Oh, man. The fatal event in my failed study of engineering was getting 99% on a first-year dynamics exam. I’d reduced the semester to a page of notes, then the page to a quarter page, then the quarter page to four equations [which of course I can’t remember now—this was 1988], which it seems had been the point of the entire semester. I then proceeded to ace it. (Got a sign wrong in one problem, hence 99%.) This was absolutely fatal to doing well henceforth, since I had no idea how I’d managed to do that well except showing up at lectures and usually doing my homework.
Damn it! I wish I still had classes like this! …I love classes where literally just understanding the material (in a deep, comprehensive way) is enough to get 100% if you don’t make stupid mistakes. I suspect this is the reason I did well in high school chemistry, physics, and bio–if you tried to really grasp the underlying concepts, the memorization required was trivial, or at least it didn’t feel like memorization.
Whereas many of my classes now are pure memorization of stuff with hardly any underlying logical structure (like pharmacology...stupid list of over 100 drug names to memorize, generic AND commercial!), or based on legal standards and “best practice guidelines” which, although they must be based on research results, don’t yield easily to my attempt to find underlying concepts. One class consisted almost entirely of memorizing the names (and acronyms, in French and English) for the various nursing regulatory organization in Ontario, and the documents they released on stuff like ethics. Gaaaah. There have been so many classes where I finished with an A- not because the class was hard, not because an A+ would have been ridiculously difficult, but because the material was so boring that I literally could not make myself study for more than a few minutes at a time, and only then by bribing myself.
Weirdly enough, I probably would have preferred doing pharmacology the hard way, i.e. learning chemistry to an advanced enough level that I could understand approximately how and why different drugs have the effects that they do. This would obviously be harder, but it would also be interesting, which would make it psychologically easier–I spend a lot more willpower on studying boring things than on studying interesting things.
Damn it! I wish I still had classes like this! …I love classes where literally just understanding the material (in a deep, comprehensive way) is enough to get 100% if you don’t make stupid mistakes. I suspect this is the reason I did well in high school chemistry, physics, and bio–if you tried to really grasp the underlying concepts, the memorization required was trivial, or at least it didn’t feel like memorization.
This is why I got frustrated with foreign language class. It’s impossible to derive one word in a vocabulary list from the others. Being able to recall the words for “red”, “blue”, “orange”, “grey”, “white”, and “black” doesn’t help you at all to remember the word for “green”.
I had the same problem when learning English; at least when I had to learn vocabulary for school (my native language is Spanish, so English would be our foreign language class). Later on I had the chance to take a couple of classes in Latin.
For the last two years I’ve been learning French with great success. I’ve not yet found out how it is it was so easy for me; I suspect it came from it being close-to-isomorphic with Spanish. The thing is, when learning vocabulary I found that knowing a little bit about the use (not necessarily the definition) of English words helped me a lot to derive the meaning of new words. Whenever I had a little knowledge of the etymology of a word (for example, from the latin course), this “logical derivation” of the meaning or usage of words (or even the less-preferred pairing with a word on another language) got a lot easier.
I think there’s a little learning curve about vocabulary, after which you get better and better about deriving meaning from context and memory of previous known uses. Actually, I believe this might be what we do with our native languages; in my case at least I know I wouldn’t be able to define or precise the meaning or definition of most words I use.
Weirdly enough, I probably would have preferred doing pharmacology the hard way, i.e. learning chemistry to an advanced enough level that I could understand approximately how and why different drugs have the effects that they do.
Is this even possible with the current best theories in medical science? It was my understanding that it was no where near that advanced.
Probably not, but it’s possible to go to a much deeper level of detail than we did, i.e. learning about receptors and physiology, to the point that all you have to memorize, pharmacology-specific, is “drug X is an antagonist for receptor Y”, and the rest (uses, side effects, etc) flows naturally from that. We did some of this, for agonists/antagonists of the sympathetic and parasympathetic nervous system. (Beta blockers, i.e. metaprolol are, let me draw out this memory for a moment...antagonists of the sympathetic nervous system, which is why they lower blood pressure, because increased heart output is something you get when you stimulate the sympathetic nervous system. I would not have remembered this if I’m just had to memorize it offhand.) I’m sure we could have done more learning of this style...it might have taken 2 or 3 semesters instead of just one, though. Also, some drugs do things that medical science doesn’t understand, i.e. anti-psychotics, and that would still have to be memorized.
