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The Most Common Errors in Undergraduate Mathematics (2009)

220 points by free0ne 6y ago

deleted 6y ago
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sparklingmango 6y ago
When I was in undergrad, I declared a math minor quite late - just before my junior year. This led to me taking up to 9 hours of math credits in a semester. The reason I declared my math minor was because I was interested in going for a PhD in economics and my econ professors all recommended that I try to take as many math classes as possible to prepare. Reader, I was not a 'numbers person' at all. However, in all of my math classes, I asked questions all. the. time. when I didn't understand. I would go to office hours and the teachers were seriously so helpful. This was the only way I survived. I learned that you don't have to be a 'numbers person' at all to be good at math. You just have to understand what's going on under the hood, and this may require lots of questions and outside help. But most importantly, you just have to practice. I ended up graduating with nearly a 4.0 for my math minor.
- jrib 6y ago
  > I would go to office hours and the teachers were seriously so helpful. As a grad student that taught undergrads I always encouraged students to attend office hours. Most instructors do want you to succeed and one-on-one time is one of the best ways to learn. Definitely take advantage of your instructors' office hours; ask questions about anything that you aren't 100% sure about!
  - nemo44x 6y ago
    > I would go to office hours and the teachers were seriously so helpful. This was the only way I survived. I learned that you don't have to be a 'numbers person' at all to be good at math. You just have to understand what's going on under the hood, and this may require lots of questions and outside help. But most importantly, you just have to practice I agree with all of this and it's how I got through extensive math. I'm not a "natural" at math and I'm not sure that really exists or possibly very people that are. I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. I figured the cost of tuition is best justified by using as many hours of one-on-one time with an expert in the subject to better understand it yourself. And to your last point about practice, I can't emphasize this enough. I would do the same problem sets over and over until I had them down. And it wasn't just memorizing the steps but rather getting to the point of being able to recognize the pattern. This made math "easy" at that point which is a weird phenomenon. Something that seemed so confusing and impossible suddenly becomes "easy". And then you can move on to the next step. I understand rote learning is not popular because it is painful. But I do believe "drilling it in" over and over gets the job done. But it takes the will and the effort. It's been a long time but I think I could still solve most the math I took back then.
    - dr_dshiv 6y ago
      > I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. I figured the cost of tuition is best justified by using as many hours of one-on-one time with an expert in the subject to better understand it yourself Truth!
      - deleted 6y ago
        [deleted]
    - randlet 6y ago
      > I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. One of my biggest regrets about university is not taking advantage of office hours. For me it was a combination of 1) social anxiety and 2) thinking that if I really belonged in this program (physics) I should really be able to figure everything out on my own. I know now that this is a stupid way to think but it's the way I felt at the time.
  - godelski 6y ago
    I TA and tell my students that if they don't come to office hours or ask questions that they might as well be getting their degree on YouTube. I don't mean this in the sense that they have to ask for every class, but the advantage of attending school (besides the piece of paper) is to get direct help. But I'm not attending a top 10 university and honestly most topics are covered by one of those universities and they post their lectures on YouTube or their own OCW.
    - vmilner 6y ago
      Out of interest, roughly what proportion of students do come to the office hours?
      - godelski 6y ago
        Really depends on the class. One class I had basically no one. This one usually about 2-3 (out of 40) but more as we're nearing the end of term. Sometimes people come to say hi (again, class dependent). Largest I've had was like 8.
  - hyperpallium2 6y ago
    https://wikipedia.org/wiki/Bloom's_2_sigma_problem https://wikipedia.org/wiki/Bloom's_2_sigma_problem the average student tutored one-to-one using mastery learning techniques performed two standard deviations better A tutor can notice gaps. You can't.
- godelski 6y ago
  I find this concept of being a numbers person weird to being good at math. Most math is about symbol manipulation and numbers are really only at the end.
  - siltpotato 6y ago
    The concept itself or the name of it?
  - magicalhippo 6y ago
    > Most math is about symbol manipulation and numbers are really only at the end. During the first semester at uni, the math prof was going through some non-trivial (at that stage) "real-world" example involving definite integrals. After what felt like 10 minutes of manipulating integrals and such we got to the final "x = ..." equation, upon which he turned away from the blackboard and said "and then you can just plug in the numbers and calculate the result, but that's not really interesting". And so the lecture continued...
    - godelski 6y ago
      Middle School / High School: "What are these letters doing in my math?!" Post University: "What are these numbers doing in my math?!"
  - andi999 6y ago
    Here numbers are always in the beginning (of each chapter as the chapter number....)
- Daub 6y ago
  Getting students to seek consultation is a major problem. I advise them to come in small groups. This helps the shy ones. They don’t realize that it is very difficult to fail a student who asks for help. Any failure of such a student is implicitly also the teacher’s.
  - Tw9875 6y ago
    Only true for the teachers that care. A lot of my professors didn't give a damn about teaching and were annoyed that I was wasting their time showing up to these office hours.
    - andi999 6y ago
      Was this math? I found usually math guys including professors like to talk about math.
      - megameter 6y ago
        In my experience, when they're top tier math researchers they tend to be oblivious to the student's perspective, grasping the problem too well to give it detailed treatment. This is neither a failing of their enthusiasm nor a social tactic, but an imbalance in their abilities. An old anecdote is the professor who was asked by a student how a certain problem is solved. The professor paused in thought for a moments, then gave the answer. The student asked again how to solve it, and the professor, again, thought for a while and gave the answer. The student persisted: "but how is it done?" To which the professor gave a frustrated look and said, "I already solved it two different ways, what more do you want?"
    - Daub 6y ago
      University lecturers in particular are a deluded bunch. They mistake their role for one of power and forget that the students are their employers. Source: 100 years as a University lecturer.
LanceH 6y ago
