I’ve probably written about this before; if so, I apologize.  I happened to think about the incident when I awoke this morning.  I slept well, which is a blessing as I approach the last year of my sixties.  Most of us my age and older don’t sleep as well as we’d like: through the night without awakening, and awakening refreshed.  It’s a good night if I’m up once, fall back asleep, and awaken on my own, without one of the cats standing on me, yowling (that was this morning, at 4:47), or barfing up a hair ball.

Anyway, for some reason, a dream maybe, I awoke thinking of one day when I volunteered  in the calculus class at Sabino High School in Tucson.  I occasionally helped out there, if for some reason the college algebra or geometry classes where I usually worked didn’t require my presence.  I liked Dave, who taught calculus; indeed, years ago, when email was new, Dave was the only faculty member with one, which is how I contacted him and began to help.

Anyway, that particular day he began by discussing the behavior of the function y=x^2, a parabola, near where x=1 and y=1.  Specifically, he was looking at the rate of change or the slope of the function as it approached the point.


What we learn by the slope is how fast the curve is changing at that point.  People get this concept wrong all the time.  If we hear the rate of growth of population is slowing, some people think the actual number is going down.  It isn’t.  It is still rising, not just as fast.  This is extremely important to know, and worth repeating: If the rate of increase is slowing, the actual number is still increasing, not falling.

Dave started by showing the slope where x=0.99 and y=0.99*0.99, or 0.9801.  The slope then was (1-0.9801)/(1-0.99), 1.99/0.01, or 1.99.  No problem. Then he let x=0.999 and, with a calculator, squared it, 0.999^2, which was 0.998001. The slope was now 0.001999/0.001, or 1.999.  He continued, saying that as x got closer and closer to 1, y would get closer and closer to 2, and the limit; that is, if we could take take x as 0.99999999— to an infinite number of 9s—and here he paused….


Notice the pattern for y—first is .81 then .9801, then .998001,then .99980001.

“You’d need a big calculator to calculate y, but the slope would be 2 at the limit.”  he stated.

I sort of blurted out without thinking, “You don’t need a calculator to get y.”  The words just appeared, I swear.  Everyone in the class turned towards me.

Dave looked at me, held the marker out in one hand, and said, “come up and write it down.”   He wasn’t at all angry.  We had known each other for several years at this point. As I walked up to the board, I asked, “How many 9s are there in your number?” I have bad astigmatism.

“Eight.”  So, I am multiplying 0.99999999 by 0.99999999.

Without a calculator.

“It is one less 9, followed by an 8, followed by the same number of 0s as you have nines, followed by 1.”  That would be 0.9999999800000001.  Take that, Texas Instruments.

I then turned to the class.  “Last year, I was here when this problem was discussed. You”—-I pointed to Dave—-“said that there was probably a pattern, and you were absolutely right.  I found it in a few minutes. The pattern is one fewer nine, an 8, the same number of 0s, and a one,” repeating myself.

Dave is good: he knows what he has taught, what a student should know, makes the student think and find answers to his questions, because they have all the information available.

I could have added that Dave, as a good teacher, didn’t take my coming to the board as showing him up; indeed, he knew that I was modeling exactly the behavior he wanted in his students.

I’ve been down that road before.  A dozen years earlier, I was in grad school getting my Masters in Statistics.  I had several professors.  One was absolutely brilliant, able to teach an entire difficult upper level graduate course in linear models without consulting notes.  There may have been one time where she made an error that another student caught, but that was it.  She was brilliant.  I’m sure I mentioned that in my post-course evaluation.

What I didn’t mention, because there were only 7 of us in the class, and I knew I would be identified, is that had she more patience with those students like me (it was the only B I got in grad school, and I worked hard to get it) who were not as brilliant, she would have been a life-changing teacher, the top of the heap.  The best teachers have patience with those who don’t have their skills.  Occasionally, I approach that.  Dave was there. So was my advisor.

My advisor didn’t need notes when he taught, either.  But he had patience with me, and that mattered a lot.  He got me out of New Mexico in 2 years, which I deeply appreciated. I haven’t seen him in about 15 years, but when I emailed him asking if he could help a friend of a friend–a free favor– he replied immediately.   He knew—damn it, I was pissed so many times when he did it, but he was dead right to do so—when I had exactly enough information to find the answer of a problem I asked him about.  He either knew the answer outright, or knew how to get it, but he was not going to tell me, but rather would give me what I needed to know—and not one hint more— to solve it.  I would then struggle for hours in some instances before having an epiphany among the papers strewn on my desk, on the floor, some crumpled and near a wall where I had thrown them in anger.

By doing that, my advisor forced me to use the new tools I had learned, to make mistakes, to figure everything out, and learn that way.  It was painful, but it was learning.  It was education, and it worked.

I’ve never gotten to my advisor’s or Dave’s level: substitutes don’t have a close relationship in one meeting, and in my brief for-profit so-called university teaching experience, students wanted everything handed to them.  But when I tutor today in the advanced math room at the community college, I occasionally encounter material far beyond what I know.  Sometimes, I try to help anyway.  And as I go through the problem with the student, asking him or her at each step how they got there, there is often a pause.  The student suddenly says, “Oh, I see what I did wrong. Oh wow, I can’t believe I did that.  Thanks.”  And walks away.  I’m still wondering what the answer was.

It just dawned on me that maybe my advisor sometimes didn’t know the answer, either, at least when I asked the question.  But he knew me well enough to know that I was capable of finding it.

Oh wow.  I can’t believe I did that. Thanks.

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