“If it ain’t broke, don’t fix it?” (Rethinking quadrilaterals)

Why on earth would we spend two whole days rewriting our Honors Geometry quadrilaterals unit? Our textbook, after all, contains a perfectly serviceable sequence of four lessons on this topic:

These lessons are adequate. In the words of the standard phrase, they “ain’t broke.” So why fix them?

The answer is that “adequate” isn’t good enough. The lessons in the textbook have a very low cognitive demand. They just don’t require honors-level thinking. And where are the Big Ideas?

Actually, there are a couple of Big Ideas buried in these four lessons:

  • The concept of a hierarchy is certainly present in the first of the four lessons. However, it’s merely taught to the students didactically, with no exploration required — it’s just a list of definitions, like “A rectangle is a parallelogram in which at least one angle is a right angle.”
  • The related distinction bettwen necessary and sufficient is implied in the second and third lessons, though it’s never made explicit.

So that’s a start. But for the most part the chapter promotes memorization rather than thinking. It emphasizes tedious lists like this one, excerpted from the middle of a list of quadrilateral properties:

There’s no sense of being a mathematician here! How does an honors math student react to such a list of 28 items (in the full list)? Well, she can simply memorize the list, or she can give up and say it’s an unreasonable task, or she can learn a few of the more important ones and ignore the rest. The textbook claims that “with some effort you will soon learn them all.”

Bleh. We can do better.

In our workshop we replaced these four textbook-based lessons with  five investigations (packaged together into a sequence of 18 problems, most of which contain multiple sub-problems):

  1. This investigation is an exploration of definitions of various quadrilaterals. Definitions can partition a larger set (integers are positive, negative, or zero); they can be hierarchical (equilateral triangles are isosceles); they can be subjects of significant dispute (the definition of trapezoid, for instance). Also, definitions have consequences. They can be equivalent (specifying the same set of objects) or not; they can make it easier or harder to prove something.
  2. The second investigation introduces the use of coordinates in this context and distinguishes necessary from sufficient conditions. In all cases it requires reasoning and defense of one’s reasoning, sometimes to the point of being a proof.
  3. The third investigation pursues the necessary/sufficient distinction by presenting a list of 12 proposed properties without any didactic teaching: each one might be necessary for a particular type of quadrilateral, or it might be sufficient, or both necessary and sufficient, or neither. Exploring these possibilities starts to lead students to the idea of moving up or down the hierarchy in the next investigation.
  4. And here in the fourth investigation we assemble a bunch of observations in order to construct that hierarchy ourselves. We also pursue coordinate reasoning some more.
  5. Finally, the fifth investigation explores the creation of one quadrilateral from another. For instance, if you connect the midpoints of all four sides of a rectangle, what do you get? Can you prove it? What if you bisect the angles and connect the intersection points? What if you construct equilateral triangles from the sides (pointing outward) and then connect the outermost vertices of these triangles?

This newly rewritten unit still needs some tweaking, but our hope is that we have a much richer sequence that truly challenges honors students to do their best.

Categories: Math, Teaching & Learning, Weston