Assigned Blog #2

Instructional Goals

During summer school, two of my objectives were as follows:
1)Students will multiply and divide exponential expressions and
2)Students will simplify square roots.
As per the assignment, one of these was met successfully and one less successfully. It does not take a genius to imagine which.
After a fairly short explanation, the students were quite capable of using exponent rules to perform multiplication and division. This process includes only one step and students have only a few things to consider in their minds before getting started. One reason we were more successful with this goal was simply that it was easier. Another, is that at least some of my students began to actually see why it has to be true. By expanding exponents and showing repeated multiplication, my students could look back at the definition of exponents and see that as long as they accepted this definition, the rules must be true. Maybe I ought to prove everything next year, and tell my kids they will have to reproduce certain proofs on the test. Because when you know that something must be true, you don't guess, you don't have to ask yourself whether you multiply or add, or whether your base multiplies too or just your exponent. You don't have to memorize anything, although you will. Also, using induction to allow students to discover these rules for themselves helped them to get a better grasp of what is going on.
Teaching students to simplify square roots was much more difficult and less successful. Partially, this is due to the increased complication of the task. There are more steps, and the process is less intuitive, but there are some obvious things I could have done differently that would have helped my students be more successful. I tried to rush through this lesson because of time constraints but if I did not have enough time to teach it well, I ought not to have taught it at all. What I should have done was illustrate why the process we were working on has to occur as it does, why root eight must equal two root two. Unfortunately, illuminating such connections requires me to spend more time talking, writing on the board, guiding through handouts, or something equally preachy. I have not yet been able to develop an inductive strategy to meet this objective, nor have I been able to find one on the internet. Working with decimal approximations on the calculator would be one way to tackle such a problem, but it would require the belief that the calculator is magical and always correct. I try very hard to dismantle the calculator myth in my classroom, so such an exercise would be highly counter-productive.

Differentiated learning in a classroom of three students just happens. It becomes obvious very quickly though informal assessment which students are not understanding the material, and the plans change accordingly. It is more difficult with more teachers, because each teacher might not see what the other teachers see, so communication is essential. Differentiation in the classroom took place in the form of assigning different students different problems, and also in questioning, when different students were asked questions relating to different depths of understanding.

In the future, to better address the learning needs of my students, I think it is important to seek out inductive strategies when possible, and to be patient and avoid rushing students. I need to be always committed to the vision of mathematics as a unified, interconnected web of knowledge and never forget that it cannot be understood piecemeal.