The complaint here is about not having access to solutions for problem sets. On the one hand I can see why this would be annoying, on the other hand part of learning math is about coming up with your own questions and answers, a skill that is hard to develop if all the answers are just handed to the students.
There's a pretty solid literature on math pedagogy, and it's clear from that research that students need some way to tell when their answers are right or wrong, so that they can learn from the mistakes. For homework, that could mean getting problem sets back with the wrong answers marked. But it sounds like these were just practice problem sets, so the students literally had not way to tell if they were doing them right or not.
> For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control.
Wild.
EDIT: A response (and +1) to other comments: of course, there are many solutions to different problems on the internet.
But IMO, a big advantage of taking a class is to have shared context with a group of people, and thus interactive help (from other students, TAs, etc) working through specific problems together.
Being able to know if you were "right" seems to be an important part of this process. So having solutions to the shared problems you're all looking at seems important. (Withholding solutions seems like a method for instructors to reuse questions instead of having to write more every semester.)
It's 100% because the problems come from a textbook and some other professors assign them as homework, as opposed to study, rather than writing their own problem sets.
If it's from the professor's own problem sets then actually WTF.
Textbooks written "for teachers" are the bane of every college student's existence if you dare to actually learn from them yourself. "Only solutions to odd problems -- cool guess I have to look on the internet or never be able to validate whether I got them right."
Cheaters, if they are good/hard to detect, would be more likely to be at Princeton than at a community college. By the time you get to Princeton, you're either good at studying or cheating (or being born to the right parents.)
My mother was a professor of one of my college STEM subjects, and had used my textbook in my introductory class. My teacher was horrible, and she agreed. So she gave me her teachers answer guide, which not only provided answers to chapter questions but explanations (some including comments about where students might go astray, and why, and why that was wrong).
I gave up on lecture, and taught myself the subject by doing problems out of the back of our chapters. Aced the class, changed my prof next semester, changed my major to the subject the semester after that, and graduated with honors.
To this day, I am dumbfounded by an approach to STEM education that would withhold a critical tool to iteratively learn via problem solving.
It would be like XP without writing tests. You ship your knowledge to the exam, and pray it doesn't break. Seems ridiculous.
It puts the responsibility on the student, like it will be on the job. You have all the material on this planet at your disposal. Works in other countries: lectures and classes for two years and then one final exam. No quizzes, homeworks, extra credits, etc. Just the classes, lectures, you, your self-motivation, your peers, and all the knowledge of the world in books or the internet to help you learn the topic.
> STEM education that would withhold a critical tool
How does it do that? Are students locked away? Is there a secret police that storms into dorms and burns all material that students try to learn from? Does the Dean come in to break up illegal learning groups?
To offer a different perspective a bit: For every person who desires to learn for it, there are also folks who want to do as little as possible and see no issue with just copying solutions to turn into homework. This includes groups who hold onto the solutions and will give them out to folks taking the classes. Professors had one of two ways to fix that: change questions every year, or just not allow solutions.
That being said, I seem to recall that when I took math classes in college, we had books that had even questions that the solutions were in the back, and the odd questions did not (they were only provided to the teachers). The questions were largely the same, so if a student had an issue with an odd question, they were able to just go to the cooresponding even question and work it out. I felt that was a reasonable compromise.
heh, I actually had this discussion with the professor I an TAing for. The (graduate) class is entirely a class project (it is an ASIC design class), and, if you finish the class project (you can tape out your chip), you get an A.
We have a set of homework assignments to help the students learn the tools (kind of a "hello world" script for the project), and I asked if we just tell them we don't look at them and they are entirely for your benefit. The professor said you if you do that, there will be a decent amount of students who will just not do the assignments and later on complain that they have no idea how to use the tools.
The solution we came up with is to have a set assignment due date so students feel like they have to do them.
Personally, I am surprised that graduate students need that sort of motivation to do the work.
Professors could change the numbers in the questions every year, without really changing the questions.
And then give the answers after the homework was handed in, if they want to grade the homework. Or compromise, by giving answers to half the assignments, and not grading those.
Many years ago, my univerity's choice of calculus book also had the solutions to half the problems (evens or odds, doesn't really matter).
Department policy was also one of "we won't provide you the correct answers" even after an assignment had been turned in because the book was used for multiple years. The publisher had a new edition every year, but the university stuck with the same book because it was used for Calculus I, II, and III, which for most students was 3 or 4 semesters between starting I and finishing III, usually due to a scheduling conflict requiring a semester off between them, or because they had to repeat Calc II since the math department's selection of instructors was particularly bad for that course.
Those same (usually bad) instructors were all too happy to follow the department policy and not provide any feedback other than "correct" or "incorrect".
On the other hand, the university book store carried, and put on the shelf right next to the calculus book, the publisher's "teacher's solution guide", in two very reasonably priced volumes, which had the answers for the other half of the problem sets, as well as the step-by-step process for most of them, which was the valuable part, as you could see where you were erring.
Math department policy was that you weren't allowed to have those, either.
You can guess how well that policy was followed.
Those that put the effort in and learned the material did well. Those that just copied from the solutions book and turned it in did not.
At the end of the day, if a university gives out grades, there has to be some sort of objective way of handing out a grade. What I have seen for the good courses is it is a mixture of exams, projects, and homework.
Some students are very good at taking exams, others, not so much. This can be due to a learning disability, stress over test taking, etc (or it could even be that person just is having a bad day!). Having homework and projects allows for students to have a different way of showing that they understand the coursework, and are able to apply the material.
It also gives the professor and TAs insight into the student. Why is a student doing so well on homeworks, and not the exam or project?
Some courses have it where if youre final grade on the exam is an A, you get an A (since it is a comphrensive knowledge base test of what you are expected to know of the material). But, let's say you don't do as well on the exam, you can have the homeworks average out the exam grade. Or you have a project, that can help equalize out the grades, because the application of the knowledge is important as well.
It's highly likely that there is a large pool of exam questions, and every year, they are split into "practice" and "actual test". Providing a gold standard answer would lead to memorizing solutions in a way that providing all the questions does not.
It's likely the point is to talk to the professor about the ones you're having trouble with, or where your study group disagrees.
I always believed that if a course assignment needed a student to look up anything online (provided that their background was solid enough), let alone the solution, this would be something professors would feel moderately self-conscious about.
However, I found out the hard way that also in respected universities, the actual assignments in "hard" courses were long-known, extremely challenging problems that were not really meant to be solved at this level, especially on a weekly basis.
Folks handing them in were often using solutions that were quite obviously inspired from online sources, if anything because of the weird methodology needed to solve some of them, without the course teams caring at all.
I left with the certainty that I would never take a math class again, and a lack of desire to explore other scientific fields for fear that I would have a similar experience.
That is one of the most tragic things I can imagine reading. A student who really, genuinely wants to study Maths, just for its own sake... and is dissuaded from doing so by myopic, short-sighted, inane policies. It makes you wonder "could you fuck up education any worse, if you were intentionally trying?"
On the other hand though... I believe a student with enough self motivation can overcome those inane department policies and learn the Maths by pulling in outside resources. Yes, it's more work, and yes maybe it's not "fair" that you have to do that in addition to attending (and paying for!) a class. But it's possible.
Consider the availability of free resources like Khan Academy, Paul's Online Math Notes, hours and hours of Youtube videos from people like Professor Leonard, Gibert Strang, etc. One can also often find problem sets with solutions by just Googling around and finding previous years course websites for various courses that have been taught at universities all around the world.
There are also online forums where you can go to get your answers validated, or get additional explanations about Maths problems. They vary in the extent to which they accept "do my homework for me" style questions, but almost any site will acknowledge somebody who has put in some work, come up with an (possibly incorrect) answer and says "Can somebody help me understand what's going on here?"
* https://math.stackexchange.com (note: but not so much Mathoverflow, which is more for research level Maths and is not a good place for straight up "do my homework" style questions at all)
And if one is willing to pay a bit, the Schaum's Outlines books and similar books provide a huge catalog of worked problems, along with problems and solutions. There's also Brilliant.org and other paid educational resources for Maths that can supplement ones university course(s). One can also look into hiring a personal tutor as an option.
Many (most? all?) universities also have something like a "math lab" or "learning center" or something where students can go for individual tutoring and additional support. My experience (albeit dated now) leads me to believe that these are probably drastically under-utilized.
Should universities do a better job with introductory Maths courses? Almost certainly. But I'd encourage anybody dealing with this to dig in and try to overcomes such shortcomings by using other available resources as well.
EDIT: inspired by @skywardavocado's answer, I also wanted to add that old-fashioned "study groups" are another valuable arrow in the ole quiver. If you join up with 2, 3, 5, whatever, of your peers to study together, chances are that somebody in the group will understand the thing that the others are struggling with, and can explain it. I didn't do a lot of this in college myself, probably to my detriment. But to the extent that I did occasionally join a study group, I'd say they can be wildly helpful.
> It makes you wonder "could you fuck up education any worse, if you were intentionally trying?"
Of course you could. Take a look at K-12 schooling some time, where the prevailing educational theory is that "students must learn the math by themselves", and are expected to devise "their own methods" to do so, including "guessing" and doing calculations in their heads, not on paper. Is it any wonder that even "getting the right answer" has been de-emphasized, never mind "show your work"? This is what passes for math education these days, courtesy of "educators" who have never gotten a proper education at the college level in the actual subject.
Have you experienced this personally? I ask because this hasn't been my experience as a parent of a K-12 student. They are expected to show and be able to explain their work, and the teachers use that work to gauge the student's understanding of the material.
It seems to be an issue in some school districts and because it's such a weird thing (to most of us) it ends up getting a lot of "air time", so to speak. Outliers doing bizarre things become representative when they get the majority of coverage.
The school district my children were in fiddled with it. One of my kids got actively in trouble for the fact he could do the "real" math problem in his head and wasn't "estimating", despite also being able to estimate correctly on larger problems. It is the direction math education is going in the public school system, and it is the direction it has had for a while so it's not like this is really some sort of surprise. Standards will be lowered until all students pass. There are political currents in that direction, plus there's the ever-present fact that the easiest way to make a given school's performance go up is to lower the standards, and there are many entities with incentive to do that. If your local school still has standards, they may currently be the mountains still standing above the rising seas, but the seas are still rising.
In some ways I wish they'd just get on with it and leap straight to the logical conclusion of just officially eliminated the standards and passing all students guaranteed, so we can all get on with the task of dealing with the fact that such a credential would be worthless, instead of this long, drawn-out process of lowering standards while trying to pretend the standards aren't being lowered.
(I think the second derivative of this process has turned away from dumbing down. Pushback is really coming up in earnest. But it'll be a while before it so much as turns the first derivative back in the correct direction, let alone get to the point where the problem is largely fixed.)
I wish! I regularly got marked down for doing mental math or not following the exact sequence of steps and writing down every last petty instruction. Busywork was 50% of the grade for "Pre-Calc", so I had to take it 3 times...
> Consider the availability of free resources like Khan Academy, Paul's Online Math Notes, hours and hours of Youtube videos from people like Professor Leonard, Gibert Strang, etc. One can also often find problem sets with solutions by just Googling around and finding previous years course websites for various courses that have been taught at universities all around the world.
As a university professor, given the availability of all these resources, I'm not sure why you'd want to take advantage of them and take a university course if you're not interested in the credentialling. Universities are a place to learn the deepest knowledge of content experts, and I don't want to downplay that; but these content experts are usually not pedagogical experts, and they often tend to be less skilled at teaching less advanced material. That's not to say that there are no good introductory-level teachers out there—there are lots, and they do heroic work—but that a random university professor, even a very good professor, probably won't be as good at teaching introductory material. (With that being said, I don't regard Linear Algebra as introductory material—but, then, I'm not teaching at Princeton.)
Certainly, if I were advising a non-math major who wanted to take an introductory math class, I would encourage them to think very carefully about an auto-didactic approach to see if they like it well enough to continue. If you want to learn lots of mathematics, then a university math department is the place for you; but, if you just want to dip your toe into it, then it may well not be, because so many of those courses are set up as 'service courses' for people who don't want to be there but have to be, and that inevitably shapes the tenor of those classes.
As a university professor, given the availability of all these resources, I'm not sure why you'd want to take advantage of them and take a university course if you're not interested in the credentialling.
Fair enough. FWIW, my post was written from the perspective of being targeted at someone who has already chosen to take a university class, for whatever reason. But I agree with your point, and that approach is, in fact, my own. I mean, yes, I took some university maths classes in the past. But now as I want to learn new maths or re-learn maths I've forgotten, I prefer to just study on my own using mostly the exact resources I called out above. I wouldn't go pay to take a university class at this point in my life.
Does anyone have good book recommendations on Math? I've always deeply enjoyed the subject, and I think I'd like to study on my own but I'm unaware of where to even delve into the subject.
My username comes to mind. Spivak's Calculus is hands down the best intro Calc book if you want to learn math for its own sake. His book on manifolds is also amazing but definitely not intro.
If you want stats I highly recommend Statistical Inference by Casella & Berger -- it's extremely dry but so many stats books out there try to "make it easy" but the simplification means that you can't actually grok what's really going on. HOWEVER if you want to actually apply anything in this book you'll need to grab something more practical as well. Going through an applied stats book after having done SI is like having superpowers.
Spivak’s Calculus is hands down an excellent book for a first course in real analysis, and I’ll die on that hill(it’s only real competition is Analysis 1 by Tao, in my opinion).
Anyone that says “it’s just a calculus book; it’s not good for introductory real analysis” is invited to go solve every problem in, for example, the chapter which defines integration, and compare the difficulty with problems in “traditional” analysis books.
If they know how to do (rigorous) proofs, then Spivak would be good. Otherwise it would be far too advanced. It's essentially an introduction to analysis.
That's fair. I don't really know what counts as advanced anymore. There just aren't that many books on theoretical math that aren't "graduate level" (which is way scarier than it sounds) -- Spivak was my Real Analysis 101 book as a freshman math major.
Spivak at least has the nice property that it doesn't assume you know much.
I've made it a hobby of reading beginner Calculus texts. I believe it to be a fascinating thing to explain and teach. Recently, the Teach Yourself series re-released their 1992, Calculus: A Complete Introduction. It isn't clever like Things Better Explained, with the visualizations, but does share excellent examples you already understand as unusual and then provides a pathway to using the integral. Like most good Calc books, just bring algebra!
There are multiple entry points, depending on your goals.
However, if your goal is to learn mathematics for the sake of art, I recommend Kolmogorov's Elements of the Theory of Functions and Functional Analysis.
I'd like to augment request one step further: Is there anything that kinda goes through the fundamentals all over again, but is programming aware?
I constantly assert that I'd have actually been successful with mathematics throughout school if I were able apply it in code (which was not a thing in my educational environment); not with my broken brain where I fuck up numbers on paper and use my fingers for arithmetic.
And somebody who posts here on HN recently published a book with a title something like "Mathematics for Computer Programmers" or something to that effect. I forget the username and the exact title though. If you search around you can probably find it.
Edit: here's that last one. A Programmer's Introduction to Mathematics
> I'd like to augment request one step further: Is there anything that kinda goes through the fundamentals all over again, but is programming aware?
This is the second time today I'm recommending Knuth's "The Art of Computer Programming". It really is a math book, and it includes answers to ALL the exercises. For example, the last 150 pages of Volume 1 are solutions to the exercises.
No one can have an enjoyable and helpful reading experience by picking a textbook or applying for a random course. Instead, reading the history of mathematics to learn about topics and their background would be a great way to capture the idea and a good starting point to find your favorite area. I recommend searching for Paul Lockhart's books, Mathematics in Western Culture by Morris Kline, and Famous problems of geometry and how to solve them by Benjamin Bold.
Not a book. But I'd recommend this website :- https://www.mathsisfun.com/ website. Everything is broken down into to smaller chunks. I stumble upon it when I was looking to revise my algebra concepts. But now I'm learning physics on it.
Why not both? A particularly bad class demonstrating typical problems in an unusual intensity. Also an individual who had a particularly intense mismatch between her learning style and the course design.
