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 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.
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.
> 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.
> 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.
> 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
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.
> 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...
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.
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.
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?"
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.
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 write the times as two crossing lines, and an X as two vertical arcs that touch in the middle. Now they are easy to distinguish. It isn't until I went looking for an image of it that I figured out how weird this is.
I also draw a y with one stroke so that it has a visible curve. That keeps a bad y and bad times from looking like each other.
I had an English calc teacher in high school who drew his 'x' as approximately ")(" (With the arcs touching-ish) and the cross in the normal way, which made it much easier to distinguish the two.
I blew several minds at my first job when I pointed out the ampersand is just a mirrored capital cursive S. Suddenly it wasn't such a mysterious character to any of them anymore.
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?
I mostly use an Amazon Fire 10" for math. It shows that it is IPS, but it really looks good, imo.
Do you have a big contrast between your screen and your room? That's a big deal. I run a blue light filter at night.
I do prefer eink and anything oled looks fantastic, but the fire looks really good, well beyond its $150 price tag. Better than my computer monitors (which are nice).
Are you reading tiny, detailed print? Is it just a resolution thing?
Lastly, are you getting older (40ish) and need reading glasses? That happened to me when crossing 40. I could still focus sharply at reading/screen distance, but I got tired. I went through that a couple years and got reading glasses -- actually just weaker versions of my nearsighted glasses. I had the doctor dial in the focal distance for screen distance, then got another pair for 12 inches are so (for a reading a book up close).
So I have three pairs of glasses, one for distance (everything sharp from 3 feet outward). These strain my eyes from 3 feet inward. You don't need reading glasses if your arms are long enough...
I have computer classes dialed in at roughly 28 inches. I can read up close, and see nearly perfectly at distance. I'm probably 20/20 at distance, but with my regular prescription I'm around 20/10.
Then the reading glasses for up close things. These are fine for computer as well, and just slightly more blurry at distance.
Thanks! I hadn't considered room lighting - though I also need it bright enough for pen and paper.
Resolution seems ok - I zoom in when needed, which is rare. Some pdf's have lines too long to fit on the screen, for the text to be large enough - but they are also rare. I think a 10" would solve that (held lanscape, 10" is basically A4 portrait width, if you zoom out the margins). So it doesn't feel like resolution is a problem - although some people say fuzzier text tires eyes.
I have 0.28 and 0.38 and I probably choose based on the paper. If I choose one pen, it's the Uniball Signo DX 0.38. That seems enough for cheap paper and precise enough that everything is legible without extreme care.
If I were doing math I would prefer to use the 0.28 version and make sure my paper is good. Anything smaller and it feels like I'm drawing and not writing. I think the 0.2's are mostly used for art.
Pens are fine for math. I often used a fine point gel pen in college. However, you have to have a high degree of confidence in what you're writing. If you make a lot of mistakes (either actual mistakes or just wrote down something incorrectly or with a misspelling), then pencils are definitely better.
I got to a point where I would just write down all my solutions on a final paper to turn in for my math assignments.
I did everything in pen, bc of handwriting issues and pencils. I simply burned through sheets and sheets of paper with my preliminary work- then copy everything to something both legible and presentable.
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.
I had a few professors require LaTeX solutions. After a little practice it was only slower than handwritten answers for some symbol-heavy assignments. So much time was spent comparatively on solving the problems that it was a negligible overhead regardless.
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 thought this was what university did. For instance, there's a little stats in calculus courses, then there's stats in mathematical stats courses, then in queuing theory course, then in networking course, and of course in all kinds of statistical learning and machine learning courses. By the time you finish those courses, you will get a pretty good grasp of basic stats.
As long as we are talking math, you can take a bunch of books and learn the same without need for university. You can even go to talk to somebody who will tell you what books to take and in what order.
What university does is help you connect with other people who study the same, to observe or work with them when solving problems and to have access to people working the cutting edge of the discipline you are studying who you would not normally have access to.
Also, it gives a little bit pressure which might be important for some or most people.
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.
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.
I took Physics as an elective in University, but it really didn’t sharpen my Calculus skills. Which is sad, but probably because I did so bad in Calculus as an undergraduate. But, one thing I absolutely did take away from Physics was the idea of units and their conversions. It has come in handy so often though in and outside my degree of computer science. If I had to simplify it, it really helps you not to compare apples to oranges. At the beginning of Street Fighting Mathematics it brings up that point. For example it isn’t a good comparison to compare the wealth in assets of a given company to a countries GDP. Because, GDP is happening over every year in time whereas the wealth is merely effected by what point in time it’s observed. Units help one see that in order to be truly and more fairly comparable they have to be the same type of units.
