Michael Pershan’s Exponential Post part 1 (of 3 maybe)

Last week Michael Pershan wrote an interesting article about exponentials that has had me thinking for days:


Fortunately I’ve got about 90 minutes on the bike back and forth to work each day, so that leaves plenty of time to think through talking about math with my kids.  This time around I needed all of that time and more, and I hope that writing about introducing exponential to kids will clarify some of my own thoughts which are presently pretty fuzzy.

But I want to start pre-exponentials, though.  My hope is that thinking through my approach to teaching some basic ideas of arithmetic will lead to some clarity on exponentials.  Thank goodness that hope is a strategy.

I’m sure that my own views on teaching arithmetic to kids lean towards the formal side much more so than most people’s views do.  Here, for example, is my first talk about division with my younger son where, following Art of Problem Solving’s Prealgebra book, we define division by  X as multiplication by 1 / X:

Even what what we talked through today (May 7th, 2014) – trying to connect decimal division and fraction division – I’m sure would be regarded on the formal end of the spectrum:

Though I surely lean a little formal, that approach isn’t all consuming.  In fact, studying Fawn Nguyen’s teaching / examples has had a tremendous influence to how I approach teaching math with my kids.  In addition to the more formal examples from above, we’ve looked at fraction division with snap cubes:

and place value by building a “binary adding machine” out of duplo blocks:

and, of course, some  Fawn Nguyen-like pattern problems (this specific one was actually from Kate Nowak):

In an attempt to gather my thoughts about exponentials I spent a lot of time thinking about why my instincts were to teach arithmetic so formally.   Ultimately I think the answer, which I admit is somewhat unsatisfying, is that with arithmetic the formality is reasonably accessible.   It isn’t super difficult to formally introduce division as multiplication by a reciprocal, or introduce subtraction as adding a negative, or even introduce imaginary numbers.

To be clear, though, I’m pretty happy with the decision to introduce arithmetic this way.  There were obviously many false starts, but I think that (i) both of the boys have developed a pretty good number sense over time, and (ii) later introducing several concepts formally and then connecting to more tangible object that they were familiar with (duplo blogs, or snap cubes)  seems to have been a reasonably good way to build on their existing mathematical knowledge.  That’s more or less all I was ever aiming for.

However, as I mentioned above, the decision to take this approach was made because I think the formality in arithmetic can be made accessible to kids.    But with exponentials the formality is not accessible.   Not even close. Almost any formal definition of e^x would be far out of reach for young kids (though I give one a try).  Even more trouble in thinking about how to introduce the topic comes from the fact that the common uses of exponentials – for me those are bond / mortgage math, the Poisson distribution, and Black-Scholes type formulas – are also well out of reach.    There’s no analogy to sharing cookies here, at least none that I know of, so the combination of the difficult formality and the difficult common examples led me to approach teaching exponentials very differently than I taught arithmetic.

I’m not even sure I realized that I’d made a different choice until I started thinking about Pershan’s blog post.    I’ll write about how I approached exponentials in the part 2.



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