9 ms·

Common Errors in Undergraduate Mathematics (2009)

180 points by vinchuco 9y ago

okket 9y ago
Previous discussion: https://news.ycombinator.com/item?id=8181101 https://news.ycombinator.com/item?id=8181101 (3 years ago, 110 comments)
tankenmate 9y ago
One point that wasn't noted is that division is anticommutative of the multiplicative variety. It always intrigued me as to why commutativity is seen as being applicable to all operators and functions that provide the necessary property, but seemingly only applies to the prefix arithmetic inverse for the anticommutative variety.
Kenji 9y ago
For instance, contrary to the belief of many students sin(x + y) is NOT sin(x) + sin(y) (x+y)^2 is NOT (x^2) + (y^2) sqrt(x+y) is NOT sqrt(x) + sqrt(y) 1/(x+y) is NOT (1/x) + (1/y) Wow, seriously? Especially the one with the fraction is painful. I'm not sure if university is the right place for people who still make mistakes like that at that age (if they had more than a decade of public school behind them). Certainly not STEM fields.
- douche 9y ago
  It's really a travesty how much of the twelve or thirteen years of compulsory schooling we waste, and how little return is generated from it
- cmrx64 9y ago
  I have a story that is too long here about my struggles with math in high school, and while I'm maybe not the model CS student for a variety of related reasons, I think that I turned out mostly fine... None of the symbolic manipulation really made sense to me until I started learning programming (and, much later, mathematical foundations) and how to define these operations. The answer to "Why isn't sqrt(x+y) = sqrt(x) + sqrt(y)" shouldn't be "It's not in the table of 'laws' we're allowed to use". Sure, you can (easily) provide examples where that purported identity is false, but trying to understand why was really hard for me, and I wouldn't accept it until I understood why. /ramble
  - douche 9y ago
    Similarly, very little of the trig and pre-calculus stuff that I had to learn in math class made any sense to me, until I either had to use it for drawing graphics with QBasic in my programming class, or for a plethora of things in my shop classes.
  - Kenji 9y ago
    It's just that you could literally substitute 1 for both x and y and it would be immediately obvious that the equations cannot hold.
    - fghvbnvbnfe 9y ago
      That's not really relevant. You only need to check if the equations hold if you don't understand the math involved. You've already lost at that point. Checking is the smart thing to do then, but ideally what is important is the understanding rather than just the diligence to make sure a guess isn't trivially shown incorrect.
      - sdenton4 9y ago
        I strongly disagree. Brains are fallible; a quick check at the end lets you unit test your understanding. I find the ability to identify one's own mistakes and correct them to be a sign of an excellent student.
        fghvbnvbnfe 9y ago
        I agree with you in general, but not when only talking about the mathematical understanding. If you want to add fractions by individually adding the numerators and denominators and the only reason you don't is one of these checks, you have some kind of fundamental lack of understanding about fractions. It's great that you are wise enough to verify, but that's irrelevant to your mathematical understanding.
        sdenton4 9y ago
- S4M 9y ago
  I remember in first year of University (Math/Physics) someone saying that arctan(x+y) = (arctan(x) + arctan(y))/(1-arctan(x)arctan(y)). (Basically he thought that the expansion of arctan(x+y) was just like the expansion of tan(x+y))
- aisofteng 9y ago
  I've graded freshman and sophomore calculus homeworks where students do things like sin(x)/x = sin/1 = 1. That's an exact example.
  - alphydan 9y ago
    What? That's clearly wrong. The answer is "sin", right? :)
- mjw1007 9y ago