There are memory courses that claim to teach one to remember large quantities of somewhat arbitrary information. Anything by Harry Lorayne, for example. (One day I shall bother with one of these courses. [i.e., I probably won’t.])
I found spaced repetition systems easier to use on a regular basis than visualised-association systems such as the peg system and mnemonic major system, which were interesting to learn, but a bit cumbersome to practice regularly. Possibly I could become more fluent with practice of the latter but it’s been procrastination-inducing so far.
I think spaced repetition is for natural amount of memory (e. g. 10 new terms each class) whereas visualization techniques are for unnatural amounts (e. g. 20 digits in 5 minutes).
Related: I used a mnemonic system for a good part of my pharmacology studying, mostly by using silly phrases to match generic and commercial names. This has stuck in some surprising ways; for example, whenever I think of the drug spironolactone, a diuretic with the side effect of gynecomasty (breast development in men), I have a vivid mental image of a man in Viking war armor with large, milk-oozing breasts (the “lactone”), holding a trident (unsure what this was a mnemonic for.)
I used spaced repetition (Anki decks) to study for the RN certification exam, and probably overshot-it was quite easy.
I suspect this is the reason I did well in high school chemistry, physics, and bio–if you tried to really grasp the underlying concepts, the memorization required was trivial, or at least it didn’t feel like memorization.
This was my experience in physics, but didn’t feel at all true in chemistry and bio. I understood the concepts fine, but nothing about the concepts seemed to let me derive anything on the fly or avoid rote memorization.
One of the interesting things about taking both an intro to materials science course (basically solid-state engineering) and a more traditional introductory chemistry course targetted to roughly the same academic level was seeing the difference in the underlying approach. The solid-state chem focused on deriving the macroscopic behavior from the physical properties of the atoms much more than the trad chem.
I don’t think it was taught that way in chem or bio, but I tried to understand it that way… My parents have always bought me science books, and I had already read most of my high school library’s science section, so most of what I was learning wasn’t new. The concepts I was learning didn’t necessarily let me predict the other concepts, but they all fit together in a logical, meshed framework where they relied on each other, and I could use that to trigger my memory to retrieve particular concepts. Which is much harder in something like “nursing theory”, which a) I didn’t spend most of my childhood reading books about, and b) doesn’t hold together in a logical framework, except in some superficial ways.
I found this was a common theme in good engineering courses. The whole course would boiled down to three or four new concepts. By the time the final came around, it just felt like thinking hard about common sense even though at the beginning of the term, everything had seemed counter-intuitive.
Where are you from that the school system is sane enough to assume calculus for undergrads?
There are places that don’t? Australian here too and calculus is assumed (for undergrads doing appropriate degrees). It is assumed because the applicable subjects are either an outright requirement of the course or there are remedial level subjects required as prereqs (that most students in the course will skip) for those that somehow managed to avoid learning calculus in school.
In canada, almost nobody learned calc in high school, it was an advanced placement course. We started learning calculus in first year engineering but didn’t really start using it in other courses until third year.
Partially this was because the 4 year degree program piggy backed on a 2 year technologist program. So we learned the basics and applied practicalness of everything without calculus in the first two years, then went back and learned all the deep theory in last two years.
Ahem. I took calculus as a freshman. In highschool. I only had to retake the second half because I was so horribly sleep deprived during the final, and so lazy about homework. I then got a 5 on the AP test (score maxes out at a 5).
Now I’m not typical, but I suspect that a school system that cannot teach calculus to the top ~10% or so of its math-inclined students when they’re in high school is failing somewhere along the way.
The question is not whether high schools can teach calculus, but rather whether they should. I believe the value of calculus education at that level is not very high relative to classes on, e.g., personal finances, logic, probability, economics, civics and etc.
That’s what was meant by “sane” in the grandparent: a non-efficient use of classroom time.
“not very high relative to classes on, e.g., personal finances, logic, probability, economics, civics and etc.”
But high relative to, say, teaching a class on postmodernism.
what I do know is this: the grade school that fed into my highschool had a nifty policy for math: you go to the math class at the level your ready for, independent of your grade level. This is on the assumption that (innate?) mathematical aptitude is relatively uncoupled from other “classroom skills”.
As a result, those who don’t have a super-duper knack for math don’t generally hold back those who do have a knack for math, so the upper end of the curve is already extending itself by the time those kids get into high school.
I’m sure at least 40% of high school freshmen could learn Mandarin, but you don’t see that happening. Therefore, merely because students are capable of learning a subject, does not imply that that subject is or ought to be taught.
Seriously. I’m talking about opportunity costs.
