Exploit Parallelism

I was reading the old VerizonMath meme recently and began thinking about it in terms of a teaching moment. George Vaccaro was clearly trying very hard to teach the Verizon employees a little something about math, and they just weren’t getting it. Part of the problem is surely, as everyone points out, the lack of math common sense of the Verizon employees involved; a trait all too common in American today.

But from another perspective, George himself contributed to the problem. If you read the (unverified) transcript of the exchange, you’ll notice that there several points where George tries, with good intentions, to lead the employees through a line of reasoning that can lead to only one possible conclusion: that 2/1000 of a dollar is not the same currency value as 2/1000 of a cent.

George: Yes, do you you recognize there’s a difference between those 2 numbers [“point zero zero two dollars and point zero zero two cents”]?

[pause]

M: No.

G: Okay, is there a difference between 2 dollars and 2 cents?

M: Well, yeah, sir..

G: Well okay, is it.. is there a difference between .002 dollars and .002 cents?

M: .002 dollars and .002 cents.

G: Yes, is there a difference between..

M: Sir, sir, they’re.. they’re both the same if you, if you look at ’em on paper-wise

In that moment, George and the Verizon employee (Mike or just “M”) had a meeting of the minds. They probably shared a common mental model of the value of currency. I say “probably” because we can’t know, from the outside, what George and Mike were actually thinking in that moment nor can they, post hoc, likely reconstruct what they were actually thinking in that moment. Still, George seems to have had Mike at a point of common understanding:

G: Okay, is there a difference between 2 dollars and 2 cents?

M: Well, yeah, sir..

But then George lets the moment slip away.

G: No.. they’re not, actually. It.. is .5 dollars the same as .5 cents?

M: Is .5 dollars..?

G: Is half a dollar..

M: That would be.. That would be 50 cents.

G: A half a dollar.. is it the same as a half of a cent?

M: No.

By changing the values involved, George confused Mike… You can see the moment it happened: when Mike struggled to think about the different quantities being given. As a learner in that moment, Mike was probably trying to glean from the examples being presented just what George’s point was. He was trying to find order in the pieces of the puzzle George had so far shared with him but, as many learners do, Mike ended up confused, instead of enlightened.

In my view, what George should have done at that point was to exploit the parallelism of not only the ideas, but also the precise language that he and Mike shared. What would that look like? Well, the ideal exchange would have been:

G (fictional): Okay, is there a difference between two tenths of a dollar and two tenths of a cent?

G (fictional): Okay, then, is there a difference between two hundredths of a dollar and two hundredths of a cent?

G (fictional): Okay, well, one way to represent “two hundredths of a dollar” is to say “zero point zero zero two dollars” and to represent “two hundredths of a cent” we can say “zero point zero zero two cents”. Now, you’ve just agreed that 0.002 dollars is different from 0.002 cents, correct?

I believe, had George done that, that Mike would have had a better chance of seeing the problem with his reasoning. Mike would have been able to see clearly that 0.002 dollars and 0.002 cents are different currency values.

There are several points in the transcript that reflect this issue: when Mike gets confused, George injects a different example in an attempt to build some critical mass of evidence. But Mike isn’t able to organize the evidence and, so, the valuable distinction that George understands and is trying to convey to Mike isn’t understood by Mike.

The problem, of course, is that “cents” itself hides a multiplier. “Cents” means 1/100 of a dollar, in this context. So, what George is really arguing is that

0.002 * 1 dollar does not equal 0.002 * (1/100) dollar

But, of course, I’m now guilty of the same massing of evidence.

I see in public policy debates, as well as educational settings, that the desire to present a coherent collection of evidence is instead interpreted as disjoint examples.

My solution is a scientific one: hold all things constant, except one. In other words, exploit the parallelism inherent in the issue being considered.

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