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.
> 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!
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.
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.
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.
This right here was such an infuriating discovery for me. Math never made sense to me until I had a kindly old teacher who insisted she watch me write out every step of a problem (in community college). She pointed out that the logic was fine, I was just being sloppy with my notes which led to errors.
It was like a dam broke over night. Everything clicked and I was so pissed off that it took so long to find such a stupid issue. Better late than never I suppose...
I had a soccer teammate who was a grad student in mechanical engineering. I was having problems in my Trigonometry class and asked for his help. I was absolutely shocked at how precise his handwriting was when he was writing stuff down and explaining it to me. It looked like he had typed out his notations, it was crazy.
He said something similar to you, "I'm not a doctor writing prescriptions, I need to be precise. If my writing is precise, its easier to read, easier to understand now and later."
When I was 14 my math teacher requested a conference with my parents. I was mortified. All he said was that I needed to write neatly. He was correct. That's all it took to go from a B student to A+. I make a ton of mistakes if I don't write out every step (turns out, the very mistakes on this list).
In the Intro-to-C class I TA'd, we dedicated several minutes in lab to instructing students on how to properly draw an ampersand (&) character, and the other ways of drawing it so potentially confusing (especially since a + means something completely different).
Personally, the +/t handwriting issue has never been an issue for me: make sure you get a good tail on the t, and it's pretty easy to distinguish from a +. It's the x/× distinction that was always the most painful to distinguish for me.
I took a course taught by a nice lady with a PhD in pure mathematics, no English skills, and handwriting that made differentiating between mu, u, and w very difficult. It was not a good time.
FWIW, if you use a pencil you should get to know your tool. Investigate different pencil options. There are quite a few cool features on mechanical pencils for instance. Try different lead hardnesses and thicknesses. Get a quality eraser.
I'm 45 and finally discovered I like softer lead. It's easier to read and smudging isn't a problem for me. I took drafting in high school but never retained any of the knowledge or put it to good use when writing.
\tangent can you recommend what to look for in a tablet for mathematics? I've got a lenovo M8 - only 8", 800p, but I think that's OK; it's the insufficient contrast of IPS screen that makes my eyes tired (suggesting AMOLED or epaper) - but I don't know the cause for sure... What did you find?
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.
The problem exists in the educational methods, in my opinion. It seems most, not just in mathematics, feel that learning is the same as if the student can simply be played a tape that cannot be paused or rewound but can sometimes be sped up. Nearly nothing works like this, and it’s just that the penalties in mathematics are greater.
I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions.
A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting.
One of the best examples of this I know of is the book Advanced Calculus: A Differential Forms Approach by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.
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.
I sort of agree, but your advice is targeted at an entirely different group of students than the article's. If someone is confused about how to interpret sin^2(x) vs sin^{-1}(x), there's no amount of "understanding" that's going to help them (it really is just idiosyncratic, inconsistent notation), but a checklist like the article might clue them in.
I think students often underestimate what "understand" means in a math class. The progression you're describing ("the first couple of semesters are foundational...") is definitely not universal in US undergrad programs, though. There's just not that much coordination between professors.
> the first couple semesters are foundational and if you don't understand something it is like a pyramid
I was taking a math/cs hybrid major in college, and the upper level math courses just felt like they were getting harder and harder in ways that didn't make sense in my previous math experience. Tests more and more were weighted towards verbal proofs, and I hadn't yet developed the vocabulary to deal with the precision necessary to use English to prove something mathematically. It wasn't until after graduation that I learned that there are some sophomore math courses at my university that Math majors usually take that were unofficially known as "intro to proofs". I have no doubt this was missing foundation that made 400 level math courses so painful for me.
There is no absolute here. While I agree that a great degree of understanding is desirable, lots of concepts and techniques don’t sink in until hours, days or even years after being presented.
I still remember powering through lots problems without much clarity and achieving the right answer. Every attempt, failed or successful, got me closer to really understand what I was doing.
It’s ok not to understand a topic the first or nth-time time you come across it. If you follow through, something will stick, and sooner than later you’ll get a good enough picture.
Mathematics is high on the list of "you can't fall behind and cram".
This is true not only within a single class but from class to class.
