A guide to learning math

Math is objectively a difficult subject. Aside from the amount of information to learn, problem-solving strategies are very difficult. It is relatively easy to experiment, but it is much more mentally challenging to use existing equations to build something novel, something that no one has thought of before. That being said, mathematics at the undergraduate level is much easier to grasp and more concrete in application than higher-level math, so it is important to understand it. Here are some insights that will help develop math skills.

• Start from the very basics. Although this might seem obvious, it is important to go to the foundations of what you study and examine them from the ground up. Suppose one is learning Calculus III, and is starting vector calculus. Their understanding is compromised and they can’t keep up in class. Therefore, they should start from the beginning - making sure they have a fundamental understanding of vectors and two-dimensional derivatives/integrals. Only then should they attempt to start vector calculus. Just as a shaky house requires a check of its foundation, a struggling student should revisit and solidify their grasp of fundamental concepts.
• But what exactly is “fundamental understanding”? Make sure to do all of these to get a fundamental understanding:
• Being able to explain the topic in as simple terms as possible
• Being able to re-derive formulas from scratch, construct proofs for a theorem
• Understanding the core concepts of a subject
• Develop fundamental understanding in the following order:
• Attempt problems, then take notes (this will draw attention to why a problem can’t be solved/what strategies will be needed to solve it)
• Try to find multiple ways to solve the same problem
• Attempt more challenging problems (such problems will need two or more steps or a shift in perspective, and care will be required in not make silly mistakes; a couple of sufficiently hard problems will suffice; if a problem can’t be solved, understand the solution)
• Teach the problem-solving process to others

By using this approach, one can excel in math and apply it in more difficult scenarios, such as competitive math tournaments. Doing problems deliberately and attempting challenging problems will not be easy, but attempting problems this way will allow much more depth of understanding and will be tremendously beneficial long term.