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## Main Question or Discussion Point

Hi Folks!

I'll start by introducing myself a little bit, as it leads into my question. I'm finishing my first year of a program called Engineering Science at the University of Toronto, it differs from other engineering programs in its emphasis on rigorous mathematics and basic science as well as engineering. Coming from a highschool where an entire semester was devoted to single variable differential calculus, and fast forwarding to now where I've now dealt with differential equations and multivariable calculus as well as linear algebra (abstract vector spaces and such) has really opened my mind to just how limited a view of mathematics one can leave high school with.

To me, in high school, it was all about finding the answer to the problem at hand. Find these co-ordinates, find this rate, etc. In university it seems it's all about the general case (prove for any vector in Rn, show for any constant c that n must always be... you get the idea). Ideas that were minimized as unimportant in high school such as induction, or summing an infinite set, or indeed set theory itself are all now central topics in my math courses. And my physics, statics, circuits, etc courses all rely on math that considers my level to be only the most basic level. It's left me feeling a tad small, I'll admit. It's exciting too, and I really feel like I have developed my mathematical abilities significantly this year. But, I can't help but feel like I'm on the cusp of a new level with respect to my thinking and reasoning, similar to the change from arithmetic to algebra so many years ago. I don't think I've made it yet, but I feel like it's within my ability to do so.

So here it is: in general what do you folks consider to be key realizations/concepts/"ahas"/etc in making the leap from high school math level thinking to becoming a player in the game of "higher math"? Any tips for someone attempting to make this change?

I appreciate any responses you may have!

-Paul

I'll start by introducing myself a little bit, as it leads into my question. I'm finishing my first year of a program called Engineering Science at the University of Toronto, it differs from other engineering programs in its emphasis on rigorous mathematics and basic science as well as engineering. Coming from a highschool where an entire semester was devoted to single variable differential calculus, and fast forwarding to now where I've now dealt with differential equations and multivariable calculus as well as linear algebra (abstract vector spaces and such) has really opened my mind to just how limited a view of mathematics one can leave high school with.

To me, in high school, it was all about finding the answer to the problem at hand. Find these co-ordinates, find this rate, etc. In university it seems it's all about the general case (prove for any vector in Rn, show for any constant c that n must always be... you get the idea). Ideas that were minimized as unimportant in high school such as induction, or summing an infinite set, or indeed set theory itself are all now central topics in my math courses. And my physics, statics, circuits, etc courses all rely on math that considers my level to be only the most basic level. It's left me feeling a tad small, I'll admit. It's exciting too, and I really feel like I have developed my mathematical abilities significantly this year. But, I can't help but feel like I'm on the cusp of a new level with respect to my thinking and reasoning, similar to the change from arithmetic to algebra so many years ago. I don't think I've made it yet, but I feel like it's within my ability to do so.

So here it is: in general what do you folks consider to be key realizations/concepts/"ahas"/etc in making the leap from high school math level thinking to becoming a player in the game of "higher math"? Any tips for someone attempting to make this change?

I appreciate any responses you may have!

-Paul