Previously, I thought certain math topics were "hard" (e.g. category theory) while others were supposed to be "easy" (e.g. Calc I). I beat myself up for struggling with the "easy" topics and believe this precluded me from ever tackling "hard" topics.
I was thirty-something years old when I finally realized math has a well-documented maturity model, just like emotional maturity or financial maturity. This realization inspired me to go back and take a few math classes that I had previously labeled as "too hard," with the mindset that I was progressing my math maturity.
My point is that choosing an "age-appropriate" (in terms of math maturity, not actual calendar age) textbook is important. I also find it extremely helpful to chat with people who are more mathematically mature than I am, in the same way it's helpful to seek advice from an older sibling.
This was very much my experience with computer science. When I first studied computer science in middle school at age 13, I could only understand simpler algorithms like quicksort. I simply couldn't grasp dynamic programming. When I studied it again at age 19 (after having learned a couple of more programming languages like C++ and Python and Haskell, as well as taken some classes in mathematical proofs), it became much easier to understand. And then it was around age 22 when I could solve competition-style dynamic programming problems with ease.
This sounds somewhat similar to my experience in mathematics. I realized part way through college that I would only gain an innate understanding of a topic about two semesters into a harder one - ie I only had a firm grasp of Calculus I once I took Calculus III. I would still do well in those courses, but I would have to follow the motions at times. Alas, this meant the tail end of my education was doomed to some courses I only sort of understood. Fortunately I didnt major in it.
I am not sure what you are trying to say because your message and your posted link seem to be an odds with each other.
Mathematical maturity has all to do with practice and experience and nothing to do with age.
Category Theory is easy because it starts from nothing, literally. You can learn it at any age and with no almost no prior education. Same with various formal logics.
You can't study or use in any way the theory of Calabi–Yau manifolds unless you have mastered all of its prerequisites.
Certainly the advice of not choosing textbooks you don't understand is spot on, however. Unfortunately (?) most textbooks assume quite a bit of background, so you don't often have much choice in this regard.
> Category Theory is easy because it starts from nothing, literally. You can learn it at any age and with no almost no prior education. Same with various formal logics.
I disagree, you need to have capacity for abstract thought for abstract topics, most especially with category theory. Normal people have quite difficulty understanding abstractness, you either have the aptitude for it, or work your brain hard enough that it becomes somewhat easier. Children and younger people especially have difficulty understanding non-concrete topics.
> Mathematical maturity has all to do with practice and experience and nothing to do with age.
That's OP's point.
"Age-appropriate" was put in quotes because it was referring to a metaphorical "mathematical age" to tie it to the concept of mathematical maturity.
And, indeed, certain books require more mathematical maturity to get through than others, even though the prerequisites may be minimal. You'll see this often explicitly described in the preface of textbooks and reviews of those textbooks.
They mentioned studying certain books to develop mathematical maturity because that's where the practice and experience happen. Calculus is one such course used as part of this process, as are many courses intended as a first exposure to proofs like linear algebra and discrete math. Some might use the Moore Method with point-set topology.
While beginning calculus students often pick up derivatives and integrals (and the associated formulas) easily, the delta-epsilon definitions of limits and continuity are a well known stumbling block for many. I've been told that the difficulty stems from that being the first place math beginners really see nested quantifiers: (forall epsilon)(exists delta)(...). In logic though, nested quantifiers are fundamental. I don't know what happens if someone tries to study logic without first having studied calculus. Maybe it's a good idea, but few people do it that way.
> I don't know what happens if someone tries to study logic without first having studied calculus.
When I was in college, the Philosophy department offered this course. It was considered an easy way to get a general education math credit without needing to be good at math. It was a really enjoyable course[0] that put me on the path to becoming a computer programmer. It occasionally comes in handy[1].
Delta epsilon is just an annoying unenlightening technicality, not the essence of real analysis. Surreal numbers (infinitesimals)solve the problem more elegantly.
To each his own, but epsilon-delta is my go-to example of formalizing an intuitive concept ("gets closer and closer"), which is a high-level mathematical skill.
The intuition and the formalism are presented together (at least, they should be!). To learn the role of epsilon and delta, the student needs to jump back and forth, finding the correspondences between equations and the motivation. This is a skill that needs practice; this was one of the first places I found the equations dense enough that I couldn't just "swallow them whole".
(The earlier I remember is the quadratic formula, which I first painfully memorized as technical trivia. It took me a couple of years to grasp that it was completing-the-square in general form. Switching between the general and the specific is another skill that you develop)
Surreal analysis is sort of a thing but it is quite far out there (e.g. you can have transfinite series instead of merely infinite ones). Maybe you meant nonstandard analysis (NSA), which is real analysis done with infinitesimals, but the machinery justifying it is way outside of what you'd see in even a theory-oriented intro calculus class. There was an intro calculus text (Keisler, 1976) that used infinitesimals and NSA. I don't know how it dealt with constructing them though.
