# How to Math

August 14, 2015 7:10 AM Subscribe

I'm a computer science major in college. I'm not a freakin' genius, but I do well-- get A's in my classes, good with abstraction (e.g. pointers), etc. However, I seem to have missed... my entire high school pre-Calculus math curriculum? I don't know.

I've never gotten less than an A in a math class, but mathematical notation and reasoning just do not stick with me. I can't really explain it, but I'm good at tests, not so good at creative mathematical thinking. Would do poorly on the Putnam. It's a huge hole in my self-esteem and I really want to change this.

Tiny example: During a programming class, an instructor scribbles down a formula involving factorials. I stare at the board and try to remember how factorials work, even though I aced my Honors Analysis course (which was mostly proof writing). It's like I have to learn everything from first principles to remember any of it.

Further examples are hard to come by as it mostly relates precisely to things I can't remember or explain. I've never taken a class on probability and subsequently I absolutely hate it, because it seems like mostly arcane formulae and non-intuitive reasoning about weird, hypothetical events. Combinatorics is another important field that I suck at. I wouldn't say I'm super skilled at Calculus, but I had very little trouble in my Calc class since by the time I took Calc I was actively interested in math and soaked it up better, and the principles of real numbers are familiar. (I have a feeling most of this lack of proficiency comes from not being a natural math whiz and having a rather poor high school math/science education. I'm self-taught at most things, but math wasn't my focus or main talent in high school.)

I recently took the GRE and studied a lot (well, moderately-a lot) for the math section and did not do all that stellar (78th percentile). Sad! But it highlighted my weaknesses-- data analysis is easy but sometimes I go too fast and make dumb errors, I have no fucking idea how probability/combinatorics work, even ALGEBRA eludes my slippery grasp when I'm making dumb leaps of logic.

Is there just like, a GREAT book for hammering down these kinds of concepts that I should obviously have under my belt by now? I know what it feels like to be the smart kid in the class-- not necessarily the superior intellect but the one that just has the hang of things-- and I want to be that, now, for math. Pretty soon I'm going to be facing more discrete math and algorithms classes and I want to be better prepared.

Note: Also coming in to play is the fact that I'm female and so I constantly have this voice in the back of my head saying "why are you such a STUPID GIRL why are you such a MATH IS HARD BARBIE DOLL" and it's driving me crazy

I've never gotten less than an A in a math class, but mathematical notation and reasoning just do not stick with me. I can't really explain it, but I'm good at tests, not so good at creative mathematical thinking. Would do poorly on the Putnam. It's a huge hole in my self-esteem and I really want to change this.

Tiny example: During a programming class, an instructor scribbles down a formula involving factorials. I stare at the board and try to remember how factorials work, even though I aced my Honors Analysis course (which was mostly proof writing). It's like I have to learn everything from first principles to remember any of it.

Further examples are hard to come by as it mostly relates precisely to things I can't remember or explain. I've never taken a class on probability and subsequently I absolutely hate it, because it seems like mostly arcane formulae and non-intuitive reasoning about weird, hypothetical events. Combinatorics is another important field that I suck at. I wouldn't say I'm super skilled at Calculus, but I had very little trouble in my Calc class since by the time I took Calc I was actively interested in math and soaked it up better, and the principles of real numbers are familiar. (I have a feeling most of this lack of proficiency comes from not being a natural math whiz and having a rather poor high school math/science education. I'm self-taught at most things, but math wasn't my focus or main talent in high school.)

I recently took the GRE and studied a lot (well, moderately-a lot) for the math section and did not do all that stellar (78th percentile). Sad! But it highlighted my weaknesses-- data analysis is easy but sometimes I go too fast and make dumb errors, I have no fucking idea how probability/combinatorics work, even ALGEBRA eludes my slippery grasp when I'm making dumb leaps of logic.

Is there just like, a GREAT book for hammering down these kinds of concepts that I should obviously have under my belt by now? I know what it feels like to be the smart kid in the class-- not necessarily the superior intellect but the one that just has the hang of things-- and I want to be that, now, for math. Pretty soon I'm going to be facing more discrete math and algorithms classes and I want to be better prepared.

