I can't teach anyone anything. I have doubts that anyone else can do it, either; but can only speak to my own experience.
I can provide lots of assistance to those who are trying to learn.
We tell people "Edumacation" is going to happen to them whether they want it or not; and then they have to sign up for more "or you'll never have a good life" etc. It seems that we've lost the reason why we do these things. It's just empty ritual.
Things are this bad in famous, expensive upper class schools; imagine how bad they are in poor to normal elementary schools. We got kids with failing grades because they never got a (theoretically school provided) chromebook, never got a login to "the homework sites", or can't get reliable connectivity from home or school. They may or may not know the material they were supposed to learn but who cares about that? As long as the attendance records are good the school gets paid.
An interesting take and, yes, in the west it is hard to instill in a student that it is up to them to earn their way to success by making the maximum use of the teaching resources and not to sit back and get fed what they need.
In poorer countries, this is better understood, especially since many people do not go to school so the alternative to working hard is plain to see.
Not sure how you fix it though, perhaps some out of the box ideas to reward those that work hard in a way that reinforces that hard works pays off. I don't know.
That's got nothing to do with west vs. poor. Have you seen the 2-year French math requirements before you are even admitted to an engineering school? Have you seen German universities where there is zero advisement and where your two years worth of math courses in your engineering program are tested in a single final exam, as the very first feedback (no quizzes, no homework, no midterm, not split into 10 math micro courses, no projects, no extra credit, etc.)? It's sink or swim.
No offense, but I think you're being a bit dismissive: if a students doesn't learn a thing it is because they didn't try.
There is a middle road you are not calling out: if you make math (or any subject) approachable in your teaching style, you can get students to want to learn, try to learn.
I'd say that is generally true for pretty much any subject. Why does someone try to learn history? Probably because they somehow became interested in it. A good teacher will try to make it interesting.
I disagree. I think learning to learn, and, first, learning to motivate yourself to learn is a huge part of a college education with the goal to prepare you for a life in which boundary conditions change all the time.
What happens to those students who are then in a job, tasked to solve a problem, but now expect their boss to dance around them to make it `interesting'?
You may be right, I was speaking more broadly about education, not focused on college education. Although that does suggest a wider question, why education at all?
Would you apply your same approach/attitude to elementary school education? At what point is it that we shift the burden to learn to the student?
I suspect society is worse off if we "give up" on educating a large segment of the population. (And I don't mean to imply everyone needs to learn higher math, I am speaking more broadly of education.)
> Would you apply your same approach/attitude to elementary school education? At what point is it that we shift the burden to learn to the student?
I'd say the ideal is that a person is a curious self-starter throughout their life. But we have to differentiate between different degrees ( as in `levels') of education. I think that everyone needs some fundamental understanding of the the world (history, physics), needs to be able to read and calculate simple things.
And there are indeed schools who do it from an early age, like Montessori, in which students essentially pick their own projects.
Also, we do give up on educating a large segment of the population eventually: not everyone has a PhD. But we're not worse off as a society, because the returns are diminishing quite a bit.
From my personal experience doing homework with my son I think a big issue that is not addressed is that kids/people are different, some don't instantly get it and most f the time the kids will not have the courage to tell the teacher that they do not understand. I have a lot of difficulties working with only 1 child, imagine now working with 25 at a time, and it happens that I am convinced I explained stuff super clear and my son understand it but later I discover that still there was something unclear.
I empathize with teachers and with parents that also stuggle with making the children really understand math/programming and not only solve the homework and memorize some rules. I also wish there were less classes in school, general knowledge is nice too have but IMO is too much(my son is in a public high school in Romania , a good one but not elite)
> From my personal experience doing homework with my son I think a big issue that is not addressed is that kids/people are different, some don't instantly get it and most f the time the kids will not have the courage to tell the teacher that they do not understand.
You are absolutely right. I have a unique experience with my children. Two of them are homeschooled, two attend public school. The two who are homeschooled learned to read around age five, are excellent readers and are about two years ahead in math/algebra[0]. My public school kids were a little above average students, as well. That all changed over the last two years -- partly due to pandemic/remote learning, but not because we became more actively involved in their education. My two homeschooled kids get three hours of formal education, per day, on a busy day. When you're one-on-one with two students, and the teacher understands the student better than any other human being on the planet, it's really efficient[1].
Having worked remotely for a while, I told them that they have to get to know their teachers. They attended a "remote first" school which we transferred them to -- this was not a normally in-class school that had been forced to go "online only" in an unprepared manner, but one had existed for years prior and taught classes completely differently than the in-person schools. We forced them to schedule time every time it was offered to meet with their teachers. The result was an increased comfort with speaking up when they got lost. Returning to in-person schooling this year, they have kept that pattern; all of their teachers know them well -- when we pop up on video for parent-teacher conferences, we hear "Oh, you're (name omitted)'s parents!" Both are honor-roll students, now, two years in a row.
[0] That's an unscientific assessment; I use it because I regularly caught my son helping his two-years-older stepson with math homework; they were either working on the exact same concepts or they my son had done it the prior year.
[1] This works in the exact opposite direction, as well. My daughter, for a solid year and a half, would break down crying over her inability to understand her math work and would become very defiant about learning -- giving up before trying, etc -- because "it's Mom, and I can get away with these things with Mom". It's manageable, though, when you recognize that the "giving up before trying because she feels she can" is the way she expresses how she's struggling. The advantage of home schooling is that we don't have to move on to the next thing until they understand the previous enough to restore their confidence. We don't have to start summer time-off or even have it at all (it really doesn't feel any different, anyway!) if they need it.
It's a structural problem in conventional physics classes (I'd think similar to math) that the median grade is awful (sometimes even like a 30) but it gets graded on a curve so that's considered a C, an acceptable grade.
In a class like that students walk out with a passing grade but often a weak understanding of the subject.
I was involved in teaching the PHYS 1101-1102 sequence at Cornell
which is an auto-tutorial class where students are required to take each test multiple times until they get a grade that's high enough to indicate mastery. Some people find this course to be nervewracking, often they fail the test horribly two times, sit through debugging sessions with the tutors, come back and do pretty well on the last try and walk out with a B+ or A- grade.
The course is unique in a lot of ways because it is focused on the needs of pre-meds to take the MCAT so it is heavy on fluid mechanics and all the tests are multiple-choice similar to the MCAT. Students who have less aptitude for the subject work harder in this class than a conventional physics class but they walk out with more mastery and they do much better on the physics part of the MCAT than students who take the conventional class that Cornell offers.
This mastery based approach makes the most sense to me. The difference between grades and percentages is really weird. In some sense, we expect a 90% grade to mean you were able to solve problems on 90% of the material, or with a 10% error rate.
However, in some of the higher level classes, the tests are not on whether you have learned and can apply the material, but whether you can solve new types of problems that you haven't seen before. I've always found these sort of tests to be quite bizarre, as you really can't study for them, and whether I would come up with the required insight in the time limit was an inconsistent indicator. They seem better suited as problem sets.
That seems like a really interesting approach. I experienced exactly what you describe taking the introductory physics for engineers class. I got something like a 27% on the final exam, leaving entire problems blank because I did not even know how to begin. I clearly did not understand the material and would have dropped the class early if I was getting poor grades. But that didn't happen; with the curve, I received an AB in the class (halfway between A and B). They sent me on my way and to this day, I really only understand the physics that we learned in high school.
I relate to her quite a bit.
What helped me understand math was realizing, Math was a language that required an lot of effort to read, understand, write.
These days I find doing complex math with programming far far more understandable. I personally think my biggest blocker was the mathematical notation.
Same here! I've been studying the mathematical notation Wikipedia page, and being able to "pronounce" or verbalize symbols has made things much more comprehensible.
My current internal tussle is squaring the declarative nature of math with the imperative nature of programming, which I find much easier to reason with.
Studying the Greek alphabet was also massively helpful for me. In college I stuck a huge poster of all the Greek letters with their pronunciation above my computer monitor. It makes a huge difference reading an equation when you can say "omicron" or "tau" in your head instead of "weird squiggly thing".
"As a teacher I hand out problems people can work on but will not provide the solutions."
That is completely BS and my degree is in Math and I labored through the same during my undergraduate degree. It is unacceptable in my opinion in any field and I hope that other fields like engineering and medicine don't take this elitist approach.
Ever try or even see a "Leet-Code" style question that stated the problem without providing example results? Ever try to help someone with a problem on Stack Overflow who can state their problem but will not provide example data? How do you ever know you got the correct result? You don't. Perhaps worse, you are confident in a "solution" that includes an "off by 1" bug.
There is such a thing as an 'open book exam' where you literally have access to the textbook in the exam.
The purpose of such exams is not to test your memorization of the subject, but how well you understand it. The questions are more challenging, and are explicitly designed to see if you can take the information in front of you and synthesize or interpret it.
If you don't already understand the material, having access to it in the exam will not help as much - when the questions are properly designed - as you can't just copy down a bunch of disconnected facts. Instead you have to build an answer that relies on those facts and the framework that surrounds them.
Most my CS exams in college were open notes / open book, but Google was banned.
> The purpose of such exams is not to test your memorization of the subject, but how well you understand it. The questions are more challenging, and are explicitly designed to see if you can take the information in front of you and synthesize or interpret it.
The obvious difference is the existence of Stack Overflow. Synthesizing a correct answer from my hand written notes is a proof of understanding. Being able to google a question and having the answer produced for you is absolutely not a proof of understanding.
Its the difference between letting a math student have a formula sheet, so the don't need to memorize all the different formulas they need to apply, and giving them access to wolfram alpha. The later makes the exam an exercise in copy / paste.
Perhaps certain subjects are better suited to having broader questions. I still think it would be _possible_ to create CS questions that would not be helped by having access to a search engine.
As an analogy, it's a little like the difference between a unit test and an assembly (integration, etc) test. The unit test question is just a simple one like "When did this King die?" or "What is the area of this triangle?".
