Has Math given up on me?

Question

Last fall I took Real Analysis and Modern Algebra courses but I could not complete them and dropped out because :-

1. I found them to be non intuitive especially Algebra (groups, rings, fields).

2. I found it hard to do proofs in assignments or write my own.

3. Maybe my teachers screwed it up or I am not intelligent enough

4. Lots of things did not make much sense, especially Algebra

I still think these courses are fun because many people these days are researching in field of pure math. Maybe I am not doing it right.

Any suggestions regarding how to start over..any resources(books, videos, notes and so on) or whatever you can help

Thanks

Answer 1

I found them to be non intuitive especially Algebra (groups, rings, fields).

Well, there's a quite common misconception that intuition is something you have. Rather, intuition is something you develop when working on a subject; that's one of the reasons to have exercises, to develop intuition. Intuition is nothing than a set of (often unconscious or semi-conscious) heuristics.

Note that this point is not restricted to mathematics, but is true for anything. For someone who has never seen anything flying (be it a bird or an airplane), it would seem unintuitive that something can fly. Because we know such things, we have a pretty good intuition when objects can fly, even if we cannot do the aerodynamics to formally analyse the flight of a bird or an airplane.

You obtain intuition by working with the subject, by exploring simple cases, by exploring special cases, and by simple practice.

I found it hard to do proofs in assignments or write my own.

That's because you haven't yet developed the intuition for them.

Maybe my teachers screwed it up or I am not intelligent enough

I can't comment on whether your teachers screwed up (I didn't attend their classes, after all). But I don't think it is a lack of intelligence (your post indicates that you are able to clearly identify and formulate issues, which IMHO indicates that you should have the mental requisites for mathematics). It may, however, be a lack of self-confidence: If you assume you won't understand it anyway, you may not put sufficient effort into understanding, and as result your assumption can turn into a self-fulfilling prophecy.

If you think it may be because of your teachers, you might consider to find some course on YouTube (there's lots of material there) and see if that helps you.

Lots of things did not make much sense, especially Algebra

Then try to find out why the rules are that way. In particular, consider what they mean if you consider numbers; most rules get obvious when you consider numbers, and for those which don't work with numbers, finding out why they don't work with numbers will generally be instructive as well.

Answer 2

The skills for doing proofs should be done early -- before college. In the old days, those skills were taught in High School, but currently, at least in the U.S., the teacher base at the High School level is a lot weaker than it used to be, so teaching the technology of proofs is something the teachers themselves can't do too well, hence for the most part, they avoid trying to teach it.

The students in High School on the Math Team do work on developing proof skills, but most other students, when they enter College, have no such background. For those students, their first exposure to a pure math course is bound to come as a severe "culture shock", which just gets worse as the course proceeds to pick up pace.

To do real proofs, you have to start with "baby proofs", but those courses don't do that -- they don't have time. So the solution is to try to develop those early, prerequisite proof skills on your own, working on problems where the proofs are easy, and checking against reliable sources (e.g., MSE) to see if your proof attempts are correct, and even if correct, could be improved.

Answer 3

Before I talk about getting back in the saddle, there are a few misconceptions about mathematics I would like to comment on.

First, you are almost assuredly intelligent enough to do well in these courses. This is not to say that they are easy -- quite the contrary. However, oftentimes "being good" at mathematics is coupled with innate intuition and skill. While natural talent certainly has its place, skill and understanding in mathematics comes largely from practice and study.

I have not yet met a successful student in mathematics who does not practice their skills often in some way. Each person does it differently, but they all do it. This is due to something you have already come across -- the formalism and abstractness that mathematics requires in order to be precise. Without becoming intimately familiar with these abstractions, intuition can only carry you so far.

Getting back into the game

Before I mention anything about self-study, or specific textbooks, it's important to note that, in my experience, the most useful way to learn mathematics is to look at it from various perspectives -- especially if something isn't making sense. Sometimes, individuals or authors will explain a concept or a proof in a certain way that, to you, may not be very clear. Instead of suffering through their explanation and attempting to understanding their way of looking at math, you should search around for an explanation that makes sense to you. Mathematics is personal, and your way of looking at math may be very different from someone else -- this is absolutely okay.

To "start over," I would suggest finding resources on your work that make sense to you. Try out different books, and find the ones with perspectives you like. Try and use multiple books, and read them only for the strengths they provide.

For example,

• Gallian's Abstract Algebra has a plethora of examples, and practice problems. If you're looking to practice (which you should be!), this book has a wealth of useful questions to look at.
• Nathan Carter's Visual Group Theory really stresses the intuition behind algebra. This book was useful for me in my first semester of algebra, since pictures and shapes made more sense than notation and symbols.

$$(n_{1},h_{1})\bullet (n_{2},h_{2})=(n_{1}\varphi (h_{1})(n_{2}),\,h_{1}h_{2})=(n_{1}\varphi _{h_{1}}(n_{2}),\,h_{1}h_{2})$$

How is that intuitive? :P

To start up again, I would suggest a few things:

1. Try and learn some discrete mathematics, as @quasi suggests. Oftentimes, this section of mathematics can be more intuitive than other areas, and so it's a good jumping off point to hone your ability to couple intuition and abstraction.

2. Don't be afraid to spend time on a single question! Convincing yourself that you can't do a problem is the best way to stunt your learning. Give the problem space if you've spent a good amount of time on it, and try doing something else, or work on a different problem.

3. Use all possible resources! Talk to your friends, and talk to your professors! I can guarantee you that more than one of your professors has had a "Am I right for this field?" moment, so if you need help, don't be afraid to reach out.

4. In that vein, use Math StackExchange! Seeing the way other people solve problems can be very helpful, and this site has many talented people who love answering questions.

Answer 4

I'll share what was the biggest oversight in my own math training (became critical a year or two after where you are): Trying to do everything alone. For many of us, we may have cruised through all our math fairly easily without ever learning how to get help from others. Depending on cultural upbringing, this may be more or less a point of moral pride. There's even a name for this: "John Henry Syndrome".

Yet (almost) all of us will reach a point in math where we can't do it alone. So part of the work is developing soft skills: working with others, forming a study group, being able to go to office hours and ask for help. In some sense you have an advantage now because a site like Stack Exchange exists (both more accessible, and a bit lower-stakes as far as face-to-face interaction, which may be important for some personalities).

My suggestion would be: I assume that you have the Algebra book, so start working through it carefully at your own pace. First step: Post a question here about this book by name; is it quality, are there better books anyone would recommend? The book should be readable by a normal being, if read slowly and carefully with a lot of side-scratch work. As you do exercises or run into problems post here on SE and show how far you're getting. Best of luck.