Summer School Goals

What can I possibly teach this group of students tomorrow / next week / next month? This question has been one that troubled me throughout the year, and all too often I found myself asking about tomorrow rather than thinking as far ahead as I would like, but with summer school, things have been a little easier.

I have four students this summer. Some failed pre-algebra, some failed transitions to algebra. Luckily for me, and unluckily for the students, the pre-algebra and transitions are in fact the same course. So is eighth grade math. And seventh grade math. Most of sixth grade is devoted to the same concepts as well, which are then re-taught throughout the high school math sequence.

This is not just a rant about the absurdity of the system (though such absurdity is significant). If affects the goals that I choose for my students. These are not always the objectives I write on my lesson plan and on the board. I do that because I get observed. The goals that I have been trying to chose for my students involve deeper levels of learning, connectivity and understanding than most of the objectives in the frameworks target.

These goals are much more appropriate for my students (in terms of development, past mathematical experiences, and student needs). Recognize patterns. See mathematics as something that stems from fundamental truths about the way that numbers interact, and begin to see that the rules governing these interactions are not arbitrary, that in fact numbers could come together no other way. Build confidence. Create a framework of language that will allow each student to process thoughts about math in logical ways. My students have been "taught" the tricks of math so many times. They've been told that when they multiply exponents they add the exponents, and other equally nonsensical things, but they do not see the connections, they do not see the why, and so the rules all get jumbled when there are no reasons supporting them.

My instructional strategies so far have been very different than they were last summer, and throughout last year. Last year, my goal was to survive. This year, I have loftier ambitions. I have a vision of math as I want them to see it. To that end, I have changed the way that I speak and give instructions. I pay very close attention to the language that I use, defining and re-defining math terminology, and making my instructions much clearer and more specific. I try to provide more definite structure to my lessons, and place enormous stress on the continuity of ideas, the connectivity of concepts, and the multiple paths to solutions. I stopped saying the word "answer" so that I can always be sure that both I and the students know what it is we are talking about. We are never trying to find the answer to a math problem, we are always trying to find something, and if we cannot name that thing for which we are searching, we may never recognize it when we find it.

One inductive strategy that I created this summer was used to introduce exponent rules. I gave students several products and quotients of exponential expressions to simplify, without mentioning the exponents rules, and offered them "clues" from a clue jar to help them solve the problems. The clues showed similar products and quotients that had been simplified correctly, and the students were left to deduce the rules for themselves. The activity was appropriate in that it forced all my students to think and will help them remember the exponent rules much better in the future. However, it did not do enough to make these rules more than simply rules, even though they may be more memorable since they discovered them themselves. I followed the activity up with a formal definition of the rules and an explanation of the symbolic language used in the definition, then a lecture and discussion about why the rules, especially the multiplication rule, must be true as long as we understand the definition of exponents. At that point, it was obvious that the definition of exponents had not completely sunk in yet, but I think that may be something I'll have to keep plugging away at.

Although not exactly relevant to the assignment, I feel I ought to mention confidence. Confidence in a summer school class is bound to be low. Everyone failed regular class. Yet confidence is essential to learning, and I feel that it is my responsibility as a teacher to build the confidence of my students back up, and so I've made that one of my goals. We're on the right track on that one, but it comes slowly, and I hope that our first round of quizzes will not crush that.

I haven't yet spoken much about how to assess these goals, mostly because that is a question that I am still struggling with. But I'm working on it.