An assembly test is more like "What was the legacy of this King's reign?" or "What is the area of this shape that is the intersection of various other shapes?". I mean, Wolfram Alfa is only so powerful, right? Of course websites add features all the time, so it's always possible that the day the exam comes out they add a new "What is the area of this shape?" feature.
A big part of exams for math majors is proving stuff, not applying formulae. Wolfram|Alpha won't be able to help you there.
(That said, there's stuff like proofwiki or math.stackexchange, so unless you really spend time on crafting a novel problem, a student that is good at "researching" might be able to find a solution to a similar problem or to a substep of the proof, etc.)
How often do any of those subjects require instant perfect recall of knowledge/topic/concept with high stakes?
That's my criterion on whether external references should be allowed in tests. A surgeon should know their specialty as they won't have time to consult a reference during a surgery. A physicist will always have the time to refer to something when performing their work, as will an editor or teacher.
That's entirely different from saying they don't need to learn it however. If I drop a list of 100 formulas and equations on a person who doesn't know how to use them, they're useless. That background knowledge on what's available and how to use the tools you have is what the class should be teaching you.
Or as my physics prof used to say, "I want the engineer building the bridge I drive over to have looked up a reference on the static weight load of the materials selected, not pulling figures out of their head".
We were always allowed to use 1 standard sheet of paper, front & back, for our tests. Most of the time you could answer most questions without needing them beyond referring to a specific equation to make sure you weren't leaving off a constant (damn factors of 2....).
I graduated in 2018 from a state school in the USA.
No helping other students on assignments.
All exams are: closed book(this means no notes or textbooks with you during the exam) and done with a pencil and a BLANK white sheet of paper. I still managed a 3.75 CS GPA, but we were literally writing syntactically correct C and C++ on blank paper. Why oh why?
Early on before I had learned C++ well, I would get marked off something to the tune of -30 for missing a curly brace.
No one will miss that IRL when the IDE shows you the visual prompt that it's not there, OR you waste maybe a minute of your life if you forgot it when using VIM... -30 man oh man.
It depends on the content of the class. I had a class in Android programming in college where google was fair game because the API / libs were so large that google was a necessity to access all the documentation you need to write android apps.
My low level systems class was closed-book, no google, etc. At some point you have to learn something. Pointer manipulation is not something you should have to look up every time you do it.
> Ever try or even see a "Leet-Code" style question that stated the problem without providing example results?
This style of question asks the examinee to devise a procedure for producing those results, not to come up with the individual results themselves. If a student is given a bunch of systems of linear equations to solve, an answer key isn't especially useful since checking a candidate solution is pretty straightforward.
> Ever try or even see a "Leet-Code" style question that stated the problem without providing example results? [...] How do you ever know you got the correct result? You don't. Perhaps worse, you are confident in a "solution" that includes an "off by 1" bug.
Yes. Well, no--even worse. When I took intro CS in college many years ago we were given a problem with incorrect example results due to an off-by-one error in the code written by a TA. After many hours of banging my head against the desk and eventually working several cases out by hand on paper, I pointed this out to the professor. Rather than correct the error he emailed the class that the example results were only intended to demonstrate the formatting of the input and output, and should not be relied on to be correct.
I had a similar problem. Where we were supposed to do some sort of optimization (loop unrolling or something) to a function. We had done this several times throughout the course. But this time loop was slightly different (unintentionally) and I managed to construct some answer that I wasn't happy with and turned in the exam last. We got the papers back and of course mine was wrong in spite of producing correct results, while everybody who blinding applied the wrought memorized mechanization we had learned got it right. It took forever for that professor to understand how bad the question was.
Some people believe the purpose of education is "weeding-out" weaker students akin to a kind of tournament, or a harsh initiation ritual that members of a profession go through. These kinds of rituals enhance loyalty/camaraderie among those who survived (sunk cost) while also minimizing the number of new entrants/competitors (aka gatekeeping).
People holding those views would believe it's kind of "unsporting" to help students or make it easier in any way, that it would defeat what they see as the purpose (that being, the difficulty itself).
I think some degree of "weeding out" is required. Just because you got into Fancy University College it doesn't mean you are automatically great at all academic endeavors. If you sail through Math 101 and 102, and run into a brick wall in Math 202, maybe you aren't as great at math as you think you are. Or maybe you need to work harder.
We tend to think of higher education as some sort of inalienable right to all peoples across the world, but it just isn't. Everybody has a ceiling. Higher education, and especially elite universities, should be for those whose ceiling is higher than the average bear.
That said, until you get into the later years of higher education, "weeding out" doesn't have hard and fast boundaries. Atypical thinkers may have a blind spot that needs a different teaching method, or possibly "departmental policy" is just garbage ideas ossified into departmental culture. It's always worth revisiting those Chesterton Fences to see if they need to be altered or moved, or even torn down.
High standards are quite distinct from the indiscriminate "weeding out" effect that's most often the outcome of bad teaching. If you make every effort to teach Math 202 in the most effective way but also expect stellar results from everyone, that's not the same as what the Princeton student experienced.
I agree and it amazes me. We are implicitly asking students who want to learn something outside of their area of focus to do so at a community college or for "no credit". Rather than risk their GPA on a course where the philosophy might be "it was hard for me so I'm going to make it hard for them".
The first day of my math undergraduate, in a lecture theater of probably 300 students, the professor said confidently and with joy rather than remorse.
"look at the person to your left now your right. They will fail out of this program."
A compromise approach would be to provide numerical solutions so students know they're correct, but will still require them to put effort into working on the problem. This way instructors won't be afraid that students will simply look at the solutions before solving the problem.
I will add though that there are courses that use the same graded assigments year after year (my OS161-based OS course comes to mind, where multiple universities basically teach the same content). As a result full solutions are all over Github, and the honest students who work hard on completing the problems on their own are penalized over those who look over the solutions first before writing any code.
When you filter on a bell curve because fewer spaces exist next year you see this approach because being helped gives you an advantage over one of the other hundreds the professor cannot help directly.
It's part of the elite status colleges give out with degrees. Only the top 10% make it through is great advertising.
When doing my MSc, my class submitted long-form essays on several topics. I asked when we would get the marked papers back, but that was not available - some excuse that the marking scheme was technical. I was bemused.
Some problems should be given without solutions, because it will help a student identify when their use of solutions is becoming a crutch to help them solve problems. But as you point out, some problems need to come with solutions so a student can check their work. I assume this is a reason why so many textbooks have answers for every other question instead of answers for no questions or all questions.
With technology we can improve this by having an answer that is available only after the problem has been attempted, but the online answer checkers seem to introduce a host of other problems that make it unclear if there is any net advantage.
In my maths BSc, I've never had this happen. I've seen it in textbooks, yes, but there I kind of understand it—it would otherwise just bloat the textbook, and many people use textbooks in class with a teacher anyway.
I've usually either had practice questions with solutions directly provided or assignments to hand in (usually this was optional). The solution to these assignments would be posted after the due date and of course, you would get your submission checked by a TA.
MY first class of calculus 1 we were about 80 to 90 students in the class, all ready to study and learn.
The teacher walked in look at us and said: The department only allows me to pass 7 or 8 of you, so either you are a genius, or you are doing this class for the Nth time.
At the end of the class, we pass 10, which makes it roughly a 10%. I talk to the teacher after the class about it and he told me that the department want it that way to filter out the people that does not make themselves responsible of their own learning.
I did pass, and I don't agree with the brutality of the rule, but I do agree with the principle.
I could maybe see their point if it was just a really hard exam and only 10% happen to be able to pass it. But artificially restricting to 10% of the class is ridiculous. The goal should be to pass students that have the requisite knowledge, whatever the prof decides that should be. Judging only relative to peers is a lazy way for the prof to not have to make that distinction. It also creates a toxic environment.
That doesn't make sense to me. In a group of 90 people there can be more than 9 of them who strive to learn. I get not making classes an "easy A", but making an intro class that difficult arbitrarily seems against the entire point of education.
Others might call it "gate keeping", but that implies an intent that I don't think is really there. I think instead that the math professors think everyone else is like them, learn like they did.
I'm kind of dumb and so things as abstract as higher math have come to me with difficulty. I find I have to truly understand a thing before I can possibly expect to remember it or use it. And apparently I am not quick to understand things like differentiation, integration.
I think I only grasped matrix multiplication, linear algebra when I tried to write a flight simulator from scratch (well, from a book actually, but it had you start from scratch with your own Bresenham algorithm for line drawing, write your own polygon clipping code, face culling, etc.).
I have done pretty good (I must say) in helping my daughters get through high school math though since I do understand it (now). I think it helps that I probably approach teaching the concept as though it is to someone, like me, who is not going to make the mental leap to abstraction. I suspect many professional math teachers/majors don't suffer my shortcomings and don't teach from that vantage point.
I agree that there's a difference between learning and understanding, in a similar vein I wrote my own 3D renderer and really understood matrix multiplication.
But say you and I were in the same class, but only one was allowed to pass, even though we both now understand how things work. One of us has to take it again or change paths for no reason.
>I talk to the teacher after the class about it and he told me that the department want it that way to filter out the people that does not make themselves responsible of their own learning.
This was my big mistake with college. I walked in expecting to be taught things, only to realize that teaching has nothing to do with a college class. It is simply a forum for you to express what you already know.
I think there is a major distinction between "we have a standard of expectations that roughly only 10% have managed to meet in the past" and "only the top 10% will pass". One pits student against standard, the other pits student against student. These create different environments and different levels of consistency in outcomes. Using the latter as a simple substitute for the former is a very disagreeable action.
My wife took a child development course last semester and her instructor said the opposite: the school doesn't want me to fail anyone, so here's a bunch of makeup assignments and extra credit options.
“the mechanism is that the student works all the problems they can and then asks in office hours or review session about the problems they couldn’t do”
Wait, really? You want all 150 students in all your classes to come to your office every time you give out a homework assignment?