But high relative to, say, teaching a class on postmodernism.
Philosophy is not even a blip on the secondary education subject radar.
due to my experiences as a young kid (I could challenge Authority (my parents) on a math problem and sometimes WIN, and I learned how to factor quadratics in first grade) and the fact i see the application of math almost everywhere (computers in particular, but bridges and buildings as well) means that I see math as our best tool. time to start applying it recursively.
I believe that math is usefull for a human with ~99.99+% probability. (that number discards the probability of an AI for the sake of speed)
I believe, with about a 75% confidence (for now) that having calculus as an available, but not required, course, should be possible in at least 80% of high schools.
even an understanding of a simplified calculus is usefull in many other subjects.
if (#2), then a high school unable to offer calculus is either a particularly small highschool, or is being fed by an underperforming grade school system.
There is plenty of evidence that SOMETHING in the gradeschool system is underperforming regarding mathematics.
-Which belief elements do you want me to try to expand on the most when i get back?
Which belief elements do you want me to try to expand on the most when i get back?
None. You’re still in a “can = must” frame of mind, after three attempts to explain my position. Your one engagement with the idea of replacing high school calculus with something else was, as far as I can tell, facetious.
Conversely, I took calculus in high school, didn’t understand what the heck was going on with most of the concepts, got a 5 on the AP test and an A in the class, and forgot all the material immediately.
Then, years later, I took calculus in college and understood everything.
Brown University (where my brother went) assumes that incoming freshman have already had math equivalent to the first two semesters of college calculus. My high school’s calculus course only covered Calc I.
My first-year courses in engineering (in Canada) made basic use of calculus without assuming any real understanding of it. By second-year, the calculus was assumed and for linear ODE’s and similar. Third-year, we moved to Laplace and Fourier transforms and the final year finally started to get into applications and standards and “real” things.
I’ve always wondered how different other engineering school curricula are.
Also canadian engineering school. First year we learned calculus but did not use it. Second year we learned even more calculus (up to PDEs, inear alg, and vector calc), but still didn’t use it. Calculus started being assumed in third year. We suggested to the school that they make more of an effort to use the skills that they teach between classes, so this might change soon.
Oh, man. The fatal event in my failed study of engineering was getting 99% on a first-year dynamics exam. I’d reduced the semester to a page of notes, then the page to a quarter page, then the quarter page to four equations [which of course I can’t remember now—this was 1988], which it seems had been the point of the entire semester. I then proceeded to ace it. (Got a sign wrong in one problem, hence 99%.) This was absolutely fatal to doing well henceforth, since I had no idea how I’d managed to do that well except showing up at lectures and usually doing my homework.
Damn it! I wish I still had classes like this! …I love classes where literally just understanding the material (in a deep, comprehensive way) is enough to get 100% if you don’t make stupid mistakes. I suspect this is the reason I did well in high school chemistry, physics, and bio–if you tried to really grasp the underlying concepts, the memorization required was trivial, or at least it didn’t feel like memorization.
Whereas many of my classes now are pure memorization of stuff with hardly any underlying logical structure (like pharmacology...stupid list of over 100 drug names to memorize, generic AND commercial!), or based on legal standards and “best practice guidelines” which, although they must be based on research results, don’t yield easily to my attempt to find underlying concepts. One class consisted almost entirely of memorizing the names (and acronyms, in French and English) for the various nursing regulatory organization in Ontario, and the documents they released on stuff like ethics. Gaaaah. There have been so many classes where I finished with an A- not because the class was hard, not because an A+ would have been ridiculously difficult, but because the material was so boring that I literally could not make myself study for more than a few minutes at a time, and only then by bribing myself.
Weirdly enough, I probably would have preferred doing pharmacology the hard way, i.e. learning chemistry to an advanced enough level that I could understand approximately how and why different drugs have the effects that they do. This would obviously be harder, but it would also be interesting, which would make it psychologically easier–I spend a lot more willpower on studying boring things than on studying interesting things.
This is why I got frustrated with foreign language class. It’s impossible to derive one word in a vocabulary list from the others. Being able to recall the words for “red”, “blue”, “orange”, “grey”, “white”, and “black” doesn’t help you at all to remember the word for “green”.
I had the same problem when learning English; at least when I had to learn vocabulary for school (my native language is Spanish, so English would be our foreign language class). Later on I had the chance to take a couple of classes in Latin.
