My recent article on teaching calculus generated some pushback (crossposted here as well) from Jeffrey Shallit, a computer science professor who seems to consider himself a guardian of academic thought. I am saddened by such displays because they demonstrate that academia has stopped trying to educate students and is instead in the business of simply making them repeat terms and fill out forms correctly.
His post displays most of the reasons why ordinary humans have stopped respecting academics and credentialed experts. That is sad because society needs experts and credentialing serves an important function. Unfortunately, what makes you an expert today is not your clarity of thought but rather your ability to conform your thoughts entirely to the constraints of your profession’s vocabulary.
If you haven’t yet read my original article, “Doing the impossible: A step-by-step guide,” I’ll give you a quick summary: The goal is to show how mathematics can teach people deeper insights about the world than how to manage numbers. Doing mathematics can be thought of as a training ground for developing insights into other aspects of life. Most people who learn mathematics will never, ever again need the specific mathematical knowledge conveyed. What they will need, however, are the mental tools that the study of mathematics creates. The entire point of the article was to demonstrate how you can use mathematics to stretch your students’ imaginations, offering as a specific example, how insights from mathematics can help overcome literally impossible barriers.
For academics like Shallit, none of that is important. Teaching students how to think through problems, how to re-imagine difficulties in new ways, and how to overcome barriers? Not for him! Apparently, for him, the only thing that matters is precision of vocabulary.
Shallit does not suggest a better way to expand a student’s mind. He does not in any way show how using imprecise terminology led the student to lower-quality thinking. He does not take into consideration the fact that, for the vast majority of students, talking about a polynomial with an infinite number of terms is actually more understandable than using the formal term “power series.”
Because the article was discussing the development of new ideas, talking about a polynomial with an infinite number of terms actually more closely matches the way that the idea developed.
History shows that Newton’s own development of the idea of infinite series was exactly as I described. In Newton’s day, polynomials were known to be imperfect stand-ins for transcendental functions. Newton showed, however, in De Analysi, that when polynomials are extended to infinite terms, they became perfect stand-ins for these transcendental functions. Newton did not use the term “power series” (it did not arise until the 1890s), but instead used the Latin “aequationes numero terminorum infinitas” which means “equations with an infinite number of terms.”
This attitude exemplifies the contagion affecting the academic and expert classes—instead of engaging on the substantive issues and providing value to students and society at large, the academics and experts instead engage in a game of pedantic one-upmanship, with the primary goal not of advancing the conversation, but merely providing themselves with the gold star affirmations that their egos need.
So, congratulations to Jeffrey Shallit—you get a gold star on your vocabulary test. I hope it is more worthwhile for you than teaching students to think.
Also by Jonathan Bartlett on calculus:
Jonathan Bartlett: Often, in life as in calculus, when our implicit assumptions as to why something can’t be done are made explicit, they can be disproven Calculus textbooks are the most dry and boring presentations of mathematics I have ever seen, even though calculus offers some of the most amazing insights. Unfortunately, most mathematics texts teach only the mathematics, never the insights.
Walter Bradley Center Fellow discovers a longstanding flaw in an aspect of elementary calculus. The flaw doesn’t lead directly to wrong answers but it does create confusion.
Featured image: Seeing mathematics in everyday life/Reuben Tao, Unsplash