The problem is that epsilon deltas have very little practical use outside of theoretical proofs in pure mathematics. Even for cutting edge CS/statistics fields like high level machine learning, most of the calculus used are existing formalisms on multidimensional statistics and perhaps differential equations. Aside from Jensen's inequality and the mean value theorem, I have never seen any truly useful epsilon delta proofs being used in any of the ML papers with significant impact. It's perhaps mentioned once in passing when teaching gradient descent to grad students.
> Even for cutting edge CS/statistics fields like high level machine learning, most of the calculus used are existing formalisms on multidimensional statistics and perhaps differential equations.
If you mean experimental work, then sure, that's like laboratory chemistry. You run code and write up what you observe happens. If you're trying to prove theorems, you have to understand the epsilon delta stuff even if your proofs don't actually use it. It can be somewhat abstracted away by the statistics and differential equations theorems that you mention, but it is still there. Anyway, the difficulty melts away once you have seen enough math to deal with the statistics, differential equations, have some grasp of high dimensional geometry, etc. It's all part of "how to think mathematically" rather than some particular weird device that one studies and forgets.
I agree, and including delta-epsilon proofs in calculus 1 seemed like a way for the curriculum authors to feel good that they were “teaching proof techniques” to these students, when in reality they are doing no such thing. I later did an MS in math, and loved the proofs, including delta-epsilon proofs…after taking a one semester intro to proofs class that focused just on practicing logic and basic proof techniques
If you want to do "exact" computation with real numbers (meaning, be able to rigorously bound your results) you just can't avoid epsilon-delta reasoning. That's quite practical, even though in most applied settings we just rely on floating point approximations and try to deal with numerical round-off errors in a rather ad-hoc way.
I am not aware of any good one, but I realized you could probably mechanically extract such a map from Lean's mathlib[0][1].
Since Lean builds everything from scratch, this should be doable, albeit Lean builds everything on top of type theory which is not the only choice possible. Different foundations will result in a different graph.
Also the best way to learn math is probably not by following this sort of graph, it would be far too abstract and disconnected from both the real world and usual practical applications.
Garrity's All the Math You Missed book, mentioned elsewhere in the comments, draws a nice map of subjects, along with little introductions and book recommendations. The map is good for continuous mathematics, but IMHO fails to consider logic and type theory, which is a bit odd given the separate chapter for category theory. It also does not do proper justice to computation and clumps everything together under the label "algorithms".
Good alternatives are The Princeton Companion to Mathematics by Gowers and Mathematics: Its content, Methods and Meaning by Alekxandrov, Kolmogorov, et al. Those present much more detailed maps so YMMV.
> Category Theory is easy because it starts from nothing, literally.
It has virtually no prerequisites, at least in classical mathematics. But I wouldn’t call it ‘easy’ (indeed, many proficient in elementary calculus and so on find it very hard). If you study category theory with no knowledge of any of the concepts it’s designed to abstract it’s not going to make any sense and the whole exercise is pointless. You may be able to follow it and complete exercises, but you won’t actually grok it.
It's sufficiently general as to be approachable from all angles but to actually understand why anything is being discussed I think category theory requires a certain amount of background material.
Yes, mathematical maturity is a consideration, but working carefully through just one mid-college math book that is based on theorems and proofs is a reliable cure. The consideration is not really big since the main goal remains: Just prove the theorems.
(2) Something missing: As a grad student
studying the Kuhn-Tucker (KT) conditions
and the constraint qualifications (CQ),
there was interest in implications among
the CQs. Two of the famous CQ were (a)
the Zangwill and the (b) KT, but the
implications between them were "missing".
So, that was a problem, a theorem "need to
prove". My approach was to look for a
counterexample among wildly goofy sets,
e.g., the Mandelbrot set or Brownian
motion. As appropriate for the KT work,
both sets were closed. Hmm .... So,
needed an optimization objective
function to be minimized. So, ..., soon
enough, for each closed set there is a
real valued function zero on the closed
set, strictly positive otherwise, and
infinitely differentiable. Then I had a
counter example. Two weeks of work in a
reading course. Published it.
Was doing some AI for monitoring but
wanted a better approach, the "need".
Used the probabilistic concept tightness
to get another approach, the basis of the
"proof", widely applicable because still
distribution free (i.e., made no
assumptions about probability
distributions, e.g., Gaussian). Published
it.
the FedEx BoD wanted some revenue
projections, wanted so much it nearly
killed FedEx. So, as in that Hacker News
URL, got some "intuition", ..., got a
simple differential equation (the
theorem), solved that (the proof).
