Review of Pea Soup, Tripe and Mathematics

Full article can be found here

Topics are taught without motivation behind them, and most students find this uninteresting.

I agree with this. I've always been concerned that this was what made me "an engineer" but I believe simpler examples could really
motivate the topics. I also find this point to be somewhat untrue, but I will metion that later

Topics are taught 'in the air' so the course has no apparent common
thread other than the trivial one

This is still particularly true, we've broken our mathematics down into a series of courses that have lovely labels (such as "Algebra" or "Calculus") and you need fundamentals from them to build more complex parts of the other - but the truth is these systems developed in parallel and built on each other. Much of the material isn't relevant without the other topic motivating it.

Topics that are not relevant to further study should be ruthlessly culled.

It's a great idea, but I think this is unreasonable. The depth he pursues this is that say astronomy majors and biology majors should be taught different sets of mathematics. Unfortunately breaking people up into smaller discrete, I'm afraid would overpower the math departments ability to staff and teach them.

"Trade mathematics is to mathematics as plumbing is to hydrodynamics
and as electrical wiring is to electromagnetic theory."

This is an interesting point, and I think I like it. We separate
theory from practice in many other fields, but fail to do so in math.

Mathematical texts need work

This seems to be less relevant now. I find most modern texts to be fairly complete and interesting. Particularly in the calculus books I have encountered, many of them make efforts to introduce real problems that the mathematics in the chapter is capable of solving, and also introduce short biographies of the people that discovered these ideas. However, most instructors ignore this. Is it the fault of the text that the instructor fails to utilize it fully?

Material presented is over the heads of students

Sometimes true, because there seems to be a need to show proofs of every topic introduced, but the proofs aren't relevant to the homework or application (trade vs. theory)

"... a thorough understanding of the concrete, must come before the abstract."

Not possible. The way our minds work and the amount of time we have to teach the material, requires months of coming back. Our timelines are often so compressed that a concrete understanding is not possible. Graph. This graph is interesting. The left axis is memory retention. A concrete understanding is simply not reasonable in the amount of time alloted. The topics must be repeated in later coursework for this to be effective.

"Abstraction is not the first stage but the last stage of
development. It may give new insight but only into concrete subject
matter already learned."

Good quote, but in a way advocate the bottom up approach. If you present the abstracted big picture and build down you may gain a deeper interest. I can argue both directions on this one though.

Mathematics is a physically derived topic

Yes, so use more physical insight! More people have it!

The capacity to appreciate mathematical rigor is a function of the age of the student and is independent of the age of mathematics.

But we must tailor our mathematics to the typical ages of those in our program, say 18-22. So, perhaps toning down the rigor is in order?

"Thirdly, rigorous proof is the polish on mathematics. It is the
last stage of a development. ... Today we are trying to emphasize the
gilt and the polish and we are leaving out the substance."

Also true. Based on many professors here, I would say that their understanding of mathematics is limited, and hand-waved (or gilt and polish) and they retain very little substance. I would even say some of my own background has been the same. Re-learning for applications is often necessary

"This discussion could even enter into some problems of philosophy,
then extend perhaps dependant upon the interests of the group. With a
liberal arts group the subject should be pursued further than with a
group of mathematics majors."

Again, great idea, almost impossible to implement without infinite resources

"Incidentally we should teach the algebra we need as we need it and
thereby take care of some of the topics in the usual college algebra
course."

Right. Topics were developed together, and motivate each other, they didn't spring forth fully formed.

"Math is an abstract formulation of ideas suggested by the physical
world; it is the artful use of reasoning processes to infer and deduce
new facts about the physical world; it is a series of significant
assertions about the physical world; and it contains implications
about almost all the arts and sciences. All four aspects of
mathematics must be taught. Techniques are boring but necessary
details which must be properly subordinated to the ideas, the
reasoning and the conclusions. Moreover they should be taught when
needed for some larger goal."

In conjunction with culling out less relevant topics, we could add in some of the more humanistic sides. The most engaging instructors I have found in mathematics were able to bring out tidbits of the personalities of the people who created these ideas in the first place.