Sometimes I feel so obtuse...

December 19, 2008 2:26 PM

Please help me rectify my biggest regret from my high school and college days - - - not taking enough math!

Although I've always been fairly gifted mathematically (when I was in elementary school, I helped grade my fellow students' math tests; I also got a 770 on my quantitative SATs) I've never taken any math class higher than pre-calculus.

This was in part because of a depression that caused me to drop out of high school, and in part because I became interested in political science and social psychology, two areas I didn't need math for. And so I dismissed the idea of taking any more math as "unnecessary".

Anyway, now I'm involved in cognitive neuroscience and while I get along fine without knowledge of higher math, I definitely regret not having it. For instance, I wish I could understand the way fMRI data is adapted to make it more analyzable - people toss around phrases like "fourier transform" like I'm supposed to know what that means.

My current frustration is computational modeling/structural equation modeling of psychological processes. I find this area absolutely fascinating but I'm worried that I'm out of my depth trying to understand it. A coworker was trying to explain the math behind Bayesian modeling to me (I get the theory fine) and got frustrated by having to explain basic concepts to me. (My brain is like a chinese doll. "Okay, I understand your explanation - except for that. Can you explain it to me?")

I am willing to devote a fair amount of time and effort to this, but I don't want to do any unnecessary lifting. I'm already working 40+ hrs a week and learning several programming languages and more advanced statistics. Part of my problem with getting started is not knowing where to start - wishing that someone could draw me a straight line of concepts I need to know instead of just saying, "Well, read a calculus textbook. And then a linear algebra textbook. And then a..."

The university I work for lets me take classes for practically nothing. I'm also pretty good at teaching myself things from textbooks - or I could even hire a tutor, although that could get kind of expensive. But sometimes I get discouraged and feel like I'm already way too behind and I should stick to what I already know.

So, AskMe, what do you think I need to know in order to "get" things like computational modeling and bayesian probability? How hard will it be to learn? And how should I go about doing it?

(I get the feeling I'm being too vague with this question, so I will definitely be hanging around to reply if you need clarification!)

posted by shaun uh to education (17 comments total)

Part of my problem with getting started is not knowing where to start - wishing that someone could draw me a straight line of concepts I need to know instead of just saying, "Well, read a calculus textbook. And then a linear algebra textbook. And then a..." ... The university I work for lets me take classes for practically nothing.

It seems to me that the second sentence here solves the first — most mathematics departments have pretty well-defined "tracks" that you need to take, at least for the intro classes. At my alma mater, it was a semester of calculus, a semester of linear algebra, and a semester of multi-variable calculus, in that order. Fourier transforms would be covered in another course, often one on differential equations. Alternately, you could see if the physics department has a "Mathematical Methods for Physicists" course; if it does, it almost certainly covers Fourier transforms (though I'd check first to make sure.
posted by Johnny Assay at 2:50 PM on December 19, 2008

wishing that someone could draw me a straight line of concepts I need to know instead of just saying, "Well, read a calculus textbook. And then a linear algebra textbook. And then a...

That is exactly the problem. Math is not a straight line of concepts; it's a highly interconnected network of different ideas that all relate to each other in complicated ways. I'm not sure of the exact background material that you need, but calculus and linear algebra are probably the minimal background necessary to make any progress. And a course in probability wouldn't hurt either.

I'd recommend that you take courses to learn this stuff, rather than studying on your own or hiring an expensive tutor. That way you have somebody experienced to guide you through the material and emphasize the relative importance of different concepts, which is often quite difficult to get through self-study. You can probably do calculus and linear algebra at the same time, then take probability afterwards.

And if you really want to learn the stuff, do the homework problems. It's not helpful to just learn the 'theory' as you put it, without being able to use it on examples.
posted by number9dream at 2:51 PM on December 19, 2008

ok. calm down. what do you want to know? forget the question of what is required of you, or where you should be. What do you want to know?
Maths, I think, helps thinking. When you move into an area like cognitive neuroscience, you need all the critical skills you can get. Its not a cock fight. Its armor. Can you stop flapping, and say where you want to go? What questions do you want to ask?
posted by stonepharisee at 2:53 PM on December 19, 2008

I don't know your field well enough to tell you exactly what you'll need. I'm a computer programmer, though, and think mostly like you. I took a number of math courses in high school and college. Here's my take.

Your question is a hard to answer not because it is vague, bit because math is so interconnected. You just flat out cannot learn calculus without knowing algebra. It's like trying to learn ballet before you can even walk--you just don't have any tools or frame of reference.

