6 ms·

You never did math in high school

69 points by DavidChouinard 13y ago

deleted 13y ago
[deleted]
gboudrias 13y ago
Well, I sure wish this article had more explaining of what we should actually teach, instead of "go read this other article". I was waiting for the point but it wasn't there.
pbiggar 13y ago
See Lockhart's Lament for an in-depth view of the same argument: http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...
- Zarel 13y ago
  He actually summarizes and links to Lockhart's Lament in the first six paragraphs. More interesting is his link near the end, about how he teaches graph theory to college students: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/ http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...
  - pbiggar 13y ago
    Whoops, I read to the end but missed that link!
thrush 13y ago
I think that you're getting at a great point, but another contributing factor may be that high school has a different scheduling structure. High School: Classes/Week = 5 Time/Class = 45 min HW Frequency = Every Day or Every few Days Exam Frequency = Every Week or Every Couple Weeks University: Classes/Week = 2 or 3 + Discussion Time/Class = 90 or 120 min HW Frequency = Every Week or Every Couple Weeks Exam Frequency = Every Month or 3 per Semester All I am trying to say is that difficulty of material is not always the problem. Sometimes the system can be the problem. It's clear that university expects you to spend more time doing independent study given there is more time between sessions, yet many new students aren't accustomed to spending their free time this way.
- adamnemecek 13y ago
  Indeed. For me, the worst part about college was how everything was spaced out. Once you got into the zone, you didn't have that much time before you had to go to another class or something.
- zephjc 13y ago
  That's the sort of high school experience and college experience I had, where the college experience was much more conducive to really understanding the material. My sister-in-law is in high school, at a school that has fewer courses per term, and fewer per day, and going more in-depth in each class; I'm really glad she gets that experience. More context switches only hurts the student's ability to get into a subject.
hawkharris 13y ago
I remember reading a fun fact in an MIT faculty newsletter a few years ago: the amount of money that American students spend failing calculus — paying for the course and tutoring, then redoing it — is greater than the budget of Avatar (over $300 million). Every year, our students spend the budget of a James Cameron film on failing to understand calculus...
shalmanese 13y ago
I remember my dad giving me the following problem on a long distance car ride: An A4 sheet of paper, when folded in half, becomes an A3 sheet of paper with the exact same ratio of long side to short side. Given this information, can you figure out the what the ratio is? I must of spent 3 hours of that car ride doodling on a piece of paper trying to nut out that problem with no additional information or tools, ultimately unsuccessfully. To this day, I think that was when the light switch went off for me about math as a creative problem solving endeavor and not just a rote series of calculations.
- potash 13y ago
  Interesting, I'd never thought about this before. In case you (or anyone else) are still wondering... If an A3 sheet is X units by Y units (with X the long side) and A4 is 2Y units by X units then their ratios are X/Y and 2Y/X, respectively. If these are equal we have 2Y/X = X/Y cross multiplying gives 2Y^2 = X^2 rearranging and taking a square root (X and Y are positive) gives X/Y = sqrt(2).
- adnrw 13y ago
  I think you mean A5, not A3 (A3 is bigger than A4)
- minikomi 13y ago
  Definitely don't want to detract form the sentiment of this post with pedantry but just as a point - the numbers count up as you get smaller ;) [1] [1] https://www.hobbyking.com/hobbyking/forum/uploads/56/A_size_illustration.png https://www.hobbyking.com/hobbyking/forum/uploads/56/A_size_...
6d0debc071 13y ago
Any good books, or online resources, people can recommend on actually doing maths for people from a non-maths background? I mean this is an old refrain, and not one I disagree with, but it's missing the positive side of the argument.
- tomsthumb 13y ago
  Pick through the stuff by Dover publishing. They have some very intense math books, but there are definitely some books about math for "lay people". Many of them can be had cheaply, and there are usually plenty of reviews on amazon to figure out what is both interesting and accessible.
  - crander 13y ago
    I really like the Dover book Mathematics for the Nonmathematician by Morris Kline. It adds historical perspective, makes you think, and teaches basic and understandable proofs.
- pseut 13y ago
  Polya's "how to solve it" is a classic.
- j2kun 13y ago
  Try reading the article linked at the end.
- trentmb 13y ago
  BOOK OF PROOF: http://www.people.vcu.edu/~rhammack/BookOfProof/index.html http://www.people.vcu.edu/~rhammack/BookOfProof/index.html Foundations of Higher Mathematics: http://www.amazon.com/Foundations-Higher-Mathematics-Peter-Fletcher/dp/053495166X http://www.amazon.com/Foundations-Higher-Mathematics-Peter-F... How to Solve It: http://en.wikipedia.org/wiki/How_to_Solve_It http://en.wikipedia.org/wiki/How_to_Solve_It EDIT: Mathematical Reasoning: Writing and Proof: https://sites.google.com/site/mathematicalreasoning3ed/ https://sites.google.com/site/mathematicalreasoning3ed/
cperciva 13y ago
Yes I did. Also: when a freshman college student tells me that he was always good at math, it translates to “I was very good at following obscure steps to manipulate mysterious symbols, without any real understanding of what they mean.” The most common complaint I hear about undergraduate students is that they aren't able to blindly manipulate symbols, and consequently get hopelessly confused when they have to deal with "unintuitive" concepts.
