Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.
I've got Mathematics for the Nonmathematician by Kline and that's kinda heading the right way, but what about whole courses of study? More books? It's more of an introduction than a thorough resource or course, and feels like it needs another four or five volumes and a lot more exercises to be really useful.
I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.
My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult.
The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses.
I still remember freshman year, showing up to the standard intro calc course, and dropping it as quickly as I could in favor of my uni's equivalent of Math 55 (i.e. the hardest u/g intro math course) because of how asinine I found the content...
I'm really wary of this. House wiring would be more useful than Shakespeare. That's not to knock house wiring, or statistics. I'd love to know more about both.
But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a few curves with a compass. Before Calculus, you might actually believe that numbers that can be expressed as the ratio of two integers are typical, and that numbers like pi and the square root of two are "irrational" rarities (and until calculus, you probably don't know about Euler's constant unless it was introduced in precalc as another one of those odd and rare numbers).
Look out at nature, where are the triangles, rectangles, and circles? Maybe a wasp nest? Nah, not really. Try to draw a cloud, a tree, a tiger, or a human face. How useful is that straight line or compass? How useful is a line at all, other than to hint at something you can't actually draw, maybe by implying it exists as an ever vanishing limit from above and below? Math required calculus the instant humans decided to describe the world as it is, rather than by the limits of what we impose on it.
Also - in stats, how do you know what the area is under the probability density function?
Education is a Hard Problem TM because the knowledge of those links isn't useful without sufficient existing knowledge, and you need to introduce things in a way such that the things you hand-wave away are tolerable knowledge gaps. Knowing that a CDF is an integrated PDF isn't useful for someone who doesn't even have a notion of what a PDF is. Describing a state machine that satisfies conditions X Y Z as an Abelian group is similarly not useful for someone who doesn't have an intuition for groups or fields.
My claim is that, given the finite amount of time allotted to mathematics in a secondary school curriculum, it's better to spend that time learning about medians/means, regressions, statistical tests, and so forth, instead of memorizing power laws and derivatives of trigonometric functions.
What would someone other than a scientist use regressions for?
I've used one once. I wanted to make a volume control that I thought sounded subjectively even across the range and gave the right amount of control. So I used a polynomial regression calc on a random website and gave it some data points I wanted it to go through.
But that's not actually doing a regression, just knowing it exists and computers can do them.
Is there any math that has any use whatsoever unless you know a whole lot of it and plan to do some technical projects? I thought the whole point of learning any math(Even arithmetic, since phones exist now) is just so you can learn other more advanced math, and maybe someday be a solid state chemist or something.
Statistics lets you read a scientific paper, but you don't actually need to understand it, unless you're actually going to be reviewing their raw data. Most everyday people just trust the p values and move on to wondering about confounders they forgot.
Seems like you could cover all the statistics people will actually use it about 4 hours.
Basically every person who writes papers using statistics has taken a course in statistics. Medical doctors do it, psychologists do it etc. And the statistics course you'd replace calculus with would be even lower level than that, basically worthless to everyone. At least calculus trains you to think about rate of change and slopes, statistics gives you nothing practical. It isn't like people remember those formulas so they can apply them in their daily lives.
People don't need to remember and apply formulas. They just need an intuition that they could be applied and how that might change the appearance of the statistics, as a kind of gauge of reliability. A tool for rationalization.
The problem with this approach is your basically teaching this as a side effect and hoping they're smart enough to actually connect all of that together vs. teaching those concepts directly and being far more understandable.
If you want people to understand that numbers are concepts / descriptions of something potentially infinite, and that it's ok to totally work with non-finite describable numbers without ever really writing out their values, just fucking start with that. I honestly think the start of most undergrad math curriculums should be a logic & foundations class with proofs, sets, number theory and the whole "numbers are more logical concepts, not really specific values" vs. calculus. Then teach calculus, with a big dose of 'what is infinity, really?'.
I think it's actually a great disservice and the hand wave that most calculus math classes do about those exact concepts really fucks over a lot of people. It makes calculus a weed out class because the fundamentals are not explained properly so a good amount of people just go into it as yet another thing they have to ape without real understanding. And for the people who don't work well with things they half understand, they really struggle, like I did. It really made my computer science degree a lot worse, because of the instance of starting all undergrad math with hand wavy calculus, for 3 or 4 classes.
I even wrote blog articles about it, it was probably the worse part of my computer science degree, and if math education was done differently where they didn't handwave, I probably would've had a much better time.
Well, AP statistics covered the area under a probability distribution function and people seemed to understand that -- you look up the answer in a table (or use the TI-83 function). Presumably they'd do the same for a cumulative distribution function.
Teaching people that there are lots of problems where solving them numerically is the best way of solving them is often just the truth. Also, you then get to discuss some of the really cool numerical techniques which tend to get short-changed in traditional math classes.
We already do that in elementary and high school with logarithms, square roots, sin, cos, tan, pi, radians, e, irrational & imaginary numbers, etc. The tables are just built into pocket calculators instead of paper. Even Pythagoras is a bit weird.
House wiring and Shakespeare are completely different subjects. (I’d expect someone professing Shakespeare to avoid such a straw man, but I see your point!)
Most people don’t need mathematical statistics, but far more will find meaning/interest in the immediate applications of lighter statistics than the another-math-class-full-of-equations that calculus feels like to many.
Anecdotally, most universities are scrambling to add lighter data science courses to their humanities majors. These all teach basic stats, and many ignore the low-level calculus required for those methods (again, speaking of the humanities versions here).
We dont need proofs, calculus, reasons, process, principles and theorems actually. Just results and facts are in demand for our lives. But it may make us subordinate to people who have invented theories of statistics or calculus, because we have no choice but refer to and based on their theories, I mean, depend on them even if the theories are absolutely correct or truth of the universe.
In my experience, neither intro calc classes nor intro stats classes are that useful at teaching the underlying principles. The classes mostly just test whether students can learn an algorithm to solve a certain class of problem, whether it's differentiating a function or running a statistical test for scenario X.
I was a math major and didn't understand calc all that well until I took advanced calc, and I didn't understand statistics all that well until I took the upper-level mathematical stats courses.
But Shakespeare is incredibly useless. People don't even talk or write like that anymore. Any imagined benefit of Shakespeare is just fanciful wishful thinking.
When I was younger, I thought like you did. Now that I get older, the value of culture and an understanding of where it came from, how it developed, makes my life just so much richer. That does not necessarily mean that I need to have memorised the whole corpus of Shakespeare, but understanding a reference when it comes around is adding more layers of meaning to other works.
This does not only extend to literature. I have had similar experiences with religion (which I thought of as utterly useless and potentially dangerous, as it "deactivates critical thinking and creates sheep-people which will follow whatever their shepherd/priest/guru tells them"), creative arts (both painting as well as music), and the basic sciences (biology, chemistry, physics).
Now, I don't genetically engineer the stuff in my garden, but understanding Mendel was useful. I don't speak in iambic pentameters, but I can appreciate when it is being used as a stylistic choice. I am in no way a church-going, devout Christian, but I have found meaning in some of the deeper wisdom enshrined in the Bible (and the Koran, and the Bhagavad Gita, and about half a dozen Sutras.)
Would I have come to that if I didn't have primers in school? Maybe. But the primers certainly helped.
On the other hand, I didn't learn plumbing in school, or laying electrical wires, or "doing my taxes", but these are things I can simply have someone do for me who is a lot better equipped and trained to do so, or - for the small stuff - I can figure them out on the fly.
There is a ton of stuff that makes life richer. That doesn't mean we should be teaching it at schools. I can think of movies, games, novels, TV shows, blogs, and music that have had orders of magnitude more impact on my life and thinking than Shakespeare. Should we be teaching any of that stuff at school?
I'm willing to believe that some people like Shakespeare, but it's a small minority. You can tell by the number of people who read Shakespeare for pleasure - is that number larger or smaller than the number of people who read JK Rowling? Why should we teach the entertainment that a small number of people prefer in schools? I believe the only reason we actually do is tradition.
You mention that you can simply have someone learned in plumbing or sundry skill do those tasks for you. I can do one better. I can simply have no one read Shakespeare for me and I can not read it at all and nothing is lost. That is, of course, because unlike plumbing or laying wire there is no reason to need Shakespeare.
It's good that you enjoy Shakespeare, but some people enjoy plumbing. Plus, plumbing has a practical purpose, unlike Shakespeare. There is no real reason to teach Shakespeare, other than tradition, and people trying to seem smart or educated. There are many other subjects that make a much stronger case for deserving to be in school curriculum.
I read a quote somewhere that goes something like "A society that separates warriors and scholars will have an army led by fools and thinking done by cowards." Similar logic, with different vocabulary, applies, I think, to a society where scholars can't do manual labor.
I like to believe that school works as a catalyst here, showing you stuff you wouldn't normally encounter - you have no trouble accessing popular culture, so there's no need to teach you about that (that's being said, one of my more memorable music lessons was the teacher analysing the composition of Pink Floyd's "Shine on you crazy diamond" with us. I can absolutely see Harry Potter becoming an object of academic study and literature teaching in 50-70 years, it is happening with Tolkien already).
On the other hand, less-than-popular culture still is the foundation for our popular culture today - the amount of times "Romeo and Juliet" or "Macbeth" has been adapted, referenced and deconstructed in (popular) media is astonishing, and without knowing the original, you wouldn't fully get the modern references.
You don't need to love Shakespeare (I don't exactly, though I like to see the "Scottish play" every few years) to reap these benefits.
School is not about what people like. If you go by that measure, there's no real need in math, because you will find a lot less people doing calculus for fun, as compared to people who play fantasy football. Of course, you can't really understand the ideas behind fantasy football if you don't know maths, but who cares, it's not popular...
Please note that I did not claim physical labour has no value and one should not know about it. I said that it's relatively easy to pick up, and that it may be more worthwhile to get a professional when the job demands it (a professional, I would like to add, who learned his craft after school, ideally in some kind of apprenticeship). In my country, to get a driver's license, you need to know basic first-aid procedures (and yes, there's a one-day course you need to attend). That doesn't make you a neurosurgeon, and if you have persistent headaches, it's probably better to ask a professional than to depend on me with my first-aid course - or some guy who "learned about surgery techniques" in high school.
Shakespeare may have a profound impact in the English speaking world. But from the perspective from any other language than English, Shakespeare is an esoteric subject with limited impact. I do agree that the Christian Bible is of immense importance to European and American culture. Today we blithely dismiss all things religious, but I would argue 99% of modern European/American culture has been affected in some way by Christianity. Pretending it doesn't exist has been a major disservice to students.
I think it is because we have gutted the idea of a liberal arts education.
Shakespeare is supposed to be read along with a massive amount of other literature going back to the Greeks. We have ditched most of that though besides Shakespeare for some reason so it doesn't make sense and makes him stick out more than he should.
Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie.
What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO.
And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).
> meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory?
Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite.
Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.
I think the problem is that students don't know these fields exist unless they took an especially strong liking to the standard school math curriculum. Going into college I couldn't have told you what most of the fields mentioned were about even broadly. Truly I had no idea what math was, because we barely talked about proofs through high school. The only reason I swung back around to learning some of it is because my upper level college CS classes introduced me to ideas that then made me want to revisit my math curriculum.
There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum.
Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.
Calculus and linear algebra continue to dominate in applied mathematics. "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed.
> "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed.
And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!
Interestingly the ascent of linear algebra is a relatively recent thing. I have an engineering degree but while I was certainly exposed to basic matrix stuff, never took a linear algebra class. When I was an undergrad, you took differential equations in addition to basic calculus for engineering. (You needed for system dynamics among other things but it was very cookbook.)
Linear algebra became a lot more interesting once you had cheap computers and Matlab.
When did you do your Engineering degree? When I did my BE Degree (2004 to 2008) Linear Algebra was included. We were introduced to the basics of linear Algebra in year 12 of high school (simple matrix manipulation, intro to vectors, dot and cross products that sort of thing). I'm Australian if that makes a difference.
The Math in my 4 year engineering degree was structured something like this:
First Semester Year 1 two math courses: Calculus 1, Linear Algebra
Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem)
First Semester year 2 two math courses: Differential Equations, Statistics for Engineers.
From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized.
A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)
I fully agree with your general point, but in the US I've found calculus to be much more dominant than linear algebra. I went to an "elite" high school that still does not offer any course in linear algebra, despite having enough juniors in AP calc to fill a room. Senior year options for those students are AP stats or multivariable calc. At my college, everyone was (and still is) required to take through multivariable calculus, but there's no such requirement for linear algebra.
I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.
Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra.
And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at.
Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.
In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others.
The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory.
And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.
I think a layperson's outlook on these topics in high school could be very interesting though. Use this outlook to help motivate learning calculus and linear algebra. With a mix of engineering survey and math survey there are plenty of kids out there that would find parts of the course that stimulate their natural interest. Then the "calculus has many practical applications" thing would feel way less contrived. I mean of course I know now that it is true, but in high school I didn't feel it because I didn't have enough broader context.
A decade+ ago when I was in school, the policy was to push honors and AP classes for "college-bound" students. Stats (not offered as honors or AP in my school) was one of the fun courses that co-op kids got to take while we were stuck in pre-calc.
In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.
Not sure I agree with that - you also have to take into account that calculus is a requirement for many other sciences. I suppose you could just of course make it a pre-req for things like that but I found that in highschool rarely did they go that deep. Physics becomes a hell of a lot easier with basic calculus for example.
If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again.
You can certainly make significant headway with some basic combinatorics and basic tables for things like the normal distribution, no calculus needed. This is in fact how most people currently learn statistics.
I mean no one integrates to find the area under subsections of the normal distribution because you have to use numerical methods for that anyway. And it only gets more complicated from there with chi-squared and various degrees of freedom etc. Tables are a perfectly fine way to deal with distributions…
Most real life integrals and differential equations are not analytically solvable anyway, so you run into tables like you do with most real life applications of logarithms.
What happens is you learn numerical approximation methods and then recreate the tables with human 'computers', like we used to do before electrical computers came into play. Or create mechanical analog computers like they did in the 1800s or greek times.
That’s question is akin to someone saying what happens when you don’t have a calculator. Everyone has one in their pocket at all times.
How would the typical person go about calculating the normal distribution by hand? Are we going to call everyone blind because they haven’t memorized e to arbitrary precision?
Yes and no. You have to understand the fundamental concepts of integration, differentiation, differential equations etc. and what they mean and represent.
All the time spent learning how to do symbolic integration by hand is however time that could be much better spent.
I also think you could and should start by teaching probability and statistics and using that to introduce calculus as and when it shows up naturally, rather than teaching calculus in the abstract first and then showing the applications much later.
> If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again.
What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?
Pretty much all the classic Newtonian formulae you memorize in high school physics classes can be derived (relatively) easily once you bring calculus in. A general problem with high school classes in particular--still somewhat true in certain university classes but much fewer--is that there are dependencies across classes. So you end up with a lot of "Memorize this thing because you don't have an advanced enough background to understand why $XYZ is the case."
In my high school experience, this was avoided if you had the desire and aptitude to just take Honors Physics instead.
To work around the fact that many of us were still in precalc, our teacher just taught us the power rule, the relationship between slopes of tangent lines, etc., without diving into the "why" of why those things worked.
That said, yeah maybe for the most basic of university of physics the whole "derive it on the fly" strategy works, I guess? But when you get to more advanced courses like mechanics of materials, you'll do yourself the favor and take to memorizing at least a few of the commonly used equations.