I never went to office hours unless there was some kind of highly unusual situation going on, like when I had a migraine with aura during an exam and couldn’t read the questions. Or when my wife had a life threatening accident and I had to tend to her and needed an extension on an assignment. Were other students just going in and out of there constantly?
>Were other students just going in and out of there constantly?
YES. I didn't learn this until after college but apparently other people were going to office hours for all their classes whenever they could. It also helped them form deeper relationships with professors for recommendations.
I never would have guessed people would spend much time with teachers/professors outside of class. If I had to schedule more than 1 or 2 sessions I'd drop the class; clearly I'm missing something and may need to retake prerequisites instead of wasting time failing.
I wish my profs had made that clear! I always felt like it was a big imposition to come in there and take up their time on what must have felt to them like a trivial problem. I would have been too embarrassed, I didn't want to be viewed as a needy or problematic student.
Odd. In all my math and physics classes the homework counted for less than 5% which basically made it all optional. After the assignment was done all answers were posted somewhere.
>Were other students just going in and out of there constantly?
Not constantly, but fairly frequently, yes. But judging from how frequently I'd meet other students waiting in the hall, it was probably only 1/5th of any class who'd bother to do this.
For math specifically, if the prof I wanted to see wasn't available and I thought the question was quick, I'd sometimes pop in to another professor's office who'd taught me before. e.g., For Calc3 I'd sometimes pop into my Calc2 professor's office if his door was open.
I hate this mentality. If you have two hours of office hours a week and they're both during other classes I have, then I can hardly be held accountable for not showing up to your office hours.
Let's call it what it is: bad teachers trying to offload culpability for their inability to teach the material adequately during lecture by pretending the students are too lazy to come in after-hours for additional instruction.
Sorry, I didn't mean that in a pejorative way, probably poor choice of words. I just meant that it wasn't that common for students to come in. Some students I'd see more frequently at the university's tutorial centre than office hours, maybe because they preferred that format.
One interesting thing I remember though is that profs in science/math/engineering tended to have better attended office hours than humanities courses. Humanities profs usually seemed surprised to have someone show up during office hours, and I'd rarely meet anyone in the halls apart from humanities grad students.
I was three years deep into my engineering degree before learning that many of my peers would attend office hours regularly. I was shocked. Like you, I was under the impression that you only attended office hours if you missed a lecture or something similar. I had always assumed "I don't get it" wasn't a valid reason for bugging the instructor.
> You want all 150 students in all your classes to come to your office every time you give out a homework assignment?
A class with 150 students should have like 2 or 3 TA, preferably more, so the workload is split. If it's a obligatory homework, the profesor and the TA may have to grade it anyway. It it's optative homework, many students will not write it down, but it would be nice if all of them do it.
Checking an exercise in the first years of the university is easy. If it's correct, you must only check 3 or 4 key points. It's harder if it has a silly numerical mistake and you must propagate the error to see if the answer is "correct" or there are more hidden errors. Also, the important part is to identify why the student was wrong and try to get actionable advice, like study the rule of a derivative of a division, or study the rule of the multiplication of the signs, and perhaps add a 5 minute personalized class on the spot.
We have a list optative homework, and it would be nice if everyone do them. It would increase the workload a lot, so we would have to imagine how to solve that. In particular in a long list the profesor/TA doesn't have to check every exercise, only the ones the student had a difficult time or the ones the profesor know that are prone to error. If there are 10 integrals of polynomials and the first 2 and the last one are correct, the other 7 are probably correct too. In case there is an error, the advantage is that the profesor can explain one or two and ask the student to redo the other and come again later or another day.
The long list of exercises is useful, but in my opinion it's more helpful to write in the blackboard an additional optative homework each class. There are more solutions attempt than just leaving the list to be downloaded from the web page. (And it's easier to check because all the students are doing the same exercise. Also I can check if they are writing correctly the solution, sometimes the students get the correct result but don't explain the intermediate steps with enough details.)
> making nice prepared slides like the professors in all their non-math classes do, instead of just showing up and writing on the blackboard.
I mostly agreed with the conclusions in this article, but this one stood out as something not so good. I much prefer a dynamic lecture to a boring slide presentation.
My favourite teacher in high school was my biology teacher. His teaching style was to sit on his desk facing us, next to his overhead projector which had a reel of acetate on it, and just write and draw as he spoke. By the end of a typical lesson he'd have a few feet of acetate filled with notes and beautifully drawn diagrams (I always wondered if he kept them, should've asked). It was engaging, and his attention to detail showed how much he cared about the subject.
> the mechanism is that the student works all the problems they can and then asks in office hours or review session about the problems they couldn’t do
As someone from outside that world this line strikes me as truly weird. Yes, if I struggle with a problem I may well be aware that I'm struggling. But what about the ones where I think I'm doing fine, but I in fact have the wrong answer? Short of bringing ALL solutions to be checked in office hours how would someone even know?
Does it really mean that the answers are given but the working of the solution is not? That would seem more reasonable at least - is that what's meant here?
I think much of the time, correctness of the answer can be checked easily. E.g., if you have to solve for x and find an answer, you can fill it in in the original equation and see if it works.
This does not sound like a proof heavy class, but this approach unfortunately does not work as well for more advanced classes. Especially for problems where you can lose points for not writing the proof like a mathematician would even if you get the steps right (there was one prof in the math department who would deduct points for inelegant but otherwise correct proofs)
These come with answers. It makes me wonder if "solutions" in the original article has been misinterpreted to mean "answers" in which case the complaints are much less warranted as you say.
Regardless, from my experience, people who studied solutions often failed to learn the requisite verification skills necessary and ended up doing poorly on the exam. I believe the student from the original article would have performed the same relative to her peers if they were all given solutions. The only difference is that the exam would have decided everything. No additional learning would have occurred with the solutions. Unfortunately, students expect attendance to warrant an automatic A in the course. From my experience, the #1 predictor for a student's rating of a course was their perceived standing in the course and how much they perceived the instructor cared about their success. If you are an instructor today, put an additional 2 homeworks in the syllabus and then drop two of them over the course of the semester to accommodate you students' busy schedules and that should boost your ratings. Extra points for the theatrical democratic process during class.
You hand out practice problems for an exam because you want students to do the problems, not because you want them to read the solutions; the mechanism is that the student works all the problems they can and then asks in office hours or review session about the problems they couldn’t do.
That's so ridiculous it's hard to believe anybody could say that with a straight face. First of all, to a first approximation nobody goes to "office hours". I mean, this is Real Life we're talking about here - your instructor's "office hours" are basically never (ever) going to be at a time when it's actually possible for you to go even if you wanted to.
And if you do go, is your instructor going to be capable of explaining things in a way that makes sense anyway? How many of us have known math teachers who are simultaneously A. perfectly competent to solve a problem and B. completely incapable of explaining how to solve the same problem to somebody else?
Never mind that all of this requires that you find your instructor approachable / friendly / whatever enough that you'd feel comfortable going to their office and interacting with them individually in the first place.
Basically, "office hours" are not a realistic solution to anybody's problem, with rare exceptions.
Review sessions? Great, except for now the feedback loop between "do homework problem and get stuck" to "get explanation" is on the order of days. Yeah, that's effective.
I don’t think it’s bad to include solutions, but I would never say that not doing so makes it “impossible to study.”
That's because you're too myopic and unimaginative to consider that not everyone studies and learns the same way you do.
> I mean, this is Real Life we're talking about here - your instructor's "office hours" are basically never (ever) going to be at a time when it's actually possible for you to go even if you wanted to.
Why would it be hard to go to office hours? These are schools where most students are full time students who live on or near campus and will have largely similar schedules. It is not hard for an instructor to have office hours at a time when most students in the class will be able to come.
> And if you do go, is your instructor going to be capable of explaining things in a way that makes sense anyway? How many of us have known math teachers who are simultaneously A. perfectly competent to solve a problem and B. completely incapable of explaining how to solve the same problem to somebody else?
If that is the case then how likely is it that it would have been helpful if the instructor had provided a written solution with the problem in the first place?
I'm not speaking in regards to any specific schools, but rather to the general principle. In my experience, many students (myself included) had plenty of obstacles to attending "office hours" - jobs, other classes, volunteer activities, etc. I won't dispute that there are some circumstances where some students can (and do) take advantage of this, but it doesn't seem to me that - broadly speaking - "office hours" are particularly effective strategy for most people.
If that is the case then how likely is it that it would have been helpful if the instructor had provided a written solution with the problem in the first place?
Good question. I suppose it depends in part on whether the instructor themselves made up the problem & solution, or it it came from a 3rd party. In either case, at least if you have a written solution you tighten up the cycle time to finding out that your understanding is flawed, which opens up more opportunity to pursue alternate explanations - maybe an online forum, a private tutor, or just consulting a different book, etc.
I think this illustrates a disconnect between what the teacher wants to accomplish and how the educational system works.
At the end of the day, the goal of the students is to pass the exam, not to learn mathematics. If they have learned some by the end of the course, that's a bonus, but the way the system is structured this is inherently a side-effect.
Some tests require that the students understand the topic in depth, but the map will never be the territory and I think framing this as "students and teachers need to work together to make better classes", misses the actual problem of exams being detrimental to learning.
Exams may sometimes be disruptive to learning, but they exist for a compelling reason: they help ensure that students exiting the class have actually learned the material.
All students want the credential of having passed the course, but not all students are willing and able to learn the material. If we care who learned the material, we have to verify who learned the material.
To eliminate exams, we would have to either (1) change the world so it doesn't matter who actually learned what they earned credentials for in school, or (2) make all students so intrinsically motivated that they all reliably learn the course material without any need for verification.
Does anyone think that either of these goals is realistic?
It's also conceivable (theoretically) to just decouple teaching from examination. Anyone may participate in any class (no entry requirements), but there's a certain bar for passing exams and getting a degree.
To a certain extent, this possibility already exists with textbooks, openly accessible video lectures, etc., but there's also a lot of people who would still benefit from in-person teaching even if they'd never get to the same level of understanding as people who are going to earn a degree.
