Re-Learning Math: A Comprehensive Approach

[Aufbau]

Hello all,

I feel as though I never learned mathematics the right way which severely impacted me from pursuing other science/math careers involving higher mathematics. I feel as though I need to approach mathematics from a proof based manner in order to properly learn. The way that mathematics is taught in the public school system is very memorization based to an extent. However, I am trying to mitigate this effect and re-learn everything I was ever taught (sciences too) in a more rigorous format. So far I have Spivak for Calculus I and II along with Courant Differential and Integral Calculus volumes I and II. For Multivariable Calculus I will be using Shifrin's Multivariable Calculus with linear algebra. Further I will be using Ross and another text for differential equations and later Real Analysis by Ireland and Artin for abstract algebra as well as a discrete mathematics-Lovatz. I wanted to use Feller for Statistics and a few other mathematics texts.

In addition, I will be starting my high school mathematics sequence over again (i.e., starting from algebra). I thought that starting out with number theory along with algebra followed by geometry and algebra II and finally with precalculus. Then I figured my math would be strong enough to get into Spivak and the others. I am not sure where to fit discrete math in with the other mathematics courses. The high school math list is as follows:

Modern School Mathematics Geometry-Jurgensen
Modern Algebra and Trigonometry (Book 2) Structure and Method-Dolciani
Modern Algebra Structure and Method (Book 1)- Dolciani
Modern Introductory Analysis-Dolciani
College Algebra-Charles H. Lehmann
Analytic Geometry by Steen and Ballou (3rd Edition)

We used the Holt, Rinehart, Winston series I believe for every math class I ever took with the exception of college and everyone knows that the Reform Calculus books aren't that great.

-I have noticed that working only difficult problems and taking time on those is better than working a ton of easy problems because the extremely difficult ones test your knowledge and ability to apply/synthesize the material.

Any commentary would be greatly appreciated. What I am basically asking is whether or not this approach is feasible and what are the flaws you currently see in it? What are some improvements that could be made?

Thank you-

 

[homeomorphic]

I'm not familiar with most of those books, so I can't say too much, but I would point out that "proof-based" is not the opposite of memorization-based. Proofs can help your understanding, but they can also hinder it in some cases. Ideas are more important than formal proofs.

It might take too much time to review all of high school math. It might be a good idea to review some things, but it would probably be better to review as needed by studying calculus, etc. Having gaps in your knowledge can be dangerous, but my hunch is you can figure out the gaps as you go along.

 

[oneamp]

This was a help for me:

https://www.amazon.com/dp/039304002X/?tag=pfamazon01-20

 

[Aufbau]

Thank you.I was hoping to really figure out a way to fully understand mathematics so that I don't get into much trouble when taking mathematically intensive science courses. I wanted to find a way that was not strongly rooted in memorization but rather in proofs and other forms of mathematical logic. I feel that I have always learned better that way and am just now figuring out how I learn best. I never really questioned the conventional way to learn anything because I didn't know any better. I still don't know everything but I am trying to learn so that I can better myself.

 

[Matrix0]

Hello,

I understand the importance of a strong foundation in mathematics for pursuing careers in science and math. It is commendable that you are taking the initiative to re-learn math in a more rigorous and proof-based manner.

Your approach of starting with number theory and algebra before moving on to more advanced topics such as calculus and abstract algebra is a good one. This will help you build a strong conceptual understanding of the fundamentals before diving into more complex topics.

I would suggest incorporating some practice with real-world applications of mathematics, such as in physics or engineering, to supplement your learning. This will help you see the practical applications of the concepts you are learning and strengthen your problem-solving skills.

In terms of resources, the textbooks you have chosen are reputable and will provide a solid foundation in their respective topics. However, I would also recommend seeking out additional resources such as online lectures, problem-solving workshops, and practice problems from different sources to supplement your learning.

Overall, your approach seems feasible and well thought out. The only potential flaw I see is the possibility of feeling overwhelmed with the amount of material you are planning to cover. Make sure to pace yourself and prioritize understanding rather than rushing through the material. And don't forget to seek help from teachers, tutors, or peers if you encounter any difficulties.

I wish you the best of luck in your re-learning journey. With dedication and perseverance, I have no doubt that you will achieve your goals. Keep up the good work!