For tests and the like, when I was an undergrad, it was fairly common to be able to bring in a one-page "cheat sheet." The idea was that you probably could derive a lot of the stuff given time and maybe references, but you almost certainly couldn't in a 1 hour exam in addition to solving the actual problems on the test.
I'm trying to remember when I had one of those last!
I actually don't love the idea of cheat-sheets but maaaybe if it's a cumulative final I could see it being helpful? If you're taking chapter exams on some material, I think you ought to have worked enough problems so that formulas/constants are drilled into your head. But I guess too, where do you draw the line at engineering appendices? I sure don't have the MoI of every common body memorized.
> If you're taking chapter exams on some material, I think you ought to have worked enough problems so that formulas/constants are drilled into your head.
Well, sure, but, if you don't, then what's the point of punishing you? As a teacher, frankly, I'd be happy to have my students bring in any static resources they wanted to consult—I say 'static' to emphasise not, e.g., consulting a cheating site, although it's fine with me if they've pre-compiled solutions in advance to any problems they think might be interesting or important—except that (1) I think that would encourage bad study habits, and, more importantly, (2) it would be unwieldy in a packed classroom to try to have adjacent people juggling multiple textbooks, notebooks, etc.
In fact, I loved the freedom to give extended-time, fully open-book, open-note exams during the fully remote classes. I wish I could still do that; if cheating weren't so endemic under those conditions, then I would.
Generally for stem test for application of material not memorization (Medicine you need both, etc). My lower level undergrad math courses all allowed us to bring a crib sheet with equations. It helps you go back over and study/relearn. In the graduate math / physics courses a crib sheet was provided and standardized. Memorizing final PDE solutions or things like Laplace Transform tables is no bueno.
Okay I'm just a systems engineer so certainly my material isn't as intense--fair points. I just didn't care for the explicit lack of rigor I encountered in my linear algebra class. I got a 94% and I can't remember anything.
Knowing the basics of integration/derivation makes all sorts of very common concepts in physics much more intuitive - velocity, acceleration, area etc.
FWIW (and I know you're not making this claim) I don't believe Math 55 is a good example of how mathematics should be taught at large, nor do I think calling it an intro class accurately conveys what it is. It's effectively the compression of an entire undergraduate degree into a single freshman course. I say "effectively" because the material covered depends heavily on who's teaching it, but certainly anyone coming out of Math 55 can pick up any undergraduate math content trivially. For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out.
It's strongly self-selecting and as a result can afford to cover a truly insane amount of ground. The overwhelming majority of people who take it drop out (the usual dropout rate from people who take it in the first week is probably > 90%), but the people who stay almost all get As. And you will need to be almost entirely self-motivated because a lot (maybe most) of your waking hours will be thinking about math.
There's a very small minority of students for whom this is an optimal way of learning. For most students this is the quickest way to make them run screaming away from mathematics even faster than they already do.
> For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out.
Not at all, you are only prohibited from taking "freshman courses"[1]—that's what 55 is supposed to cover (definitely not all of the undergrad math curriculum), though some professors go beyond that. Many students go on to take at least some undergrad courses, with those in the 140's range being mathematical logic gems with no real counterpart in the graduate department.
[1]Students from Math 55 will have covered in 55 the material of Math 122 and Math 113. If you have taken 55, you should look first at Math 114, Math 123 and the Math 131-132 sequence. https://legacy-www.math.harvard.edu/pamphlets/courses.html
I wonder if the pamphlet's changed or I'm misremembering. I distinctly remember Math 123 and Math 114 both explicitly excluding Math 55 (It's also worth stating that Math 122 and Math 113 are definitely not freshman courses). But regardless I'd be very surprised to learn a Math 55 student took Math 114 or Math 123. I would also be surprised (although less so) to learn of a Math 55 student in the 130 series.
The Math 140 series have only become more serious courses in the last 10 - 15 years or so IIRC. The 240 series was generally where to go for serious mathematical logic courses (and generally is where you would go after e.g. a first 140 series course in set theory anyways).
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult.
Strong agree, but engineers probably need both. I'm currently watching a course on causal inference, and the tools are very much calculating gradients. And even if you just use someone else's MCMC, even in the models a differential equation or integral can randomly appear usefully.
In retrospect I should have taken a stats class in high school when I had that 1 hour gap for 1 semester, just to build a better intuition around the basic concepts.
I wonder how much statistics someone can understand without calculus. For example, how do you explain what a continuous probability density function (such as the Gaussian) is without calculus?
I teach introductory quantitative research methods to communication undergrads, most of whom have no calculus. (Which is essentially optional in the UK as it's taught at A Level after most students have specialised away from maths).
They don't seem to have a problem intuiting what a plotted PDF is showing. I think that's because in some sense it can be read analogously to a histogram. Of course, they don't have the tools to generate or manipulate one. But that's honestly not something that applied social science researchers have to do often when using traditional methods.
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult.
I think math people may have a view of this question that is skewed in an interesting way.
Statistics is very useful and its common techniques are not difficult to apply.
But they seem to be very difficult to apply correctly. We have entire academic fields that are built mostly on the spurious application of statistical methods in contexts that make the whole project invalid.
And this sort of "techniques without theory" approach is what's being advocated for upthread and represents a failure mode that math people are unlikely to consider -- because they know that part of being able to apply a technique is being able to tell whether or not applying it is valid. Math people are unlikely to fall into the trap this approach sets. But the same people who want this approach are also likely to end up being hurt by it.
My thoughts exactly. What you want to teach ultimately before anything else is mathematical maturity, which in my opinion, calculus is currently doing the best job at providing.
I’ll admit I’m biased on this issue, but I’m very wary of this type of reasoning. There are plenty of courses that are more useful than calculus, but kids/young adults won’t be convinced to learn a subject purely out of its utility in adult life (same reason I’m against replacing math with a “life skills” class); in g TV fact Calculus was the course that changed my whole college and career trajectory. Up until I took Calculus, I recognized the importance of math, but hated every dull second of it - there was no motivation that appealed to me, and it seemed that math could only be used to solve the most contrived problems (word problems that resulted in linear systems, simple roots of trig or polynomial functions, etc).
The only reason I took calculus is because it was the next math course after pre-calc and I still had a year left in high school. I didn’t even realize that there was a type of math that could be used to exactly calculate quantities related to continuously changing processes, but I was absolutely fascinated by how many real world problems calculus could solve - I realized that the problems given in algebra were contrived out of necessity to resolve in a neat manner. Now I had access to an entirely new vocabulary that allowed me to describe the world as it is, not as it needed to be to neatly fit in a 10th grade word problem.
Until that point I was interested in psychology (and had a vague notion that I’d drop out of school to make a career in music somehow), but I immediately dropped everything to take as many math and physics courses as I could. 14 years later and I work in a very math heavy engineering oriented field.
I know my story is probably atypical, and I have no clue what I’d be doing right now if I hadn’t taken calculus, but it’s one of those things that I look at in hindsight and think that I was that close to giving up on STEM, but for being forced to take Calculus. Instead, I earned my Ph.D. in applied math, and there’s not a day where I don’t use calculus of some kind.
There is one problem that I have not seen mentioned. Statistics might be much more useful but except for the first chapter (discrete probability) they are much harder. As someone who wanted to work in data I have tried hard to understand probability distributions, I have watched 10s of videos by different professors, done exercises and sure I was able to pass but the truth is I never really understood it. The intuition behing statistics is non trivial. Sampling is very hard to understand. Even on HN during COVID there were mathematically literate people trying to make some back of the enveloppe probability calculation and all they were able to do was show their misunderstanding of how statistics work (to be fair I would not have been able to do the task myself but I could feel something in their reasoning was wrong).
The ideas behind Calculus are much more useful IMHO. The idea of rate of change, or change over time, converging and diverging series, these all have practical application to the real world.
And the basics of physics makes next to no sense without calculus, and even parts of intro chemistry make more sense after having taken calculus.
I remember in one economics class, which didn't have calc as a pre-req, the prof said "alright, for those of you who have taken calculus, this is an integral, you can now leave the classroom and come back tomorrow. Everyone else has to stay."
The ideas of calculus just seem so fundamental to me. It is sad that the American schooling system is so slow, and expectations so low, that it isn't taught to everyone. Meanwhile in other countries, everyone, artists to engineers, learns calculus.
I'm a high school math teacher, and I think statistics education is desperately lacking in graduating seniors. I think the space for it should come from cutting pre-calculus as a yearlong course though, with some of the concepts backfilled to Algebra 2.
There's nothing a basic Stats 1 course needs calc grounding for (perhaps Riemann sums under a normal curve but that's more for deriving than the concept itself). As it stands now, AP Stats is used for kids that don't want to progress to an AP Calc, but want a math or need to hit a school or county requirement.
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult.
> The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses.
A statistics course would be a much more brutal weed-out course.
I think I'd rather we lose geometry than calculus. Geometry made more sense as an intro to the idea of 'proving stuff' when everybody's dad had a workshop, compass, etc.
Agreed, also the way my high school taught it made proofs seem like a boring subset of Geometry. The unit was poorly connected to other units and the problems I found by far the easiest in the course as we got very formulaic ones. It's entirely possible my teacher was just awful but I definitely walked away with some misconceptions about what proofs are. Perhaps it would have been more appropriate at an earlier stage of development.
Or the math covered in finite math classes which is a mix of combinatorics, stats, probability and linear algebra. I remember doing my time in the math tutorial room during my grad school days and helping the kids from the business calc classes who were learning math they weren't going to use from books written by people who didn't understand the domain that they were trying to teach math for. Seriously, what business use is there for f(x) = x^1.3 or the indefinite integral thereof?
When I tutored in business school--mostly related to math-related things (which itself says something about the level of math knowledge among those who weren't engineering undergrads)--pretty much the only calculus that came into play was finding maxima and minima of curves in economics which was both simple differentials and mostly pretty academic anyway. A little later I did some more complex optimization problems but that was done by software (LINDO at the time) anyway.
Statistics without calculus relies pretty heavily on memorization to get the right answers. I'm remembering my AP Statistics class (high school) where most of the time students got lower grades because they looked up numbers in a table wrong, or used the wrong table. Never mind when you start using regression models and need to start understanding axioms of the models you are using like homoscedasticity. There's some intuition there, but generally speaking, calculus is where most people "mature" into mathematics. I would argue that most people (NOT engineers/doctors/etc.) don't really get the point of math until around calculus where it starts to really open up for them. While statistics would certainly be useful, and is a must for modern day ever-connected society, it would just be another math subject that needs to be memorized like trigonometry. Some people mature in math a lot younger, but it entirely depends on the quality of their math teachers when growing up (and the curriculum/testing focus in which they are subjected to).
A few years ago USC’s MBA program replaced statistics with data science, which is further down the pragmatic chain.
Stats is not a generally useful skill (the concepts are, but can be taught in a data science course) but understanding how to work with data is.
Discrete math seemed to be the most applicable math class to my CS curriculum. I never took graphics so didn’t use linear algebra, and definitely never touched anything related to DiffEq.
I took calc 1 and stats 1 & 2. Much preferred the stats and it set me up for understanding all kinds of science lingo in articles and papers. I also indirectly use stats fairly often at work.
There are different types of stats. I took a stats course in my grad engineering program. Not being the best person at math I think I probably struggled with the math sufficiently to get distracted from the concepts. When I breezed through a stats course when getting an MBA I understood the concepts much better--though the professor was almost certainly better as well.
Strong agreement here. I'd go further and say that introductory stats should replace trigonometry too. An understanding of statistics is simply required to function optimally as a citizen in modern society, whereas trigonometry and calculus are niche.
I'm all for giving tasters of as many different "beautiful ideas" as possible in school, but I think we should be elevating practical statistics into the top tier of subjects that we require kids to go through.
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult.
I'm questioning your math pedagogy.
Calculus is the single most important math anyone, of any field, can learn as it's the first "practical math" you actually learn. Life behaves like calculus and in order to think about real life you need the concept of limits, derivatives, integrals, differentials, etc. It's patently absurd to say this should be replaced by statistics, which done to any rigor requires up to 2 years worth of calculus (through diff eq.) to even appreciate.
I'm shocked that you're a math major and didn't take away the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly. While you may not be asked to vomit cantor's diagonalization onto paper for an interview the ability to think about problems you learned from doing these proofs translates to so many different fields, jobs, and life skills that I take the complete opposite view. If you want to understand anything you need to learn how to proof. I don't care if you're a nurse or an accountant. A rigorous proof based math course will change your life.
If by "learn statistics instead of calculus" you mean being able to mindlessly vomit today's new machine learning paradigm without understanding a thing then I think I can understand where you are coming from. Otherwise, I think this is some absurd parody of someone who studied math.
Unless you are dealing with subjects that would actually use calculus, people in their day to day lives would do much better to understand probabilities, statistics and experiment design. Most of our lives are dealing with the results of empirical studies, in business, science and policy and humanity could use a big reminder that life is not a binary black or white, but a huge pot of maybes.
Otherwise it just gets absorbed and forgotten like another 'useless information' class. Ask how many CS majors post college use or remember calculus vs understanding stats for the seven millionth badly designed A/B test they ran at work today.
> Unless you are dealing with subjects that would actually use calculus, people in their day to day lives would do much better to understand probabilities, statistics and experiment design.
Probability theory is applied analysis. For example a measure density is a derivation of a measure with respect to another measure (Radon–Nikodym derivative). Or how do you often assign probabilities to measurable set: Lebesgue-Stieltjes integral.
So you better have a very good understanding of calculus before starting with probability and statistics.
> the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly
I was a math major and this is not exclusive to calculus or analysis. If anything, algebra classes in things like group theory were much more instructive on this point. Everything from my third and fourth years (when I specialized into combinatorial optimization) was built on a cornerstone of proofs and theorems.
I think calculus and trigonometry shouldn't be taught to non-math majors. You have the perspective of someone who is passionate about calculus, which is great, but considering many people aren't as passionate, what you're suggesting (status quo) doesn't work. Or maybe you have ideas on teaching calculus better than how it's currently done? Calculus is obviously extremely important, but I don't think it's useful to drill a few things that most people will quickly forget after the test.
I do think logic is important, and teaching that instead of focusing on math proofs might be a good alternative.
No, most of STEM should learn calculus at some point. In engineering and physics they are useful; in CS, chemistry and biology they are reasonable foundational knowledge and used in some specializations. Trigonometry is used even in some of the trades.
For science-oriented high school curriculums, calculus is necessary, and should only be optional in a few cases where the student already really knows what they want to do.
By the time a student wants to specialize at say 20, they should have already had some foundational ability for that specialty, otherwise they wouldn't be able to compete internationally. For people who don't end up using calculus, it's a waste, but there is a trade-off. As a result, much of the high school curriculum is to build a broad foundation to prepare for multiple possible specializations.
Most people rarely seem to encounter anything well defined enough to use math or any kind of formal reasoning on, except stuff that has an app or a professional service for.
Maybe it would help them not get scammed by keto diets and unsafe drugs and such though.
I did a degree in applied math. You’d think this would be “math you’ll use,” but the fact is that despite my program having a CS concentration, most of the stuff I did was not really applicable in practice.
However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered.
I really wish I’d spent a lot more time on proofwriting than say, vector calculus.
Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.
Proofs are indeed very practical. For example, I believe Leslie Lamport mentioned somewhere that he only came up with the final version of Paxos once he tried to prove it, and noticed that some condition he assumed wasn't necessary at all.
