Time for Reflection

I really enjoyed Zull's discussion in chapter 9 about the need for reflection in order to have an aha! moment. I am beginning to understand how important reflection is, but how little time we set aside for it in our curriculum. We are rushing through so much material with our students that when we feel like we are a little bit ahead we continue to rush on instead of just slowing down and taking some time for reflection. I completely understand the feelings of the teacher on page 161 who complained that his students have no idea if their answers are reasonable or not. So often, I forget that I wasn't always successful in reflecting on an answer and determining whether it made sense or not. I felt that I was lucky if I picked the correct formula from my hat and got an answer at all! Understanding reasonableness is a skill that takes time (in two ways). As students take more and more math courses, they will become better and better at judging and interpreting their answers. In addition, this skill also requires time in the classroom. If it is something we think is important for the education of our students, we must devote time to it in class. With the way things are currently going in the education system, I can see that teachers don't always feel that they have sufficient time to deeply focus on things like interpreting answers. Sometimes we feel successful if we are able to teach our students all of the formulas they are supposed to know for testing purposes. I think it is really sad that we are sort of taking the math out of math education. Students need to understand that sometimes you won't know the answer right away, but that does not mean you are bad at math or that you should just give up! Mathematicians dedicate their lives to solving one problem. Can you imagine the reflection process that must occur for these mathematicians? It's okay to take time to reflect on problems and let them turn over and over in your mind. During all of the time you spend thinking about the problem, your brain is trying to make connections and put together solutions. This is mathematics, and I hope that I can show this to my students.

Zull also has a great discussion in chapter 8 (similar to the discussion in Zadina chapter 6) about how math is made up of two processes: estimation and calculation. Calculation activates areas mainly in the front cortex; interestingly, arithmetic calculations most actively engage the same area that is associated with production of language. We use a different part of our brain, however, when we try to estimate answers. We use the where pathway in the upper back cortex. I hadn't thought about these two processes, estimation and calculation, in these terms before now. Estimation uses the where pathway because in order to estimate you have to be able to compare, which means you need to know where things are with respect to one another. For example, if I ask my students to estimate the number of gumballs in a jar of 100 gumballs, they have to be able to compare numbers on a number line to one another. A student knows that 2 is not the correct answer because it's too small of an estimate and they can clearly count more than 2 in the jar. So much of math depends on this ability to estimate and determine the reasonableness of our answers, but again I feel that we focus so heavily on the calculation aspect that we are taking away from our students' math experience. In my classroom, I hope to strike a good balance between the calculation and estimation aspects of mathematics, and in doing so I hope to help those students who are weak in one area or the other, or possibly both!