- baddox 13y ago
  It's even worse than that. It's more like "I was very good at doing enough of the practice/homework problems to memorize all the possible combinations of problems that will end up on the test."
  - Pxtl 13y ago
    Exactly. You have to remember who the loudest complainers in a school are - not the nerds, not the losers, but the keeners. The kids who play the game of earning marks instead of learning to think. That's who the current system panders to, because they're the ones who scream loudest when it doesn't. And they find it morally offensive if you ever ask them to solve a problem that isn't extremely similar to an example they've already seen. And they're the ones who think they're "good at math". That's who highschool caters to because that's what standardized testing caters to and that's what the noisiest parents and students want... and teachers that have higher ideals are swimming up-stream if they want to actually challenge the kids.
  - pseut 13y ago
    Ah, if only...
- wfunction 13y ago
  Shouldn't you understand what you're doing?
  - taeric 13y ago
    Maybe? Really depends on what you are doing, and what level of understanding you mean.
mullingitover 13y ago
This just made me remember the whole class struglling with Riemann sums in Calc II, and Russian professor yelling "This is grade school calculus!" at us. Apparently Russians do Calculus in grade school.
- cperciva 13y ago
  My mother went through high school in England in the mid-60s. I recently saw her A-level exam papers; they had a lot of questions which I would be surprised to see anywhere prior to third year college.
  - eru 13y ago
    Was that a grammar school?
    - cperciva 13y ago
      Yes.
  - billyjobob 13y ago
    In the 60s only a tiny percentage of the top students went to university and so the standard at university was much higher. A-levels are university entrance exams and so similarly were only taken by the best. Even the old O-levels, taken at age 16, were equivalent to university level courses today. This fact is often used as proof that kids today aren't as smart at kids in the 60s, but in fact most kids in the 60s didn't take any exams whatsoever and just left school at age 15 with no qualifications.
- alex-g 13y ago
  Your comment reminded me of this story: "A certain well known mathematican, we'll call him Professor P.T. (these are not his initials...), upon his arrival at Harvard University, was scheduled to teach Math 1a (the first semester of freshman calculus.) He asked his fellow faculty members what he was supposed to teach in this course, and they told him: limits, continuity, differentiability, and a little bit of indefinite integration. The next day he came back and asked: What am I supposed to cover in the second lecture?" (from Spiro Karigiannis at http://mathoverflow.net/a/53238 http://mathoverflow.net/a/53238 - comments below refer to a version of the story where the professor is from the USSR specifically)
xname 13y ago
I don't like this kind of claim. Even calculation of 1+1 is math. To be constructive, instead of claiming what students learnt is not math (which actually is math), please articulate what you think needs to be added to high school math. However, please be aware that not everyone is expected to learn calculus after high school.
- lotsofmangos 13y ago
  Calculation of 1+1 is arithmetic, proving that it is equal to 2 within a given logical framework, would be maths.
- DerpDerpDerp 13y ago
  A lot of people separate "math" (the process of reasoning about structures or relations) from "arithmetic" (the process of calculating with numbers). I think this is reasonable, as "math" is essentially just another name for "meta-arithmetic", and we draw similar boundaries between science and engineering, for example.
- baddox 13y ago
  I actually don't think high school math is too much of a problem. For me, my mental rejection of math education began at calculus. It was never taught to me with any semblance of mathematical formality; instead it was almost entirely rote symbolic manipulation. Of course, elementary school arithmetic is symbolic manipulation (made faster with memorization of small integer products), but between arithmetic and calculus I don't see many problems. Geometry, algebra, trigonometry, and basic statistics in secondary school didn't cause the mental rejection for me that calculus did.
nayefc 13y ago
The assumption that's explaining why high schools in America SUCK: "freshman calculus course". In my school and country, we take calculus 1 & 2 in grade 11; that's two years before we go to college. In grade 12, we take differential equations.
- RogerL 13y ago
  Calculus is taught in high school. However, it is not mandatory/standard - you have to be on the college track, and it is skippable. (these are all general statements, different school districts make different choices). For example, in my high school, in a fairly small rural school, we were taught calculus in senior year. I was bored, and did it sooner, but that was the general rule. We also had things like advanced placement biology, physics, and so on. Generally speaking, at the end of those courses you can take the optional AP tests, which allow you to skip the Physics I, Calc I&II, etc., that are the normal fare for freshmen. I can't recall anyone who had diff eq taught in high school, though we really did end up using it anyway in advanced physics. To roughly sketch the limits of what we did, I remember needing Green's theorem, and having to prove things using elliptic integrals, but nothing much beyond that.
  - nayefc 13y ago