Currently my startup needed some progress,
and I formulated a suitable theorem and
proved it.
A concern about such theorems and proofs,
for the published ones, the check has yet
to arrive.
Was eating lunch with some well known
mathematicians, and they asked what I was
working on. I explained, "scheduling the
fleet at FedEx, which airplanes go to what
cities in what order". Immediately one of
the mathematicians with contempt scoffed
and said "the traveling salesman problem"
as if that was the "theorem" to be proved,
i.e., P = NP.
Nope: I was just trying to save FedEx
some money. So, my approach was 0-1
integer linear programming (ILP) set
covering; that this is in
NP-Complete was to me next to irrelevant; I
just wanted feasible solutions that would
save money. Maybe over a year the savings
would be some $millions, but each feasible
solution might be $1000 above an optimal
solution. At 365 days a year, I'd leave
$365,000 on the table. Fine with me!!!
To the mathematician, all that
consideration of money was irrelevant --
he wanted to focus on P = NP and regarded
that as too difficult (it still is) and I
was foolish for working on it (I wasn't
working on it). In short I was counting
the millions to be saved, not the
thousands of saving to be missed.
Later there was a 0-1 ILP with 40,000
constraints and 600,000 variables. I used
the IBM OSL (Optimization Subroutine
Library) and in 900 primal-dual iteration
for Lagrangian relaxation got a feasible
solution within 0.025% of optimality.
Lesson: O-1 ILP can be a good tool in
practice, sometimes can save a lot of
money.
So the well-known mathematician and I
disagreed on your "which theorems you need
to prove"!!!
Of course there is the now famous
Garey and Johnson, Computers, and
Intractability, Bell Labs, 1979.
The authors were trying to find a least
cost design for some Bell network.
On pages 2-3 we see some cartoons with
"I can't find an efficient algorithm, I
guess I'm just too dumb."
and
"I can't find an efficient algorithm, but
neither can all these famous people."
It turns out by "an efficient algorithm"
they meant (a) getting least cost
solutions, least down to the last tiny
fraction of a penny, (b) to worst case
problems, (c) guaranteed, (d) with
computer time growing no faster than some
polynomial in the size of the problem.
I just wanted to save FedEx some $millions
a year.
The famous mathematician insisted on
(a)-(d) or no savings at all.
It's a common issues with self-learners, mathematics or not: there is no perfect course out there, and switching from courses to courses can be wasteful.
In my experience, focusing on a single, good-enough course (when in doubt, go for a famous/respected author/field contributor) and looking for other sources once in a while, has been the best approach.
Applies the same to job search too. Find one that is good enough and then work from there for future prospects. Often, the definition of “good” changes over time as priorities in life change.
We do this with all demanding endeavours. Circling around the tactical perimeter is easier than knuckling down and getting it done. But it's close enough that we can fool ourselves into thinking we are being productive.
Many examples:
- It's easier to research the "best textbook for me" than it is to study and do problems.
- It's easier to read about the optimal periodization cycle while sitting on the couch than it is to go sweat in gym.
- It's easier to read about dieting (it must be the best diet for meee!) than it is to just stop ordering pizza.
- It's easier to order business cards and redesign your logo than it is to find customers.
- It's easier to fiddle with your vim config than it is sit down and write code.
Unless you are already in top 10%, focusing on optimality is a distraction.
And if you pick up the wrong one, you might just end up dropping the whole ordeal. It's not so black and white, it makes sense to spend a bit of time and figuring out a good resource. At the least you'll get a sense of the domain's main trunk of knowledge, get into the jargon, etc.
Yeah, this is it. A year ago if I tried to find the perfect textbook to learn Linear Algebra, I would still be looking.
There are certainly good and bad textbooks, and a book good for many people might be unsuitable for your style, your goals, and your background. But there are plenty of good enough textbooks, trudging through any of them will yield far more benefits than getting that ideal book.
I like that he leaves determinants to a later chapter and doesn't _start_ with them, I never understood why they were useful or made sense. His view, represented on the cover, is great for learning
They're fine where they are useful, I guess, but my undergrad put way too much emphasis on them when they're not intuitive, don't help (me) much with comprehension, and aren't useful in that many cases compared to the other techniques.
Surely the optimal solution would be to spend a few hours / days in the first week picking the textbook, then 51 weeks studying it, as opposed to literally picking the first one you see and studying it for 52 weeks.
Not equally, better. It's intended as a book for learning the concepts of Linear Algebra intuitively and with some introductory rigor, before doing it "right" in a professional way.
