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Archive of posts filed under the Technology category.

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.

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An Introduction to Desmos

An Introduction to Desmos

I’ve presented on Desmos many times to teachers, administrators, and students.  So I was excited to bring that experience to the Math for America community through my workshop, An Introduction to Desmos, at the MfA offices in New York City.

Nearly 50 MfA teachers attended, and it was a very active and engaged bunch.  Most attendees were familiar with Desmos, and many were using it in their classrooms.  But I got the sense that everyone’s eyes were opened a bit wider to the power and possibility of this mathematical technology.

Participants began by working through a document I’ve put together that functions as a guided tour of Desmos.  I’ve used this document many times with both teachers and students:  it provides a quick overview of the power and breadth of the functionality of Desmos, and it allows me to circulate and answer, and ask, questions.  [You can find the document here: Introduction to Desmos]

The second part of the workshop had participants working on a series of content-specific challenges.  The goal was to use get teachers using Desmos to build mathematical objects.  For example, some teachers worked through these parabola challenges:

           Construct an arbitrary parabola
                  (a) with vertex (2,3)
                  (b) with vertex (x_1, y_1)
                  (c) with roots 2 and 3
                  (d) with roots r_1 and r_2 
                  (f) with focus (a,b) and directrix y = c

There were similarly structured challenges for LinesTransformations, Regions, and several other areas.  Participants could choose what to work on based on what they taught or what they were interested in.

As I circulated the room, I answered lots of good questions.  And I listened in as teachers talked about how they were already using Desmos in their classrooms.  I was especially gratified to hear several teachers tell me that they learned something in the workshop that would have made yesterday’s lesson better.  It felt good to deliver immediate impact to my colleagues, and I’m excited to know that many teachers have already integrated Desmos into their instruction.

Throughout the workshop I emphasized that the real power of Desmos is not as a presentation tool, but as a creative tool.  I often describe Desmos as a mathematical makerspace:  a place where we can design and build using the tools and techniques of mathematics.  As teachers, it’s tempting to see Desmos primarily as a tool for demonstrating mathematics to our students, but it’s true power lies in how it can help us all, teachers and students alike, make mathematics.

You can find more of my work with Desmos here.  And you can see pictures of the workshop here.


Student Desmos Projects

Desmos, the free, browser-based graphing utility, has quickly become an indispensable tool in the mathematics classroom.  It provides easy, intuitive access to graphs of functions and relations, and creates unique opportunities to understand mathematical relationships dynamically.

But to me, its greatest virtue may be that Desmos provides opportunities to use mathematics to create.  I like to think of Desmos as a mathematical makerspace, where the tools at our disposal are exactly the tools of mathematics.

To that end, when I introduce students to Desmos, we always work toward the creation of something mathematical.  Below are some beautiful examples of student work from our latest round of Desmos projects.


Varignon’s Theorem Vector Projections
Angle Bisector Theorem Three Lines Intersecting

You can find more of my work with Desmos here.

3D Printed Surfaces

I’ve been enjoying getting familiar with our new 3D printer.  I’m not sure these parametric surfaces were the best pieces to start out with, but I have learned a lot!

3d Surface Collage

From left to right, we have a figure eight torus, a Fresnel surface, and a cyclide.  Not pictured:  the many, many failures.


When Technology Fails

when technology failsAt Math for America’s most recent Master Teachers on Teaching event, I presented “When Technology Fails”, a short talk about how my personal and professional experiences have shaped the way I view and teach technology.

The failure of technology has been a consistent part of my personal and professional computing experience.  These failures have served as excellent learning opportunities, and perhaps more importantly, they have instilled in me a healthy distrust of technology.

As a teacher, I find students far too trusting of technology.  Often, they accept what their calculators or computers tell them unthinkingly.  In my talk, I discuss how we can make students conscious of the shortcomings of technology in ways that create meaningful learning opportunities.  And hopefully, by confronting the failures of technology head on, students will develop a healthier attitude about what technology can, and can’t, do.

A video of “When Technology Fails” can be viewed here.  And a talk I gave at a previous MT^2 event, “g = 4, and Other Lies the Test Told Me”, can be seen here.