In 2020, I wrote No One Way, a blog post which I used to explain my website’s motto (“There is no one way.”) I argued that it is the math itself that demands that we approach important topics in multiple ways. As it turns out, this is a favorite topic of mine: in 2016, I challenged those who believe they have found the way to teach math in Eclectic and the subsequent posts. And in 2015, in How To, I critiqued the idea that there is one best way to teach any given topic, using equation solving as the key example.
Today, I return to this with two more examples, both of which came up recently on NCTM’s email discussion list.
The first is the familiar and evergreen topic about the definition of a trapezoid. The choice is between the exclusive definition (a quadrilateral with exactly one pair of parallel sides) and the inclusive definition (a quadrilateral with at least one pair of parallel sides.) The first excludes parallelograms, and the second includes them.
There are valid arguments on either side, and I won’t get into all of them here. Still, I lean towards the inclusive definition, and I’ll mention two reasons why:
- Geometry Labs 6.2 is an activity where students write instructions for “walking” parallelograms (not unlike turtle geometry programs.) From that point of view, a parallelogram is a trapezoid, since with the right numerical inputs for sides and angles, a trapezoid-walking algorithm can yield a parallelogram.
- Likewise a trapezoid construction in GeoGebra (or other interactive geometry application) can be dragged into a parallelogram.
But to me what is more interesting than choosing one or the other definition is that this question provides an excellent opportunity for a worthwhile discussion of definitions in mathematics. Different mathematicians, different textbooks, and different teachers sometimes disagree on the best definition for a given entity. Criteria vary: consistency within a book or a course, for example, or what theorems are implied by one choice or another.
In Geometry Labs I suggest that the trapezoid question should be part of the discussion about the classification of quadrilaterals — a discussion to be had in the classroom, not just among teachers. In Transformational Proof in High School Geometry, the book I co-authored for teachers and curriculum developers, the late Lew Douglas and I define quadrilaterals by their symmetry properties. From that point of view, only isosceles trapezoids belong in the standard hierarchy. General trapezoids have no symmetry. We define them by the property that one side is a dilation or translation of another — which turns out to be an inclusive definition. If we had not mentioned translation, then it would be exclusive.
Would students be confused by having that discussion? Quite the opposite! It is not particularly important that they understand or remember the arguments, but having that conversation helps to clarify what a trapezoid is. In the end, the teacher can ask the class to vote on this, or (more convenient) say something like: “In this class we will stick with the definition used in our textbook.”
Another question that came up in the NCTM email list is “Why are slope triangles drawn beneath the line? Wouldn’t it be better to put them above the line?” My response is that neither is better. If students are stymied by one or the other, it is because they don’t understand slope. Both choices are mathematically valid. We should use both, show they yield the same value for the slope, and seize the opportunity to get at the underlying ideas about rate of change.
In fact, I used GeoGebra to create an applet which makes it possible for students to draw “stairs” that consist of consecutive slope triangles. That setup yields various puzzles which help see that there are many possibilities for slope triangles: smaller vs. larger steps, above vs. below the line. (The accompanying worksheet suggests specific challenges.)
Certainly, you’re entitled to have an opinion on the trapezoid definition, and you might feel like you just don’t have time to discuss it with your students. Fair enough. That question is of interest to geometry teachers, and not that crucial in the grand scheme of students’ education. But the concept of slope is so foundational to so much mathematics, all the way to calculus, and it connects to so many other important ideas (proportion, fractions, similar triangles…) that you cannot afford to teach it in just one way. There is no best way!