Bad handwriting. At age 40 I started taking a masters in stats and had situations where I had exponents on exponents. This lead me to buy a higher resolution tablet for reading pdf's with tiny math. I also bought a finer point pen and this helped me improve my handwriting a lot. Closing loops on "o's" or backtracing the upward line of a cursive "t" to not make a loop. With the finer point, I was able to see my imprecision and improve it. I used to rewrite my finished work in undergrad and now my writing was improved to the point where the first draft was entirely legible.
- whimsicalism 6y ago
  I stopped handwriting math to turn in my freshman year of college, LaTex was just too good looking.
  - smabie 6y ago
    that seems like a ton of work, especially for problem sets
    - whimsicalism 6y ago
      You get very fast at it, especially if you start early in college - I actually live tex'd the lectures I went to starting sophomore year. Nowadays I mostly use markdown with inline latex as formatting with latex can be a pain.
    - dbcurtis 6y ago
      It’s a matter of practice. My kid takes lecture notes live using LaTeX. Of course that reflects a ton of practice on PSets and papers over the years.
    - MereInterest 6y ago
      For the most part, just a matter of practice. One advantage that it has is that you have less incentive to make multiple simplifications in a single step. Copy a line, make a rearrangement. Copy a line, combine two like terms. Since you aren't re-writing everything that is unchanged, there isn't as much of a cost to showing your steps. I did get a rather funny look from a professor in grad school when I turned in a fully typeset document on a timed take-home exam.
tgb 6y ago
The "working backwards" one was what I always noticed students doing. It's super frustrating as a teacher since they're simultaneously so close to being right (when they manage to avoid an irreversible step) and yet also have completely misunderstood a basic idea in what it means to even attempt to prove something.
- faceplanted 6y ago
  Do you factor that into the teaching now?
  - tgb 6y ago
    I no longer teach. I'm also skeptical of how useful this page is except as therapy for teachers - students have to make mistakes to learn from them. Though it could be a good reference to point to after the fact instead of explaining the problem from scratch.
- voxl 6y ago
  "Working backwards" is a completely valid method of proof, used all the time in automated reasoning. If a student successfully reasons backwards then there is no issue, it is a proof.
  - tgb 6y ago
    It's a completely valid work to do to figure out how to write a proof, but is not a proof itself. The section in the article states all my thoughts on that matter more clearly than I can. https://math.vanderbilt.edu/schectex/commerrs/#Backward https://math.vanderbilt.edu/schectex/commerrs/#Backward
    - voxl 6y ago
      As a PhD student myself, I would die on this hill, and you would lose. Moreover, the first sentence: "This is an unreliable method of proof" implies that it can be a proof if done correctly.
konjin 6y ago
Mathematical notation is terrible. I don't mean this in the usual "we need to invent new symbols to make it clearer" way. I mean it in the way that it has implicit typing that gets coerced constantly. Using Haskell types D :: (R -> R) -> (R -> R). Yet it gets used on things like D 2 = 0 which implies D :: (R -> R). What you've actually done is an implicit conversion of 2 :: R to 2(x) = 2 :: R -> R and 0 is not 0::R it is 0(x) = 0 :: R -> R. The Q_0 system described in [0] is a good start for a mathematical system that is both sane and as expressive as the one we have now. With term rewriting you can even keep the ridiculous notation we have today if you love it so much. Though I have grown incredibly fond of typing out extended typed lambda calculus using scheme notation in Emacs with a home grown automated theorem proving mode that was trivial to implement. [0] https://www.springer.com/gp/book/9781402007637 https://www.springer.com/gp/book/9781402007637
- euiq 6y ago
  That's not a very convincing example, given that numeric literals are polymorphic in Haskell.
  - jy14898 6y ago
    Not all Haskell is GHC, though
  - konjin 6y ago
    I never said Haskell was a good language. Q_0 is a minimally acceptable math notation.
- bunje 6y ago
  So you say we should specify when 2 means number two and when it means the constant function? How does it help?
  - konjin 6y ago
    Derivatives then are a higher order function that follow it's definition in all cases. You can't exactly apply it's definition as a limit in cases where it's applied to non-fictions.
ivan_ah 6y ago
Math images are not loading for me. Here is an archive.org link where the images are loading correctly: https://web.archive.org/web/20200305180920/https://math.vanderbilt.edu/schectex/commerrs/ https://web.archive.org/web/20200305180920/https://math.vand...
bpyne 6y ago
My biggest problems as a math major at the undergraduate level were proofs. I could muddle my way through abstract algebra proofs but real analysis just didn't click. The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.
- kevmo 6y ago
  Highly recommend the book "How to Prove It" by Velleman.
- bitxbitxbitcoin 6y ago
  I had a similar experience doing my math minor. All the way through Caluclus classes and Diff EQ and up until the first half of Linear Algebra, everything is plug and chug. After the first half of Linear Algebra when proofs started making an appearance, I came to the realization that proofs are another type of mental activity entirely. Survey of Algebra, Basic Real Analysis, and even the dedicated proof writing course I took were all exponentially harder to pass.
  - bpyne 6y ago
    My university realized the need for a proof writing course a little too late to help me. Students who did well in the more abstract math classes, aside from the outlier "gifted" mathematicians, formed study groups. I wasn't mature enough at the time to realize how valuable those groups were so I went it alone and my grades reflected it.
    - bitxbitxbitcoin 6y ago
      I hear you! I wasn't mature enough to join study groups or go to office hours during undergraduate, either, and my grades also reflected it.
  - vharuck 6y ago
    >the dedicated proof writing course What was yours like? The one I took didn't cover much on the actual writing of proofs. The professor accepted reasonable essays with high-school level notation. Instead, he gave us a toolbox for proving things: pairing terms in a series to find the sum, rewriting recursive equations, etc. It was mostly to show that clever tricks are how mathematicians prove new things. But that might've been because the professor was a guy who reveled in clever solutions.
waldrews 6y ago
I'd love to find such a clear and comprehensive list covering errors in my own area, probability - here errors usually blur the line between math and interpretation, and plenty of undergraduate errors persist among those of us old enough to know better.
Cyder 6y ago
My undergrad math classes were in lecture halls with a couple hundred students. The labs taught by grad students were better but would have been more helpful without the exponential confusion added by the lectures that had required attendance. That ended the pursuit of an engineering degree. Thanks George Mason. The economics courses and professors were amazing however.
chapium 6y ago
Most of these appear to be errors caused by stress or tiredness.
- commandlinefan 6y ago
  That's what my (junior in high school) son keeps saying when he makes simple arithmetic errors that cost him points on tests. I disagree, actually - these mistakes are made by lack of practice. If you've practiced solving enough integrals, it doesn't really matter how tired you are, you're going to get the right answer just as you won't read words incorrectly if you're tired.
  - mattbk1 6y ago