Note: Also coming in to play is the fact that I'm female and so I constantly have this voice in the back of my head saying "why are you such a STUPID GIRL why are you such a MATH IS HARD BARBIE DOLL" and it's driving me crazy

Also: I am with you on hating probability. One book which was a surprisingly big help to me in understanding some probability concepts was the weirdly-titled 1982 book Aha! Gotcha: Paradoxes to Puzzle and Delight. The author wrote a monthly math column for Popular Science for like 25 years, so he's very good at explaining the concepts to a lay audience. The goal of the book is not to present you with numerical problems - it really isn't a "math book" at all - but to explain the concepts behind various mathy things, one of which is probability. I love this book.

posted by showbiz_liz at 7:32 AM on August 14, 2015 [1 favorite]

posted by showbiz_liz at 7:32 AM on August 14, 2015 [1 favorite]

I found Precalculus Mathematics in a Nutshell very helpful. In addition, have you tried out Kahn Academy yet? It's free and would be a great resource for making sure you have all the basics covered and also reviewing precalculus. In particular, I found Kahn good for getting back up-to-speed with probability/combinatorics.

As a former math teacher, I just wanted to add that your small errors are a sign that you are close to really learning the material and not that you can't do this. A lot of people who are amazing at math need to really understand how a problem is solved and have difficulty with rote memorization. Although it sounds counterintuitive, struggling with elements of precalculus is often a sign that you have the ability to be awesome at math.

You should also check out the Women in Science and Engineering (WISE) group on your campus. It may go by a different name. Sometimes forcing yourself to identify as a CS/engineering person versus someone who is trying to become a CS person can make a world of difference when it comes time to power through learning tough concepts.

posted by JuliaKM at 7:36 AM on August 14, 2015 [5 favorites]

As a former math teacher, I just wanted to add that your small errors are a sign that you are close to really learning the material and not that you can't do this. A lot of people who are amazing at math need to really understand how a problem is solved and have difficulty with rote memorization. Although it sounds counterintuitive, struggling with elements of precalculus is often a sign that you have the ability to be awesome at math.

You should also check out the Women in Science and Engineering (WISE) group on your campus. It may go by a different name. Sometimes forcing yourself to identify as a CS/engineering person versus someone who is trying to become a CS person can make a world of difference when it comes time to power through learning tough concepts.

posted by JuliaKM at 7:36 AM on August 14, 2015 [5 favorites]

With the disclaimer that I'm not nearly as advanced as you are, I wanted to suggest doing an insane number of practice problems, if you haven't already.

Since I grasp a lot of mathematical concepts relatively easy, at least on the surface level--and it sounds like you do too--I could always learn stuff quickly and do well on tests. But I was not actually storing the concepts and step-by-step processes in the part of my brain that would allow me to remember them and apply them 3-4 months later to novel problems. The biggest difference I noticed between me and the people who remembered everything months later was simply practice.

So when I started doing chemistry and needing various math that I understood but never "cemented", I did practice problems like it was my job. I would do problems until I thought I had the concept down and then do 100 more practice problems. I did way, way more than were assigned because I know that my high school education in math was not as good as my professors assumed it was.

Ideally, I spaced this practice over the course of a week or longer, both for my sanity and to make sure I had time to forget the parts I was weaker on and re-learn them. I didn't schedule the practice using time; instead I would pick a certain number of problems to get through in a certain session and then do them all until they were right.

I made myself redo problems from the beginning when I got them wrong, even if was obvious to me what I did wrong the first time. This, combined with having a certain number of problems to get through before I was done, gave me a lot of motivation to slow down and get every problem right the first time. It also got me into the habit of checking all of my work before declaring myself "done" with a problem--much easier to go back and correct a small mistake than to start everything all over. Anything I struggled with and wanted more practice with, or anything I got wrong, I marked in the book and came back a few days later to do them again.

This obviously reinforced processes that I was weak on, which is important. It also gave me an easy reference to use for what my weak points were. That way I was prepared to go to office hours/TA hours/study groups and get the most out of being there, because I knew what I was having problems with. (It also has the effect of being painfully boring if I made a rushed mistake the first time--again, though, this is good if it promotes a more careful approach to "easy" problems.)