I don't know if it's true of all schools, but I had a very similar experience at another Ivy League school. Took the first semester of linear algebra, got a very respectable grade, but really didn't learn much because the class wasn't structured to do that. It was there to make STEM majors prove they were good enough. I wasn't a STEM major, so didn't need to keep going with that kind of hazing.
In my undergrad years every major seemed to have what students referred to a "weed-out course." These courses were intentionally made difficult and often has inane non-standard grading policies (for example, in one, anything under 85 was considered an F). They were often taught on odd schedules (like one semester a year or something) and at unusual times. And they were often taught by difficult professors who did little, if anything, to help students succeed. If you were lucky, there might be a TA that might be able to help you, but students were often expected to form independent study groups and assist one another.
Basically if you couldn't pass this course - and in some cases department policy limited you to a single try - your chances of success in that major were slim to nonexistent, and you were encouraged by your advisor to find a different major. If you could pass the weed-out course, you could be reasonably assured you were ready to face the rest of your coursework, because nothing ahead would be as difficult as that course was. And, that you were reasonably certain this was what you wanted to do.
They weren't necessarily directly related to the major. For some of the engineering and science programs, a math course was often the weed-out course.
I can't say for sure that this was the author's experience, but her description of it sounds very much like a weed-out course.
Unfortunately it seems like no one has been willing to teach the author what mathematics is. Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work; it isn't mathematics.
Now it's possible I'm misunderstanding the problem here: I am assuming that during lessons students have been given answers to problems worked in class, so that they can see what an answer looks like, which is probably one of the most important parts in learning introductory mathematics. Indeed, learning to validate one's own answers is not only the most important lesson in this stage of learning mathematics; it's also the only way to actually learn mathematics as it's the only way actual math gets done.
(ETA:) So I guess what I'm saying is that the author is right that no one has taught her mathematics, but as a symptom of this she doesn't even understand what form actually learning mathematics would take.
I think there are shades of correctness to your answer (at the upper echelons of math study, as it were, there will be no answer book for you to check), but the act of confirming your answers should be transitioned in and taught; which from this author's perspective hasn't happened. Further, there should be education, refinement, and strengthening of learning to validate one's own answers before leaving them to flounder in a 200 level STEM-focused course.
(Honestly, I'd probably have a better relationship with math if it were taught to me in the method you seem to require as "actual math".)
> Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work
This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math. I remember the agony I experienced as a 3rd grader for not being able to memorize "multiplication tables". Instead I took the approach of trying to understand what the mathematical meaning of multiplication is. This was to my extreme academic & mental detriment as we were not graded on understanding, but instead graded on how fast we could answer math questions.
I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles.
Obviously, my concern over this was totally pointless as I've never been in a career environment where I was told "just memorize the answer, don't bother trying to understand anything".
> Instead I took the approach of trying to understand what the mathematical meaning of multiplication is.
There's not much to think about tho, is there? You take 3 boxes of 4 apples each, how many apples is there?
Or were you thinking about groups, rings and fields of integers? I highly doubt it at that age ;)
Ultimately you need to know multiplication tables instinctively to be able to do algebra, so they make you learn them. I don't think there's any way to skip it, just like you can't be good at football without doing a lot of cardio.
For the record - I also hated memorizing multiplication tables and thought I will be a writer because clearly math isn't for me. Fortunately in the next semester we had word problems and I loved that part of math.
I think there is. For example, multiplication is not visually intuitive on a number line, but it is very intuitive shown as a grid of row x columns. Doing the math out helps build an intuition for things like commutative properties and what not.
Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times.
> Doing the math out helps build an intuition for things like commutative properties and what not.
Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway.
I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.
> Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times.
Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.
Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.
> Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.
That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa.
I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.
That is entirely the point. The number line is awkward for multiplication.
But mapping the relationship between multiplication and the area of rectangles is very visually intuitive. Geometric reasoning is powerful. Being able to use your understanding of multiplication rules to derive the formula for the volume of a cube, or a cylinder, or whatever is probably more insightful than realizing some arithmetic tricks.
> That is entirely the point. The number line is awkward for multiplication.
Strong disagree. The number line is one of the best tools for understanding multiplication of numbers (not just whole numbers, but any decimal). The problem is that it is no longer taught. Example: American rulers have both metric (cm) and Imperial (inch) units. You can use it to visually multiply and divide by 2.54.
To generalize multiplying positive numbers x and y, (1) Mark x on a number line, (2) create another number line with the number 1 placed where x would be on the original line, (3) find y on the new number line, and (4) the corresponding point on the original line is the product x*y. The point: Multiplication is scaling (stretching, shrinking, whatever you want to call it). The Common Core tries to address this.
As a kid, the row x column thing never made sense to me. I was horrible at understanding visual metaphors, I don't think I "got" the row x column thing until well into high school, and even then it was "I have to think about it".
I always had better luck with just manipulating the numbers mentally.
Visual explanations of geometry, trig, and calculus made sense.
But as a kid I never got the multiplication thing. Even now as an adult I understand what they are trying to do, but it isn't "obvious" to me.
Then again I am one of those people who doesn't read comic books because all the pictures get in the way of the words. Heck as a kid I read illustrated books and didn't realize until I had finished the book that it even had pictures.
>I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles.
and this is why kids are taught to memorize these lookup tables. When doing algorithms such as long division or multiplication these lookup tables can make the process much faster.
I don't know what the answer is, but we're currently going through this with our kids, but the opposite. There is now a push to not memorize the multiplication table, but treat multiplication as repeated addition. So to get 6 * 7 you do 6+6+6+6+6+6+6 or 7+7+7+7+7+7 (I forget which is 'correct' as it's been a few years).
That's good because it gives understanding, but when you're trying to multiply 2 4 digit numbers, it's really useful to have your tables memorized!
We're now going into a new school and their biggest advice to new students is to memorize their multiplication tables, as they'll have trouble otherwise. Maybe they should do understanding followed by memorization?*
Unfortunately it is according to some of the common core grading rubrics (in the US).
5 x 3 can be understood as five groups of three items, or three groups of five items, and students at a particular grade level are "encouraged" to use one interpretation over the other to the point of taking off points when they naturally use the commutative property (and or choose to chunk differently). I forgot what it was exactly but I read about a parent (rightfully) complaining about this.
I don't think this is anywhere in the common core standards, but probably the teacher/school was using some pre-packaged rubric and the teacher didn't bother to override using better judgment.
I picked up this trick intuitively as a very young kid, perhaps 7-8
I break multiplication and division up into chunks of 1's, 5's, and 10's so that it's mostly just adding a zero to the end, or doing that and then cutting it in half.
I never learned multiplication tables, long division, lattice multiplication or anything else. just did this in my head.
So say you want to find 86 * 32
86 * 10 = 860
860 * 3 = (2400 + 180 = 2580)
2580 + (86 * 2) = 2752
It starts to break down for bigger numbers depending on how "uneven" they are but it's been pretty useful in general
Multiplication tables memorized in school usually stop at 12x12. They're super useful for doing the exact thing you demonstrated, in your head, very fast, using some tricks to turn the problem into smaller problems (with memorized answers, so they're solved almost instantly).
Memorizing that multiplication table was a more valuable use of time than the vast majority of time I spent learning mathematics. It's useful daily. Can't say that for much else.
The principles of multiplication are pretty straightforward. However, especially before calculators were commonplace (one can probably debate if there's sufficient value in memorizing multiplication tables today), there was a lot of benefit in doing multiplication and other simple operations quickly and mentally. There are a ton of basic things that we memorize without necessarily knowing how to get there from first principles off the top of our heads.
This is true with programming as well. Sure you may look up APIs or specific syntax but if you had to look up every detail every time you put fingers to keyboard, you'd be really slow.
My feeling is that what you say is true to a varying degree, dependent on the course in question. Maybe more particularly the level of the course. For proofs based math where the goal is to write a proof, I'm more inclined to agree with you. Especially since there can be more than one "correct answer" and seeing one may or may not help you at all in understanding why/how another one would be constructed.
But for lower level "grind and chug" Maths classes (High-school algebra, intro Calculus, non-proofs-based Linear Algebra, etc) I think there is value in having access to "the answer". Certainly at least for a subset of the problems one works on, IMO.
> But for lower level "grind and chug" Maths classes (High-school algebra, intro Calculus, non-proofs-based Linear Algebra, etc) I think there is value in having access to "the answer". Certainly at least for a subset of the problems one works on, IMO.
I think there is a value in introducing the concept of verifying your answers early, as your ability to perform calculations is beyond useless if that isn't something you are thoroughly familiar with.
If you're just doing the calculation but not the verification, you're skipping half the exercise.
Sure but those two positions are not mutually exclusive. I don't think I ever took a Maths class that didn't introduce the idea of verifying your answers. But at the lower level, most of them used books that provided answers for some but not all of the exercises.
If nothing else, this is an advantage from a time efficiency viewpoint, as you can more quickly identify the problems you got wrong, and focus your review time on those. Those presumably being the ones most likely to highlight some misunderstanding of the material.
If time were infinitely available maybe the distinction wouldn't matter, but things being what they are...
Definitely agree with this. At least for me when taking undergraduate math, once you get to the proof based stuff, often times seeing the answer was in no way helpful. You either knew you proved it correctly or you weren't able to do the mental gymnastic it required to get there and seeing an answer was likely not going to help. I usually just had to re-read the book and try different things, take a break maybe, or something before I would finally get the intuition or whatever to really grasp it.
For calculation based math, which this course sounds like and based on a sample test I looked at, more or less is, having some subset of answers would definitely be helpful. I am not sure why they would make it more obtuse than it needs to be. I thought the course I took on multivariable calculus and linear algebra were quite easy but the number of people who dropped math entirely after those was pretty drastic. I don't think you really need to make these classes unnecessarily hard to weed people out. They'll do it themselves anyway.
Learning by example is a thing in math too.
Learning theorems and methods is not sufficient, they are tools and you need training to master them.
Not everyone is able to generalize from one class example to other situations. For that you need to work with other examples. Not only the problems, the answers too.
Also if no one liked this course there is certainly a problem that can't be dismissed in a handwave like you did in your answer.
Richard Borcherds, a field medal winner, stated that he approaches new topics by doing a lot of examples before trying to understand the theorems and such. There is no "right" way to do math, contrary to what OP claims.
> Not everyone is able to generalize from one class example to other situations.
I found the same thing in my tutoring experience. I had several students in physics whom I couldn't "reach" no matter how hard I tried to present "theory" using the equations in full generality (symbols instead of numebrs). I taught them this way because they this is the way I learned and because it seems OBVIOUS to me that learning the abstract thing is more powerful/efficient way to get through it.
Imagine spending dozens of hours with them and they still didn't "get" anything to the point I was feeling discouraged and was like this person will never pass this exam that they have next week. I felt bad because these were friends and I really wanted them to succeed (more friends in STEM, more STEM conversations with friends).
Lo and behold, these same people whom I thought were "physics dumb," were able to pick up all the material in literally minutes, as soon as we started the exercises (one person) or having tried some exercises on their own (the other person). I don't mean "imitate the steps"-know, I mean completely solve 80% of the problems on the practice exams and adapt the use of equations as needed for each problem. I mean it's still PHYS101, so there are only 10 or so equations, so they didn't turn savants or anything, but the transformation was so sudden and drastic that I realized people simply have different learning approaches.
Some people like the abstract/general approach. Some people like the hands-on approach. The latter kind of people often think they are "not good at X" but really they would have no problem picking up X if they had learning resources suitable for them (often lacking in traditional textbooks and courses). I think that's why Khan Academy videos are so popular. Sal does a fair bid of hands-on approach, and many people need that!
Imagine learning to code again. You're shown one for loop, a class header file, and a cartoon of a data structure. You write programs by hand on paper, submit them, and never get any feedback. And you're forbidden from looking at open source code. Some amount of struggle and self validation is fine, but unnecessarily wasting absolutely all your time on dead ends is a waste.
This type of bullshit is normal in crappy STEM classes. Universities are so concerned with keeping secrets that they won't give you the information in the first place. It's a sign of bad course design that's unfortunately common.
I took a couple classes that were like that, but I swapped the class for another subject those semesters and waited until it was offered again by a better professor. Same material, except I actually learned it instead of getting stuck at the starting gate.
Grading the homework is not the same thing as providing feedback. A big red X tells you that something is wrong, but not what you can do to fix the problem or even why that thing is wrong.
Okay, so go to office hours or a math center (if available, apparently quite common).
I certainly don't want to be misquoted as claiming that mathematics must be learned in a vacuum. Just that access to a large collection of questions and answers is not helpful. Your own argument goes through just as well to show that access to the answer isn't helpful either: you know you got the wrong answer but you don't know why.
For proof based homework typically a grader will tell you which step failed with a short note saying why ("this mapping is not well defined", "here is a counterexample", etc)
It is cryptic at times bit ultimately developing rigor and intuition is something that in my experience will always involve a certain amount of struggle and confusion that no professor can or should alleviate
Right, but at the pace that many of these courses go, and with how the material continually builds throughout the semester, not getting answers until a week after the quiz, and hence two weeks after the lecture taught the material, is setting up a lot of people for failure.
By having access to multiple worked examples of problems, students at least have a fighting chance of learning how to solve problems _in parallel_ with the course material, rather than having to wait days or weeks to even know what they did wrong, while getting more and more lost.
Remember, these are Princeton students, some of the most brilliant and motivated minds of their generation, and the averages are _abysmal_. That indicates there's a greater systemic issue.
Unstated premise: the most brilliant and motivated minds of their generation ought to all be able to grasp mathematics. I for one don't believe this premise. University level mathematics requires a level of abstract thinking that most people are simply incapable of.
I respectfully disagree. Learning mathematics is just as much a craft as learning an instrument, or how to paint, or how to write prose, or how to layout a PCB; it requires a lot of hard work, but if you're consistent AND have the right tools, you can learn it just like any other subject.
Believe it or not, the number of practicing mathematicians and PhD students who have the innate grasp that a Terry Tao does are few and far between. The vast majority have to work hard to earn their skill.
Someone who is tone deaf will never be proficient with a violin, or a theremin.
No amount of hard work would have allowed Stephen Hawking to succeed at sports.
Some pursuits are simply unavailable to those without the requisite immutable attributes. This is not politically comfortable, but it is true. I think the only legitimate question is whether mathematics is such a pursuit, and I'd like to be proved wrong on that point, but trying to convince me that anyone can do anything they put their mind to is wasted effort as it is so plainly contradicted by the available evidence.
Note that the contradictions you mention are _physical_ in nature; it is true that there are some functions that some bodies can perform that others cannot. Mathematics is not one of those things.
Is the brain not a physical thing? Do you believe that all human brains are physically and functionally equivalent such that one brain should be able to learn to do anything that another can?
What about those people in this world with genetic abnormalities that affect brain function? Would you expect someone with Prader-Willi syndrome to be able to become competent at mathematics if they only worked hard enough at it? At what point does “hard enough” become “virtually impossible” or even “definitely impossible”?
While I feel your point is incredibly pedantic, perhaps I should have qualified with some form of "at least X% of people that do not have rare genetic disorders can learn math, X >> 50". However, I think such statements are usually implicit when talking in generalities.
Got it. Your previous statement didn't really seem like you were speaking in broad generalities and instead seemed more absolute, like math is something that could always be learned by any human brain. Clearly that's not the case, which was the point of gpp, I think.
Why, sure - from the physics standpoint all the human brains are essentially the same, they all consist of the same protons, neutrons, electrons... (Also, they are close enough to being spherical.)
I had an organic chemistry professor who would not return anything until the end of the semester, homework, quizzes, test, nothing. I had the impression she was a procratsinator and just didn't do the work. It was beyond frustrating. She also got her ph.d from Princeton coincidentally!
The most lenient late policy I've experienced in my (ongoing) university experience has been that you can submit 1 day late for a 30% penalty. Most of my classes lock the dropbox immediately at midnight.
Simply returning work shortly after the late work window closes should suffice, no?
I did this at Loyola New Orleans (an intro physics lab class) and I'll tell you why. I didn't trust the kids to not lose their work. They could come to my office and collect it and if they challenged their grade, I had all the material to hand with me. Online reports were graded on Canvas at any rate.