    Interesting. When I came to college here in the US (CS, engineering department), I was rather shocked that most, if not all, engineering students came in with zero calculus or advanced physics.
- deleted 13y ago
  [deleted]
asperous 13y ago
This may be true but personally I have found that the mathematics I was taught was good enough for most purposes. People learn how to drive a car by doing it. They aren't encouraged to discover the laws of combustion or shown the magic of kinematics. They just learn how to press the petals.
- jimmaswell 13y ago
  Well yes, but understanding the math is the important part for most people unless they use it daily, while in driving a car it doesn't matter much as long as you can drive it.
baddox 13y ago
Interestingly, I observed that the inverse seemed to be true in college (I was a computer science major). People (mostly math majors) who were very good math students, meaning they breezed through the college calc courses, struggled when we met again in more CS-relevant classes like Discrete Mathematics and Algebraic Structures. It should be troubling to any educator to see people excel at Calc 2 integrations, yet struggle to accomplish much simpler (at least to me) tasks like proving that the product of two even integers is an even integer.
- cowsandmilk 13y ago
  > struggle to accomplish much simpler (at least to me) tasks like proving that the product of two even integers is an even integer. I always see articles that claim I didn't do math in high school, and then people talk about things like this. I did this in high school. In the US. In 2001. Literally, one of our test questions was: (a) prove the product of two even numbers is even (b) prove the product of two odd numbers is odd (c) prove the product of an odd and even number is even And this wasn't regurgitating a proof we learned in class or had on homework, it was a question we had not seen before. Blanket statements about what people see and don't see in high school are worthless.
  - baddox 13y ago
    I didn't make any blanket statements. I simply didn't encounter any proofs in high school math courses, other than "geometric proofs" which weren't formal proofs.
- j2kun 13y ago
  This is the problem: the way calculus is taught is not very mathematical.
benihana 13y ago
I wish I would have known this in high school and college. I was convinced I sucked at math because I did poorly on math tests. It wasn't until I started programming that I realized I was good at math; I was just bad at the chickenshit busy work they made us do in high school algebra and geometry.
skywhopper 13y ago
He's spent 20 hours doing one-off lectures to high school math students, and he believes he's an expert teacher? He seems to have an interesting approach and can get the students intrigued, but giving an interesting hour-long presentation and teaching a high school class for a full year are very different things. I'm not trying to be cynical or discouraging, but dial back the certainty in your genius methods a couple of notches until you've done more than just gotten kids interested for an hour.
ruswick 13y ago
The OP comes across as very fond of deriding the US schooling system, but proposes nothing beyond an egotistical assurance that he can teach math well even though no one else can. Math education is not homogenous. People are not homogenous. Everyone experiences math in different ways both in school and in everyday life. Some people will simply memorize the steps necessary to solve problems. Some will actually grasp the theory behind the problem solving. Some will experience very little of what is considered "classic" high school math. (At my school, those who are not tracked into calculus or statistics as Seniors take a course that deals with things like formal logic. It actually seems really interesting.) The OP's blanket assertions are simply wrong. Moreover, math is a massive field. The OP's claim that calculus and algebra are not math but that "pattern recognition" is is just absurd.
- primitivesuave 13y ago
  He puts high-quality teaching notes online for free, and it is quite callous to call his proposal an egotistical assurance or to deride his definition of math.
- m_ke 13y ago
  Jeremy is a phd student at UIC right now and I'm pretty sure he taught a few classes there. As someone who moved to the states at 13 I have to agree with him. Most teachers in America are forced to prep their students for exams instead of teaching them how to think critically. Instead of building an intuition, students are forced to memorize rules and equations so that they can finish their standardized exams in time. The problems done in class and for hw are of the same format as the exams, with the numbers used being the only difference. I've seen plenty of college students get offended when a professor puts a problem on an exam that requires some critical thinking. I did my math degree at a school that had a special program for future math teachers and they were consistently the worst performing students in my classes. Abstract Algebra was a disaster for them, most couldn't put together even the simplest proofs. With exams approaching, they all begged the professor to tell them which proofs they were required to memorize. Somehow I got by with perfect scores by understanding the content while they had a tough time pulling off anything above 20%.
joshontheweb 13y ago
I definitely felt like my grade school education did me a disservice in the math department. I liken it to learning grammar for 12 years without ever writing an essay. This is why I think programming should be taught starting in elementary school. It gives you an application for the math you need to learn. I never cared about learning algebra, calculus, or trigonometry until I started programming. Now I wish I had paid attention and now I am going back and self educating on the subjects in order to empower my programming.
