How can I improve my problem solving/critical thinking skills and learn higher math?

Question

I'm a rising sophomore in high school. So far, I've taken Algebra One, Two, and Geometry in school. I want to learn higher math such as precalculus/trigonometry, calculus, linear algebra, and more, so I can go into topics such as cryptography, advanced computer science, and possibly take the AMC and other olympiad tests (I'm not too interested in that).

The only problem, though, is that my abilities in problem solving and other stuff in math aren't that good. I do pretty well in my classes (high As) but that doesn't mean anything. The U.S. system doesn't seem too good in actually teaching math.

For example, I can do whatever is on my homework or tests. But, if I'm given a more difficult problem than usual concerning a topic I learned (say logarithms or something), I can't solve it.

I feel like this is going to be a hindrance to me learning higher math, doing well in more difficult subjects like calculus and linear algebra, doing well on olympiad tests, and going into math-heavy fields like computer science and cryptography.

So, how can I change all of this and improve my skills? Are there any books that teach problem-solving, mathematical thinking, and higher math (or something like precalculus)? Again, I want to better these skills so I can do well not only in math, but other fields.

Any help is really appreciated.

Answer 1

"For example, I can do whatever is on my homework or tests. But, if I'm given a more difficult problem than usual concerning a topic I learned (say logarithms or something), I can't solve it."

Did you mean you can't solve it or that you can't solve it right away? It's been my experience teaching college math that students often become quickly frustrated if they can't get the correct answer in a relatively short period of time. I have a masters in math and, when I was in grad school, I would struggle with problems for hours or sometimes days before they finally started to make sense. This kind of persistence is an important part of both advanced learning and research. So, to answer your question, I would start with persistence. There's nothing wrong with struggling with problems - that's a sign that learning is happening.

On a more immediate note, don't be shy about asking questions. If you ask your teachers, just ask if they can help you get started. Don't look for a complete solution to the problem or even for them to help you walk through the problem. Again, it's been my experience that this is what students often expect, i.e. that they should walk away from a conversation with a complete solution. Instead, get just enough help to get you moving again then go back to working on the problem on your own. The more of the answer you develop yourself, even if it takes time, the better your understanding is going to be.

Answer 2

You probably have reasonable facility if you get high As, even in courses that are not super hard. The simple answer to your question is to work more problems. Try to do some harder ones, but don't jump off the deep end. Have some sense of progression (from easy, to medium, to mind stretcher). Some students overestimate their mastery of the basics and then it holds them back on the trickier problems. Look at other books and work problems in them. Check out Schaum's Outlines or look for older texts where there is more emphasis on learning lots of tricks, I mean techniques. Hart College Algebra is a good one.

Math is a lot like sports. You need to build your muscles up. Do problems. Don't try lifting 1000 pounds if you are just learning to squat. But do have some sense of progression so you are increasing the weight on the bar. And get the reps in. It helps.

Also, look at where your issues are, where you are weaker (logs more the issue than exponents, etc.) And target some exercises where they are needed.

I would also add that there is more and more research on the brain showing that we are not wired for math the way we are for reading. For a computer it is normal to calculate and to hold abstractions. But evolution has trained our minds to have much more facility for the nuances of the verbal than of the logical. This means that math can not be learned passively.

Like complicated gymnastics, you will never just grow up around math stuff and have it rub off on you and start doing it. You have to work the progressions and learn to do things that are possible but not something we are so much wired for. In contrast some physical tasks are things we pick up easily (gait for instance is amazingly complex from a mechanical standpoint, much more than a backflip). Also just exposure somewhat pleasantly and observationally to verbalisms can lead to ready incorporation.

But we are complicated and have lots of capabilities. So you can learn to do tricky problems, even building some ability to do new ones or ones that are combinations, just from learning other tricks along the way.