You might be surprised. My daughter took first year university linear algebra last year and graded assignments and tests were often returned weeks later, in some cases well into the exam preparation period. I told her this was unacceptable and she should complain, but you can imagine why a young, shy first year student might not be able to get over that potential barrier.
Another thing that one might not have considered is the perception of women and math. Of course your daughter might not want to protest against unfair mathematics teaching structures given that her struggle is societally used to represent that all women struggle at math. <relevant xkcd comic here>. I would also hesitate to protest unfair practices in situations where I was a minority because I know anything short of quietly excelling is a negative representation of my entire demographic.
Teaching yourself math by producing answers and then validating them seems like a great way to possibly completely fail at learning math. What if you are making some fundamental mistake, and now all that time you've spent studying reinforces that mistake? We see this in software all the time; people will latch on to bad patterns and stick with them for years.
I think that is the whole point of not giving answers.
You should be able to solve a problem in multiple ways and prove correctness.
It takes loads of time, but that is the only way to truly learn.
Most of students given answers will solve a problem in one way see that answer matches the correct solution and stop there. That is not the way to learn anything.
It is the same with software, it takes a lot of time to get good at.
In software it is easy to get correct solution with wrong approach, but often people just stop at "make it work" where they should step back and "make it work, make it right, make it fast".
> Attempting to learn mathematics as some sort of pattern matching, [...] does not work; it isn't mathematics.
It's possible the author was in a "math for engineers" class
Where a math student may see a technique as a swiss army knife that can be used for loads of things, an engineer might only care for the one blade that lets them check the stability of their boost converter and pass their power supply design class.
>Brief Course Description: More abstract than calculus, this course aims to develop basic algebraic tools for work with problems involving many variables. Starting from systems of linear equations and vectors in 2-space and 3-space, this course develops ideas about length, angles and resolving a general vector into useful components, identifying features of linear systems or processes in order to choose a basis that is well-adapted to studying a particular phenomenon and move between different points of view to reveal the essential underlying structure. Companion course to 201 (Multivariable Calculus). Discusses matrices and linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvalues and their applications to quadratic forms and dynamical systems.
No. She was in a 202 class. "math for engineers" is usually locked behind a course of study or prerequisite classes so people who need those classes to graduate get them. Everything 200 and below is usually tagged as sophomore or below and is intended to be used cross major. 300 and up is when specialized topics start coming up.
You could argue it was "math for engineers" in that Liberal Arts / Social Studies don't require them and only the STEM programs do, I guess. You can graduate with a non STEM degree with just an Algebra course in most Universities in America. And those Algebra courses are usually just high school refreshers if you went to a good high school. Some of the non STEM require Statistics. But even there some schools do "Statistics for STEM majors" and "Statistics for non STEM majors"
> Attempting to learn mathematics as some sort of pattern matching,
The math I did before university was doable with basically pattern matching ; I then still had a close to photographic memory (I could skim materials and recall them exactly) and got A’s in math because of mostly that. I could do exams by filling in methods I memorised. There was no understanding. Then in uni I got a good scare as none of that worked anymore; now I had to actually know what I was doing. That was obviously much better but I was not prepared for that at all.
Finding alternative proofs is not a valid way to confirm the original proof; so... no?
You appear to be absolutely correct that no one has taught you what mathematics is. There's definitely a failing here in mathematics education; unfortunately this is not a surprise at all.
Validating the proof requires understanding what it is you're trying to prove, understanding what are valid proof techniques, and understanding the premises. You then check every step against this knowledge. It isn't easy, and tbh I usually couldn't be bothered after spending hours on a problem, but this is where results are obtained in proportion to effort.
Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work; it isn't mathematics.
You're making the student out to be some kind of simpleton. She's not.
She just wants what any decent textbook has: An answer key to the practice problems, so she knows whether her answers were correct or not. Being as they are, after all, practice problems. Meanwhile her instructor is telling her, in so many words, to buzz off.
That's what is at issue here. Not (as you presume) her being some kind of a lunkhead, who thinks it's all about "pattern matching".
Very few of the maths textbooks I've read have had the answers to the exercises in them. If the author wants to put in a worked example, that will be in the chapter itself, rather than the exercises.
In maths, you know when you've got an exercise right, and so you don't need the answer written out for you. If you're looking up the answer before you get that feeling, you're cheating yourself. You won't learn the material.
It would seem weird if someone put the answers in the textbook, like having the answers to a newspaper crossword on the same page as the crossword itself. You'd have to exert willpower in order to not look at them.
The whole point of the exercises is to play with the ideas until you understand what they're for. If you don't at least partly invent the subject yourself, you never really 'get' it.
Textbooks are really guides to how to reinvent the field, what order to think about things in, what are the cleanest ways to break up the patterns, what are the best notations to use, so you can reinvent it all at a reasonable speed rather than having to spend a whole lifetime on each problem.
If you literally can't do the exercises despite having read the chapter, then it's a bad textbook. Throw it away and get a better one.
Very few of the maths textbooks I've read have had the answers to the exercises in them.
Well it's been a while - by my recollection is that (at fresh and sophomore level at least), the better textbooks always did have answer keys in the back (at least for a significant subset of the problems). But as we all know -- the better textbooks are few and far between. And college is filled with lousy textbooks that get thrown at students for who knows what reasons.
To the extent that they think lousy textbooks are the norm.
> In maths, you know when you've got an exercise right, and so you don't need the answer written out for you.
No, I might think I have the correct answer but I cannot know until I have seen the correct answer. I can justify any false answer that's not obviously false for myself which is why I need to compare.
> I am assuming that during lessons students have been given answers to problems worked in class, so that they can see what an answer looks like, which is probably one of the most important parts in learning introductory mathematics
As someone deeply familiar with/scarred by courses just like this one, in-classroom introductory mathematics pedagogy at elite institutions can vary wildly because mathematics graduate students and professors are incredibly specialized in their fields of study and accustomed to very specialized types of intuition - with perhaps more of a disconnect from the introductory student's learning experience than in any other field. Language barriers can exacerbate this. Proofs are thrown on a board, time management on the lecturer's part is nonexistent, and examples are inevitably skipped over as "an exercise for the student."
In contrast, many computer science programs recognize this tendency and optimize for pedagogy. I remember my university's CS department hiring and paying upperclass CS students to guide other students through their first times debugging code outside of the authority structures of official teaching staff. No similar program existed for the mathematics department, and arguably it should have - but who would volunteer? Once a mathematics major, you're pulled into a realm that's quite isolated from the broader student experience!
> Unfortunately it seems like no one has been willing to teach the author what mathematics is.
You're assuming there's only one answer to "what mathematics is" and that it coincides with your view. You appear to be a fan of doing math, but far more people use math.
I'm not a mathematician. To me, it's a tool that can be used to solve the problems I care about. I don't get joy from spending hours proving results just to say I did it - but I understand that others do. In spite of that, if I can understand the conditions in which an algorithm will converge to the solution, I can benefit from studying math.
From what I understood, the HW/exams were graded, but solutions were not provided. I've studied math at two universities, and every professor provides solutions after they've been graded. And if they didn't, they would give you the solution to a particular problem in their office.
They may not give it as a handout, but more likely would solve the problems in class for everyone to see. But one way or other, they would give it.
The course is "Linear Algebra with Applications" - and the OP explicitly wanted a course to learn linear algebra for non-math majors. It's not about "doing mathematics" in the sense of doing novel math, it's about learning basic linear algebra and being able to apply it to a non-math field.
IMHO for everyone who wants or needs to "learn actual mathematics" there are at least ten people who need "applied mathematics" in the sense of being able to apply well-known theorems from a well developed field (like linear algebra) and with zero need to ever prove anything novel, even trivial things; they need to understand the concepts and know how and where to look up the appropriate formulas to do the linear algebra calculations they need for their non-math domain. This OP wanted a course for the latter need and this course failed at this goal.
For a random example I heard recently; there's applications of differential equations in pharmacokinetics, so people studying pharmacokinetics need a sufficient math background for that. However, the appropriate math background for these students involves the general concept of differential equations and being able to understand a couple specific equations which characterize their physical domain, and do calculations involving them. For that use case there is no need for a skill in "doing math" - all the mathematical properties of those particular equations are known and they can and should look them up instead of learning how to derive them from first principles, much less making any new assertions.
> Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work; it isn't mathematics.
You're right, but you need to know the basic mechanics pretty well in order to start to understand the deeper philosophies and ways of thinking that math _is_.
It's like learning to ride a motorcycle without ever having ridden a bike; Riding a bike is not riding a motorcycle, but you're going to really struggle to make progress past a certain point of learning since you're mastering the absolute basics of _both_ things at the _same_ time.
It's the same thing with math. If I never see what well-formed answers to problems look like, never gain familiarity with the symbology or jargon, then I can never start to even learn proper math because the professor can't communicate with me.
Attempting to learn mathematics as some sort of pattern matching... does not work
China (and other countries who put American math education to shame) disagrees.
For you, math may have been a deduction from first principles, but to us mere mortals, math is a decade-long slow reveal through wax-on-wax-off repetition.
> Unfortunately it seems like no one has been willing to teach the author what mathematics is. Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work; it isn't mathematics.
That is exactly how I got through university math so I think you are wrong. I worked on what felt like millions of problems the teachers gave out and then on some more I found in books. Then I looked on the answers and tried to figure out what my errors were. Then I worked on more problems. What didn't work for me was looking at lecturers writing out proofs for various theorems on the blackboard. I had almost zero use for that.
> Attempting to learn mathematics as some sort of pattern matching, which is where having large sets of questions and answers to study is useful, does not work
This is how mathematics works for most students.
They get a set of problems, then a (poorly explained) set of tools and they try to apply those tools to solve the problems during tests. Then they leave school and never use those tools again - they often completely forget about them. In Eastern Europe they even call it the "memorize->get drunk->forget" cycle.
Arguably, this is the experience of most people, in most fields of study. This pattern matching does not only happen in maths. Most Bachelor's and Master's thesis are basically a compilation of quotes from various books and papers - without any new ideas, or original research. Maybe I am cynical, but in my opinion you are discouraged to make any original research even while getting a PHD as well. Your thesis is more about quoting the "right people" (= quoting papers of the people who will be promoting you), than doing anything new, or original.
But coming back to the question of actual education: the article clearly shows the experience of most people at Universities. If they want to 'learn for the sake of learning' you will get very discouraged fast, your experience will be very miserable - the university is not there to broaden your horizons, it is just a combination of a testing facility and a diploma mill.
What is even sadder is that if you are laser focused to learn something that will be useful at work.. then you will get disappointed as well. The things taught at universities are often not even applicable at academia, not to mention "real" work. Writing a master's thesis teaches you how to write a master's thesis (assuming someone even properly teaches you how to do that, for me they just told us to write one, without even bothering to show few other ones as examples - I had to get those myself). It still teaches you something, but that something is not good enough for real academia or real work.
In addition, since lots of lectures are graded on basis of tests, universities often feel like a glorified highschool. And you know what? Probably we cannot have it better: most people wont make original research - because it is hard, so universities are stuck with the tests and papers (that are a collection of unoriginal quotes). Also you need to learn to walk before you run, how are you supposed to write a good original thesis, if you didnt write a single non-original thesis, what is much easier?
The main reason why the blog's author was disappointed is the systematic problem of rewards for university professors. They arent really rewarded for doing more than the minimum. They have their own rat race, with their own reward structure, that rewards "research", understood mostly are publishing quotable papers, sometimes getting grants. Changing the reward structure would help a lot - those who focus on teaching, should be rewarded better. Universities supposedly are for students, but most of the time it feels like students are an unwelcome afterthought for the research professors.
Also, professors seem to like to say that "if you want to learn a trade, go to college - university is there to broaden your horizons" - what is obvious bullshit, college seems to be the same as university, but just with a worse diploma at the end. [as a thought experiment: would the professors teach better, if their pay would be related to the salaries achieved by their students - ignoring GDPR, PII for a moment? if the professors could get a "share" of their students incomes, would they teach better? now in USA the students are obliged to pay their tuition with 0 impact on the quality of the received education, or in Europe receive free education]
Then we have those "good universities". USA has this corrupted system of admissions (probably so that students with best results dont t...
I wonder if some universities should pay more attention to the impact of discouragement on some students.
I remember attempting a grad class called "Hard problems in combinatorial optimization". I was struggling so I asked the professor for advice on getting my footing in the class.
His answer was that I may just not have the "mathematical maturity" for the material. I was so discouraged that I dropped out of the program.
What I didn't realize at the time was the particular meaning of that term. His advice was well intended and accurate.
But what I heard was that I was hopeless in this topic area because I wasn't smart enough.
So much pain could have been avoided if we just extended the conversation by a few minutes, or he invited me for a discussion over coffee.
You think Mathematicians in general have people skills? Most Mathematicians i met were good people but they definitely kinda lived in their own world, they were super helpful, but only if you explicitly asked for their help.
People skills? The informal prereqs/expected background for a class should be listed in the syllabus. Lack of "mathematical maturity" is a non-answer, they should make it clear what sort of math knowledge that refers to.
Maybe the class did have pre-req? I mean i think it is obvious that a class called "Hard problems in combinatorial optimization" is aimed more at future Mathematicians/Computer Scientists than people who want to take it for the fun of it.
I disagree. 'Mathematical maturity', to me, means:
There is no specific advanced prerequisite knowledge, but you need to be able to follow something technical.
As an example: you'd reasonably expect a student with mathematical maturity to know what a set, the Cartesian product of two sets, and a function are. That's high school level math.
From there you can define a ring and a module over a ring, in like 5 mins.
So at that point the Prof, strictly speaking, told you everything you need to know to start studying modules.
Of course, many students will be like whaaaaaa?! When a ring is defined. But here's the point: Mathematical maturity does not mean that you already know what a ring is, but when the Prof defines a ring, you are following along.
Anyway, my point is that there is nothing specific that you need to know.
Mathematical maturity should be well defined if you are using it as a prerequisite. Mathematicians, of all people, should be good at coming up with such precise definitions.
Some people are better with math than others. Undergrad math majors thought the major was pretty easy. Many of us in engineering majors--who weren't that bad at math (it was a good school)--could no more have graduated with a degree in math (or physics) than have flapped our arms and flown.
The issue today, IMHO, is that every university is looking for professors who are "good at research" (meaning, they can win grants), and teaching is not even a secondary or tertiary thought (if a thought at all). The last four professors we hired in our STEM department could barely speak English (I am in the states), let alone get points across or create examples. You know what they can do well? Write grants that get awarded.
Research and teaching are both important, and a smart university tries to use its professors' talents most effectively.
I studied at the VU, where Andy Tanenbaum, writer of a ton of famous CS books teaches. Of course I followed his classes. His books are excellent, but his classes are basically him reciting his books from memory. The jokes are literally identical.
At some point, more and more of his classes would be taught by Maarten van Steen instead; he's an excellent teacher who really knows how to engage the students and make them think about problems, instead of merely telling them the answer.
There was another professor who, as far as I know, didn't do any research at all; he just taught a ton of classes.
A professor who is great at one thing is not necessarily good at something else.
I felt this in Spain. In my little local university it seemed like they were gatekeepers. Some professors were hostile and treated you like if you were dumb.
It didn't help that I went to uni at 24 IIRC while I was working my ass off, so I wasn't that happy to put up with the BS.
It seems less like gatekeeping and more like "I'm struggling for my life to climb this mountain on my own, and I'm not strong enough to drag up your dead weight up behind me. I'll train you if you seem to have the aptitude and drive to eventually become a useful partner in my struggle, but I'm not going to waste my life energy on people who will never be anything more than dead weight."
Calling it gatekeeping seems like a fish complaining that the birds are gatekeeping the sky. The sky is open for anyone, but only if you have wings to fly.
I obviously don't know your specific situation, but often this is the result of idiotic cost cutting measures by universities.
As a lecturer, I've had several job interviews where half of the questions were some variety of "are you ok to put up with having way too many students and not enough time?". I had the luxury of not having to take these jobs, but obviously someone ended up taking them.
Maybe they should teach self esteem and resiliency in the face of criticism and failure, because you will definitely be criticized and you'll definitely fail. You will face setbacks and discouragement at many points in your life. More often than not, your biggest critic will be yourself. That much is certain.
