Mind Your Language (2)

Ask a group of students (or adults come to that) to draw a parallelogram and see what the most common answer is. When I have done this (with students, educators and parents) by far and away the most common shape is the example A below. I will even have a drawing like this prepared and reveal it to the 'audience' and announce that I read their minds to such an extent that I knew that the longest side of the parallelogram would be horizontal and that it would slope left-to-right.

Rarely will folk draw a parallelogram like B and even more rare will they draw a rectangle or square. This might seem innocent enough but can be a big clue to a huge misconception that often goes unnoticed. This was evident in a Grade 6 class I saw this week. We had given them this question from the 2012 Ontario Junior EQAO test:

 

 

It's a nice question as there are a variety of ways to think about solving this. One student solution was shared with the class and provoked some great discussion:

Some students argued that the shape on the left wasn't a parallelogram. One student argued that it was as parallelograms are shapes that have "...two pairs of parallel sides." Years ago I would have left this statement unchallenged. Now, I jumped at the opportunity it gave and asked if the shape below was a parallelogram:

Probably half the students said yes it was; they understood that a parallelogram is any shape with two pairs of parallel sides. The remainder of the class seemed unsure. We then got in a debate as to whether or not a regular hexagon is a parallelogram(!); some said no as it had three pairs of parallel sides, others said yes as it had at least two pairs of parallel sides.
As we mused how to deal with this, a student asked if she could look up the definition of parallelogram. This she did, and there it was: "A parallelogram is a quadrilateral with two pairs of parallel sides." This was news to a lot of students. However with this new knowledge they were now OK with saying that the rectangle in the solution is also a parallelogram.
On reflection, I now realise that giving insufficient examples and using imprecise language  restricts students' understanding of what a parallelogram is. It would be better for students to construct their own understanding of what a parallelogram is by showing them something like this (from Ontario's MOE's Guide to Effective Instruction Grades 4 to 6: Geometry) and asking them to define 'parallelogram'.