RFC: Replacing the current Calculus and Differential Equations courses with proof based versions

[ernest-mm]

Problem:
The current Calculus 1, 2, 3, and Differential Equations courses in the OSSU math roadmap focus too much on doing calculations instead of teaching proofs.

Duration:
This should most often be 1 month from the date of posting.

Background:
Right now, the calculus and differential equations courses are set up in a way that works best for engineering students. They focus on using theorems to solve problems by calculation.
However, students who want to study math as a major need courses that help them learn how to build rigorous proofs and think abstractly (yes, even if those courses are first year courses. Take for example this math curriculum from the University of Toronto, they start with a proof based course in Linear Algebra their first year)
In short, while the current courses are good for computation, they don’t match what a top pure math (even applied math if we take the University of Toronto as an example) curriculum should provide.

Proposal:

• Replace the current Calculus courses with Calculus with Theory and Multivariable Calculus with Theory from MIT Open Courseware.
• Make Calculus 1A: Differentiation, Calculus 1B: Integration and Calculus 1C: Coordinate Systems & Infinite Series part of high school math and prerequisites for Calculus with Theory.
• Swap out the Differential Equations course with Honors Differential Equations.

Alternatives:

• Keep the current computational courses for those interested in it, but create a separate track with proof-based courses for people pursuing a pure math major roadmap.

[ericung]

This is cool. Have you looked at Calculus by Spivak?

[ernest-mm]

This is cool. Have you looked at Calculus by Spivak?

Yes, but I can't find a free online ressource that uses it and has solutions to problems

[waciumawanjohi]

students who want to study math as a major need courses that help them learn how to build rigorous proofs and think abstractly

True

In short, while the current courses are good for computation, they don’t match what a top pure math... curriculum should provide.

Other mathematicians have similarly pointed out that the math curriculum is computation heavy.

At the same time, I don't want the curriculum to over correct. For example, calculus is not part of the high school math curriculum. OSSU has a pre-college math curriculum, which refers to an external curricular guideline, the Common Core Mathematics Standards.

Since Calculus with Theory requires Single Variable Calc as a pre-req, the OSSU curriculum should position Calc with Theory as a follow on course in the curriculum.

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the current courses are good for computation, they don’t match what a top pure math... curriculum should provide.

This proposal swaps out MIT's standard Differential Equations course (18.03) for its Honors Differential Equations course (18.034). I notice that MIT's requirements for a Math Major in the Pure Math program are satisfied by 18.03.

Should 18.034 be an alternative option for learners?

[ernest-mm]

For example, calculus is not part of the high school math curriculum. OSSU has a pre-college math curriculum, which refers to an external curricular guideline, the Common Core Mathematics Standards.

Yes it's not a high school subject but when you study pure math, many universities expect you to have taken the computational part of single variable calculus in high school (at least to know how to work with limits, derivatives and integrals).
Since people will be self-studying, I don't understand why they wouldn't use the best approaches possible if they're interested in math.

Since Calculus with Theory requires Single Variable Calc as a pre-req, the OSSU curriculum should position Calc with Theory as a follow on course in the curriculum.

Something like this?
- Calculus 1A: Differentiation
- Calculus 1B: Integration
- Calculus 1C: Coordinate Systems & Infinite Series
then
- Calculus with Theory
- Multivariable Calculus with Theory

I don't know, it's too much Calculus.

Should 18.034 be an alternative option for learners?

In my opinion it should be a replacement of 18.03. 18.03 is great but from its description and course goals we see that it focuses on equations and techniques:

This course focuses on the equations and techniques most useful in science and engineering.

By the end of the course students will be able to:
1. Model a simple physical system to obtain a first order differential equation.
2. Test the plausibility of a solution to a differential equation (DE) which models a physical situation by using reality-check methods such as physical reasoning, looking at the graph of the solution, testing extreme cases, and checking units.
3. Visualize solutions using direction fields and approximate them using Euler’s method.
4. Find and classify the critical points of a first order autonomous equation and use them to describe the qualitative behavior and, in particular, the stability of the solutions.

On the other side 18.034's description clearly states that:

This course covers the same material as 18.03 (Differential Equations) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems. It shares in its spirit with the two semester sequence Calculus with Theory I and II (18.014 and 18.024).

[waciumawanjohi]

I don't know, it's too much Calculus.

If it's too much calculus, we should jettison the extra calculus (18.014). But I think your point is that 18.014 is a course worth including. If it is worth including, it's worth being up front about the pre-requisite work, which is 18.01.

[waciumawanjohi]

I agree, MIT's 18.034 clearly states that it includes more theory than 18.03. What I'm asking for here is clearer evidence that this extra theory is necessary for math majors. Again, MIT allows undergrads on the pure math track to take either 18.03 or 18.034. Walk us through why OSSU math should require 18.034.

Something to note that isn't mentioned in the Background above, the course after Differential Equations in the OSSU curriculum begins with proofs.

[ericung]

What if it's too big of a jump for someone not prepared? What are supplemental materials for someone to get up to speed. To me, it's an Olympic sport competing with people trained to be global champions taking a class of that level - entry and introductory.

[ernest-mm]

If it is worth including, it's worth being up front about the pre-requisite work, which is 18.01.

Then we can have a three course Calculus sequence:

• Single Variable Calculus
• Calculus with Theory
• Multivariable Calculus with Theory

The prerequisites for 18.01 are high school algebra and trigonometry.

But we should also state the 18.01 is optional for someone who has mastered high school Calculus at the level of AP Calculus BC

[waciumawanjohi]

But we should also state the 18.01 is optional for someone who has mastered high school Calculus at the level of AP Calculus BC

What do you think is the best way to do so? We could put it in the pre-req section of Calc with Theory. Something like "Single Variable Calc or equivalent (AP, Clep, etc)"

[waciumawanjohi]

Then we can have a three course Calculus sequence:

• Single Variable Calculus

What's the reason for changing the recommendation from the OpenLearningLibrary version of single variable calc to the OCW version? Is it a better learning resource for the topic? Or is it just fewer links on the page?

[ernest-mm]

We could put it in the pre-req section of Calc with Theory. Something like "Single Variable Calc or equivalent (AP, Clep, etc)"

We can do that.

What's the reason for changing the recommendation from the OpenLearningLibrary version of single variable calc to the OCW version? Is it a better learning resource for the topic? Or is it just fewer links on the page?

Fewer links and the fact that it is only a one semester course instead of three 13 weeks courses.
It can be completed much faster than the Open learning library version.
We don't lose in quality because it's a SC version.

[waciumawanjohi]

Fewer links and the fact that it is only a one semester course instead of three 13 weeks courses.
It can be completed much faster than the Open learning library version.
We don't lose in quality because it's a SC version.

This is a reasonable statement. It's also separate from the question of proof based courses. How do you feel about opening a separate RFC for swapping the MIT Calc 1 offering (SC vs OLL)?