I used an analogy of units in physics recently to help a teammate with types and generics, sadly I can’t remember exactly what I said haha other than don’t compare apples to oranges, but I think I had further insight than just that.
Absolutely! internalizing dimensional analysis has been one of the most useful things I learned in grad school -- even though converting units was something many of us learn, in one form or another, before college.
It is interesting how some of the things just feel like common knowledge when they are not.
I frequently see people comparing things that aren't really comparable and it always seems to me it should be obvious this can't be done.
Most common example is mixing watts and watt*hours. Which should be pretty obvious. Nobody mixes by accident the speed at which you travel with the distance you have to travel. Nobody says "My car can pull off 200kms" or "I have 120km/h to cover" yet the same happens for energy vs power on a daily basis, even in publications.
At Berkeley, for instance, there were multiple Physics sequences: engineering + physics + chem majors took the 7 series ("Physics for Scientists and Engineers"), which used calculus; for most other majors (e.g. premed), the 8 series ("Introductory Physics") sufficed for degree requirements.
(Also, note that this is similar to the split between the AP Physics B and C courses that high school students might take. Physics B has apparently been replaced with Physics 1 / 2, both of which are algebra- rather than calculus-based.)
I think ultimately this is a function of where you went to school, more than anything.
I think tech-oriented schools, e.g., Caltech, MIT, Georgia Tech, etc., are exceptions rather than the rule. Most large state universities in the US are likely to have different physics / chemistry / math courses for students heading in different directons. Not all intro physics courses will be calculus-based, yet some of the students in those courses will still have to take calculus.
Many US medical schools, for example, silll require 1 semester calculus and some physics, but I don't think the physics course needs to be calculus-based. Many used to require a second calculus, but I think that has largely shifted to biostatistics instead.
Oh yeah, I'm with you there. Teaching quality at both my undergrad math program and my graduate compsci program was abysmal, with the exception of a handful of dedicated teachers. Both at well-ranked research institutions, which is the problem.
The whole system is crazy. Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!
> Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!
Because the culture of innovation that part-time researchers/part-time teacher bring is far more important to society in the long run than the short term gains gotten by full-time teaching staff.
Most fields in the world are rapidly changing, and what's taught in them changes every decade or so. Most full time teachers will atrophy because they don't have any real incentive to keep their knowledge and skills at the cutting edge. After a decade or so, your full time educational institute will be teaching worse than a research+teaching institute.
Have you experienced student life at a large public research university recently?
The well-documented trends of administrative bloat, publish-or-perish, exploitation of adjuncts and TAs for cheap labor, obsession with luxury campus amenities, etc etc. have all slowly degraded this "idealized" version of academia that many people have in mind (assuming such a thing ever existed in the first place).
I run into the same wall talking about this with my parents, who don't really seem to understand just what a racket the whole university system as become.
> Most full time teachers will atrophy because they don't have any real incentive to keep their knowledge and skills at the cutting edge.
Tenured professors also have no incentive to keep their knowledge and skills at the cutting edge. An oh boy do I have some stories about lazy professors, both as a student and TA.
> Because the culture of innovation that part-time researchers/part-time teacher bring is far more important to society in the long run than the short term gains gotten by full-time teaching staff.
Research and education are both important. Both have long-term gains. Research is hard, yes, but many people fail to recognize that teaching is also hard. We should fund both, and use a transparent pricing model.
If an 18-year-old takes on tens of thousands of dollars of personal debt, the benefit should be directly to them. The whole "coLLegE lOAnS aRe An InVEstMEnt in YoURselF!!" argument against free university education is naive from the start, but becomes completely absurd when that money isn't even used to directly fund education.
Yes, basic research is fundamental to the advancement of human society. So let's fund it publicly, through business or personal taxes on all members of society. Let's also recognize that higher education has massive long-term benefits to society as well, and make high-quality college education cheaper and more accessible by hiring dedicated teaching faculty who consult with research experts to design their curriculums. We can do both.
The current system is extremely inefficient at distributing the time and resources of research faculty. We don't need research faculty to regurgitate the same intro linear algebra lecture in-person over and over again every year.
