old-timer (aporia) checking back

Hi guys, I used to post under the name aporia. That account seems to have been deleted for inactivity over the past year, so now I’ve got this one.

Life has been busy… I’m publishing research papers in solar astrophysics. I teach calculus to freshman college students, and they’re doing pretty well. I got married to the girl I posted about last year. I’m a little embarrassed now about how worried I was about physical attraction, and how wonderful things have become since I made the simple choice to look past the shallow “imperfections” and follow my heart. Being married seems to have made me a little less standoff-ish, more relatable, and more sensitive to appearances… in short, more normal. I regard it as a very positive development, something that would probably shock myself of two years ago.

In terms of my philosophical development… I think the focus of my reflections has shifted from “big questions” to the particular circumstances of my life, my family, and my community. I wonder about how to be a good husband, a good citizen, a good teacher, a good researcher, when life is so very busy and short. It seems to me that genuine progress in humanity’s condition comes from each of us doing our own small part, and doing it well, with our full attention and engagement – not so much from vast reflections on the nature of things, as fun as those can be.

In no particular order, shout-out to Uccisore, xanderman, Ed3, faust, Xunzian, felix, tentative, Tab, Old Gobbo, and many others whom I enjoyed interacting with in past years. Hope all your lives are full of curiosity, love, engagement, and meaning.

…and then there’s the people I missed before I even left, and who are probably still gone: most of all, Dunamis (where IS that guy???) and gamer. At a crucial time in my life, Dunamis broke my tunnel-vision of the world and introduced me to a wide world of philosophical possibilities – possibilities I had assumed utterly untenable on account of mere lack of imagination. And I still remember the honky-tonk piano songs gamer wrote about ILP.

I probably won’t be back for more than a few days, but I thought it’d be nice to check in with you guys. Anything new going on in your lives or thinking?

Hi Hetro,

It is nice to here from you, and I’m glad that things are going well. Young love and teaching math, it seems to me like you have died and gone to heaven!

I’m a little curious about your teaching style. For example, are you weighting your class toward a theoretical Real Analysis basis or a more practical approach favoring the physicists and engineers?

The very little that I taught was pretty much right down the middle, but I regret that slightly.

Hopefully you will post now and again!

Best of luck.

Ed

My Calc I course is pretty unusual - it’s almost certainly very different from how you were taught. Its content is dictated by a bureaucracy and I don’t have much say in the content, but I do control how it’s taught. The objectives are first to get the basic concepts of calculus intuitively, then learn the rules of computing derivatives, then apply the conceptual and computational knowledge to solve practical problems (e.g. optimization, related rates). The course definitely favors application over theory, and I would not consider it adequate preparation for a physicist or other heavy-duty math user (they take honors sequences which are above my pay grade, for now). But it does give the students a pretty good idea of how to think about calculus and apply it to real-world modeling problems.

The major strength of the course, in my opinion, is an emphasis on presenting material from several points of view (verbal, graphical, numeric, and algebraic). The older presentation of math tended to be too centered on algebraic and symbolic manipulation, although the best texts have always stressed a balance of these four approaches. Over the last couple years I’ve spent teaching this course, I’ve discovered that it’s much easier to understand certain topics from one or two of these points of view first, then bootstrap that into an understanding from the other points of view. For example, derivatives are easiest to understand from a graphical perspective at first; then some numerical work clarifies how limits are calculated; a verbal interpretation of the derivative in terms of rate of change helps you to use it in modeling the world; and an algebraic understanding allows you to find all of those beautiful and powerful derivative rules, which make it possible to compute clean analytical answers to difficult questions.

How was your calculus class back in the day? I bet you learned l’Hopital’s rule and how to compute limits algebraically. These kids don’t, which is a bit sad but may be a better pedagogical choice in the long run.

Nice to have you back… even if it is only for a few days - at least you have a good excuse why you don’t have the time for ILP… being busy with such high-profile projects. Try and come back from time-to-time :slight_smile:

It all won’t last long

Yeah, the good and the bad are all temporary, it seems. I feel pretty good about having managed to make a couple months of my life very satisfied and happy, thanks in no small part to my lovely wife… we’ll see how long we can keep it that way! :slight_smile:

Hi Hetro,

The description of your approach to Calculus seems pretty reasonable for the students you have.

“Back in the day" for me, consists of three phases:

  1. The Institute of Technology course work at the University of Minnesota

  2. Teaching accelerated high school students who had past all their course work and needed/wanted additional mathematics

  3. Writing a mathematics textbook, which while it encompasses 132 typed pages, is still at least ½ unfinished.

  4. In my first experience we used a textbook entitled “Calculus with Analytic Geometry” by Angus Taylor

Basically, after some fundamental review, the author talks about a physics problem of finding the rate of change in distance with the corresponding rate of change in time at a point. He never does define what a limit is and leaves it relatively intuitive and largely algebraic until some where around page 450. (You would have won your bet).

  1. When I taught calculus, I introduced the idea of a slope of a curve at a point by drawing line segments on what appeared* to be an exponential curve. And while I gave a token delta epsilon explanation of the definition of limit, most of the rest of the course dealt with the rules for finding derivatives of compound and composite functions. Near the middle of the year we did Integration and near the end I introduced a little Topology (though the students were clearly suffering from senioritis by then).

  2. In my own textbook, which I entitled “Mathematics A Socratic/Historic Approach”, after some fundamental background, I again started by trying to find the slope of a curve at a given point. However, this time I went into much more detail on the nature of limits and their sometimes problematic existence. The derivatives of compound and composite functions were done rigorously using the definitions of limits and the historical and logical importance of limits was emphasized.

I like your idea of using a two prong approach and later adding one or two different approaches.

Thanks Ed

  • The problem with the graphical approach is that it is basically reasoning by analogy and at least theoretically you can bring along faulty ideas. Continuous functions with this graph approach look to be differentiable, but the Weierstrass function is continuous everywhere and differentiable nowhere. The Bourbaki hated pictures for this reason (I believe). Still I think that it is a good learning tool.

P.S. I wish you all the best.

Nice to see you pop in Aporia; wonder what happened to Uccisore?