Math Talk and Definitions
This is from an interaction with the children of year 6 where I had asked them to do a flipped class on geometry. They had prepared something on lines and angles and were discussing transversals and angles formed when the transversals intersect the lines. Let's look at the points that they shared about their understanding of transversals.
The student-teacher (ST) (while explaining the concepts): So a line which intersects two or more lines is called a transversal.
Other children (OC): So is a line a transversal even when it meets all at the same point.
ST: No it must meet at separate points.
OC: what if we have three lines and two of them intersect and then the transversal meets two of them at that point of intersection and the third line at a distinct point. Will it still be a transversal.
ST: no I don't think so and (then they looked at me)
Me: what do you think you read about the transversal?
ST: that it meets two or more lines at distinct points.
I took out the phone and asked them to type transversal and look at at least 5 different definitions.
The entire conclusion ended by us defining the transversal as a line that passes through two or more lines at distinct points.
In the second instance, we discussed what are different types of angles formed when a transversal cuts two lines. When the students who were taking the class started teaching they immediately started with the definition of the corresponding lines which according to them were two angles of which one is in the interior and one in the exterior. To this, there were counter questions about them being linear pairs as well.
I interrupted and asked what does the class understand by interior and exterior. Few children tried to explain by pointing to some region vaguely. so we realized that the first step is to make it clear what we understand by the interior and exterior when we talk about the corresponding angles in a plane.
Then finally we saw that for something to be called corresponding pairs there must be three conditions that we must have.
- One angle in the interior and the other in the exterior.
- On the same side of the transversal
- Not having the same vertex.
These conditions led to an interesting discussion as to what happens when we relax any of these conditions.
These above instances reflect how important mathematical division is for the development of any concept and how important it becomes to have the most refined definitions for the conceits to avoid ambiguity.
Sanjay Raghav
7 August 2019