Math is hard.(fourthcheckraise.blogspot.com)
fourthcheckraise.blogspot.com
Math is hard.
http://fourthcheckraise.blogspot.com/2011/11/math-is-hard.html
9 comments
I want to say that while I did very well in math in high school, it didn't hold my interest until I got into calculus in university, and am now deeply interested in advanced vector analysis and other applied mathematics (I still hate proofs). I very much agree that much of high school is spent drilling useless topics into students heads that are completely unnecessary and never even explained beyond a few ridiculous "real-life" applications. All the interesting math is locked away, available only to those who suffer through meaningless tests and bureaucratic madness. It would be much better to introduce higher level topics first, showing students the incredible usefulness of high level math, and then using their curiosity to dig into the lower level concepts that make everything work, instead of forcing them to try and be interested in boring low-level rigor that , without a high-level structure, is of limited use.
It's like we're teaching kids mathematical assembly language instead of Python.
It's like we're teaching kids mathematical assembly language instead of Python.
Math at the lower level never holds anybody's interest (even those who go on to be very good at a STEM field). It is exactly (half of) what you said it is: boring low-level rigor. It is absolutely not of "limited use".
Your comparison to assembly/Python doesn't fit at all. You cannot start in on vector analysis without an understanding of geometry and algebra. You can start Python without understanding assembly at all. You can go back and learn assmebly after you've learned Python but this isn't true in most cases with higher level math. How do I explain the chi-squared distribution without wasting the entire semester on basic maths? Math isn't like an Apple product, it doesn't "just work."
Some make the argument that how we teach STEM based courses can be adjusted so that we start from high-level stuff and teach low-level mechanics along the way. While this approach is intriguing, it will never work. It relies on an assumption about students that isn't true: they are actually interested in the subject. If I had to guess the percentage of students with a deep interest in their subject I would put it at a little bit below the actual graduation rates. Most students are in degree programs nowadays because it is the next step, all of their friends are doing it, it is expected of them, and/or they are trying to make their parents happy. Many of these students will find out what it is they really want to do (whether it is in their first field or their second field), many will recognize that they hate their field and still slog through for social reasons, and many will fail. College is a life changing experience for most people, no matter what category they fall in.
Your comparison to assembly/Python doesn't fit at all. You cannot start in on vector analysis without an understanding of geometry and algebra. You can start Python without understanding assembly at all. You can go back and learn assmebly after you've learned Python but this isn't true in most cases with higher level math. How do I explain the chi-squared distribution without wasting the entire semester on basic maths? Math isn't like an Apple product, it doesn't "just work."
Some make the argument that how we teach STEM based courses can be adjusted so that we start from high-level stuff and teach low-level mechanics along the way. While this approach is intriguing, it will never work. It relies on an assumption about students that isn't true: they are actually interested in the subject. If I had to guess the percentage of students with a deep interest in their subject I would put it at a little bit below the actual graduation rates. Most students are in degree programs nowadays because it is the next step, all of their friends are doing it, it is expected of them, and/or they are trying to make their parents happy. Many of these students will find out what it is they really want to do (whether it is in their first field or their second field), many will recognize that they hate their field and still slog through for social reasons, and many will fail. College is a life changing experience for most people, no matter what category they fall in.
Algebra is important. Knowing sum-to-product trig identities, limits to 0/0 and obscure trigonometric integral identities are not. I am not saying none of it is required, I am saying some of it is not required, and that there is an unnecessary emphasis on it.
My comparison to assembly/Python was a comparison, but of course I should have expected people to take it to an extreme hyperbole. But then, your entire argument is based on the assumption that most people are not interested in mathematics, so I guess circular logic is totally fine in these arguments (and hilariously ironic).
No really, let's look at this. I'm saying that teaching high level math alongside low level math (or at least putting less emphasis on low level math) will make more people interested in math. You're saying this won't work because people aren't interested in math. That doesn't work.
My comparison to assembly/Python was a comparison, but of course I should have expected people to take it to an extreme hyperbole. But then, your entire argument is based on the assumption that most people are not interested in mathematics, so I guess circular logic is totally fine in these arguments (and hilariously ironic).
No really, let's look at this. I'm saying that teaching high level math alongside low level math (or at least putting less emphasis on low level math) will make more people interested in math. You're saying this won't work because people aren't interested in math. That doesn't work.
Upvoted for similar experience. It is much more interesting to solve arbitrary math problems than a specific arbitrary math problem.
In order to study statistics you need calculus. In order to do basic calculus you need trigonometry.
The idea that you can just toss "more interesting" concepts around without the basics won't work.
The idea that you can just toss "more interesting" concepts around without the basics won't work.
This is deeply incorrect.
If it were correct, mathematics education would have to begin with set theory and physics probably could not begin with F = ma. Our pedagogy is cumulative but it is tellingly built on some hand-waving and "you'll have to trust me on this part".
This means that we can take a different approach to it pretty easily. For example, why does set theory often wait until college and probability until... never? These concepts are much more analytically useful to the average person than is the quadratic equation (which you can look up as needed) or (heaven help me) angle-side-angle.