primitivesuave 13y ago
Jeremy has put together some of the best graph theory educational notes, and I refer to his website a lot when I'm trying to show teachers how to infuse something abstract and meaningful into their traditional mathematics education. I highly recommend checking out his work, he really exemplifies modern mathematical education.
mamcx 13y ago
I have tough that education must be more integrated. Take for example, the Pythagoras theorem. Is a boring fact, with null use... Wait! If somebody tell me before that I can use it for get out of a jungle... I could have listen better. The class could have started like this: "You were traveling in plane, when suddenly, it crashed. Nobody else survived. You don't know where you are. It look like a jungle. Not civilization around, no cellphone, nothing. You will die in 1 week if not reach civilization. How can you escape?" And if in history (at the same time that in maths) we talk about the man (Pythagoras), and about the compass. Then in social about the problems in traveling in the ancient cultures. Geography about maps. In artist class ("Art" class was more about technical diagrams for me, we never ever do oleo or similar stuff), how draw maps and in spanish build histories about it. Then all the clases related to each other. Probably the math class must be the last of it, to make this build-up effective. In short, all the classes connected around the theme of the week or what are we doing at the moment.
xupybd 13y ago
I love this article. I hated "Math" in school, loved in at university. Why don't they teach real Math at school?
Pxtl 13y ago
I have to disagree with this. For example, he attacks geometric proofs. Obviously, geometric proofs are useless and pointless, except that they represent a very simple and intuitive sandbox for learning proofs. Which is fantastic. How better would you teach kids to understand what math means beyond arithmetic? You give them a bunch of simple rules that make obvious sense, and then show them how you can use those to prove non-obvious things. Our curriculum has all but dropped geometric proofs and the kids are poorer for it. I think our curriculum here in North America (I'm in Ontario, but I imagine Americans have very similar classes) is fine - the problem is that people are afraid to make the problems really hard and force the students to really stretch their skills beyond just "same as the example but change the numbers". That's what's missing, not some esoteric subject matter, just taking the existing tools in the existing toolbox and pushing the kids to build higher instead of more.
- j2kun 13y ago
  It's the way geometry is taught, not the content, that is the problem. This is essentially the argument for all types of math; they aren't "math" because they're taught in a way that removes all the math and leaves... what exactly?
analog31 13y ago
I taught a college pre-calc course for a short time. We had "show your work" exams, so I got to see how students solved problems. I think that many of them had been taught "test taking skills" in high school, including a method of finding answers to math problems by guess-and-try or process of elimination. Basically, their teachers had hacked the standardized testing process. (Probably including the AP exam). Honestly, college math wasn't much better. Rather than confronting them with critical thought, we simply replaced the old hack with a new one: 1. Recognize the "form" of the problem, corresponding to a section in the textbook. 2. Plug the parameters of the problem into an equation solvable by the method of that section. 3. Apply the method and write the answer. When I realized this, I told my students about it. There was a chapter on maxima and minima. But the only function they had learned was the quadratic, so all optimization problems boiled down to arranging things into a quadratic. The text had them graphing each quadratic. I decided it would be more interesting for the kids to see how we make our own formulas, so we derived one for the optimum of a quadratic function, and memorized it. Now is this math or not? Well, I think there's a place in math that involves classifying the forms of expressions and equations, sort of like taxonomy in biology. But it shouldn't be the only thing.
Imagenuity 13y ago
My college experience was that teachers taught how to pass exams, as described in the article. But when asking the teacher how the math worked, they were unable to because they learned and were trained through the same system. They didn't 'do' math or know math. They didn't possess the deeper understanding I was looking for someone to explain.
graycat 13y ago
Nonsense. I did four years of math in high school, and it really was 'math'. It wasn't all research or 'critical thinking', but I didn't memorize "steps" either -- I have an awful 'rote' memory, at least without some oral hints. And I did fine with math later on, in including my Ph.D. in applied math (stochastic optimal control) and peer reviewed publications in applied math -- with theorems and proofs. The OP has a small point but takes it way, way too far.
brg 13y ago
Other's have pointed this out already, but there is a need for everyone to learn symbol manipulation, geometric representation, and 5 line arguments before mathematics mastered. Saying that this is not part of a complete mathematics education is saying that grammar and vocabulary are not part of an English education. And yes, my school did mathematics in high school. We went through much of Euclid's Elements.
andrewcooke 13y ago
wish i'd known about this earlier. when i didn't get something i assumed i had to work harder. didn't realise i was supposed to blame the teacher. damn.

