Fractions, Tilings, and Geometry

Ostensibly designed for precollege teachers, this teach-the-teacher course (based on one offered in the Summer School Teacher Program at the Park City Mathematics Institute) has applicability to teaching undergraduates as well. This introduction to non-periodic tilings in two dimensions and space-filling polyhedra is built around engaging and enlightening group activities to benefit of the secondary education instructor. The content can also be used as “classroom capsules” to enliven first-year college courses.

The topics covered range from tessellations, including Penrose dart and kite tilings and Voronoi diagrams, to the golden ratio, packing problems, and more. Very well designed to offer participants an authentic experience of discovery through an orderly succession of motivation and revelation through guided, hands-on investigation of patterns, the finished material can be directly applied to aid students in honing reasoning, learning to formalize ideas, and grow in mathematical sophistication while having fun solving problems. Here is an example problem with representative tone:

Draw a circle, then plot four points. Use those points to create quadrilateral FRED.

1. Find all the points that are the same distance from F and R, and draw the shape.
2. Do this for each side of the quadrilateral.
3. Notice anything? Compare to others’ work. Can you
4. explain why this happened?

Arranged by increasing complexity, the problems offer various options of entering and exiting depending on the audience.

The material comes in three overlapping sections: problem sets, “facilitator notes”, and solutions. Key to the value here is the facilitator notes, which cover goals, prerequisites, advice, and background for each problem. These notes include observations on typical sticking points encountered by participants and direction on moving groups toward the next problem efficiently. Augmenting the text in these chapters are illustrations and photographs, some culled from actual sessions where participants cut, folded, drew, and arranged. Some of this material is online, including animations and spreadsheets, with URLs mentioned in context. One set of such examples can be reviewed at http://projects.ias.edu/pcmi/hstp/sum2014/morning/.

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Tom Schulte feels geometry needs more widespread presence in college curricula in the United States and agrees with the observation that geometry “in addition to being a splendid example of deductive reasoning, important and interesting in itself, can also serve the purpose of creating a critical attitude of mind toward deduction and thinking in general…” (From The Place of Mathematics in Secondary Education: The Final Report of the Joint Commission of the Mathematical Association of America and the National Council of Teachers of Mathematics, 1940).