    I read words incorrectly when I'm tired. Is this abnormal?
    - ac42 6y ago
      At the least, you're not alone.
  - throw9128928 6y ago
    Which is it? Stress? Or tiredness? Both? Capacity to deal with either actually grows over time, you know that?
  - bigger_cheese 6y ago
    Stress/pressure has an effect on your performance for sure. In high school I didn't have many problems but at University I found I would often choke during exams despite knowing the material and being well prepared. I would get extremely nervous in the lead up to my university exams, I'd have difficulty falling asleep the night before, I'd be tossing and turning all night going over everything in my head and I'd feel anxious and sick (my stomach would be incredibly queasy) to the point I'd have trouble eating anything in the morning. So I'd go into my exams tired and irritable (because I hadn't eaten much) and I'd make stupid mistakes. Eventually I got over this but it took me a year or two and it definitely had an effect on my grades.
cgriswald 6y ago
I’ve gone so far as to explain in my answer why a question is ambiguous and to give answers for the multiple interpretations. I had one class that I dropped for many, many reasons, and one of the reasons was frequent ambiguous questions when the tests were all online and multiple choice. Some questions were ambiguous in a way that made the T/F answer invert depending on how you read the question. (To be clear, these were not trick questions that required precise reading. They were just bad questions.) Students never knew why they got the answer wrong unless they asked, which few did. A fair number of students started believing the opposite of the truth as a result of getting a question wrong on the test due to the ambiguity.
lmilcin 6y ago
I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion about that specific part of course material. If you don't understand something today, it is very likely the material tomorrow will refer to it and you will not understand that either. Even if you try to make up for it in couple of days, it will be rushed, will require more effort and your brain connections will not be the same quality. Every person will have a different way to achieve that goal. Some people need to ask questions, some like to figure out by themselves, some will want to read, solve exercises for as long as necessary. I have observed many people drop off and the reason was almost invariably the same. A bit of material passes by or maybe the load is too much, and the slippery slope of not understanding starts. Especially when you start to study mathematics, the first couple semesters are foundational and if you don't understand something it is like a pyramid, the further in time the more connections with the material that you did not understood and the more trouble you are in. I have seen people drop off because they thought it is the same as in their past or in other study areas. It is not. Mathematics is extremely connected internally and extremely intolerant of ignorance.
- benrbray 6y ago
  I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts. A particularly bad or lazy professor can be worse than no professor at all.
  - kkylin 6y ago
    Not arguing with your point, because it has merit, but do want to add that part of the problem with math education (at least here in the US) is structural: for most students, to start with what they know when they enter and to graduate in 4 years with a engineering / science / math / CS degree that means something, there is just too much to "cover." So we tend to cover "methods and tools," e.g., calculus first, with the motivation (e.g., electrical circuits or statistical regression or what have you) later. This lack of context makes it very hard to see how things fit together. For my own part, I usually learned the material from one course when I needed to use it in a subsequent course -- it's in going back to review the material when I really learn it. Edit: also want to add that providing context in early courses isn't always easy, because students are much more diverse now in their career plans. It used to be (this tells you how old I am) that many of the students taking calculus would also be taking (or have taken) a calculus-based physics course. Those days are long gone.
throwaway2245 6y ago
This hit(s) people of a certain age: Touch-typing on a calculator from experience of a phone keypad.
k__ 6y ago
To me the biggest problem was conversion of terms/formulars. I knew all the rules, but when I needed to reach a specific form I somehow ended up running in circles. Calculus felt more like learning chess than learning math. You needed to think three conversions ahead to get a result and that had to be practiced.
- arduinomancer 6y ago
  Yup, I think the subject is cool and interesting but I hated how the questions were almost like LeetCode where if you don’t know “this one simple trick/identity” to rearrange the equation then you won’t have a chance of solving it.
  - ubercow13 6y ago
    I think that unlike with LeetCode, in maths you usually do end up using those 'one simple tricks' a lot later on.
hyperpallium2 6y ago
I understood the mathematics, but made a lot of silly errors, e.g. misreading -/+. Messy handwriting, crammed layout and too many steps at once all contributed, but I also just made many mistakes. It felt impoosible to fix and didn't seem worth it since the understanding was what I valued. But since I never felt confident in my results - checking the answer was always suspenseful - my mathematics was useless, to actually use, without an answer to check. So I decided to methodically notice where I made errors, and to verify those particular places. "It might take me twice as long", I thought, "But at least I'll have the skill to do it, when needed, and be able to have results I can actually use!" So I began this, and it did take twice as long, and it worked. That sense of suspense started to disappear, because knew I had the correct answer. I became more skilled at checking, so it look less time. And then a very strange thing started to happen... I started to make those mistakes less often. I became attentive to them as I was making them. Then stopped making them. I didn't antipicate or account for this at all... I thought it was impossible for me to change - it was like magic. In hindsight, this was "deliberate practice", with the key quality of not just repetitive practice (which eventually plateaus), but practice focussing on appropriate aspects, changing as needed. An expert coach really helps here, both for identfying issues and psychological encouragement. Although I managed this particular one on my own, it was hard. But finding such a coach seems even more difficult! And I'm so obstinate, I'm not sure I would pay attention anyway...
dkarl 6y ago
But the part that I really want to discuss here is the other part -- i.e., the phrase "if k is any constant." To most teachers, that additional phrase doesn't seem important, because in the teacher's mind "x" usually means a variable and "k" usually means a constant. Most teachers get this right, using these conventions as a redundant booster to their verbal and written communication. Some are grumpy and lazy and want as much as possible to go unsaid. "For the next couple of weeks, when I say 'ring' I'm going to mean a commutative ring." Just say commutative ring. The time you save by glossing over these details doesn't add up to anything.
- aspaceman 6y ago
  I completely agree. A good mathematician says things like "this variable x and this constant k interact with our Transfer function T" etc. It's a lot of words but it turns into a sort of drill for the students about the notational norms.