The reasoning behind having geometry be the standard high school sophomore math class is that that’s the age where kids would be ready to do proofs. Except that curriculum designers seem to have forgotten this and except in honors classes, most sophomores don’t get taught proofs in geometry and instead get a set of inert rules about shapes that they have no use for.
Geometry uses a deduction-only basket of proof techniques that don't prepare students for proofs done afterwards. I would like to see it replaced by elementary number theory which naturally uses a lot of induction/recursion and has some uses for proof by contradiction.
Geometry has really all that is needed for proofs:
* Axioms
* Substitution
* Modus Ponens
* Universal Quantification
Induction or proof by contradiction are just special cases of this.
But yeah, geometry for introducing proofs is difficult, because it is so easy to confuse visual intuition with proof. At the very least, you need a capable teacher who knows the difference. But nobody expects children to understand it all from the get go. A healthy struggle to disentangle intuition and proof, and then to entangle them again later on once you know the difference, that's the path to understanding mathematics.
I got a math degree, but I definitely failed at understanding that difference when learning geometry in high school. My teacher was good, but not good enough in separating that, and with classroom sizes getting larger by the decade, I think it's not the best approach to require that kind of tip-toeing. I'm not sure what would be better though, I've thought about maybe combinatorics could be a good replacement, but I also don't have the brain of a teenager anymore and I don't teach them either, so I don't know how far of a reach that would be.
The thing about geometry is that it does not take long before you've taught those four things, and then you start teaching stuff that is specific to plane geometry.
There are worse things to learn than plane geometry. It's actually a good thing to have a fixed topic to really learn those 4 things. Because once you really understood those 4 things, you are done, and you know everything about proofs in general there is to know.
Applied-focused courses (like modeling, optimization, etc.) are very broad, and I think that's where some application gets lost. For me, learning proofs within an applied-subject-domain (like partial differential equations) helped me a lot, because it was a different "flavor" of proofs, where you could understand that applied domain much much deeper.
I'm actually currently writing a book with exactly this emphasis!
The working title is "Practical Math for Programmers," and the idea is to build a collection of 60-75, 3-page long, _compelling_ demonstrations of mathematics used in production settings, biased toward stuff a generalist programmer might find useful. Not going into proofs or foundations, but providing lots of references to further reading.
I'm aspiring for it to be like a Hacker's Delight, or Programming Gems, but just for math that is genuinely useful.
I have some sympathy for this sentiment, and there is no doubt that the producers of mathematics could and probably should spend more time making life easier for the users and consumers of mathematics. This is true even /within/ the discipline, for very theoretical stuff. In the end this takes vastly more work than people expect. That's not an excuse, but it's true.
However, I also have a fundamental objection. I don't see how you can be an intelligent tool user without at least a little curiosity about how your tool functions. Maybe you can apply your tool, even be highly effective, in certain instances. But this is inherently brittle knowledge. When the parameters of your problem change and you don't understand your tool well enough to adapt, you're lost.
"Math for people who just want to use it" is very broad. What do you want to use it for? Physics, biology, chemistry, computer science? Sociology? Economics? There might be some shared stuff, but for all of these disciplines there is a vast space of mathematics that might be relevant.
I think Eliezer Yudkowsky's idea of a book (series) covering "The Simple Math of Everything" is fantastic. I would love to read that book.
I do not have the right background to be an authority on this, but from going through school myself and paying a fair amount of attention to math reformers over the years, it seems like they usually believe:
1) The problem with math in school is that there's not enough "real math".
2) Relatedly, insufficient exposure to "real math" in compulsory schooling is also (a major part of) why people think they don't like math.
My suspicion (again, without the actual background to make this claim with any authority) is that they are dead wrong on point 2—the "real math" parts probably contribute strongly to most folks' dislike of the subject, and the parts the mathematicians didn't like are probably relatively popular among people who don't go on to become mathematicians. This puts point 1 on some shaky ground (though it could still be true and well-justified, for other reasons).
I think it's a kind of rope that frays on both ends for systemic reasons:
1. Students in math courses and their parents grow to prefer(through the overall institutional constraints) to have a simple exercise that guarantees them credit - while actually doing math is a matter of crossing the Rubicon into tough puzzle-solving, and it needs some guidance for unexceptionable students to start enjoying.
2. Math teachers, particularly in the lower grades where qualifications are lower, have a harder time teaching concepts than they do exercises. And they are also incentivized to hand out a grade, preferably one that satisfies the parents.
So no matter how the high level is set up, everything converges into giving the kids a worksheet to "plug 'n chug." Which is just a confusing, badly paced grind, and therefore an easy reason to hate math. Either you get it completely and are just sitting there chugging through the problem set, or you have no idea what's going on and it's due tomorrow so your grade rests on something you feel defeated by.
I actually think that for the parts that are currently treated as rote memorization work, the curriculum should lean into it and treat it like learning the alphabet, with worksheets where you literally fill in the dotted lines repetitively; hand them out to everyone as a portion of the homework. And then the logic and critical thinking aspects need to proceed like a philosophy course, with interaction through a step by step process, not "get the answer in the back of the book". This element is something I've long thought could be automated in some degree with computer systems that let you play with the concepts, and therefore correct your thinking.
My suspicion (again, without the actual background to make this claim with any authority) is that they are dead wrong on point 2—the "real math" parts probably contribute strongly to most folks' dislike of the subject, and the parts the mathematicians didn't like are probably relatively popular among people who don't go on to become mathematicians. This puts point 1 on some shaky ground (though it could still be true and well-justified, for other reasons).
So you're saying you know nothing yet are sure "experts" are wrong, based on no evidence. OK.
No expert either, but I can speak as a non-traditional student (didn't go to college right out of high school join the Navy and started a family first) about to graduate with a degree in engineering. I was decent in math in grade school but did not like it. I didn't learn to actually enjoy math until my college calculus classes. And it goes back to the real math you are referring to. Looking back, Algebra in highschool just felt like, "memorize this type of problem and the steps," but didn't do anything to build intuition in actually understanding the why. Then calculus comes and I felt like that gave me the 'why.'
Its like how people use tools, like gears, knowing when to use what type of gears without needing to understand the microscopic material structure that provides its strength, or the mathematical definition of it's kinematics.
I think if you use gears, it would be well to know some Newtonian mechanics. The material structure might not be relevant, but in mathematics you also don't need to go down to the foundational level for every problem. You don't need to go deep into set theory to understand the proofs in calculus (though it is true that Cantor's investigations into set theory started from the question of convergence of certain infinite series, which is a calculus problem).
> Are there any "math for people who just want to use it" tracks in math pedagogy?
Use it for what? That is the question. If you pursue the what, you will inevitably be exposed to genuine ways that mathematics may be employed by it.
The academic standard for "learning math" is like "learning programming" by reading the C++ language/STL spec from front to back. No one productively learns programming that way, and even if someone did, they would hardly be well off when faced with a real-world production C++ codebase that follows $BIGCORP's inhouse programming style.
I think statistics is by and large the most proportionally underrated subject proportional to its utility. A good command of stats and probability expands your power to use data to reason about answering questions. The channel author, Ben Lambert, has an alternative playlist where he uses some of the techniques taught in this playlist to solve problems in econometrics. However, a lot of what is taught here builds a great foundation for other domains, on everything from bioethics to data journalism to computer vision.
Another great channel that focuses a bit more on the machine learning side of things is StatQuest with Josh Starmer: https://www.youtube.com/c/joshstarmer
Math education has become torturously miserable in the US by moving extraordinarily slowly. You have multiple years of working on only slightly more complex equations and concepts and naturally people get sick of it.
You have generations of teachers who barely know math and view it as a punishment, teaching kids and instilling the same views in them.
And then you have outsider still saying "can't we have condense it and simplify it further so we won't have to learn all these useless abstractions" and the curriculum bends further this way. But these actual situation of math is that not understanding what's happening is the thing makes it an empty and unpleasant activity.
Edit: Also, yeah, 90%-99% of math can be accomplished with some math software. It's just for the remaining small percentage of stuff you need some understanding and for a small percentage of that you need lots of understanding. So most of this seems useless but 99% correct is actually not enough in some significant number of technical situations, etc.
Well you can think of it like going to the gym. You don't exactly see people doing squats during their daily life, but you can see the results of people having a good physical core.
This needs to be explained further during education and motivated appropriately. We have a short-term utilitarian perspective, and we need to take a step back at times and recall that it takes time and lots of sculpting to transform a wood log to a art piece.
As you can jog everyday for fun and/or for the challenge you can also jog to improve your physical health. And not doing proofs is like declaring a guy can weight-lift by just watching videos on youtube and never lifting a weight. Or a guy can "code" without writing a line of code.
(I'd even replace Strang's "linear algebra" recommendation with this book.) Imo, proofs are useful in so far as they are enlightening (e.g., the proof that a problem has a minimum is often useful in so far as it tells you how to solve it!) but in many cases they are less so.
Math is pretty fun, though, proofs and all, and I'd recommend trying your hand at it as a cool little side hobby! It can often help with "clarity of thought" :) (In many cases, proofs are just one or two lines that tell you something interesting, too, not page-long arguments that are mostly definitions chasing.)
> Math is pretty fun, though, proofs and all, and I'd recommend trying your hand at it as a cool little side hobby! It can often help with "clarity of thought" :) (In many cases, proofs are just one or two lines that tell you something interesting, too, not page-long arguments that are mostly definitions chasing.)
I find proofs and identities very hard to read. I assume it's a bit what having dyslexia feels like. I have to turn them, glyph by glyph, into something more algorithmic to make any sense of them. The only math-for-fun I've ever enjoyed are simple recreational math puzzles—proofs, reading or writing, are torture. I actually enjoyed the parts of math classes that mathematicians insist are bad and are the reason kids don't like math—the parts heavy on memorization and drilling the application of an algorithm—far more than anything that came later. Perhaps not coincidentally, those have also proven to be by far the best bang-for-the-buck of all my time spent in math classes over the years. I use that stuff every day.
I just want a place I can go to weekly for 15min to practice math. It keeps track of what I know and gives me exercises of stuff so I can retain the skills.
Khan Academy. Finish a course, retake the course exam once a year afterwards, fill in knowledge gaps as needed. Doing this should keep all undergraduate math fresh in your mind.
I wanted to like Khan Academy, but the pacing of the exercises is so frustrating. They become too easy too fast and then it's an extremely repetitive slog of useless computations.
I wish for a service just like them but with many more difficulty levels and less (non-spaced) repetition.
If you have a solid background in calculus, I'd recommend Zill's Advanced Engineering Mathematics, which is pretty much basic math for physicists and engineers (aka for people who need to "use it").
In that case, I recommend starting out with Zill's Precalculus with Calculus Previews and then working through Stewart's Calculus: Early Transcendentals!
If you just want to learn math for purely practical reasons, Khan Academy (http://www.khanacademy.org) is great. It might have added more lessons, but when I went through it, it went up through 1st year college calculus.
The thing that makes it VASTLY better than most self study math programs or books is that there are hundreds of exercises that you can do, and see if you got the right answer. If you didn't, it will in most cases explain how to do the problem so you can try again with a completely different problem, so you're not just memorizing the answers.
Another thing that makes it great is you can do a little bit a day, start and stop, and come back to it and it will remember your progress and where you left off.
Khan is also a gifted teacher. Unlike a lot of math teachers, he has great pronunciation and handwriting and you can watch his lessons as many times as needed.
They don't have exercise for some of the more advanced college level topics. You can see the exercises on the right side of the screen under "practice" for some of the more introductory topics:
The list presented here, calculus, odes + pdes, linear algebra, is essentially that, mathematics for people who just want to use it. It's all undergrad level. There's several layers on top of this - set theory, rigorous probability theory, algebraic geomeetry, topology, that are less useful but interesting to mathematicians.
YES! This is exactly my complaint about math. I find the abstract proof-oriented math kinda interesting (although How to Prove It is an exception that I discuss below) but I _really_ want practical real-word applications of these maths.
Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that perspective in many other fields. Not once have I applied a proof to solve a programming problem.
I've thought of publishing, i.e.blogging, examples that I've come across but that would just be a mish-mash of stuff I've read elsewhere with no overarching theme/framework. Besides, someone else must have done this, no?
> Not once have I applied a proof to solve a programming problem.
There are people in this very thread insisting that proofs are extremely useful in programming. I dunno if I just picked up the same skills elsewhere (Logic? Philosophy? Just... IDK, thinking and developing an absolute shitload of heuristics through years of experience?) or am entirely missing out and in fact don't have a clue how to program, but I don't see it (outside some rare niches where it probably is useful—coq exists, after all).
Sure, the word "exhaustive" can apply both to accounting for all (reasonably) possible problems in a block of code, and also to proofs, but the former doesn't feel at all like working on proofs, to me, to pick just one example (and some posts have seemed to imply that accounting for e.g. edge cases is exactly one case in which experience with proofs come in handy, but man, they feel like very different and barely-related activities to me).
In a way, writing types is proof-esque. It establishes some basic correctness guarantees regardless of the programming language. Some concepts like refinement types make it seem more "mathy", by requiring quantifiers within programs: https://ucsd-progsys.github.io/liquidhaskell/blogposts/2019-...
Someone without any background in the subject would probably find Dafny interesting.
In my experience, being familiar with inductive proofs is pretty useful for programming. For most non-trivial code involving recursion or loops the way I personally "know it works" usually has the flavour of an inductive proof.
For example, I might not remember how to code bisection search but I can figure out the loop invariant and from there it's easy enough to code a working binary search. And even if you have right bisection search down to muscle memory, you can modify the loop invariant to create left bisection search if you need it, while if I tried to keep two binary search algorithms in my head I feel like it would be more error-prone.
I dunno, at that point, are you really doing math? Math is proof. Now, granted, nobody expects you to know exactly how everything is proved. But there is an expectation that you can prove most of the stuff. Otherwise, it's very easy to stray from the truth into the plausible-sounding, but incorrect.
That said, you're absolutely correct that more justification and motivation is important. So much of math can be taught with problems from physics, computer science, etc. Perhaps a good book for you would be Concrete Mathematics by Knuth? I haven't read it but people swear by it.
We're now at a point where college math departments have to offer an "introduction to proofs" for students who have never seen one. Even high school geometry is now taught with few or no proofs.
The catch is that I myself don't actually understand this well. It's just "What I learned googling stuff while working as a programmer. I've had help reviewing it but there could be errors.
I'm trying to cover all areas of math that a nontechnical person, or typical non math focused programmer would need to know, but I treat actually doing any of it by hand on paper as an arcane thing for the really dedicated, so there's not really any excercises.
Instead of actually teaching a real understanding of math, which I can't do, because I don't know it well, I just explain what people who do understand it use it for and why you might want to go actually learn it.
I also have any historical math related stuff that I find interesting.
I can't tell you how to derive or prove Euler's equation , but you don't need to know math to understand the emotional impact from a humanities perspective, and be amazed that all those constants fit together like that, and that someone could discover it.
Ultimately, I think traditional math education has it totally right. My life would be so much better if I knew it, because there's jobs that seem to require exactly what they tech in math class.
It's not directly useful for non STEM types, but the idea seems to be give everyone a head start since so many do want STEM jobs.
I think you really do have to get to the being able to do proofs level to make use of it in the real innovative applications.
The common everyday applications math people like to cite can usually have a dedicated software package. It's not like we still need to add two numbers on paper. If you want to build something, we have RealThunder's FreeCAD.
Excel's Goal Seek and CAS systems do the stuff people say we will use algebra for.