For the last two years I’ve been learning French with great success. I’ve not yet found out how it is it was so easy for me; I suspect it came from it being close-to-isomorphic with Spanish. The thing is, when learning vocabulary I found that knowing a little bit about the use (not necessarily the definition) of English words helped me a lot to derive the meaning of new words. Whenever I had a little knowledge of the etymology of a word (for example, from the latin course), this “logical derivation” of the meaning or usage of words (or even the less-preferred pairing with a word on another language) got a lot easier.
I think there’s a little learning curve about vocabulary, after which you get better and better about deriving meaning from context and memory of previous known uses. Actually, I believe this might be what we do with our native languages; in my case at least I know I wouldn’t be able to define or precise the meaning or definition of most words I use.
Is this even possible with the current best theories in medical science? It was my understanding that it was no where near that advanced.
Probably not, but it’s possible to go to a much deeper level of detail than we did, i.e. learning about receptors and physiology, to the point that all you have to memorize, pharmacology-specific, is “drug X is an antagonist for receptor Y”, and the rest (uses, side effects, etc) flows naturally from that. We did some of this, for agonists/antagonists of the sympathetic and parasympathetic nervous system. (Beta blockers, i.e. metaprolol are, let me draw out this memory for a moment...antagonists of the sympathetic nervous system, which is why they lower blood pressure, because increased heart output is something you get when you stimulate the sympathetic nervous system. I would not have remembered this if I’m just had to memorize it offhand.) I’m sure we could have done more learning of this style...it might have taken 2 or 3 semesters instead of just one, though. Also, some drugs do things that medical science doesn’t understand, i.e. anti-psychotics, and that would still have to be memorized.
There are memory courses that claim to teach one to remember large quantities of somewhat arbitrary information. Anything by Harry Lorayne, for example. (One day I shall bother with one of these courses. [i.e., I probably won’t.])
I found spaced repetition systems easier to use on a regular basis than visualised-association systems such as the peg system and mnemonic major system, which were interesting to learn, but a bit cumbersome to practice regularly. Possibly I could become more fluent with practice of the latter but it’s been procrastination-inducing so far.
I think spaced repetition is for natural amount of memory (e. g. 10 new terms each class) whereas visualization techniques are for unnatural amounts (e. g. 20 digits in 5 minutes).
Related: I used a mnemonic system for a good part of my pharmacology studying, mostly by using silly phrases to match generic and commercial names. This has stuck in some surprising ways; for example, whenever I think of the drug spironolactone, a diuretic with the side effect of gynecomasty (breast development in men), I have a vivid mental image of a man in Viking war armor with large, milk-oozing breasts (the “lactone”), holding a trident (unsure what this was a mnemonic for.)
I used spaced repetition (Anki decks) to study for the RN certification exam, and probably overshot-it was quite easy.
This was my experience in physics, but didn’t feel at all true in chemistry and bio. I understood the concepts fine, but nothing about the concepts seemed to let me derive anything on the fly or avoid rote memorization.
One of the interesting things about taking both an intro to materials science course (basically solid-state engineering) and a more traditional introductory chemistry course targetted to roughly the same academic level was seeing the difference in the underlying approach. The solid-state chem focused on deriving the macroscopic behavior from the physical properties of the atoms much more than the trad chem.
I don’t think it was taught that way in chem or bio, but I tried to understand it that way… My parents have always bought me science books, and I had already read most of my high school library’s science section, so most of what I was learning wasn’t new. The concepts I was learning didn’t necessarily let me predict the other concepts, but they all fit together in a logical, meshed framework where they relied on each other, and I could use that to trigger my memory to retrieve particular concepts. Which is much harder in something like “nursing theory”, which a) I didn’t spend most of my childhood reading books about, and b) doesn’t hold together in a logical framework, except in some superficial ways.
I found this was a common theme in good engineering courses. The whole course would boiled down to three or four new concepts. By the time the final came around, it just felt like thinking hard about common sense even though at the beginning of the term, everything had seemed counter-intuitive.
I got 100% on that exam. No homework, no notes. I rarely did as well as that, but the strategy worked and I’m graduated.
I’m now trying to remember what the four equations were. One was how to transform between Cartesian and polar coordinates …
dynamics? assuming algebra but no calculus.
you say one was polar/cartesian
one was
x = x0 +v0*t + 1/2*a*t^2one was
f = m*aone was either quadratic formula to reverse #2, or reversed #2.
Doing dynamics at a tertiary level without calculus? o_O
Where are you from that the school system is sane enough to assume calculus for undergrads?