This echoes my own experience. I made an attempt at andrew Ng's revised machine learning course last year. I could understand well enough at a high level how gradient descent worked but its been 20 years since I did any calculus and the level to which my ability to even do algebra had atrophied greatly. To their credit, they derive the equations for you but I did't see how I was going to be able to really apply any of what I learned when the caclulus "handwaiving" is probably the most crucial step.
I'm a fulltime CTO so finding textbooks that can fill in the gaps and finding endless problem sets to solve was just not going to work. Luckiy, A good friend of mind from hack reactor clued me in to mathacademy. I would argue thats its probably one of the biggest underated resources for getting back in mathematical shape. I've been setting aside an hour a day to just grind through the lessons and problem sets that it throws at me. it uses spaced repetition along with an inital placement test to figure out what you're weak at and just hits you with those problems as you improve.
echoing the sentiment in the article, you'll get better just grinding though different problem sets consistently each day with the occasional metaphorical boss battle. Once you realize that, actually getting better at math is more of a logistical challenge (having to track down skill appropriate problems to cut your teeth on) Mathacademy basically automates that completely for you. I've gone from giving up on ever getting into this machine learnign stuff to looking forward to spending next year taking on deep learning.
PS: not paid by mathacademy.com. just an incredibly pleased custoner
Also PS: didn't realize you worked at math academy. any plans on expanding into physics problems? would LOVE these ideas to delve back into phsyics. (especially circuits.)
Why not work with a tutor? Also I found the fast.ai videos good for explaining stuff like gradient descent.
I don't see spaced repetition being useful for theoretical math though maybe it is ok for calculation. Main thing as you say is grind out problem sets, and that's more a question of logistics and motivation than finding the right textbook.
I've always been skeptical of sites like mathacademy but I'll take a look at it.
I had considered working with a tutor. but honestly, its not hard to understand math from the textbooks. The main value of the tutor is that they figure out what you're weak at. however cost and time come into the equation. with a tutor, you have to plan ahead. you also can't have a tutor by your side everytime you practice math.
> I don't see spaced repetition being useful for theoretical math though maybe it is ok for calculation.
what Ive learned over the years over multiple disciplines is that really understanding theory is built on a strong intuition and a strong intuition is built on LOTS of practice. if you're solving the problem a lot, you are engaging with it on multiple levels which is going to force your brain to understand it on a level you will simply not get from simply watching a video on the subject. In much the same way that watching a video on react does not make you a web developer.
A tutor can give you an opinionated approach to a subject. They can offer you what they believe are the best (well, good) references. They can give you intuitive explanations for why things are true - the kind of explanation that is missing in a typical “rigorous” book. They can tell you what lines make sense to explore at your level and what things you should save wondering about for later. A tutor is another name for a professor, if you expect enough.
Studying statistics / ML, I absolutely found that there were “truths” from the text that I would try to prove with simulations and … couldn’t really reproduce. Having somebody tell you that “that chapter is good but those particular statements are controversial / wrong / not exactly saying what you think they are saying” is really valuable.
In the end, my opinion is that if you find it easy to learn from a text, probing the boundaries of your understanding could be good. Some stuff is easy. Other stuff just sounds easy but there is a deeper understanding to acquire.
Wow! Love running into MA users. Your comment totally made my day. Thanks for the kind words and I'm so happy that the system is working out for you. After all the work we've put into building this thing, it's the best feeling ever to hear about positive impact it's having on people's lives.
> any plans on expanding into physics problems? would LOVE these ideas to delve back into phsyics. (especially circuits.)
Our grand plan is to completely fill out out math courses, then expand to other related fields such as computer science and physics.
I've been wanting to read books like Algorithms for Optimization (https://www.amazon.com/gp/product/0262039427/) and Probabilistic Machine Learning (https://www.amazon.com/gp/product/0262046822/) but when I open them up I immediately think "wow I really need to refresh my calculus because I have no idea what's going on." It sounds like I should look into Math Academy for relearning old Calculus and learning some new stuff as well.
2. We designed a Mathematical Foundations course sequence specifically for adults who want to get up to speed or relearn math skills they have forgotten as preparation for Mathematics for Machine Learning and other university math courses. More info here: https://www.mathacademy.com/adult-students.
3. When you start on the system we assess not only your knowledge of the course you're placing into, but also lower-level foundational topics. If we detect that you have any gaps in your foundations (most learners do, especially adults who have been out of the game for a while), then we'll automatically add them to your learning plan and make sure you learn them before we give you any more advanced topics that depend on them. More info here: https://www.mathacademy.com/how-our-ai-works#diagnostic-algo...