You can certainly self-instruct in mathematics. However, I, and many others, find it difficult to actually learn the math without doing the arithmetic* a few times. It helps so immensely to see the answer to a few questions mainly because it allows you to extrapolate some properties of the answers to future questions without doing all the work.

I would recommend that you take classes instead of trying to self-instruct. Make sure to ask around about which professors are the best (because a bad math prof is of no use to you whatsoever). The structure of the class makes learning the math much easier. They will, hopefully, walk you through a series of topics that build in a logical progression. You might look for computer-assisted courses in the topics you need, as much of the unpleasantness of the arithmetic will be handled by the machine.

The most important thing that a professor can do, almost no matter how bad (s)he is, that you cannot do for yourself is tell you what you're doing wrong. A person can watch you solve a problem, and spot your mistake. The best you can hope for without help is for the solution manual to tell you that have made a mistake. You'll have to keep reworking your arithmetic until you find the error--and by the time you get to integral calculus, the arithmetic can take up five to fifteen sheets of paper per problem.

If you decide on self-instructing (for time or personal reasons) I insist that you invest in a piece of software like Mathematica. Even if you take courses, I still suggest it. You want something that allows for symbolic manipulation as opposed to just numeric. The software will allow you to concentrate on the theory of what you're doing, while minimizing the number of arithmetic errors you make.

While I said I don't know exactly what you'll need, I can tell you what you'll need to know to know what you need. That is, the basic concepts that you'll need in order to understand most math generated by non-physicists and non-mathematicians. It will definitely give you the background necessary to read a paper and determine what new, specialized math you'll need to learn in order to understand that paper.

This list is of the mathematical concepts you'll need. You can learn the arithmetic lazily... that is, you'll need to know what the root of a parabola is from the start, but you can save learning to solve it until you need to. But, I suggest that you play with the arithmetic in your software so that you can get a feel for how each concept "behaves".

You need, at minimum (with what my schools have called them):

• High-school algebra (Algebra I and II). Variables, equations, polynomials, systems of equations.
• Differential Calculus (Calculus I). Limits, the derivative (in all its guises), lines tangent to curves, characteristics.
• Integral Calculus (Calc II). The anti-derivative, the integral, areas under the curve, rotations, convergence/divergence, Fourier analysis.
• Linear Algebra (The Most Useful Math Course on the Planet, I Absolutely Fucking Swear). Vectors, matrices, basis vectors and spaces, transforms, kernels, eigenvectors and eigenvalues.
• Probability and Statistics (Probability and Statistics). You need a deep understanding of prob-stat to do Bayesian modeling. I recommend you take a course, or buy a book and go cover-to-cover. If this is where your work lies, you will need all of it.

That list is absolutely not complete. But, in order to understand scientific math, I guarantee that you will need advanced algebra (through linalg) and at least basic calculus. Multivariate calculus is also very useful, but not mandatory... until you find that it is, in which case you'll be spending a couple months teaching yourself that.

*Most math folks make a distinction between "math", which is the theory and the reasoning, and "arithmetic", which is the mechanism by which you manipulate symbols and arrive at solutions.
posted by Netzapper at 3:25 PM on December 19, 2008 [10 favorites has favorites]

There's a pretty detailed explanation of Bayes' Theorem here, that you might want to look at. You could probably dive right into that without needing anything else.
posted by losvedir at 3:25 PM on December 19, 2008

Johnny Assay & number9dream - I was kind of hoping you wouldn't say that. I can only take one class at a time. If I just take math classes, and ignore the other courses I want to take, it will still take me over a year just to get the background done. I guess if I really want to learn this, I can commit to that, but I was hoping there might be a way of targeting more precisely what I want to learn.

Stonepharisee - I'm most interested in how people create equations that model human behavior by taking into account lots of different variables that affect it, including variables that are uncertain or learned. Things like this and this and this.
posted by shaun uh at 3:25 PM on December 19, 2008

Have you considered asking a math professor to let you audit a class off the books? You'd still have to put in the time, but it wouldn't cost you anything or keep you from taking other classes.
posted by Etrigan at 3:32 PM on December 19, 2008

Have you considered asking a math professor to let you audit a class off the books?

This is absolutely a valid, expedient way to go about this. You don't care about the credit, or the grade. All you want is the knowledge.

Sit-in, off-the-record auditing does, in my experience, often mean that the professor, imo justly, shows preferential treatment to "real students" with regard to their time. In some cases, it even means that your homework and tests are not graded, or not commented/corrected.

