Teaching

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.

By MrHonner, ago

Writing in Math Class: Favorite Shape

favorite shapesI love giving short writing assignments to my math classes.  It’s a great way to get students engaged in mathematics in a different way, and it gives me a different window into how my students think and who they are.

Here’s a prompt I gave them recently.

Draw a shape that you like.  Write 1-2 complete sentences explaining what this shape is and why it appeals to you.

This simple prompt was something of an afterthought on a recent assignment, but as usual, the students surprised me with their responses.  Here are some of my favorites.

I like the square because of how organized it is.

The triangle appeals to me because it is the shape of things I love to eat, like a slice of pizza or a piece of cake or pie.

A rhombus is different from other shapes, but so alike.  It shows how a little change can make a new shape.

Circles appeal to me because they have infinite lines of symmetry.

This is a circle.  It has no corners and is symmetrical.  This shape is appealing because it feels open, and since there are no corners, there are no sharp edges to hurt you.

This shape is a circle.  It appeals to me because, as weird as it sounds, I want my life to be as perfect and as well-rounded as a circle.

A triangle is simple.  It has the least number of sides a polygon needs.

I like the square because I like things to be equal, not different.

I’m always smiling, laughing, and thinking after reading what students have to write.  There are lots of great reasons to get students writing in math class, so give it a try!  You can find more resources here.

By MrHonner, ago

3D Printing in Calculus Class

I’m looking forward to exploring 3D printing in Calculus class this year.  We don’t have a printer in our classroom (yet!), but some students have enough experience and access to work on modest projects outside of class.

Here’s a print of an interesting surface in xyz-coordinate space.

Beautiful Surface and Printed SurfaceIt’s always exciting to find a new way to represent or experience a mathematical idea, and physical representations can be especially powerful.

And perhaps more importantly, 3D printing gives students an opportunity to use mathematics to create.  Mathematics is a creative endeavor, and whatever helps promote this idea will ultimately help change attitudes about math.

 

By MrHonner, ago

George Hart Workshop on Symmetry

Through Math for America, I had the pleasure of participating in a one-day workshop on symmetry led by well-known mathematician/computer scientist/sculptor George Hart.  The workshop featured some great math and some excellent hands-on projects that really had us exploring some deep mathematical ideas.

We began the day by talking a bit about what symmetry is and the types of symmetries we’re accustomed to thinking about.  Then we explored how the symmetries of a given object, when thought of as actions (like reflections or rotations), form a group, which creates an interesting mathematical structure to work with.

After the introductory mathematics, George led us through three hands-on activities meant to explore different symmetry groups.

The first project was building a Tunnel Cube from a set of pre-cut playing cards.  The 12 cards were notched in such a way that the piece could be assembled without any glue or tape.

Tunnel Cube

It did, however, require a great deal of dexterity and patience!  You can see George’s explanation of the Tunnel Cube here, and watch a video in which he assembles it here.

The second project was building a ruled hyperboloid using kebab skewers and rubber bands.

Ruled Hyperboloid

The last project was a group build, where we assembled a George Hart original sculpture.  This was a bit harder than I imagined, but the process was full of the small frustrations and successes that good collaborative work entails.

George Hart Sculpture

In addition to the fun project ideas, the big takeaway for me was using symmetry as a design parameter.  While we assembled, and then admired, the final sculpture, George talked a little bit about his creative process.  By thinking first of symmetries, and symmetry groups in particular, he outlines a design space for a particular piece, and then starts playing around in that space until he finds what he’s looking for.  Each of the projects emphasized that idea with a different symmetry group.

Many thanks to George Hart, and Math for America, for an enriching day!  You can see more pictures from the workshop here.

By MrHonner, ago

Regents Recap — June 2014: Common Core Scoring

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

June, 2014 saw the administration of the first official Common Core Regents exam in New York state,  Algebra I (Common Core).  Roughly speaking, this exam replaces the Integrated Algebra Regents exam, which is the first of the three high school level math Regents exams in New York.

One of the biggest differences in the two exams is how raw scores were converted to scaled scores.  In the graph below, Integrated Algebra is represent in blue, Algebra I (Common Core) in orange.  Raw scores are on the horizontal axis, and scaled scores are on the vertical axis.

ia vs cc scaled score plot

The raw passing score (a scaled score of 65) is roughly the same for both exams:  30/87 for IA, and 31/86 for CC.  But notice the divergence in the plots after a raw score of 30.  This is because the raw “Mastery Score” (a scaled score of 85) is quite different for the two exams:  65/87 on IA, vs 75/86 on CC.

It’s curious that the exams could be evaluated in such a way that passing requires the same raw score on both, but mastery requires a much higher score on one than the other.  If the exams were of equal difficulty, this would mean the same percentage of students would pass the exam, while dramatically fewer students would achieve mastery on the Common Core exam.

This is especially curious, since the tests don’t really seem that different to me apart from some substantial changes in content emphasis.  It’s hard not to see this as merely a deliberate decision to lower the mastery rate.

Furthermore, the consensus is that raw scores on the Common Core exam are substantially lower than on the Integrated Algebra exam.  Based on the exams and on conversations I’ve had, I wouldn’t be surprised if scores on the Common Core exam were lower than scores on the Integrated Algebra exam by 10 raw points on average.  This would lead to an even larger drop in mastery rates, as well as a drop in passing rates.

There’s an interesting opportunity here, though.  I’m certain that a large number of students across New York state took both the Algebra I (Common Core) exam and the Integrated Algebra exam.  This offers a unique opportunity to directly compare the tests and the conversion charts the state decided upon.

I think the Department of Education should release that data so we can see for ourselves just how different student performance was on these two exams, and judge for ourselves the consequences of these very different conversion charts.

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By MrHonner, ago
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