>it's the best feeling ever to hear about positive impact it's having on people's lives.
you guys deserve it! its a great product. admittedly its a bit spartan/plain in terms of ui but I respect that you guys focus on substance over useless shit that emphasizes edutainment over actual actionable knowledge cough brilliant cough
Yeah, we realize the UI is a bit spartan. No disagreement there; it's a valid critique. As you say, we're focusing on functionality and content first since that's what really moves the needle on people's learning -- but yes, we're also going to be adding some bells and whistles in the future.
I take classes at community college to up/revise skills in foundational mathematics. What I love is the small class sizes, the presence of adult students and the teacher engagement. This alongside the need to finish hw, keeps me on track
I signed up for mathacademy and I'm extremely overwhelmed by the courses. I'm not exactly sure what course I am supposed to start with. I went with Linear Algebra and I'm going through the diagnostics but I'm struggling with all the questions so far.
Hey! So, sometimes adult learners sign up for a university-level math class not realizing how serious our courses are or how much foundational knowledge is necessary (or how much of it they never actually learned during school, or how much of it they've forgotten since then).
That's totally fine and all it means is you may need to start off in a lower course to shore up your missing foundations [1].
There have been so many people in this situation that we actually designed a Mathematical Foundations course sequence specifically for adults who want to get up to speed or relearn math skills they have forgotten (from fractions through calculus) as preparation for upper-level university math courses. More info here: https://www.mathacademy.com/adult-students.
Please do let me know if you have any follow-up questions about that or if anything is unclear. I'm always happy to chat with people who are serious about learning math.
---
Footnotes
[1] Note that we do check for missing foundations during diagnostics, and any that we find we'll add to your learning plan -- but currently there's a limit to how far we look back. For Linear Algebra, we only look back to the beginning of Algebra 2. So if you're rusty on any Algebra 1 stuff -- factoring, quadratic equations, systems of linear equations, etc. -- or arithmetic stuff like working with fractions/exponents, then you'll need to drop back to a lower course to shore up those foundations.
Sounds like a plan! If you run into any issues at all, feel free to ping me on X/Twitter (https://x.com/justinskycak) or shoot me an email (justin@mathacademy.com). The first piece of the puzzle to learning math is just getting on the rails at the appropriate level and I want to make sure we help you get over that hump.
Faithfulness to a single source is the biggest reason I see for failure In students. Be promiscuous. If a page, chapter, or even whole book bores you, scan ahead, put it on trial for a bit, and if it doesn't redeem itself quickly, replace it. The same goes (to the extent possible) for courses, teachers and even whole media. Only once you've tried the whole universe do you have reason to lower your standards and try something again from that universe that didn't meet your earlier ones. A book isn't a friend. There are no brownie points for completion.
Also most subjects are like that too. If you really want to know a natural language and hate the verb rules, focus on the rest of the language. If you soak up the verbs more slowly you'll still be hnderstandable, and you'll have fun, and most importantly you won't give up.
And programming languages are especially like this. Don't like class methods? Good! They suck anyway. Keep your functions pure. Don't like generics? Well that's a shame but it didn't stop the first many generations of Go programmers who couldn't use them if they wanted to. Etc.
because the topic, textbooks, is pedagogic, when i read your "Don't like class methods?" I thought the follow on was going to be "find a new professor!"
In all seriousness, this seems to carry risk of never doing anything deep or hard. In particular, I've been programming for long enough, that I can be casual about many programming languages until I hit something which is actually new, such as in Rust or Prolog.
Promiscuous doesn't have to mean having a low tolerance for difficulty, but everything else you wrote seems to support that. So, are you saying that enduring difficulty is unnecessary, or did you mean something different?
Not the person you responded to, but I actually take a similar approach as them so maybe I can explain.
Firstly, difficulty and fun are not always directly correlated. Something can be difficult but fun, or difficult and unfun.
Following on my first point, different activities or goals usually have aspects that are more or less fun. It’s better to start with the easy and fun stuff. You don’t have to swallow the ocean.
Now this last part really is dependent on the type of person you are. For me, once I’ve gotten into a subject, I just become more curious about it. The aspects of the subject that were unfun at first are now interesting. I’m more invested and I’m more curious. And if I’m more curious, it’s more fun.
To sum it up, it’s not that one can avoid enduring difficulty. It’s more about harnessing your own strengths and curiosity.
There needs to be more resources for self-learning. Solutions need to be provided for problems, with clear explanations. It's a different mindset from a formal academic setting, where there's a strong focus on cheating prevention.
Funny you say that. I recently picked up a textbook on Corporate Finance for self-learning purposes. Going through the problem sets, it's not really that useful if you have no idea if you got the answer right or not. Looked all around online for where to buy the solutions manual, ended up just calling the publisher to ask. Turns out they refuse to sell the solutions manual to anyone not a registered instructor at a University.
