Why Won’t Anyone Teach Her Math?
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quomodocumque.wordpress.com
"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.
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.
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)
> 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.
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.
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.
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.
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.
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.
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.
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.)
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.
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...
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.
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...
Why won’t anyone teach me math? - https://news.ycombinator.com/item?id=30302079 - Feb 2022 (670 comments)
Edit: formatting
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 students, and the purpose is to simplify the assignment of letter grades: So, if student A does better in the ranking than student B, then student A will get the same or better letter grade than student B.
But WHAT letter grades? My standards are, A means that know the material well; B, knows the material; C, knows some of the material; D, doesn't know much; F, doesn't know much and none of us want the student to get credit for the course. So, I look at the work on the tests, judge A-F and decide what final scores deserve what letter grades. Students are not in competition: E.g., I'm perfectly willing to assign As to all students who know the material well; would be pleased to do so; the situation would represent a big success. The department has no rules against my assigning As to everyone.
I'm available in class, after class, and in office hours. Generally the office hours are not heavily used.
The US colleges and universities have some really big heartburn about the comparative roles of teaching, applications, and research.
Research: Can argue that academic research is by a wide margin the most powerful contributor to civilization, health, wealth, understanding, etc. So, with lots of research universities where the profs are required to publish in peer-reviewed journals of original research and hopefully get prizes for work they have done and grants for the work they are doing will, year by year we get some astounding results. Quite broadly the most respected research is theory based on math, theory that mathematizes the field. Applications and teaching are regarded as much less desirable.
Applications. Schools of medicine take applications fully seriously -- that is, a lot of the teaching and research are done in research-teaching hospitals with patients at risk of dying. The field of medicine has a major and respected role for clinical, professional practice. The MD degree, to be a "doctor", is about practice, not research.
In schools of engineering, there can be some significant attention to applications. Still, generally engineering schools do not try nearly as hard as medical schools to purse practice, be clinical or professional.
Teaching. There usually is a lot of good teaching in colleges that emphasize teaching over research. But generally the research universities and their profs look down on teaching -- bad for the field, students, profs, schools, and the US.
Generally there is a belief, attitude, assumption, expectation, tradition, etc. that school is to be a competition, pick winners and losers, be challenging, be a filter, have the students put up with being dumped on, get lessons in going without sleep, etc. As a prof, I refused to play that game -- it was in the interest of all my students to work together to learn, come to my office hours, confess what they didn't know, and get additional explanations, problem solutions, etc.
My view now, from this thread and more, is that the US research universities are on thin ice, about ready for their "comeuppance". Via taxes, tuition, etc., the public is paying the bills and wants results -- contributions to health and wealth, national security, and especially good careers for the students.
Bad teaching, dumping, contempt, generally hurting the students, are creating some really bad PR for the research.
Next, for the teaching and a lot of what the customers want now is arriving via the Internet, e.g., YouTube. I did some teaching here in the "Part II" I wrote.
A lot of that teaching is NOT good, but the best of it is getting good. When there is some really good teaching on the Internet for free or nearly so, bad teaching, hurting the students, etc. will come to hurt the universities, even the Ivy League.
It's possible for a really good student, PBK, etc., brilliant, dedicated, to be so dumped on, treated with so much contempt, that their anxiety level gets really bad, they lose sleep, have physical symptoms, get depressed, clinically depressed, don't recover, and commit suicide. Some education. We're expected to pay for such abuse, via taxes, tuition, our health, our lives?
Bad teaching, hurting the students -- NOT acceptable, grade F.
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-leading-the-blind scenario.
And here's the worst of it: I'm not bad at math! This class made me fear every future maths course I had to take. I got through two Calculus classes many years later due to a desire to fill in the gaps I had learned from books -- they were easy by comparison. I think the way it was taught was specifically because it's a class that many fear; we were gently introduced to the concepts and they covered much less by comparison than linear algebra covered. They were also taught from a more practical standard -- while we were working with odd terms, functions, formulas and theory, they were often brought back to "and here's where it applies in engineering/life" which made it a lot more interesting. The problems we faced in linear algebra, despite having many interesting practical applications, were rushed through and were taught as though "this is just something that Maths majors learn so that they can fill whiteboards for use as props in movies and The Big Bang Theory to make the characters look smart".
My experience in higher education was, in general, not a good one. I left this school a semester later and went elsewhere mostly because of this class, though the final nail in the coffin was an experience in my first programming class[2] where a test -- with very few answers -- responsible for half of the grade had a bug in it -- a failure to dereference a pointer ... a missing star, caused more than half of the answers to be "F - None of the above" or didn't have an answer. I chose to treat the unanswerable as "undefined behavior" (it was C, after all). Luckily, in an attempt to provide answers that might "look right", I found that if I pretended the function call didn't exist, I'd find an answer among the choices and explained to the professor that my fictional compiler optimized away the call since there wasn't an answer among the list that worked without it. I spent an hour arguing with the guy but he ended up re-scoring it to perfect.
[0] One was Indian, the other was Pakistani and I know I would have done better with either of them given that I worked with folks from those regions regularly who had pretty thick accents; like anything, you learn to understand the phrasing and tempo and you stop hearing the accent pretty quickly. I suspect many of the younger students coming from American high schools struggled having encountered it for the first time. It's nothing on the professors or their country of origin, though.
[1] I have never visited a professor for "Office Hours" outside of this class; I don't know if it was because this professor was on his first semester at my school or if this is common but no part of it had the scent of "open office hours" -- it was done in a conference room about the size of my kid's bedroom with a single round table and four chairs cramped under it, one of which couldn't be pulled out due to its proximity to the wall. The professor tended to explain things while searching for materials to help me ... that he didn't have on him that day, so he'd scribble on a piece of paper.
[2] I agonized over this during the 45 minutes we had to finish the exam. Worse, it was proctored by a TA or assistant -- someone I'd never seen, before, and not someone who would know if it was a mistake on the professor's part or not. I'm also pretty sure the guy was playing Descent on the laptop he sat in front of while the test was being taken. It was pure luck that I found a way to answer the questions accurately which also could be defended when an accurate answer wasn't a choice. My concern was that by ignoring the bug, had the professor intended it, I'd have no defense that wouldn't sound like "I didn't notice it and now that I know I got it wrong, I'm trying to weasel my way to a better grade". It was still a tough hill to climb.