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Math Photo: Manhole Math Art

Manhole Math Art

Ever since I noticed this pothole pentagon, I’ve been looking down a lot more.

The lovely mathematical designs on this this manhole cover immediately put me in mind of the labyrinths of Robert Bosch.  And then later I realized this work resembled another piece from a Bridges artist:  Tim Locke’s Starburst.

 

Did No One Care About Seth Godin?

In his typically direct style, Seth Godin’s “Good at Math” purports to rebuke the common belief that if you’re not a math person then you’re destined to never be good at math.  This is indeed a destructive attitude, and one we should work to dispel.

Unfortunately, Godin’s piece takes an all too familiar turn.  If not genetics, Godin wonders, then what has prevented you from learning math?

If you’re not good at math, it’s not because of your genes. It’s because you haven’t had a math teacher who cared enough to teach you math.  They’ve probably been teaching you to memorize formulas and to be good at math tests instead.

To Seth Godin, the answer is simple:  bad teachers.  And not just incompetent bad, but uncaring bad.

This claim is ridiculous.

First, most teachers care quite a lot about what they do, and whom they serve.  Saying that students don’t learn because teachers don’t care is not only insulting, but it demonstrates a fundamental disconnect with the reality of who teachers are and what they do.

Second, there are many reasons why someone might not master math in school.  Math is hard.  Learning is hard.  Teaching is hard.  And even when teacher and student both care deeply, learning doesn’t always happen on schedule.

And if you want to criticize teachers for teaching students to be good at math tests, fine, but know that this is often exactly what teachers are told to do, directly or indirectly.  This can be completely consistent with a teacher caring about their work and their students.

Lastly, there’s no point in telling people not to blame their genes if you’re just going to tell them to blame something else that’s largely out of their control.  Blaming teachers won’t empower anyone to learn math; it just shifts the blame to a more convenient target.  If anything, this argument reinforces the sense of powerlessness that struggling students often feel.  At least Godin makes his attitude explicit:  it’s far more common in today’s discourse to merely imply that teachers are an obstacle to improvement.  Often, it’s simply an unstated assumption.

What would Seth Godin tell a struggling piano student who feels they simply aren’t a “music person”?  Is this student not a good piano player because no teacher cared enough to really teach them piano?  I suspect anyone who knows how hard it is to learn to play the piano would laugh at such a response.  Is anyone laughing at this characterization of math teachers?

The work of a teacher is hard, and teachers work hard.  And they care.  Blaming teachers for all learning failures is simple-minded and impractical.  No attempt to improve education will succeed if it is based on the premise that teachers are incompetent or uncaring, and that students are passive or powerless.

You can read Seth Godin’s piece here.  And math educator David Coffey has written a nice response here.

Math Photo: Pyramid Projection

DSC09181-001

Naturally, the geometry of this simple piece of playground equipment caught my eye, but the shadows really sparked my interest.

The shadows are the projections of the edges of this pyramid, and they form a set of angles on the ground.  Notice immediately that the largest angle (the shadow formed by the “back” face) is the sum of the other three angles.

DSC09183-001

There are many other interesting questions to ask, and relationships to explore.  What I was most curious about, however, is how accurately we could locate the sun in the sky based only on this information.

 

A Sidewalk Paradox

A serendipitous sidewalk sign, given a recent unit on mathematical logic.

Sidewalk Paradox

So, what’s the truth value of this statement?

 

Demonstration of Linear Independence

vectors spanning the plane

I’ve put together a simple Desmos interactive that demonstrates the basic ideas of linear independence.

If two plane vectors are linearly independent, then every vector in the plane can be written as a linear combination of those two vectors.  Those two vectors span the plane.

By playing around with the sliders in this interactive, you can see how every vector in the plane can be expressed as a linear combination of the two original vectors.

Moreover, if you make the two original vectors parallel, they no longer span the plane.  That’s because the two original vectors are now linearly dependent!  Each is a linear combination (in this case, a scalar multiple) of the other.

You can see this Desmos interactive here, and you can find more of my Desmos-based demonstrations here.