What if it changed the game so that instead of exams causing stress and discouraging learning, people were incentivized to learn at their own pace and eventually pass the exam?
This situation already exists with many professional credentials, such as CPA, CFA, FSA, etc. They tell you what will be on the exam, provide practice problems, lots of study materials are available from different sources, and you take it when you're ready.
There's also college exams that have a similar function (CLEP is the acronym I think?) but they're not really highlighted as a preferred path to students.
> There's also college exams that have a similar function (CLEP is the acronym I think?) but they're not really highlighted as a preferred path to students.
That's correct. But not all schools accept it, and if a school accepts it they may not accept all the different exams. (CLEP exams mostly cover general-ed subjects.) Most schools also have an in-house option to test-out of classes, but it may be up to the individual departments to determine whether they want to allow that. Schools may also disincentivise testing-out in other ways... When I wanted to test out of the intro-to-computers course which was, at the same, required of all students, I had to pay the normal per-unit fee (as if I had taken the course normally) plus a special "challenge fee".
I wish this was how college worked; everything is free, except to sit for an exam and then to obtain the degree itself.
Colleges could structure degrees and majors by examination and interview, rather than by "credit hour", which itself is a meaningless assessment of comprehension.
As an example:
"The University of College offers a Degree in Electrical Engineering, offered at the following levels:
- Associate (At least a 60% in the following examinations: Classical Electricity, Circuit Design, ...)
- Bachelor (Meets Associate requirements plus at least an 80% in the folowing examinations: Signal Analysis...)
- Master (Meets Bachelor requirements plus at least a 90%...)
- Doctor (Meets Master requirements plus three original research publications...)
"
HOWEVER--this would require the total dismantling and reform of the student loan industry, academia itself, and the secondary market that's sprung up around college being a "part of life" for young adults. "College" is a many-billion dollar industry and very few peoiple want it disrupted to the degree that it would require to move to an examination-based format, with the exception of builders and technicians (engineers, etc.) who are actually trying to filter on skill when they hire.
The German system goes with a different take: it is almost free because much of the burden is placed on the student, no TAs to grade homework that isn’t graded anyways, just one final exam to determine your grade for the course. No fancy sport centers or dorms or other facilities that Americans find mandatory but don’t really matter that much.
I know a few Germans--the only issue I have with that system is that they're required to take the class before sitting for the examination. This has two effects; one, it makes a degree vanishingly difficult for anyone that doesn't have a lot of time, and two, it makes it impossible to use your degree to accurately reflect your skill level once you're out of college.
I think it ought to be appropriate for a degree to take decades, with examinations taken and passed at leisure. Most entry-level jobs that require degrees are far easier than college degrees, anyway.
> Most entry-level jobs that require degrees are far easier than college degrees
Germany handles that with their apprentice programs. But for a SWE, I think having gone through the class in a limited amount of time and forced to take an exam at the end, that can be reassuring to some people. Anyways, it still beats the American system, IMHO.
I live in Germany. Yes, education is (almost) free but still not anyone can just enroll, although I think some universities have programs where you can just attend lectures and not do exams. I don't think that's possible for some more prestigious universities (like KIT or TU Darmstadt), though. I might be wrong.
Yes, I do think both of those goals are realistic, although certainly difficult to achieve.
I typed six or seven drafts of a comment trying to approach you from a convincing angle and realized that it was probably hopeless, because we likely have too many conflicting assumptions about society, morals, human nature, etc. to come to an agreement in a web forum.
I would like to extend to you an invitation, and that is to read the following essay:
If you find this essay accurate and intriguing, consider sending me an email (address in profile) and I'll send you a large set of essays I've collected that discuss the same principle (i.e. that modern education is harming society more than it's helping).
If you find the essay naive and inaccurate, then there's likely very little that we would agree on at all, and it's not worth a 10-message reply chain where we argue over semantics. (Although I admit I enjoy a comments section rumble, as my post history reveals.)
I'll have a look. We will definitely agree that there is a lot wrong with US schooling but the devil's in the details.
Broadly, there's a lot to like about human nature, but I don't believe in it to that degree.
I like the other commenter's idea to decouple learning from exams, but I'm not sure it would work at all ages, and it mostly just solves the problem that I, personally, had in school: being forced through things I already knew or learned extremely quickly. Others have different problems I'm sure.
I will say Montessori has solutions to some of the problems in this essay and my kids go to a Montessori school.
I am in disagreement with the statement that exams provides proof that the student has learned the topic.
In my opinion, exams show which students have figured out how to beat the exam.
There might be some that do this by learning the topic, but in my experience the path of least resistance to a good grade has been to study the exam - not the topic of the course that it's based on.
As an example, I was one of the three people (out of 250 students) in my physical chemistry class that got the highest possible grade. I gained no understanding of the topic at all during that time - what I learned was to recognise which formula to use for specific problems on the test. I don't know what any of the results meant or what said result could be used for, simply that it was correct.
This correlates well with the observations made by Google and other large companies that analyse how to identify valuable hires. Turns out that there's no correlation between grades and workplace performance.
So if the exam result offers no information of the students ability to do a job in "real life" and at the same time introduce incentives that conflict with actually learning a topic, then I think it's better to significantly limit their importance or outright get rid of them.
I haven't been able to find any positive gains from having exams, but if someone has some resources on this, I'd be very curious to see it :-)
I agree that exams don't always do a great job of measuring actual competency, but I'm not sure they are worthless either.
> This correlates well with the observations made by Google and other large companies that analyse how to identify valuable hires. Turns out that there's no correlation between grades and workplace performance.
These observations probably suffer from a confounding effect known as range restriction. Basically, if Google generally only hires people with a GPA of 3.0 or above, they may have some idea of how GPA variation above that level correlates with job performance, but they don't know about it below that level. So they won't know if people with GPAs of 1.5 will do just as well at Google as people they normally consider.
Suppose Google started hiring people with much lower GPAs than they do today. Anyone want to make a bet what the outcome of that experiment would be?
Well, Google knows, and that's why they don't do the experiment.
Google has no requirements for GPA - here's a quote from 2013 by Lazlo Bock who served as Senior vice President of People operations at Google for several years:
"One of the things we’ve seen from all our data crunching is that G.P.A.’s are worthless as a criteria for hiring, and test scores are worthless — no correlation at all except for brand-new college grads, where there’s a slight correlation. Google famously used to ask everyone for a transcript and G.P.A.’s and test scores, but we don’t anymore, unless you’re just a few years out of school. We found that they don’t predict anything."
This says nothing about whether Google has a range restriction effect. It just suggests that their HR doesn't know or care what range restriction is.
Google probably filters out low-GPA candidates based on their poor performance by other criteria. It doesn't mean GPA isn't predictive of how such candidates would perform at Google.
If this is untrue, let them bring out their parade of successful low-GPA engineers.
Where's your evidence that they do have a range restriction? Seems to me that you're moving the goal post without providing any justification for it.
There's plenty of sources online stating Google doesn't have a GPA requirement, that you're welcome to look up on your own time.
Besides those I can also tell from my own (and a couple of friends) experiences, interviewing with Google, that indeed they do not care about GPA - which is a plus when you consider, how many potential hires are not American and have used a completely different grading scheme.
The quote literally states the opposite of what you're claiming, according to Google - GPA does not in any way predict job performance, so why is it so important to you find a narrative, where it does?
What seems more likely - that several Google engineers all made an erroneous conclusion based on the available data or that they found that GPA has zero value in real life?
Ignoring range restriction is a very common and very consequential statistical fallacy. I'm not going to blindly assume that Google engineers are above statistical fallacies. Especially ones that embody universal human cognitive biases, like thinking that what you see is representative of everything that's out there.
There's plenty of evidence that grades matter in other contexts, so I'm not going to throw all of that out because some Google guy made a vaguely general claim based on his limited and very biased sample.
As I said, all Google has to do to refute my doubts is point out its huge population of very low GPA, very successful engineers. If GPA is meaningless, this should be a trivial thing for them to do.
The office hours idea sounds crazy to me, and that is despite the fact that I actually had a professor's attention for an hour for each sheet. (2/3 on 1 tutorials). Some place doing mass lectures is going to inundate the professor's office, surely? Or perhaps in modern unis the students themselves share solutions on an online board?
There's just so much self-learning involved it can be quite hard to narrow down your questions to where it can be discussed in an hour. That's especially if you actually tried to dig deeply into the concepts. You might have gotten stuck for days on a single problem, which is made much worse if there's no hints about how to find the solution.
Another problem seems to be the opposite: you come in for the tutorial, you solve all the problems, and you're no wiser about the issues, despite having learned a mechanism for solving each problem on the sheet.
> There's just so much self-learning involved it can be quite hard to narrow down your questions to where it can be discussed in an hour.
I had this problem while taking grad courses. I needed time to produce follow up questions and office hours are just a misfit especially when I know there are a bunch of people waiting in line for time.
What we ended up with is a Piazza where TAs mostly do the ritual. In popular areas, I wonder why I am paying when its no better than an internet forum, probably worse.
I feel learning experience is just poorly structured. Teachers get away from the feedback received by invalidating it instead of introspecting it. I can't imagine any other field where feedback is just thrown out no matter how noisy it is.
> And of course, most people taking Math 202 are not taking it for intellectual broadening, as Rabieh admirably was; they are taking it because somebody told me they had to.
This is an unacceptable response as well. Why would the students get told that they had to if not for the purpose of intellectual broadening?
In my college experience, symbolic logic served this purpose. It was taught by a clearly dismayed senior member of the Philosophy department, who used plenty of class time to berate the general idea of the "gifted computer whiz kid who doesn't do the work" while clearly running the material on autopilot from the publisher provided syllabus. It didn't make me less interested in philosophy, but it made me a lot less interested in formal philosophy as taught at that school.
I have a lot of thoughts on this from my own personal experience and was honestly surprised to find that it was "Linear Algebra" that was the class being discussed -- I thought I was the only one who suffered through that!