  I don't think that's as bad as it looks. Part of the point of inventing concise notation for mathematics is so that you can "let the formalism carry you through"; a great deal of the time you can write down the next line of a proof by looking at the previous line or two and simplifying something without thinking too hard about what it means. One of the things that the students are practising is performing a longish series of operations while thinking about the important stuff and letting the simple stuff take care of itself. So I don't think it should be too concerning if a student starts to make mistakes like this: what's gone wrong may not be "the student doesn't understand what sin means", but "the student has gone astray in converting their understanding to an internalised rule of formal manipulation". (It's a bit like if a younger pupil mistakenly gets it stuck in their head that 6x7 is 49: it doesn't mean they've fundamentally misunderstood what multiplication is.)
- omaranto 9y ago
  On the other hand, sin(x-y)sin(x+y) = sin(x)^2 - sin(y)^2.
- roberto 9y ago
  My algebra teacher in college did the first mistake. I still remember it after 21 years: 1. She was reducing some formula, and got it down to cos(a - 0), which is obviously cos(a). The problem is, her cheat sheet had the solution as sin(a). 2. She "expanded" to cos(a - 0) = cos(a) - cos(0) 3. We know cos(0), so this leads us to cos(a) - 1 4. This is where she got stuck. After thinking for a long time she wrote on the board: "(from trigonometry) cos(a) - 1 = sin(a)" 5. Final answer: sin(a) I went up to the board and explained step 2 was wrong and how she should've done it, and I explained that step 4 was wronger, because that would mean that for some angles cos/sin would be outside the [-1, 1] range. She said, "let's try your way then, young man", and doing the expansion on cos(a - 0) got it to cos(a). She then proceed to say I was wrong, because the answer had to be sin(a). I still thank her for teaching me to question authority.
- KGIII 9y ago
  Mathematics is usually taught as rote. Many, even at higher levels of academia, never realize that mathematics is a language. Truth be told, I'm not sure most are cut out to fully grasp much beyond rote. I hated math. I hated it, with a passion. I was figuring out what area of right triangles was, when the teacher spoke from behind me. He said, "You know, you're just squaring the triangles and dividing that in half." I realize how trivial and silly that is. But, it was that step that made me realize that the notation was telling me something. I realized that it was a language, one which I didn't understand and still hated. I'd eventually get my Ph.D. in Applied Mathematics, all because I learned you could square a triangle, find the area of that, and divide it in half. I'm not sure how many folks can point to the exact moment their life changed? I can, and I am glad. I don't know the solution, because math really can be hard. Not many folks want to learn that much. I am reluctant to say this, but maybe they don't really need to?
bluenose69 9y ago
It seems unlikely that students who need this advice will be willing to read all the words in this essay. But, so long as they continue to pay tuition...
tgb 9y ago
These are primarily for algebra mistakes. Teaching a proof based course, the most frustrating error I saw was people doing a fine proof but in reverse (this is related to the article's point about irreversible steps). If I wanted them to prove that the square of any number was nonnegative, they'd start by assuming that x^2 >= 0 and then driving equations from that until they got something obviously true like 0>=0. This was the proof written "upside down" if every step were reversible. They should have just taken the last step and started from there until they derived their first step. So how was I supposed to grade that? It showed a significant misunderstanding but also often literally had every step necessary for a correct proof. Usually I just wrote overly long notes about how they should write proofs in the future that I'm sure the students never read. The other fun one was unnecessary proof by contradiction. They'd start by assuming not P. Then they'd prove P directly, not using that assumption. But they, that contradicts the assumption of not P. Hence not P is false, so then they conclude P. Nothing was wrong there, they just had a couple extraneous sentences that overcomplicated things.
- ucarion 9y ago
