Today, I returned to school from a very much needed spring break. From my jolting return from winter break, I knew to expect things to be bumpy coming back, and the students did not disappoint. However, I think I was better prepared this time around.
I can’t say that we accomplished a whole lot of any of my classes today, besides (hopefully) getting the kids focused on doing work again. I hit them with a quiz that they probably weren’t well prepared for, which generally is not a very good idea. However, I think most students actually took it fairly seriously, and I hope it was a bit of a wake up call to them that I’m not wasting any time getting them back to work.
Twenty-two instructional days remain before the HSA, and things are not looking particularly pretty. During quarter 3, I think I tightened up my routines a lot, but I still wasn’t seeing much, if any, gain in results. This last quarter, I need to try to change up my approach.
Some things are going to have to wait until next year. Speaking of which, my fellow TFA first year algebra teachers and I have started talking about some things we want to iron out over the summer so that hopefully we can avoid the debacle that this year has been for all of us. One of the toughest questions is how to approach the teaching of algebra as a subject, given the students we have and the way they are assessed by the HSA.
The HSA is actually a very rigorous test of students’ abilities to apply the skills of algebra and data analysis to real-world problems. Nowhere on the test do you see problems simply demanding the students to “solve for x”. Most of the questions require the students to analyze a word problem carefully to find out what it’s asking for, then to synthesize skills from different parts of the curriculum to answer that question.
The problem in preparing students to do this is two-fold. First of all, my average student is shaky on basic arithmetic and not at all comfortable with the concepts and skills he/she should have mastered in pre-algebra. Secondly, the curriculum provided by the City teaches the concepts in isolation. Whereas the HSA questions usually require several skill to be applied at once, the assessment questions I’m provided in the curriculum are usually very one-dimensional. Even more problematic is the fact that the materials provided by the curriculum are even more simplistic.
So my dilemma is that in my limited time, with little appropriate material provided, I’m supposed to be teaching the students to the complex process of assembling skills to solve problems, when the students struggle to perform each skill itself.
Faced with this situation, the holy grail is to be able to teach the students the concepts, from which the skills will follow. If a student truly understands equations and the rules of algebra, they don’t need to know the step-by-step process of solving an equation, that follows naturally from their own understanding. Teaching concepts is extremely hard, especially when faced with an extremely diverse set of very frustrated students, who aren’t used to being made to think critically. And so the temptation is to teach the skills one-by-one, because most students can follow a step-by-step process. But the eventual issue is that the students will never truly understand the why behind what they’re doing, and there’s simply no way to teach an algorithm for every single type of problem the student might happen to see on the HSA.
Teaching concepts directly is simply not realistic. I really believe that only the brightest math student have the abstract thinking capabilities to work that way. I think I’m a reasonably sharp mathematician, and even I struggled in high school math when it got too abstract, and I had fantastic math teacher my entire life.
What I’m looking to do is find a happy medium. I’m envisioning a curriculum that focuses on teaching students the “tools” of algebra. By tools, I mean the things we write on paper to solve algebraic problems–the expressions, the graphs, the patterns tables, and so on–and the rules for using them. Instead of focusing on the abstract theory of negative numbers or on specific methods of subtracting them, I want to show the students what a number line is and how to use it to see the difference between two numbers. I want to show them how two number lines form a graph, and how they can use that graph to see the relationships between two numbers. I’m hoping that by knowing the tools inside and out, the concepts will follow as the student sees how the different tools–function tables, graphs, and equations, for example–relate to each other and show the same underlying concept in different ways. And with that understanding of the tools, I’m hoping the students will have an easier time attacking these complex HSA problems and be much better prepared for higher math. We’ll see I guess, but pretty much anything’s better than what I’ve had going this year.