The current system also forces research faculty (and graduate student TAs) to teach, who more often than not have no interest whatsoever in teaching. It is also extraordinarily difficult to get a well-paid college teaching job without a research PhD.
I would advocate for a system where research faculty are in charge of designing the curriculum and modernizing lecture materials as the times change. Rather than repeating the same lecture hundreds of times, have them make lecture recordings, which can be updated gradually over time. This frees them up to design better assignments and projects, rather than repeating the same lecture over and over.
It is well documented that students benefit from small classrooms. Hire qualified full-time teaching staff to lead discussions / labs / projects. Pay them enough that they don't need a second or third job, so they have the time to stay up to date. Personally, I would love love love to teach undergrad-level compsci or math if such an opportunity presented itself.
This is the role that TAs normally fill, but it is extraordinarily rare to find a graduate student who has passion and skill for teaching, as well as the luxury of enough time to do it well.
I have not studied as an undergrad at a large institution, but I have taught at one, and I have taught as a prof at teaching-only institute. There is very little difference in the quality of teaching. Guess what, when teaching only staff is asked to teach 6 courses in an year, they teach the same as a research prof who spends half their time in research and half their time teaching 3 courses.
I think there are many things wrong with academia and with university structures, and teaching quality is one of them. But the problem is not what you think it is. The problem is capitalism, and money-optimizations being the final decision maker rather than quality of teaching or research (which is also much worse than 50 years ago).
This is the same reason school education is so mediocre. You overwork school teachers, and don't pay them enough. And so even the best can't do much good.
You need more money from the state to go into education. And yes, full time teaching staff [1] to conduct recitations and office hours would help a lot. Smaller classrooms would help as well.
I don't agree at all with lecture recordings. As much as you want people in society learning, you also want a lot of people in society learning to teach. The long term intellectual benefits to society where people learn to teach others are enormous. It creates a culture of intellectual inquiry, which is different from attending a pre-recorded lecture or engaging in research or all the other things students/faculty at universities do.
[1] I have been told in German unis these are people with Masters who are waiting to go for a PhD. But they are essentially kicked out after 2 years because the uni recognizes that long-term full-time teaching staff degrades in quality.
> Guess what, when teaching only staff is asked to teach 6 courses in an year, they teach the same as a research prof who spends half their time in research and half their time teaching 3 courses.
> ...
> The problem is capitalism, and money-optimizations being the final decision maker rather than quality of teaching or research (which is also much worse than 50 years ago).
> ...
> You need more money from the state to go into education.
Absolutely, agreed on all points. Find people who are passionate about education and give them the time and resources to do it well. Build a culture of learning, mentorship, and open discussion. Let students and teaching faculty mingle as much as possible with research faculty while still keeping priorities straight.
Sure, in an ideal world, every student would get one-on-one tutoring from a brilliant researcher. But this doesn't scale well, and isn't necessarily a great use of the researcher's time. Teaching-only faculty are more than good enough until the student approaches the research level.
> I don't agree at all with lecture recordings. As much as you want people in society learning, you also want a lot of people in society learning to teach.
Oh I agree, perhaps I misrepresented myself. I'm not suggesting we replace traditional lectures with a big movie screen that plays pre-recorded lectures.
However, I think the current way of doing things -- where a professor inherits some slides she didn't create herself and reads them off in front of the class with no preparation whatsoever -- isn't the best either. This just goes back to finding passionate teachers and giving them the resources they need to be successful.
I do think pre-recorded lectures have their place alongside traditional textbooks, lecture notes, etc.. One downside of traditional lectures is that lecturers get very little feedback about their teaching style, and it's difficult to diagnose how students are really doing.
A handful of courses in my undergrad math program were run in an "Inquiry-Based Learning (IBL)" format. Rather than a traditional lecture, the professor breaks the class into small groups and asks a series of leading questions designed to help students discover a new concept on their own. The professor can adjust the pace and offer explanations as needed.
Here's an example [1] of some class handouts from a topology class. I borrowed them from a friend who took the class, and going through all the exercises on my own brought me a sense of clarity that I was never able to achieve from the standard lecture-based topology course I took. My friend felt the same way, and according to surveys done by the department, students overwhelmingly prefer the IBL format to traditional lectures. This format benefits professors, too, who get immediate feedback about their teaching methods and how well students are doing.
