Keep Two Thoughts

Personal essays

Understanding - Essay from Newsletter 55

There are the things we do every day and there are the things that we are.

Find It

Before the pandemic, if you’d asked me what I do, I would most likely have told you I write books and deliver training on writing iOS apps.

But that wasn’t really true.

Mostly what I did was meet friends for coffee, hang out with Annabelle the wonder dog, and spend way too much time on the couch scrolling the internet.

During those moments I trusted that my mind was working in the background so that when I sat at my keyboard my books and talks would have taken shape.

Different people write different sorts of books and give different sorts of talks.

Mine try to take you on a journey.

I try to show you connections along the way and sometimes I delay details until I think you need them.


Whether or not you’ve taken Calculus, stay with me a moment.

In your mind or on a piece of paper, draw a smooth curve and pick a point on that curve. I’m going to call that point P.

Pick any other point on the curve and draw a straight line from P to that other point. That line has a slope. You might remember dividing the change of y by the change of x or the rise over run to get that slope.

Now choose a point on the curve closer to P. Honestly, I don’t care if the new point is to the right of P or to the left of P, it just has to be closer to P in the x direction. Calculate the slope of the line connecting that point to P.

Keep choosing points that get closer and closer to P and take a look at the pattern of the slopes. As the x values of the points get closer and closer to the x value of P, do the slopes get closer and closer to some number?

If they do, this is the derivative of the function describing the curve at the point P.

And because this is the truth, this is how we teach derivatives in calculus.

But at least a third of the class knows that there’s a shortcut.

They know that after you make them use some long version with limits, there’s going to be a quick version that allows them to find the derivative of the curve at the point.

And they tell their classmates this.

And soon the class is doing what you make them do, but they know an easier way is coming and that you’re just being a jerk for not showing it to them.

Flipping the Order

And so I used to flip the order.

I’d show the entire class the short cut.

And then I’d show them cases for which the shortcut may not work.

And then I’d show them the long way.

They’d still get to the truth but now the truth is the punchline not an obstacle on the way to the punchline.

Now limits become a tool we can come back to to discuss many topics.

I saw an interesting talk the other day at the UK’s 150th conference of the Mathematical Association.

Colin Foster asked if we can teach understanding.

He looked at Pythagoras and said that we can certainly teach students to be successful at using Pythagoras but can we teach them to understand it. If so, how?

He had many examples which seemed to help us understand the theorem - but really it enriched the experience of those of us who already understand it. None of them conveyed understanding.

It brought me back to this example.

Is our world simple

Take another look at our curve and the point P and imagine a vertical line through the curve at P.

Remove P from the curve and place it anywhere else on that vertical line.

There are infinitely many places you can move P – and, for those of you who know what I mean by this, the big infinity not the small infinity.

So for every smooth, nicely behave graph like our first curve, there are way more curves with jumps and holes.

And yet, we spend most of our time in school looking at the nicely behaved graphs which are very rare.

We ask students to factor polynomials with integer coefficients that can easily be factored when there are way more polynomials like this that can’t be factored.

It’s like the old joke of the man coming upon his friend at night looking for his keys under a street light. The man had lost his keys over there somewhere, but it’s easier to look where it’s light.

Foster’s examples might not explain Pythagoras but they helped anchor the memory.

These demonstrations are street lights that get the students close to where they should look. Maybe in the morning they’ll find what they’re looking for.

It was always thus

When I was a young math teacher, a college professor of mine had funds to bring me to a retreat for math and science teachers.

There were moments that stick with me nearly forty years later.

One memory is of Joan Countryman who was head of the math department at Germantown Friends in Philadelphia.

She held her TI Calculator up high and said, “this calculator could get a B+ in our Algebra II classes. Shouldn’t we change the way we teach?”

That was in the early 1980s.

I think about this a lot.

Now that I’m beginning the edits on my bread baking book, I’m fighting the urge to write books on statistics and calculus. There are plenty of books on math. Then again - there are plenty of books on baking bread.

Foster and Countryman didn’t ask their questions in sniping, “tear it all down” ways. They put considerable thought into what we can do better and they inspired those of us that heard them.

I’ve been flying high since I heard Foster’s challenge and it took me all the way back to Countryman’s forward looking challenge.


I’m in a transition right now as my business has totally changed.

In the past decade I’ve moved from writing books and articles for publisher to delivering them directly to you.

In the past year, I’ve not been able to travel to teach in person. Much of my strength as an instructor comes from my personality in the room. Those opportunities have disappeared.

We have to look beyond what it is we do.

That can change.

We need to look at what we are.

I am a teacher and a storyteller.

Where and what I teach changes in the face of a changing world, just as the math I teach had to change in the face of Countryman’s challenge.

Our story should end there but I have a small coda.

I recently searched online for information about Joan Countryman. She had been the first African American to graduate from Germantown Friends and then had returned as a member of their faculty. I was looking for her because Maggie has just been hired to teach Latin there in the fall.

In Maggie’s essay on her teaching philosophy, she is holds up her own version of Countryman’s calculator in her own field and wonders how teaching Classics must change in the context of the modern world while still respecting the traditions and the canon.

Essay from Dim Sum Thinking Newsletter 55. Read the rest of the Newsletter or subscribe

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