It took like 5 minutes on the phone to even explain to them that I'm reading the book for self learning purposes. Like they'd never encountered such a thing. Even after explaining, they wouldn't let me have the solutions.
I ended up just going on the black market, and finding some anonymous person to sell me the solutions on WhatsApp for $25.
I just googled, and by the way when you want to google for something with qustionable legality, it's best to use search engines not based in western countries, such as Yandex. There ended up being a lot of sites advertising the solutions manual for sale. But also, there are entire subreddits where these people post threads advertising that they have whatever ebook/solutions for sale with a whatsapp phone number included in the content. I messaged on Whatsapp and sent a paypal payment (yes, I had no recourse, it was a leap of faith) but it's not that much a risk since it's only $25.
Agreed. It's frustrating to solve a problem and being unable to check if it is correct. It's even more frustrating knowing that if you had the answer, it would help you to solve the problem. Sometimes the answer pushes you in the right direction to figure out how to solve the problem.
> with clear explanations.
Hopefully separate from the answers.
For programming exercises, we should be given datasets so that we can tests whether our code works or not. Heck books should provide links to unit tests.
I'd go a bit deeper: as we developed, in particular with the advent of the Internet, we went from scarcity of information to spectacular opulence. This demands different studying habits that what we had 30 years ago or so.
For example, we need to find ways to filter out noise from signal, or to connect scattered bits of knowledge from various sources to get intelligible solutions to problems (most problems can be solved by googling around, especially in maths/physics, because people of all levels have been asking/answering questions for Internet points e.g. on Stack Exchange & cie for many years now, but — take it as a feature — you have to work a little to get there).
EDIT: regarding solutions, it's not just about preventing cheating, it's because teachers wants you to do the work. The point isn't necessarily to succeed in solving problems, but more to have you try, get creative, etc.
Perseverance is crucial to move forwards. But they could still provide clear and/or progressive solutions, I fully agree.
I'd really like a Khan Academy-like site, maybe with explanations from different textbooks for each concept. Of course then you'd need a good set of diverse problems or a way to generate such problems.
It's a shame that Khan has gotten less diverse in subjects. They used to have medicine and bio stuff among others, but now it's mostly maths. The maths lectures and exercises are still great, but even those seem limited (still no discrete maths last time I looked).
A Textbook is good enough for self learning. Almost all university learning is "self learning", at least that has been the case for my mathematical training.
> It's a different mindset from a formal academic setting, where there's a strong focus on cheating prevention.
What? Who cares about cheating prevention, most of my classes had oral exams, you can't cheat there.
These were master and late Bachelor courses, so 30 people at most. Exams lasted around 60 minutes, of which around 45 were questions.
>how many questions are asked?
Totally depends on the subject and how the exam goes.
>What's it like in general?
Your professor is poking you with questions. Usually he has prepared some general questions and then asks follow ups. It might go something like this. "What is X Theorem? What does it represent geometrically? Does conditions Z need to be true for the Theorems to hold? Can you name a counter example? How does the proof (discussed in lecture) look like? How exactly do you construct that part? Where do you need that condition? Here is a similar theorem (not discussed in class), can you outline a proof for this?"
I agree with you. It's been my experience though that tracking down solutions manuals for textbooks is very hard. Presumably because they want them out of the hands of students (to prevent cheating).
Have you tried to read any of the literature on the Risch algorithm? If you haven't, you might want to get started by taking a look at the paper "Integration in Finite Terms" by Rosenlicht [1] and chasing down some of the references mentioned in [2].
Of course, in the real world we don't give up on integrals just because they can't be expressed in terms of elementary functions. Usually we also check if the result happens to be a hypergeometric function, such as a Bessel function. If you want to get started on understanding hypergeometric functions, maybe try reading [3] (as well as the tangentially related book "A = B" [4]).
I hate when I buy an interesting math book and then it's like "oh wait, no solutions." And then I end up going on GitHub hoping for community-worked solutions.
For a business I had to learn how to design parts for mass production injection molded plastic. It’s simple in concept but the devil is in the details, of which there are a great many.
I couldn’t find a general non-fiction book with the information I needed, so I found and ordered the best textbook I could find on the subject.
Teaching yourself from textbooks, I think you just have to be prepared for a serious grind, involving lot’s of looking up math and other terms that you either forgot or never knew, trips down the Wikipedia rabbit hole, etc.
Those books are, for the most part, designed as teaching tools to accompany classroom learning — sometimes the whole class is going to come and not have a clue what they’ve read, and it’ll be via class or office hours they figure out WTF is going on. These books are not designed for autodidacts.