Aren't fluent in manipulating equations? Calculus is going to be misery. Don't know calculus like the layout of your bedroom? Differential equations aren't going to happen.
All the way back to `2+3 = 5` in first grade. If that doesn't go well, the rest will likely not, either.
I agree with this as the father of two kids in high school.
I routinely force them to fully understand some things they learn, and I am more liberal with others.
Mathematics is one of these subjects where there is almost nothing you can afford not to understand. It will absolutely bite you someday.
When learning math there are things that are really new (such as differentials, or operations on fractions) that are conceptually different from other things you just learn (Pythagoras theorem for instance) and I spend a lot of time with them to help them understand the "why" of these operations. Otherwise i know they will have a very, very hard time to follow up.
Physics is lighter - you can have trouble understanding thermodynamics and it will not mean much when doing mechanics. There are parts of physics I never really understood despite having a PhD in physics.
This is in sharp contrast with, say, history where missing a bit does not impact the next things one learns (at least at high school level - and this is from someone who loves history).
Maybe college-level math is different from math taught in a maths depart? Either way, the common errors mentioned in the article appear to be, well, elementary. I'd imagine 6 or 7 graders make those mistakes, but college students? Shouldn't they make mistakes like "there is no function that is everywhere continuous but nowhere differentiable", or "isomorphism of factors implies isomorphism of quotient groups"? The webpage says a lot about the quality of the K12 education in the US.
I did well in my college math exams (top 5 percent of class) by just memorizing proofs and memorizing the methods to solve common problems without understanding much of anything about what I was doing. The exams were very much hackable.
I wouldn't be able to pull this off in grad level maths but undergrad it's feasible.
It was a total waste of time though and I regret that approach, but I did what I was given an incentive to do (least effort for max reward)
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.
> 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.
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.
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.
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). 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)
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...
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.
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.
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.
More lack of preparedness. My approach in setting exams was always that someone who actually understood the material well would find most of the exam quite easy - i.e. you'd be able to pass without focusing hard. Getting top marks would require deeper understanding. The basic idea here is that any student who genuinely understood the course material should definitely be able to pass even on a bad day, or a bad exam schedule, etc. If you can't meet that mark, you shouldn't advance.
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.
"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.
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.
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.
[+] [-] sparklingmango|5 years ago|reply
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|5 years ago|reply
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!
[+] [-] godelski|5 years ago|reply
[+] [-] Daub|5 years ago|reply
[+] [-] LanceH|5 years ago|reply
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.
[+] [-] beepboopbeep|5 years ago|reply
It was like a dam broke over night. Everything clicked and I was so pissed off that it took so long to find such a stupid issue. Better late than never I suppose...
[+] [-] woliveirajr|5 years ago|reply
At that age it took me a while to understand where the problem was and too late to ask my teacher to reconsider it.
Well, was so intense that I remember it until today, 30 and some years later.
[+] [-] at-fates-hands|5 years ago|reply
He said something similar to you, "I'm not a doctor writing prescriptions, I need to be precise. If my writing is precise, its easier to read, easier to understand now and later."
[+] [-] viburnum|5 years ago|reply
[+] [-] jcranmer|5 years ago|reply
Personally, the +/t handwriting issue has never been an issue for me: make sure you get a good tail on the t, and it's pretty easy to distinguish from a +. It's the x/× distinction that was always the most painful to distinguish for me.
[+] [-] beervirus|5 years ago|reply
[+] [-] LegitShady|5 years ago|reply
[+] [-] 01100011|5 years ago|reply
I'm 45 and finally discovered I like softer lead. It's easier to read and smudging isn't a problem for me. I took drafting in high school but never retained any of the knowledge or put it to good use when writing.
[+] [-] hyperpallium2|5 years ago|reply
[+] [-] barrenko|5 years ago|reply
[+] [-] whimsicalism|5 years ago|reply
[+] [-] lmilcin|5 years ago|reply
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.
[+] [-] bmitc|5 years ago|reply
I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions.
A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting.
One of the best examples of this I know of is the book Advanced Calculus: A Differential Forms Approach by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.
[+] [-] benrbray|5 years ago|reply
A particularly bad or lazy professor can be worse than no professor at all.