This was 20 years ago, for me, and my experience was nearly identical. I was not a great student in High School but ... now having to pay for my own classes ... did well in every class in college except for this one. I attended sessions with tutors offered through my class, I attended office hours, I failed ... hard. The pattern I kept running into was one of learning the material after the exams for each unit were returned to me. In fact, it was so ridiculous, I stopped going to class, resigned to the fact that I was going to fail and simply showed up for the exams. I re-took the class and managed a near-4.0 on the same material. The school I attended had a bad reputation among its math department as being ridiculously difficult staffed with professors who had a basic understanding of English. Of course, this came from students who had recently failed, and they'll have the loudest and often most defensive voices on the subject.
None-the-less, out of the three professors, two were in the "avoid" category pretty quickly due to feedback I received about how difficult it was to understand the words they were speaking[0]. The third was a recent hire -- the rumor being that they picked someone from the US because of the frequent complaints about understanding the set of maths professors they currently employed. But I suspect a lot of people played the game of "pick the guy with the American sounding name" and found out the same thing that I did -- a deep south Alabama accent is much farther from English as it is spoken where I live than any ESL professor's native accent. I picked up about every fifth word coming out of this gentleman's mouth with his very thick drawl and unusual word choices and lousy grammar. In the north, we'd be surprised to find out that he's a maths professor, stereotyping him as a farmer
I suspect, after reading this, that there was more to it than just my struggling to understand what was being said. My pattern was that despite all of the extra energy put into attending tutoring sessions and scheduling office hours, I found myself understanding the material just after we'd taken an exam on the portion we were studying. As this was too long ago, I don't remember if we were given solutions or not, but it fits with that pattern -- the tests were reviewed pretty thoroughly after they were taken and I could see what I did wrong -- it clicked. It clicked so much that despite getting a hard fail in the class, I ended up sailing through with -- I think -- a 4.0 or somewhere near the second time around, attending only on exam days -- with the same professor.
The tutoring/office hours which are brought up as "the way you teach without the solution" were totally inadequate. The staff provided for assistance were not professors, or even particularly skilled tutors. They were advanced maths students who found the work being done to be easy and didn't know how to teach things that were very basic to them. Open office hours were difficult for me to get because I had a full-time job in the field I was seeking my degree in; they were always booked and by the time I knew I needed them, the earliest I could meet with the professor was after the exam. I ended up just scheduling them as soon as I was allowed and cancelling if I didn't need it. Many, many others did this -- it resulted in a group forming by the smallish meeting space that he reserved for these[1] in hopes that he'd finish early and grab the first student waiting. I never did this, but on days that I had a formal appointment, I'd noticed the waiting students would sit on the floor with their materials trying to help one another -- it was a blind-lead...
BTW, if one wants to see more problems with worked out answers than their professor and/or textbook provides, there are books you can buy for this. For linear algebra for example you could get Schaum's Outline of Linear Algebra [1] for $13.44 which includes 612 practice problems with step-by-step solutions.
There are such books for most of the major first and second year undergraduate STEM and STEM-adjacent courses.
I’m surprised by the outrage to the teaching style, specifically people taking issue with the expectation that students take questions to office hours. Perhaps many who have studied math have forgotten, but “so and so is left as an exercise to the reader” is basically a trope at this point. I’d even go as far to say that you can’t really learn all the content in a math textbook without talking it over with someone. Math text books aren’t very dense, yet it can take a day or more to fully grasp a few pages. Simply adding more exercises, even with answers, isn’t necessarily going to deepen one’s intuition. Which brings me the specific class this article is about: linear algebra. LA has to been one of the worse taught classes, and that seems to be universal. Things like determinants and Eigen values are drilled in students’ heads without developing the intuition for the subject first. They’re given examples without context, which is what the girl described in the article seems to be struggling with. She probably still would struggle to learn LA even with answers to the examples.
The two parts are needed because of length
limits of HN posts.
The OP (original post)
"Why won’t anyone teach me math?"
is about a coed at Princeton in a badly
taught course in linear algebra.
It turns out, I did really well in linear
algebra, and "Part I" explains how well I
did and how I did that.
In "Part II" I give an outline of an
introduction to linear algebra and matrix
theory.
For my background I've taught applied math
and computer science at three well known
research universities and have a Ph.D. in
pure/applied math from a fourth.
From how I learned math I have to say that
there is a lot of having contempt for the
students and dumping on them.
This dumping on the students, treating
them with contempt, is unnecessary,
unjustified, inexcusable, and bad for the
students, researchers, teachers, schools,
math, and the US.
The dumping isn't just in famous research
universities: E.g., I went in grades 1-12
to the unique college prep school in the
city. I got dumped on from the 4th grade
on and maybe also in the first three
grades but simply don't remember much
about grades 1-3.
I wanted to learn but got dumped on so
much that I just gave up trying to please
the teachers and did what I could with the
books for the courses. Turns out, that
was a good thing for me to do, especially
for learning math.
There were a lot of standardized tests,
both aptitude and achievement, and
apparently I did well on those, much
better than how the teachers viewed me.
Point: The opinions of teachers who dump
on students can be highly inaccurate; some
more objective measures are needed.
Due to the objective measures, in the 9th
grade I got sent to a math tournament and
in the 11th grade, to an NSF math &
physics summer program. Point: The
objective measures helped. Turns out,
that was true throughout my education.
But for college I went to a private school
that took lots of pride in doing well at
teaching. Some of the teaching was quite
good. Still, however, there was some
dumping: E.g., the college got a new prof
fresh from a famous research university,
and he brought his dumping/contempt
propensities with him. At one point I
took a reading course from him from a
then famous, advanced, notoriously
difficult book in topology. So, for each
chapter one week I gave a lecture to the
prof on the math and the next week, on the
exercises. I did well. Once another prof
showed up. But even with that good
self-teaching success, the prof at best
only slightly relaxed his dumping/contempt
mode. Point: In that reading course I
was a spectacularly good student but the
contempt still continued.
My Summary View:
I see no good reason or excuse for the
dumping and contempt. I refused to do any
such things in my teaching.
Generally I told the students: The
material is in the book. Class is to help
you learn. So, I'm here to explain and
answer questions. There is homework which
is to work as many of the exercises as you
need to to learn the material, in
particular to get good at working the
exercises. The tests are problems similar
to the exercises -- if you are good on the
exercises, you should be good on the
tests. From the numerical scores on the
tests, I rank the s...
Never took a math class that didn't provide a solution index for practice problems. University level students are expected to be able to self-regulate their study ethic and this sort of paternalist bullshit is turning undergraduate programs into trades learned by rote.
108 comments
[ 5.2 ms ] story [ 84.1 ms ] threadI can provide lots of assistance to those who are trying to learn.
We tell people "Edumacation" is going to happen to them whether they want it or not; and then they have to sign up for more "or you'll never have a good life" etc. It seems that we've lost the reason why we do these things. It's just empty ritual.
Things are this bad in famous, expensive upper class schools; imagine how bad they are in poor to normal elementary schools. We got kids with failing grades because they never got a (theoretically school provided) chromebook, never got a login to "the homework sites", or can't get reliable connectivity from home or school. They may or may not know the material they were supposed to learn but who cares about that? As long as the attendance records are good the school gets paid.
In poorer countries, this is better understood, especially since many people do not go to school so the alternative to working hard is plain to see.
Not sure how you fix it though, perhaps some out of the box ideas to reward those that work hard in a way that reinforces that hard works pays off. I don't know.
There is a middle road you are not calling out: if you make math (or any subject) approachable in your teaching style, you can get students to want to learn, try to learn.
I'd say that is generally true for pretty much any subject. Why does someone try to learn history? Probably because they somehow became interested in it. A good teacher will try to make it interesting.
What happens to those students who are then in a job, tasked to solve a problem, but now expect their boss to dance around them to make it `interesting'?
Would you apply your same approach/attitude to elementary school education? At what point is it that we shift the burden to learn to the student?
I suspect society is worse off if we "give up" on educating a large segment of the population. (And I don't mean to imply everyone needs to learn higher math, I am speaking more broadly of education.)
It seems I addressed that in my post?
> Would you apply your same approach/attitude to elementary school education? At what point is it that we shift the burden to learn to the student?
I'd say the ideal is that a person is a curious self-starter throughout their life. But we have to differentiate between different degrees ( as in `levels') of education. I think that everyone needs some fundamental understanding of the the world (history, physics), needs to be able to read and calculate simple things.
And there are indeed schools who do it from an early age, like Montessori, in which students essentially pick their own projects.
Also, we do give up on educating a large segment of the population eventually: not everyone has a PhD. But we're not worse off as a society, because the returns are diminishing quite a bit.
I empathize with teachers and with parents that also stuggle with making the children really understand math/programming and not only solve the homework and memorize some rules. I also wish there were less classes in school, general knowledge is nice too have but IMO is too much(my son is in a public high school in Romania , a good one but not elite)
Having worked remotely for a while, I told them that they have to get to know their teachers. They attended a "remote first" school which we transferred them to -- this was not a normally in-class school that had been forced to go "online only" in an unprepared manner, but one had existed for years prior and taught classes completely differently than the in-person schools. We forced them to schedule time every time it was offered to meet with their teachers. The result was an increased comfort with speaking up when they got lost. Returning to in-person schooling this year, they have kept that pattern; all of their teachers know them well -- when we pop up on video for parent-teacher conferences, we hear "Oh, you're (name omitted)'s parents!" Both are honor-roll students, now, two years in a row.
[0] That's an unscientific assessment; I use it because I regularly caught my son helping his two-years-older stepson with math homework; they were either working on the exact same concepts or they my son had done it the prior year.
[1] This works in the exact opposite direction, as well. My daughter, for a solid year and a half, would break down crying over her inability to understand her math work and would become very defiant about learning -- giving up before trying, etc -- because "it's Mom, and I can get away with these things with Mom". It's manageable, though, when you recognize that the "giving up before trying because she feels she can" is the way she expresses how she's struggling. The advantage of home schooling is that we don't have to move on to the next thing until they understand the previous enough to restore their confidence. We don't have to start summer time-off or even have it at all (it really doesn't feel any different, anyway!) if they need it.