mcguire 6y ago
"Loss of invisible parentheses. This is not an erroneous belief; rather, it is a sloppy technique of writing. During one of your computations, if you think a pair of parentheses but neglect to write them (for lack of time, or from sheer laziness)..." "The great number of sign errors suggests that students are careless and unconcerned..." Sounds like someone thinks they should have a better class of students.
- Izikiel43 6y ago
  I had a recurrent error to add extra parenthesis at the end I think
BrandoElFollito 6y ago
> Lack of clarity often comes in the form of ambiguity This is sometimes useful. Mathematicians have precision ingrained but this often repeals pupils. When you start with 'f is a continuous function defined in R of x defined in R, and for each x ..." - well, I am lost. This was the introduction of differentials in my son's high school. I am a physicist, so I started the other way round, by talking about speed, how it is calculated, how one can get more precise by shortening time and that, eventually, we get to the exact momentary speed. My son started to ask all kind of question such as how to "get closer" on my wavely drawing, to which I told him "good question - this is possible only when we know the function d(t)", etc. It is only when he understood the general reason for differentials to exist that we went back to the conditions (continuity, planes, etc.). He actually deduced the continuity constraint himself because he understood the "why". My math told told us once "I will show you a neat trick that you will not understand this year, but when you understand it next year it will be way less useful to you. Just know that it works only when this and that". So lack of perfection is sometimes useful for people to understand something at all and not wander off after the first two introductory sentences.
- aspaceman 6y ago
  Somewhat tangental but I am very jealous of a high school that introduces differentials. My high school didn't have calculus - lack of interest and no one qualified to teach it.
  - BrandoElFollito 6y ago
    This is interesting - so what (roughly) did you have in high school? In the few Europeans schools I know of, calculus is brought in though all the 3 or 4 years in high school because it allows, ultimately, to draw functions. (I am not a math teacher, just someone who went though French school and was very interested in the school system in some other European countries). It comes in pieces: you get to learn differentials and their meaning, usually then used in physics curriculum to denote speed and force. Then you get limits and how to calculate them for infinity or when approaching a non-continuous function from both sides. And l'Hôpital rule, etc. Again, it has direct implications in physics, and sometimes in biology (I saw that only once, when the teacher introduced enzyme kinematics). Then second differentials to analyze convexity. All this takes a huge chunk of math in high school, and you cannot du much in functional analysis without using differentials - so I am genuinely interested what was the pressure put on in your school (would you mind naming the country?). Trigonometry for instance? Or probability? (these are the two others I can think of that are the other important pieces they learn). Some of the schools ended by introducing integrals, usually though the concept of calculating surfaces. It was quite useful to have a rough idea of integrals before going to university (where they were introduced anyway)
deleted 6y ago
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dang 6y ago
If curious see also 2017 https://news.ycombinator.com/item?id=14628803 https://news.ycombinator.com/item?id=14628803 2014 https://news.ycombinator.com/item?id=8181101 https://news.ycombinator.com/item?id=8181101 2010 https://news.ycombinator.com/item?id=1108161 https://news.ycombinator.com/item?id=1108161
dan-robertson 6y ago
Regarding arrogant teachers, I’ve heard an anecdote about Dirac: if anyone ever asked a question, he would pause for a moment thinking, find the relevant part of the lecture, and the repeat it word-for-word. I guess he didn’t get many questions. I’m often a bit surprised by the handwriting problems but here are some tips: 1. Just write things bigger. Paper is cheap. Ink is cheap. I never liked the lines getting in the way so I just wrote on printer paper. 2. People often talk about superscripts and subscripts but outside of random calculus exercises, superscripts are either only 2 or sometimes 3 (so make them look different), a few letters like a, b, ab, 2a, a + b, etc (not hard to differentiate), or can be inferred from position (eg they go a_0 + a_1x + a_2x^2 + ...). Subscripts are usually either positional, something simple like n, n+1, n+2 (easy to differentiate), or some subset of the letters i,j,k,l (give j a decent descender to differentiate it from i; make sure the dot and curve on the i are obvious, write an l like a \ell). Mostly, you get to choose the letters so you don’t need to worry about differentiating \xi^{x^t} from \zeta^{\chi^\tau} (a close case is a pair of substitutions x -> f(\xi), t -> g(\tau), but this isn’t so hard when you realise that the Greeks go together and the Latins go together so you don’t need to differentiate t from tau) 3. A few simple handwriting modifications can help a lot (See https://news.ycombinator.com/item?id=22989703 https://news.ycombinator.com/item?id=22989703). I found larger ascenders and descenders helped (but make sure you know what they are in Latin and Greek), especially for i vs j, g vs p vs q, a vs d, u vs y, f vs s, z vs 7. I also modify a rho to sit on the baseline and slant more than a p. Curves and bowls help eg to differentiate a gamma from a y, or a k from a K from a kappa (write k with a bowl, kappa a bit like an x a la \varkappa), an l from a i, or a nu/u/v. I found italics unhelpful. My normal handwriting is italic but I write mathematics with an upright italic. 4. It doesn’t really matter if your xi and zeta are ugly so long as they don’t look like eachother or anything else. I had a lecturer who didn’t like to draw or say xi, he called it squiggle and would read aloud “so d squiggle d t is ...” (also twiddle is a much better name for \sim than tilde)
Tomminn 6y ago
The sin^n x notation is bad. I don't blame the undergrads there. So we learn the following fact: sin^{-1} x = y => x = sin y Being enthusiastic new algebra students, we presume know this must work by applying sine to both sides: sin^{1} sin^{-1} x = sin^{1} y => sin^{0} x = sin y here sin^{0} is zero applications of sine to x => x = sin y Noting we could start at line 2, and apply sin^{-1} to both side also, we have now learnt that: sin^{a} sin^{b} x = sin^{a+b} x if a = +/- 1 and b = -/+ 1. Presumably, if notation is at all sane, the rule applies to other values of a and b so: sin^2 x = sin sin x (? Surely!) Right? No. sin^2 x = (sin x)^2 and sin sin x = (has no other name) No wonder students get confused. The notation is trying it's darnedest to confuse them.
- Tainnor 6y ago
  Yes, this is definitely a problem. I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin". It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2]. Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.
  - Tomminn 6y ago
    This is a very good point. The sine function obviously doesn't have a technical inverse on any interval where it has two values. The notation does make me forget this sometimes. Of course, the situation is different from many other functions without inverses, because the set of all valid inversions can be trivially generated from one solution. Just put a mirror at pi/2 and -pi/2.