[+] [-] grayclhn|5 years ago|reply
I think students often underestimate what "understand" means in a math class. The progression you're describing ("the first couple of semesters are foundational...") is definitely not universal in US undergrad programs, though. There's just not that much coordination between professors.
[+] [-] milesvp|5 years ago|reply
I was taking a math/cs hybrid major in college, and the upper level math courses just felt like they were getting harder and harder in ways that didn't make sense in my previous math experience. Tests more and more were weighted towards verbal proofs, and I hadn't yet developed the vocabulary to deal with the precision necessary to use English to prove something mathematically. It wasn't until after graduation that I learned that there are some sophomore math courses at my university that Math majors usually take that were unofficially known as "intro to proofs". I have no doubt this was missing foundation that made 400 level math courses so painful for me.
[+] [-] dzolob|5 years ago|reply
I still remember powering through lots problems without much clarity and achieving the right answer. Every attempt, failed or successful, got me closer to really understand what I was doing.
It’s ok not to understand a topic the first or nth-time time you come across it. If you follow through, something will stick, and sooner than later you’ll get a good enough picture.
They are like seeds, you know.
[+] [-] JJMcJ|5 years ago|reply
This is true not only within a single class but from class to class.
Aren't fluent in manipulating equations? Calculus is going to be misery. Don't know calculus like the layout of your bedroom? Differential equations aren't going to happen.
All the way back to `2+3 = 5` in first grade. If that doesn't go well, the rest will likely not, either.
[+] [-] BrandoElFollito|5 years ago|reply
I routinely force them to fully understand some things they learn, and I am more liberal with others.
Mathematics is one of these subjects where there is almost nothing you can afford not to understand. It will absolutely bite you someday.
When learning math there are things that are really new (such as differentials, or operations on fractions) that are conceptually different from other things you just learn (Pythagoras theorem for instance) and I spend a lot of time with them to help them understand the "why" of these operations. Otherwise i know they will have a very, very hard time to follow up.
Physics is lighter - you can have trouble understanding thermodynamics and it will not mean much when doing mechanics. There are parts of physics I never really understood despite having a PhD in physics.
This is in sharp contrast with, say, history where missing a bit does not impact the next things one learns (at least at high school level - and this is from someone who loves history).
[+] [-] hintymad|5 years ago|reply
[+] [-] hnracer|5 years ago|reply
I wouldn't be able to pull this off in grad level maths but undergrad it's feasible.
It was a total waste of time though and I regret that approach, but I did what I was given an incentive to do (least effort for max reward)
[+] [-] konjin|5 years ago|reply
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
[+] [-] BrandoElFollito|5 years ago|reply
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|5 years ago|reply
[+] [-] bpyne|5 years ago|reply
The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.
[+] [-] ivansavz|5 years ago|reply
Here is an archive.org link where the images are loading correctly: https://web.archive.org/web/20200305180920/https://math.vand...
[+] [-] dan-robertson|5 years ago|reply
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). 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)
[+] [-] k__|5 years ago|reply
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.
[+] [-] Tomminn|5 years ago|reply
So we learn the following fact:
Being enthusiastic new algebra students, we presume know this must work by applying sine to both sides: Noting we could start at line 2, and apply sin^{-1} to both side also, we have now learnt that: Presumably, if notation is at all sane, the rule applies to other values of a and b so: Right? No. No wonder students get confused. The notation is trying it's darnedest to confuse them.[+] [-] hyperpallium2|5 years ago|reply
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...
[+] [-] tgb|5 years ago|reply
[+] [-] cgriswald|5 years ago|reply
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.
[+] [-] dkarl|5 years ago|reply
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|5 years ago|reply
"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.
[+] [-] chapium|5 years ago|reply
[+] [-] ska|5 years ago|reply
[+] [-] commandlinefan|5 years ago|reply
[+] [-] dang|5 years ago|reply
2017 https://news.ycombinator.com/item?id=14628803
2014 https://news.ycombinator.com/item?id=8181101
2010 https://news.ycombinator.com/item?id=1108161
[+] [-] mcguire|5 years ago|reply
"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|5 years ago|reply
[+] [-] Cyder|5 years ago|reply
[+] [-] unknown|5 years ago|reply
[deleted]
[+] [-] waldrews|5 years ago|reply