This wasn't even really intentional discouragement, but it was perceived as such. Hopefully, you learned the higher lesson.
You can have plenty of self esteem and resiliency and also decide based on evidence that an activity may not be the best use of your time. If you respect a teacher's evaluation of your performance in a subject and the teacher's evaluation is that you don't have the maturity to learn from them, it may make much more sense to try other things than insist on forcing them to teach you anyways. People don't have to intend to say cruel things in order to be cruel-- for example, you don't have to intend to hurt someone who recently miscarried by talking about your successful pregnancy and its definitely not considered the fault of the person who is grieving. Similarly a professor doesn't have to intend to be discouraging in order to be discouraging.
I think we can reasonably argue that the statement is overdetermined. Charitably, perhaps the professor did mean mathematical maturity in the sense of "try learning some background first, and give yourself the time to let your skills and intuition develop." Uncharitably, perhaps the professor did not, and meant mathematical maturity in the sense of, "you have an inherent deficiency that makes this an inappropriate undertaking for you."
I think where the fruit of this discussion may be is deciding to what degree an authority figure like a professor has a responsibility or an obligation to their students and advisees to make themselves clear which way they intend their statements to be interpreted. Personally, this begets a discussion of the nuances between the obligations and responsibilities of communication between (1) strangers, (2) peers/colleagues, and (3) authority figures and their charges.
How did you interpret “immature” as “not smart enough”? A baby is not mature, which doesn’t mean he won’t grow up. “Immaturity” means that there’s room for improvement and that that improvement is attainable.
What I mean to say is that this issue stems from your misunderstanding of the meaning of the word “mature” and nothing else.
It's been a long time, but IIRC I recognized that my understanding of the term was vague but that all interpretations boiled down to, if I wasn't ready now I never would be.
I'm not claiming that my response was entirely rational or measured. I'm not blaming the professor, and I'm not justifying my own reaction. I was just saying what my reaction was, in case it shows a pattern of student experiences that's worth addressing.
If you have reasonable people skills, you may well have correctly inferred what the lecturer was communicating: overly questioning the words can be damaging. As I have matured, I have learnt better to trust my intuitive reading of what someone says, especially when I detect that the meaning of their words is divergent from what I believe they think beneath the surface.
Stubbornness, tenacity and self-belief are useful skills for success - perhaps you gave up too quickly? Unfortunately, soft skills are very rarely taught well. We all rely upon our subconscious learning and a good amount of luck to gain necessary skills, and those skills are often learnt by osmosis while very young. I also think the meta-skill of being able to teach yourself soft skills is not common either and it - although we can make an effort to improve that meta-skill even though it is somewhat self-referential.
Assuming OP was an undergrad, it’s totally fair to assume that the prof in question was trying to euphemistically say that they weren’t good enough for the class, especially since “mathematical maturity” is something of a term of art in the field. I don’t think it’s worth putting down teenagers for not understanding the nuances of every single bit of terminology in a field they’re dipping their toes into, the expectation should be that professors be more empathetic to that.
"mathematical maturity" would be a combination of "not smart enough" and "haven't studied enough". Highly talented people can reach high levels of maturity very quickly, people who are just a bit above average can reach a reasonable maturity through hard study, and below average talent means it is very unlikely that the person will be able to comprehend university level theoretical math.
A teacher who doesn't know you will not know if you're not smart enough or just didnt study hard enough, and probably doesn't care much, especially for an undergrad. For a phd student, they may care if they think they've identified a "diamond in the rough" that can be turned into a gem with some work, but that is quite a bit beyond basic linear algebra.
> “mathematical maturity” is something of a term of art in the field
Fascinating, it sounds like "MM" is tacit knowledge, which begs the question why it isn't explicitly taught.
In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right.
That sounds like either an absolutely terrible teacher, or a terrible program. Why were so many people enrolled in a class that less than 25% would pass? I'm not American so I don't really understand the course names but I guess a class just called "Algebra" shouldn't be too hard? Stuff like basic groups/rings/fields or is it a more advanced class?
The title "College algebra" in the US indicates material that is well pre-groups/rings/etc.; it is about the basic skills of computations involving manipulation of variables, often involving such topics as solving linear and quadratic equations, and perhaps inequalities.
At least at my university, the reason that such classes have a high failure rate is that the university is incentivized to have high enrollment numbers by admitting students who do not have the background to succeed at college mathematics at the level of calculus or above, and so need some pre-calculus courses; but someone who does not have a solid grounding in pre-calculus mathematics will often struggle to learn it when it is taught at a rapid pace at the college level—especially by professors most of whose teaching experience is with college-level math.
But what incentivises the students to take a class that so many won't pass? I don't really understand the American university system but surely the sensible choice would be to take a course you're most likely to pass?
I guess its possible to learn stuff from a course you fail, but you should be able to find a course that you're likely to pass and learn something from?
It's mandatory for the degree, that's the motivation. It's typically required for all students who didn't score high enough on entrance exams or take enough math courses in high school. They also likely don't talk to their advisor and say, "I want to take something else, like prob/stat or calculus, to substitute for this requirement." STEM majors will find it easier to substitute another course than non-STEM majors (because they likely have the background to skip it from HS course work). Engineering universities and the like usually send their first year students straight to calculus, though.
I've never heard of so many failing a college algebra course, at my second university it was around 25% failing and mostly because they'd relaxed the entry requirements in order to increase enrollment.
However, for non-STEM and non-Business majors college algebra is rarely a prerequisite for any other courses so you can still continue your major without it, you do need to pass a math course before graduating though. They also have other math courses that can be used that aren't college algebra and may be easier.
At my university, CS was called math-CS, it required intenses calculus courses with a similar failure rate.
Of those who failed : ¼ retook the course, ¼ pivoted to the bussines-CS program (given by the CS department) , ¼ pivoted to bussines-IT (given by the business and management school) and as far as I know, the rest disappeared from the university.
It's a "weed out" class. When a huge number of people want to be STEM, but there is no way they're going to last, it's better to have a tough class that sets that expectation early, so they have time to pursue a more achievable major. In my case, huge State university, Freshman year Physics was the weed-out class. It was not advanced or (to me) particularly difficult, but there was a LOT of content, we moved very fast, and the workload was punishing. If I recall correctly, our first class was over 1000 people in a huge auditorium forum. By the time we took final exams, we were down to around 100 people.
My high school AP calc course started out somehow getting behind in the first week. After that, the teacher kept assigning homework for material we wouldn't even cover until the day it was already due to be turned in. It was pretty brutal for someone like me who was accustomed to coasting through all of my other courses.
I took a bit more math in college. It wasn't easier; the professor simply kept pace with the material. If you needed help, you had to go to a TA or the professor out of class hours (an option that wasn't really available for a high school setting).
I can easily imagine that many students who were exposed to college level math for the first time in one of those courses would flounder and drop out (though I didn't witness much of it myself).
Maybe I'm massively out of touch but I'm pretty sure a good teacher could teach that stuff to almost anyone.
Like this stuff isn't particularly complicated, I'm not a very good teacher but I did teach some undergrad seminars during my PhD and I think even my worst students were capable of understanding functions, graphs and trig.
That's not the issue. College algebra courses are super compressed, they're basically trying to teach the whole of junior high + high school math in a single semester. That's barely teaching, and the natural outcome of such a cursorily taught course is to act as a 'weedout' course.
Yup. My observation (having not had to take College Algebra myself, but knowing people who did) is that it's a class to make sure people who fell off the math train back at operations on fractions or factoring or whatever, don't get any farther.
I mean, what do you want them to do? Invest resources in splitting up material into multiple courses when they should have learned it before even coming to college? It's not their fault public schools suck.
All public schools do not suck but especially urban schools can have a lot of problems and many of the students don't have a great home environment either,
State universities kind of do have this responsibility. My school is doing more or less this, restructuring some of the math sequences that have a high failure rate, like precalc/calc 1.
Also state universities are where most of the public school teachers and administrators trained, so they probably deserve a little of the blame for the state of public schools.
IMO the only stuff a student "should have learned" is stuff that they can't get onto the course without knowing. Allowing someone onto the course, for which "x" is a prerequisite, who doesn't know "x" is entirely the university's responsibility.
I was a tutor in business school on behalf of the MBA program. Basically, there was a small group of students who were... not prepared with respect to math. We're talking really simple algebra, maybe the simplest of differentiation (e.g. maxima of a simple quadratic function).
One of them told me one day "I don't understand graphs." I tried but it's impossible at that point to make up for an apparent total lack of at least high school level arithmetic/math.
You are in touch with your kind of people, the kind who graduate from university.
If you were doing a PhD you were at the kind of university that has a PhD programme. Probably it was one of the 200 or so US universities out of a thousand that are selective, rejecting more students than they accept. I doubt there was a single student in your seminar who wasn’t in the top 30% by academic aptitude of their age group.
> According to the U.S. Department of Education, 54% of adults in the United States have prose literacy below the 6th-grade level.
> His answer was that I may just not have the "mathematical maturity" for the material. I was so discouraged that I dropped out of the program.
There's a problem with making a statement like that and not following it up with what you need to do gain the needed "maturity". Even adding "maybe you should take this course over here and come back to this one in a semester..." could make a world of difference.
I think usually it boils down to teachers in advanced schools were the kind of pupils who got most things easily and have no clue how to explain things more than <list of set comprehension>. Got me to think that these universities are more magnets from strong ex-pupils to similar minded new pupils that will fit in without much pedagogical efforts.
The average IQ of math majors is absurdly high, more than two sigma above the mean[1]. In an organization like that the smartest person is likely going to be a breathtaking five sigma above the general population mean[2]. I get the impression that people in math departments like it that way, so weeding out persons of more ordinary intelligence is more of a feature than a bug. I doubt many would admit to that, but it fits the observables.
[2] Assuming gaussian distribution with no skew, which is probably a bad assumption, but I'm spitballing here. Suffice to say math departments at elite colleges have many extremely bright people.
IQ has no skew by definition. That said, I think there are a number of IQ boosting mutations and innate mathematical aptitude is likely correlated to possession of one or more of these rare(ish) mutations.
The more selection criteria depends on cut-off (which it is in the university setting), the more likely selection from normal distribution is going to be skewed. Then Fields medalists are selected from long tail of already skewed population, all bets are off.
Let's say every mathematician draws an IQ from the IQ distribution and a luck score from the luck distribution. Then, the highest sum of numbers wins a Fields medal.
That would be distributed according to the distribution of the maximum of N Gaussian-distributed random variables, and according to this[0] stack overflow answer, it's skewed downwards, below its mean. (Its mean, of course, is much higher than the mean IQ+Luck).
I misread the point they were trying to make, I took the 5 std to be a sign they were pulling from general population IQ. On second reading it’s clear that is not their intended point.
In any case the resulting intermediate distribution from selection criteria bias would be a Chi distribution (I think - it has been a while.) 5 standard deviations on a normal distribution is 175 IQ or top 1/700K which is insane.
A few additional biases, Physics majors have higher average IQs, and not everyone smart enough to do the math has the desire or the opportunity.
The point I was making is that IQ score has been normalized, when imagining the intermediate distribution of IQ it would make more sense to look at composite probabilities of factors that led to the IQ score and consider the probability that those same factors lead to a sufficient innate ability in math.
I wouldn't say feature, but they're probably not naturally inclined toward spending more time raising the average level instead of enjoying the few genius newcomers.
There's a small sadness in this but I can't word it out perfectly.
ps: I mean I would understand very much that these people just want "new buddies to play hyperstimulating math games with no drag whatsoever" but from school I expect just a few pointers. The rest is on me (us).
I think this happens a lot. For some reason people have a really strong tendency with math to attribute it to innate talent. Of course talent matters, but people get hung up on it and discount other factors. The word "maturity" is supposed to help indicate that it's a matter of having and digesting the right experiences, but it honestly isn't great, because maturity can be driven by experience, but it can also be dictated by biology. If someone is isolated from human contact between the ages of ten and twenty, they will emerge fully mature in some ways and immature in others. We need a less ambiguous way of saying that somebody hasn't yet had the exposure and practice that will prepare them to tackle certain material.
I thought about this a lot. In college I often failed and teacher would look at me dumbfounded (followed by the traditional "it's trivial"). Very discouraging indeed. Years later I revisit some things, and without struggling too much, I get insights and ideas and overall easier time actually solving things. So in the end it does look like some "maturity" was lacking. Now it's bothersome that schools can only operate on people with magic maturity and other people are left on the side (the school as semi-comb theory) and I wish we could have some pedagogical light on what is that missing maturity and how to grow it explicitely.
I’m not sure that college students - or, to call them by their age group, teenagers - are really appropriately assumed to be “adults” at all times. Obviously we don’t think that when it comes to alcohol and partying - universities step in and make heavy-handed decisions we’d never apply to full adults all the time (example, shutting down a house & evicting residents for having a party where an assault happened). So why not apply that same “not yet fully mature adults” logic when it comes to something with massive social benefits, which society has in interest in promoting: more mathematically literate university graduates?
> [...] may just not have the "mathematical maturity" for the material.
LOTS of non-math-major science/engineering students face that problem as soon as they start grad school.
It's a consequence, IMHO, of math curriculums failing to focus on the fundamentals rigorously enough. There's too much focus on what we used to call "plug-n-chug" mathematics where students learn just enough to apply formulae to get their problems solved. It might seem like "just enough" but it ends up pushing out higher mathematics and keeps students from being able to apply more advanced mathematical concepts that they would have gained from a much deeper dive into Real Analysis, abstract algebra, differential geometry, etc, and the grind of the theorem-proof cycles. Many of us felt this right away upon taking graduate level courses.
I wish that I had been counseled to take additional math courses before (or even at the beginning) of grad school. It would have reduced my suffering A LOT and I would have gotten more out of the degree.
I thought I had it - lots of real analysis and abstract algebra proofs - but then in grad school I hit a 400 level numerical methods class that I simply lacked the mathematical sophistication for. Just completely at sea - an experience which I lacked the academic maturity to deal with.
> What I didn't realize at the time was the particular meaning of that term. His advice was well intended and accurate.
I had a similar experience - and while it was a blow to my confidence I’m happy that he didn’t tiptoe around it with feel-goodisms. What I was doing as a student, and what had ben successful so far, was not working and doing more of it or doing it harder wasn’t going to help.
Honestly leaving that program was the right thing to do, I probably shouldn’t have been in it, though I’m confident a few years later I would have been successful, but I lacked the maturity and mathematical sophistication to get it then.
I wonder if the whole "weeding out" theory extends to college education in general, and like you ask, how constructive to society that is.
I, very briefly and quite honestly, attended a lower tier state school that was an absolute joke. Shortly after I started, there was an outreach program for local high school kids where if they showed a high school ID, they got admission for the upcoming year. The head of my dept dismissed it and quipped, "It's ok. Anyone that wasn't meant to be here will just drop out after a year anyway and we have their money."
I, thankfully, ended up at a much bigger P5, well-known university. One of the controversies was the engineering departments were trying to get as many students as possible to the point of exceeding infrastructure and admitting lower quality students into the program. Upperclassmen I talk to said, "that's why everyone hates <1st semester math for engineers> and <2nd semester math for engineers>. They immediately weed out those that don't qualify."
"Weeding out" is what college admissions should be doing, not the departments. When the departments weed someone out, they've wasted at least a year of their life and a year in tuition, and demoralized them.
This is what really the "college for everyone" mentality has led to. If the kid shouldn't be there, or should have started in junior college first, we need to be more upfront about that earlier.
>"Weeding out" is what college admissions should be doing, not the departments. When the departments weed someone out, they've wasted at least a year of their life and a year in tuition, and demoralized them.
Admissions can't weed people out, they can only copy the weeding that highschool has done. If you rephrases the question that way, as in, "should highschool teachers be responsible for college admissions," then the idea of letting people join a major they haven't proven themselves capable of doesn't sound so bad after all.
I disagree with that. Even if someone fails all their classes in their first year the experience will teach them much more about life than any job would. The real problem is that US universities cost a lot of money so the cost of failure is insane.
