5 ms·

No way high school or college level classes are teaching intro calc with delta-epsilon proofs, it just won't make any sense. It's generally introduced in Real A

by Hasz 8y ago

No way high school or college level classes are teaching intro calc with delta-epsilon proofs, it just won't make any sense. It's generally introduced in Real Analysis, which is generally a sophomore level class. Develop the intuition, then crystallize and formalize the idea with a proof. Otherwise, it's just not going to make any sense.

nextos 8y ago
My last year high school math class had this approach, and it was a regular math course taken by every student. Same thing at university, in a CS degree. Freshman Calculus taught at a level somewhere between Spivak and Rudin. In fact, both books where on the official course bibliography. All this took place in a fairly big EU country, ~10 years ago.
chongli 8y ago
I was taught the Epsilon-Delta definition of the limit in Calc 1, right off the bat in first year (University of Waterloo). Real Analysis here is a third year course run by the Pure Math department and from what I've heard it's one of the hardest undergrad courses the university offers, period. I'm nearing the end of my second year as a math student, finishing up calc 3 and linear algebra 2. When I take a look at a few of the questions on assignment 1 from a past real analysis course I have no idea how to do any of it, it's way out of my league.
- Ntrails 8y ago
  Real analysis is not actually that hard, you do however have to learn a bunch of notation, concepts and processes before you can grasp things. The people who struggled with it at my course (different country, similar content I suspect) were, entirely unsurprisingly, the people who didn't attend all the lectures.
  - chongli 8y ago
    I don't think the difficulty is due to anything inherent to real analysis. I think it's due to the fact that most of the professors in the pure math department participate in the development of the math contests for CEMC [1]. Thus they tend to love creating extremely complicated and challenging homework problems. If you have time, click around on that page and check out some past Euclid (grade 12) math contests. Even if you've come through real analysis I bet some of the later problems in those Euclids will give you trouble. [1] https://www.cemc.uwaterloo.ca/contests/contests.html https://www.cemc.uwaterloo.ca/contests/contests.html
rhizome 8y ago
I took Calc I at a community college last fall and epsilon-delta was about halfway through.
sevensor 8y ago
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.
- Izkata 8y ago
  Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.
  - sevensor 8y ago
    Yeah, exactly. I honestly don't know how else you'd teach the derivative either, but I guess it's done.
  - giornogiovanna 8y ago
    The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.
  - nextos 8y ago
    Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century. It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s. To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book. [1] https://www.macalester.edu/aratra/ https://www.macalester.edu/aratra/
- fatnoah 8y ago

by Hasz 8y ago

nextos 8y ago

My last year high school math class had this approach, and it was a regular math course taken by every student. Same thing at university, in a CS degree. Freshman Calculus taught at a level somewhere between Spivak and Rudin. In fact, both books where on the official course bibliography. All this took place in a fairly big EU country, ~10 years ago.

chongli 8y ago

I was taught the Epsilon-Delta definition of the limit in Calc 1, right off the bat in first year (University of Waterloo). Real Analysis here is a third year course run by the Pure Math department and from what I've heard it's one of the hardest undergrad courses the university offers, period. I'm nearing the end of my second year as a math student, finishing up calc 3 and linear algebra 2. When I take a look at a few of the questions on assignment 1 from a past real analysis course I have no idea how to do any of it, it's way out of my league.

Ntrails 8y ago
Real analysis is not actually that hard, you do however have to learn a bunch of notation, concepts and processes before you can grasp things. The people who struggled with it at my course (different country, similar content I suspect) were, entirely unsurprisingly, the people who didn't attend all the lectures.
- chongli 8y ago
  I don't think the difficulty is due to anything inherent to real analysis. I think it's due to the fact that most of the professors in the pure math department participate in the development of the math contests for CEMC [1]. Thus they tend to love creating extremely complicated and challenging homework problems. If you have time, click around on that page and check out some past Euclid (grade 12) math contests. Even if you've come through real analysis I bet some of the later problems in those Euclids will give you trouble. [1] https://www.cemc.uwaterloo.ca/contests/contests.html https://www.cemc.uwaterloo.ca/contests/contests.html

rhizome 8y ago

I took Calc I at a community college last fall and epsilon-delta was about halfway through.

sevensor 8y ago

Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.

Izkata 8y ago
Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.
- sevensor 8y ago
  Yeah, exactly. I honestly don't know how else you'd teach the derivative either, but I guess it's done.
- giornogiovanna 8y ago
  The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.
- nextos 8y ago
  Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century. It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s. To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book. [1] https://www.macalester.edu/aratra/ https://www.macalester.edu/aratra/
fatnoah 8y ago