Redesigning the math/physics curriculum, take 2

[Will Flannery]

I saw the topic and have given it a lot of thought over the past few years, and my take is so different from the other thread I'm starting another.

Problems with the beginning math curriculum:
1. Too abstract and difficult
2. Totally unmotivated

These problems were unavoidable 50 years ago, but now computers have changed everything. They have totally changed how math is used in the real world, but this hasn't been reflected in the educational system.

Why is math so difficult? The quick answer is that physical laws are typically written as differential equations, so differential equations are the starting point of mathematical physics, and most differential equations are unsolvable. E.g. Newton's first differential equation, for the trajectory of a falling apple, p''(t) = C/p(t)*p(t), is unsolvable.

The result has been that the math curriculum (in my experience) avoids differential equations for as long as possible. And when you finally take a course in DEs it is a hodge podge of methods you will likely never use of even think of again. At least I didn't in 20 yrs. as an engineer.

What has happened is that computers have totally changed how differential equations are analyzed in the real world, where numerical methods are used extensively. The simplest numerical method for computing solutions to ordinary differential equations is Euler's method which boils down to the equation distance equals velocity times time, and can be taught to juniors in high school in a day, no previous calculus experience required.

Computational calculus, unlike analytic calculus, is easy to learn, and it immediately opens up the world of physics where all sort of interesting and challenging problems can be presented.

In fact, I have a series of video lessons that do just that, go to YouTube and search for wdflannery and you'll find two playlists, let's see if I can remember the topics ... nope .. got to look ...

1 - Computational calculus basics, MATLAB, Newton's model, Orbits, More orbits, Apollo, Juno, Accuracy of Euler's Method, Electrical circuit basics, Capacitors, Inductors, Oscillators, Filters, SPICE, Rigid Body Dynamics, A bouncing spinning ball, A smooth ride over a bumpy surface, An airplane simulator, Rocket launch to orbit, VEX robot sim

The second series uses the finite difference method, that is Euler's method applied to partial differential equations, to analyze ...

2 - Heat transfer 1, 2, 3, 4, An epic battle between a sine wave and a square wave, Analysis of a square drum, Tensors in twenty ... minutes, Stress and strain, the party's over, The stress tensor, The strain tensor, Stress and strain case studies 1 and 2, The Navier-Stokes Equations, The stress and strain rate tensors, A Poisson equation for pressure, Cavity flow, flow over a backward step, Maxwell's equations in integral form, Maxwell's equations in differential form, Computational electrodynamics in 1-D, Computational electrodynamics in 2-D, The Yee/FDTD algorithm

It's a hell of a list, but I actually went thru most of this material in a 1-yr informal course with two very bright high school students. The projects are really very simple up to the last three topics, starting with tensors and stress and strain in materials, but even here the calculus is still the FDM, but the derivations of the model equations are more involved. (By the way, I claim and I hope demonstrate that tensors are straightforward once you decide to bypass the specialized notation (up and down indices, pre and post multiplication, suppressed summations) and use conventional notation.

 

[martane]

Wow, thank you for sharing your perspective on the problems with the beginning math curriculum. I agree that the traditional approach to teaching math, with a heavy emphasis on abstract concepts and unsolvable differential equations, can be daunting and unmotivating for many students.

Your use of computers and numerical methods to teach calculus and differential equations is very intriguing. It seems like a much more practical and engaging way to approach these topics, especially for students who may not have a strong interest in pure math.

I also appreciate that you have created video lessons and playlists for others to access and learn from. It's great to see someone taking a different approach to teaching math and making it more accessible and relevant to real-world applications.

I will definitely check out your videos and playlists on YouTube. Thank you for sharing your knowledge and ideas with the community. I hope more educators and students will explore this alternative approach to learning math.