I am finding I rarely have time to blog while school is in session. So during the semester break, I am attempting to capture some of my thoughts from the past semester... the good, bad and ugly about the changes I am making. This post is part of that series of reflections.My colleague across the hall, Blair Miller, was doing a little light reading... some PhD thesis. In the paper, the author imagined a math class where every answer is 6. Hey, most math problems have contrived answers anyway... why not 6?
It got us thinking... what would this look like? What benefit could this possibly have?
It removes the need to get the right answer. Even if you didn't tell your class that every answer is 6, they would figure it out by the end of the first week (second week tops). Without needing to find the right answer, the focus shifts from answer to process. Sure, the answer is 6, but how do we get there?
Students could try a bunch of random approaches to see what works. Let's face it, some of our students use random approaches anyway. This problem has 2 numbers... I think I will multiply. But they don't understand why multiplying works. But if the answer is 6, they immediately know if they are correct. It would sort of self-teach and auto-correct common mistakes. Hmmm, multiplying didn't work, let's try adding instead.
Consider an example. The Pythagorean Theorem. The 2 most common mistakes are not squaring the values and not identifying the hypotenuse correctly for substitution into the formula. But, if every answer were 6, these deficiencies would show up immediately. Wait a minute... I added the 2 legs but I didn't get 6. What went wrong? Oh, yeah, I need to square the values! Or, I squared the values but still don't get 6. Oh yeah, the longest side is the hypotenuse. I substituted the values in the wrong place.
A chapter test might have 2 problems involving the Pythagorean Theorem. For the first one, the answer is 6, just like during practice time in class. For the second one, the answer could be any random number. This allows the test to truly assess mastery of the concept. After all, if you really get the concepts, you should be able to figure out a problem where the answer is not 6.