In a class like that students walk out with a passing grade but often a weak understanding of the subject.
I was involved in teaching the PHYS 1101-1102 sequence at Cornell
https://physics.cornell.edu/physics-health-careers
which is an auto-tutorial class where students are required to take each test multiple times until they get a grade that's high enough to indicate mastery. Some people find this course to be nervewracking, often they fail the test horribly two times, sit through debugging sessions with the tutors, come back and do pretty well on the last try and walk out with a B+ or A- grade.
The course is unique in a lot of ways because it is focused on the needs of pre-meds to take the MCAT so it is heavy on fluid mechanics and all the tests are multiple-choice similar to the MCAT. Students who have less aptitude for the subject work harder in this class than a conventional physics class but they walk out with more mastery and they do much better on the physics part of the MCAT than students who take the conventional class that Cornell offers.
However, in some of the higher level classes, the tests are not on whether you have learned and can apply the material, but whether you can solve new types of problems that you haven't seen before. I've always found these sort of tests to be quite bizarre, as you really can't study for them, and whether I would come up with the required insight in the time limit was an inconsistent indicator. They seem better suited as problem sets.
These days I find doing complex math with programming far far more understandable. I personally think my biggest blocker was the mathematical notation.
My current internal tussle is squaring the declarative nature of math with the imperative nature of programming, which I find much easier to reason with.
https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...
"As a teacher I hand out problems people can work on but will not provide the solutions."
That is completely BS and my degree is in Math and I labored through the same during my undergraduate degree. It is unacceptable in my opinion in any field and I hope that other fields like engineering and medicine don't take this elitist approach.
Ever try or even see a "Leet-Code" style question that stated the problem without providing example results? Ever try to help someone with a problem on Stack Overflow who can state their problem but will not provide example data? How do you ever know you got the correct result? You don't. Perhaps worse, you are confident in a "solution" that includes an "off by 1" bug.
As a physics student, there were some exams where I was able to "google," but almost every exam I took allowed me to bring some notes.
The purpose of such exams is not to test your memorization of the subject, but how well you understand it. The questions are more challenging, and are explicitly designed to see if you can take the information in front of you and synthesize or interpret it.
If you don't already understand the material, having access to it in the exam will not help as much - when the questions are properly designed - as you can't just copy down a bunch of disconnected facts. Instead you have to build an answer that relies on those facts and the framework that surrounds them.
Most my CS exams in college were open notes / open book, but Google was banned.
> The purpose of such exams is not to test your memorization of the subject, but how well you understand it. The questions are more challenging, and are explicitly designed to see if you can take the information in front of you and synthesize or interpret it.
The obvious difference is the existence of Stack Overflow. Synthesizing a correct answer from my hand written notes is a proof of understanding. Being able to google a question and having the answer produced for you is absolutely not a proof of understanding.
Its the difference between letting a math student have a formula sheet, so the don't need to memorize all the different formulas they need to apply, and giving them access to wolfram alpha. The later makes the exam an exercise in copy / paste.
Perhaps certain subjects are better suited to having broader questions. I still think it would be _possible_ to create CS questions that would not be helped by having access to a search engine.
As an analogy, it's a little like the difference between a unit test and an assembly (integration, etc) test. The unit test question is just a simple one like "When did this King die?" or "What is the area of this triangle?".
An assembly test is more like "What was the legacy of this King's reign?" or "What is the area of this shape that is the intersection of various other shapes?". I mean, Wolfram Alfa is only so powerful, right? Of course websites add features all the time, so it's always possible that the day the exam comes out they add a new "What is the area of this shape?" feature.
(That said, there's stuff like proofwiki or math.stackexchange, so unless you really spend time on crafting a novel problem, a student that is good at "researching" might be able to find a solution to a similar problem or to a substep of the proof, etc.)
That's my criterion on whether external references should be allowed in tests. A surgeon should know their specialty as they won't have time to consult a reference during a surgery. A physicist will always have the time to refer to something when performing their work, as will an editor or teacher.
That's entirely different from saying they don't need to learn it however. If I drop a list of 100 formulas and equations on a person who doesn't know how to use them, they're useless. That background knowledge on what's available and how to use the tools you have is what the class should be teaching you.
Or as my physics prof used to say, "I want the engineer building the bridge I drive over to have looked up a reference on the static weight load of the materials selected, not pulling figures out of their head".
We were always allowed to use 1 standard sheet of paper, front & back, for our tests. Most of the time you could answer most questions without needing them beyond referring to a specific equation to make sure you weren't leaving off a constant (damn factors of 2....).
No helping other students on assignments.
All exams are: closed book(this means no notes or textbooks with you during the exam) and done with a pencil and a BLANK white sheet of paper. I still managed a 3.75 CS GPA, but we were literally writing syntactically correct C and C++ on blank paper. Why oh why?
Early on before I had learned C++ well, I would get marked off something to the tune of -30 for missing a curly brace.
No one will miss that IRL when the IDE shows you the visual prompt that it's not there, OR you waste maybe a minute of your life if you forgot it when using VIM... -30 man oh man.
My low level systems class was closed-book, no google, etc. At some point you have to learn something. Pointer manipulation is not something you should have to look up every time you do it.
This style of question asks the examinee to devise a procedure for producing those results, not to come up with the individual results themselves. If a student is given a bunch of systems of linear equations to solve, an answer key isn't especially useful since checking a candidate solution is pretty straightforward.
Yes. Well, no--even worse. When I took intro CS in college many years ago we were given a problem with incorrect example results due to an off-by-one error in the code written by a TA. After many hours of banging my head against the desk and eventually working several cases out by hand on paper, I pointed this out to the professor. Rather than correct the error he emailed the class that the example results were only intended to demonstrate the formatting of the input and output, and should not be relied on to be correct.
People holding those views would believe it's kind of "unsporting" to help students or make it easier in any way, that it would defeat what they see as the purpose (that being, the difficulty itself).
We tend to think of higher education as some sort of inalienable right to all peoples across the world, but it just isn't. Everybody has a ceiling. Higher education, and especially elite universities, should be for those whose ceiling is higher than the average bear.
That said, until you get into the later years of higher education, "weeding out" doesn't have hard and fast boundaries. Atypical thinkers may have a blind spot that needs a different teaching method, or possibly "departmental policy" is just garbage ideas ossified into departmental culture. It's always worth revisiting those Chesterton Fences to see if they need to be altered or moved, or even torn down.
Sorry, but that's just silly. Extremely silly.
You don't weed students out through the <<teaching process>>.
You weed them out through exam results...
The first day of my math undergraduate, in a lecture theater of probably 300 students, the professor said confidently and with joy rather than remorse.
"look at the person to your left now your right. They will fail out of this program."
I will add though that there are courses that use the same graded assigments year after year (my OS161-based OS course comes to mind, where multiple universities basically teach the same content). As a result full solutions are all over Github, and the honest students who work hard on completing the problems on their own are penalized over those who look over the solutions first before writing any code.
It's part of the elite status colleges give out with degrees. Only the top 10% make it through is great advertising.
With technology we can improve this by having an answer that is available only after the problem has been attempted, but the online answer checkers seem to introduce a host of other problems that make it unclear if there is any net advantage.
I've usually either had practice questions with solutions directly provided or assignments to hand in (usually this was optional). The solution to these assignments would be posted after the due date and of course, you would get your submission checked by a TA.
The teacher walked in look at us and said: The department only allows me to pass 7 or 8 of you, so either you are a genius, or you are doing this class for the Nth time.
At the end of the class, we pass 10, which makes it roughly a 10%. I talk to the teacher after the class about it and he told me that the department want it that way to filter out the people that does not make themselves responsible of their own learning.
I did pass, and I don't agree with the brutality of the rule, but I do agree with the principle.
I'm kind of dumb and so things as abstract as higher math have come to me with difficulty. I find I have to truly understand a thing before I can possibly expect to remember it or use it. And apparently I am not quick to understand things like differentiation, integration.
I think I only grasped matrix multiplication, linear algebra when I tried to write a flight simulator from scratch (well, from a book actually, but it had you start from scratch with your own Bresenham algorithm for line drawing, write your own polygon clipping code, face culling, etc.).
I have done pretty good (I must say) in helping my daughters get through high school math though since I do understand it (now). I think it helps that I probably approach teaching the concept as though it is to someone, like me, who is not going to make the mental leap to abstraction. I suspect many professional math teachers/majors don't suffer my shortcomings and don't teach from that vantage point.
But say you and I were in the same class, but only one was allowed to pass, even though we both now understand how things work. One of us has to take it again or change paths for no reason.
Yeah, that is it. The year is about right too. I was programming Macs so I skipped over a lot of the PC-specific stuff. But the math is all there.
This was my big mistake with college. I walked in expecting to be taught things, only to realize that teaching has nothing to do with a college class. It is simply a forum for you to express what you already know.
Wait, really? You want all 150 students in all your classes to come to your office every time you give out a homework assignment?
I never went to office hours unless there was some kind of highly unusual situation going on, like when I had a migraine with aura during an exam and couldn’t read the questions. Or when my wife had a life threatening accident and I had to tend to her and needed an extension on an assignment. Were other students just going in and out of there constantly?
YES. I didn't learn this until after college but apparently other people were going to office hours for all their classes whenever they could. It also helped them form deeper relationships with professors for recommendations.
Not constantly, but fairly frequently, yes. But judging from how frequently I'd meet other students waiting in the hall, it was probably only 1/5th of any class who'd bother to do this.
For math specifically, if the prof I wanted to see wasn't available and I thought the question was quick, I'd sometimes pop in to another professor's office who'd taught me before. e.g., For Calc3 I'd sometimes pop into my Calc2 professor's office if his door was open.
I hate this mentality. If you have two hours of office hours a week and they're both during other classes I have, then I can hardly be held accountable for not showing up to your office hours.
