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New York State Master Teacher Program

NYSMTPI’m happy to announce that I am now officially a member of the New York State Master Teacher Program.

The NYSMTP is designed to connect great math and science teachers from around New York State through networking, professional development, and professional service.  The program is inspired, in part, by the Math for America Master Teacher program in New York City, which I have been actively involved in for the past 9 years.

This past summer I was fortunate to attend a NYSMTP retreat in upstate New York, and I had a fantastic time.  I talked with teachers from all over the state, and learned a great deal about the many different, and similar, things going on across New York.  I also ran a workshop on using Twitter for professional development, which I think is a natural medium for connecting teachers in a program like this.

I’m looking forward to working more with great colleagues from across New York State!

 

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.

Math Photo: Curvilinear Coordinates

Curvilinear Coordinates

Looking through this system of parallel curves makes me think about the many different ways we can impose coordinate systems on spaces.  An ordered pair of coordinates specifies a unique location on this curved surface just as a pair (x,y) locates a point in the flat Cartesian plane.

This image also reminds me of the role of context in geometry.  From our perspective, this coordinate system looks curved, but if we lived on this surface, it would all seem perfectly flat!  Maybe our world looks really curved to someone standing outside it.

 

 

Expanding Cylinders

In class, and on Twitter, I posed a question that led to lots of great conversation.

There are many reasons I love this particular question.  It’s familiar, accessible, and usually counterintuitive.  And it bridges algebra and geometry in a very natural way.

A reasonable response is to argue that an increase in radius will be better.  The volume of a cylinder is V(r,h) = \pi r^2 h, so an increase in radius seems to have a squared effect on the volume, while the effect of an increase in height is only linear.

If you think about the grey cylinder shown below, this argument seems to make sense.  Increasing the height a little bit adds a small blue disk of volume to the top, but increasing the radius a little bit adds a large blue shell.  The additional volume of the shell clearly appears to be more than that of the disk.

Expanding Cylinders tallHowever, this argument is a lot less convincing if you start with a different cylinder.

Expanding Cylinders wideIt’s not obvious which additional volume here is larger, which suggests some further thinking is in order.  At this point, some multidimensional extreme-case thinking usually leads to an appropriate conclusion:  namely, that the answer depends on the dimensions of the cylinder.

This problem is my standard introduction to partial derivatives.  It creates great context for computing and comparing

V_r = 2\pi r h      and      V_h = \pi r^2

But its versatility is another reason I like this problem so much.  Geometry and Calculus students can both engage in this problem in a meaningful way, using the tools available to them to analyze the situation.  And it always results in great conversations!

 

 

 

Math Photo: Riemann Shadows

Riemannian Shadows 2

The shadows fall like approximating rectangles under a sine curve.  This brings to mind a basic approach in computing the area of curved regions:  approximating the curved region with a series of flat regions, whose areas are easy to compute.

Riemann Sum Sine blue

The angle of the sun makes the shadows more like approximating parallelograms, though.  So we’d probably need a change-of-coordinates to complete our calculation!