220 points by free0ne 6y ago

deleted 6y ago

[deleted]

When I was in undergrad, I declared a math minor quite late - just before my junior year. This led to me taking up to 9 hours of math credits in a semester. The reason I declared my math minor was because I was interested in going for a PhD in economics and my econ professors all recommended that I try to take as many math classes as possible to prepare. Reader, I was not a 'numbers person' at all. However, in all of my math classes, I asked questions all. the. time. when I didn't understand. I would go to office hours and the teachers were seriously so helpful. This was the only way I survived. I learned that you don't have to be a 'numbers person' at all to be good at math. You just have to understand what's going on under the hood, and this may require lots of questions and outside help. But most importantly, you just have to practice. I ended up graduating with nearly a 4.0 for my math minor.

jrib 6y ago
> I would go to office hours and the teachers were seriously so helpful. As a grad student that taught undergrads I always encouraged students to attend office hours. Most instructors do want you to succeed and one-on-one time is one of the best ways to learn. Definitely take advantage of your instructors' office hours; ask questions about anything that you aren't 100% sure about!
- nemo44x 6y ago
  > I would go to office hours and the teachers were seriously so helpful. This was the only way I survived. I learned that you don't have to be a 'numbers person' at all to be good at math. You just have to understand what's going on under the hood, and this may require lots of questions and outside help. But most importantly, you just have to practice I agree with all of this and it's how I got through extensive math. I'm not a "natural" at math and I'm not sure that really exists or possibly very people that are. I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. I figured the cost of tuition is best justified by using as many hours of one-on-one time with an expert in the subject to better understand it yourself. And to your last point about practice, I can't emphasize this enough. I would do the same problem sets over and over until I had them down. And it wasn't just memorizing the steps but rather getting to the point of being able to recognize the pattern. This made math "easy" at that point which is a weird phenomenon. Something that seemed so confusing and impossible suddenly becomes "easy". And then you can move on to the next step. I understand rote learning is not popular because it is painful. But I do believe "drilling it in" over and over gets the job done. But it takes the will and the effort. It's been a long time but I think I could still solve most the math I took back then.
  - dr_dshiv 6y ago
    > I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. I figured the cost of tuition is best justified by using as many hours of one-on-one time with an expert in the subject to better understand it yourself Truth!
    - deleted 6y ago
      [deleted]
  - randlet 6y ago
    > I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. One of my biggest regrets about university is not taking advantage of office hours. For me it was a combination of 1) social anxiety and 2) thinking that if I really belonged in this program (physics) I should really be able to figure everything out on my own. I know now that this is a stupid way to think but it's the way I felt at the time.
- godelski 6y ago
  I TA and tell my students that if they don't come to office hours or ask questions that they might as well be getting their degree on YouTube. I don't mean this in the sense that they have to ask for every class, but the advantage of attending school (besides the piece of paper) is to get direct help. But I'm not attending a top 10 university and honestly most topics are covered by one of those universities and they post their lectures on YouTube or their own OCW.
  - vmilner 6y ago
    Out of interest, roughly what proportion of students do come to the office hours?
    - godelski 6y ago
      Really depends on the class. One class I had basically no one. This one usually about 2-3 (out of 40) but more as we're nearing the end of term. Sometimes people come to say hi (again, class dependent). Largest I've had was like 8.
- hyperpallium2 6y ago
  https://wikipedia.org/wiki/Bloom's_2_sigma_problem https://wikipedia.org/wiki/Bloom's_2_sigma_problem the average student tutored one-to-one using mastery learning techniques performed two standard deviations better A tutor can notice gaps. You can't.
godelski 6y ago
I find this concept of being a numbers person weird to being good at math. Most math is about symbol manipulation and numbers are really only at the end.
- siltpotato 6y ago
  The concept itself or the name of it?
- magicalhippo 6y ago
  > Most math is about symbol manipulation and numbers are really only at the end. During the first semester at uni, the math prof was going through some non-trivial (at that stage) "real-world" example involving definite integrals. After what felt like 10 minutes of manipulating integrals and such we got to the final "x = ..." equation, upon which he turned away from the blackboard and said "and then you can just plug in the numbers and calculate the result, but that's not really interesting". And so the lecture continued...
  - godelski 6y ago
    Middle School / High School: "What are these letters doing in my math?!" Post University: "What are these numbers doing in my math?!"
- andi999 6y ago
  Here numbers are always in the beginning (of each chapter as the chapter number....)
Daub 6y ago
Getting students to seek consultation is a major problem. I advise them to come in small groups. This helps the shy ones. They don’t realize that it is very difficult to fail a student who asks for help. Any failure of such a student is implicitly also the teacher’s.
- Tw9875 6y ago
  Only true for the teachers that care. A lot of my professors didn't give a damn about teaching and were annoyed that I was wasting their time showing up to these office hours.
  - andi999 6y ago
    Was this math? I found usually math guys including professors like to talk about math.
    - megameter 6y ago
      In my experience, when they're top tier math researchers they tend to be oblivious to the student's perspective, grasping the problem too well to give it detailed treatment. This is neither a failing of their enthusiasm nor a social tactic, but an imbalance in their abilities. An old anecdote is the professor who was asked by a student how a certain problem is solved. The professor paused in thought for a moments, then gave the answer. The student asked again how to solve it, and the professor, again, thought for a while and gave the answer. The student persisted: "but how is it done?" To which the professor gave a frustrated look and said, "I already solved it two different ways, what more do you want?"
  - Daub 6y ago
    University lecturers in particular are a deluded bunch. They mistake their role for one of power and forget that the students are their employers. Source: 100 years as a University lecturer.