Another thing that has helped me cement these kinds of processes is writing out step-by-step approaches to solving a given type of problem. I rarely-to-never used these as references after I wrote them out, but writing them serves a similar purpose to teaching someone else to do the problems (another good trick). It also helped me to learn what the vital characteristics of different problems were, which made it easier to identify them in the wild, so to speak.

Finally, between college and going back to school, I stopped giving a fuck if people thought I was an idiot. I know it's hard, and I was freeeeaking out when I first started class, especially as an older student, a single parent, a woman of color...But what I told myself is that

Hope that this helps, and if you're already doing this stuff, apologies :)

posted by internet fraud detective squad, station number 9 at 8:46 AM on August 14, 2015 [8 favorites]

Since I grasp a lot of mathematical concepts relatively easy, at least on the surface level--and it sounds like you do too--I could always learn stuff quickly and do well on tests. But I was not actually storing the concepts and step-by-step processes in the part of my brain that would allow me to remember them and apply them 3-4 months later to novel problems. The biggest difference I noticed between me and the people who remembered everything months later was simply practice.

So when I started doing chemistry and needing various math that I understood but never "cemented", I did practice problems like it was my job. I would do problems until I thought I had the concept down and then do 100 more practice problems. I did way, way more than were assigned because I know that my high school education in math was not as good as my professors assumed it was.

Ideally, I spaced this practice over the course of a week or longer, both for my sanity and to make sure I had time to forget the parts I was weaker on and re-learn them. I didn't schedule the practice using time; instead I would pick a certain number of problems to get through in a certain session and then do them all until they were right.

I made myself redo problems from the beginning when I got them wrong, even if was obvious to me what I did wrong the first time. This, combined with having a certain number of problems to get through before I was done, gave me a lot of motivation to slow down and get every problem right the first time. It also got me into the habit of checking all of my work before declaring myself "done" with a problem--much easier to go back and correct a small mistake than to start everything all over. Anything I struggled with and wanted more practice with, or anything I got wrong, I marked in the book and came back a few days later to do them again.

This obviously reinforced processes that I was weak on, which is important. It also gave me an easy reference to use for what my weak points were. That way I was prepared to go to office hours/TA hours/study groups and get the most out of being there, because I knew what I was having problems with. (It also has the effect of being painfully boring if I made a rushed mistake the first time--again, though, this is good if it promotes a more careful approach to "easy" problems.)

Another thing that has helped me cement these kinds of processes is writing out step-by-step approaches to solving a given type of problem. I rarely-to-never used these as references after I wrote them out, but writing them serves a similar purpose to teaching someone else to do the problems (another good trick). It also helped me to learn what the vital characteristics of different problems were, which made it easier to identify them in the wild, so to speak.

Finally, between college and going back to school, I stopped giving a fuck if people thought I was an idiot. I know it's hard, and I was freeeeaking out when I first started class, especially as an older student, a single parent, a woman of color...But what I told myself is that

**I might very well have been an idiot, but I was an idiot who was going to learn the material and get an A.**Ultimately, you're paying to learn and your grades will demonstrate what you've learned to people who matter.**You aren't paying to limit your learning and limit your use of your school's resources because your classmates may be sexist jackasses. Hard work is what makes you good at math.**That's why entire cultures are about a billion times better at it than we are--because they treat it like something you work hard to learn. Do what they do, do the work and don't worry about the natural talent--because it's not like you can change it anyway! What you can change is the amount of work you put in and the way you use the resources you have available to you.Hope that this helps, and if you're already doing this stuff, apologies :)

posted by internet fraud detective squad, station number 9 at 8:46 AM on August 14, 2015 [8 favorites]

First of all, the Putnam is a ridiculous stick to measure yourself by. Most mathematicians would do poorly on the Putnam.