This is a really deep point. A math professor at The University of Chicago wisely pointed to me as this being a super important thing to keep in mind when teaching (I've gone into secondary education). Often the material you are and how well the student is receiving has a much smaller impact on the student than what the emotional experience of it communicates to them about their own capabilities, including their chance at continue to struggle through the material in the hopes of fully grasping it. Especially in the latter years in school, it seems you find a lot of teachers who almost revel in making their courses "impossible" for kids, which can sometimes be very motivational for extremely high-achievers but absolutely devastating to a well-intentioned student who is having a challenge with the material.
Another UChicago student here (former math major). I think the UChicago math department took education seriously in a unique way that I wish they got more credit for; the Inquiry-Based Learning classes, the summer classes for Chicago public school teachers, and the Research Experience for Undergraduates were all pretty special, I think.
Also, the math classes all had a shared policy of "you can work with anyone you like on a problem set, as long as everyone's name is listed on there when you hand it in." I loved that, and I did all my problem sets in groups while I was there, and I think I learned more because we were always explaining things to each other and arguing about solutions. I heard a rumor at one point that this policy was the direct product of Peter May's bad experience at Princeton, which required students to work alone when he was there.
TFA does seem like a pretty stinging indictment of Princeton's math department.
Maybe they are interested only in future Ph.Ds and Professors. Anybody else is a burden to be sent somewhere else as soon as possible. Maybe they think they're doing a favor to those students because they won't be wasting their time on subjects that are too difficult for them to grasp in the short time of a university class.
> His answer was that I may just not have the "mathematical maturity" for the material.
What a clueless, absolute piece of shit..
Clearly he lacks the emotional maturity to deal with students (or just people in general) and unequivocally should be fired as a teacher and restricted to doing research.
Sounds like she would have enjoyed a stats class more. Applied math without as much emphasis on first principles and formal proofs. I didn’t like university level math (first year lvl) because I preferred programming and doing math without computer assistance seemed to involve a lot of wasted effort. Maybe I’ll revisit it with Lean.
What I've found, is that often teachers (really anybody who's understood the subject and is willing to teach) will try to teach both the subject AND how they have learned; i.e. they do not only tell you about the math but also the way in which they have arrived at understanding it (how they've related to it).
to use an analogy, when describing the math they tell one about a place where they are (a 'place' where the math is there, already understood) and how they got there (the analogue of route they took to get there e.g. "took public transport line 4, get off at station Example and walk until you're there").
This leaves little room for those who may prefer to ride a bike all the way there (the analogy has been stretched to the breaking point), cuz the teacher will knock you off the bike and insist that you ride the line 4 then walk, same as they did.
For most maths, at least the stuff you'd learn before or during an undergrad degree the interesting maths is the journey. The teacher is attempting to teach you to ride a bike, and the student is like "this is stupid, I already walked to the end of the road".
The teaching of different ways of making the journey is (often) the destination.
According to the author, the course is "for underclassmen who are majoring in engineering or sciences", but she is a humanities student who wants to "explore STEM fields."
Isn't it obvious that the course is not designed for her learning purpose?
"A humanities student"? She's a humanities student who had taken up through linear algebra! I have a CS degree and I never learned linear algebra in high school or college--I learned about the topic later!
The idea that a 200-level course should be unapproachable for an interested person with a stronger mathematics background than I have, with an ostensibly mathematics-based degree, is absolutely silly. I took 300 and 400-level classes in economics, in English, in history and philosophy. I never felt unwelcome or incapable.
> I have a CS degree and I never learned linear algebra in high school or college
Really? I mean, you probably did have linear algebra in high school, it was just "disguised" (you may not have seen matrices and vectors, but probably did solve linear equations of multiple variables). I'm surprised about the college thing, though. I thought every CS degree (in the US at least) required at least through linear algebra (linear algebra with applications, probably not linear algebra with proofs and theory).
I'm assuming the issue is that there isn't a math pathway for humanities students who want to take math electives. This is the closest thing she could find. This was the case at the Ivy I attended. There were classes designed to fill the one-course math requirement, but they were all below the level of the math I'd done in high school. There wasn't anything for someone who was decently good at math and genuinely interested in it, but not planning on a math-related major.
I’d have more sympathy if this were someone from the inner city or rural America. But they’re in Princeton crying about not getting answers to problem sets. As if the first time they’re not spoon fed something, it’s time to trash the department. If you can afford Princeton, buy a linear algebra book with solutions. It’s as if a Math major said, “Oh I would love to read Hemingway but there is nobody in the English department to tell me what questions are on the test. How can a Math major learn to read? Where in the world might I find a clue about Hemingway?”
Yes - math education can be awful, but this snowflake needs to lose the entitlement.
I'm shocked with how inarticulate the author is. I'm not getting any idea of what math class they are taking or what they expected to get about it. You'd expect a humanities student to be able to explain themselves better.
The one well-developed complaint is the one about not getting the answers. On one hand that sounds almost calculated to provoke the reaction that you had, but on the other hand for every math class I've taken I turned in the problem sets and had them get graded with corrections that were useful.
Honestly I read the whole article and didn't pick up on which field of math either despite it being clearly stated several times. I think my brain couldn't accept the idea that someone was complaining publically about Linear Algebra being difficult.
Princeton is 100% free if your parental income is below $65,000; you only have to pay for room and board (not tuition) if parental income is below $160,000.
Regardless, you're not paying for instruction. You're paying for networking, access to world-class experts in your field of study, a name-brand, etc.
If you read the article they say that they did achieve success in the class, but the experience could have been much better/easier. Only if you believe that an ivy league institution is there to provide a kind of S&M experience for its students does your comment ring true.
So the author took one class that was unnecessarily difficult. That’s unfortunate and worth following up with the Math department on. But expanding that into a full-length article with the thesis of “Why won’t anyone teach me math?” seems like a stretch.
I don't think she had an issue with the difficulty of the course, but rather the pedagogical approach. It seems she walked away feeling like she didn't actually learn very much, despite passing the course.
The article is from the Princeton university student newspaper, so I assume the author meant "Why won't anyone teach me math at Princeton?". It's specifically about the experience of liberal arts majors at Princeton, not some general thing.
It seems like the issue was more that the class was not only bad for her but poorly structured in general - it would be a good thing for non math majors to learn topics like linear algebra but the department could set up classes differently to better teach them.
I've been terrible at math in hs and then I started doing programming and realized that math ain't hard and actually liked doing it
I managed to improve and then I had math at higher edu institution
and oh boi, I instantly had no interest in doing it 4fun, just learn, pass and move on.
idk, I feel like there's too much stuff to be done (a lot of courses + you gotta learn other stuff like "real world" computer's related stuff) and there's lack of time for 4lulz messing with fancy topics like maths, but on the other hand I've been working + studying, so maybe that's why
There isn't any supporting material in the article (the links to course review comments are behind an authentication wall) so it's hard to be sure, but it sounds like the answer is "because you had abusive teachers". By that I mean psychologically abusive people who derive pleasure from seeing students struggle in pain. It makes them feel better about themselves.
That isn’t what the article says. It says the teacher’s hands were tied because of department policy. Why does the department have that policy? What set of circumstances led to that policy being set and what would be a better policy to both address the set of circumstances and to not hinder teachers?
A class doesn't become bad enough to write an op-ed about for that alone. There must've been multiple other issues for this class to have a 2.7 rating.
Seems that is exactly what the article says. A professor that can get a job teaching undergrad math at Princeton can likely get the same job at any other university. So this professor is actively choosing to enforce this policy (it's their revealed preference).
No, those aren't abusive teachers. That's the standard approach to teaching STEM subjects in most of the worldwide. I'd say the abusive part is the often-unspoken expectation that students are going to sneak around department policies to find ways to check their work other than handing it in.
Sadly, in my experience, these was the case in introductory STEM courses. They could care less whether you understood or not. There were plenty of people in line behind you, so your personal experience didn't matter.
This is a big reason why I encourage people to take as much STEM as possible at community college instead of public university. Faculty are focused on teaching, not research and publishing. You have a better chance of actually learning something.
Reading the article am I the only one who had difficulty understanding what exactly the subject matter of MAT 202 was? She's indicated that she's taken courses up to linear algebra and then states that "she wanted to learn linear algebra". Is this a linear algebra course?
I mean, she said she had taken math classes "up to linear algebra" and she took the course to learn linear algebra so I would guess that it was linear algebra. Google confirms: https://www.math.princeton.edu/undergraduate/placement/MAT20... - Linear Algebra with Applications.
Thanks. Maybe I'm being pedantic but leading off with "so I selected MAT 202 - Linear Algebra with Applications" would have provided the article with some clarity.
I believe there is a major misalignment of goals/incentives in undergraduate academia. As students we think the model is, we pay [quite a lot of] money to be taught useful knowledge in a rigorous way that means we will leave with real knowledge. Universities, on the other hand, seem to think the model is to churn out well qualified individuals in their field. There are two ways to accomplish that goal: doing the hard work of providing a real education to everyone; or doing the much easier work of filtering out those who don’t already "get it". This is why Universities don’t care that a class is reviled by all the students.
In my experience undergrads are almost universally seen as a nuisance by professors, especially at the most prestigious schools where professors see themselves as godly researchers. It also sounds like this class isn't even for students in the department, which usually means no one wants to teach it. Basically becoming a professor has almost nothing to do with ability to teach, so I'm not surprised when the least desirable classes to teach are taught by people who have next to 0 ability to do so.
> doing the hard work of providing a real education to everyone; or doing the much easier work of filtering out those who don’t already "get it"
I wonder how much of a problem this is in intro CS. Everyone knows that the camel has two humps, but are we inadvertently gatekeeping by not even trying to teach the low-scoring "hump" about the fundamentals of the discipline?
I thought it was common knowledge that this is the problem with intro CS. There are places that get much better results by making sure that either freshmen test out of the intro class or they all know equal amounts of nothing when they take it.
In college they did the same thing, so I pirated the book the teacher was using for the class so I could get the answers to the questions we were being assigned. It helped me learn the material with multiple examples and solutions. I think it should be required to give the solutions to homework; especially, with classes like differential equations and vector calculus.
>For example, though we were provided with practice problems to prepare for our exams, we were never given solutions.
Strange. But it also may no longer be true, or not to the extent that it was when this student was enrolled in the course. From the course website:
"Try some old quiz problems for this course, but don’t just read the questions and solutions. Instead see if you can produce correct solutions to most of the problems in the allotted time."https://www.math.princeton.edu/undergraduate/placement/MAT20...
So clearly, now, there are archives of problems w/ solutions that can be reviewed after attempting a solution.
Apart from lack of solutions, I don't see anything "off" here. I don't see other criticisms of the teaching approach. I am also not surprised that one of the best math & physics universities in the world does not cater even its lower level courses towards students for whom they are optional and not part of their intended academic career.
In my school (which was not a top-tier program) courses for majors in those areas were also similarly difficult. The difference was that my Uni also offered a few "Math for non STEM majors" courses to choose from because they still did have requirements for all students in those areas. Basically a survey course of mathematical concepts and how they show themselves in the world & everyday life, along with practical applications of how they may need to use math in any way throughout their lives, things like how basic level probability & statistics will be useful no matter what they end up doing in life. And things like fractals, or the golden ratio, and other concepts along those lines that show up everywhere in life.
The course was designed to answer the question, with concrete examples, "Why should I care about math if I'm barely going to have to use it in life & have a calculator?" I think Princeton simply expects their students to understand the answer to that question without putting their students through the trouble of an entire semester on the topic.*
*Though clearly should address the "no solutions" issue if it has not been fully resolved already.
My entire undergrad schedule was arranged according to getting the best-ranking math teacher on ratemyprofessor.com. I had enough painful experience and had to early drop a math class or two to learn this fact. Bad math teachers essentially forced me to self-teach, and I did not have the ability to do that with math, as I wasn't a natural in math beforehand. It got so bad that I happily took 8am classes despite being a night owl.
What it looks like now as an adult is just a failure of pedagogy and institutional incentives. I suspect the people teaching it had little reason to take the teaching part seriously, it was just something that got in the way of their research/grad studies.
I went to a well-known state school. There were other departments like this.
> This struggle was reflected in our exam averages, which were, respectively, in the 50s, the 60s, and the 30s.
Is there some prior knowledge about university exams I should have in order to understand what these figures relate to? Standardized set of 3 exams maybe?
The grades are out of 100%, so these averages mean that the professor has to z-score the grades in order for any significant portion of their students to pass at all (the pass line in the USA is usually 65%). I'd consider this a strong sign of bad teaching.
This is a very vague article. The only fact I see is not providing solution to practice problems. That's not ideal but shouldn't be a dealbreaker. More details might be helpful.
I do find it surprising that the whole class could have such a low grade in linear algebra, which is one of the easier introductory math classes. But I've heard of grade deflation in Princeton is pretty severe, so I don't know what standard grades are there. IIRC Princeton also has one of the premier math departments in the entire world, so I would expect the program to be very difficult.
On the other hand, I think the interest of tenured professors in math at universities is almost entirely on research. This doesn't mean they can't be good teachers but the ones who are I think in general are thinking more of cultivating potential phd students and the like. A practical class for non math students I can easily believe would be completely blown off.
Well as a physics grad who struggled with first year LA and a parent of a current university student I can fill in some blanks. Linear Algebra specifically is a superpower these days and I was very excited to do the best I could to help by daughter get past the barriers I experienced and learn it well.
I failed. I found her course to be not only sink or swim, but actively hostile to student learning. As mentioned in another post, assignments and quizzes were returned weeks late with little helpful coaching. Assignments very often worked with higher levels of abstraction than presented in the lectures, for example the first time she saw a matrix of functions was on a homework assignment.
Furthermore there seemed to be no additional help for Linear Algebra. To my great surprise the math department had no facility for matching students with tutors. The university math help center was almost exclusively oriented towards calculus and had no resources for Linear Algebra. After weeks of searching online we finally got a call back from a third party tutor, only to be informed that this person (a grad student in math) couldn't help us because the assignments were "too specific to the particular course."
The _only_ place I've ever seen someone show any creativity and ingenuity in teaching linear algebra is 3Blue1Brown, but obviously even though he's got a whole linear algebra series it's not a complete university course, nor is it meant to be.
So yeah, it's a damn shame, and IMO inexcusable. Linear Algebra is the foundation of modern data science and machine learning. Math departments should consider it a sacred duty to bring as many students as possible to at least some level of comfort and familiarity, but instead they seem to treat it as an annoying distraction at best, or a weeder class at worst.
You're totally right that LA is a superpower! I wrote a book on Linear Algebra that might be of interest to you and your daughter in case y'all want to revisit these topics. In particular it covers a lots of applications.
I first learned from Lax's book which I felt was a bit too difficult for an undergraduate course. But in a graduate level class I learned from the lecture notes of Professors Sophie Marques and Fred Greenleaf which I was a huge fan of that you can find on this website:
It's a much higher level of abstraction than an introductory Linear Algebra course but I found them amazing for developing intuition. I believe they have also released them in textbook form which is probably more polished.
Don't most universities have TAs around, especially for lower year courses, to help students falling behind? I remember in my undergrad when we had difficult courses, half the class basically camped outside our TAs' office before each assignment deadline to ask for help or clarifications.
I agree. I'm a profesor in the first year of the University of Buenos Aires. We have no official list of answer for the exercises (but there are a few unofficial ones).
The students can ask me or the TA abut the exercises. We prefer to see what the student has attempted, and find the exact spot where the student made the mistake. It takes more time, but it's more helpful than a nice ideal solution.
We also may recommend to make another exercise from the official list and come back in 10 minutes with a solution attempt, or make some custom exercise on the spot that is about the same subject. (Exercises about derivatives are easy to invent, integrals and linear algebra is harder.)
It's difficult to know, because sometime the course have 100 students, and each day different students make questions to each TA. My guess is 10%-50%, but it depends a lot on the major the students are.
One of the useful trick is to write an optional homework exercise in the blackboard. I get a lot more of answers than just selecting an exercise from the official list. Sometimes it's just an exercise from the list with different number and sometime it has an intermediate step to make it easier. But writing it in the blackboard increase the chance the students will write it down.
Another trick is to write an old midterm and then go to each desk to talk to the students about what they are doing. We usually write the solution at the end of the class, and have some discussion about it, but the useful part is the discussion with each student (or small group of 2-3 students).