LanceH 6y ago

Bad handwriting. At age 40 I started taking a masters in stats and had situations where I had exponents on exponents. This lead me to buy a higher resolution tablet for reading pdf's with tiny math. I also bought a finer point pen and this helped me improve my handwriting a lot. Closing loops on "o's" or backtracing the upward line of a cursive "t" to not make a loop. With the finer point, I was able to see my imprecision and improve it. I used to rewrite my finished work in undergrad and now my writing was improved to the point where the first draft was entirely legible.

whimsicalism 6y ago
I stopped handwriting math to turn in my freshman year of college, LaTex was just too good looking.
- smabie 6y ago
  that seems like a ton of work, especially for problem sets
  - whimsicalism 6y ago
    You get very fast at it, especially if you start early in college - I actually live tex'd the lectures I went to starting sophomore year. Nowadays I mostly use markdown with inline latex as formatting with latex can be a pain.
  - dbcurtis 6y ago
    It’s a matter of practice. My kid takes lecture notes live using LaTeX. Of course that reflects a ton of practice on PSets and papers over the years.
  - MereInterest 6y ago
    For the most part, just a matter of practice. One advantage that it has is that you have less incentive to make multiple simplifications in a single step. Copy a line, make a rearrangement. Copy a line, combine two like terms. Since you aren't re-writing everything that is unchanged, there isn't as much of a cost to showing your steps. I did get a rather funny look from a professor in grad school when I turned in a fully typeset document on a timed take-home exam.

tgb 6y ago

The "working backwards" one was what I always noticed students doing. It's super frustrating as a teacher since they're simultaneously so close to being right (when they manage to avoid an irreversible step) and yet also have completely misunderstood a basic idea in what it means to even attempt to prove something.

faceplanted 6y ago
Do you factor that into the teaching now?
- tgb 6y ago
  I no longer teach. I'm also skeptical of how useful this page is except as therapy for teachers - students have to make mistakes to learn from them. Though it could be a good reference to point to after the fact instead of explaining the problem from scratch.
voxl 6y ago
"Working backwards" is a completely valid method of proof, used all the time in automated reasoning. If a student successfully reasons backwards then there is no issue, it is a proof.
- tgb 6y ago
  It's a completely valid work to do to figure out how to write a proof, but is not a proof itself. The section in the article states all my thoughts on that matter more clearly than I can. https://math.vanderbilt.edu/schectex/commerrs/#Backward https://math.vanderbilt.edu/schectex/commerrs/#Backward
  - voxl 6y ago
    As a PhD student myself, I would die on this hill, and you would lose. Moreover, the first sentence: "This is an unreliable method of proof" implies that it can be a proof if done correctly.

konjin 6y ago

Mathematical notation is terrible. I don't mean this in the usual "we need to invent new symbols to make it clearer" way. I mean it in the way that it has implicit typing that gets coerced constantly. Using Haskell types D :: (R -> R) -> (R -> R). Yet it gets used on things like D 2 = 0 which implies D :: (R -> R). What you've actually done is an implicit conversion of 2 :: R to 2(x) = 2 :: R -> R and 0 is not 0::R it is 0(x) = 0 :: R -> R. The Q_0 system described in [0] is a good start for a mathematical system that is both sane and as expressive as the one we have now. With term rewriting you can even keep the ridiculous notation we have today if you love it so much. Though I have grown incredibly fond of typing out extended typed lambda calculus using scheme notation in Emacs with a home grown automated theorem proving mode that was trivial to implement. [0] https://www.springer.com/gp/book/9781402007637 https://www.springer.com/gp/book/9781402007637

euiq 6y ago
That's not a very convincing example, given that numeric literals are polymorphic in Haskell.
- jy14898 6y ago
  Not all Haskell is GHC, though
- konjin 6y ago
  I never said Haskell was a good language. Q_0 is a minimally acceptable math notation.
bunje 6y ago
So you say we should specify when 2 means number two and when it means the constant function? How does it help?
- konjin 6y ago
  Derivatives then are a higher order function that follow it's definition in all cases. You can't exactly apply it's definition as a limit in cases where it's applied to non-fictions.

ivan_ah 6y ago

Math images are not loading for me. Here is an archive.org link where the images are loading correctly: https://web.archive.org/web/20200305180920/https://math.vanderbilt.edu/schectex/commerrs/ https://web.archive.org/web/20200305180920/https://math.vand...

bpyne 6y ago

My biggest problems as a math major at the undergraduate level were proofs. I could muddle my way through abstract algebra proofs but real analysis just didn't click. The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.

kevmo 6y ago
Highly recommend the book "How to Prove It" by Velleman.
bitxbitxbitcoin 6y ago
I had a similar experience doing my math minor. All the way through Caluclus classes and Diff EQ and up until the first half of Linear Algebra, everything is plug and chug. After the first half of Linear Algebra when proofs started making an appearance, I came to the realization that proofs are another type of mental activity entirely. Survey of Algebra, Basic Real Analysis, and even the dedicated proof writing course I took were all exponentially harder to pass.
- bpyne 6y ago
  My university realized the need for a proof writing course a little too late to help me. Students who did well in the more abstract math classes, aside from the outlier "gifted" mathematicians, formed study groups. I wasn't mature enough at the time to realize how valuable those groups were so I went it alone and my grades reflected it.
  - bitxbitxbitcoin 6y ago
    I hear you! I wasn't mature enough to join study groups or go to office hours during undergraduate, either, and my grades also reflected it.
- vharuck 6y ago
  >the dedicated proof writing course What was yours like? The one I took didn't cover much on the actual writing of proofs. The professor accepted reasonable essays with high-school level notation. Instead, he gave us a toolbox for proving things: pairing terms in a series to find the sum, rewriting recursive equations, etc. It was mostly to show that clever tricks are how mathematicians prove new things. But that might've been because the professor was a guy who reveled in clever solutions.

waldrews 6y ago

I'd love to find such a clear and comprehensive list covering errors in my own area, probability - here errors usually blur the line between math and interpretation, and plenty of undergraduate errors persist among those of us old enough to know better.