Funny, you link these notes. Because the prof is a full time researcher and seems to be teaching the same couple of courses over and over again [1]. Which is why you get the quality of the notes that you see.
Sure, some professors manage to do well at both research and teaching! It's great that some professors like her care about education and are experimenting with new classroom formats! My point was that traditional lectures are not always the best way of teaching.
And, I did not say it is impossible or even uncommon for research faculty to be good at teaching. However, being good at research does not automatically make one good at teaching, and there are other factors like time that prevent those who do care about education from giving their students a good experience in class.
In my own personal experience, and that of my classmates, the vast majority of research faculty simply do not have either the interest or the time to teach well. Out of the 30 or so math courses I took as an undergrad, I would say only about 4-5 of my professors put real effort into teaching.
I'm now interested in a list of transactions ordered by participant dissatisfaction, like, paying for gym memberships and not getting the outcomes they want, therapy or medical attention to mostly be ignored by the provider, ISPs in the US, online shopping to get something wildly different.
Is education especially bad compared to these things? or just that it's such a large expenditure we just-world-fallacy ourselves into thinking it ought to work better?
Is it really a logical fallacy to assume that we should have better outcomes from spending tens to hundreds of thousands of dollars on academia vs e.g. spending thirty dollars on amazon?
At least naively, that money could (should?) be going towards paying for staff and a support system/hierarchy to ensure a higher quality. That's at least nominally why accredited, non-profit education is so expensive, because it's nominally supposed to be capable of paying decent salaries to intelligent people dedicated to teaching, compared to say amazon or the gym where the incentives are to cost cut and maximize profit.
In reality, I suspect the same applies to colleges, but I don't think it should, in some ideal world.
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.
Those upper division math classes are pretty painful regardless. The material is just innately hard in a way that's dissimilar to most other hard things.
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.
I guess what you mean by "sink in" is a form of deeper more intuitive knowledge. Like when you suddenly figure out how everything is connected.
If that's what you mean, then yes -- this is something that is going to happen over time.
What I mean is that your teacher will want you to understand some concepts and learn some skills before you move to next topic. Very deep understanding is probably not on the list and it is expected to only form over time. But there are topics that are explicitly expected and you have to understand because otherwise what is going to be on the board the next day is not going to make whole lot of sense.
Yes and no. What I meant was more like, you don’t need to fully understand rigid body mechanics to build a table, but if you build enough tables, you might win some knowledge on rigid body mechanics that you may have missed in class.
Of course, this is not always true, but a concrete example that comes to mind is integration. It’s very hard conceptually, but not so in technique. If you learn how to do it, for the most part you can skip the concept (to some degree, obviously). I didn’t understood Riemann integral until I learned Lebesque.
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.
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.
I honestly never found this an issue. Shorthand and type coercion is used a lot but at least when taught, it's usually made very explicit what notation means and when things are being excluded. After that point, the notation is the least challenging part.
It is not an issue until you start trying to teach any sort of fix point theorem, at which point the wheels fall off because the notion can't deal with higher order functions at all.
I guess I just don't get your point. D 2 = 0 doesn't imply anything, it's just shorthand to make working on something less tedious for people who are already familiar with it. It seems perfectly expressive and sane: mathematicians have successfully studied differentiation using standard math notation. 2 and (λ (x) 2) aren't 'treated the same', that's just a strawman.
Funny how its also used in every textbook for people who aren't. This is getting to be apologia on the levels that I'd only seen in history books about why Roman numerals were superior.
The premise of your argument is that this notation is confusing people, or it's impossible to make explicit what it means. I don't buy it, you even explained the type elision using standard mathematical notaion in your original comment.
So if there is some superior notation that would benefit all of maths, that sounds very interesting but I don't think you've made that point very well. Maybe a better example is needed, because on its own, eliding repetitive information from notation does not seem like a problem.
Because I'm not going to write a few dozen pages of latex in a text comment. If you're interested thrown me $2k over bitcoin and I'll spend the 8 hours it would take to make this rigorous at below my usual rates.
I agree that your example isn’t very convincing, for an additional reason. You can treat 2x as a function, or you can treat it as an expression, where x is its own special kind of thing: a symbol.
With how the notation is typically used, it’s at least closer to my mental model.
Treating the expression as a function under substitution is one way to look at it, but not the only way.
The point is that 2 and (λ (x) 2) are vastly different objects, yet in standard math are treated the same.