But if your in tech eventually you run into the wall and need to do something like a Kalman filter or calculate stresses in a bridge and you're screwed.
It depends on which department you take math course in. Any course offered in a math department, even statistics and probability will be treated formally there because that's what mathematics is. What you probably need is to take math courses from an applied mathematics department like physics or engineering.
But math is a broad field so you're going to have to pick specific courses. For example, partial differential equations are quite common. But if you learn it in a math department it'll have proofs and full rigour just like a course in set theory. While some techniques for solving them will be covered, we mostly study the underlying mathematical structures, why certain techniques work, etc. If you take it in a physics department they'll teach you techniques and numerical methods to solve a certain class of problems like heat equations or fluid dynamics. But learning through direct applications will inevitably limit you to those techniques while learning things in full generality tends to make it easier to pick up specific techniques when needed.
> Are there any "math for people who just want to use it" tracks in math pedagogy?
If you're talking about college level, "math for people who just want to use it", is basically all it is (outside of math departments and perhaps outlier curriculums in some elite places) and that's a problem.
Learning just the applications without picking up the theorems and without a true understanding of the concepts makes more advanced work in whatever discipline one chooses more difficult. Why? Because you'll have to follow mathematical arguments and it's so much easier to do that when you got the background to fall back on.
I think students before college need more focus on mastering the basics. They're rushed so much, and tragically, math is very cumulative. Any one who has tutored before will notice that it's disturbing how many people don't really understand how to manipulate fractions as they enter a calculus course.
It's still a bit much to absorb, but you can't really ever say the author didn't attempt to describe each lesson in real world terms. You memorize or learn to graphically/logically derive a few things, like how log(1+x)≈x for small x or how you can sorta guess for most quantities or plots. (One book example figures the data capacity of a CD if you know its music storage using informed guessing of each conversion factor from seconds of music to bits.)
As another example: powers of cos x from -pi/2 to pi/2... you can sketch cos x and then roughly sketch the second power, and then it's clear that the more times you do that, you get a bit more of a bell curve. One in the middle will always stay one in the middle, but you get less area with each iteration as the rest of the function approaches zero. If you wanted an integral, you eyeball where the plot is at 0.5, draw a full-height rectangle from the left 0.5 value and the right 0.5 value. Because the height is 1, the width is pretty close to the true area. You can decide at this point---if you want more precision---to visually guess if the tails or the main part of the function should have more area and adjust your answer.
The audio CD thing is pretty clever. Even if you don't know important factors like the maximum frequency, you can get a great guess based on what you already know... like knowing Freddie Mercury could sing four octaves, starting from probably somewhere above "transformer hum sound".
You'd have to know each octave doubles in frequency.
Side quest: When you play the bugle, the played frequency increases or decreases by MULTIPLES of the base frequency---NOT powers of 2. Suppose this base frequency is 250 Hz. There is an octave from 250 to 500, but there's a note between the octave from 500 to 1000 at 750 Hz, and a few notes between 1000 and 2000 Hz, which is the part of the musical scale something like Reveille is played. If Reveille jumped from octave to octave, it would just sound like the intro to Justin Hawkin's cover of This Town Ain't Big Enough.
So, if you know transformer hum is 50 or 60 Hz and Queen's frontman starts his singing at 100 Hz, then he can sing up to 1600 Hz, or four octaves. Mentally recalling what his falsetto sounds like, you can imagine a really high-pitched guitar solo an octave above this, and you can still imagine what an octave above that would sound like. (Maybe you're getting close to dog whistle territory in your imagination.)
This, then, is 6400 Hz you are imagining. The top of each sound wave to the top of the next is 6400 Hz. To record this, you'd need the top AND bottom of each sound wave, because the speaker cone moving from maximum to minimum displacement is how the sound is made. If you want to make sure you aren't accidentally recording the middle (zero crossing) of each wave, you can even take three or four or five samples per sound wave instead of two. It's a lot of thought, but you can reasonably decide that 25000 Hz is a good sampling rate for capturing much of the range of human hearing. Going too far beyond that, you're wasting storage space.
A CD holds a bit more than an hour of music, or 3600 seconds. If you've listened to Dire Straits, Eagles, Cyndi Lauper, Metallica, David Bowie, Led Zeppelin, ELP, or nearly any other band, you're probably aware the recordings have independent left and right channels.
Finally, each sample is going to be somewhere between "speaker fully retracted" and "speaker fully extended". With 5 bits, this gives 16 "stops" from the middle point to fully extended. But we know that music can get really quiet when it fades out, and a lot of volume knobs can go from zero to thirty and sometimes higher. When you have the volume at one, you can still tell the difference between loud parts and quiet parts, so you'd need an extra 5 bits just to get good dynamic range at loudest and quietest volume settings, or 10 bits. What happens when you double this? If you have 20 bits, you are probably close to wasting bits. You have a million places where the speaker coils can move to. For a speaker that moves a few millimeters, this means 20-bit resolution allows steps of a few nanometers. This is the scale of computer chips and color wavelengths. If you took the color blue and shifted its wavelength by a few nanometers, it would still be practically the same shade of blue! Without knowing about bit depth, you can reasonably assume 16 bits is good because it's a power of two and will give a lot of dynamic range. 8 would be too low. 32 is just wasteful.
With 32 bits, a speaker capable of moving 1 cm end-to-end will have 10 carbon atom diameters of linear resolution. The ears are impressive, but I don't know they can differentiate the air displacement of (speaker cone area) x (ten carbon atoms). Even having 0 to 100 on the volume knob, this leaves 25 bits of range at each volume setting. This is audiophile (and arguably, snake oil) territory.
So then, you can say 3600 seconds is pretty close to 3000 seconds, 2 channels is close to 3, 16 bits is close to 10, 25000 Hz is close to 30000 Hz... 3 x 3 x 3 x 10 x 1000 x 10000 ≈ 3,000,000,000. Since a byte has about 10 bits, divide by ten, and this yields a first approximation---based on logic reasoning of what we know---of 300 MB. It's wrong, but it's not "very" wrong. (It's off by a factor of two, not a factor of ten! Not bad for 4 rounded, intermediate conversion terms...)
(The idea is to round each term to a value starting with 1 or 3, because multiplying 3 and 3 is close to 10. The reason 2 is close to 3: 10^(1/2) = 3.16. This states that a good midpoint of 1 and 10 is 3.16, because if you square each term, you get: 1, 10, 100. Now, 10^(1/4) = 1.78. This means that any value less than 1.78 would be closer to 1 after squaring, and any value higher will be closer to 10.)
You can even take the analysis further and back-calculate things like how fast the CD might spin by guessing the track width and bit area, how long a track skip would be, whether the size limitation of the CD is due to optical or material properties, how far the laser would need to be to converge at one bit while being close enough that any deviation in the surface flatness doesn't send the return beam away from the sensor, etc. (This is all the info you'd probably use to begin the approximation if you weren't aware an audio CD holds an hour of music, like if you were asked in 1975 to "back of envelope" whether a compact, non-contact, vinyl-like, LP-length recording medium was possible.)
I have 2 kids who are in college now. The beauty of math that inspired me to study math (as an undergrad math major) was treated as an aside in K-12 math, and has now been replaced with Grind. I learned to love math thanks to just the enthusiasm of my teachers, and stuff that I did outside of class. Both of my kids got great math scores, good enough to skip college math altogether, which one of them has done.
Today, kids need to be told: Math is nothing like what you learned in high school.
I think K-12 math should be divided into 4 quadrants, taught in a soft of spiral:
1. Arithmetic (symbol manipulation, through algebra and calculus)
2. Computation (numeric and symbolic)
3. Learning from data
4. Theory (sets, proofs, etc.)
Some of these things could be blended with the science curriculum.
The proofs can help immensely in remembering theorems and knowing their needed assumptions and application area. Mainstream math education is also already light on difficult proofs in introductory (undergraduate) texts. You can always skip the proofs anyhow.
This is more of a university perspective, but......
> Are there any "math for people who just want to use it" tracks in math pedagogy
It's called Industrial Engineering /s (but only kind of)
It is tame as compared to a pure math or physics B.S. But, you pretty much cover the gamut of every math tool that is useful in the real world. From stats, supply chain, search, calculus to combinatorics and so on. Each concept is squarely grounded in a field where it is used.
I was surprised at how well my mechE (usually the closest undergrad degree to Industrial) degree prepared me for applied math and ML coursework for my CS masters. In comparison, a lot of CS undergrad peers struggled in those courses.
Mathematics is a social activity, in the sense that mathematicians create proofs to convince others that their findings are valid. Some proofs are easier to verify than others. Some proofs are "believed to be true" because no one has yet managed to prove them wrong yet.
So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented.
And just to be clear, there is little to no "real world usage" math without academic math.
Well, it is not only about people. You would also want to "convince" Nature (so that the aircraft you designed does not crash right after a takeoff), your own or, often, someone else's wallet (so that money is not spent on nothing).
But the distinction is generally not as clear as you may think. (1) Much of the mathematics came from real world problems (so, in particular, one may get drawn into some kind of mathematical research and even end up discovering new mathematical facts). (2) Sometimes when applying mathematics one still needs to employ deduction (derive a formula, prove a statement one wants to rely on, etc.).
> I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.
The problem is that the answer will depend heavily from person to person and from field to field and often the most sensible answers require a mathematical maturity that creates a chicken-and-egg kind of difficulty.
Do you care about engineering? Well you'll need some calculus for that. Do you care about prediction modeling? Well there's some stats you'll need for that. Do you care about finding patterns in the world? Well there's abstract algebra for that. Do you care about reasoning itself? Have fun with mathematical logic. But because humans have different motivations, there's no one-size-fit-all motivation-based approach.
This is especially painful for mathematics because I think most people who have learned some amount of pure mathematics will relate heavily to what helpfulclippy says in a sibling comment: "However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?)."
The skills of rigorous and abstract thinking that pure mathematics provides is both nearly-universally helpful, but also simultaneously as a result very difficult to motivate. "This will help you think better across everything you do" is lofty-sounding, but generally not a convincing sell unless someone is already curious. But it's true that being able to wrap one's mind around pure abstraction (after rattling off a rigorous definition for an abstract question: Question: "But what is X really?" Answer: "X is just that. No more, no less.") has ramifications for all that one does.
And the most painful part of all of this is if you try to start by teaching the wonders of pure mathematics instead of all the messy, boring rote stuff, students' eyes are liable to glaze over even more because of the aforementioned chicken-and-egg issue with mathematical maturity.
In this way it's similar to trying to motivate someone to read and write. The key that unlocks that interest for everyone is going to be different and it's very hard to explain the near-universal benefits that reading and writing bring to one's way of thinking (but I can always just have a computer transcribe it or read it aloud to me!) without some inherent curiosity in them.
I didn't take any of these classes personally, but I do remember in college there was a whole host of Calculus for Business and Economics type courses that were much more focused on practical application than theory. Maybe picking up some of those textbooks would be the way to go.
Proofs certainly aren’t the only aspect of what one might call “understanding mathematics.” Aren’t proofs generally limited to very specific mathematical courses (at least until you get to the cutting edge where new results inherently require proofs)?
Theoretical physicists use probably more math than other subjects. Generally the litarature to use math is called 'mathematical methods for physicists'. The book by Arfken is considered standard by many. Did you mean something like that?
History of mathematics is a cool and relatable subject in so far as you can imagine the problems of ancient Egyptian worker paid in bread, and land survey/architecture. problems to Nazi code breaking, yeah?
Yes, all tracks of math involve people using math. Proof writing is a part of use for a lot of it. There are also very many tools within the use of proof writing where a lot of education is around learning a bunch of tools so that you can better "recognize which tool to apply, then apply tool", in a future real-world use where you want or need to prove something.
I'm a little snarky, but you have a broken idea of what math is. It's not even your fault. I don't even claim an unbroken idea for myself, I went through public education too, though I do think it's less broken. Somehow compulsory education has managed to get near universal basic literacy, but seems to have failed on whatever equivalent some sibling comments have hinted at exists for math or at least mathematical reasoning. A lot of algebra work taught for junior high can be understood as just a foundation to be able to understand later things (though you can of course use some of it directly as taught without having to learn more for every-day things like some boy scouting activities, or helping with putting together a garden or a fence, or programming -- and some of course is entirely useless). But instead of pushing algebra even earlier, states are instead moving to push it even later. (Let alone trying to spread awareness of even a hint of the subtle divide between more general algebra and analysis that a lot of STEM undergrads don't even really get a whiff of except maybe knowing it's often said to be a thing.)
To try and be more helpful, I'll suggest you don't actually want to learn math at all. So don't! At least, not directly. Instead, find something you want to learn more about in science, engineering, or technology/programming, and dig into it until you start hitting the math being used. For many things, especially at the introductory level, it's fundamentally no more complicated than being able to read a junior-high-school level equation. Occasionally you'll need to know about some functions like square root, or sine, or exponentiation, or some other new functions that will be explained (like a dot product) in terms of those things. When you don't understand something, you may need to find an outside reference (or a few) for it, if the book itself doesn't cover it enough or to your liking. Even then, you can often find outside presentations of that thing which are still motivated by the general field and are thus not proof-heavy.
However sometimes the best explanation may still be found in a "pure" book just about the thing, and if you can get over whatever problem you have with proofs you can learn to see how they can be used to build your understanding of the thing in smaller pieces, not just as tools to say whether this or that is true or false. In other words, proofs can serve the same function as repetitive problem-solving exercises, and are often given as exercises for that reason.
I'm a fan of the Schaum's Outlines series of books just for the sheer amount of exercises available in them, I just wish I had better self-discipline to actually do more exercises. Though they maybe aren't the best resources for a brand-new introduction to something.
To give a small example, maybe you're interested in game programming, and eventually want to dive into studying 2D collision detection more specifically so you can implement it yourself instead of using someone else's library, so you might stumble on a copy of "2D Game Collision Detection: An introduction to clashing geometry in games". Its explanation of the dot product comes early (its whole first chapter is on basic 2D vectors), consisting of 2 diagrams and two code examples (the first mostly defining dot_product(), the second using it as part of a new enclosed_angle() function) and some text all over 2.5 pages. It gives things in programming notation instead of mathematical notation, apart from some ² squared symbols occasionally. It gives a few equivalences like a vector's dot product with itself is its length squared, shown as dot_product(v, v) = v.x² + v.y² = length², without proving them, and points you to wikipedia of all places if you want to know more about how that or another detail are true. Why learn it? It's used immediately after in explaining projection, and then later in collision detection functions. Generally that book is structured as: learn the bare minimum of vectors, use them to implement collision detection for lines, circles, and rectangles.
I'm not saying this is a great book but it's representative of what you'll find that I think you're really after, which is motivated use of some bits of math. If you don't like that book's treatment of vectors, there are a billion other game programming books that cover the same thing as a sub-detail of their main topic, and maybe even better for you because it'd be grounded in e.g. a graphical application you've already got setup and running to see results rather than a standalone library. Or there's special dedicated math books like "Essential Mathematics for Games and Interactive Applications". Or you can go find dedicated "pure math" books on linear algebra if you want. Or maybe your junior high / high school math education was good enough you can more or less skip most of this and move on to something more interesting, like physically based rendering (https://www.pbr-book.org/) which also of course has vectors and dot products with brief explanations. Or maybe you don't care at all about game programming, and want to learn about chemical engineering, or economics, or the mechanics of strength and why things don't fall down, or...
> I'm a little snarky, but you have a broken idea of what math is.
