In Fundamentals of Geometry, 6th and 7th graders are using hands-on tools to explore geometry while discovering the clear line between conjectures and proven results.
As a mathematician whose area of expertise is "probabilistic phenomena in high-dimensional convex geometry", I was nervous when I taught Fundamentals of Geometry for the first time last year. Would I be able to make angles and polygons and parallel lines interesting to my 6th and 7th graders? Would they struggle to understand what I've taken for granted for so long that I can't remember learning it?
It turned out that I needn't have worried: put a rich enough puzzle in front of Proof School kids, and they will make the discoveries all on their own. The hardest part for me was to stay out of the way! This year, teaching Fundamentals for the second time, I built on that basic approach, while adding more mathematical rigor along the way.
Whenever possible, we used hands-on tools to explore the geometry around us. Students used patterns of wooden tiles to work out the area of a dodecagon; drew triangles with a compass and straightedge to figure out which combinations of sides and angles were enough to uniquely determine a triangle up to congruence; and took a scavenger hunt in search of everyday geometry in Nob Hill, measuring the length of the labyrinth at Grace Cathedral and observing how the design of a fire escape depends on the flexibility of a rhombus. Conversely, much of our pencil-and-paper work had immediate relevance to the world around us. We used similar triangles and reflections to improve our pool skills—at least in theory!—and figured out how to measure the distance to a far-off building with the help of an outstretched thumb or the moon on the horizon.
Some of the best activities were those which went where I didn't expect. To understand where the formula for the sum of internal angles in an n-sided polygon comes from, my students divided polygons into triangles and observed how the number of triangles varied depending on how many auxiliary points they added in the interior and on the boundary. (I borrowed this activity from my colleague, Henri Picciotto.) When some students got results that didn’t conform to the formula I was expecting, we looked closer—and discovered that the instructions for “triangulating” a polygon came with an unstated assumption built in. My class figured out how to spot violations of this assumption, and were able to adapt their formula to correctly predict the number of triangles even in the presence of such violations. Henri was impressed enough to request an erratum, in my words and the words of Proofnik geometers; you can read it here.
If you do, you’ll note that the students didn’t just discover the extended formula: they also stated the reasoning behind it. In spite of our experimental approach to geometric discovery, my class learned early that "you can’t always trust your eyes". I am impressed by how well they have taken to that maxim of geometry which bemuses so many first-time learners of the subject, that appearances merely suggest the truth and reason must certify it. On one of our last labs of the block, students investigated what kinds of shapes we can get by connecting the midpoints of the sides of various types of quadrilateral: a rectangle, a rhombus, a parallelogram, and so on. (If you don’t know the answer, this is a fun activity to try for yourself!) After trying a few examples, it is hard to resist making a few guesses. But I was struck by the clear line my students drew between their conjectures and their proven results. One student created a special stamp to commemorate the latter. Geometry, as they are discovering, is not just about lines, angles, and circles; it’s about how we make sure that we know what we know.
-- Austin Shapiro