  I've noticed this (backward steps) with some friends. I think it's because people don't think logically with math notation--they just play with symbols until they get the sentence they want. To get around this, I just get people to argue aloud. People wear their logic hats again when they have to talk to a human and convince them of something.
  - sdenton4 9y ago
    Alternately, it's a very useful technique to identify the penultimate step; it's often a useful rephrasing of the problem, and gives something easier to prove. And then just repeat... (It's probably usually a hangover from poorly taught rote proof exercises in a lower grade, though.)
    - arielb1 9y ago
      This is how you do high-school algebra, after all. As long as it is clear that all steps are reversible, it is ok.
ezequiel-garzon 9y ago
I think it ultimately comes down to the level of primary school teachers, and good luck reversing that. I taught at a community college some classes for future elementary school teachers. When I found things like 1/2+1/3=2/5 I tried my best to get everybody's attention as to how important fixing this misconception was, especially given they'd be teaching children in a few years... Unfortunately, I mainly received questions about how much of an impact an error like that would have on their grades. Major sigh...
- throwawayjava 9y ago
  > Unfortunately, I mainly received questions about how much of an impact an error like that would have on their grades. Major sigh... "You will fail the entire assignment on which the error was made." If only... :)
- dmix 9y ago
  This a merely a symptom of bad schooling, not the students (well maybe a little bit the students and the math/learning-hostile culture they grow up in). There should be no reason why kids who spent such a large part of their childhood all day long, 5 times a week, in a learning institution and can't demonstrate basic learning despite getting good enough grades to get in college. Don't most college courses requiring math have an algebra recap course to start in first year? Likely as a result of this lack of basic math literacy. I honestly don't blame a lot of kids who enter college not knowing how to add fractions. Math classes are pretty awful in most schools and they likely just memorized enough stuff before tests then subsequently forgot it. They didn't have any deep grasp of the fundamental rules, such as finding a common denominator. Although there is no excuse for answering that by guessing. When I started a journey down learning advanced math I bought 'Forgotten Algebra' [1]. It's a great recap over everything taught in elementary/high schools and can be finished pretty quickly. Any college course involving math should make this required reading the summer before school or as part of first year for those who fail a basic math test. I'm convinced this wouldn't been nearly as necessary if we had classrooms with the Khan Academy model with homework in class in group settings with the teacher walking around helping and lectures watched online, with interactive teaching tools. This is particularly useful for math. Something built on top of Sage (now called Vocal) [1] or Python notebooks would be great for this. [1] https://www.amazon.com/gp/aw/d/1438001509/ https://www.amazon.com/gp/aw/d/1438001509/ [2] https://cocalc.com https://cocalc.com
madez 9y ago
The author claims > And for some purposes, an ellipsis is not just a convenience, it's a necessity. For instance, "1, 2, 3, ..., n" represents all the integers from 1 to n, where n is some unspecified positive integer; there's no way to write that without an ellipsis. This claim is wrong. Let n be a positive integer. If one wants to denote the set of integers between 1 and n, then one can do so by just formally writing what one wants: {k in Integers | 1 <= k <= n}. If one wants the (ordered) list of integers between 1 and n, then one can write (k)_{k=1}^n.
- posterboy 9y ago
  > For some purpose e.g. without going into set theoretic fundamentals to build up ordered sets.
  - madez 9y ago
    That is not necessary, at all. One can always just literally write "the list of ascendingly ordered integers that are greater or equal to 1 and smaller or equal to n". The introduction of formal notation is just a shortcut.

