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My Favorite Theorem

It was an honor to appear on the latest episode of My Favorite Theorem, the podcast hosted by mathematicians Evelyn Lamb and Kevin Knudson.

Evelyn and Kevin invite mathematicians to talk about their favorite theorem, and I chose Varignon’s theorem: I love sharing and exploring this theorem with students because it’s so each to start playing around with and it constantly defies expectations and intuitions!

To find out more, you can listen to the podcast at Evelyn’s Scientific American blog or download it from iTunes. You can also find a full transcript of our conversation at Kevin’s website.

I had such a blast talking about mathematics and teaching! Many thanks to Evelyn and Kevin for having me, and for putting on such an excellent podcast. I’ve been introduced to a lot of great people and math through My Favorite Theorem. I highly recommend it, and you can catch up on all the episodes here.

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Regents Recap — January 2018: Isn’t this Algebra?

What constitutes an algebraic solution? Let’s find out.

Here’s number 37 from the January, 2018 Common Core Algebra Regents exam. Only the final part of this six-point question is presented: Determine algebraically the number of cats and the number of dogs Bea initially had in her pet shop.

The key phrase here, apparently, is determine algebraically. If you don’t determine the answer algebraically, you can’t receive full credit. Here’s an example from the official Model Response Set in which the student loses a point for using “a method other than algebraic to solve the problem”.

In this solution, the student manipulates multiple equations, proportions, and equivalencies, determines the values of the variables that made the equations simultaneously true, and then applies substitution to verify their correct answer. That sure sounds like an algebraic solution to me.

Apparently it wasn’t the algebraic solution the test makers were looking for. But so what? We should be rewarding students for applying tools and techniques flexibly, not penalizing them for failing to adhere to a narrow, and secret, definition of what mathematics is.

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Regents Recap — January 2018: How Do You Explain That 2 + 3 = 5?

This has quickly become my new least-favorite kind of Regents exam question. (This is number 32 from the January, 2018 Algebra 2 Regents exam.)

What can you say here, really? They’re equal because they’re the same number. Here’s a solid mathematical explanation. Right?

Wrong.

According to those who write the scoring guidelines for these exams, this is a justification, not an explanation. Because students were asked to explain, not justify, this earns only half credit.

This is absurd. First of all, this is a perfectly reasonable explanation of why these two numbers are equal. This logical string of equalities explains it all. This clear mathematical argument demonstrates what it means to raise something to the power 3/4.

Second, whatever it is that differentiates an “explanation” from a “justification” in the minds of Regents exams writers, it’s never been made clear to test-takers or the teachers who prepare them. A working theory among some teachers is that “explain” just means “use words”. Setting aside how ridiculous this is, if this is the standard to meet, students and teachers need to be aware of it. It needs to be clearly communicated in testing and curricular materials. It isn’t.

Third, take a look at what the the test-makers consider a “complete and correct response”.

In this full-credit response, the student demonstrates a shaky mathematical understanding of the situation (why are they using logarithms?) and writes a statement (“81 with four roots gives you 3”) that, while on the right track, is in need of substantial mathematical refinement. Declaring this to be a superior response to the valid mathematical argument above is an embarrassment.

These tests are at their worst when they encourage and propagate poor mathematical behavior. We deserve more from our high-stakes exams.

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02/18/2018 — Happy Permutation Day!

Today we celebrate a Permutation Day! I call days like today permutation days because the digits of the day and the month can be rearranged to form the year.

We can also consider today a Transposition Day, as we need only a single transposition (an exchange of two numbers) to turn the year into the day and date.

Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!

Regents Recap — January 2018: Is it Better to Justify or Explain?

On question 32 of the January, 2018 Common Core Algebra 1 Regents exam, students were asked to explain why a quadratic whose graph is given might have a particular set of factors. Here are two sample student responses from the state-produced Model Response Set.

On the left, the student says “Yes”, sets each factor to 0 and solves, and produces the roots x = -2 and x = 3. On the right, the student says “Yes, because the x-intercepts are (-2,0) and (3,0).”

One of these responses received full credit, the other half credit. I posted this to Twitter and invited people to guess.


According to the official scoring guide, the response on the right earned full credit: it is a “complete and correct response”. The response on the left earned half credit, because the student “gave a justification, not an explanation.”

It seemed as though the majority of respondents on Twitter favored the response on the left; a few even specifically said it offered a better “explanation” than the full-credit response. Many did choose the response on the right, especially those familiar with how New York’s Regents exams are scored.

To me, both answers are unsatisfying. The full-credit response offers an “explanation” but is devoid of justification: the student doesn’t make the connection between the x-intercepts and the roots. The half-credit response derives the roots algebraically, but fails to explicitly connect the roots to the intercepts. It’s hard for me to accept that one of these responses is substantially better than the other: both responses expect the reader to fill in an equally important gap.

It’s also hard for me to accept what counts as “explanation” here. Several teachers familiar with New York’s Regents exams commented that, in this context, “explain” simply means use words. And we’ve seen example after example of ridiculous “explanations” on these exams. It sends the wrong message to students and teachers about what constitutes mathematics, and since the message is transmitted via high-stakes exams, it can’t be ignored.

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