Hi kalid,
I have arrived at a conclusion that your approach towards learning maths basically encompasses these areas:
Philosophy : Observing world as it is without any bias would help us to see shapes, changes in figures that do not naturally tally with our innate logical conclusions or calculations.
Abstraction : Once a person identifies a real world problem scenario then he can break it down to simpler and easier to tackle diagrams or figures ( for e.g. straight lines instead of actual continuosly changing landscape).
Pattern Finding : This would require collecting abstracts from multiple problem scenarios and finding common link or relationship.
Language : Finding or labelling abstractions,patterns and relationship with proper word, and best possible phrases with example for consumption by other people… i can say communication.
Symbolizing & Reduction: Replacing general language statments with symbols as much as possible to reduce ambiguity (Formalism).
Rigor : working with Formal methods on huge chunk of data or repeating above steps to make sure proof is strong and is factually correct.
But considering vast requirement of educating childern worldwide, plus geo-political turnmoils that we face; Providing such proper math education seems overwhelmingly mammoth task and extremely difficult to plan and execute.
However, If we find historical concepts difficult to grasp or imagine me thinks that we can start rightway with following generalized method:
Listing out problems that we face today like social, political, infrastructural (we can easily get correct data which is in abundance these days) and applying above listed methods on them… may be that would bring in faster cognition for 14+ age students (even for adults like me who have missed out an opportunity).
I just encountered this site tonight, in search of a way to convince a friend that the magic number ‘e’ essentially always represents proportionality when it appears in physical equations, due to the fact that powers of ‘e’ are their own derivatives. Mission accomplished in that regard, but what I found interesting, and that I’d never thought of before (and I’m fifty-something), is the connection of the individual terms in the infinite series for ‘e’ to the integrated compound interest mechanism. Thanks!!
Thank you for succintly listing out a method to go about intuitively thinking about and understanding math
I’m taking a math senior seminar and I have to present on the Euler-Lagrange equation and give a short intro. into the history, derivation, and basics of calculus of variations…I am pretty lost still but your site (though it doesn’t deal with the topic) has helped me immensely in better understanding fundamental math ideas and helped me find a way to focus the paper I have to write too.
I especially like how you mention looking into the history of an idea to find its central theme, and your example with e. Seriously, this takes away at least a little of the anxiety I’m feeling about the paper and the presentation.
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Kalid: thank you, so much, for writing this article.
I’m a second year university student in, gasp, the liberal arts.
As you may infer from that statement, I’m not exactly good at mathematics.
Thing is, I want to be good at mathematics. I want to be able to see a proof and be able to understand it and tell that it’s beautiful, or whether it’s not elegant and so on.
I think the approach here will be useful in my attempt to self-educate myself in mathematics. First thing I’m looking at is Euclid’s Elements; hopefully, attempting to use the approach that you have here will help me in my understanding. Once I’ve got Euclid, I’m moving into trigonometry and calculus, and so deeper and deeper until I’m at least competent in number theory, which is where I’m interested.
Who knows, I might end up as a Fermat- he didn’t put a lot of focus into his mathematics study until he was around my age; who knows?
@Dave: Thanks for the comment! Glad it was helpful.
To be honest, I don’t think many people who are “good at math” really have a grasp of the beauty and elegance. It’s a bit like saying some who aces spelling bees would be a good writer, since they have a great vocabulary. Maybe, maybe not. Real understanding comes from seeing a lot of the subtle connections, not mechanical techniques.
I think your interest in finding insights and real understanding will really help you – re-learning math now, I’m finally starting to see connections I completely missed in college and high school. You’re lucky that you’re able to start so early :).