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EDIT: A response (and +1) to other comments: of course, there are many solutions to different problems on the internet.
But IMO, a big advantage of taking a class is to have shared context with a group of people, and thus interactive help (from other students, TAs, etc) working through specific problems together.
Being able to know if you were "right" seems to be an important part of this process. So having solutions to the shared problems you're all looking at seems important. (Withholding solutions seems like a method for instructors to reuse questions instead of having to write more every semester.)
Man, what a lame excuse.
As if there were not hundreds of books with solutions.
If it's from the professor's own problem sets then actually WTF.
Textbooks written "for teachers" are the bane of every college student's existence if you dare to actually learn from them yourself. "Only solutions to odd problems -- cool guess I have to look on the internet or never be able to validate whether I got them right."
I gave up on lecture, and taught myself the subject by doing problems out of the back of our chapters. Aced the class, changed my prof next semester, changed my major to the subject the semester after that, and graduated with honors.
To this day, I am dumbfounded by an approach to STEM education that would withhold a critical tool to iteratively learn via problem solving.
It would be like XP without writing tests. You ship your knowledge to the exam, and pray it doesn't break. Seems ridiculous.
> STEM education that would withhold a critical tool
How does it do that? Are students locked away? Is there a secret police that storms into dorms and burns all material that students try to learn from? Does the Dean come in to break up illegal learning groups?
That being said, I seem to recall that when I took math classes in college, we had books that had even questions that the solutions were in the back, and the odd questions did not (they were only provided to the teachers). The questions were largely the same, so if a student had an issue with an odd question, they were able to just go to the cooresponding even question and work it out. I felt that was a reasonable compromise.
We have a set of homework assignments to help the students learn the tools (kind of a "hello world" script for the project), and I asked if we just tell them we don't look at them and they are entirely for your benefit. The professor said you if you do that, there will be a decent amount of students who will just not do the assignments and later on complain that they have no idea how to use the tools.
The solution we came up with is to have a set assignment due date so students feel like they have to do them.
Personally, I am surprised that graduate students need that sort of motivation to do the work.
And really, shouldn’t this be how we approach most problems?
I know, I was referring to the homework.
And then give the answers after the homework was handed in, if they want to grade the homework. Or compromise, by giving answers to half the assignments, and not grading those.
Many years ago, my univerity's choice of calculus book also had the solutions to half the problems (evens or odds, doesn't really matter).
Department policy was also one of "we won't provide you the correct answers" even after an assignment had been turned in because the book was used for multiple years. The publisher had a new edition every year, but the university stuck with the same book because it was used for Calculus I, II, and III, which for most students was 3 or 4 semesters between starting I and finishing III, usually due to a scheduling conflict requiring a semester off between them, or because they had to repeat Calc II since the math department's selection of instructors was particularly bad for that course.
Those same (usually bad) instructors were all too happy to follow the department policy and not provide any feedback other than "correct" or "incorrect".
On the other hand, the university book store carried, and put on the shelf right next to the calculus book, the publisher's "teacher's solution guide", in two very reasonably priced volumes, which had the answers for the other half of the problem sets, as well as the step-by-step process for most of them, which was the valuable part, as you could see where you were erring.
Math department policy was that you weren't allowed to have those, either.
You can guess how well that policy was followed.
Those that put the effort in and learned the material did well. Those that just copied from the solutions book and turned it in did not.
If the person teaching a maths course needs to look the answers up, you are just fucked.
Some students are very good at taking exams, others, not so much. This can be due to a learning disability, stress over test taking, etc (or it could even be that person just is having a bad day!). Having homework and projects allows for students to have a different way of showing that they understand the coursework, and are able to apply the material.
It also gives the professor and TAs insight into the student. Why is a student doing so well on homeworks, and not the exam or project?
Some courses have it where if youre final grade on the exam is an A, you get an A (since it is a comphrensive knowledge base test of what you are expected to know of the material). But, let's say you don't do as well on the exam, you can have the homeworks average out the exam grade. Or you have a project, that can help equalize out the grades, because the application of the knowledge is important as well.
You're right, of course, but then it seems like the question is the following: which of the two should the class serve?
It's likely the point is to talk to the professor about the ones you're having trouble with, or where your study group disagrees.
However, I found out the hard way that also in respected universities, the actual assignments in "hard" courses were long-known, extremely challenging problems that were not really meant to be solved at this level, especially on a weekly basis.
Folks handing them in were often using solutions that were quite obviously inspired from online sources, if anything because of the weird methodology needed to solve some of them, without the course teams caring at all.
That is one of the most tragic things I can imagine reading. A student who really, genuinely wants to study Maths, just for its own sake... and is dissuaded from doing so by myopic, short-sighted, inane policies. It makes you wonder "could you fuck up education any worse, if you were intentionally trying?"
On the other hand though... I believe a student with enough self motivation can overcome those inane department policies and learn the Maths by pulling in outside resources. Yes, it's more work, and yes maybe it's not "fair" that you have to do that in addition to attending (and paying for!) a class. But it's possible.
Consider the availability of free resources like Khan Academy, Paul's Online Math Notes, hours and hours of Youtube videos from people like Professor Leonard, Gibert Strang, etc. One can also often find problem sets with solutions by just Googling around and finding previous years course websites for various courses that have been taught at universities all around the world.
There are also online forums where you can go to get your answers validated, or get additional explanations about Maths problems. They vary in the extent to which they accept "do my homework for me" style questions, but almost any site will acknowledge somebody who has put in some work, come up with an (possibly incorrect) answer and says "Can somebody help me understand what's going on here?"
Examples:
* https://www.reddit.com/r/learnmath/
* https://www.reddit.com/r/cheatatmathhomework/
* https://www.reddit.com/r/MathHelp/
* https://www.physicsforums.com/
* https://math.stackexchange.com (note: but not so much Mathoverflow, which is more for research level Maths and is not a good place for straight up "do my homework" style questions at all)
And if one is willing to pay a bit, the Schaum's Outlines books and similar books provide a huge catalog of worked problems, along with problems and solutions. There's also Brilliant.org and other paid educational resources for Maths that can supplement ones university course(s). One can also look into hiring a personal tutor as an option.
Many (most? all?) universities also have something like a "math lab" or "learning center" or something where students can go for individual tutoring and additional support. My experience (albeit dated now) leads me to believe that these are probably drastically under-utilized.
Should universities do a better job with introductory Maths courses? Almost certainly. But I'd encourage anybody dealing with this to dig in and try to overcomes such shortcomings by using other available resources as well.
EDIT: inspired by @skywardavocado's answer, I also wanted to add that old-fashioned "study groups" are another valuable arrow in the ole quiver. If you join up with 2, 3, 5, whatever, of your peers to study together, chances are that somebody in the group will understand the thing that the others are struggling with, and can explain it. I didn't do a lot of this in college myself, probably to my detriment. But to the extent that I did occasionally join a study group, I'd say they can be wildly helpful.
Of course you could. Take a look at K-12 schooling some time, where the prevailing educational theory is that "students must learn the math by themselves", and are expected to devise "their own methods" to do so, including "guessing" and doing calculations in their heads, not on paper. Is it any wonder that even "getting the right answer" has been de-emphasized, never mind "show your work"? This is what passes for math education these days, courtesy of "educators" who have never gotten a proper education at the college level in the actual subject.
In some ways I wish they'd just get on with it and leap straight to the logical conclusion of just officially eliminated the standards and passing all students guaranteed, so we can all get on with the task of dealing with the fact that such a credential would be worthless, instead of this long, drawn-out process of lowering standards while trying to pretend the standards aren't being lowered.
(I think the second derivative of this process has turned away from dumbing down. Pushback is really coming up in earnest. But it'll be a while before it so much as turns the first derivative back in the correct direction, let alone get to the point where the problem is largely fixed.)
Fair enough. That was a VERY long time ago for me, and I don't have children, so that world is pretty closed off to me.
As a university professor, given the availability of all these resources, I'm not sure why you'd want to take advantage of them and take a university course if you're not interested in the credentialling. Universities are a place to learn the deepest knowledge of content experts, and I don't want to downplay that; but these content experts are usually not pedagogical experts, and they often tend to be less skilled at teaching less advanced material. That's not to say that there are no good introductory-level teachers out there—there are lots, and they do heroic work—but that a random university professor, even a very good professor, probably won't be as good at teaching introductory material. (With that being said, I don't regard Linear Algebra as introductory material—but, then, I'm not teaching at Princeton.)
Certainly, if I were advising a non-math major who wanted to take an introductory math class, I would encourage them to think very carefully about an auto-didactic approach to see if they like it well enough to continue. If you want to learn lots of mathematics, then a university math department is the place for you; but, if you just want to dip your toe into it, then it may well not be, because so many of those courses are set up as 'service courses' for people who don't want to be there but have to be, and that inevitably shapes the tenor of those classes.
Fair enough. FWIW, my post was written from the perspective of being targeted at someone who has already chosen to take a university class, for whatever reason. But I agree with your point, and that approach is, in fact, my own. I mean, yes, I took some university maths classes in the past. But now as I want to learn new maths or re-learn maths I've forgotten, I prefer to just study on my own using mostly the exact resources I called out above. I wouldn't go pay to take a university class at this point in my life.
https://www.forbes.com/sites/michaeltnietzel/2021/09/13/us-n...
(U.S. News & World Report's Best College rankings)
If you want stats I highly recommend Statistical Inference by Casella & Berger -- it's extremely dry but so many stats books out there try to "make it easy" but the simplification means that you can't actually grok what's really going on. HOWEVER if you want to actually apply anything in this book you'll need to grab something more practical as well. Going through an applied stats book after having done SI is like having superpowers.
Anyone that says “it’s just a calculus book; it’s not good for introductory real analysis” is invited to go solve every problem in, for example, the chapter which defines integration, and compare the difficulty with problems in “traditional” analysis books.
Spivak at least has the nice property that it doesn't assume you know much.
However, if your goal is to learn mathematics for the sake of art, I recommend Kolmogorov's Elements of the Theory of Functions and Functional Analysis.
I constantly assert that I'd have actually been successful with mathematics throughout school if I were able apply it in code (which was not a thing in my educational environment); not with my broken brain where I fuck up numbers on paper and use my fingers for arithmetic.
https://www.manning.com/books/math-for-programmers
There is also the coding based Linear Algebra course that is available online (there's an accompanying print book).
https://codingthematrix.com/
And somebody who posts here on HN recently published a book with a title something like "Mathematics for Computer Programmers" or something to that effect. I forget the username and the exact title though. If you search around you can probably find it.
Edit: here's that last one. A Programmer's Introduction to Mathematics
https://www.amazon.com/gp/product/B088N68LTJ/
http://billsix.github.io/modelviewprojection/intro.html
Most of it is done but I’m updating the content in web form weekly
This is the second time today I'm recommending Knuth's "The Art of Computer Programming". It really is a math book, and it includes answers to ALL the exercises. For example, the last 150 pages of Volume 1 are solutions to the exercises.
Basically if you couldn't pass this course - and in some cases department policy limited you to a single try - your chances of success in that major were slim to nonexistent, and you were encouraged by your advisor to find a different major. If you could pass the weed-out course, you could be reasonably assured you were ready to face the rest of your coursework, because nothing ahead would be as difficult as that course was. And, that you were reasonably certain this was what you wanted to do.
They weren't necessarily directly related to the major. For some of the engineering and science programs, a math course was often the weed-out course.
I can't say for sure that this was the author's experience, but her description of it sounds very much like a weed-out course.
Now it's possible I'm misunderstanding the problem here: I am assuming that during lessons students have been given answers to problems worked in class, so that they can see what an answer looks like, which is probably one of the most important parts in learning introductory mathematics. Indeed, learning to validate one's own answers is not only the most important lesson in this stage of learning mathematics; it's also the only way to actually learn mathematics as it's the only way actual math gets done.
(ETA:) So I guess what I'm saying is that the author is right that no one has taught her mathematics, but as a symptom of this she doesn't even understand what form actually learning mathematics would take.
(Honestly, I'd probably have a better relationship with math if it were taught to me in the method you seem to require as "actual math".)
This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math. I remember the agony I experienced as a 3rd grader for not being able to memorize "multiplication tables". Instead I took the approach of trying to understand what the mathematical meaning of multiplication is. This was to my extreme academic & mental detriment as we were not graded on understanding, but instead graded on how fast we could answer math questions.
I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles.
Obviously, my concern over this was totally pointless as I've never been in a career environment where I was told "just memorize the answer, don't bother trying to understand anything".
There's not much to think about tho, is there? You take 3 boxes of 4 apples each, how many apples is there?
Or were you thinking about groups, rings and fields of integers? I highly doubt it at that age ;)
Ultimately you need to know multiplication tables instinctively to be able to do algebra, so they make you learn them. I don't think there's any way to skip it, just like you can't be good at football without doing a lot of cardio.
For the record - I also hated memorizing multiplication tables and thought I will be a writer because clearly math isn't for me. Fortunately in the next semester we had word problems and I loved that part of math.
> Doing the math out helps build an intuition for things like commutative properties and what not.
Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway.
I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.
Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.
Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.
That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa.
I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.
But mapping the relationship between multiplication and the area of rectangles is very visually intuitive. Geometric reasoning is powerful. Being able to use your understanding of multiplication rules to derive the formula for the volume of a cube, or a cylinder, or whatever is probably more insightful than realizing some arithmetic tricks.
Strong disagree. The number line is one of the best tools for understanding multiplication of numbers (not just whole numbers, but any decimal). The problem is that it is no longer taught. Example: American rulers have both metric (cm) and Imperial (inch) units. You can use it to visually multiply and divide by 2.54.
To generalize multiplying positive numbers x and y, (1) Mark x on a number line, (2) create another number line with the number 1 placed where x would be on the original line, (3) find y on the new number line, and (4) the corresponding point on the original line is the product x*y. The point: Multiplication is scaling (stretching, shrinking, whatever you want to call it). The Common Core tries to address this.
I always had better luck with just manipulating the numbers mentally.
But having that capability is pretty important for understanding other concepts. For example, many concepts in calculus.
But as a kid I never got the multiplication thing. Even now as an adult I understand what they are trying to do, but it isn't "obvious" to me.
Then again I am one of those people who doesn't read comic books because all the pictures get in the way of the words. Heck as a kid I read illustrated books and didn't realize until I had finished the book that it even had pictures.
and this is why kids are taught to memorize these lookup tables. When doing algorithms such as long division or multiplication these lookup tables can make the process much faster.
God I hope they're not teaching one of them is correct and one of them is incorrect.
5 x 3 can be understood as five groups of three items, or three groups of five items, and students at a particular grade level are "encouraged" to use one interpretation over the other to the point of taking off points when they naturally use the commutative property (and or choose to chunk differently). I forgot what it was exactly but I read about a parent (rightfully) complaining about this.
I don't think this is anywhere in the common core standards, but probably the teacher/school was using some pre-packaged rubric and the teacher didn't bother to override using better judgment.
I break multiplication and division up into chunks of 1's, 5's, and 10's so that it's mostly just adding a zero to the end, or doing that and then cutting it in half.
I never learned multiplication tables, long division, lattice multiplication or anything else. just did this in my head.
So say you want to find 86 * 32
86 * 10 = 860
860 * 3 = (2400 + 180 = 2580)
2580 + (86 * 2) = 2752
It starts to break down for bigger numbers depending on how "uneven" they are but it's been pretty useful in general
Memorizing that multiplication table was a more valuable use of time than the vast majority of time I spent learning mathematics. It's useful daily. Can't say that for much else.
8 x 300 = 2400
Add two zeros
8 + 8 = 16
16 + 8 = 24
Also university-level math for everyone who doesn't get up to taking proof-based analysis.
The principles of multiplication are pretty straightforward. However, especially before calculators were commonplace (one can probably debate if there's sufficient value in memorizing multiplication tables today), there was a lot of benefit in doing multiplication and other simple operations quickly and mentally. There are a ton of basic things that we memorize without necessarily knowing how to get there from first principles off the top of our heads.
This is true with programming as well. Sure you may look up APIs or specific syntax but if you had to look up every detail every time you put fingers to keyboard, you'd be really slow.
