- 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:
- How much information do I need to know before I can start making assumptions?
- Are there 'good' assumptions and 'bad' assumptions, and what is the criteria for making that judgement?
- 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.