Ten Weeks of Summer: To Teach It or Not To Teach It (Week 4)

"To teach it or not to teach it. That is the quesiton. Whether 'tis nobler in the mind..." Whoops, slipped into Shakespeare's Hamlet for a second.

I am part of an email group of math teachers and last week there was a question going around about should we teach and/or why don't we teach concepts about alternate exterior angles formed by two parallel lines and a transversal.  Some argued via email that it was important to teach that concept for future engineers.  Others explained that they did teach the concept and that they taught it through discovery.  I said that it is probably not seen in many textbooks, because interior angles have more importance, especially when considering quadrilaterals.  So, to teach it or not to teach it.  Here are two stories I shared with my email group.

Story #1: I got my master's degree at Iowa State and for one of my projects, I used a theorem about secants and tangents to circles.  It is a theorem based on similar triangles.  The professor (with a doctorate degree) questioned me on the use of the theorem.  Either he had never heard of it or he couldn't recall it.  However, he had enough understanding of geometry that when I explained why it was true, he understood.  

Story #2: Over the past 10 years as department chair, I have to speak to groups of prospective parents/students and describe what is offered in the mathematics department at my school.  Do I speak about specific theorems or concepts?  No - they assume that I am teaching the math content that is typical of a high school mathematics curriculum, and let's be honest, a list of topics would be yawn-inducing.  Instead, I say that in math class students make and test conjectures, combine concepts in new ways, form logical arguments and critique the reasoning of others (verbally and in writing), notice and describe patterns, and make connections between different ways to represent a concept.  What I have just described is something that ALL students need no matter what major or career they go into.

Of course there are big ideas that matter in the teaching of mathematics and we can't gloss over those - function, inverse, transformations, proportionality, and rate of change are just a few.  However, if we leave off some concepts, will our students be somehow lacking?  Think back to Story #1. Was the Ph.D. mathematician lacking?  Some might say yes and that he should have learned and remembered that concept.  I would argue no, because he had a firm understanding of the big ideas and the ability to reason and apply his understanding.  His ability to do this, and not recalling a specific theorem, is what makes him, and I hope my students, mathematicians.