I do not, which is why I explicitly acknowledged that what I want is not "real math". I don't care about math for math's sake, even a little. Not my thing, never will be.
Basically I want to learn to apply useful results from hundreds-of-years-old "advanced" math the same way I learned math in grade school: memorization, pattern recognition, heuristics, intuition, and drilling, all with a focus on application. Keep the proofs far, far away unless there's some excellent reason I have to know them (and perhaps there's actually no way around that, but I suspect the current situation has more to do with the interests and world-view of people who design math curricula, i.e. mathematicians, than strict need, if you're mainly focused on application). Ideally, almost every single problem set would consist mostly of so-called word problems, drawn from realistic circumstances.
I don't think it's possible to divorce "proof" and "application" as cleanly as you'd like.
For instance, let's take a very common "application" of statistic: given a test that has false positive probability p, you can apply the test n times, and the false positive rate of the composite test is p^n. (For e.g. this is used to analyze bloom filters, or the Miller–Rabin primality test, or hash tables). However this is only true if the tests are independent. If you have to apply this result to analyzing (for e.g.) some new data structure you wrote, you'd first have to prove that the tests are independent. And maybe it's just me, but I find that if I use heuristics to check if some events are independent, I really often get it wrong.
Another example: the uniform limit theorem (https://en.wikipedia.org/wiki/Uniform_limit_theorem) is quite useful, but to properly apply it, you have to understand the difference between uniform convergence and pointwise convergence, and maybe it's just me, but I found the difference very unintuitive when I first encountered them (in a standard proofs-based analysis class). Even now, if you gave me some random series of functions, I can't really imagine using heuristics to check it uniformly converges to its pointwise limit, I'd want to try to write down a proof to be sure. So this useful tool is gated behind understanding some (to me) subtle proofs.
Haha, you were so careful and still got the classic "That's not serious mathematics!" response.
You might want something less applied, but I highly recommend Burden & Faires "Numerical Analysis" and Trefethen "Numerical Linear Algebra". The various interpretations provided by Trefethen for understanding a matrix, eigenvectors, singular value decompositions, etc. are amongst the most insightful descriptions I've come across.
What you're describing as "real math" is no more "real" than the kind of "math" you do claim to want. They're both math. What you think of as math for math's sake is not necessarily any more for math's sake than for the sake of something else. As other commenters have tried to explain, the two go hand in hand often enough anyway. (And you're right they sometimes don't, go to some random math journal and pick a random paper and it'll probably be something neither you nor I can hope to understand any time soon, with application seemingly nil in any way we could see. A lot of high level math is like that. Maybe some of it is best termed 'exploration'. Nevertheless, it has no claim to "real" math, and I don't think those guys are responsible for US math curriculum.)
Drilling proofs is a valid kind of drilling and can be an effective way to learn something. Not necessarily the only way, sure. But there's nothing fundamentally different or "more real" about drilling proofs vs drilling grade school multiplication problems. You'll memorize things, you'll see patterns, develop heuristics, gain intuition.
Application can sometimes be tricky; did those grade school multiplication drills have application? Are they granted more application by phrasing things in terms of word problems around counting apples or whatever rather than the compressed a times b = blank expression? Well, sometimes the application of proofs will be more direct, sometimes less, and can be phrased better or worse, more realistically (and necessarily more complexly) or less so, like any other exercise, whether it wants a proof or not. CS proofs about big-O complexity are applicable to analysis of algorithms, which is pretty important if that's your focus. Though most problems you could find to drill specifically on big-O (as opposed to other parts of algorithm analysis, like recurrence relations) would likely take the form "find the complexity of this" or "given the complexity is such, estimate..." rather than "prove that...". There are many things no one knows how to prove that are still an area of study, clearly proofs aren't the be-all-end-all. Anyway, the mental processes involved between something like "find x, the hypotenuse of the triangle" and "prove the Pythagorean theorem" often aren't that different. There are multiple ways to prove it, you could drill on them.
And technically, computer programs themselves can be thought of as proofs (Curry-Howard correspondence) so if you've ever written a program that terminates you've written a proof... Proofs don't necessarily have the form or flow "by axiom 1, axiom 2, theorem 34, modus ponens on this, proof by contraposition on that which we'll name lemma 8, and by induction over the integers here, we have proved blah, QED".
And if you grant simple algebraic symbol manipulation as something you would do to solve a word problem, well, that itself is a style of proof. (There's a whole automated proof engine written entirely on the basis of substitution, the same process you use in a simple algebra problem of substituting x + 3 = 10 with x + 3 - 3 = 10 - 3 and reducing to x = 7.)
But fine, no proofs, not even in disguise! What is it that science, engineering, and technology focused subject books that use math only as needed without bothering to prove things when unnecessary (some having exercises and drills of word problems from realistic circumstances) don't do to solve your craving?
I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to revise or learn Algebra, then I would do Stroud “engineering math” first which is designed for self-learners with lots of solutions and feedback.
At some point, it would be good to get a a copy of Lyx and start to learn to write math in LaTeX - Then you can get feedback on your proofs online at math.stackexchange.com if you don’t know any math people locally.
Feel free to get in touch with me if you want to discuss further, happy to help!
I looked up the Chartrand Mathematical Proofs book and it's been a while since I had to buy a textbook, but $175 for hardcover and $75 for paperback or ebook? That's nuts. If I were a student today, I'd pirate that and feel absolutely no remorse for doing so.
Maybe, but this is not so bad, if you realize that the time and the effort you would spend on learning the material from such textbook is in some ways incomparable with the price of a couple of dinners at a restaurant.
If you have the means, try printing them little by little: chapter by chapter plus any back-of-the-book material. This way you can focus on one thing at a time, tactile paper in hand, without expending money on topics you don't yet need. You can annotate it, solve problems in the margins, file it, and reserve it for reference if/when you move on to successive chapters.
Yes! Combinatorics, graph theory, linear algebra, and even some parts of theory of computation and grad-level abstract algebra and algebraic number theory can be learned without calculus (I think!).
N.b. This is not to say that these are all easier than calculus, and I wouldn’t even really recommend learning say Galois theory before calculus, I’m just saying it seems that one could.
How old were you when you started into maths past the highschool level, and how many hours a week were you working in the unrelated field? I’m 29 and have a sincere interest in math, but I’m relegated to self-learning because I earned on.y a 2.3 GPA in undergrad. I’ve been working threough Hammond’s proof book as well as some vector calculus, and while I seem to completely understand the material as I read about it, the exercises leave me pretty bruised up.
I don’t know anyone who knows or values math much past algebra. “I don’t need it for my career, so why would I spend tome on it?” I don’t plan to switch to engineering, so I often find myself distracted & wondering what value math will add to my life other than making my interests more obscure and distant from the everyday people I meet. All I have to go on is “I’m interested & I trust that I’ll find it helpful once I know it. Also some people who are good at math make pretty good money.” But when I get 60% on a set of exercises, it’s challenging to keep faith.
I feel like one can easily get a bunch of "Really you should start with X" statements concerning math. Really you should start with proofs, really you should start with problems, really you should start with these concepts. I started with concepts rather than proofs or problem and I too went to a MA and various study. I tackled both proofs and problems but I don't think I'd have done as well if I'd jumped on these immediately.
So, altogether for someone wanting to get into advanced math, I'd say to look at the variety of advice out there and follow the kind that seems to help your progress.
On the other hand I sometimes wonder why Calculus is made a big deal out of. It was probably the easiest among all the parts of maths. You just have to imagine an small unit and how to extrapolate for integration and imagine how to break it down to small parts for differentiation. When I originally learnt Calculus in high school it was couple of days of learning the concepts and the a week of deriving everything from that and move on!
I read a Murakami novel in high school, 1Q84. The protagonist is a math teacher who talked about math in a way that I had never seen before. I'd been told I was "good at math" beforehand(for whatever that means, I'm not a fields medalist or anything), but for ~6 months after reading that book, I was _really good_. Like, suddenly I did not have to do any homework in my sr. year calculus class. I loved sitting in class and watching my teacher work through problems, and it seemingly imprinted directly into my brain, because while doing no homework I could still ace the exams while writing with a pen (no erasing and re-do'ing with a pencil). All because of the way this fictional teacher from 1Q84 talked about math.
Has anyone else had an experience like that? (With math or other things?)
I'm doing a MSc in mathematics. I tried to read that book once and I thought it was terrible. One of the few books I dropped before finishing it and also the reason why I have been reluctant to pick up other Murakami books. With respect to your phenomena of leveling up your mathematical maturity without doing mathematics, that has happened to me a few times and I would say yoga and psychadelics are the primary source of inspiration. Of course you need to do a lot of reading and studying (or if you have courses then you can sit back and have them feed it to you). One thing is making the information known and the other is to understand it, this is where the things that can spark random enlightenments come in.
I loved last year being able to take university courses online. I knocked out analysis, topology, and quantum mechanics as a non matriculated student. I'd had those books for years but never could get through them alone. (The main thing being, you really don't have anything to gague whether you know it well enough or not).
I really wish there was more opportunity for that. I'd love to take a few more classes, mostly in pure math, but there's simply nothing on offer for remote study past the 200ish level. (There are some remote masters programs in applied math, but nothing for pure).
I don't think I'd enjoy doing a PhD full-time. One or two classes per semester while working seems just about right. But the closest university is an hour away, so in-person isn't a realistic option.
If doing math is essential to conceptual understanding and application, could the interface of math and physics be made more human-centered? For instance, the shift from Roman numerals to Arabic numerals made doing math easier. Based on your experience, might it be possible to increase accessibility by revising some of the arcane conventions of math and physics?
My favorite accessibility-increasing tool is the computer. Doing math shouldn't involve so much circus math, i.e. doing things just for show, since a computer does so much immediately and accurately. We already use graphing calculators, and notation-wise graphs can be a useful tool in themselves in elaborating an idea (see also Feynman Diagrams, or electric circuits), but there's so much more calculators can do, let alone actual PCs, cell phones, and web apps. By chance in 9th grade "Intermediate Algebra/Algebra 2" I had a teacher not wholly opposed to modern technology and so he only had us do a small amount of those "solve this system of equations using a 3x4 matrix by hand, showing each matrix transformation to reach the row reduced form, taking up some pages of paper" problems before he brought in a classroom set of chonky TI-92 calculators and showed us the rref() function. That Christmas I asked my mom to upgrade me from my non-graphing scientific calculator that had served since elementary school to a TI-89 Titanium that served me even through college until I learned and got used to various PC programs. The lesson that there were powerful tools around stuck with me pretty fast though, and I wrote some simple programs on the calculator for that and other classes throughout HS (mainly just automating calculator input steps, no fundamentally new algorithms the calculator couldn't already do); in HS physics I also had learned more programming and did a little simulation with pygame and it was fun to enter numbers in the program, run it, see the mass trajectory animate and show some computed values, and then do the actual experiment and get the same results. I only wish I had been shown some PC programs earlier.
I met a friend many years later who sadly was still forced to do that rref()-by-hand for even larger systems of equations in university! That left no time to actually learn anything useful in linear algebra. Madness.
https://theodoregray.com/BrainRot/ has some nice ranting about this (though it does go a bit off the rails when it starts talking about video games).
> Why is it that every time any subject about mathematics comes up there is always a complaint about notation?
The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.
I really only think calculus notation is an issue. Calc 1 books absolutely demand that you view d/dx as an operator and not a ratio, likewise with the dummy variable and integration symbol. Then later the book teaches about unit normals and it’s implicitly acceptable and treat dx/dz and dy/dz as ratios.
i.e.
|u X v| = (1 + (dx/dz)^2 + (dy/dz)^2)^(1/2) = (dz^2 + dx^2 + dy^2)^(1/2).
This was extremely frustrating to me for a while until I accepted that this was how Leibniz did it, so if it’s good enough for him it’s good enough for me.
Sure, an introductory notation could be created to bridge the gap, and that could be more human friendly.
That said, as someone who uses notation for math regularly, I want to keep using notation. It’s a helpful tool, and it is an efficient and precise language.
I've been on a Math journey since I retired a couple of years ago and I agree with all the books mentioned that I know and look forward to picking up some of the one I do not know. I agree baby Rudin is essential, but I find it tough going.
Some books I liked for self study because they have answers:
Introduction to Analysis, Mattock.
Elementary Differential Geometry, Pressley.
There is also recently Needham's Visual Differential Geometry and Forms, which is great.
I think learning Real Analysis from baby Rudin is like learning Probability Theory from Wikipedia. It's so encyclopedic that if it's your first look at real analysis, it will be too dense to understand, but if it's your second or third look, you will find beauty in its brevity.
Very pretty book (Needham's), will check it out! I think over 20 years ago I actually attended a house party that Needham was giving in SF. It's a small world.
Agree that Baby Rudin is VERY difficult to study on its own. I recommend only studying it alongside the other two books I listed: Abbott's Understanding Analysis and Spivak's Calculus (which has a solutions manual). Abbott in particular is very straightforward (at least in comparison with baby Rudin haha)
Another point for Abbott is that it was one of the ~400 books Springer made available for free download near the start of the pandemic. I remember there were a few scripts here on HN back then to grab all those books, so many here probably already have a copy.
This felt like it was written by a physicist or engineer.
Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.
I think it is important that anyone who wants to study math understand that real math is not at all like what you learn in a physics or engineering department. In these departments you will always hear people say things like
>"proofs are not useful, all you have to do is memorize the 'trick' they use. Once you know which trick to use, it is easy"
or you will hear them say.
>"Math isn't about understanding, it is just about learning rules and symbols on paper".
This is not mathematics. These things do happen.... in a physics and engineering department. It is, in fact, a descriptions of a physics education, not a description of a mathematics education.
For this reason I would be careful taking mathematics advice from physicists too seriously as they may, unintentionally, lead you very far astray.
For what it’s worth, the curriculum in this guide is modeled after the math major maps of many universities, including the one I attended (Penn). I would be curious to know what part of an undergraduate math curriculum will lead people very far astray…
It's just that it is very much a "mathematics for engineers" style course. I think very few of the subjects outlined there give you a flavor for what "real" mathematics is really all about at all (except for algebra, which you do mention).
Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap
Group A:
number theory, graph theory, combinatorics
which shares concepts with
Group B:
Algebra, Topology, complex analysis, differential geometry, metric spaces...etc
which shares concepts with
Group C:
Functional analysis, measure theory
which shares concepts with
Group D:
probability and statistics.
As for the applied math that you mention, you should really need to add vector calculus and I'd highly encourage anyone to take a course on fluid mechanics (from a mathematics department instead of an engineering department) to get a real feel for vector calculus in action.
I suggest taking another look at the list and comparing it to the required courses of the undergraduate math majors at the top 20 universities in the USA.
Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses.
It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).
I have never met a physics student that even knows what functional analysis even is, despite it being at the core of quantum mechanics. I can't even imagine why someone in a physics department would learn about measure theory. True that any student learning General Relativity will get an introduction to differential geometry.
Except for the course ordering, it lines up almost identically with the math major at my university. I'm not sure what the other posters are going on about. Most of their preferred topics that they feel you missed are either upper division electives or graduate level.
I suspect that people forget that undergraduate programs don’t really cover very much. This true not just for math, but for pretty much every other major. I mean, think about how little of physics you learn if you only take undergraduate courses!