We teach math through loaded down problem sets instead of heading right after the most conceptually useful (and therefore) meaty chunks first. We don't even teach children mathematically-grounded science for the most part; that's a missed opportunity to take those problem sets and make them analytically powerful and memorable.
If it were correct, mathematics education would have to begin with set theory and physics probably could not begin with F = ma. Our pedagogy is cumulative but it is tellingly built on some hand-waving and "you'll have to trust me on this part".
This means that we can take a different approach to it pretty easily. For example, why does set theory often wait until college and probability until... never? These concepts are much more analytically useful to the average person than is the quadratic equation (which you can look up as needed) or (heaven help me) angle-side-angle.
We teach math through loaded down problem sets instead of heading right after the most conceptually useful (and therefore) meaty chunks first. We don't even teach children mathematically-grounded science for the most part; that's a missed opportunity to take those problem sets and make them analytically powerful and memorable.
I agree completely with this article. Math and physics are hard. It requires a certain disposition to be willing to sit with a problem for minutes or hours and have faith that you will be able to find the solution. The fields are also littered with geniuses and people with significantly more talent than yourself. The gap between those who tried and those who were gifted was severe at my alma mater(USC).
I remember being in an intermediate mechanics course with only physics majors when our professor asked us how long it took us to complete the homework sets. I was regularly spending 15-25 hours a week on them. Some of my classmates were completing the sets in 2-3 hours. What shocked me wasn't that there were students who could complete the assignments that quickly but that: 1) students were either taking 2-5 hours or 10+ hours to complete the assignment. It was very bimodal. 2) that the gap in efficiency between the best and worst students was almost 10x.
In comparison my estimate of the gap in efficiency between the best and worst medical students is 1.5-2x at the most. No medical student can claim that they can accomplish in 2-5 hours what would take another medical student 15-25 hours to learn.
In the end I made the choice to go in to medicine where I was consistently in the top 3% of my class. There were a handful of 'superheroes' in medicine but not many. Most of us started in the same place and where we ended up ranking in our class was a function of effort. It was a refreshingly level playing field in comparison to physics.
I remember being in an intermediate mechanics course with only physics majors when our professor asked us how long it took us to complete the homework sets. I was regularly spending 15-25 hours a week on them. Some of my classmates were completing the sets in 2-3 hours. What shocked me wasn't that there were students who could complete the assignments that quickly but that: 1) students were either taking 2-5 hours or 10+ hours to complete the assignment. It was very bimodal. 2) that the gap in efficiency between the best and worst students was almost 10x.
In comparison my estimate of the gap in efficiency between the best and worst medical students is 1.5-2x at the most. No medical student can claim that they can accomplish in 2-5 hours what would take another medical student 15-25 hours to learn.
In the end I made the choice to go in to medicine where I was consistently in the top 3% of my class. There were a handful of 'superheroes' in medicine but not many. Most of us started in the same place and where we ended up ranking in our class was a function of effort. It was a refreshingly level playing field in comparison to physics.
I struggled through high school never managing to get my head around statistics. I changed majors from chemistry to microbiology my first year of my science degree (because thats where all the girls where), and found statistics really easy. So what was the difference? the teachers thats all, some of the best tutors I every experienced where in the science area.
I had one tutor explain to me why he threw curve ball problems at the class. He would use it to weed out which ones gave up quick, which ones powered on, which ones just threw tantrums etc... he would then only really teach the percentage of the class that wanted the degree. The rest he said he'd get to passing, some just weren't worth teaching, but he'd make sure the real scientists walked out of the class with the best training they could get (albeit at the expense of others).
I'd hazard a guess that type of teaching behavior is endemic, so drop out rates for people who can't get the initiative to put in the extra effort get left behind pretty quickly and hence drop out.
I had one tutor explain to me why he threw curve ball problems at the class. He would use it to weed out which ones gave up quick, which ones powered on, which ones just threw tantrums etc... he would then only really teach the percentage of the class that wanted the degree. The rest he said he'd get to passing, some just weren't worth teaching, but he'd make sure the real scientists walked out of the class with the best training they could get (albeit at the expense of others).
I'd hazard a guess that type of teaching behavior is endemic, so drop out rates for people who can't get the initiative to put in the extra effort get left behind pretty quickly and hence drop out.
A Mathematician's Lament
"TEXTBOOK PUBLISHERS : TEACHERS ::
A) pharmaceutical companies : doctors
B) record companies : disk jockeys
C) corporations : congressmen
D) all of the above
We have millions of adults wandering around with “negative b plus or minus the square root of b squared minus 4ac all over 2a” in their heads, and absolutely no idea whatsoever what it means."
https://docs.google.com/viewer?a=v&q=cache:qcqr-s07yTgJ:...
"TEXTBOOK PUBLISHERS : TEACHERS ::
A) pharmaceutical companies : doctors
B) record companies : disk jockeys
C) corporations : congressmen
D) all of the above
We have millions of adults wandering around with “negative b plus or minus the square root of b squared minus 4ac all over 2a” in their heads, and absolutely no idea whatsoever what it means."
https://docs.google.com/viewer?a=v&q=cache:qcqr-s07yTgJ:...