That said, I think that Alcumus on Art of Problem Solving would be a good resource for you to try out (possibly their textbooks as well). Alcumus is an online system like Khan Academy, but with far more interesting problems that require actual thinking.

posted by ktkt at 10:18 AM on August 14, 2015 [3 favorites]

That said, I think that Alcumus on Art of Problem Solving would be a good resource for you to try out (possibly their textbooks as well). Alcumus is an online system like Khan Academy, but with far more interesting problems that require actual thinking.

posted by ktkt at 10:18 AM on August 14, 2015 [3 favorites]

For actual precalculus material, I really like ALEKS. You can purchase a month access for $20 or so, even if you're not enrolled in a class. It's like practice problems where they tell you if you got the right answer. Some really interesting problems.

(Also---just ask about the formulas. Your instructor will be happy to walk you through it. But she doesn't know that folks are confused if no one asks. I guarantee you're not the only confused one. Or go talk to her during office hours. She'll be willing to walk through stuff with you.)

posted by leahwrenn at 11:36 AM on August 14, 2015

(Also---just ask about the formulas. Your instructor will be happy to walk you through it. But she doesn't know that folks are confused if no one asks. I guarantee you're not the only confused one. Or go talk to her during office hours. She'll be willing to walk through stuff with you.)

posted by leahwrenn at 11:36 AM on August 14, 2015

*Would do poorly on the Putnam. It's a huge hole in my self-esteem and I really want to change this.*

Also, just to reiterate. I am an actual mathematician. I am still proud that I have a nonzero score on the Putnam. But equating "good at math" with "good at Putnam" is ... not a good idea, and probably not true. Sure, some folks who get good Putnam scores are good mathematicians, but the inverse statement (not good at Putnam implies not good mathematician) is not true.

posted by leahwrenn at 11:39 AM on August 14, 2015 [1 favorite]

*Note: Also coming in to play is the fact that I'm female and so I constantly have this voice in the back of my head saying "why are you such a STUPID GIRL why are you such a MATH IS HARD BARBIE DOLL" and it's driving me crazy*

I wanted to address this bit. I'm a (male) college physics professor, and it's absolutely true that there are ugly stereotypes in our society that drive women out of the sciences. Of course, I've never seen any real difference in mathematical aptitude between genders; instead, what I

*have*seen is a strong societal tendency for male students not to admit their weaknesses. I assure you that even if the male students you see are acting like they know everything, many of them also secretly feel that they barely have a grasp on the material and are just faking it. (Or, even worse, they have only the most tenuous grasp on the material but truly believe that they know it all.) This contributes to a spirit of "intellectual machismo" (as one of my female students memorably put it) that I think is ultimately harmful to physics as a field — and I don't doubt that the same problems crop up in computer science, engineering, and math as well.

But here's the thing: a big part of becoming a scientist is realizing that you don't know everything, and being brutally honest with yourself about what you do and don't know. In this sense, you're ahead of the game; you have the ability to identify gaps in your knowledge, and that is a

**huge**part of the type of intellectual maturity that's necessary to become a scientist. It's 100% normal to have trouble with this stuff; it's complicated! The important part is that you be honest enough with yourself to "know what you don't know", rather than just skating past it like so many other students would.

posted by Johnny Assay at 11:40 AM on August 14, 2015 [9 favorites]

I was a CS major. I've had a successful and satisfying career as a software engineer for the past 15 years. I cannot STAND probability, big-O notation, DFAs/NFAs -- basically anything on the more theoretical side of CS (which I don't use in my job anyway!).

Grit your teeth and get through college -- which can't be for much longer given that you've taken your GRE (in my experience, grad school was WAY easier, but that may be because I was out in the so-called "real world" for almost three years before I went back to school for my Master's). Ask for help from professors and TAs whenever needed. Realize that once you are a professional CS person, whenever you need to do probability, big-O, et cetera -- which will be rarely, if ever, in spite of what your professors want you to think --

posted by tckma at 2:57 PM on August 14, 2015

Grit your teeth and get through college -- which can't be for much longer given that you've taken your GRE (in my experience, grad school was WAY easier, but that may be because I was out in the so-called "real world" for almost three years before I went back to school for my Master's). Ask for help from professors and TAs whenever needed. Realize that once you are a professional CS person, whenever you need to do probability, big-O, et cetera -- which will be rarely, if ever, in spite of what your professors want you to think --

*you will be able look up how to do it*.posted by tckma at 2:57 PM on August 14, 2015

If you got an A in real analysis, I think you're probably good at math. But being able to guarantee you'll get the right answer under pressure in a highly-detailed problem is something else.

