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How Many Circles Pass Through Two Given Points?

The impact of technology on education is often overstated.  However, some applications of technology are unequivocally transformative in mathematics teaching.

The question “How many circles pass through two given points?” is a wonderful prompt for a geometry class.  It’s simple, it provokes debate, it can be explored in a variety of ways, and it connects to many important geometric concepts.  And in the end, it requires some imagination on the part of the student to truly comprehend the answer.

And after all that classroom work, it is so powerful and satisfying to see something like this.

circles through two points


A simple demonstration that elegantly captures the essence of the problem, and leads to new compelling questions.  That shows students that mathematics is beautiful and inspiring.  And that takes just a few moments to put together in Geogebra.

And what’s truly transformative is how easy it is to get students using technology to create their own mathematics like this!  This is the real promise of technology.

Math Photo: Stealth Cone


This cone sits in front of the engine of the SR 71 Blackbird, a retired long-range reconnaissance aircraft.  Apparently this “spike” retracts into the plane at high speeds, to favorably alter the plane’s aerodynamic properties.

Regents Recap — January 2016: Reused Exams

I’ve looked at many New York State Math Regents exams over the years, and in that time I’ve definitely seen a few recycled problems appear on tests.  But I’ve never seen anything like what happened this past Regents cycle.

Most current high school students in New York are required to take the Common Core Algebra exam, which replaced the Integrated Algebra exam in 2014.  However, some students are still eligible to take the old Integrated Algebra exam, and so to accommodate those students, a special administration of the Integrated Algebra exam was offered this past February.

What was perhaps most “special” about the exam given in February, 2016 is that it was the exact same Integrated Algebra exam that was given in August, 2009.  The exact same, as in every single question on the February 2016 exam was identical to the corresponding question on the August, 2009 exam.

duplicate regents exam

It’s hard to believe that this was allowed to happen.  Yes, it takes a lot of effort to make a state exam, and this one likely doesn’t affect that many students.  But simply copying-and-pasting an old exam seems to undermine the state’s assertion that these high-stakes exams are critically important to education and should be taken very seriously by schools, teachers, students, and parents.

As of writing this, the state still has not made public the February, 2016 exam, which is quite unusual.  Perhaps they think it unnecessary, since they already published it 7 years ago.

To see for yourself, you can find copies of the exams posted at  here’s the February, 2016 exam, and here’s the August, 2009 exam.

You can find more of my critiques of New York State mathematics exams here.

Computer Science Education in NYC

CS in NYCI’ll be participating in a panel discussion on Computer Science education in New York City on Monday, April 18th.  The event will be hosted by Math for America at the Simons Foundation, and is open to the public.

There are many new initiatives promoting the teaching of computer science in K-12 classrooms here in NYC and across the country.  Like many ambitious educational programs, these initiatives often create more questions than answers.

For example, who will teach the newly proposed computer science courses?  Where will the technological infrastructure come from?  And perhaps, more fundamentally, what do we mean when we say computer science?

The purpose of the event is to provide a variety of perspectives on what computer science teaching currently looks like in New York City.  After the moderated panel discussion, there will be a number of informal conversations about issues in computer science education facilitated by MfA teachers.

For my part, I’ll be talking about the mathematical computing course I’ve been developing over the past few years, and how my personal and professional experiences working with technology shape the ways I think about computer science, and how to teach it.

You can learn more about the event, including how to register, here.

UPDATE:  A recap of the event has been posted at the Math for America blog here.

4/9/16 — Happy Square Day!

Happy Square Day!  It’s quite rare for the numbers of the date to all be perfect squares, and even rarer for them to be consecutive!

20160409 Square Day 3
April, 2016 has been a fun month for mathematical dates!  So far this month we’ve had 4/2/16 (an Exponent Day), 4/4/16, and 4/8/16 (a Geometric Mean Day, or alternately, a Powers of 2 Day).  And there are more fun mathematical dates to come!