“In mathematics you don't understand things. You just get used to them.” - Johann Von Neuman
I so related to that statement in college. I took all the necessary math and engineering (fancy word for more math...) classes to get my EE degree. I was able to apply the math well enough to be able to stick around in my engineering program (although there were SEVERAL different points in time I almost changed major, and I did dabble in getting a philosophy minor.) In college, my mindset was: "study enough to get by. beer. boobs. football" (yeah yeah, I went to a Big 10 football school) But I didn't really care about the math too much, as long as I got my degree and a paycheck at the end.
Now, out in the wild, I find enjoyment in the application of math (I do signal processing) and I only wish someone was there to help me see the "fun", or applicable side of things back in school, so I could understand even more now. I think the big problem with my college courses was that the course material simply existed just to exist, whereas now in my job, it exists to perform a function, and it's kinda cool to see how stuff works. But make no mistake, now that I'm though with my studies, I'm more than thankful I had to take all those ball beater math classes, so I actually (at one point at least) knew the 'math' behind the math i'm using, I knew more than the math, I knew how it was derived, and THATS every course building on each other, and it's tough, and there's distractions, but it's necessary.
I so related to that statement in college. I took all the necessary math and engineering (fancy word for more math...) classes to get my EE degree. I was able to apply the math well enough to be able to stick around in my engineering program (although there were SEVERAL different points in time I almost changed major, and I did dabble in getting a philosophy minor.) In college, my mindset was: "study enough to get by. beer. boobs. football" (yeah yeah, I went to a Big 10 football school) But I didn't really care about the math too much, as long as I got my degree and a paycheck at the end.
Now, out in the wild, I find enjoyment in the application of math (I do signal processing) and I only wish someone was there to help me see the "fun", or applicable side of things back in school, so I could understand even more now. I think the big problem with my college courses was that the course material simply existed just to exist, whereas now in my job, it exists to perform a function, and it's kinda cool to see how stuff works. But make no mistake, now that I'm though with my studies, I'm more than thankful I had to take all those ball beater math classes, so I actually (at one point at least) knew the 'math' behind the math i'm using, I knew more than the math, I knew how it was derived, and THATS every course building on each other, and it's tough, and there's distractions, but it's necessary.
I hear a lot of the first half of your post but not a lot of your second half of your post (I agree with both). Most people look back and remember how hard/dumb/boring/useless that one class was and how they never use it in real life. Barely anybody looks back and remembers how something that they learned in their math class was integral to their ability succeed later on in their career. For example, almost everybody complains about how hard/useless calculus was. These same people are able to take a volume integral in cylindrical coordinates no problem. There is a fundamental disconnect here.
And anything you don't remember? Well, being familiar with a concept makes it a hundred times easier to learn the second time.
And anything you don't remember? Well, being familiar with a concept makes it a hundred times easier to learn the second time.
This post strikes me as absurd and offensive. First of all, the author comes off as exceedingly arrogant. That aside, there is an implicit judgment made in the post that STEM subjects are inherently more intellectual, more legitimate, and more important than social fields and issues like feminism.
Unfortunately, having glanced at a few other posts on that blog, I feel like the world would be a better place if the author had studied something like sociology.
Unfortunately, having glanced at a few other posts on that blog, I feel like the world would be a better place if the author had studied something like sociology.
There is something seriously wrong with the 'weeding out' mentality, in the hard sciences. I can't tell if its hazing, or some superiority complex. You end up with very hard boring work & teachers not being able to teach or willing to help, and everyone thinking 'If you are good you will figure it out yourself'. Hey, maybe math is hard, but you're not making it any easier & if I see a better opportunity I'll take it in a heartbeat.
I think Math is easier than Science. The formula is there, you just have to solve it, done.
It's always been my favorite subject.
That isn't how math is done at a high level. That is mechanical rote computation, which in higher-level courses and in the real world is done by computers.
Real math is all about, roughly, creating the formulas that the software will use, which means you have to prove that the formula is both logically correct and does what you want to do. It's a creative act.
If you remember nothing else, remember that high-level math is creative.
Real math is all about, roughly, creating the formulas that the software will use, which means you have to prove that the formula is both logically correct and does what you want to do. It's a creative act.
If you remember nothing else, remember that high-level math is creative.
I excelled in all subjects in high school except for math, which I detested and felt as difficult and, for the most part, uninteresting.
Fast forward to third year of Uni and I was looking at set theory, graph theory, and logical proofs. They were all difficult, yet they sucked me in. It was hard but I wasn't dissuaded from studying. And the reason why was because thinking of interesting proofs and devising ways of solving puzzles is far, far more interesting than doing arithmetic from rote memory.
It really doesn't surprise me at all that kids do badly in math when they're introduced to such boring and dry topics. Kids are taught useless trigonometric identities when they could be studying applied statistics, which is not only substantially more useful, but it opens a lot more doors for reasoning about the world. Learning statistics, graph theory and logic have literally helped to mold my thinking. I can't really say the same about trig of high-school level calculus (not that there's anything wrong with calculus, but i think that in order to get something out of it you need to dedicate much more time than what is available in high school).