It just sounds like you need practice with symbol manipulation. Solve a lot of easy problems, rather than a few hard problems.

You might understand the conceptual meaning of an equation, but quickly arriving at the right answer will come after solving a bunch of easy problems. Probabilities, especially conditional probabilities, are notoriously counterintuitive (even Paul Erdös was fooled by the Monty Hall problem!), but if you practice mechanically manipulating the symbols it will start to sink in. Combinatorics actually does require some creativity, it's not just brute force, but if you practice the brute force parts, the creative part of your brain will be able to grab onto the problem better.

Just do it. When a football player gets tired out halfway through a game, they don't worry that they're inherently weak or doomed to failure, they just train until they're stronger. Adopt the same mindset. This is not a matter of talent.

posted by vogon_poet at 7:37 PM on August 14, 2015 [3 favorites]

It just sounds like you need practice with symbol manipulation. Solve a lot of easy problems, rather than a few hard problems.

You might understand the conceptual meaning of an equation, but quickly arriving at the right answer will come after solving a bunch of easy problems. Probabilities, especially conditional probabilities, are notoriously counterintuitive (even Paul Erdös was fooled by the Monty Hall problem!), but if you practice mechanically manipulating the symbols it will start to sink in. Combinatorics actually does require some creativity, it's not just brute force, but if you practice the brute force parts, the creative part of your brain will be able to grab onto the problem better.

Just do it. When a football player gets tired out halfway through a game, they don't worry that they're inherently weak or doomed to failure, they just train until they're stronger. Adopt the same mindset. This is not a matter of talent.

posted by vogon_poet at 7:37 PM on August 14, 2015 [3 favorites]

You're already good at programming, so maybe it would help to write programs that enumerate things? I bet writing a program that lists all the permutations of a sequence would help with understanding the combinatorial "meaning" of the factorial function, for example.

You could try something similar with probability: use a random number generator and some code to turn the "weird, hypothetical events" into something more concrete and executable that you can play around with.

posted by panic at 8:24 PM on August 14, 2015 [2 favorites]

You could try something similar with probability: use a random number generator and some code to turn the "weird, hypothetical events" into something more concrete and executable that you can play around with.

posted by panic at 8:24 PM on August 14, 2015 [2 favorites]

It sounds like you feel anxious about not knowing probability and combinatorics, despite never having formally studied probability or combinatorics? If you've never spent time messing with permutations and combinations or binomial coefficients, it's not a surprise that factorials slow you down. There are lots of algebraic tricks for dealing with quotients of factorials that really do come with practice.

Have you ever taken a formal statistics course? If you enjoy calculus and data analysis, you'd probably find stats pretty fun; and for what it's worth, statistics is high on my list of "things I wish I'd taken as an undergrad." Many institutions offer a combined probability and statistics course. That might be a way to combine formal practice with the symbols that trip you up with a type of math you genuinely enjoy.

posted by yarntheory at 12:28 PM on August 16, 2015

Have you ever taken a formal statistics course? If you enjoy calculus and data analysis, you'd probably find stats pretty fun; and for what it's worth, statistics is high on my list of "things I wish I'd taken as an undergrad." Many institutions offer a combined probability and statistics course. That might be a way to combine formal practice with the symbols that trip you up with a type of math you genuinely enjoy.

posted by yarntheory at 12:28 PM on August 16, 2015

I want to add that mathematical creativity depends a great deal on persistence and experience. The more math you do, the more likely it becomes that any given problem will remind you of a situation you've faced before.

posted by yarntheory at 12:44 PM on August 16, 2015

posted by yarntheory at 12:44 PM on August 16, 2015

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posted by showbiz_liz at 7:21 AM on August 14, 2015 [9 favorites]