A month into school, and I’m exhausted! I kept my
stay-up-late summer schedule but get up early for school. Luckily, math classes
have been pretty fun so far, so I’m still smiling.

I had a great class a few days ago.
Before the students got into the classroom, I rearranged the desks into islands
and then, as they came through the door, asked them all to sit in a new spot. It
took only a minute, but the combination of a new configuration with new people
seemed to create a tantalizing air of excitement. Perfect!

The sheets I had worked on the
evening before (up late, again) asked straightforward but abstract questions. One
sheet, for instance, asked students to “explore” the limits of sin(*x*)/*x*
and sin(1/*x*) as *x* approaches zero. I asked for *compelling evidence* that the limit
does or does not exist. I like that word *compelling*.
It’s great for starting discussion, inviting different perspectives, and
weighing evidence, all Common Core ideals for mathematical practice.

Students found the visual evidence
pretty compelling. They showed me successive calculator-screen graphs and also
suggested looking numerically as *x* →
0 in a table.

I see the question of what
constitutes *compelling* as an emergent
property of discussion. Collecting evidence is pretty straightforward, making
access for students pretty easy. There is no “perfect” answer, but there are
certainly better and worse arguments, and students are pretty quick to sort
through them.

My favorite part of a class like
this is simply listening. Monitoring several conversations at once is
relatively easy, and I can move from island to island and compliment a neat
argument or help a group that’s struggling with consensus.

A few students asked me to look at
the differences in behavior between the graphs of *y* = sin(*x*)/*x* and *y* = sin(1/*x*). They wanted
to say that their graph of the latter (see **fig.
1**) looked as if it were tending toward zero, but their numerical analysis
would not support that statement.

**fig. 1**

We switched to a different graphing technology—desmos.com—on
my Chromebookä (see **fig. 2**), and the difference became clear: There is no sense of
convergence, and, in fact, the limit does not exist.

fig. 2
I heard rich but very different
discussions around the question from another sheet, “Is *y* = 2.4*x* – 1.6 tangent to
*y* = *x*^{3} – 2*x* +
1.95?” It’s hard to use a graphing calculator to tease out the behavior, but students
came up with ingenious solutions. One group set up the equality 2.4*x* – 1.6 = *x*^{3} – 2*x* + 1.95
but could not solve the cubic. They turned to Wolfram Alpha (www.wolframalpha.com), using an app on their phones. They used the results to
argue convincingly that a point of tangency should result in a root with
multiplicity two. Their equation clearly showed three solutions (see **fig. 3**); therefore, the line could not
represent a tangent.

**fig. 3**
I had not expected this line of attack, but the students made
a great argument. They were just as correct as groups that showed that the
derivative of the cubic was similar to but
clearly different from the slope of the line at each point of
intersection.

I really try to keep changing the
dynamics of my classes. Students will come up with surprising connections and
interesting arguments if I just make sure that I’m ready to listen. They’re
still arguing about evidence in class many days later, and I’m still smiling
and enjoying school!

Best wishes for the rest of the school year.

Greg Stephens, stephensg@hohschools.org,
is a high school mathematics teacher, department chair, and instructional
leader for the Hastings on Hudson School District in New York. He just
rotated off the *Mathematics* *Teacher* Editorial Panel but is keeping
busy in a doctoral program at Fordham University in New York City. At the
moment, his thesis topic is the impact of digital literacy on the high school
mathematics classroom, but the hardest thing of all is picking just one topic
to focus on!