I have a question that I have been wanting to ask for quite some time. I am currently a senior Computer Science student who will be graduating in a few months. I have taken all my required math courses (statistics, calculus 1-3, discrete mathematics, linear algebra, etc.) despite doing fairly well I feel like I lack any real understanding of mathematics. What I want to know is if there are any resources out there to learn math completely from the beginning starting at basic algebra and progressing into advanced topics. Right now I feel like I just know how to follow instructions and plug in formulas and numbers. I would like to eventually have a real understanding of exactly why I am doing what I am doing and how to use mathematics to solve future problems. Thanks for any advice!
[Math] Relearn Math From The Ground Up
computer scienceeducationself-learningsoft-question
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Original author's note: I am deleting my profile and would appreciate it if someone could maintain this answer going forward.
I know this post is very old but I'd like to contribute my likely-too-late response in case it'll help someone else in the future. Plus I'm in a similar position so I have a few relevant things to say, and I'm sure the landscape has changed significantly since 2010.
First, I see in the comment thread on the question that you were planning on a bachelor's with a potential future interest in a graduate degree. Since you already have the MSc in comp sci and a clear interest in math then I imagine you have at least a decent math background. Getting a bachelor's might not be the best idea, especially if you end up being unable to transfer a lot of previously earned credits. If ultimately you're wanting to pursue a graduate degree then I'd recommend at least trying to start with a graduate program. It's possible you'd need to take some prerequisite courses but surely this would be better than going through an entire bachelor's program again. But if you're thinking a bachelor's is all you want then by all means go for it.
I completely understand the feeling of wanting to pursue a degree rather than just self-studying, because I feel the same way and I'm in the same position as you right now. I do a lot of self-studying and it is certainly enjoyable and fulfilling. But I share the feeling of wanting to have something to show for it, even if that doesn't mean anything to anybody else. At the very least, having that degree is a mark of completion. And with that mark comes that sense of satisfaction from having successfully completed something challenging. And for me, that's all the motivation I need. It shows that you individually put yourself on a difficult path and successfully navigated through it. What would successful completion of an independent study even be? Finish reading the book? Work every problem in the book? Correctly work every problem in the book? Yes, we can all try to do this navigation with independent studying but there's no accountability there and no clear definition of successful completion. If you fail at independently studying something (e.g., gave up halfway through, or finished the book and learned nothing, etc.) then no one ever needs to know you tried. If you fail out of a graduate program then it's quite the blemish on your record and could be difficult/impossible to hide. My opinion is that pursuing an actual degree, especially if it's just for fun, shows a level of dedication higher than independent studying. Some may wonder what's the point of showing this level of dedication, especially if it's a just-for-fun pursuit. I address this in the next paragraph.
As I mentioned, I'm currently in a similar position. I have two bachelor's (math and computer science) and two master's (math and applied math) degrees and I've been tossing around the idea of getting more graduate degrees, possibly in statistics, computer science, electrical engineering, and even more in math. Some may (and some already did) scoff and condescendingly tell me the degrees are useless, I'm wasting my time, etc. Depending on their level of derision, to them I would say something to the effect of, "Cry me a river! I'm not living my life for you and I'm not obligated to make decisions about my life based on what's appropriate for your own personal world view." It really surprises me how many people there are who can't comprehend the idea of another person following pursuits and interests that don't line up with their own. Also, who's to say the degree is completely useless? We don't know what the future holds. Maybe you'll find during your studies that you enjoy it more than you ever thought you would, and you want to make a career change. Maybe you won't want to make a career change but at some point you unexpectedly find yourself in a situation where you need a new job in the same field. Having pursued a graduate degree from a respectable institution (i.e., not a diploma mill and preferably a regionally accredited school if in the U.S.) while holding down a full-time job should look great to any prospective employer. And as long as you left your old job on good terms, why shouldn't it look great? Sure, independently studying while working full time is also challenging and rewarding, but anyone can say they independently studied something, and even if they did we come back to the issue of how to measure success with independent studying.
A more legitimate question I get is, "Why get more math degrees?" And my answer to that is (1) because I love it and I want to, (2) different schools have different programs and curricula, so doing the same degree at different places doesn't mean I'm doing all of the same courses over and over again, and (3) I have been contemplating pursuing a PhD and I believe I've been out of formal academia too long for that to be possible otherwise.
I'll put the soap box away and go get the list of online programs in math that I've found.
