Teaching

Regents Recap — June, 2016: Simplest Form

“Simplest form” is a dangerous phrase in math class. Whether a form of an expression is simple or not depends on context. For example, while $\frac{3}{8}$ and $\frac{21}{56}$ are representations of the same number, the first fraction is likely to be seen as simpler than the second. But if the goal were, say, to determine if the number was greater than $\frac{17}{56}$ , then the expression on the right might be considered simpler.

Despite the wide and varied uses of the phrase “simplest form”, I have never heard it used in the context of complex numbers. So I was surprised by this Common Core Algebra 2 Regents exam question.

I don’t know what the author of this question means here by “simplest form”. I asked around, and someone suggested that the natural interpretation of “simplest form” here is a + bi. That seems reasonable, but since none of the answers are in a + bi form, the author of this question could not have meant that. [It is also worth noting the implicit assumption here that y is a real number, an issue that has come up before on these exams].

What’s most bothersome about this imprecise use of language is that it is completely irrelevant to this question. Whatever “simplest form” means here, it is of no consequence: there is no answer choice which is otherwise correct but in some improper form.

The question should simply ask which expression is equivalent to the given expression. The use of “simplest form” here not only obfuscates the mathematics of the problem, but models imprecise use of mathematical terminology. We should expect our high stakes exams to do better.

Teaching

Empire State Excellence in Teaching Award

I was quite surprised at our school’s graduation ceremony this week when New York State Lieutenant Governor Kathy Hochul unexpectedly presented me with the Empire State Excellence in Teaching Award.

The award was created by Governor Cuomo to recognize outstanding K-12 teachers across New York. Teachers from all ten regions of the state will be honored, serving as teacher ambassadors for New York and receiving opportunities and support for continued professional growth.

It was especially meaningful for me to be recognized with our senior class at graduation, though I was definitely a bit stunned to find myself on stage in front of 5,000+ people at the Barclay’s Center! I appreciate the indulgence of all the students, families, faculty, and guests, and a special thanks to Lt. Gov. Hochul, who made the trip from Buffalo to Brooklyn to personally present the award.

Lt. Gov. Hochul’s announcement of the award is here, and a short news article about the presentation can be found here.

Appreciation Student Work Teaching

Math Haiku — Geometry Class, 2016

I personally enjoy writing, and as a math teacher I love getting my students writing about math.

One of my favorite writing assignments for students is math-themed haiku. The rigid constraints of haiku make it an easy exercise, it allows students to access and interact with mathematical ideas in a different and creative way, and the elegance and efficiency of the form evoke the character of mathematics itself.

Here are some selections from this year’s Geometry class. Enjoy!

A rhombtangle A rhombus and rectangle Also known as square	Surface area A polyhedra’s paint job An unfolded thing
Two, the same as two A number equals itself Reflexivity	No definition For a point, a line, a plane At the base of math
As some things, unknown The things we can’t do alone Mathematics lasts	Geometry’s weird Some things just seem to work out So we find out why

Art Geometry Resources Student Work Teaching Technology

Using Mathematics to Create — Geogebra

One of my guiding principles as a math teacher, as I articulate in this TEDx talk, is to provide students with tools and opportunities to create with mathematics. Few things are as aligned with that principle as well as Geogebra, the free, open-source, dynamic geometry environment.

I’ve integrated a lot of Geogebra work in my Geometry class this year. I use Geogebra assignments to assess basic geometric skills, to connect old ideas to new, and to explore geometry dynamically.

But much like geometry itself, once you master a few elementary rules in Geogebra, you can create amazing and beautiful works of mathematics.

Below is an example of some wonderful student work from this year. After an introduction to polygons, students were given two simple ideas for creating new objects from polygons: constructing diagonals and extending sides. I gave students some technical tips on how to color and polish their final products, and invited them to be creative. As usual, they did not disappoint.

Students, and teachers, need more opportunities to create with mathematics. We’re fortunate to have technologies like Geogebra that offer us those opportunities.

Teaching

Why Are We Listening to Andrew Hacker?

I wasn’t planning on attending the math education debate hosted by the Museum of Mathematics. I have read, and written, enough about Andrew Hacker and his arguments for ending compulsory mathematics education that I didn’t feel it necessary. But in the end, I decided to go. After all, there’s something inspiring about hundreds of people attending a public debate about mathematics!

As Andrew Hacker laid out his position, he shared his one visual aid with the audience:

Photo Credit: MoMath (link)

He said his argument boiled down to one question: “Why?” As in, “Why does every student in the country, regardless of interest, ambition, or ability, have to take a full sequence of advanced mathematics in school?” It’s not an unreasonable question.

But for me, the real “Why?” question is this: “Why are we listening to Andrew Hacker?” And this question inspired my essay, “When it Comes to Math Teaching, Let’s Listen to Math Teachers“, which I wrote for Math for America’s Teacher Voices blog. Here’s an excerpt:

Andrew Hacker isn’t an expert on mathematics. And he isn’t an expert on math teaching, either. He has every right to voice his complaints, some of which are worthy of consideration, but why has he been given such an enormous platform – high profile Op-Eds, interviews, lectures, a book deal – to address the public about how to “fix” math education?

The fact that Andrew Hacker has such an outsized and undeserved role in steering this conversation is itself one of our problems: we aren’t listening to the right people. If we are really interested in identifying and addressing the problems facing math education today, we should be listening to math teachers.

You can read the entire essay here.

I also live-tweeted the event, along with a few other attendees, using the hashtag #MoMathEdTalk. You can find the tweets here; several interesting conversations ensued.

And for more of my writing on Andrew Hacker, you can start here.

Regents Recap — June, 2016: Simplest Form

Empire State Excellence in Teaching Award

Math Haiku — Geometry Class, 2016

Using Mathematics to Create — Geogebra

Why Are We Listening to Andrew Hacker?

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