Cyder 6y ago

My undergrad math classes were in lecture halls with a couple hundred students. The labs taught by grad students were better but would have been more helpful without the exponential confusion added by the lectures that had required attendance. That ended the pursuit of an engineering degree. Thanks George Mason. The economics courses and professors were amazing however.

chapium 6y ago

Most of these appear to be errors caused by stress or tiredness.

commandlinefan 6y ago
That's what my (junior in high school) son keeps saying when he makes simple arithmetic errors that cost him points on tests. I disagree, actually - these mistakes are made by lack of practice. If you've practiced solving enough integrals, it doesn't really matter how tired you are, you're going to get the right answer just as you won't read words incorrectly if you're tired.
- mattbk1 6y ago
  I read words incorrectly when I'm tired. Is this abnormal?
  - ac42 6y ago
    At the least, you're not alone.
- throw9128928 6y ago
  Which is it? Stress? Or tiredness? Both? Capacity to deal with either actually grows over time, you know that?
- bigger_cheese 6y ago
  Stress/pressure has an effect on your performance for sure. In high school I didn't have many problems but at University I found I would often choke during exams despite knowing the material and being well prepared. I would get extremely nervous in the lead up to my university exams, I'd have difficulty falling asleep the night before, I'd be tossing and turning all night going over everything in my head and I'd feel anxious and sick (my stomach would be incredibly queasy) to the point I'd have trouble eating anything in the morning. So I'd go into my exams tired and irritable (because I hadn't eaten much) and I'd make stupid mistakes. Eventually I got over this but it took me a year or two and it definitely had an effect on my grades.

cgriswald 6y ago

I’ve gone so far as to explain in my answer why a question is ambiguous and to give answers for the multiple interpretations. I had one class that I dropped for many, many reasons, and one of the reasons was frequent ambiguous questions when the tests were all online and multiple choice. Some questions were ambiguous in a way that made the T/F answer invert depending on how you read the question. (To be clear, these were not trick questions that required precise reading. They were just bad questions.) Students never knew why they got the answer wrong unless they asked, which few did. A fair number of students started believing the opposite of the truth as a result of getting a question wrong on the test due to the ambiguity.

lmilcin 6y ago

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion about that specific part of course material. If you don't understand something today, it is very likely the material tomorrow will refer to it and you will not understand that either. Even if you try to make up for it in couple of days, it will be rushed, will require more effort and your brain connections will not be the same quality. Every person will have a different way to achieve that goal. Some people need to ask questions, some like to figure out by themselves, some will want to read, solve exercises for as long as necessary. I have observed many people drop off and the reason was almost invariably the same. A bit of material passes by or maybe the load is too much, and the slippery slope of not understanding starts. Especially when you start to study mathematics, the first couple semesters are foundational and if you don't understand something it is like a pyramid, the further in time the more connections with the material that you did not understood and the more trouble you are in. I have seen people drop off because they thought it is the same as in their past or in other study areas. It is not. Mathematics is extremely connected internally and extremely intolerant of ignorance.

benrbray 6y ago
I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts. A particularly bad or lazy professor can be worse than no professor at all.
- kkylin 6y ago
  Not arguing with your point, because it has merit, but do want to add that part of the problem with math education (at least here in the US) is structural: for most students, to start with what they know when they enter and to graduate in 4 years with a engineering / science / math / CS degree that means something, there is just too much to "cover." So we tend to cover "methods and tools," e.g., calculus first, with the motivation (e.g., electrical circuits or statistical regression or what have you) later. This lack of context makes it very hard to see how things fit together. For my own part, I usually learned the material from one course when I needed to use it in a subsequent course -- it's in going back to review the material when I really learn it. Edit: also want to add that providing context in early courses isn't always easy, because students are much more diverse now in their career plans. It used to be (this tells you how old I am) that many of the students taking calculus would also be taking (or have taken) a calculus-based physics course. Those days are long gone.

throwaway2245 6y ago

This hit(s) people of a certain age: Touch-typing on a calculator from experience of a phone keypad.

k__ 6y ago

To me the biggest problem was conversion of terms/formulars. I knew all the rules, but when I needed to reach a specific form I somehow ended up running in circles. Calculus felt more like learning chess than learning math. You needed to think three conversions ahead to get a result and that had to be practiced.

arduinomancer 6y ago
Yup, I think the subject is cool and interesting but I hated how the questions were almost like LeetCode where if you don’t know “this one simple trick/identity” to rearrange the equation then you won’t have a chance of solving it.
- ubercow13 6y ago
  I think that unlike with LeetCode, in maths you usually do end up using those 'one simple tricks' a lot later on.

hyperpallium2 6y ago

I understood the mathematics, but made a lot of silly errors, e.g. misreading -/+. Messy handwriting, crammed layout and too many steps at once all contributed, but I also just made many mistakes. It felt impoosible to fix and didn't seem worth it since the understanding was what I valued. But since I never felt confident in my results - checking the answer was always suspenseful - my mathematics was useless, to actually use, without an answer to check. So I decided to methodically notice where I made errors, and to verify those particular places. "It might take me twice as long", I thought, "But at least I'll have the skill to do it, when needed, and be able to have results I can actually use!" So I began this, and it did take twice as long, and it worked. That sense of suspense started to disappear, because knew I had the correct answer. I became more skilled at checking, so it look less time. And then a very strange thing started to happen... I started to make those mistakes less often. I became attentive to them as I was making them. Then stopped making them. I didn't antipicate or account for this at all... I thought it was impossible for me to change - it was like magic. In hindsight, this was "deliberate practice", with the key quality of not just repetitive practice (which eventually plateaus), but practice focussing on appropriate aspects, changing as needed. An expert coach really helps here, both for identfying issues and psychological encouragement. Although I managed this particular one on my own, it was hard. But finding such a coach seems even more difficult! And I'm so obstinate, I'm not sure I would pay attention anyway...

dkarl 6y ago

But the part that I really want to discuss here is the other part -- i.e., the phrase "if k is any constant." To most teachers, that additional phrase doesn't seem important, because in the teacher's mind "x" usually means a variable and "k" usually means a constant. Most teachers get this right, using these conventions as a redundant booster to their verbal and written communication. Some are grumpy and lazy and want as much as possible to go unsaid. "For the next couple of weeks, when I say 'ring' I'm going to mean a commutative ring." Just say commutative ring. The time you save by glossing over these details doesn't add up to anything.

aspaceman 6y ago
I completely agree. A good mathematician says things like "this variable x and this constant k interact with our Transfer function T" etc. It's a lot of words but it turns into a sort of drill for the students about the notational norms.

mcguire 6y ago

"Loss of invisible parentheses. This is not an erroneous belief; rather, it is a sloppy technique of writing. During one of your computations, if you think a pair of parentheses but neglect to write them (for lack of time, or from sheer laziness)..." "The great number of sign errors suggests that students are careless and unconcerned..." Sounds like someone thinks they should have a better class of students.