180 points by vinchuco 9y ago

Previous discussion: https://news.ycombinator.com/item?id=8181101 https://news.ycombinator.com/item?id=8181101 (3 years ago, 110 comments)

tankenmate 9y ago

One point that wasn't noted is that division is anticommutative of the multiplicative variety. It always intrigued me as to why commutativity is seen as being applicable to all operators and functions that provide the necessary property, but seemingly only applies to the prefix arithmetic inverse for the anticommutative variety.

Kenji 9y ago

For instance, contrary to the belief of many students sin(x + y) is NOT sin(x) + sin(y) (x+y)^2 is NOT (x^2) + (y^2) sqrt(x+y) is NOT sqrt(x) + sqrt(y) 1/(x+y) is NOT (1/x) + (1/y) Wow, seriously? Especially the one with the fraction is painful. I'm not sure if university is the right place for people who still make mistakes like that at that age (if they had more than a decade of public school behind them). Certainly not STEM fields.

douche 9y ago
It's really a travesty how much of the twelve or thirteen years of compulsory schooling we waste, and how little return is generated from it
cmrx64 9y ago
I have a story that is too long here about my struggles with math in high school, and while I'm maybe not the model CS student for a variety of related reasons, I think that I turned out mostly fine... None of the symbolic manipulation really made sense to me until I started learning programming (and, much later, mathematical foundations) and how to define these operations. The answer to "Why isn't sqrt(x+y) = sqrt(x) + sqrt(y)" shouldn't be "It's not in the table of 'laws' we're allowed to use". Sure, you can (easily) provide examples where that purported identity is false, but trying to understand why was really hard for me, and I wouldn't accept it until I understood why. /ramble
- douche 9y ago
  Similarly, very little of the trig and pre-calculus stuff that I had to learn in math class made any sense to me, until I either had to use it for drawing graphics with QBasic in my programming class, or for a plethora of things in my shop classes.
- Kenji 9y ago
  It's just that you could literally substitute 1 for both x and y and it would be immediately obvious that the equations cannot hold.
  - fghvbnvbnfe 9y ago
    That's not really relevant. You only need to check if the equations hold if you don't understand the math involved. You've already lost at that point. Checking is the smart thing to do then, but ideally what is important is the understanding rather than just the diligence to make sure a guess isn't trivially shown incorrect.
    - sdenton4 9y ago
      I strongly disagree. Brains are fallible; a quick check at the end lets you unit test your understanding. I find the ability to identify one's own mistakes and correct them to be a sign of an excellent student.
      - fghvbnvbnfe 9y ago
        I agree with you in general, but not when only talking about the mathematical understanding. If you want to add fractions by individually adding the numerators and denominators and the only reason you don't is one of these checks, you have some kind of fundamental lack of understanding about fractions. It's great that you are wise enough to verify, but that's irrelevant to your mathematical understanding.
        sdenton4 9y ago
S4M 9y ago
I remember in first year of University (Math/Physics) someone saying that arctan(x+y) = (arctan(x) + arctan(y))/(1-arctan(x)arctan(y)). (Basically he thought that the expansion of arctan(x+y) was just like the expansion of tan(x+y))
aisofteng 9y ago
I've graded freshman and sophomore calculus homeworks where students do things like sin(x)/x = sin/1 = 1. That's an exact example.
- alphydan 9y ago
  What? That's clearly wrong. The answer is "sin", right? :)
mjw1007 9y ago
I don't think that's as bad as it looks. Part of the point of inventing concise notation for mathematics is so that you can "let the formalism carry you through"; a great deal of the time you can write down the next line of a proof by looking at the previous line or two and simplifying something without thinking too hard about what it means. One of the things that the students are practising is performing a longish series of operations while thinking about the important stuff and letting the simple stuff take care of itself. So I don't think it should be too concerning if a student starts to make mistakes like this: what's gone wrong may not be "the student doesn't understand what sin means", but "the student has gone astray in converting their understanding to an internalised rule of formal manipulation". (It's a bit like if a younger pupil mistakenly gets it stuck in their head that 6x7 is 49: it doesn't mean they've fundamentally misunderstood what multiplication is.)
omaranto 9y ago
On the other hand, sin(x-y)sin(x+y) = sin(x)^2 - sin(y)^2.
roberto 9y ago
My algebra teacher in college did the first mistake. I still remember it after 21 years: 1. She was reducing some formula, and got it down to cos(a - 0), which is obviously cos(a). The problem is, her cheat sheet had the solution as sin(a). 2. She "expanded" to cos(a - 0) = cos(a) - cos(0) 3. We know cos(0), so this leads us to cos(a) - 1 4. This is where she got stuck. After thinking for a long time she wrote on the board: "(from trigonometry) cos(a) - 1 = sin(a)" 5. Final answer: sin(a) I went up to the board and explained step 2 was wrong and how she should've done it, and I explained that step 4 was wronger, because that would mean that for some angles cos/sin would be outside the [-1, 1] range. She said, "let's try your way then, young man", and doing the expansion on cos(a - 0) got it to cos(a). She then proceed to say I was wrong, because the answer had to be sin(a). I still thank her for teaching me to question authority.
KGIII 9y ago
Mathematics is usually taught as rote. Many, even at higher levels of academia, never realize that mathematics is a language. Truth be told, I'm not sure most are cut out to fully grasp much beyond rote. I hated math. I hated it, with a passion. I was figuring out what area of right triangles was, when the teacher spoke from behind me. He said, "You know, you're just squaring the triangles and dividing that in half." I realize how trivial and silly that is. But, it was that step that made me realize that the notation was telling me something. I realized that it was a language, one which I didn't understand and still hated. I'd eventually get my Ph.D. in Applied Mathematics, all because I learned you could square a triangle, find the area of that, and divide it in half. I'm not sure how many folks can point to the exact moment their life changed? I can, and I am glad. I don't know the solution, because math really can be hard. Not many folks want to learn that much. I am reluctant to say this, but maybe they don't really need to?

bluenose69 9y ago

It seems unlikely that students who need this advice will be willing to read all the words in this essay. But, so long as they continue to pay tuition...

