Inquiry-Based Calculus Taught Me I Am Not John Nash (And Humility)
Rounded down, I have nearly three decades experience as a human being. Nearly one-third of this experience has been in in the world of education, a milestone that comes with pride and a certain level of humility, a concept totally amiss at milestone 2 of 3 on life’s journey.
In reflecting, taking inquiry-based calculus course at the University of Chicago was among my first truly humbling experiences, and while I may not have mastered the art of proof by induction, I learned other lessons that were invaluable when I became a special education teacher in Chicago Public Schools and throughout out my career.
Instead of “a beautiful mind,” I learned to appreciate that I might have an average one… a mind that is both below and above-average on occasion.
Adulthood Achievement Unlocked: Humility.
Proposition
Despite the positive affirmation (and incremental advantages) I received throughout elementary and high school, I am not John Nash. I do not succeed at everything I try.
Definitions
In true proof format, I should probably start with a few definitions and/or “givens.”
Humility Defined
Brief Google research produced the following definition that I believe will be suitable for purposes of proving my hypothesis.
hu·mil·i·ty
(h)yo͞oˈmilədē
noun
a modest or low view of one’s own importance; humbleness
Interestingly, when it comes to references in English-language texts, humility has not been making a major appearance in written language for some time.
Inquiry-Based Calculus
According to the University of Chicago Department of Math:
“Students who receive a sufficiently high score on the mathematics placement test may receive an invitation to enroll in MATH 16100 Honors Calculus I /MATH 16110 Honors Calculus I (IBL).”
The course is designed to give “a rigorous axiomatic treatment of the continuum and its topological properties.”
Unlike “regular math,” where the formulas you need are readily accessible, IBL requires us to return to fundamental math “truths” and prove why the rules we have taken granted for so long continue to work.
Proof by Contradiction
I recall my favorite type of proof was proving that a statement is untrue by finding one condition/instance where said statement is true.
In this case, to prove that I am not John Nash and not as smart as my mother would have had me believe, I only need to find one example where I am not successful.
To make a strong argument, however, I need to a VERY strong example of something I am very unsuccessful in. Fortunately, inquiry-based calculus supplied this strong contradiction.
Background
Having graduated from a rural public high school in small town (pop. ~1000) South Carolina, I still assumed proof was a concept with applications exclusive to the legal world.
What’s there to left to prove in math, that hasn’t been proven before?
My understanding of math until college had been that only a few things ever happen to numbers: You add, subtract, multiply, and/or divide them… and on rare occasion raise them to some power or take the square root of these.
Every now and then things would get really tricky when you weren’t given all the numbers, and you had to figure out what some missing letter was instead (usually, said missing letter was an x).
Having somehow done well enough to have the option of skipping 151 and starting 152 — which would have allowed me to credit for 151 — I decided to take challenge myself and start at the very beginning with 161, Honors Calculus!
I came to the University of Chicago to challenge myself, right? I mean, if I was smart in South Carolina to be accepted to UofC and smart enough to place into honors calc, I must be smart enough to do well in it?
I remember there were ~8 students in my section and one gentleman in his 30s or 40s who was sitting in to brush up. (I believe he was otherwise part of some Masters degree program in physics, maybe?)
The first week or two were a bit of a blur, but I took good notes and read, and re-read, the first several pages of Michael Spivak’s Calculus. I was determined to understand why the basic rules of math I had taken for granted (a + 0 = a, a*1=a, etc.) were true and how to prove said “truthiness,” despite how intuitive they may have seemed.
From the second week onward, the problem sets became much “harder.” I learned that this word has a very subjective meaning when it comes to mathematics, a meaning different than any of my prior academic experiences.
Before this course, “harder” meant that x activity took more time than y activity. A “really hard” book would take a lot more time to read than other books. A “really hard” set of math homework might consist of four worksheets instead of one.
With inquiry-based calculus, however, there was no clear linear correlation between the time I spent on a problem set and my success in completing it… I could as easily spend 10 hours on a problem and make no progress as I might happen to spend 1 hour and “get it.” (Granted, part of the time lost was due to my insistence on submitting problem sets formatted in LaTeX, an unnecessary extravagance I hoped would make me appear more intelligent.)
I kept thinking that I would find some inspiration and begin seeing numbers à la A Beautiful Mind on the windows of my dorm in Pierce Tower, but the closest I got was drawing smiley faces with my finger after fogging up the windows.
The other students in my class were inspiring, and we did sometimes work on problems sets together. While I’m not sure that working in a group really led me to better understand any particular problems, I believe we developed a shared admiration for one another’s struggles and endurance.
Ultimately, I passed with a C-.
When I met with my instructor to ask about plans for the second quarter, he did not offer the type of encouraging speech I had so often gotten throughout my prior academic career.
Conclusion
I recall genuine fear of not passing the course and catastrophizing the outcomes.
Will I be sent back to South Carolina? Will they know I am a fraud? Will they think I somehow cheated on my placement test?
While this is the I recall earning in a course, I’m more proud of that C than at least one of my Masters Degrees (In the spirit of saying nothing if I can’t say something nice, I will leave institution of said Masters degree unnamed).
I ultimately decided that pursuing a degree in Mathematics might go hand-in-hand with a <=2.0 GPA, I moved to the non-honors 152 and 153 sections to complete my year of calculus (which thankfully did feel much more manageable after 161).
All of that said, even if much of my mathematics struggle wasn’t entirely productive, I
- Developed an appreciation for an entirely different way of thinking;
- Learned that hard work does NOT always guarantee results — at least not always in a predictable/linear fashion;
- Began to appreciate more the different types of intellect in the world, in particular those who who continued down the track of more advanced mathematics;
- Became more resilient as a person and a learner: I didn’t drop out of the course mid-way through (despite oft- considering this).
Taking 161 helped broaden the “upper” and “lower” bounds (bad pun) of what “hard” means.
Seriously though, I think about inquiry-based calculus before I begin to complain about a intellectually-demanding task: I think of hours staring at problems without any progress or sense of achievement/growth.
Inquiry-based calculus also helped my develop an appreciation for how important principles can be in any field: Understanding foundational concepts very deeply can make it easier to recognize broad patterns and to see individual decisions more quickly. (See Ray Dalio’s Principles, for instance, as an ode to this general sentiment.)
This appreciation has influenced both the approaches I take in teaching as well as my leadership style: I try to help others develop an appreciation of principles that have broad application, since there’s a great deal of “noise” in education when the “signal” is often simple.
More importantly, this experience helped me reflect as a teacher about students who either might be going through this experience at any given moment, or who may have gone through this feeling of failure throughout their careers: How could I supply them with the confidence I had taken for granted? How could I help them feel a sense of progression/accomplishment? How could I help them take their moments of perceived failure and use these as opportunities to develop resilience… and as opportunities to appreciate their unique abilities and strengths?
So, what is my advice to someone who has never done proofs before and wants to take an inquiry-based calculus sequence?
Go for it!
At worst, you’ll fail — or get a C- like me — but at least you’ll have something to write about ten+ years from now!
At best, you’ll challenge your intellect, make some new friends, and develop new, innovative ways to approach problem-solving. (I suppose the best, best case might involve winning the Fields Medal and solve some global problems along the way through applications of whatever new knowledge you discover and share.)
In the end, I doubt any mind is any one adjective all the time, be it beautiful, ugly, average, or exceptional and perhaps such categorizations are misleading.
Perhaps instead of considering the beauty of a given mind, as some fixed attribute or trait to be measured, it’s more productive to look at our what we do with the minds we have.
You can be beautiful without being John Nash.
Be beautiful.
Q.E.D.