I could be less charitable and talk about a lack of competitive pressure and perverse incentives for selection of academic books, but I’ll leave it at that.
Worked out for me and the manufacturer I was working with said we were the most professional part designers he’d worked with (we were helped tremendously by software I’d written), he wasn’t a bullshitter generally, so I’m inclined to believe it.
You can be successful but it’s going to take a lot more energy than it would with a nice trade book with an animal on the cover.
Not sure if it's needed for you anymore, but I'll post it anyways, here are some links to documents from Bayer MaterialScience I have collected for designing injection molded parts. Pretty good general resources for the subject:
I have a PhD in math and read many textbooks. The tried and true approach: gather 20 books on the subject and read the first couple pages of each. It should jump out to you right away which one is the best for you.
Unless you have access to a particularly mathy library, are fond of piracy, or have an exceptionally well stocked second hand bookshop nearby: that's going to be an expensive undertaking.
(Says the guy who has dozens of untouched maths books lying around)
3. Must have many exercises AND a solution book for at least some of them.
Context: prepraing to study all undergraduate Math and Physics courses to get a holding of General Relativity. Since I graduated as a Math Master but forgot most of it, I have to start from Calculus and Linear Algebra. I count about 8-10 courses for the journey.
I download the most textbooks I can, for evaluating which appeal to me, and then use that one. If I don't like a chapter on it, I look for another one.
Don't machine learning books kind of fill that gap? e.g. Bishop uses probabilistic reasoning, Elements of Statistical Learning seems to be heavy on frequentist stats (haven't read it though), etc.
My 2 cents on the topic is that for the most part I've had a lot of success choosing what to be interested in based on good book recommendations on the internet rather than looking for good book recommendations on the internet based on what im interested in
If you're learning for fun, probably every topic in the history of the universe can be interesting given the right approach
My experience with ex-USSR published books in mathematics and physics was very pleasant -and maybe the most pleasant ever. They rather provide simple examples and easy to solve exercises which thing always boosted my confidence psychologically to the point when I faced exams I was thinking: "This is just another set of exercises which I will succeed to solve, like I did multiple times".
I also had the same experience with the American published Schaum's books.
Apart from these, I almost got PTSDs with other publishers: they give you very hard to solve exercises to the point you would feel you are useless and incompetent to face even the easiest exams.
The track record of math teaching in Russia has been impressive, Kolmogorov, Dynkin, etc.
Too much in the US, there is secret culture: Math teaching is to filter, and the students have to prove themselves against challenges presented and deliberately constructed to cause a significant fraction of the students to fail.
Eventually I had enough in accomplishments just to quit making an effort to prove myself again. Then when people started to attack me, I'd let them go too far and then use some accomplishment, old or new, to shoot them down.
One of the best ones: At the end of 8th grade arithmetic, the teacher pulled me aside, alone, and with care and trying to be good, gave me a D in her class, said I could take High School Arithmetic, and should take no more math.
It's true: I was no good in much of her course, the part that needed careful writing for the arithmetic, say, 1234 times 5678. Why? I was an 8th grade male with not so good manual dexterity -- the girls were MUCH better (standard). My 6th grade teacher saw the same. But as a senior, the SAT scores came back, and the 6th grade teacher read them to me: For the verbal score, trying to be nice, and with her low expectations, said "Very good". Not really! Then for the Math SAT she stopped. Afraid. "There must he something wrong." Yup, there was, and had been for 12 painful years. Of 1-2-3 in the class, I was #2. #3 was voted "Most Intellectual" and went to MIT. Right, since there was no Yeshiva in town, 1 and 3 were Jewish!!!
It continued that way: Get attacked and attack back with some accomplishment and win.
But, net, a dumb situation.
Solution? Own a successful startup! Working on it!
This is the wrong way to look at it. If a textbook is bad, or is good but too difficult (i.e. bad) you will stop reading it. The alternative to self-studying a good textbook isn't studying a bad one, but studying none.
The best time to start studying mathematics was when you were 4, with multiple private tutors and supportive-yet-not-overbearing parents who are also math educators.
The second best time is right now, with whatever materials you have in front of you.
I found the text and workbooks for my mathematics courses at the Open University in the Netherlands absolutely fantastic. They are created / supervised by a famous (educational) mathematician in NL named Jan van de Craats.
The method was designed for self study, and the absolute best I had ever worked through. Perhaps material from other similar institutes are of similar quality?
I'm currently working through calculus. I picked up Spivak's and Apostol's books-- probably the most recommended calc books on the internet. Aaaand... they're ok. There are many parts that are confusing, not because calculus is "hard", but because the authors didn't do any user testing. If they actually reworked the books to minimize real students struggling, the books would have been much much easier to self-study from.