deleted 13y ago

[deleted]

Well, I sure wish this article had more explaining of what we should actually teach, instead of "go read this other article". I was waiting for the point but it wasn't there.

pbiggar 13y ago

See Lockhart's Lament for an in-depth view of the same argument: http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...

Zarel 13y ago
He actually summarizes and links to Lockhart's Lament in the first six paragraphs. More interesting is his link near the end, about how he teaches graph theory to college students: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/ http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...
- pbiggar 13y ago
  Whoops, I read to the end but missed that link!

thrush 13y ago

I think that you're getting at a great point, but another contributing factor may be that high school has a different scheduling structure. High School: Classes/Week = 5 Time/Class = 45 min HW Frequency = Every Day or Every few Days Exam Frequency = Every Week or Every Couple Weeks University: Classes/Week = 2 or 3 + Discussion Time/Class = 90 or 120 min HW Frequency = Every Week or Every Couple Weeks Exam Frequency = Every Month or 3 per Semester All I am trying to say is that difficulty of material is not always the problem. Sometimes the system can be the problem. It's clear that university expects you to spend more time doing independent study given there is more time between sessions, yet many new students aren't accustomed to spending their free time this way.

adamnemecek 13y ago
Indeed. For me, the worst part about college was how everything was spaced out. Once you got into the zone, you didn't have that much time before you had to go to another class or something.
zephjc 13y ago
That's the sort of high school experience and college experience I had, where the college experience was much more conducive to really understanding the material. My sister-in-law is in high school, at a school that has fewer courses per term, and fewer per day, and going more in-depth in each class; I'm really glad she gets that experience. More context switches only hurts the student's ability to get into a subject.

hawkharris 13y ago

I remember reading a fun fact in an MIT faculty newsletter a few years ago: the amount of money that American students spend failing calculus — paying for the course and tutoring, then redoing it — is greater than the budget of Avatar (over $300 million). Every year, our students spend the budget of a James Cameron film on failing to understand calculus...

shalmanese 13y ago

I remember my dad giving me the following problem on a long distance car ride: An A4 sheet of paper, when folded in half, becomes an A3 sheet of paper with the exact same ratio of long side to short side. Given this information, can you figure out the what the ratio is? I must of spent 3 hours of that car ride doodling on a piece of paper trying to nut out that problem with no additional information or tools, ultimately unsuccessfully. To this day, I think that was when the light switch went off for me about math as a creative problem solving endeavor and not just a rote series of calculations.

potash 13y ago
Interesting, I'd never thought about this before. In case you (or anyone else) are still wondering... If an A3 sheet is X units by Y units (with X the long side) and A4 is 2Y units by X units then their ratios are X/Y and 2Y/X, respectively. If these are equal we have 2Y/X = X/Y cross multiplying gives 2Y^2 = X^2 rearranging and taking a square root (X and Y are positive) gives X/Y = sqrt(2).
adnrw 13y ago
I think you mean A5, not A3 (A3 is bigger than A4)
minikomi 13y ago
Definitely don't want to detract form the sentiment of this post with pedantry but just as a point - the numbers count up as you get smaller ;) [1] [1] https://www.hobbyking.com/hobbyking/forum/uploads/56/A_size_illustration.png https://www.hobbyking.com/hobbyking/forum/uploads/56/A_size_...

6d0debc071 13y ago

Any good books, or online resources, people can recommend on actually doing maths for people from a non-maths background? I mean this is an old refrain, and not one I disagree with, but it's missing the positive side of the argument.

tomsthumb 13y ago
Pick through the stuff by Dover publishing. They have some very intense math books, but there are definitely some books about math for "lay people". Many of them can be had cheaply, and there are usually plenty of reviews on amazon to figure out what is both interesting and accessible.
- crander 13y ago
  I really like the Dover book Mathematics for the Nonmathematician by Morris Kline. It adds historical perspective, makes you think, and teaches basic and understandable proofs.
pseut 13y ago
Polya's "how to solve it" is a classic.
j2kun 13y ago
Try reading the article linked at the end.
trentmb 13y ago
BOOK OF PROOF: http://www.people.vcu.edu/~rhammack/BookOfProof/index.html http://www.people.vcu.edu/~rhammack/BookOfProof/index.html Foundations of Higher Mathematics: http://www.amazon.com/Foundations-Higher-Mathematics-Peter-Fletcher/dp/053495166X http://www.amazon.com/Foundations-Higher-Mathematics-Peter-F... How to Solve It: http://en.wikipedia.org/wiki/How_to_Solve_It http://en.wikipedia.org/wiki/How_to_Solve_It EDIT: Mathematical Reasoning: Writing and Proof: https://sites.google.com/site/mathematicalreasoning3ed/ https://sites.google.com/site/mathematicalreasoning3ed/

cperciva 13y ago

Yes I did. Also: when a freshman college student tells me that he was always good at math, it translates to “I was very good at following obscure steps to manipulate mysterious symbols, without any real understanding of what they mean.” The most common complaint I hear about undergraduate students is that they aren't able to blindly manipulate symbols, and consequently get hopelessly confused when they have to deal with "unintuitive" concepts.