Yeah I was a physics major and honestly it felt like we barely scratched the surface. An undergraduate degree in physics is sufficient to be a high school physics teacher, but not to be a physicist of any sort.
though I am a little surprised that they have 1 course of differential equations in there instead of complex analysis as a required topic, as I think the latter is a better pure math topic. But it's MIT, so be it. Whether directly or indirectly, many of us learned to view math the MIT way by patiently working through foundational books like Artin and Munkres.
That said, my mention of non-introductory algebra topics probably is more of a personal idiosyncracy/interest.
As someone with a keen interest in learning Engineering part time, I found your write ups really helpful though! I enjoy learning math but like to have an angle towards a practical and useful application. It keeps me a little more motivated than pure math learning. With ADHD the concept of being able to build cooler things always keeps me going. But somewhere along the way of learning purely theoretical things for too long my brain just loses interest (not enough reward), even though I enjoy it in the moment it is hard to get to the starting line and take the first step after a while :)
As someone studying teaching math, I found it interesting to compare her suggestions:
- Calc I-IV
- Intro to proofs
- Linear algebra
- (Abstract) algebra I-II
- Real analysis
- Complex analysis
- Ordinary differential equations
- Partial differential equations
- (Others)
to my program plan:
- A couple teaching courses (including one for roughly grades 5-8)
- Calc I-IV
- Statistics (one without calc, one with)
- Linear algebra
- Discrete
- Geometry
- Number theory
- History of math (apparently not just a history class, I haven't taken it yet)
- Abstract algebra and into to topology
Overall a pretty decent list, although I would suggest considering some tweaks.
For real analysis it recommends as essential Abbott's "Understanding Analysis" and Rudin's "Principles of Mathematical Analysis". If you "haven't gotten your fill of real analysis" from those it recommends Spivak's "Calculus".
I'd consider promoting Spivak to essential, but using it for calculus rather than real analysis, replacing their recommendation of Stewart's "Calculus: Early Transcendentals".
By doing calculus with a more rigorous, proof-oriented introductory calculus book like Spivak, there is a good chance you won't need a separate introduction to proofs book so can drop the recommended Vellemen's "How to Prove It: A Structured Approach".
I'll second this. "How to Prove It" gets recommended a lot, but I couldn't get through it. I found it terribly boring and unmotivated. Some people can power through dry material but I'm not one of them. I found it much easier to learn to write proofs when they were related to topics I was interested in.
I loved How to Prove It. Not for the proofs - which are interesting in a gazing-at-your-navel kinda way - but rather all the little _practical_ tidbits. "So THAT'S what a partially ordered set looks life in real life!"
And the last(?) chapter where he uses induction to determine how to place an L-shaped figure on a grid...I never knew how to even approach that kinda' problem.
So yeah, I want actual practical applications ("exercises" != "applications") for math.
Modern calculus (analysis) was invented because people shot themselves in the foot working with topology and wondering exactly what is a "curve" ? I am a big fan of this approach to learning mathematics, just forge ahead and when (if) things fall apart then go back and fix up the foundations. To this end I recommend a couple of books. "The Knot Book" by Adams is a very interesting exploration in topology (without requiring all the years of study at university before you are allowed to learn exactly what a topology is). And in another direction, group theory was invented because the study of symmetry gets very tricky! But if you want to dive in anyway then have a look at Conway's "The symmetries of things". It is a lot of fun. Most modern group theory (or algebra) books don't actually have any pictures of symmetric things, just endless formulas and lemmas. If you want to be a pro, then you gotta learn that stuff, but there's definitely pathways into higher mathematics that don't require you to learn that.
Speaking of group theory, I can recommend "A book of abstract algebra". I think that it's a very approachable introduction to the topic. As a person with a CS degree doing ML, it changed my perspective on so many different topics, I can't recommend it enough.
I am bit surprised there is nothing about graph theory in there. Also nothing about combinatorics or knot theory to mention two other subjects. If you want to make people dive into mathematics, it might be a good idea to show a broad range of subjects instead of focusing on the traditional subjects.
It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute.
You listed some of my favorite stuff. Weirdly, when I was 11, my math tutor told me I’d probably really like finite mathematics. She turned out to be right.
> It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute.
I think it’s amazing how connected the fields are. It’s almost like “pick any two of analysis, algebra, geometry, number theory, topology, turn one into a adjective and you’ve got a new subject area”.
I think they’re both true. To the musician, the drum and the flute have a sameness. “You make notes in time with them”, the unconscious mastery. To the beginner (me) it’s worthwhile to see their separateness. I shouldn’t think “I don’t like playing instruments.” because of my experience with recorders in the third grade. Don’t let the trauma of high school trig keep you out of graph theory. They certainly feel different.
This is not hollow advice. Yesterday I bought a flute!
As a mathematics major, I find it encouraging to see non-mathematicians sing the praises of the subject. So I applaud that. But I think the author is overstating/slightly wrong about a few things, perhaps because her exposure to mathematics has been through the lens of physics.
Maybe a more humble rewording of some of her statements e.g., "Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics." would be helpful.
Her suggested curriculum doesn't include anything from Number Theory, which is a foundational part of an advanced mathematics education. It is also one of, if not the most, beautiful topics one can study in mathematics.
I find it odd to call out "Introduction to Proofs" as a topic in and of itself. Proofs aren't really a topic in the way analysis or number theory is. At advanced levels, devising theorems and theirs proofs is what mathematics is.
I have a decades-old math degree and ended up working in tech as an engineer. Are there options, like a "Math Camp for the Middle-aged" where I could get a chance to re-learn everything I've forgotten?
Going from Strang to D&F seems like a steep jump. The former is an applied textbook for non-mathematicians and the latter is a proof-based text for advanced undergraduate / graduate-level math students.
I would suggest working through a proof-based linear algebra book in between to ease the transition. Axler's is a good one. Alternatives include Hoffman and Kunze and the more modern Friedberg, Insel, and Spence.
Both Strang and D&F are extra-relevant for cryptography (I was struck by how much the earliest parts of D&F --- which I haven't gotten much further beyond --- read like the mathematics background chapter of a cryptography book), and I've been in study groups for both of them with non-mathematicians that went OK. But the D&F study group fell apart for logistical reasons, so maybe it would have hit a wall after a couple more months.
read like the mathematics background chapter of a cryptography book
A lot of maths-related books, especially ones intended as textbooks will read like that in part because they aren't kidding about the 'abstract' in the title - they're trying to teach/re-summarize key concepts of mathematical abstraction. It's a good and true thing to notice.
That being said, Axler is an excellent book. I don't know if I would replace Strang with it, but I should add it as a supplement to the next edition of this guide!
Second Axler. "Linear Algebra Done Right" is probably the pure mathematics textbook I've most enjoyed reading ever (but be warned you will learn very little about applied methods from it if that's what you care about).
My university course in linear algebra taught me how to manipulate matrices. It was super uninteresting, and easy. I aced every test, but got a B in the course, because the professor assigned an asinine amount of homework (that I either aced or didn't do), perhaps holding the article author's view that:
> solving problems is the only way to understand mathematics. There's no way around it.
...without also understanding that doing problems is not a substitute for understanding.
(I'm still salty about that course. I've been doing linear algebra based puzzles nearly every day of my life and this professor somehow made the topic a boring chore.)
I complained about this to a friend who had also taken the course and he turned me on to Axler. I read through the first chapter, nodding along as I went. I got to the problem questions and couldn't believe what Axler was asking was even related to the material I had read through. I really struggled at first to understand. Axler was heavily juxtaposed to my previous experience. However, when I did understand, I didn't just understand, I grokked.
It was just such an awesome experience, and I credit that book in particular with breaking me out of a mathematics plateau and with liberating my mathematics education from a strict reliance on academia. The text is almost magical.
> I got to the problem questions and couldn't believe what Axler was asking was even related to the material I had read through.
I think this is a common first experience when first hitting pure mathematics. Mathematics often feels like very rote applications of rules drilled into one's mind, and then you hit a pure mathematics textbook and the questions become a step change in difficulty where you're expected to derive novel insights on your own that the text doesn't hold your hand in showing. A single problem can easily occupy days of your time before the "aha!" moment, but as you say, once you get the "aha!" you realize your understanding is quite profound as opposed to a shallower understanding of just how to apply a given set of rules.
Strang's latest book DE&LA is disappointing, it is linear algebra and its applications with the abstraction taken out and mushed together with supplementary notes from ODE videolectures. Mattuck's ODE course is good.
I should say his penultimate minus one book, DE&LA is from 2014 and I think he has published a couple more since then. Linear Algebra and its Applications is his 1988 book.
For many years MIT students would go from Strang in year 1 to Artin in year 2. Artin != D&F of course though many would say it does less hand holding than D&F
I’d be hard pressed to recommend D&F to a self study audience. Perhaps it’s just me, but the endless enumeration of examples and counterexamples at the beginnings of most chapters seemed to destroy my intuition rather than reinforce it.
I find it hard to believe that the author started to appreciate physics by reading The Feynman Lectures on Physics before any exposure to physics or even algebra, and in less than three years went from barely knowing high school math to enjoying advanced mathematical physics and graduate-level quantum physics. It looks this is one-in-a-million level brilliance as learning the sheer amount of requirement knowledge in such a short time is amazingly challenging: analysis, functional analysis, complex analysis, linear algebra, abstract algebra, differential equations, mathematical statistics, and all the physics: mechanics, electromagnetism, thermodynamics, optics, statistical mechanics, relativity, and of course quantum physics, all in less than three years.
I agree that this does not on its surface seem possible, but I can think of a few explanations.
1. I recently spent a week on one section of one chapter of a math book. I was able to follow it within an hour on the level of "these are the rules and this is the sequence of their application," but I have stuck with it since then because I wanted to understand it well enough that the proof they chose to use would seem obvious to me. If you saw "understanding math" like the peak of a mountain, you'd get there a lot more quickly, but if you want to try out every permutation of every device and condition anything can take forever.
2. Algebra seems simple in retrospect, and my teenage self was kind of dumb. Maybe with my complete adult brain I'd be able to finish highschool starting from scratch in a few months. Evidence to that point is the pacing of college remedial math classes. Maybe, to a certain extent, people have an innate math setpoint that they will snap to very quickly when given the chance.
3. Intelligence is equally distributed between genders, but most professional physicists are men, which means that for every professor there is almost exactly one corresponding woman who has equal potential but isn't in the system. If you heard that the department chair at a university sat down and read a book about topology without a lot of trouble you wouldn't be surprised at all. In other words, it's not surprising that someone can do this, it's surprising that someone who can do this is not in the social bucket for people that do it, but if you think about the other things you've heard about that, you realize you already knew.
I am inclined towards #3 out of all these explanations but all may be true at once.
> Intelligence is equally distributed between genders, but most professional physicists are men,
Why limit yourself to gender? Why not white vs other skin color? Why not the US vs another country? Why not democrat vs republicans? Why not western culture vs whatever other culture? Seriously, this kind of categorization is just ridiculous, especially when you speculate instead of showing evidence.
No, I won't be surprised if a STEM professor is reading topology. I will be surprised if a gender-study professor is reading topology. I will be also surprised if some stranger (i.e. I don't know the background of this person) who could only do pre-algebra in high school says Topology without Tears is the first book on Topology that they read and they immediately fall in love with topology. Possible, for sure. Surprising, of course. It's just a matter of probability.
I'm not sure what objection is being made. We know that there are lots and lots of women who could be physicists but decided not to. You don't stop existing when you don't get labeled, but you do start surprising people who expect you to have been.
>The researchers say that as last author is usually associated with seniority, based on this data, their model predicts that it will be 258 years before the gender ratio of senior physicists comes within 5% of parity.
> I find it hard to believe that the author started to appreciate physics by reading The Feynman Lectures on Physics before any exposure to physics
I'm a physics major. My first exposure to physics was Feynman, and it made me want to become a physicist. I think that's a statement more about Feynman than me though, as I'm not particularly talented. It was a common story among other physics majors too.
Spivak’s Calculus reignited my interest and appreciation in math. Sad to discover the author passed away quite recently. The way of explaining principles and making you do the hard work via problems which I believe is a must with this book, is profoundly astonishing. There’s a lot of mathematical insight packed into those problems, it almost feels you can build up the entire high school and the early uni curriculum from the ground up, for instance there are a number of popular formulas you’d arrive at and derive accidentally while working on those problems. Furthermore it really works your brains by making sure you can reason within the established framework and exercise great doubt. I’m taking this book very slowly.
The website mentions some good courses, personally I love Richard Borcherds' YouTube channel[1] for both undergraduate and graduate courses. No frills, exceptionally clear, (mostly) bite-sized lectures that cover a good range of material (especially in geometry).
Something that might interest HN's demographic is Kevin Buzzard's Xena Project[2], centered around proof systems (in Lean). The natural numbers game [3] is particularly fun IMHO. I don't know if it counts as learning materials per se but it's certainly instructive.
To the section "Popular Math Books" I would add almost anything by Julian Havil. John D. Cook referred to him as a writer of "serious recreational mathematics" [1]. I would probably put his "Nonplussed!" and "Impossible?" books in the "Level: Easy" group, with the others at least in "Level: Medium". "Gamma" is one of my favorite serious recreational math books.
If you like getting into the nitty-gritty of problem solving then check out the books of Paul Nahin. They vary between "Level: Medium" and "Level: Difficult", with many of them reveling in the solution of equations, and integrals in particular. Although he recognizes the need for proofs, he makes a point of avoiding them in his books.
At a first glance that book completely fails for an undergrad lin alg course, and looks weak in other areas too.
Examples: the words nullity and kernel don't appear, rank of a matrix is not in it, and, well, every topic I can think of for undergrad linear algebra is simply not in the book.
It's equally bad for stats: no mention of many common distributions a student would learn for example.
Is that such a bad thing? I'm half-convinced that the “rank of a matrix” is just an artefact of a particular algorithm for inverting matrices. And distributions aren't everything; I can look up any distribution I want on Wikipedia, just as soon as I need it, but a proper foundation in what statistics means is much harder to come by. (I have enough of a foundation to know when it's being taught very wrong, but not enough to actually be very useful in day-to-day life.)
>I'm half-convinced that the “rank of a matrix” is just an artefact ...
The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space". It becomes more fundamental if the person goes on to study deeper math. It tells you how to compute the size of a basis for the target space. The uses go on and on.
>And distributions aren't everything; I can look up any distribution I want on Wikipedia
Yes, you can look up anything on wikipedia, but not in this book, which is why this book will not teach you the things the OP claimed it would.
>a proper foundation in what statistics means is much harder to come by
It's very hard to get that proper foundation from a book that does not cover those foundations. One is better served by using a proper book with a consistent and well laid out presentation of the needed concepts. Saying one can look fundamental stuff up elsewhere means the book is lacking.
> The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space".
That's more to do with the space of the coefficients, I think.
> Saying one can look fundamental stuff up elsewhere means the book is lacking.
A list of distributions is not fundamental to statistics. Discovering a new distribution doesn't meaningfully expand the fundamentals of the field of statistics: all the basic principles are the same. Anyone familiar with those basic principles can pick up a new distribution quite easily – and can probably derive new distributions when they're needed.
>That's more to do with the space of the coefficients, I think.
I don't think that phrase is a thing in linear algebra.
I've gotten a PhD in mathematics - I know both these fields quite well. I stand by my assessment of linear algebra - I've been using pieces form it nearly daily for decades.
>Anyone familiar with those basic principles can pick up a new distribution quite easily – and can probably derive new distributions when they're needed.
