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.

Show your work!

One of my colleagues recently said to me "You know, nobody does proofs anymore." Seriously? I think that would be a catastrophic move in the wrong direction, and it reminds me of a nagging problem that I have avoided.

Since the beginning of my teaching career, there are a handful in every class: those students who, despite numerous pleas to "show work," refuse to write anything but the answer on the page. I give what I think is a thoughtful (and lengthy) practice set as homework, only to find that student who has successfully reduced it to a smattering of digits that fills up only a quarter of a notebook page!

"You need to show your work," I'll say.
"I did it in my head," he replies.
"You did what in your head?" I'm not letting him off, I don't care if he's a genius or not. "You need to write down what you did."

At this point, blank stares or shrugs are about all I ever get, but depending upon my mood, I might launch into one of the following attacks:

• "You need to know what you've done, so you can correct yourself if mistakes are made." (You're not as clever as you think.)
  
• "A good mathematician must show diligence in precision, and exercise good habits of mind." (You'll never amount to anything.)
  
• "It may seem easy now, but it won't always be and showing steps now will make the steps more natural in the future." (Pleeeease... I need to validate my existence.)
  
• "You only get partial (or no) credit. You need to show your work to get full credit." (You will do it because I SAID so!)

The truth is, I don't really understand this student. It shouldn't surprise you to learn that I never was this student. I was the neat girl, with the 5mm mechanical pencil, gum paste eraser, and tiny, Times New Roman handwriting. My geometry problem sets looked like architectural journals and my calculus tests got photocopied and passed around as examples. The aesthetic appeal alone of those neatly written problems in nicely aligned columns with balancing symbols and digits was enough motivation for me. Those other students have always frustrated me to no end!

So then comes the blow: the thing I've been forcing myself into these past months. I ask myself "WHY?" And I don't mean "Why do these kinds of students frustrate me?" I know the answer to that question. The real question is "Why is it important to show your work?"

That particular question has evaded me many times, probably because of a lingering fear that the answer might be "It isn't." Thankfully, I don't think I'll need to go there.

At the heart of the issue is the realization that there is a quintessential difference between an answer and a solution to a problem. A solution illustrates, generalizes, communicates, and verifies the results. An answer is just a number. In mathematics it is important to find the correct answer, sure, but more importantly I want my students to gain skills in writing accurate and convincing solutions. It's not enough to just know the answer, you need to be able to prove it beyond a doubt to yourself and others. Take this example:

Student A has successfully shown that 4 is AN answer to this problem (I took the liberty of writing what I believe is going on in Student A's head, because usually this version of the solution is simply a scrawled 4 on the page) Student B, on the other hand has successfully shown (and verified) that 4 is THE answer to this problem. The difference may seem subtle and unnecessary for a typical beginning algebra student, but it quickly becomes essential, like in the case where more than one solution exists or when extraneous solutions occur.

The result of treating 'work' as 'proof' could very well be an improvement in reasoning and logic in students across the board. Certainly it would make a difference to my geometry students when they are first confronted with conditional statements, laws of reasoning, and proofs. It seems that more and more, students (and people in general) are forgiven for their inability to write a logical and convincing argument.

So, is my colleague right? ARE we moving away from proof? Are YOU? Seriously, I want to know.