Let's call it what it is: bad teachers trying to offload culpability for their inability to teach the material adequately during lecture by pretending the students are too lazy to come in after-hours for additional instruction.
One interesting thing I remember though is that profs in science/math/engineering tended to have better attended office hours than humanities courses. Humanities profs usually seemed surprised to have someone show up during office hours, and I'd rarely meet anyone in the halls apart from humanities grad students.
A class with 150 students should have like 2 or 3 TA, preferably more, so the workload is split. If it's a obligatory homework, the profesor and the TA may have to grade it anyway. It it's optative homework, many students will not write it down, but it would be nice if all of them do it.
Checking an exercise in the first years of the university is easy. If it's correct, you must only check 3 or 4 key points. It's harder if it has a silly numerical mistake and you must propagate the error to see if the answer is "correct" or there are more hidden errors. Also, the important part is to identify why the student was wrong and try to get actionable advice, like study the rule of a derivative of a division, or study the rule of the multiplication of the signs, and perhaps add a 5 minute personalized class on the spot.
We have a list optative homework, and it would be nice if everyone do them. It would increase the workload a lot, so we would have to imagine how to solve that. In particular in a long list the profesor/TA doesn't have to check every exercise, only the ones the student had a difficult time or the ones the profesor know that are prone to error. If there are 10 integrals of polynomials and the first 2 and the last one are correct, the other 7 are probably correct too. In case there is an error, the advantage is that the profesor can explain one or two and ask the student to redo the other and come again later or another day.
The long list of exercises is useful, but in my opinion it's more helpful to write in the blackboard an additional optative homework each class. There are more solutions attempt than just leaving the list to be downloaded from the web page. (And it's easier to check because all the students are doing the same exercise. Also I can check if they are writing correctly the solution, sometimes the students get the correct result but don't explain the intermediate steps with enough details.)
I mostly agreed with the conclusions in this article, but this one stood out as something not so good. I much prefer a dynamic lecture to a boring slide presentation.
My favourite teacher in high school was my biology teacher. His teaching style was to sit on his desk facing us, next to his overhead projector which had a reel of acetate on it, and just write and draw as he spoke. By the end of a typical lesson he'd have a few feet of acetate filled with notes and beautifully drawn diagrams (I always wondered if he kept them, should've asked). It was engaging, and his attention to detail showed how much he cared about the subject.
As someone from outside that world this line strikes me as truly weird. Yes, if I struggle with a problem I may well be aware that I'm struggling. But what about the ones where I think I'm doing fine, but I in fact have the wrong answer? Short of bringing ALL solutions to be checked in office hours how would someone even know?
Does it really mean that the answers are given but the working of the solution is not? That would seem more reasonable at least - is that what's meant here?
https://www.math.princeton.edu/sites/default/files/2018-04/M...
These come with answers. It makes me wonder if "solutions" in the original article has been misinterpreted to mean "answers" in which case the complaints are much less warranted as you say.
Regardless, from my experience, people who studied solutions often failed to learn the requisite verification skills necessary and ended up doing poorly on the exam. I believe the student from the original article would have performed the same relative to her peers if they were all given solutions. The only difference is that the exam would have decided everything. No additional learning would have occurred with the solutions. Unfortunately, students expect attendance to warrant an automatic A in the course. From my experience, the #1 predictor for a student's rating of a course was their perceived standing in the course and how much they perceived the instructor cared about their success. If you are an instructor today, put an additional 2 homeworks in the syllabus and then drop two of them over the course of the semester to accommodate you students' busy schedules and that should boost your ratings. Extra points for the theatrical democratic process during class.
That's so ridiculous it's hard to believe anybody could say that with a straight face. First of all, to a first approximation nobody goes to "office hours". I mean, this is Real Life we're talking about here - your instructor's "office hours" are basically never (ever) going to be at a time when it's actually possible for you to go even if you wanted to.
And if you do go, is your instructor going to be capable of explaining things in a way that makes sense anyway? How many of us have known math teachers who are simultaneously A. perfectly competent to solve a problem and B. completely incapable of explaining how to solve the same problem to somebody else?
Never mind that all of this requires that you find your instructor approachable / friendly / whatever enough that you'd feel comfortable going to their office and interacting with them individually in the first place.
Basically, "office hours" are not a realistic solution to anybody's problem, with rare exceptions.
Review sessions? Great, except for now the feedback loop between "do homework problem and get stuck" to "get explanation" is on the order of days. Yeah, that's effective.
I don’t think it’s bad to include solutions, but I would never say that not doing so makes it “impossible to study.”
That's because you're too myopic and unimaginative to consider that not everyone studies and learns the same way you do.
Why would it be hard to go to office hours? These are schools where most students are full time students who live on or near campus and will have largely similar schedules. It is not hard for an instructor to have office hours at a time when most students in the class will be able to come.
> And if you do go, is your instructor going to be capable of explaining things in a way that makes sense anyway? How many of us have known math teachers who are simultaneously A. perfectly competent to solve a problem and B. completely incapable of explaining how to solve the same problem to somebody else?
If that is the case then how likely is it that it would have been helpful if the instructor had provided a written solution with the problem in the first place?
I'm not speaking in regards to any specific schools, but rather to the general principle. In my experience, many students (myself included) had plenty of obstacles to attending "office hours" - jobs, other classes, volunteer activities, etc. I won't dispute that there are some circumstances where some students can (and do) take advantage of this, but it doesn't seem to me that - broadly speaking - "office hours" are particularly effective strategy for most people.
If that is the case then how likely is it that it would have been helpful if the instructor had provided a written solution with the problem in the first place?
Good question. I suppose it depends in part on whether the instructor themselves made up the problem & solution, or it it came from a 3rd party. In either case, at least if you have a written solution you tighten up the cycle time to finding out that your understanding is flawed, which opens up more opportunity to pursue alternate explanations - maybe an online forum, a private tutor, or just consulting a different book, etc.
At the end of the day, the goal of the students is to pass the exam, not to learn mathematics. If they have learned some by the end of the course, that's a bonus, but the way the system is structured this is inherently a side-effect.
Some tests require that the students understand the topic in depth, but the map will never be the territory and I think framing this as "students and teachers need to work together to make better classes", misses the actual problem of exams being detrimental to learning.
All students want the credential of having passed the course, but not all students are willing and able to learn the material. If we care who learned the material, we have to verify who learned the material.
To eliminate exams, we would have to either (1) change the world so it doesn't matter who actually learned what they earned credentials for in school, or (2) make all students so intrinsically motivated that they all reliably learn the course material without any need for verification.
Does anyone think that either of these goals is realistic?
To a certain extent, this possibility already exists with textbooks, openly accessible video lectures, etc., but there's also a lot of people who would still benefit from in-person teaching even if they'd never get to the same level of understanding as people who are going to earn a degree.
What if it changed the game so that instead of exams causing stress and discouraging learning, people were incentivized to learn at their own pace and eventually pass the exam?
This situation already exists with many professional credentials, such as CPA, CFA, FSA, etc. They tell you what will be on the exam, provide practice problems, lots of study materials are available from different sources, and you take it when you're ready.
There's also college exams that have a similar function (CLEP is the acronym I think?) but they're not really highlighted as a preferred path to students.
That's correct. But not all schools accept it, and if a school accepts it they may not accept all the different exams. (CLEP exams mostly cover general-ed subjects.) Most schools also have an in-house option to test-out of classes, but it may be up to the individual departments to determine whether they want to allow that. Schools may also disincentivise testing-out in other ways... When I wanted to test out of the intro-to-computers course which was, at the same, required of all students, I had to pay the normal per-unit fee (as if I had taken the course normally) plus a special "challenge fee".
Colleges could structure degrees and majors by examination and interview, rather than by "credit hour", which itself is a meaningless assessment of comprehension.
As an example:
"The University of College offers a Degree in Electrical Engineering, offered at the following levels:
- Associate (At least a 60% in the following examinations: Classical Electricity, Circuit Design, ...)
- Bachelor (Meets Associate requirements plus at least an 80% in the folowing examinations: Signal Analysis...)
- Master (Meets Bachelor requirements plus at least a 90%...)
- Doctor (Meets Master requirements plus three original research publications...)
"
HOWEVER--this would require the total dismantling and reform of the student loan industry, academia itself, and the secondary market that's sprung up around college being a "part of life" for young adults. "College" is a many-billion dollar industry and very few peoiple want it disrupted to the degree that it would require to move to an examination-based format, with the exception of builders and technicians (engineers, etc.) who are actually trying to filter on skill when they hire.
I think it ought to be appropriate for a degree to take decades, with examinations taken and passed at leisure. Most entry-level jobs that require degrees are far easier than college degrees, anyway.
Germany handles that with their apprentice programs. But for a SWE, I think having gone through the class in a limited amount of time and forced to take an exam at the end, that can be reassuring to some people. Anyways, it still beats the American system, IMHO.
I typed six or seven drafts of a comment trying to approach you from a convincing angle and realized that it was probably hopeless, because we likely have too many conflicting assumptions about society, morals, human nature, etc. to come to an agreement in a web forum.
I would like to extend to you an invitation, and that is to read the following essay:
> The Six-Lesson Schoolteacher by John Taylor Gatto: https://www.cantrip.org/gatto.html
If you find this essay accurate and intriguing, consider sending me an email (address in profile) and I'll send you a large set of essays I've collected that discuss the same principle (i.e. that modern education is harming society more than it's helping).
If you find the essay naive and inaccurate, then there's likely very little that we would agree on at all, and it's not worth a 10-message reply chain where we argue over semantics. (Although I admit I enjoy a comments section rumble, as my post history reveals.)
Broadly, there's a lot to like about human nature, but I don't believe in it to that degree.
I like the other commenter's idea to decouple learning from exams, but I'm not sure it would work at all ages, and it mostly just solves the problem that I, personally, had in school: being forced through things I already knew or learned extremely quickly. Others have different problems I'm sure.
I will say Montessori has solutions to some of the problems in this essay and my kids go to a Montessori school.
