirely appropriate.