Your school may have an official audit process, however, that grants you all the rights of a student, costs money (shock!), and goes on your transcript as an audited course--sometimes, they'll even take courses audited this way as prereqs.
posted by Netzapper at 3:42 PM on December 19, 2008

actually, learning math is pretty much a straight line or at least polygonal, it's just a question of where you start, which direction you head and where you end up.

there's really no reason to start with calculus except that the textbooks are pretty much idiot proof. they aren't deep (in fact aggresively shallow) but they are very pedagogical (coming from someone who has taught calculus to college students for years.) I would recommend starting by yourself, you might find you can go through two semesters or more if you are motivated, so:

Get the calculus book the university you work at uses for scientists and engineers (it's actually going to be easier in a lot of ways than the business and/or biology text):

Step 1)

a) read chapter 1 (usually a pre-calc refreshed), every last word
b) do every 5th exercise.

Question: can you still do algebra, do you know what coordinates are
If the answer is no, go back to pre-calc.

Step 2)

a) read chapter 2, every last word of it.
b) do every 5th exercise.

Step N+1
a) read chapter N +1...

Question: are you losing motivation?
If the answer is yes:
a) how much do you really want to learn this stuff?
If not really, learn something else, as you know there's more to learning than just math
b) if you still really want to learn it, try enrolling in the calculus class that uses this book, don't just sit in, the psychological pressure of grades attendance etc. is what you are looking for.
posted by geos at 4:49 PM on December 19, 2008 [1 favorite has favorites]

I am a math professor who, by chance, is learning about Bayesian models of decision-making for a joint project with some psychologists: so I'm sort of in the position with respect to cognitive psychology that you are with respect to math. I also have some idea of what math is needed for the kind of problems I've encountered, which might be similar to yours.

I'd say the most important thing you can do is take probability/statistics and the least important thing (for your particular purposes) is calculus. In probability and statistics you'll run up against some background you don't understand; that will tell you _exactly_ which notions from linear algebra, etc. you need to learn from books in parallel with your course. Of course, best of all would be to have a deep knowledge of calculus, linear algebra, discrete math, etc -- but there's lots of things it would be good to know, and I'm trying to steer you in a direction that's best possible under your constraints. And just about every single you learn in a good probability and statistics course is going to be relevant to cognitive science; this is far from true of calculus.

A few additional comments: a) Asking to audit a course is a great idea. But yeah, don't expect to have your homework or tests graded; not only would it be kind of unfair for me to ask my TA to grade more papers than they were paid for, it would violate union rules at my university, and probably lots of others. b) You say "A coworker was trying to explain the math behind Bayesian modeling to me (I get the theory fine)." One point I want to emphasize really strongly -- it's really easy, and really dangerous, to think you "get the theory" of some piece of mathematics. Unless you can carry out the computation yourself and get answers, I promise you that you don't really understand it, even in theory.
posted by escabeche at 4:55 PM on December 19, 2008 [2 favorites has favorites]

people toss around phrases like "fourier transform" like I'm supposed to know what that means.

Well, if you want to go down this particular topical road, you might want to read Who is Fourier. It's a math book like no other. But this is if you want to jump into the practical math behind signal processing, won't help so much with probability/stat related stuff.

Math is not a straight line of concepts; it's a highly interconnected network of different ideas that all relate to each other in complicated ways.

This is a really good insight. I do think it's possible to pick out small subsets of the discipline ("straight lines", if you will) and acquire some skills in that specific area, but you'll sacrifice some depth of understanding unless you really sink time into it. (Heck, I put four years of college study into it and I feel like my understanding is shallow.)

I'd compare it to the process of learning a foreign language -- you can develop a working knowledge of phrases and limited vocabulary that will get you by for a variety of everyday practical tasks, but fluency requires immersion, quick recall of wide swath of vocabulary, and near reflexive application of a significant body of rules. No mean feat. Being part of a community of people who practice together really helps.

If I just take math classes, and ignore the other courses I want to take, it will still take me over a year just to get the background done.

With everything else you're doing besides taking classes, unless you have a genius for this, it will probably take you a long time anyway.

I guess if I really want to learn this, I can commit to that, but I was hoping there might be a way of targeting more precisely what I want to learn.

The virtue of the traditional course structures tends to be that over time, the people who've shaped them have considered the various dependencies in the conceptual network number9dream talks about and worked out some viable paths. You absolutely can leave those well-beaten paths, but if you do, you're probably going to want a guide (either a good and unusual book that takes a different tack, or a good tutor) or you're likely going to have to spend some time being lost, looking for the missing piece of information you skipped, which sounds like it's what's happening in your conversations with your friend about Bayesian modeling.