In my opinion, exams show which students have figured out how to beat the exam.
There might be some that do this by learning the topic, but in my experience the path of least resistance to a good grade has been to study the exam - not the topic of the course that it's based on.
As an example, I was one of the three people (out of 250 students) in my physical chemistry class that got the highest possible grade. I gained no understanding of the topic at all during that time - what I learned was to recognise which formula to use for specific problems on the test. I don't know what any of the results meant or what said result could be used for, simply that it was correct.
This correlates well with the observations made by Google and other large companies that analyse how to identify valuable hires. Turns out that there's no correlation between grades and workplace performance.
So if the exam result offers no information of the students ability to do a job in "real life" and at the same time introduce incentives that conflict with actually learning a topic, then I think it's better to significantly limit their importance or outright get rid of them.
I haven't been able to find any positive gains from having exams, but if someone has some resources on this, I'd be very curious to see it :-)
> This correlates well with the observations made by Google and other large companies that analyse how to identify valuable hires. Turns out that there's no correlation between grades and workplace performance.
These observations probably suffer from a confounding effect known as range restriction. Basically, if Google generally only hires people with a GPA of 3.0 or above, they may have some idea of how GPA variation above that level correlates with job performance, but they don't know about it below that level. So they won't know if people with GPAs of 1.5 will do just as well at Google as people they normally consider.
Suppose Google started hiring people with much lower GPAs than they do today. Anyone want to make a bet what the outcome of that experiment would be?
Well, Google knows, and that's why they don't do the experiment.
"One of the things we’ve seen from all our data crunching is that G.P.A.’s are worthless as a criteria for hiring, and test scores are worthless — no correlation at all except for brand-new college grads, where there’s a slight correlation. Google famously used to ask everyone for a transcript and G.P.A.’s and test scores, but we don’t anymore, unless you’re just a few years out of school. We found that they don’t predict anything."
Google probably filters out low-GPA candidates based on their poor performance by other criteria. It doesn't mean GPA isn't predictive of how such candidates would perform at Google.
If this is untrue, let them bring out their parade of successful low-GPA engineers.
There's plenty of sources online stating Google doesn't have a GPA requirement, that you're welcome to look up on your own time.
Besides those I can also tell from my own (and a couple of friends) experiences, interviewing with Google, that indeed they do not care about GPA - which is a plus when you consider, how many potential hires are not American and have used a completely different grading scheme.
The quote literally states the opposite of what you're claiming, according to Google - GPA does not in any way predict job performance, so why is it so important to you find a narrative, where it does?
What seems more likely - that several Google engineers all made an erroneous conclusion based on the available data or that they found that GPA has zero value in real life?
There's plenty of evidence that grades matter in other contexts, so I'm not going to throw all of that out because some Google guy made a vaguely general claim based on his limited and very biased sample.
As I said, all Google has to do to refute my doubts is point out its huge population of very low GPA, very successful engineers. If GPA is meaningless, this should be a trivial thing for them to do.
Edit: formatting
There's just so much self-learning involved it can be quite hard to narrow down your questions to where it can be discussed in an hour. That's especially if you actually tried to dig deeply into the concepts. You might have gotten stuck for days on a single problem, which is made much worse if there's no hints about how to find the solution.
Another problem seems to be the opposite: you come in for the tutorial, you solve all the problems, and you're no wiser about the issues, despite having learned a mechanism for solving each problem on the sheet.
I had this problem while taking grad courses. I needed time to produce follow up questions and office hours are just a misfit especially when I know there are a bunch of people waiting in line for time.
What we ended up with is a Piazza where TAs mostly do the ritual. In popular areas, I wonder why I am paying when its no better than an internet forum, probably worse.
I feel learning experience is just poorly structured. Teachers get away from the feedback received by invalidating it instead of introspecting it. I can't imagine any other field where feedback is just thrown out no matter how noisy it is.
This is an unacceptable response as well. Why would the students get told that they had to if not for the purpose of intellectual broadening?
In my college experience, symbolic logic served this purpose. It was taught by a clearly dismayed senior member of the Philosophy department, who used plenty of class time to berate the general idea of the "gifted computer whiz kid who doesn't do the work" while clearly running the material on autopilot from the publisher provided syllabus. It didn't make me less interested in philosophy, but it made me a lot less interested in formal philosophy as taught at that school.
This was 20 years ago, for me, and my experience was nearly identical. I was not a great student in High School but ... now having to pay for my own classes ... did well in every class in college except for this one. I attended sessions with tutors offered through my class, I attended office hours, I failed ... hard. The pattern I kept running into was one of learning the material after the exams for each unit were returned to me. In fact, it was so ridiculous, I stopped going to class, resigned to the fact that I was going to fail and simply showed up for the exams. I re-took the class and managed a near-4.0 on the same material. The school I attended had a bad reputation among its math department as being ridiculously difficult staffed with professors who had a basic understanding of English. Of course, this came from students who had recently failed, and they'll have the loudest and often most defensive voices on the subject.
None-the-less, out of the three professors, two were in the "avoid" category pretty quickly due to feedback I received about how difficult it was to understand the words they were speaking[0]. The third was a recent hire -- the rumor being that they picked someone from the US because of the frequent complaints about understanding the set of maths professors they currently employed. But I suspect a lot of people played the game of "pick the guy with the American sounding name" and found out the same thing that I did -- a deep south Alabama accent is much farther from English as it is spoken where I live than any ESL professor's native accent. I picked up about every fifth word coming out of this gentleman's mouth with his very thick drawl and unusual word choices and lousy grammar. In the north, we'd be surprised to find out that he's a maths professor, stereotyping him as a farmer
I suspect, after reading this, that there was more to it than just my struggling to understand what was being said. My pattern was that despite all of the extra energy put into attending tutoring sessions and scheduling office hours, I found myself understanding the material just after we'd taken an exam on the portion we were studying. As this was too long ago, I don't remember if we were given solutions or not, but it fits with that pattern -- the tests were reviewed pretty thoroughly after they were taken and I could see what I did wrong -- it clicked. It clicked so much that despite getting a hard fail in the class, I ended up sailing through with -- I think -- a 4.0 or somewhere near the second time around, attending only on exam days -- with the same professor.
The tutoring/office hours which are brought up as "the way you teach without the solution" were totally inadequate. The staff provided for assistance were not professors, or even particularly skilled tutors. They were advanced maths students who found the work being done to be easy and didn't know how to teach things that were very basic to them. Open office hours were difficult for me to get because I had a full-time job in the field I was seeking my degree in; they were always booked and by the time I knew I needed them, the earliest I could meet with the professor was after the exam. I ended up just scheduling them as soon as I was allowed and cancelling if I didn't need it. Many, many others did this -- it resulted in a group forming by the smallish meeting space that he reserved for these[1] in hopes that he'd finish early and grab the first student waiting. I never did this, but on days that I had a formal appointment, I'd noticed the waiting students would sit on the floor with their materials trying to help one another -- it was a blind-lead...
There are such books for most of the major first and second year undergraduate STEM and STEM-adjacent courses.
[1] https://www.amazon.com/Schaums-Outline-Linear-Algebra-Outlin...
Why won’t anyone teach me math? - https://news.ycombinator.com/item?id=30302079 - Feb 2022 (670 comments)
Some people, a lot of people, will gladly "teach her math" and do a good job of it.
E.g., I tried to, here: In the thread
"Why won’t anyone teach me math?"
at
https://news.ycombinator.com/item?id=30302079
I posted
"Part I"
https://news.ycombinator.com/item?id=30307316
and
"Part II"
https://news.ycombinator.com/item?id=30307349
The two parts are needed because of length limits of HN posts.
The OP (original post)
"Why won’t anyone teach me math?"
is about a coed at Princeton in a badly taught course in linear algebra.
It turns out, I did really well in linear algebra, and "Part I" explains how well I did and how I did that.
In "Part II" I give an outline of an introduction to linear algebra and matrix theory.
For my background I've taught applied math and computer science at three well known research universities and have a Ph.D. in pure/applied math from a fourth.
From how I learned math I have to say that there is a lot of having contempt for the students and dumping on them.
This dumping on the students, treating them with contempt, is unnecessary, unjustified, inexcusable, and bad for the students, researchers, teachers, schools, math, and the US.
The dumping isn't just in famous research universities: E.g., I went in grades 1-12 to the unique college prep school in the city. I got dumped on from the 4th grade on and maybe also in the first three grades but simply don't remember much about grades 1-3.
I wanted to learn but got dumped on so much that I just gave up trying to please the teachers and did what I could with the books for the courses. Turns out, that was a good thing for me to do, especially for learning math.
There were a lot of standardized tests, both aptitude and achievement, and apparently I did well on those, much better than how the teachers viewed me. Point: The opinions of teachers who dump on students can be highly inaccurate; some more objective measures are needed.
Due to the objective measures, in the 9th grade I got sent to a math tournament and in the 11th grade, to an NSF math & physics summer program. Point: The objective measures helped. Turns out, that was true throughout my education.
But for college I went to a private school that took lots of pride in doing well at teaching. Some of the teaching was quite good. Still, however, there was some dumping: E.g., the college got a new prof fresh from a famous research university, and he brought his dumping/contempt propensities with him. At one point I took a reading course from him from a then famous, advanced, notoriously difficult book in topology. So, for each chapter one week I gave a lecture to the prof on the math and the next week, on the exercises. I did well. Once another prof showed up. But even with that good self-teaching success, the prof at best only slightly relaxed his dumping/contempt mode. Point: In that reading course I was a spectacularly good student but the contempt still continued.
My Summary View:
I see no good reason or excuse for the dumping and contempt. I refused to do any such things in my teaching.
Generally I told the students: The material is in the book. Class is to help you learn. So, I'm here to explain and answer questions. There is homework which is to work as many of the exercises as you need to to learn the material, in particular to get good at working the exercises. The tests are problems similar to the exercises -- if you are good on the exercises, you should be good on the tests. From the numerical scores on the tests, I rank the s...