Links are valid as of July 9, 2020. In no particular order:
- The Open University, based in the UK. As of the time I write this, they take online students overseas and they're actually regionally accredited in the USA. I may be mistaken but I believe this program doesn't require recommendation letters. That could be helpful for people who have been out of academia for a while.
- Applied math at Columbia if you've got that kind of cash.
- Texas A&M. This one has a few options. There's "traditional" which I guess is like a standard pure math sequence, and I think recently they also put their computational track and math for teachers track online.
- University of Washington.
- Emporia State University. This program doesn't require recommendation letters, and of course that's subject to change. I took 2 classes here for fun and they were both very well run; I recommend this for anyone interested in an online MS in math.
- University of Houston MA in math.
- University of Texas Rio Grande Valley. If I understood correctly, UT Brownsville "merged" into this school and this program is what Brownsville's was.
- University of West Florida.
- Johns Hopkins University Engineering for Professionals. There's an Applied & Computational Math option and a Financial Math option.
- University of Idaho MA in Teaching that's focused on math.
- Shawnee State University MS in Math, seems pretty well rounded and very affordable.
Updating this post to also include statistics programs, again in no particular order. Links are valid as of July 9, 2020.
- Texas A&M
- Penn State World Campus
- Rochester Institute of Technology
- California State University Fullerton
- University of South Carolina
- Colorado State University
- University of Idaho
- Oklahoma State University
- North Carolina State University
- Stanford University
- University of Delaware
- University of Louisville (Biostatistics)
- University of Kentucky
- Michigan Technological University
I haven't looked for statistics programs as thoroughly as I did with math programs so I'm sure there are at least a few others out there that aren't from online-only schools.
I should also point out that a lot of these schools, and many others not listed here, also have certificate programs which usually have more lax entry requirements. Most of these certificate programs don't require letters of recommendation. This is a good way to get back into academia if you've been gone for a while but need letters to get into the program of your choice. A lot of schools (like Texas A&M as of the time I write this) will also allow people to take courses as a non-degree student and you can even transfer a certain amount of credits if you get accepted into a degree program later.
[Added on 14/10/2020(DD/MM/YYYY)]
- List of colleges offering M.Sc Mathematics through Distance Education/Correspondence in India
- Another list of the unis providing distance learning in India
- Uni Birkbeck: MATHEMATICS BY DISTANCE LEARNING (GRADUATE CERTIFICATE)
- Certificate of Pure Mathematics
- University of North Alabama: M.S. in Mathematics Program
- University of Houston Downtown: Certificate in Graduate Mathematics
In going from high-school to, say, graduate${}^\color{Blue}\dagger$ level math, the higher math being "nowhere close to the math that appealed to" you is probably a very real threat. However, if your pleasure in learning analysis and algebra is any indication - instead, it will be math that you love even more.
A few things that I've noticed changing as I've learned more math are:
- Generality: Everyone knows about integers, rationals, reals, maybe complex numbers, but the next step up, conceptually, is to look at rings and fields and then modules and so forth. In higher math, the generality of our constructs increases a lot. Often we then narrow our focus again and end up looking at 'cousins' of the things we were originally studying. Other times we run into problems and it becomes the question of the decade precisely how to successfully go about a particular campaign of generalization and overcome the relevant obstacles.
- Branching: What a lot of people don't understand is that mathematics isn't linear, and doesn't progress in a rigid sequence. It branches out into many different areas, and in exploring these branches they can "feel" radically different. You can have a crush on one branch while hating another branch. In some cases, you have a love-hate relationship, or "it's complicated," etc.
- Reinterpretation: With a little bit of dabbling in different areas of math, it's possible that a single problem/idea can be attacked/framed from many different angles, using very different concepts. Sometimes this can seem "natural" and easily "motivated," while other times alien and bizarre. Frequently this sort of thing is a bit of a pastime for some mathematicians.
- Richness: In summary, mathematics becomes richer. In scaling conceptual mountains, we build concept on top of concept on top of seventeen more concepts until we're left studying situations that are saturated with structure, and when we hike back down the other side we come across the exotic or pathological; in branching we discover a high degree of diversity we hadn't previously imagined, each with comparable feel and texture to them; and then when we study wide and far we find that even our familiar notions have multiple sides to them.
$~~ {}^\color{Blue}\dagger$Yeah, disclaimer: I'm not actually there yet. :-)
Best Answer
Here's a possible ordered list of undergraduate-level books that could help you in your quest, but I have to warn you it will take some time. You're wanting to go all the way from understanding the 'why' to problem-solving and applications. That's quite a broad sweep of mathematics.