baddox 13y ago
It's even worse than that. It's more like "I was very good at doing enough of the practice/homework problems to memorize all the possible combinations of problems that will end up on the test."
- Pxtl 13y ago
  Exactly. You have to remember who the loudest complainers in a school are - not the nerds, not the losers, but the keeners. The kids who play the game of earning marks instead of learning to think. That's who the current system panders to, because they're the ones who scream loudest when it doesn't. And they find it morally offensive if you ever ask them to solve a problem that isn't extremely similar to an example they've already seen. And they're the ones who think they're "good at math". That's who highschool caters to because that's what standardized testing caters to and that's what the noisiest parents and students want... and teachers that have higher ideals are swimming up-stream if they want to actually challenge the kids.
- pseut 13y ago
  Ah, if only...
wfunction 13y ago
Shouldn't you understand what you're doing?
- taeric 13y ago
  Maybe? Really depends on what you are doing, and what level of understanding you mean.

mullingitover 13y ago

This just made me remember the whole class struglling with Riemann sums in Calc II, and Russian professor yelling "This is grade school calculus!" at us. Apparently Russians do Calculus in grade school.

cperciva 13y ago
My mother went through high school in England in the mid-60s. I recently saw her A-level exam papers; they had a lot of questions which I would be surprised to see anywhere prior to third year college.
- eru 13y ago
  Was that a grammar school?
  - cperciva 13y ago
    Yes.
- billyjobob 13y ago
  In the 60s only a tiny percentage of the top students went to university and so the standard at university was much higher. A-levels are university entrance exams and so similarly were only taken by the best. Even the old O-levels, taken at age 16, were equivalent to university level courses today. This fact is often used as proof that kids today aren't as smart at kids in the 60s, but in fact most kids in the 60s didn't take any exams whatsoever and just left school at age 15 with no qualifications.
alex-g 13y ago
Your comment reminded me of this story: "A certain well known mathematican, we'll call him Professor P.T. (these are not his initials...), upon his arrival at Harvard University, was scheduled to teach Math 1a (the first semester of freshman calculus.) He asked his fellow faculty members what he was supposed to teach in this course, and they told him: limits, continuity, differentiability, and a little bit of indefinite integration. The next day he came back and asked: What am I supposed to cover in the second lecture?" (from Spiro Karigiannis at http://mathoverflow.net/a/53238 http://mathoverflow.net/a/53238 - comments below refer to a version of the story where the professor is from the USSR specifically)

xname 13y ago

I don't like this kind of claim. Even calculation of 1+1 is math. To be constructive, instead of claiming what students learnt is not math (which actually is math), please articulate what you think needs to be added to high school math. However, please be aware that not everyone is expected to learn calculus after high school.

lotsofmangos 13y ago
Calculation of 1+1 is arithmetic, proving that it is equal to 2 within a given logical framework, would be maths.
DerpDerpDerp 13y ago
A lot of people separate "math" (the process of reasoning about structures or relations) from "arithmetic" (the process of calculating with numbers). I think this is reasonable, as "math" is essentially just another name for "meta-arithmetic", and we draw similar boundaries between science and engineering, for example.
baddox 13y ago
I actually don't think high school math is too much of a problem. For me, my mental rejection of math education began at calculus. It was never taught to me with any semblance of mathematical formality; instead it was almost entirely rote symbolic manipulation. Of course, elementary school arithmetic is symbolic manipulation (made faster with memorization of small integer products), but between arithmetic and calculus I don't see many problems. Geometry, algebra, trigonometry, and basic statistics in secondary school didn't cause the mental rejection for me that calculus did.

nayefc 13y ago

The assumption that's explaining why high schools in America SUCK: "freshman calculus course". In my school and country, we take calculus 1 & 2 in grade 11; that's two years before we go to college. In grade 12, we take differential equations.