And a student trying to learn where and why stats is useful should be taught a wide set of distributions so they see the nuances while they're learning. Sure, you can provide only a Gaussian, but when the student leaves completely ignorant of all the places a gaussian fails and what are some choices relevant to different situations, you can failed to teach them the fundamentals, which includes enough nuance to see when and where to apply what distribution.
>A list of distributions is not fundamental to statistics.
You may as well claim all of stats is not fundamental - just learn math principles. You can look up anything in stats and derive it yourself once given the concept if you're good at math fundamentals. Surely with enough math skills, and zero stats, you can derive all the stats knowledge needed without needing to ever see any stats in a book.
But that's a bad way to go about teaching people useful skills.
A student learning about distributions needs some examples. This book has none. You can argue all you want, but this book is crap for learning the topics the OP claimed it covers.
Find a textbook for beginning students that does not cover a multitude of distributions. Either every single author, usually writing from decades of experience, is wrong, or you are.
> I don't think that phrase is a thing in linear algebra.
You're right; sorry for the typo. I meant the span of the coefficients (which is a space).
I'm not saying the concept of “rank” is useless, or anything. I'm saying a lot of linear algebra is based on high-level techniques and algorithms, but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra, if you learn / intuit a different high-level formulation instead.
> And a student trying to learn where and why stats is useful
That's not what I mean by “fundamentals”: I mean what stats is and how it works. A solid grounding in applied statistics is just as much about human psychology as it is about mathematics, and you never need know where your distributions come from so long as you memorise the rules.
> Sure, you can provide only a Gaussian,
Is that what the book does? If so, then I completely agree. But I don't see why you can't teach statistics using a Gaussian, a Poisson, a beta (three arbitrary distributions) and a general “some distribution”. I'm more of a pure mathematician, though, so if the answer's just “that doesn't prepare statisticians well for the kind of distributions that come up in the real world” then I can well believe it.
> Find a textbook for beginning students that does not cover a multitude of distributions.
>but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra
I'd argue that without a deep understanding of rank and nullity you do not understand linear algebra. Sure, you can lean to move symbols and compute simple things, but that is not much of an understanding.
I'd guess you have not moved into deeper things - then you'll realize you're missing fundamental understanding of linear algebra needed to move on.
It's like claiming one has a fundamental understanding of calculus by being able to solve high school level integrals, but really doesn't understand deep relations between the main ideas of calculus. Sure you can write things down, but there is a major difference between that level of knowledge and what I'd call a fundamental understanding of calculus.
>span of the coefficients
????
Care to link to the thing you're misnaming? I have no idea what you're talking about, and I even googled the phrase.
This is what I mean by using a proper book to learn from.
>Aren't we discussing one?
Nope - the book in question is not offered as a beginner course stats anywhere I am aware of. Care to show one? Calling it one then claiming it is evidence of one is simply circular logic.
>Is that what the book does?
So you have not looked at the book yet are arguing what kind of book it is? That about sums this up.
> I have no idea what you're talking about, and I even googled the phrase.
A linear transformation of some n-tuple (let's call it a) to some m-tuple (let's call it b) can be thought of as the sum of an element-wise multiplication of each item of a by an m-tuple coefficient (let's call the coefficient c_i). So b = ∑ a_i × c_i.
The coefficients may or may not be linearly independent. The span (https://en.wikipedia.org/wiki/Linear_span) of these coefficients is a useful property of the linear transformation; I consider it a more fundamental, more useful concept than your "rank".
The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm, and there are lots of theorems that involve it (probably because it was discovered early on in the development of this field), but it seems a rather complicated and unintuitive concept, to me. If you're using “rank”, you need to use a lot of theorems and lemmas and conversions between different representations of things that just don't seem necessary. I am happy to be corrected.
> Nope - the book in question is not offered as a beginner course stats anywhere I am aware of.
That doesn't mean it's not a textbook for beginners. It looks like one, to me.
> So you have not looked at the book yet are arguing what kind of book it is?
It's over 500 pages long. I've scrolled through it a bit, and nothing jumped out as obviously wrong; I haven't read it. (Though I did see a few different distributions named, hence my confusion.)
You concisely illustrate my point. You misuse words so that others that have learned the material cannot understand what you're writing.
Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.
>I consider it a more fundamental, more useful concept than your "rank".
The dimension of the span is the rank. It's exactly why rank is important. The nullity is the dimension of the kernel, and rank + nullity = n.
Saying rank is not important is like saying dimension is not important. The dimension of a vector space is the first and most important invariant that describes the space. Rank is that dimension for the image of a linear transform. It's absolutely fundamental. It's why when describing some linear algebra thing, one usually starts with something like "Let V be a n-dimensional real vector space" or similar. We rarely (bordering on never) write "Let V be a vector space spanned by the following vectors", and we never write "the coefficient span of the transform".
>The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm
No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it. And no matter what choices you use in your elimination, you will always get the same rank. This is super important - that Gaussian elimination gives knowledge of the linear transformation that is independent of choice of basis or of steps performed. Changing of either vector space changes the numbers in the matrix, and different Gaussian elimination steps may have differing intermediate steps, but the rank is the rank is the rank.
It is also true that the row span = the column span = the rank, which is also not immediately obvious. These are theorems proven in basic linear algebra, and fundamental to claiming to understand basic linear algebra.
That rank shows up in Gaussian elimination is not some artifact or unique thing to Gaussian elimination. Since rank shows up everywhere, when it also shows up in Gaussian elimination shows that Gaussian elimination is doing something fundamental - it is but one of many, many ways that rank pops up over and over in linear algebra.
Rank (and nullity) are absolutely fundamental.
And no one calls that m-tuple a coefficient.
None of this is in the book above. Zero.
A final nice point to illustrate, here [1] is the index to Strang's Introduction to Linear Algebra. Rank occurs more than any just about every other entry in the index.
Here's Serge Lang's book [2]. After the word dimension, rank occurs the most in the index.
Book after book shows that after the concept of dimension, rank is probably the most important concept in linear algebra.
I could go on and on. You cannot claim to understand linear algebra without know how pervasive and useful rank is.
> You misuse words so that others that have learned the material cannot understand what you're writing.
> Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.
So… I'm not wrong; I just don't know the words? Sounds like I actually do understand the mathematical concepts. (Maybe I'd find the words easier if mathematicians named stuff sensibly… and yes, I know I'm using English to write that sentence. I'm a hypocrite, but that's no excuse for the naming conventions in abstract algebra and category theory.)
> No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it.
So why does every textbook, every lecture, and Wikipedia talk incessantly about Gaussian elimination (an algorithm for inverting matrices!) when talking about rank? Rank's only useful as a property of the vector spaces, so why treat it like a separate concept?
> It is also true that the row span = the column span = the rank, which is also not immediately obvious.
The size of the row / column span is the rank, surely? Unless I'm misunderstanding the terminology.
And that's because we have several different concepts to describe the same property, for no reason that I can see. That relationship was immediately obvious to me as soon as I worked out what “rank” was, because I'd done my own investigations into linear algebra before I ever got taught it. (Investigations that I wouldn't've thought to do had I not known about the concepts of linear functions, multi-variate functions and inverses, so I'm not claiming to have independently invented linear alegbra.) There's no way they could be different, because it's about the linear independence (which regions of n- or m-dimensional space are reachable by a linear combination of the… you don't want me to say “coefficients”).
Likewise, I don't think we should be using “row” and “column” to describe linear mappings. “We sometimes write numbers in a grid to represent the function” isn't fundamental.
> And no one calls that m-tuple a coefficient.
But it is a coefficient, if you write the transformation the way I wrote it. Why is that wronger than talking about the “rows” and “columns” of a function? Other than convention, of course.
> None of this is in the book above. Zero.
To be fair, the book does purport to teach statistics; it uses, but does not claim to teach, basic linear algebra. A solid grasp of linear algebra is a prerequisite to understanding the book, so if you understand the book you probably understand linear algebra.
>Maybe I'd find the words easier if mathematicians named stuff sensibly
Column space and row space are completely sensible.
>And that's because we have several different concepts to describe the same property, for no reason that I can see.
You do not see. If you think column space and row space are the same thing, then that's completely wrong. They have the same dimension, which is a theorem, but they are not the same space.
>But it is a coefficient,
So is everything, which is why calling this a coefficient, when there is a better word, is useless.
If you have columns m1, m2, m3, m4, and form the linear combination a1m1 + a2m2 +... Then the ai are also coefficients. And they're much more like what people call coefficients since they're scalars. If you want to call the mi the coefficients, what are you calling the ai? Numbers? Integers? Crawdads?
The mi are vectors, they are column vectors, linear combinations of them form a subspace, and the things multiplied by them to form the subspace are called coefficients.
So yes, you can call them coefficients, but you may as well call them numbers, or pointy-things, or anything else you make up, and no one will be able to talk with you, since you insist on doing things in a manner that makes your work unintellgible.
>Likewise, I don't think we should be using “row” and “column” to describe linear mappings. “We sometimes write numbers in a grid to represent the function” isn't fundamental.
... and you're off the deep end again. I'm glad you invented close but not correct linear algebra, that you missed so many important relations, that you use words in the manner you believe they should be, and on and on.
Of course, your methods clearly must be better than centuries of mathematicians - you should publish a book and clear it up for everyone.
>So why does every textbook
It's baffling to me how hard you push at simply learning. Pick up one of those textbooks I mentioned, and look at every page indexed to rank, and look at how it's used.
That you continue to debate this point is astounding and willful ignorance. I've given you why it's important. I've shown that in textbooks it indexed second only to dimension. I've given at least 5 places it shows up. I've explained how it's the dimension of extremely important items.
It becomes more and more important the deeper you go into math, and that is probably the biggest reason it is so important here. The concept of rank is the tip of an iceberg going through everything above linear algebra: Hilbert spaces, operator theory, exact sequences, homology, cohomology, topology, and on and on and on.
I'm done. You don't care to learn. You want to keep claiming your made up vocabulary and sloppy terms that miss important issues are superior. You're far too stubborn to educate. Go do it yourself.
> You do not see. If you think column space and row space are the same thing, then that's completely wrong.
You were the one who wrote column space = row space…?
> If you want to call the mi the coefficients, what are you calling the ai?
I'm calling them the coefficients because they're “part of the function” and don't change. They're the thing you'd naturally call a coefficient. Coefficient isn't as broad a term as you seem to think it is; In f(x) = x² + x + 3, the coefficients are 3, 1 and 1; not 1, x and x².
> You should publish a book and clear it up for everyone.
As soon as I have any actually original work, I plan to. But linear algebra isn't a specialism of mine, so I doubt I ever will.
And no, I don't think the terminology I've used here is an improvement over the status quo; I'm not a complete imbecile. I just don't get why terminology is considered more important than understanding in mathematics education, and why it's practically impossible to use different words for things even when you do have an improvement.)
> That you continue to debate this point is astounding and willful ignorance. I've given you why it's important. I've shown that in textbooks it indexed second only to dimension. I've given at least 5 places it shows up. I've explained how it's the dimension of extremely important items.
If you think any of those are counterarguments, you were never addressing what I was trying to say.
> You want to keep claiming your made up vocabulary and sloppy terms that miss important issues are superior.
Where did I claim this? I believe I said I was “not wrong” (a weaker label than “correct”), and I identified some problems I have with the existing terminology, but I don't think I ever said my (inconsistent, ad-hoc) terminology was better.
I give a book recommendation and ppl complain, yet no one complains when ppl recommend Khan Academy. Do people think Khan Academy is more rigorous than a textbook?
I'm not complaining about a book recommendation. You claimed this book will cover college-level concepts for linear algebra. It does not even come close to doing that. It's misleading for anyone wanting to learn linear algebra and taking your book recommendation.
As a math PhD I have to say the only way you're going to learn mathematics is if you actually have a pressing need to do so. i.e. You have a project at work that needs some math, you have a hobby that needs some math. In this case you just learn what you need. Just learning math for its own sake outside of a University STEM track is just too hard (I wouldn't be able to do it and I've tried).
I hesitate to contradict someone who has gone through the whole Math PhD process, but I have to say that the best mathematicians that I know treat the problems they're working on as games, or riddles to be solved... and they've taught their kids (and others) this same method of thinking about these problems. There's a huge mental tool set, and often a lot of grinding to get to a solution, but it's just a game (and there are often more elegant solutions)!
I enjoy provoking interest in complex numbers and exponentials among precocious teens, but I've never been more humbled than having a Galois theory joyously explained to me by an 11 year old. (p.s. I'm an engineer so I follow your technique).
To me the problem isn’t learning it, it’s retaining it.
I learned all kinds of quantitative analysis and statistics in the CFA program ten+ years ago.
I had daily sheets that I would solve equations and answer all kinds of questions. I knew them forwards and backwards. I just looked at one now on fixed income - not sure if could answer any of the questions today to save my life.
I don't think you're contradicting the GP. Solving problems for fun still counts as "pressing need".
The alternative is something like "Oh, I should learn abstract algebra because it's a fundamental course in all curricula". You'll learn it, and then forget it.
Was programming a 3-axis machine in early college back in 90s. After a few months I was mostly re-inventing trigonometry.
The I actually took trig later on. That would have been super handy at time.
I think the question needs to be asked, "What am I hoping to get from this?"
If its tools to help you solve some problem, great.
If its because you find it fun, great.
But if its because you feel you want to 'better yourself' or perhaps feel like it would somehow prove your intellectual worth, you probably won't get a lot out of it.
This is true. When I started my first job, I tried casually learning more from where I stopped after I finished school. My motivation slowly waned as I realized I vastly preferred playing video games than watching lectures and trying to do some problem sets.
Correct me if I’m wrong, but I assume you’re looking at this through the wrong lens. As a PhD you’ve learnt something hard at a considerable depth but this isn’t what I, or most people, hope to achieve by learning maths themselves. It’s normally to a far lower depth that’s far more achievable like Calculus I or some aspects of number theory or knowing what all the funky symbols such as Summation mean.
I experienced something like this recently. I struggled to grasp linear algebra when I took it in undergrad and as a result was always intimidated by the subject. Now, years later, I'm taking a course in graphics and naturally needing to relearn linear algebra and suddenly everything just clicks.
I am fairly confident that Susan Rigetti is a future president of the USA. In addition to becoming somewhat well-known as a household name early on in her adult life, she has achieved so many difficult and impressive things (publishing multiple books, studying physics and philosophy at graduate level, working for a top-tier tech company, taking down the CEO of a top-tier tech company and damaging the company's reputation, being asked to work for the New York Times, publishing curricula in graduate Physics, graduate Philosophy, and undergraduate mathematics). Furthermore, she seems to have a gift for or knack with the public eye.
I love that the author highlighted Prof. Robert Ghrist's great material. I took his calculus classes through Coursera maybe about 8-9 years ago. He just makes everything exciting and his visuals are just beautiful. For example, within the first few lectures, he made it feel like Taylor series was like the coolest thing ever. Highly recommend checking his lectures out. Check out his website: https://www2.math.upenn.edu/~ghrist/
The curriculum guides Susan Rigetti provides are an amazing resource for self-study. And the fact that she worked through all of this is truly inspiring.
Not to be greedy, but do any of you know of other thorough curriculum guides like this? I know about https://teachyourselfcs.com already -- another amazing guide. Are there others? I would love to find one for statistics especially, but really any subject would be interesting.
I'm still stuck at "wait, sets can contain other sets, and sets can contain themselves?" part of Russell's paradox, and I'm close to retirement!
I don't want to study math. I want to know enough of it to solve some well-understood problems I've wanted to solve for decades. Simply learning how to diagonalize a matrix (and how to use such a thing) meant more than understanding a bunch of complicated matrix theory.
