Monday, May 18, 2015

Teaching math as needed for physics (for nerds only)

I came across a physics textbook a long time ago that was striking to me because it interrupted the flow of the physics instruction with about 7 gray sections on calculus. As a teacher, this approach intrigues me. It could be used in the teaching of any discipline that has prerequisites.

Although I haven't finished or published examples of this approach. I have used something like it to teach biblical languages and the Seminary uses an approah something like this in its integrative approach to the practice of ministry. I believe it is far more effective than the "building block" model if it is done correctly. It is something like problem-based learning.

But I can imagine putting together the beginnings of a science curriculum this way.

1. Instantaneous velocity and acceleration (while discussing motion)
This is a great place to introduce the basics of finding a derivative, and this is how the Marion/Hornyak (MH) book pictured began. Integration can also be introduced as the inverse process in brief. Both would be introduced simply, to be expanded upon later.

2. Vector addition (motion in more than one dimension)
Vector addition begins to come into play as soon as you hit motion in more than one dimension. I find Young and Freedman's (YF) introduction of all things vector in chapter 1 a teaching problem. The cross product in particular is, I think, a difficult concept to introduce at the very beginning. Wouldn't it be more helpful to introduce the various characteristics of vector addition and multiplication as they arise in specific topics? This is also a place to review some basic trig.

3. Summation (forces)
When you get into forces, you could introduce summation notation and extend integral calculus. MH review definite integrals when they get to the application of Newton's laws.

4. Vector dot product (work)
Young and Freedman introduce dot products in chapter 1, but students don't need it until chapter 6. Why not introduce it there? There are more integrals in the treatment of work. So an integrative approach could be solidifying and extending techniques of integration as the student went along.

5. Partial derivatives, nablus (energy)
It is amazing to me that I wasn't introduced to partial derivatives until my third semester of calculus and I don't remember hearing about the nablus/gradient in four semesters. Yet these are relatively easy concepts I could have learned in a first semester. This is frequently the case. Teaching the calculus in order, you don't get to simple and useful concepts until way down the line.

Yet chapter 7 of YF already introduces these fairly straightforward concepts in their treatment of energy.

6. Integration in two and three dimensions
MH have a calculus 5 section on this before a chapter on angular momentum. YF also have an advanced section involving integration in their chapter on angular velocity.

7. Radian measurement (angular velocity)
YF review radian measurement as they begin their chapter on angular velocity.

8. Cross product
It is not until chapter 10 and the treatment of torque that YF ever use the vector cross product. Why introduce it in chapter 1, when most students will have no idea what it means or what it is for?

9. Taylor series
MH review this third semester calculus topic in between their chapters on gravitation and periodic motion.

10. Differentials
I see some differentials by themselves in YF treatment of thermodynamics.

11. Surface integrals
When you get into Gauss' law regarding electric flux

12. The gradient
More partial differentials when you get to electric potential

13. More cross product
When you get into electromagnetic induction

14. Second order differential equations
MH finally review second order differential equations, a fourth semester calculus topic, as they are digging deeper into electromagnetic waves.

I picture, perhaps, two teachers tag teaming over the course of a year or summer intensive. Problems might circle back around to earlier physics topics after new mathematical concepts were introduced. Similarly, methods of application (e.g., in calculus) could be introduced in the process of doing problems.

Just some ideas for a Monday morning...

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