Also you don't need a symbol type to deal with variables, you just need to allow type variables to be used in standard expressions. (define x R) (^ x 2) is just as easy to manipulate as (λ (x) (^ x 2)), the only issue is that your rewrite rules how aren't just looking at syntax, but semantics too.
> 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.
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)
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.
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.
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.
It has been over a decade so I don't exactly remember - I looked it up and the class was Introduction to Mathematical Reasoning.
What I personally recall was following along in class while the professor walked through classic proofs emphasizing what each new notation meant as well as the difference between Direct, indirect, and induction proofs. Tests and assignments were essentially recreating proofs cherry picked to be similar to ones we walked through in class.
I realize that's kind of how all my higher level math classes were run. It's just that once the cherry picking becomes looser, intuition doesn't necessarily catch up :(.
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.
I had a very similar experience during undergrad. I loved most of my classes through Linear Algebra but developing proofs felt like trying to learn a new language. Unfortunately, it never really clicked for me.
I'd love to know how to develop an intuition for writing proofs from scratch.
I also strugged with analysis. I always felt like epsilon-delta proofs involved pulling some absolutely strange value for delta out of your ass that happens to work out in the end, and I never developed an intuition for that. Same with integrating by parts, oh it just works out so nicely if you rewrite u in this totally obtuse way.
The tricky thing with analysis that I don't think many professors are good at conveying is that the ordering of statements in the proof isn't the same as the ordering of steps the proof writer performs to come up with the proof.
Basically you sort of write the broad strokes of the proof up front, leaving the right hand sides of statements like "Choose epsilon such that epsilon = __" blank. Then you do a bunch of scratch work to figure out what epsilon needs to be so that your proof works out in the end.
Another challenge with analysis is that inequalities are central, so fluency in their manipulation is absolutely critical. And naturally most students aren't fluent with them by the time they take analysis, so they get bulldozed by Baby Rudin and learn to hate a pretty cool (and useful) branch of math
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)
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.
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.
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.
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.
"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.
Nobody’s saying it’s totally invalid. I think you’re reading that into what they’re saying and you’re talking past each other.
The complaint is that students work backwards without understanding why working backwards is different than working forwards. So if they happen to do it correctly (e.g. by using reversible steps), then it’s only success via accident.
Done correctly, and on purpose, with the different care that isn’t required working the other way, yeah it’s legit, but that’s not what’s being complained about.
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.
Say a student is completing a problem set and is asked to prove P. If the student writes "P therefore Q therefore True since Q was proven in class. QED" then the student has committed a significant error in reasoning. If the student instead just writes "P. Q. True. QED" then they are writing the above in a more terse manner and have equally committed an error in their reasoning. If they first write that but then amend in some "iff"s between each step I'll give them full credit (assuming iffs are valid) but be slightly worried that they just learned that you need to put an "iff" in those spots to keep the teacher happy but don't understand why.
If a strategy is not guaranteed to give proof, then you need to verify afterwards that the putative proof is in fact a proof. Just as if you get a potential solution to an equation (via solving a more general equation, perhaps), you have not "solved" the original equation until you actually check that solution, even if your putative solution is the true one. If a student does not do the "check if steps are reversible" part, then they have not written a proof even if every step is reversible! That's what's lacking form their proof.
Well, it's a proof as long as you stipulate somewhere that all the logical connectives are "if and only if." This doesn't mean that a student who writes down a long series of equations beginning with some identity to be proven and ending with some known fact has proven the identity.
Sure, that's fine, but to say that it is an invalid method of proof is wrong.
If I want to prove A, and I prove A <=> B for some proven statement B, then I have proven A! There is no question! Feels like crazy pills to think otherwise. This is a kind of backwards reasoning!
It is a kind of backwards reasoning that students empirically screw up all the time.
Edit: so much so, that I would definitely recommend students rewrite these proofs on assignments as an exerecise to make sure it's correct. By the time they reach a math PhD they probably don't need to do that anyomre. :)
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.
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.
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.
Sorry, but this is absolutely wrong and potentially harmful to your son if you eventually convince him. It is absolutely possible and not that uncommon for a neurotic person's mind to go blank when under considerable stress, at the point of forgetting otherwise completely trivial information.
I urge you to not dismiss your son's complaints but to instead investigate them further, preferably with the help of a professional therapist, to save him from the possibility of escalation of his problem.
"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.
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.