Izikiel43 6y ago
I had a recurrent error to add extra parenthesis at the end I think

BrandoElFollito 6y ago

> Lack of clarity often comes in the form of ambiguity This is sometimes useful. Mathematicians have precision ingrained but this often repeals pupils. When you start with 'f is a continuous function defined in R of x defined in R, and for each x ..." - well, I am lost. This was the introduction of differentials in my son's high school. I am a physicist, so I started the other way round, by talking about speed, how it is calculated, how one can get more precise by shortening time and that, eventually, we get to the exact momentary speed. My son started to ask all kind of question such as how to "get closer" on my wavely drawing, to which I told him "good question - this is possible only when we know the function d(t)", etc. It is only when he understood the general reason for differentials to exist that we went back to the conditions (continuity, planes, etc.). He actually deduced the continuity constraint himself because he understood the "why". My math told told us once "I will show you a neat trick that you will not understand this year, but when you understand it next year it will be way less useful to you. Just know that it works only when this and that". So lack of perfection is sometimes useful for people to understand something at all and not wander off after the first two introductory sentences.

aspaceman 6y ago
Somewhat tangental but I am very jealous of a high school that introduces differentials. My high school didn't have calculus - lack of interest and no one qualified to teach it.
- BrandoElFollito 6y ago
  This is interesting - so what (roughly) did you have in high school? In the few Europeans schools I know of, calculus is brought in though all the 3 or 4 years in high school because it allows, ultimately, to draw functions. (I am not a math teacher, just someone who went though French school and was very interested in the school system in some other European countries). It comes in pieces: you get to learn differentials and their meaning, usually then used in physics curriculum to denote speed and force. Then you get limits and how to calculate them for infinity or when approaching a non-continuous function from both sides. And l'Hôpital rule, etc. Again, it has direct implications in physics, and sometimes in biology (I saw that only once, when the teacher introduced enzyme kinematics). Then second differentials to analyze convexity. All this takes a huge chunk of math in high school, and you cannot du much in functional analysis without using differentials - so I am genuinely interested what was the pressure put on in your school (would you mind naming the country?). Trigonometry for instance? Or probability? (these are the two others I can think of that are the other important pieces they learn). Some of the schools ended by introducing integrals, usually though the concept of calculating surfaces. It was quite useful to have a rough idea of integrals before going to university (where they were introduced anyway)

deleted 6y ago

[deleted]

dang 6y ago

If curious see also 2017 https://news.ycombinator.com/item?id=14628803 https://news.ycombinator.com/item?id=14628803 2014 https://news.ycombinator.com/item?id=8181101 https://news.ycombinator.com/item?id=8181101 2010 https://news.ycombinator.com/item?id=1108161 https://news.ycombinator.com/item?id=1108161

dan-robertson 6y ago

Regarding arrogant teachers, I’ve heard an anecdote about Dirac: if anyone ever asked a question, he would pause for a moment thinking, find the relevant part of the lecture, and the repeat it word-for-word. I guess he didn’t get many questions. I’m often a bit surprised by the handwriting problems but here are some tips: 1. Just write things bigger. Paper is cheap. Ink is cheap. I never liked the lines getting in the way so I just wrote on printer paper. 2. People often talk about superscripts and subscripts but outside of random calculus exercises, superscripts are either only 2 or sometimes 3 (so make them look different), a few letters like a, b, ab, 2a, a + b, etc (not hard to differentiate), or can be inferred from position (eg they go a_0 + a_1x + a_2x^2 + ...). Subscripts are usually either positional, something simple like n, n+1, n+2 (easy to differentiate), or some subset of the letters i,j,k,l (give j a decent descender to differentiate it from i; make sure the dot and curve on the i are obvious, write an l like a \ell). Mostly, you get to choose the letters so you don’t need to worry about differentiating \xi^{x^t} from \zeta^{\chi^\tau} (a close case is a pair of substitutions x -> f(\xi), t -> g(\tau), but this isn’t so hard when you realise that the Greeks go together and the Latins go together so you don’t need to differentiate t from tau) 3. A few simple handwriting modifications can help a lot (See https://news.ycombinator.com/item?id=22989703 https://news.ycombinator.com/item?id=22989703). I found larger ascenders and descenders helped (but make sure you know what they are in Latin and Greek), especially for i vs j, g vs p vs q, a vs d, u vs y, f vs s, z vs 7. I also modify a rho to sit on the baseline and slant more than a p. Curves and bowls help eg to differentiate a gamma from a y, or a k from a K from a kappa (write k with a bowl, kappa a bit like an x a la \varkappa), an l from a i, or a nu/u/v. I found italics unhelpful. My normal handwriting is italic but I write mathematics with an upright italic. 4. It doesn’t really matter if your xi and zeta are ugly so long as they don’t look like eachother or anything else. I had a lecturer who didn’t like to draw or say xi, he called it squiggle and would read aloud “so d squiggle d t is ...” (also twiddle is a much better name for \sim than tilde)

Tomminn 6y ago

The sin^n x notation is bad. I don't blame the undergrads there. So we learn the following fact: sin^{-1} x = y => x = sin y Being enthusiastic new algebra students, we presume know this must work by applying sine to both sides: sin^{1} sin^{-1} x = sin^{1} y => sin^{0} x = sin y here sin^{0} is zero applications of sine to x => x = sin y Noting we could start at line 2, and apply sin^{-1} to both side also, we have now learnt that: sin^{a} sin^{b} x = sin^{a+b} x if a = +/- 1 and b = -/+ 1. Presumably, if notation is at all sane, the rule applies to other values of a and b so: sin^2 x = sin sin x (? Surely!) Right? No. sin^2 x = (sin x)^2 and sin sin x = (has no other name) No wonder students get confused. The notation is trying it's darnedest to confuse them.

Tainnor 6y ago
Yes, this is definitely a problem. I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin". It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2]. Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.
- Tomminn 6y ago
  This is a very good point. The sine function obviously doesn't have a technical inverse on any interval where it has two values. The notation does make me forget this sometimes. Of course, the situation is different from many other functions without inverses, because the set of all valid inversions can be trivially generated from one solution. Just put a mirror at pi/2 and -pi/2.