I eventually found David Galvin's calculus notes[1] from University of Notre Dame. He basically follows Spivak closely, but reorganized the material a bit in response to user testing. The notes aren't perfect, but much much easier to follow. Same experience with Terence Tao's linear algebra notes[2].
I think book authors, even very highly respect ones, often kind of suck because they optimize for writing a beautiful book, not for minimizing student confusion. Once you struggle through the confusing parts, yes, the book is beautiful. But it's supposed to be written for people to learn, not for experts to appreciate! Notes written by professors who teach smart kids, optimize for minimizing confusion, and do real user testing are often much better than the best books, in my experience.
Exactly. It's very hard to judge how hard it is to understand something when you already understand it. This is also why video games with puzzles need extensive play testing. Because the puzzle designer, already understanding how to solve the puzzle, is terrible at judging how hard it is to understand the puzzle.
It seems that our counterfactual reasoning ability largely breaks down when it comes to understanding. For some reason we can't evaluate the question "what would I think if I didn't already understand this?"
> Notes written by professors who teach smart kids, optimize for minimizing confusion, and do real user testing are often much better than the best books, in my experience.
Yeah. Though smart kids get confused less easily, which means lecture notes written to teach dumb kids are even less confusing.
Exercises in textbooks usually focus on proofs, but mathematics isn't just about proving theorems. Mathematics is also about:
1. Understanding mathematical concepts (e.g. what is an "acyclic" relation? What is KL divergence?) and theories (several interrelated concepts, e.g. decision theory). This also includes knowing why those concepts are important in the first place, which is often neglected.
2. Knowing the meaning of mathematical notation and technical terms, e.g. to be able to read papers in some field. Papers are often full of mathematical and other jargon while otherwise not necessarily being difficult to follow.
3. Learning mathematical formulas (e.g. Bayes' rule) and algorithms (e.g. differentiation), in order to solve specific problems by calculation or computation (mostly in applied mathematics, more rarely in pure mathematics)
4. Proving conjectures (mostly in pure mathematics, less often in applied mathematics)
5. Learning how to formalize informal problems using mathematical concepts and theories (by applying conceptual understanding gained by 1) in order to understand the problem better, or to make it easier to solve, e.g. by employing calculation (2). (This is often done in engineering and science)
Problem sets in textbooks often focus on proofs (4) or some more difficult algorithms (3) but less on the other applications of mathematics.
They could also check conceptual understanding (1) by asking the reader to explain some concept in their own terms, or how two different concepts relate to each other, or which concepts various example cases have in common, or how the cases differ on a conceptual level. Though verifying the answers might require a human teacher.
5) could be taught by coming up with word problems from a scientific or engineering (or economics etc) example, where the solution is easy once the correct formalization is known.
Unfortunately it is hard to come up with such artificial word problems in which the correct formalization is unique, non-trivial, and doesn't require technical background knowledge from engineering/science etc.
Moreover, in the real world, the difficulty with formalization is often to recognize in the first place that there is some problem that could be formalized, which can't be replicated in an artificial word problem.
Overall, coming up with good exercises, especially for 5, but also partly for 1, might require the writer of the textbook to know a lot of possible practical applications. Writers of math textbooks are often mathematicians, so they probably don't know a lot about engineering, computer science, empirical science etc in order to come up with good word problems.
I avoid any book that have complicated equations on the first few pages. There is a place for that but in this case the author is just trying to show everyone how smart and complex they are rather than trying to teach people.
IMO textbooks are dead and they were never a great source of knowledge to begin with. The idea of reading some piece of text written by someone, then edited by someone else, and expressed in an "appropriate" style and language (think formal language which uses fancy jargon), and then having to robotically solve some end-of-chapter problems is just absurd. My experience tells me the "gems" of knowledge are often found when authors and experts just say whatever the fuck they want on a forum or in personal discussions. That obviously presumes those authors actually know what they're talking about. So many math, physics, engineering, etc. books are written by people who had no business talking about those topics.
Previously, I thought certain math topics were "hard" (e.g. category theory) while others were supposed to be "easy" (e.g. Calc I). I beat myself up for struggling with the "easy" topics and believe this precluded me from ever tackling "hard" topics.
I was thirty-something years old when I finally realized math has a well-documented maturity model, just like emotional maturity or financial maturity. This realization inspired me to go back and take a few math classes that I had previously labeled as "too hard," with the mindset that I was progressing my math maturity.
My point is that choosing an "age-appropriate" (in terms of math maturity, not actual calendar age) textbook is important. I also find it extremely helpful to chat with people who are more mathematically mature than I am, in the same way it's helpful to seek advice from an older sibling.