I want to start with an observation about the Common Core State Standards for mathematics.  In the past, I have been openly critical of many standards, but I find the central ideas behind this set quite appealing.  The key point is that these standards point the way toward student understanding of critical mathematical ideas, not just the memorization and regurgitation of facts quickly forgotten after the test.

Here are the key concepts:
  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

These are all wonderful ideas that directly relate to how mathematicians work.  The challenge is how these can be implemented in the teaching of mathematics.  One example I find interesting is to see how we can use 3D printers in the math classroom in support of these core ideas.  For example (and I have several), consider the construction of polyhedra using plastic struts and rubber bands.  These shapes are called tensigrity structures first popularized by Buckminster Fuller.  The idea is that a three-dimensional structure can be built with only two kinds of elements – those in compressions (the struts) and those in tension (the rubber bands.  For example, the following image shows an octahedron built with three struts and eight rubber bands.

This simple structure is easy to assemble, and numerous other polyhedra can be built the same way.  But you might be asking where the math comes in.  This polyhedron has 8 faces, 6 vertices (corners) and 12 edges.  What relation, if any, exists between the number of faces, edges, and vertices of polyhedra?  If you can find a relationship, how can you prove it works for any polyhedron?
These questions all derive from the design and construction of a kit for making polyhedra, and the questions we ask cover quite a few of the key concepts for the Common Core math standards.
Imagine how activities like this might make math exciting for kids who currently fail to see why math can be an interesting topic!