But for lower level "grind and chug" Maths classes (High-school algebra, intro Calculus, non-proofs-based Linear Algebra, etc) I think there is value in having access to "the answer". Certainly at least for a subset of the problems one works on, IMO.
I think there is a value in introducing the concept of verifying your answers early, as your ability to perform calculations is beyond useless if that isn't something you are thoroughly familiar with.
If you're just doing the calculation but not the verification, you're skipping half the exercise.
If nothing else, this is an advantage from a time efficiency viewpoint, as you can more quickly identify the problems you got wrong, and focus your review time on those. Those presumably being the ones most likely to highlight some misunderstanding of the material.
If time were infinitely available maybe the distinction wouldn't matter, but things being what they are...
For calculation based math, which this course sounds like and based on a sample test I looked at, more or less is, having some subset of answers would definitely be helpful. I am not sure why they would make it more obtuse than it needs to be. I thought the course I took on multivariable calculus and linear algebra were quite easy but the number of people who dropped math entirely after those was pretty drastic. I don't think you really need to make these classes unnecessarily hard to weed people out. They'll do it themselves anyway.
Also if no one liked this course there is certainly a problem that can't be dismissed in a handwave like you did in your answer.
I found the same thing in my tutoring experience. I had several students in physics whom I couldn't "reach" no matter how hard I tried to present "theory" using the equations in full generality (symbols instead of numebrs). I taught them this way because they this is the way I learned and because it seems OBVIOUS to me that learning the abstract thing is more powerful/efficient way to get through it.
Imagine spending dozens of hours with them and they still didn't "get" anything to the point I was feeling discouraged and was like this person will never pass this exam that they have next week. I felt bad because these were friends and I really wanted them to succeed (more friends in STEM, more STEM conversations with friends).
Lo and behold, these same people whom I thought were "physics dumb," were able to pick up all the material in literally minutes, as soon as we started the exercises (one person) or having tried some exercises on their own (the other person). I don't mean "imitate the steps"-know, I mean completely solve 80% of the problems on the practice exams and adapt the use of equations as needed for each problem. I mean it's still PHYS101, so there are only 10 or so equations, so they didn't turn savants or anything, but the transformation was so sudden and drastic that I realized people simply have different learning approaches.
Some people like the abstract/general approach. Some people like the hands-on approach. The latter kind of people often think they are "not good at X" but really they would have no problem picking up X if they had learning resources suitable for them (often lacking in traditional textbooks and courses). I think that's why Khan Academy videos are so popular. Sal does a fair bid of hands-on approach, and many people need that!
This type of bullshit is normal in crappy STEM classes. Universities are so concerned with keeping secrets that they won't give you the information in the first place. It's a sign of bad course design that's unfortunately common.
I took a couple classes that were like that, but I swapped the class for another subject those semesters and waited until it was offered again by a better professor. Same material, except I actually learned it instead of getting stuck at the starting gate.
I certainly don't want to be misquoted as claiming that mathematics must be learned in a vacuum. Just that access to a large collection of questions and answers is not helpful. Your own argument goes through just as well to show that access to the answer isn't helpful either: you know you got the wrong answer but you don't know why.
It is cryptic at times bit ultimately developing rigor and intuition is something that in my experience will always involve a certain amount of struggle and confusion that no professor can or should alleviate
By having access to multiple worked examples of problems, students at least have a fighting chance of learning how to solve problems _in parallel_ with the course material, rather than having to wait days or weeks to even know what they did wrong, while getting more and more lost.
Remember, these are Princeton students, some of the most brilliant and motivated minds of their generation, and the averages are _abysmal_. That indicates there's a greater systemic issue.
Believe it or not, the number of practicing mathematicians and PhD students who have the innate grasp that a Terry Tao does are few and far between. The vast majority have to work hard to earn their skill.
No amount of hard work would have allowed Stephen Hawking to succeed at sports.
Some pursuits are simply unavailable to those without the requisite immutable attributes. This is not politically comfortable, but it is true. I think the only legitimate question is whether mathematics is such a pursuit, and I'd like to be proved wrong on that point, but trying to convince me that anyone can do anything they put their mind to is wasted effort as it is so plainly contradicted by the available evidence.
What about those people in this world with genetic abnormalities that affect brain function? Would you expect someone with Prader-Willi syndrome to be able to become competent at mathematics if they only worked hard enough at it? At what point does “hard enough” become “virtually impossible” or even “definitely impossible”?
Why, sure - from the physics standpoint all the human brains are essentially the same, they all consist of the same protons, neutrons, electrons... (Also, they are close enough to being spherical.)
You're begging the question. Please substantiate.
Simply returning work shortly after the late work window closes should suffice, no?
Glad to hear you didn't actually do that. Returning digitally is still returning.
If she continues on to functional analysis she can just tunnel through.
You should be able to solve a problem in multiple ways and prove correctness.
It takes loads of time, but that is the only way to truly learn.
Most of students given answers will solve a problem in one way see that answer matches the correct solution and stop there. That is not the way to learn anything.
It is the same with software, it takes a lot of time to get good at.
In software it is easy to get correct solution with wrong approach, but often people just stop at "make it work" where they should step back and "make it work, make it right, make it fast".
It's possible the author was in a "math for engineers" class
Where a math student may see a technique as a swiss army knife that can be used for loads of things, an engineer might only care for the one blade that lets them check the stability of their boost converter and pass their power supply design class.
Indeed, she says exactly that:
> MAT 202 is not a course that math majors typically take, but rather for underclassmen who are majoring in engineering or sciences.
>Brief Course Description: More abstract than calculus, this course aims to develop basic algebraic tools for work with problems involving many variables. Starting from systems of linear equations and vectors in 2-space and 3-space, this course develops ideas about length, angles and resolving a general vector into useful components, identifying features of linear systems or processes in order to choose a basis that is well-adapted to studying a particular phenomenon and move between different points of view to reveal the essential underlying structure. Companion course to 201 (Multivariable Calculus). Discusses matrices and linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvalues and their applications to quadratic forms and dynamical systems.
Definitely NOT a math class
You could argue it was "math for engineers" in that Liberal Arts / Social Studies don't require them and only the STEM programs do, I guess. You can graduate with a non STEM degree with just an Algebra course in most Universities in America. And those Algebra courses are usually just high school refreshers if you went to a good high school. Some of the non STEM require Statistics. But even there some schools do "Statistics for STEM majors" and "Statistics for non STEM majors"
The math I did before university was doable with basically pattern matching ; I then still had a close to photographic memory (I could skim materials and recall them exactly) and got A’s in math because of mostly that. I could do exams by filling in methods I memorised. There was no understanding. Then in uni I got a good scare as none of that worked anymore; now I had to actually know what I was doing. That was obviously much better but I was not prepared for that at all.
You appear to be absolutely correct that no one has taught you what mathematics is. There's definitely a failing here in mathematics education; unfortunately this is not a surprise at all.
Validating the proof requires understanding what it is you're trying to prove, understanding what are valid proof techniques, and understanding the premises. You then check every step against this knowledge. It isn't easy, and tbh I usually couldn't be bothered after spending hours on a problem, but this is where results are obtained in proportion to effort.
You're making the student out to be some kind of simpleton. She's not.
She just wants what any decent textbook has: An answer key to the practice problems, so she knows whether her answers were correct or not. Being as they are, after all, practice problems. Meanwhile her instructor is telling her, in so many words, to buzz off.
That's what is at issue here. Not (as you presume) her being some kind of a lunkhead, who thinks it's all about "pattern matching".
In maths, you know when you've got an exercise right, and so you don't need the answer written out for you. If you're looking up the answer before you get that feeling, you're cheating yourself. You won't learn the material.
It would seem weird if someone put the answers in the textbook, like having the answers to a newspaper crossword on the same page as the crossword itself. You'd have to exert willpower in order to not look at them.
The whole point of the exercises is to play with the ideas until you understand what they're for. If you don't at least partly invent the subject yourself, you never really 'get' it.
Textbooks are really guides to how to reinvent the field, what order to think about things in, what are the cleanest ways to break up the patterns, what are the best notations to use, so you can reinvent it all at a reasonable speed rather than having to spend a whole lifetime on each problem.
If you literally can't do the exercises despite having read the chapter, then it's a bad textbook. Throw it away and get a better one.
Well it's been a while - by my recollection is that (at fresh and sophomore level at least), the better textbooks always did have answer keys in the back (at least for a significant subset of the problems). But as we all know -- the better textbooks are few and far between. And college is filled with lousy textbooks that get thrown at students for who knows what reasons.
To the extent that they think lousy textbooks are the norm.
No, I might think I have the correct answer but I cannot know until I have seen the correct answer. I can justify any false answer that's not obviously false for myself which is why I need to compare.
As someone deeply familiar with/scarred by courses just like this one, in-classroom introductory mathematics pedagogy at elite institutions can vary wildly because mathematics graduate students and professors are incredibly specialized in their fields of study and accustomed to very specialized types of intuition - with perhaps more of a disconnect from the introductory student's learning experience than in any other field. Language barriers can exacerbate this. Proofs are thrown on a board, time management on the lecturer's part is nonexistent, and examples are inevitably skipped over as "an exercise for the student."
In contrast, many computer science programs recognize this tendency and optimize for pedagogy. I remember my university's CS department hiring and paying upperclass CS students to guide other students through their first times debugging code outside of the authority structures of official teaching staff. No similar program existed for the mathematics department, and arguably it should have - but who would volunteer? Once a mathematics major, you're pulled into a realm that's quite isolated from the broader student experience!
You're assuming there's only one answer to "what mathematics is" and that it coincides with your view. You appear to be a fan of doing math, but far more people use math.
I'm not a mathematician. To me, it's a tool that can be used to solve the problems I care about. I don't get joy from spending hours proving results just to say I did it - but I understand that others do. In spite of that, if I can understand the conditions in which an algorithm will converge to the solution, I can benefit from studying math.
They may not give it as a handout, but more likely would solve the problems in class for everyone to see. But one way or other, they would give it.
IMHO for everyone who wants or needs to "learn actual mathematics" there are at least ten people who need "applied mathematics" in the sense of being able to apply well-known theorems from a well developed field (like linear algebra) and with zero need to ever prove anything novel, even trivial things; they need to understand the concepts and know how and where to look up the appropriate formulas to do the linear algebra calculations they need for their non-math domain. This OP wanted a course for the latter need and this course failed at this goal.
For a random example I heard recently; there's applications of differential equations in pharmacokinetics, so people studying pharmacokinetics need a sufficient math background for that. However, the appropriate math background for these students involves the general concept of differential equations and being able to understand a couple specific equations which characterize their physical domain, and do calculations involving them. For that use case there is no need for a skill in "doing math" - all the mathematical properties of those particular equations are known and they can and should look them up instead of learning how to derive them from first principles, much less making any new assertions.
You're right, but you need to know the basic mechanics pretty well in order to start to understand the deeper philosophies and ways of thinking that math _is_.
It's like learning to ride a motorcycle without ever having ridden a bike; Riding a bike is not riding a motorcycle, but you're going to really struggle to make progress past a certain point of learning since you're mastering the absolute basics of _both_ things at the _same_ time.
It's the same thing with math. If I never see what well-formed answers to problems look like, never gain familiarity with the symbology or jargon, then I can never start to even learn proper math because the professor can't communicate with me.
China (and other countries who put American math education to shame) disagrees.
For you, math may have been a deduction from first principles, but to us mere mortals, math is a decade-long slow reveal through wax-on-wax-off repetition.
That is exactly how I got through university math so I think you are wrong. I worked on what felt like millions of problems the teachers gave out and then on some more I found in books. Then I looked on the answers and tried to figure out what my errors were. Then I worked on more problems. What didn't work for me was looking at lecturers writing out proofs for various theorems on the blackboard. I had almost zero use for that.
This is how mathematics works for most students. They get a set of problems, then a (poorly explained) set of tools and they try to apply those tools to solve the problems during tests. Then they leave school and never use those tools again - they often completely forget about them. In Eastern Europe they even call it the "memorize->get drunk->forget" cycle.
Arguably, this is the experience of most people, in most fields of study. This pattern matching does not only happen in maths. Most Bachelor's and Master's thesis are basically a compilation of quotes from various books and papers - without any new ideas, or original research. Maybe I am cynical, but in my opinion you are discouraged to make any original research even while getting a PHD as well. Your thesis is more about quoting the "right people" (= quoting papers of the people who will be promoting you), than doing anything new, or original.
But coming back to the question of actual education: the article clearly shows the experience of most people at Universities. If they want to 'learn for the sake of learning' you will get very discouraged fast, your experience will be very miserable - the university is not there to broaden your horizons, it is just a combination of a testing facility and a diploma mill. What is even sadder is that if you are laser focused to learn something that will be useful at work.. then you will get disappointed as well. The things taught at universities are often not even applicable at academia, not to mention "real" work. Writing a master's thesis teaches you how to write a master's thesis (assuming someone even properly teaches you how to do that, for me they just told us to write one, without even bothering to show few other ones as examples - I had to get those myself). It still teaches you something, but that something is not good enough for real academia or real work. In addition, since lots of lectures are graded on basis of tests, universities often feel like a glorified highschool. And you know what? Probably we cannot have it better: most people wont make original research - because it is hard, so universities are stuck with the tests and papers (that are a collection of unoriginal quotes). Also you need to learn to walk before you run, how are you supposed to write a good original thesis, if you didnt write a single non-original thesis, what is much easier?
The main reason why the blog's author was disappointed is the systematic problem of rewards for university professors. They arent really rewarded for doing more than the minimum. They have their own rat race, with their own reward structure, that rewards "research", understood mostly are publishing quotable papers, sometimes getting grants. Changing the reward structure would help a lot - those who focus on teaching, should be rewarded better. Universities supposedly are for students, but most of the time it feels like students are an unwelcome afterthought for the research professors. Also, professors seem to like to say that "if you want to learn a trade, go to college - university is there to broaden your horizons" - what is obvious bullshit, college seems to be the same as university, but just with a worse diploma at the end. [as a thought experiment: would the professors teach better, if their pay would be related to the salaries achieved by their students - ignoring GDPR, PII for a moment? if the professors could get a "share" of their students incomes, would they teach better? now in USA the students are obliged to pay their tuition with 0 impact on the quality of the received education, or in Europe receive free education]
Then we have those "good universities". USA has this corrupted system of admissions (probably so that students with best results dont t...
I remember attempting a grad class called "Hard problems in combinatorial optimization". I was struggling so I asked the professor for advice on getting my footing in the class.
His answer was that I may just not have the "mathematical maturity" for the material. I was so discouraged that I dropped out of the program.
What I didn't realize at the time was the particular meaning of that term. His advice was well intended and accurate.
But what I heard was that I was hopeless in this topic area because I wasn't smart enough.
So much pain could have been avoided if we just extended the conversation by a few minutes, or he invited me for a discussion over coffee.
As an example: you'd reasonably expect a student with mathematical maturity to know what a set, the Cartesian product of two sets, and a function are. That's high school level math.
From there you can define a ring and a module over a ring, in like 5 mins.
So at that point the Prof, strictly speaking, told you everything you need to know to start studying modules.
Of course, many students will be like whaaaaaa?! When a ring is defined. But here's the point: Mathematical maturity does not mean that you already know what a ring is, but when the Prof defines a ring, you are following along.
Anyway, my point is that there is nothing specific that you need to know.
Any good definition of mathematical maturity would likely be so technical that you need mathematical maturity to understand it.
Some people are better with math than others. Undergrad math majors thought the major was pretty easy. Many of us in engineering majors--who weren't that bad at math (it was a good school)--could no more have graduated with a degree in math (or physics) than have flapped our arms and flown.
Most colleges have classes designed to foster mathatical maturity, but that is done by how the class is taught, not what is taught in it.
I studied at the VU, where Andy Tanenbaum, writer of a ton of famous CS books teaches. Of course I followed his classes. His books are excellent, but his classes are basically him reciting his books from memory. The jokes are literally identical.
At some point, more and more of his classes would be taught by Maarten van Steen instead; he's an excellent teacher who really knows how to engage the students and make them think about problems, instead of merely telling them the answer.