The solution to russel's paradox is that sets can not contain themselves. There is a carefully crafted set of rules describing what sorts of sets exist, and it is designed to avoid paradoxes such as that one. These rules are the ZFC axioms (https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html)
I never got beyond algebra/geometry in High school. I think I had to take one 100 level math class in college, but it was basically a review of HS math. Oh, and I had to take a stats class for non-technical people in graduate school. That was my worst graduate class by far. But, I would like to learn some more math, like calculus. I’m hoping to get to it when I retire in a decade or so.
I question just how realistic is is to have "Proofs from the Book" at the start. While the art is wonderful, the background required to read it is not realistically at a lower level than the textbooks listed.
The discussion here has been much more interesting than the actual list, to me. If you wanted to master all of that material, I think a master's program is the way to go, not self-study.
I would be interested in hearing from people who _do_ successfully self-study. What makes it work?
The least interesting (from my point of view) is "already successful in a related field, applied my skills". That would include CS professionals studying maths, I think.
More interesting would be "unable to attend university because of X, did Y and really enjoyed it." Are you a person who completes MOOCs and get something out of them?
I can't stress it enough. I know this sounds funny, but when I went to study math a lot of people would drop out very quickly because they did not realise that what most people call math and what is taught at high school is completely different from actual theoretical math.
What most people think math is is a collection of formulas that you need to learn to "know math".
Math is actually a dynamic activity and is 100% about solving problems.
Just like programming is not about knowing programming languages. Programming is about solving problems (and knowing programming languages is necessary but not sufficient to be programming).
It's true that a standard undergraduate curriculum in mathematics will contain a lot of analysis (including calculus), a lot of ODEs/PDEs and some algebra. But if I wanted to get someone interested in math I would also point them towards some of the more fun (and less known stuff), such as combinatorics, probability, topology, differential geometry and number theory. Some of these are also much more applicable today than, say, complex analysis.
Calculus: I suggest just forget about "precalculus" and, instead, just get a good calculus book and dig in.
There are two main parts of calculus, and both can be well illustrated by driving a car. In the first part, we take the data on the odometer and from that construct the data on the speedometer. The speedometer values are called the (first) derivative of the odometer values. In the second part we take the speedometer values and construct the odometer values. The odometer values are the integral of the speedometer values. In notation, let t denote time measured in, say, seconds, and d(t) the distance, odometer value, at time t. Let s(t) be the speed at time t. Then in calculus
s(t) = d'(t) = d/dt d(t)
And d(t) is the integral of speed s(t) from time t = 0 to its present time.
Those are the basics.
Applications are all over physics, engineering, and the STEM fields.
Linear Algebra: The subject starts with a system of simultaneous linear equation. The property linearity is fundamental, a pillar of math and its applications. The STEM fields are awash in linearity. E.g., a concert hall performs a linear operation on the sound of the orchestra. E.g., in calculus, both differentiation and integration are linear. In the STEM fields, when a system is not linear, often our first step is to make an attack via a linear approximation. E.g., perpendicular projection onto a plane is a linear operator and the main idea in regression analysis curve fitting in statistics.
Most of math can be given simple intuitive explanations such as above.
Susan, I greatly appreciate this list and will definitely come back to use it as a reference if I need a book recommendation. (I don't think I'm the target audience, although who knows what the future brings..)
That being said, I think you are missing out on an opportunity to reach a wider audience. It bugs me a bit that the requirements seem very American-centric. What I mean is the following bit:
> A high school education — which should include pre-algebra, algebra 1, geometry, algebra 2, and trigonometry — is sufficient.
And later the paragraph on "pre-calculus".
I know that many places don't have such names for courses in high school. In fact, often it's just called "Mathematics" and you either take it or you don't (obviously there is a spectrum here).
How is a prospective (non-American) student to know what is covered in Algebra 2 in an American high school?
I'm not asking you to change the article, I just hope I can nudge you into realizing that the text as it is now is more difficult than it needs to be for non-Americans.
Do we have good universal descriptors for math levels? I'm a big fan of accessibility, and I think your idea about tweaking language to reach a wider audience could be a big win for increasing the article's impact.
To update the article to include your recommendations, the author would probably need some kind of "cross-walk" which would map the American perspective to a more universally understood framework. Would you happen to know what "pre-calculus's" opposite number would be in the universal framework?
If you actually want to study math, you probably shouldn't touch calculus until you've take linear algebra and a fair amount of topology, since these are the two structures on sets that (differential) calculus is founded upon.
For other subjects, you can briefly substitute an intuition for the underlying structures with sufficient finesse in the presentation of the material (see the theory of knots and links, for an example), but calculus is not, in my experience, such a subject, and the early emphasis on it is harmful for the study of mathematics, which is supposedly what your list is for.
For some reason this is heresy, but I have honestly no idea how you are supposed to appreciate calculus from a mathematical perspective without being able to define the large stack of terms that constitute it. The situation is potentially different for a physicist, but if you want to study mathematics, the physical world is not the object of study, rather it is precisely the definitions that we have chosen.
It also comes after many semesters of calculus, which depend upon it, and before any topology, which it depends on. Even if you are just interested in these things as tools for describing physical phenomena, there is value in placing mathematical knowledge in its natural structure.
Setting aside the question of motivation, how would a working adult find the time to work through all these books? The author of this clearly leads a busy life outside of reading math textbooks.
Reading a math textbook is time consuming endeavor, regardless of underlying ability. The author herself mentions this in the introduction. I can think of a few factors that might make it possible for a busy person to go through all these books in a few years:
- They include books that were read partially while taking course.
- Consistency: allocating 1 - 2 hours per day for a few years.
- Doing exercises selectively: If you only do a handful of exercises per chapter this would dramatically increase the rate at which you go through a book. This would come at the cost of deeper understanding.
I have a large backlog of math books I'd like to read, but time is a constraining factor. If people have found strategies for reading these types of books, I'd like to hear about them.
For me, it’s all about consistency. It’s spending a few minutes to a few hours every day, forever, til the day I die.
I usually can squeeze in 30 minutes to an hour every day to study something (whether it’s math or something else — right now I’m studying cinematography). Sometimes that’s in 15-minute chunks if it’s a busy day. Usually it’s before bed or while I’m eating lunch or if I have extra time on the weekends while my kids are napping.
It’s all about just doing a little bit every day. That’s been successful for me.
I don't agree with this article, it as off-putting as the usual math eduacation it criticizes. I wonder how one can propose a curriculum to study math and not mention Euclid. One learns more mathematics from this article https://mathshistory.st-andrews.ac.uk/Extras/Russell_Euclid/ by B. Russell where he harshly criticizes Euclid than 2 years of calculus. Newton did not know calculus but he knew Euclid's Book 5, the book about ratios and proportions. Euclid's 5th Book must be the starting point for the study of math. When we say "math is the language of nature" we really mean that nature is proportional. Ratios and proportions are fundamental.
Yeah, it's more application oriented and philosophical than the pure calculation of pure math. I think it's under-taught in schools though. I think it's more useful than calculus for most people and should be taught before it.
For many, many years I thought I did. I'd have a brief surge of interest for a few weeks, and then get completely bored of it. I'm not someone who finds it inherently easy, so boredom + difficulty = failure.
When I was foolish enough to do this in university, it meant doing great in the first few assignments, and then abysmally in the exam.
So my policy now is to never study maths for its own sake. Only when there's equations in a computer science paper I don't understand.
How do you like to solve math problems in this day and age?
I'm partial to Jupyter notebooks lately - I run it locally from a docker container, and have a folder of notebooks. Mostly markdown cells, alternating between my narrative thinking and LaTeX math output.
The author doesn't seem to take her own brilliance into account when composing these self-study guides.
For the other 99.99% of us, it would take many lifetimes worth of free time to make a substantial dent in these materials. To me, guides like these are too intimidating to even begin. Maybe what's needed is a meta-guide on structuring one's time and developing the necessary focus to be able to do this within one human lifetime.
Grab a pen and paper, open up the first book in any of the guides, and start reading. Read for 15 minutes, or 20 minutes, or whatever time you have on your lunch break or before you go to bed or while you’re using the bathroom. Do it again the next day. And the next. And the next.
That’s how I did it. There’s no brilliance involved. It’s just jumping in. The more you do, the easier it gets.
I knew consistency was important, and did not place much importance on "brilliance", although that attribute has been showered upon me since I was little.
What I did not know, and still don't know is that studying only for a few minutes or half an hour regularly will make me good at something as advanced math.
You seem to know your stuff and I like your approach and attitude, and I will do now what I do very rarely and upon serious consideration- take a leap of faith.
I will start doing math everyday for at least half an hour, and I will see how that goes.
If you're interested in both mathematics and physics, does it make sense to learn both concurrently? If yes, what areas complement each other? Or is there no overlap to warrant concurrent study of the essentials? By essentials I mean what a college student must know, or really anyone who pursues self-education without a background in these areas.
Beautiful website, by the way!
Check out my physics guide: https://www.susanrigetti.com/physics. It has both the physics core curriculum AND the math essentials you need to know in order to understand the physics essentials. (And thank you!)
IME (as a math-degree-haver) the value of mathematics is in improving one's ability to mentally model and reason about complicated real-world phenomena. A lot of folks lose sight of the reality and get lost in the mysticism, especially within the academic regime.
> [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. [...] That doesn’t mean it’s easy — no, mathematics is an incredibly challenging discipline, and there is nothing easy or straightforward about it
I am always, always going to condemn this unnecessary mystification and idealization of mathematics. It's exclusive and misleading.
On the mysticism note, I want to add that math is perhaps the only subject that forces one to engage the upper "abstract" mind. The lower mind is concerned with modeling real world phenomenas, while the upper mind works with purely abstract things, aka the "true reality" in mysticism.
You cut out the middle of that paragraph, which says:
"Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a "language person” — such a profoundly ignorant and damaging statement. Here is the truth: if you can understand the structure of literature, if you can understand the basic grammar of the English language or any other language, then you can understand the basics of the language of the universe."
I'm not sure what your point is. Are you implying that you are not contributing to the mystification and idealization of mathematics?
In other words, I do not see how you are dealing with the "baggage" of learning mathematics beyond name-dropping it. In my opinion, the mysticism is the baggage. And then the rest of the blogpost reads like a conventional curriculum within the conventional academic regime with which we associate that baggage.
I don't follow. The author dismissively ctrl-V'd a paragraph with no further explanation, and my response asking for elaboration gets shadow-buried by a mod. What?
Your post wasn't simply asking for elaboration—it was written in the cross-examining style that we particularly want to avoid here because it kills curious conversation.
If you didn't intend to come across like an interrogator trying to back an opponent into the corner, then your comment needed to be written quite differently.
> My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics.
NO, NO, NO.
There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly what Linear Algebra would look like if instead of finite dimensional vector spaces we considered infinite dimensional vector spaces. It opens the rigorous path to non-linear optimization, analysis of pdes, numerical analysis, control theory, an so on. What this article mentions is a way to work around things, but nowhere near an undergraduate degree in mathematics.
I'm astonished that the PDE section has such books, they look like the calculus aspect of partial differential equations. A more appropriate book would be L. C. Evans' Partial Differential Equations. Same with ODEs, no mention of Barreira's or Coddington & Levinson's books.
I'm a fan of functional analysis, but even in my (very competitive) undergraduate curriculum, it wasn't required for a bachelor's in mathematics. I think Susan's guide covers most of what the undergraduate programs I've seen require.
No recommendation on probability. Thats strange given that the author is a physicist and fundamentals of modern physics rests on probability. My recommendation is the classic "Probability Theory, The logic of Science by E.T.Jaynes" which is a Bayesian formulation.
While I understand that the author has good intentions, I strongly disagree with the general idea of this post, which is that anyone can learn math through an almost entirely analysis-focused curriculum while other topics like topology, game theory, set theory, etc. are presented as advanced and graduate-level. This is practically equivalent to saying that anyone can learn history, and they should learn all about British history in undergrad, and then graduate-level courses might teach you more about the history of South America.
Some of my thoughts (mostly drawn from personal experience, feel free to disagree):
1. IMO "learning math" is really about learning how to recognize patterns and how to generalize those patterns into useful abstractions (sometimes an infinite tower of such abstractions!). So it really doesn't matter if one does abstract algebra or linear algebra or combinatorics or number theory or 2D geometry or whatnot at the beginning. Any foundational course in any branch of mathematics, or any book on proofs, will fulfill this need. People learn in different ways and have affinities for different topics, so some subjects will be easier and/or more interesting for them, so aspiring mathematicians should start with a topic they're at least initially entertained by. If you don't know where to start, one fun (for me) topic is the game of Nim; other combinatorics topics are also elementary and entertaining to think about. I'm fairly sure that if I had to take this suggested curriculum as an undergraduate, I would have picked a different major entirely, I personally find analysis quite difficult :(
2. One's first foray into a topic should be a one-semester course, not a textbook. Lecture notes for many courses are freely available online also, so you don't have to pirate the books you want if you aren't willing to pay $100 :P The reason is this: courses are curated by a mathematician to teach students the basics of a topic in one semester, so they will better highlight what you need to know, like important theorems, and have a more careful selection of problems. If you're confused, you can read the relevant textbook chapters. On the other hand textbooks are more like comprehensive references - reading a textbook through and doing all the problems will make you an expert at the material, but it's not as time-efficient (or interesting) as a course.
3. There are benefits to diving very deeply into a topic, but IMO one's mathematical experience is much richer if there's more consideration for breadth, especially when you're starting out. A student learning basic real analysis would benefit from understanding some point-set topology (not just the metric topology that usually begins these courses) and seeing how (some of the) pathologies of topological spaces disappear when you impose a metric and then you get things like being Hausdorff or having many different definitions of compactness coincide. After learning real and complex, of course one could move onto differential equations, but there are so many other ways to branch out, like exploring differential topology or learning about measures & other forms of integration, which also meshes very nicely with statistics. Exploring different branches emphasizes that there are so many directions you can go with math, even when you're just starting out, and gives you a better feel about how "math" is done, as opposed to just the techniques for a specific topic.
This is my first comment on HN, so please let me know how I can improve this comment!
Wrong: math is neither hard nor difficult. It is this single belief that deters many people from learning it. Math is all about logic. It is nothing but how to go from A to B. Any person who can reason should be able to be good at math.
All over the world many people try to learn French. Almost any would tell you that learning a second language is hard. Nobody will fight you on this.
Math is a language to express entirely new concepts thar have nothing to do with everyday thinking and often include things like recursion that makes them impossible to reason about without more than just a few bytes of mental RAM.
There is no step by step algorithmic process to do it. If there were, a computer would be able to do it and far fewer people would want to learn.
Even at the level where there is an algorithm, it's far too big, nobody hand executes the source code of XCas by hand.
Math seems to require not just a skill that can be learned, but an inate ability to deal with multiple connected pieces of information at once, and to see abstract patterns in things.
In programming, if you have to understand more than one tiny bit at a time, you might consider tossing it and starting over. In math it's just normal for lots of ideas to connect.
True! But sometikes getting better at something is all people really want, and that's ok. I see that most of my CS students just want to be able to not see math as an obstacle when learning new/interesting things.
I've got Mathematics for the Nonmathematician by Kline and that's kinda heading the right way, but what about whole courses of study? More books? It's more of an introduction than a thorough resource or course, and feels like it needs another four or five volumes and a lot more exercises to be really useful.
I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.