Wednesday, November 30, 2011

Making Assumptions

There's the old adage about making assumptions: If you assume, it makes an "ASS of U and ME." I might argue though, that I make assumptions all the time, and that assumptions are vital to the efficiency of my day:

  • I see an 'open' sign in a window, and I assume I am welcome to enter.
  • I hear a siren, so I assume that there is an emergency situation.
  • I smell something foul, so I assume that it is not safe to eat.
  • I see a person with blue eyes, so I assume they carry the OCA2 gene.
  • I see a square, so I assume that it has four right angles.

At this point you are hopefully feeling a bit of unease, if not downright criticism, towards my list of presuppositions... especially the last one, which I will reveal as being the original impetus for my train of thought. I have been rethinking the quadrilaterals unit (in geometry) lately. Not that it needs an overhaul or anything. In fact, it's a pretty solid unit, but my new inner voice needs to find a higher rationalization for everything.

Searching for "essential questions in geometry about quadrilaterals" brought results that seemed petty and narrow: "How do we identify various quadrilaterals?" "What types of quadrilaterals exist and what properties are unique to them?" etc. At the risk of offending, or sounding trite, I don't really care about the answers to those questions. But here's a couple questions I DO care about:

  1. How much information do I need to know before I can start making assumptions?
  2. Are there 'good' assumptions and 'bad' assumptions, and what is the criteria for making that judgement?
  3. What are the consequences of making invalid assumptions?

What I like about THESE questions is that they are not necessarily about geometry, or even math for that matter, but they DO get at the heart of something I am trying to convey over the entire geometry curriculum: truth is the result of a string of proven facts (or unanimously approved axioms). In the context of the quadrilaterals unit, this idea gets a chance to settle in a little more deeply.

How much information do I need to know before I can start making assumptions?
I cannot assume my 'blue-eyed person' carries the OCA2 gene unless I know for certain that he is indeed a true blue-eyed person (and not wearing contacts, or transplants, or cosmetic pigmentation)... and for that I might need genetic proof. When it comes to quadrilaterals (or any type of defined shape), I need to know and fulfill a precise definition. I cannot assume that my 'square' has four right angles unless I know for certain that it is indeed a square... and for that, I need to meet the definition of a quadrilateral with four congruent sides and four right angles.

Are there 'good' assumptions and 'bad' assumptions, and what is the criteria for making that judgement?
I would say a 'good' assumption is a valid or proven one. 'Bad' assumptions might be invalid or simply unproven (i.e. unsure). If I am making an assumption about the diagonals of a quadrilateral bisecting one another, I'll need to be able to demonstrate sufficient evidence that my quadrilateral is a parallelogram... or run the risk of making a bad assumption.

What are the consequences of making invalid assumptions?
Ahh, the rub. Indeed, sometimes the consequences of invalid assumptions are minor. If I assume that a foul smelling item is not safe to eat, I might miss the opportunity to taste a delicacy, but in the absence of further evidence, it just may be the safest assumption to make. In math class, the consequences of making bad assumptions are usually loss of credit/points on an assessment. Like my choices about the foul smelling item, students quickly learn what is 'safe' and fail to find the motivation for further validation.

But geometry is the study of space and measurement and the world around us. It's important that our students fully understand that the true consequences of invalid assumptions can be catastrophic, expensive, deadly, or any combination of these. It is essential that they understand that even the most generic classroom exercises have implications for the world around them. I mean, I'm not really suggesting that their attention to precision on my quadrilateral homework will save the lives of hundreds of people. Or, maybe I am.

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