Math Post

Last week we worked in partners to play a dice game (adding).  We played 10 rounds, 6 times.  I was Player A, my partner was Player B.  To determine who won each round, we followed a scoring system.  If I rolled a sum of 2, 3, 4, 5, 10, 11, or 12 then I won a chip.  If he rolled a sum of 6, 7, 8 or 9 then I lost and he received a chip. 

One of  the main questions that was posed to us was whether or not the game was fair.  Immediately I began to think of addends and I was busy writing numbers down while my partner was ready to play the game.  The results were that I won only 3 times and my partner won the other 7.  I wanted to revisit the chart that I had been creating to find a solution to the question.  What I had not anticipated is that when a person rolls, there is a chance of rolling, for example, (3,1) or (1,3).  It was explained to me that I needed to take this into consideration because the order would matter if a person were to roll the dice one at a time.  So while my model was slightly off, with that one "tip", I could have revised it and come up with the probability that I would have won and the probability that my partner would have won--and then proven the fairness of the game.  Again, this proves that individuals have different ways of reaching conclusions. 

One other thing we discussed that I appreciated was that when we use the term "understanding" that is often not a measurable objective--it is the big-picture, overarching goal.  One analogy that our professor used was learning to drive a car (like learning to do anything else...reading, cooking, etc.)  When a person learns to drive, they break the activity down--seat belt on, shift gear into reverse, depress the emergency break, look behind right and left, release break and slightly administer gas pedal.  Break.  Put car into drive, look left, look right, gas pedal.  I loved the analogy because I can remember being very paranoid about driving, no matter how many times I had read my manual.  I was afraid!  But today, I'll jump into any car (provided it isn't a manual--then I feel like that kid again who processes everything step-by-step) and off I go.  It's second nature.  For the beginner, they don't understand how to drive the way that I understand to drive.  So the nature of understanding is when things stop having to be broken down step-by-step.  Cognitive compression is what allows us to get to a point without having to break anything down.  It becomes intuitive.  I loved this point.  So when we want to measure understanding, we have to think about what is measurable.  However, I wonder--don't students have to understand the components or pieces of what they're doing, then synthesize the parts?  I feel like this is what some of us are thinking as we write our lesson plan objectives.  But given the nature of learning--that it happens at different times and rates, it must be difficult to distinguish between rote memory v. understanding.  Where does that leave us?  We have to seek explanation?  Regardless, I am thinking very carefully about measurable objectives and assessments that look for specific evidence.