RogerL 13y ago
Calculus is taught in high school. However, it is not mandatory/standard - you have to be on the college track, and it is skippable. (these are all general statements, different school districts make different choices). For example, in my high school, in a fairly small rural school, we were taught calculus in senior year. I was bored, and did it sooner, but that was the general rule. We also had things like advanced placement biology, physics, and so on. Generally speaking, at the end of those courses you can take the optional AP tests, which allow you to skip the Physics I, Calc I&II, etc., that are the normal fare for freshmen. I can't recall anyone who had diff eq taught in high school, though we really did end up using it anyway in advanced physics. To roughly sketch the limits of what we did, I remember needing Green's theorem, and having to prove things using elliptic integrals, but nothing much beyond that.
- nayefc 13y ago
  Interesting. When I came to college here in the US (CS, engineering department), I was rather shocked that most, if not all, engineering students came in with zero calculus or advanced physics.
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asperous 13y ago

This may be true but personally I have found that the mathematics I was taught was good enough for most purposes. People learn how to drive a car by doing it. They aren't encouraged to discover the laws of combustion or shown the magic of kinematics. They just learn how to press the petals.

jimmaswell 13y ago
Well yes, but understanding the math is the important part for most people unless they use it daily, while in driving a car it doesn't matter much as long as you can drive it.

baddox 13y ago

Interestingly, I observed that the inverse seemed to be true in college (I was a computer science major). People (mostly math majors) who were very good math students, meaning they breezed through the college calc courses, struggled when we met again in more CS-relevant classes like Discrete Mathematics and Algebraic Structures. It should be troubling to any educator to see people excel at Calc 2 integrations, yet struggle to accomplish much simpler (at least to me) tasks like proving that the product of two even integers is an even integer.

cowsandmilk 13y ago
> struggle to accomplish much simpler (at least to me) tasks like proving that the product of two even integers is an even integer. I always see articles that claim I didn't do math in high school, and then people talk about things like this. I did this in high school. In the US. In 2001. Literally, one of our test questions was: (a) prove the product of two even numbers is even (b) prove the product of two odd numbers is odd (c) prove the product of an odd and even number is even And this wasn't regurgitating a proof we learned in class or had on homework, it was a question we had not seen before. Blanket statements about what people see and don't see in high school are worthless.
- baddox 13y ago
  I didn't make any blanket statements. I simply didn't encounter any proofs in high school math courses, other than "geometric proofs" which weren't formal proofs.
j2kun 13y ago
This is the problem: the way calculus is taught is not very mathematical.

benihana 13y ago

I wish I would have known this in high school and college. I was convinced I sucked at math because I did poorly on math tests. It wasn't until I started programming that I realized I was good at math; I was just bad at the chickenshit busy work they made us do in high school algebra and geometry.

skywhopper 13y ago

He's spent 20 hours doing one-off lectures to high school math students, and he believes he's an expert teacher? He seems to have an interesting approach and can get the students intrigued, but giving an interesting hour-long presentation and teaching a high school class for a full year are very different things. I'm not trying to be cynical or discouraging, but dial back the certainty in your genius methods a couple of notches until you've done more than just gotten kids interested for an hour.

ruswick 13y ago

The OP comes across as very fond of deriding the US schooling system, but proposes nothing beyond an egotistical assurance that he can teach math well even though no one else can. Math education is not homogenous. People are not homogenous. Everyone experiences math in different ways both in school and in everyday life. Some people will simply memorize the steps necessary to solve problems. Some will actually grasp the theory behind the problem solving. Some will experience very little of what is considered "classic" high school math. (At my school, those who are not tracked into calculus or statistics as Seniors take a course that deals with things like formal logic. It actually seems really interesting.) The OP's blanket assertions are simply wrong. Moreover, math is a massive field. The OP's claim that calculus and algebra are not math but that "pattern recognition" is is just absurd.

primitivesuave 13y ago
He puts high-quality teaching notes online for free, and it is quite callous to call his proposal an egotistical assurance or to deride his definition of math.
m_ke 13y ago
Jeremy is a phd student at UIC right now and I'm pretty sure he taught a few classes there. As someone who moved to the states at 13 I have to agree with him. Most teachers in America are forced to prep their students for exams instead of teaching them how to think critically. Instead of building an intuition, students are forced to memorize rules and equations so that they can finish their standardized exams in time. The problems done in class and for hw are of the same format as the exams, with the numbers used being the only difference. I've seen plenty of college students get offended when a professor puts a problem on an exam that requires some critical thinking. I did my math degree at a school that had a special program for future math teachers and they were consistently the worst performing students in my classes. Abstract Algebra was a disaster for them, most couldn't put together even the simplest proofs. With exams approaching, they all begged the professor to tell them which proofs they were required to memorize. Somehow I got by with perfect scores by understanding the content while they had a tough time pulling off anything above 20%.