And it does sound to me like you're eventually going to want the contents of a first semester calculus, linear algebra, and mathematical statistics classes.
posted by weston at 5:45 PM on December 19, 2008

I might observe that even if you HAD started earlier, you'd still be almost as ignorant, so don't be too intimidated or embarassed. That's just the way it is.

Also, abandon all hope of encompassing the field in its entirety. There is no way you could digest what was published TODAY next year.

The best one can do is to decompose what you need to learn into fundamentals, assess which ones you are thin on, prioritize those, attack them with vigor/immersion, and continue until death places it merciful hands over your lifeless eyes, hopefully many, many math filled decades from now. As mathematician Paul Erdos said 'There's time for sleep in the grave. There's math do be done now!"

Learning anything is a lifelong process. It doesn't take too long to forget anything but the most obvious take-aways from any course. If you had learned all this stuff in high school, it would be stale by now unless you routinely used it. True, it gets easier to refresh when you do it a lot, but you still have to constantly refresh and remind. Just get used to it being a side pursuit and stay at it for a long time.

In practical terms, I have found that I am almost always ignorant at the beginning of a project and need to learn, explore, refresh, or investigate as I work. Your successive projects will dictate what you need to know. Most everyone else has listed a good selection of major categories to guide you. Some will never rear their heads in your work. ALso, I have found that in most cases there is always at least one worker bee on a project with a grasp of the relevant fundamentals involved who can mentor you in situ and teach by osmosis. (Helps if you have grasp of the basics, obviously!)

(Once you figure out Fourier transforms, you'll be pleased you made that particular effort, BTW. Love 'em! )
posted by FauxScot at 6:13 PM on December 19, 2008

I think from where you are it's tough to go where you'd like to be. The kind of applied math used by statisticians these days is pretty abstruse. If you want a deep understanding of what kind of transformations are done on various distributions to make them into predictive models, you probably need a few semesters of calc and applied math under your belt. Then you're ready to start taking the math classes that go under silly names like "biostatistics." I took 6 semesters of "biostats" and it was really just math, applied to a particular kind of analysis or modeling.

At the end of this process - and I started it with a fairly good grounding in multivariable and tensor calc as well as a semester of applied math - I still didn't understand enough about the models I was using to be able to derive or usefully extend them.

What eventually worked for me was making friends with a professional statistician. I strongly suggest this, save you a lot of time and heartache. And when they say "just take my word for it" learn to regard that as gospel.
posted by ikkyu2 at 8:27 PM on December 19, 2008

Thanks for the answers, folks. I will check out the books and websites recommended, and also investigate auditing some courses. Please keep up the recommendations, if you have any left unshared. ;)

I think I have to figure out for myself just how much I want to learn this and how much time and energy I'm willing to commit. Clearly going from where I am to where I want to be is not going to happen in a day. At the same time, I still have hope that having a specific goal & area of interest will keep me motivated and help me cut a clear path.

Lastly, with the number of math/language comparisons flying around here, it makes me wish someone offered a month-long program of math immersion!
posted by shaun uh at 8:52 PM on December 19, 2008

I went through a similar post-school getting up to speed in math and my favourite website where I learned a lot of cool stuff was Ask Dr. Math.
posted by GleepGlop at 9:48 PM on December 19, 2008 [1 favorite has favorites]

I love the idea of math immersion, but I don't know how you would pull it off. The idea reminds me of a story that Hulse tells about the time he spent writing his computerized data analysis in the early 1970s: occasionally, he says, he would accidentally switch to hexadecimal arithmetic while balancing his checkbook.

The language comparison cuts both ways. Sure it takes years to really get fluent in a new language. But if you learned tonight that, next month, you needed to go to Russia, would you drop everything for a Russian immersion course? Probably you'd get a textbooky book and a phrasebooky book, try to get the alphabet enough to use your dictionary, and spend the first week of your trip asking your escort to remind you the word for "please," since you forgot it again. But you could mostly get around that way.

When you take a math course, you learn a curriculum designed to take you in a reasonably coherent way through to some exams; the person writing the exams suspects that curriculum will probably prove useful to most of the students, eventually. When you are exploring some math to answer a not-obviously-related question, your needs may or may not overlap with a standard curriculum. Don't let this surprise you into misdirecting your time. But don't be surprised when you guess wrong, too. Trust your statistician friend.
posted by fantabulous timewaster at 10:46 PM on December 19, 2008

I could have written the first three paragraphs of your question to describe myself. I tend to fall back on books like this when I need to know more maths. They don't provide deep knowledge, but they're a quick way to get up to speed.
posted by PueExMachina at 11:47 PM on December 19, 2008