A Retirement Journal: I can't do high school math anymore and yet ...

I have 2 degrees in Electrical Engineering. Yet, I found myself several weeks ago struggling to complete a math problem. It was as seen above, one the teacher in an 8th Grade Honors Geometry class at Greene Middle School in Palo Alto had assigned. I was subbing that day early in the school year having signed up via my substitute app for that particular class. After all, it was 8th grade and I did fancy myself a tech veteran and science and math savvy, albeit “retired”. The teacher’s lesson plan for that class included review problems for which he left me the answer sheet with instructions to help the kids who were struggling. He did however say in his notes he would be surprised if anyone did need my help. No student came to me with questions. Getting a bit bored, I thought I would try the Challenge 2 problem above even as the kids were working. My, My!

The problem asked us to show how the diagram above showing a big cube with each side with dimensions a (and hence a volume of a3) with a smaller cube of dimensions b (and volume b3) could be used to factor the algebraic expression a3 – b3 into its common factors. I went to work along with my students. It was a struggle but I eventually managed to get to the terms shown above a3-b3 = (a-b)(a2+ab+b2). This was just 8th Grade. Forget trying to really teach trigonometry or calculus at Palo Alto High School (PALY). And, I have two degrees in engineering!

Even as I labored through the algebraic manipulation, I was deeply struck by the beauty of the situation. This algebraic expression could be visually represented as a 3-d geometrical situation. The algebra could be solved via geometry by calculating the volume of the 3 remaining solids if a cube with dimension b was cut out of a bigger cube with dimension a.? I had never in my own high school days stopped to marvel at the symbiosis between the different mathematical disciplines. I did in my electrical engineering studies pause briefly to note that Maxwell’s Equations show the symbiosis between magnetic and electrical fields creating electro-magnetism. ?But the new relationship that I was observing in September 2023 was truly marvelous- way cool as my 3-yearold grandson J would say.

My inability to manipulate algebraic polynomials contrasted with my increased wonder at the interplay between geometry and algebra is beautifully captured by social scientist Arthur Brooks in his book From Strength to Strength. Brooks asserts that there are at least two types of intelligences. Fluid intelligence is what we are born with and develop into adulthood. This is the intelligence that helps us focus sharply on a specific issue, think deeply, get profound insights, and make significant new discoveries. Brilliant music composers, scientists and mathematicians for example do their groundbreaking work by their thirties and certainly before their forties. In Brooks’ view, people trying to hang on to their fluid intelligence beyond the “use by” date usually end up deeply frustrated and unhappy. Instead, Brooks argues folks should transition to their crystalline intelligence which does not per the author atrophy. This later stage intelligence is what lets us elevate, get the big picture, do pattern-recognition, and synthesize multiple facts clearly and easily. This crystalline intelligence is ideally suited for mentoring and counseling. The wisdom of the elders as perceived from time immemorial is real per Brooks. It is this type of intelligence that when leveraged leads to increased contentment, indeed flourishing in the later part of life.

The non-profit Oak Guild Institute that is being incubated by my friends Kate Whitehead and Katie Gray Garrison held a discussion salon this past summer on Brooks’ book. We had a wonderfully rich discussion between people who were 20 such as my nephew Peter Chacko to folks in their 80s. Intergenerational conversation is a wonderful thing. It is for me in my substitute teacher gig. Kids in grade school squeal, “It is Mr. Chacko again- yeah!” when I return to a classroom I had had earlier. As for the higher grades, I am scheduled to sub in a few weeks for my daughters’ favorite teacher at PALY- Paul Kandell . I think I can handle journalism with my newfound crystalline intelligence better than I can math!

Do you find your own intelligence shifting from fluid to crystalline? And?

Jake

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