Modeling perfect answer is not modeling perfect learning

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Last week a student of mine was curious about the quadratic formula.  He had seen it before but didn’t understand why we used it.  “Where does it come from?”, he asked.

I loved this question.

I did the thing any excited math teacher would do and decided to take a couple of minutes to explain to him how we can start with ax^2 +bx + c = 0 and complete the square.  Luckily, this took place after school because five minutes quickly turned into twenty and soon he said “sorry, mister, I need to run!”  Even though it was a concept that I understood and I have taught before, I ran into bump after bump and I found myself thinking collaboratively with my student about the problem but ran out of time and couldn’t quite figure it out in time!

Luckily, I didn’t feel too defeated because I figured it out a few moments after he left, but it got me thinking…  Often as a teacher, I feel the need to be fully prepared – down to the last detail.  I feel like any example I show or any question that I answer must be done to perfection.  And yet, learning is the exact opposite.  Learning is messy and bumpy.

If I would have prepared a lesson and taught the quadratic formula explanation perfectly by detailing every step would more learning have occurred?  What is the value of modeling a perfectly solved problem?  Would it be more meaningful to model good questioning and struggle through a problem?  How do we let students solve questions like that while still allowing them the support to access it?

I don’t know the answer to it all, but definitely some questions worth grappling with.  Curious if you have thoughts or similar situations to this!