There was another professor who, as far as I know, didn't do any research at all; he just taught a ton of classes.
A professor who is great at one thing is not necessarily good at something else.
I don't think that that is an introductory course.
It also usually helps if it's an interesting problem...
It didn't help that I went to uni at 24 IIRC while I was working my ass off, so I wasn't that happy to put up with the BS.
Calling it gatekeeping seems like a fish complaining that the birds are gatekeeping the sky. The sky is open for anyone, but only if you have wings to fly.
As a lecturer, I've had several job interviews where half of the questions were some variety of "are you ok to put up with having way too many students and not enough time?". I had the luxury of not having to take these jobs, but obviously someone ended up taking them.
This wasn't even really intentional discouragement, but it was perceived as such. Hopefully, you learned the higher lesson.
I think where the fruit of this discussion may be is deciding to what degree an authority figure like a professor has a responsibility or an obligation to their students and advisees to make themselves clear which way they intend their statements to be interpreted. Personally, this begets a discussion of the nuances between the obligations and responsibilities of communication between (1) strangers, (2) peers/colleagues, and (3) authority figures and their charges.
What I mean to say is that this issue stems from your misunderstanding of the meaning of the word “mature” and nothing else.
I'm not claiming that my response was entirely rational or measured. I'm not blaming the professor, and I'm not justifying my own reaction. I was just saying what my reaction was, in case it shows a pattern of student experiences that's worth addressing.
Stubbornness, tenacity and self-belief are useful skills for success - perhaps you gave up too quickly? Unfortunately, soft skills are very rarely taught well. We all rely upon our subconscious learning and a good amount of luck to gain necessary skills, and those skills are often learnt by osmosis while very young. I also think the meta-skill of being able to teach yourself soft skills is not common either and it - although we can make an effort to improve that meta-skill even though it is somewhat self-referential.
A teacher who doesn't know you will not know if you're not smart enough or just didnt study hard enough, and probably doesn't care much, especially for an undergrad. For a phd student, they may care if they think they've identified a "diamond in the rough" that can be turned into a gem with some work, but that is quite a bit beyond basic linear algebra.
Fascinating, it sounds like "MM" is tacit knowledge, which begs the question why it isn't explicitly taught.
In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right.
https://en.wikipedia.org/wiki/Mathematical_maturity
4 hours a night 6 days of week and I got a C. Less the 25% made it to end. Don’t think many of remaining passed.
Got a lot out of it though.
At least at my university, the reason that such classes have a high failure rate is that the university is incentivized to have high enrollment numbers by admitting students who do not have the background to succeed at college mathematics at the level of calculus or above, and so need some pre-calculus courses; but someone who does not have a solid grounding in pre-calculus mathematics will often struggle to learn it when it is taught at a rapid pace at the college level—especially by professors most of whose teaching experience is with college-level math.
I guess its possible to learn stuff from a course you fail, but you should be able to find a course that you're likely to pass and learn something from?
However, for non-STEM and non-Business majors college algebra is rarely a prerequisite for any other courses so you can still continue your major without it, you do need to pass a math course before graduating though. They also have other math courses that can be used that aren't college algebra and may be easier.
Of those who failed : ¼ retook the course, ¼ pivoted to the bussines-CS program (given by the CS department) , ¼ pivoted to bussines-IT (given by the business and management school) and as far as I know, the rest disappeared from the university.
I took a bit more math in college. It wasn't easier; the professor simply kept pace with the material. If you needed help, you had to go to a TA or the professor out of class hours (an option that wasn't really available for a high school setting).
I can easily imagine that many students who were exposed to college level math for the first time in one of those courses would flounder and drop out (though I didn't witness much of it myself).
No, it's usually basics like functions, graphs, and trigonometry. It's basically a remedial class for what students didn't learn in highschool.
Like this stuff isn't particularly complicated, I'm not a very good teacher but I did teach some undergrad seminars during my PhD and I think even my worst students were capable of understanding functions, graphs and trig.
Also state universities are where most of the public school teachers and administrators trained, so they probably deserve a little of the blame for the state of public schools.
One of them told me one day "I don't understand graphs." I tried but it's impossible at that point to make up for an apparent total lack of at least high school level arithmetic/math.
If you were doing a PhD you were at the kind of university that has a PhD programme. Probably it was one of the 200 or so US universities out of a thousand that are selective, rejecting more students than they accept. I doubt there was a single student in your seminar who wasn’t in the top 30% by academic aptitude of their age group.
> According to the U.S. Department of Education, 54% of adults in the United States have prose literacy below the 6th-grade level.
https://en.m.wikipedia.org/wiki/Literacy_in_the_United_State...
"4 hours a night 6 days of week and I got a C. Less the 25% made it to end. Don’t think many of remaining passed.
Got a lot out of it though."
Actually is abstract algebra and not highschool level. Nobody teaches remedial classes in such a way that only 25% make it to the end.
There's a problem with making a statement like that and not following it up with what you need to do gain the needed "maturity". Even adding "maybe you should take this course over here and come back to this one in a semester..." could make a world of difference.
[1] https://chhaylinlim.wordpress.com/2014/09/24/average-iq-per-...
[2] Assuming gaussian distribution with no skew, which is probably a bad assumption, but I'm spitballing here. Suffice to say math departments at elite colleges have many extremely bright people.
Are you arguing that it is actually skewed towards some cutoff point of "just smart enough to get the medal"?
That would be distributed according to the distribution of the maximum of N Gaussian-distributed random variables, and according to this[0] stack overflow answer, it's skewed downwards, below its mean. (Its mean, of course, is much higher than the mean IQ+Luck).
[0] https://math.stackexchange.com/questions/2105921/pdf-for-min...
That's a huge assumption. I could easily propose a half dozen other models each with a different conclusion. You would need to look at the data.
In any case the resulting intermediate distribution from selection criteria bias would be a Chi distribution (I think - it has been a while.) 5 standard deviations on a normal distribution is 175 IQ or top 1/700K which is insane.
A few additional biases, Physics majors have higher average IQs, and not everyone smart enough to do the math has the desire or the opportunity.
The point I was making is that IQ score has been normalized, when imagining the intermediate distribution of IQ it would make more sense to look at composite probabilities of factors that led to the IQ score and consider the probability that those same factors lead to a sufficient innate ability in math.
There isn't a skew per se but possible inflation that gp intended to account for. See Flynn Effect.
There's a small sadness in this but I can't word it out perfectly.
ps: I mean I would understand very much that these people just want "new buddies to play hyperstimulating math games with no drag whatsoever" but from school I expect just a few pointers. The rest is on me (us).
edit:
Or if you had the maturity to accept criticism like an adult.
LOTS of non-math-major science/engineering students face that problem as soon as they start grad school.
It's a consequence, IMHO, of math curriculums failing to focus on the fundamentals rigorously enough. There's too much focus on what we used to call "plug-n-chug" mathematics where students learn just enough to apply formulae to get their problems solved. It might seem like "just enough" but it ends up pushing out higher mathematics and keeps students from being able to apply more advanced mathematical concepts that they would have gained from a much deeper dive into Real Analysis, abstract algebra, differential geometry, etc, and the grind of the theorem-proof cycles. Many of us felt this right away upon taking graduate level courses.
I wish that I had been counseled to take additional math courses before (or even at the beginning) of grad school. It would have reduced my suffering A LOT and I would have gotten more out of the degree.
I had a similar experience - and while it was a blow to my confidence I’m happy that he didn’t tiptoe around it with feel-goodisms. What I was doing as a student, and what had ben successful so far, was not working and doing more of it or doing it harder wasn’t going to help.
Honestly leaving that program was the right thing to do, I probably shouldn’t have been in it, though I’m confident a few years later I would have been successful, but I lacked the maturity and mathematical sophistication to get it then.
I, very briefly and quite honestly, attended a lower tier state school that was an absolute joke. Shortly after I started, there was an outreach program for local high school kids where if they showed a high school ID, they got admission for the upcoming year. The head of my dept dismissed it and quipped, "It's ok. Anyone that wasn't meant to be here will just drop out after a year anyway and we have their money."
I, thankfully, ended up at a much bigger P5, well-known university. One of the controversies was the engineering departments were trying to get as many students as possible to the point of exceeding infrastructure and admitting lower quality students into the program. Upperclassmen I talk to said, "that's why everyone hates <1st semester math for engineers> and <2nd semester math for engineers>. They immediately weed out those that don't qualify."
"Weeding out" is what college admissions should be doing, not the departments. When the departments weed someone out, they've wasted at least a year of their life and a year in tuition, and demoralized them.
This is what really the "college for everyone" mentality has led to. If the kid shouldn't be there, or should have started in junior college first, we need to be more upfront about that earlier.
Admissions can't weed people out, they can only copy the weeding that highschool has done. If you rephrases the question that way, as in, "should highschool teachers be responsible for college admissions," then the idea of letting people join a major they haven't proven themselves capable of doesn't sound so bad after all.
Also, the math classes all had a shared policy of "you can work with anyone you like on a problem set, as long as everyone's name is listed on there when you hand it in." I loved that, and I did all my problem sets in groups while I was there, and I think I learned more because we were always explaining things to each other and arguing about solutions. I heard a rumor at one point that this policy was the direct product of Peter May's bad experience at Princeton, which required students to work alone when he was there.
TFA does seem like a pretty stinging indictment of Princeton's math department.
What a clueless, absolute piece of shit..
Clearly he lacks the emotional maturity to deal with students (or just people in general) and unequivocally should be fired as a teacher and restricted to doing research.
to use an analogy, when describing the math they tell one about a place where they are (a 'place' where the math is there, already understood) and how they got there (the analogue of route they took to get there e.g. "took public transport line 4, get off at station Example and walk until you're there").
This leaves little room for those who may prefer to ride a bike all the way there (the analogy has been stretched to the breaking point), cuz the teacher will knock you off the bike and insist that you ride the line 4 then walk, same as they did.
The teaching of different ways of making the journey is (often) the destination.
Isn't it obvious that the course is not designed for her learning purpose?
The idea that a 200-level course should be unapproachable for an interested person with a stronger mathematics background than I have, with an ostensibly mathematics-based degree, is absolutely silly. I took 300 and 400-level classes in economics, in English, in history and philosophy. I never felt unwelcome or incapable.
Really? I mean, you probably did have linear algebra in high school, it was just "disguised" (you may not have seen matrices and vectors, but probably did solve linear equations of multiple variables). I'm surprised about the college thing, though. I thought every CS degree (in the US at least) required at least through linear algebra (linear algebra with applications, probably not linear algebra with proofs and theory).
I learned what a (mathematical) vector was offhandedly in college, but I think it was from talking to someone, not in a class.
Yes - math education can be awful, but this snowflake needs to lose the entitlement.
The one well-developed complaint is the one about not getting the answers. On one hand that sounds almost calculated to provoke the reaction that you had, but on the other hand for every math class I've taken I turned in the problem sets and had them get graded with corrections that were useful.
Regardless, you're not paying for instruction. You're paying for networking, access to world-class experts in your field of study, a name-brand, etc.
I managed to improve and then I had math at higher edu institution
and oh boi, I instantly had no interest in doing it 4fun, just learn, pass and move on.
idk, I feel like there's too much stuff to be done (a lot of courses + you gotta learn other stuff like "real world" computer's related stuff) and there's lack of time for 4lulz messing with fancy topics like maths, but on the other hand I've been working + studying, so maybe that's why
I wonder how much of a problem this is in intro CS. Everyone knows that the camel has two humps, but are we inadvertently gatekeeping by not even trying to teach the low-scoring "hump" about the fundamentals of the discipline?
Strange. But it also may no longer be true, or not to the extent that it was when this student was enrolled in the course. From the course website:
"Try some old quiz problems for this course, but don’t just read the questions and solutions. Instead see if you can produce correct solutions to most of the problems in the allotted time." https://www.math.princeton.edu/undergraduate/placement/MAT20...
So clearly, now, there are archives of problems w/ solutions that can be reviewed after attempting a solution.
Apart from lack of solutions, I don't see anything "off" here. I don't see other criticisms of the teaching approach. I am also not surprised that one of the best math & physics universities in the world does not cater even its lower level courses towards students for whom they are optional and not part of their intended academic career.
In my school (which was not a top-tier program) courses for majors in those areas were also similarly difficult. The difference was that my Uni also offered a few "Math for non STEM majors" courses to choose from because they still did have requirements for all students in those areas. Basically a survey course of mathematical concepts and how they show themselves in the world & everyday life, along with practical applications of how they may need to use math in any way throughout their lives, things like how basic level probability & statistics will be useful no matter what they end up doing in life. And things like fractals, or the golden ratio, and other concepts along those lines that show up everywhere in life.
The course was designed to answer the question, with concrete examples, "Why should I care about math if I'm barely going to have to use it in life & have a calculator?" I think Princeton simply expects their students to understand the answer to that question without putting their students through the trouble of an entire semester on the topic.*
*Though clearly should address the "no solutions" issue if it has not been fully resolved already.
What it looks like now as an adult is just a failure of pedagogy and institutional incentives. I suspect the people teaching it had little reason to take the teaching part seriously, it was just something that got in the way of their research/grad studies.
I went to a well-known state school. There were other departments like this.
Is there some prior knowledge about university exams I should have in order to understand what these figures relate to? Standardized set of 3 exams maybe?
I do find it surprising that the whole class could have such a low grade in linear algebra, which is one of the easier introductory math classes. But I've heard of grade deflation in Princeton is pretty severe, so I don't know what standard grades are there. IIRC Princeton also has one of the premier math departments in the entire world, so I would expect the program to be very difficult.
On the other hand, I think the interest of tenured professors in math at universities is almost entirely on research. This doesn't mean they can't be good teachers but the ones who are I think in general are thinking more of cultivating potential phd students and the like. A practical class for non math students I can easily believe would be completely blown off.
I failed. I found her course to be not only sink or swim, but actively hostile to student learning. As mentioned in another post, assignments and quizzes were returned weeks late with little helpful coaching. Assignments very often worked with higher levels of abstraction than presented in the lectures, for example the first time she saw a matrix of functions was on a homework assignment.
Furthermore there seemed to be no additional help for Linear Algebra. To my great surprise the math department had no facility for matching students with tutors. The university math help center was almost exclusively oriented towards calculus and had no resources for Linear Algebra. After weeks of searching online we finally got a call back from a third party tutor, only to be informed that this person (a grad student in math) couldn't help us because the assignments were "too specific to the particular course."
The _only_ place I've ever seen someone show any creativity and ingenuity in teaching linear algebra is 3Blue1Brown, but obviously even though he's got a whole linear algebra series it's not a complete university course, nor is it meant to be.
So yeah, it's a damn shame, and IMO inexcusable. Linear Algebra is the foundation of modern data science and machine learning. Math departments should consider it a sacred duty to bring as many students as possible to at least some level of comfort and familiarity, but instead they seem to treat it as an annoying distraction at best, or a weeder class at worst.
preview: https://minireference.com/static/excerpts/noBSLA_v2_preview....
concept map: https://minireference.com/static/conceptmaps/linear_algebra_...
intro videos and SymPy notebooks: https://github.com/minireference/noBSLAnotebooks#contents
http://math_research.uct.ac.za/marques/LA.html
It's a much higher level of abstraction than an introductory Linear Algebra course but I found them amazing for developing intuition. I believe they have also released them in textbook form which is probably more polished.
The students can ask me or the TA abut the exercises. We prefer to see what the student has attempted, and find the exact spot where the student made the mistake. It takes more time, but it's more helpful than a nice ideal solution.
We also may recommend to make another exercise from the official list and come back in 10 minutes with a solution attempt, or make some custom exercise on the spot that is about the same subject. (Exercises about derivatives are easy to invent, integrals and linear algebra is harder.)
One of the useful trick is to write an optional homework exercise in the blackboard. I get a lot more of answers than just selecting an exercise from the official list. Sometimes it's just an exercise from the list with different number and sometime it has an intermediate step to make it easier. But writing it in the blackboard increase the chance the students will write it down.
Another trick is to write an old midterm and then go to each desk to talk to the students about what they are doing. We usually write the solution at the end of the class, and have some discussion about it, but the useful part is the discussion with each student (or small group of 2-3 students).