# A question for the thinkers!

## 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

## Answers and Replies

chroot
Staff Emeritus
Gold Member
Many people, including myself, consider the beginning of "real math" to be the study of analysis, which is essentially the rigorous underpinnings of calculus. (Virtually all basic college calculus courses are taught with very little mathematical rigor.)

- Warren

symbolipoint
Homework Helper
Gold Member
Many people, including myself, consider the beginning of "real math" to be the study of analysis, which is essentially the rigorous underpinnings of calculus. (Virtually all basic college calculus courses are taught with very little mathematical rigor.)

- Warren
Was this true 25 to 30 years ago? The first three semesters of Calculus were developed for students mostly as tools for both analyzing some of the mathematics and tools to apply to real-to-life and multi-step situations. Some moderate effort was given to limit proofs, continuity, and several other rules, but as the courses progressed, the continuity and epsilon-delta exercises were diminished in favor of theoretical and applied problems.

Was this true 25 to 30 years ago? The first three semesters of Calculus were developed for students mostly as tools for both analyzing some of the mathematics and tools to apply to real-to-life and multi-step situations. Some moderate effort was given to limit proofs, continuity, and several other rules, but as the courses progressed, the continuity and epsilon-delta exercises were diminished in favor of theoretical and applied problems.
Well, in some places including the state university back home in my country, Math students on the very first year first semester do rigorous math analysis, not like calculus that we do here in usa. With this i mean they prove every single theorem, that in most universities here,they do only in real analysis or some upper division course.

Well, in some places including the state university back home in my country, Math students on the very first year first semester do rigorous math analysis

I've been wondering for a while what this country, you call home, might be

As for me..hard to say what the "key aha" was, maybe I'm stil waiting for it...:)

I've been wondering for a while what this country, you call home, might be

As for me..hard to say what the "key aha" was, maybe I'm stil waiting for it...:)
Well, the country where i come from is located somewhere in Balcan, in between Italy and Grece. So the system there is still quite based on the soviet one. Let me just briefly name the courses that a first year student have to take there:
1.Mathematical Analysis I- this starts with sequences continues with numerical series, goes up to the limit of the sequences functions, continuity, uniform continuity goes to derivatives, taylor polynomial, L'hopital rules, differentials and their conection with derivatives, and up to the graphing of functions using the properties of derivatives.

2. Mathematical Analysis II- This course deals only with Indefinite integrals, Reiman Sums, definite integrals and their application.

3. Theory of Sets and logical mathematics
4.Analytical Geometry
5.Abstract and Elementary Linear Algebra
6.Elementary Mathematics
7.Non linear Inequalities
8.Descriptive Geometry

And two other courses which names i cannot think of right now.

And all these courses are proof based courses, really really rigorous.

However i do not actually like that much this system, because it does not actually create to a student an intuitive understanding of phenomena, but rather just goes through things,because it is a lill bit hard to be ''fluent" at all those courses at the same time, especially when they are completely proof based for a first year student.

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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?l
Hi Paul,

A lot of it has to do with maturity. Maybe that isn't the best word, development might be better. In High School our brains are not fully developed yet. I think part of being able to make that jump is the ability for your brain to deal with the abstract.

The same is true for algebra around the time we are entering our teens. Some students simply aren't ready developmentally to deal with the abstraction necessary to grasp algebra.

Wow, thanks for all of the responses! I think that I might expand myself this summer with some real analysis and graph theory to complement my course content. If there's one thing that's got me about my math courses this year is the rigor involved... absolutely everything is proved as it is presented, and they expect us to do similar proofs on tests. It's all worthwhile however, when you need to solve a problem that doesn't involve plugging in values and "turning the crank".

Regarding the development comment, I see what you mean. Back in grade 10 I couldn't seem to get my head around the idea of instantaneous rates of change and such, and now that's child's play but I'm presented with a whole new array of seemingly baffling mathematical contortions now! What seems to be "the age" when people are fully developed in the sense of mathematical maturity and the abillity to deal with "high level" math?

^Any age! I knew a high school kid who took advanced calculus and abstract algebra while I was taking those courses in my junior/senior year in college!

Wow, thanks for all of the responses! I think that I might expand myself this summer with some real analysis and graph theory to complement my course content. If there's one thing that's got me about my math courses this year is the rigor involved... absolutely everything is proved as it is presented, and they expect us to do similar proofs on tests. It's all worthwhile however, when you need to solve a problem that doesn't involve plugging in values and "turning the crank".

Regarding the development comment, I see what you mean. Back in grade 10 I couldn't seem to get my head around the idea of instantaneous rates of change and such, and now that's child's play but I'm presented with a whole new array of seemingly baffling mathematical contortions now! What seems to be "the age" when people are fully developed in the sense of mathematical maturity and the abillity to deal with "high level" math?
Calculus in 10th grade? Nice.