There are places that don’t? Australian here too and calculus is assumed (for undergrads doing appropriate degrees). It is assumed because the applicable subjects are either an outright requirement of the course or there are remedial level subjects required as prereqs (that most students in the course will skip) for those that somehow managed to avoid learning calculus in school.
In canada, almost nobody learned calc in high school, it was an advanced placement course. We started learning calculus in first year engineering but didn’t really start using it in other courses until third year.
Partially this was because the 4 year degree program piggy backed on a 2 year technologist program. So we learned the basics and applied practicalness of everything without calculus in the first two years, then went back and learned all the deep theory in last two years.
Australia. (I have a little bit of culture shock now.)
I’m not very confident that a school system that teaches calculus to high schoolers is sane.
Ahem. I took calculus as a freshman. In highschool. I only had to retake the second half because I was so horribly sleep deprived during the final, and so lazy about homework. I then got a 5 on the AP test (score maxes out at a 5).
Now I’m not typical, but I suspect that a school system that cannot teach calculus to the top ~10% or so of its math-inclined students when they’re in high school is failing somewhere along the way.
The question is not whether high schools can teach calculus, but rather whether they should. I believe the value of calculus education at that level is not very high relative to classes on, e.g., personal finances, logic, probability, economics, civics and etc.
That’s what was meant by “sane” in the grandparent: a non-efficient use of classroom time.
“not very high relative to classes on, e.g., personal finances, logic, probability, economics, civics and etc.” But high relative to, say, teaching a class on postmodernism.
what I do know is this: the grade school that fed into my highschool had a nifty policy for math: you go to the math class at the level your ready for, independent of your grade level. This is on the assumption that (innate?) mathematical aptitude is relatively uncoupled from other “classroom skills”.
As a result, those who don’t have a super-duper knack for math don’t generally hold back those who do have a knack for math, so the upper end of the curve is already extending itself by the time those kids get into high school.
I’m sure at least 40% of high school freshmen could learn Mandarin, but you don’t see that happening. Therefore, merely because students are capable of learning a subject, does not imply that that subject is or ought to be taught.
Seriously. I’m talking about opportunity costs.
Philosophy is not even a blip on the secondary education subject radar.
due to my experiences as a young kid (I could challenge Authority (my parents) on a math problem and sometimes WIN, and I learned how to factor quadratics in first grade) and the fact i see the application of math almost everywhere (computers in particular, but bridges and buildings as well) means that I see math as our best tool.
time to start applying it recursively.
I believe that math is usefull for a human with ~99.99+% probability. (that number discards the probability of an AI for the sake of speed)
I believe, with about a 75% confidence (for now) that having calculus as an available, but not required, course, should be possible in at least 80% of high schools.
even an understanding of a simplified calculus is usefull in many other subjects.
if (#2), then a high school unable to offer calculus is either a particularly small highschool, or is being fed by an underperforming grade school system.
There is plenty of evidence that SOMETHING in the gradeschool system is underperforming regarding mathematics.
-Which belief elements do you want me to try to expand on the most when i get back?
None. You’re still in a “can = must” frame of mind, after three attempts to explain my position. Your one engagement with the idea of replacing high school calculus with something else was, as far as I can tell, facetious.
Conversely, I took calculus in high school, didn’t understand what the heck was going on with most of the concepts, got a 5 on the AP test and an A in the class, and forgot all the material immediately.
Then, years later, I took calculus in college and understood everything.
So, YMMV on how good an idea it is.
Agreed.
Brown University (where my brother went) assumes that incoming freshman have already had math equivalent to the first two semesters of college calculus. My high school’s calculus course only covered Calc I.
My first-year courses in engineering (in Canada) made basic use of calculus without assuming any real understanding of it. By second-year, the calculus was assumed and for linear ODE’s and similar. Third-year, we moved to Laplace and Fourier transforms and the final year finally started to get into applications and standards and “real” things.
I’ve always wondered how different other engineering school curricula are.
Also canadian engineering school. First year we learned calculus but did not use it. Second year we learned even more calculus (up to PDEs, inear alg, and vector calc), but still didn’t use it. Calculus started being assumed in third year. We suggested to the school that they make more of an effort to use the skills that they teach between classes, so this might change soon.
All engineers at my public US alma mater learned calculus up through differential equations and calculus-based physics was required.
This does not, unfortunately, mean that anybody graduates with a functional understanding of calculus...
We also learned calculus, just not in time for first year dynamics.
I should probably do something like dig up the curriculum. (University of Western Australia, 1988, bog-standard first year engineering.)