tgb 9y ago

These are primarily for algebra mistakes. Teaching a proof based course, the most frustrating error I saw was people doing a fine proof but in reverse (this is related to the article's point about irreversible steps). If I wanted them to prove that the square of any number was nonnegative, they'd start by assuming that x^2 >= 0 and then driving equations from that until they got something obviously true like 0>=0. This was the proof written "upside down" if every step were reversible. They should have just taken the last step and started from there until they derived their first step. So how was I supposed to grade that? It showed a significant misunderstanding but also often literally had every step necessary for a correct proof. Usually I just wrote overly long notes about how they should write proofs in the future that I'm sure the students never read. The other fun one was unnecessary proof by contradiction. They'd start by assuming not P. Then they'd prove P directly, not using that assumption. But they, that contradicts the assumption of not P. Hence not P is false, so then they conclude P. Nothing was wrong there, they just had a couple extraneous sentences that overcomplicated things.

ucarion 9y ago
I've noticed this (backward steps) with some friends. I think it's because people don't think logically with math notation--they just play with symbols until they get the sentence they want. To get around this, I just get people to argue aloud. People wear their logic hats again when they have to talk to a human and convince them of something.
- sdenton4 9y ago
  Alternately, it's a very useful technique to identify the penultimate step; it's often a useful rephrasing of the problem, and gives something easier to prove. And then just repeat... (It's probably usually a hangover from poorly taught rote proof exercises in a lower grade, though.)
  - arielb1 9y ago
    This is how you do high-school algebra, after all. As long as it is clear that all steps are reversible, it is ok.

ezequiel-garzon 9y ago

I think it ultimately comes down to the level of primary school teachers, and good luck reversing that. I taught at a community college some classes for future elementary school teachers. When I found things like 1/2+1/3=2/5 I tried my best to get everybody's attention as to how important fixing this misconception was, especially given they'd be teaching children in a few years... Unfortunately, I mainly received questions about how much of an impact an error like that would have on their grades. Major sigh...

throwawayjava 9y ago
> Unfortunately, I mainly received questions about how much of an impact an error like that would have on their grades. Major sigh... "You will fail the entire assignment on which the error was made." If only... :)
dmix 9y ago
This a merely a symptom of bad schooling, not the students (well maybe a little bit the students and the math/learning-hostile culture they grow up in). There should be no reason why kids who spent such a large part of their childhood all day long, 5 times a week, in a learning institution and can't demonstrate basic learning despite getting good enough grades to get in college. Don't most college courses requiring math have an algebra recap course to start in first year? Likely as a result of this lack of basic math literacy. I honestly don't blame a lot of kids who enter college not knowing how to add fractions. Math classes are pretty awful in most schools and they likely just memorized enough stuff before tests then subsequently forgot it. They didn't have any deep grasp of the fundamental rules, such as finding a common denominator. Although there is no excuse for answering that by guessing. When I started a journey down learning advanced math I bought 'Forgotten Algebra' [1]. It's a great recap over everything taught in elementary/high schools and can be finished pretty quickly. Any college course involving math should make this required reading the summer before school or as part of first year for those who fail a basic math test. I'm convinced this wouldn't been nearly as necessary if we had classrooms with the Khan Academy model with homework in class in group settings with the teacher walking around helping and lectures watched online, with interactive teaching tools. This is particularly useful for math. Something built on top of Sage (now called Vocal) [1] or Python notebooks would be great for this. [1] https://www.amazon.com/gp/aw/d/1438001509/ https://www.amazon.com/gp/aw/d/1438001509/ [2] https://cocalc.com https://cocalc.com

madez 9y ago

The author claims > And for some purposes, an ellipsis is not just a convenience, it's a necessity. For instance, "1, 2, 3, ..., n" represents all the integers from 1 to n, where n is some unspecified positive integer; there's no way to write that without an ellipsis. This claim is wrong. Let n be a positive integer. If one wants to denote the set of integers between 1 and n, then one can do so by just formally writing what one wants: {k in Integers | 1 <= k <= n}. If one wants the (ordered) list of integers between 1 and n, then one can write (k)_{k=1}^n.

posterboy 9y ago
> For some purpose e.g. without going into set theoretic fundamentals to build up ordered sets.
- madez 9y ago
  That is not necessary, at all. One can always just literally write "the list of ascendingly ordered integers that are greater or equal to 1 and smaller or equal to n". The introduction of formal notation is just a shortcut.