joshontheweb 13y ago

I definitely felt like my grade school education did me a disservice in the math department. I liken it to learning grammar for 12 years without ever writing an essay. This is why I think programming should be taught starting in elementary school. It gives you an application for the math you need to learn. I never cared about learning algebra, calculus, or trigonometry until I started programming. Now I wish I had paid attention and now I am going back and self educating on the subjects in order to empower my programming.

primitivesuave 13y ago

Jeremy has put together some of the best graph theory educational notes, and I refer to his website a lot when I'm trying to show teachers how to infuse something abstract and meaningful into their traditional mathematics education. I highly recommend checking out his work, he really exemplifies modern mathematical education.

mamcx 13y ago

I have tough that education must be more integrated. Take for example, the Pythagoras theorem. Is a boring fact, with null use... Wait! If somebody tell me before that I can use it for get out of a jungle... I could have listen better. The class could have started like this: "You were traveling in plane, when suddenly, it crashed. Nobody else survived. You don't know where you are. It look like a jungle. Not civilization around, no cellphone, nothing. You will die in 1 week if not reach civilization. How can you escape?" And if in history (at the same time that in maths) we talk about the man (Pythagoras), and about the compass. Then in social about the problems in traveling in the ancient cultures. Geography about maps. In artist class ("Art" class was more about technical diagrams for me, we never ever do oleo or similar stuff), how draw maps and in spanish build histories about it. Then all the clases related to each other. Probably the math class must be the last of it, to make this build-up effective. In short, all the classes connected around the theme of the week or what are we doing at the moment.

xupybd 13y ago

I love this article. I hated "Math" in school, loved in at university. Why don't they teach real Math at school?

Pxtl 13y ago

I have to disagree with this. For example, he attacks geometric proofs. Obviously, geometric proofs are useless and pointless, except that they represent a very simple and intuitive sandbox for learning proofs. Which is fantastic. How better would you teach kids to understand what math means beyond arithmetic? You give them a bunch of simple rules that make obvious sense, and then show them how you can use those to prove non-obvious things. Our curriculum has all but dropped geometric proofs and the kids are poorer for it. I think our curriculum here in North America (I'm in Ontario, but I imagine Americans have very similar classes) is fine - the problem is that people are afraid to make the problems really hard and force the students to really stretch their skills beyond just "same as the example but change the numbers". That's what's missing, not some esoteric subject matter, just taking the existing tools in the existing toolbox and pushing the kids to build higher instead of more.

j2kun 13y ago
It's the way geometry is taught, not the content, that is the problem. This is essentially the argument for all types of math; they aren't "math" because they're taught in a way that removes all the math and leaves... what exactly?

analog31 13y ago

I taught a college pre-calc course for a short time. We had "show your work" exams, so I got to see how students solved problems. I think that many of them had been taught "test taking skills" in high school, including a method of finding answers to math problems by guess-and-try or process of elimination. Basically, their teachers had hacked the standardized testing process. (Probably including the AP exam). Honestly, college math wasn't much better. Rather than confronting them with critical thought, we simply replaced the old hack with a new one: 1. Recognize the "form" of the problem, corresponding to a section in the textbook. 2. Plug the parameters of the problem into an equation solvable by the method of that section. 3. Apply the method and write the answer. When I realized this, I told my students about it. There was a chapter on maxima and minima. But the only function they had learned was the quadratic, so all optimization problems boiled down to arranging things into a quadratic. The text had them graphing each quadratic. I decided it would be more interesting for the kids to see how we make our own formulas, so we derived one for the optimum of a quadratic function, and memorized it. Now is this math or not? Well, I think there's a place in math that involves classifying the forms of expressions and equations, sort of like taxonomy in biology. But it shouldn't be the only thing.

Imagenuity 13y ago

My college experience was that teachers taught how to pass exams, as described in the article. But when asking the teacher how the math worked, they were unable to because they learned and were trained through the same system. They didn't 'do' math or know math. They didn't possess the deeper understanding I was looking for someone to explain.

graycat 13y ago

Nonsense. I did four years of math in high school, and it really was 'math'. It wasn't all research or 'critical thinking', but I didn't memorize "steps" either -- I have an awful 'rote' memory, at least without some oral hints. And I did fine with math later on, in including my Ph.D. in applied math (stochastic optimal control) and peer reviewed publications in applied math -- with theorems and proofs. The OP has a small point but takes it way, way too far.

brg 13y ago

Other's have pointed this out already, but there is a need for everyone to learn symbol manipulation, geometric representation, and 5 line arguments before mathematics mastered. Saying that this is not part of a complete mathematics education is saying that grammar and vocabulary are not part of an English education. And yes, my school did mathematics in high school. We went through much of Euclid's Elements.

andrewcooke 13y ago

wish i'd known about this earlier. when i didn't get something i assumed i had to work harder. didn't realise i was supposed to blame the teacher. damn.