Learning and Teaching of School Algebra

Again, algebraic thinking is a student's ability to think abstractly about numbers and symbols, as well as manipulate equations in order to come to conclusions about a once unknown quantity. Kieran suggest that algebra has been taught more simply as a procedure, and teachers crossed their fingers hoping that those that enjoyed working with numbers got it, and those who didn't were able to figure out how to survive. However there is hope for the future as Kieran suggest algebra is more of a generalized, problem solving tool, able to be modeled, and applied to the work at hand. Being a math major and having taken abstract algebra, I think the most important thing about algebraic thinking revolves around the use of functions and variables. The key to algebraic understanding then lies within a student's ability to make sense of the operations they see every day, (+, -, x, /) and how they act on quantities, whether known or unknown. And that understanding comes from a beginning introduction to mathematics and a students notion about what it means to add and multiply (as well as their inverses, subtractions and division). Thus, what I want my students to understand is that algebra is not merely a procedure that can be applied under certain circumstances, but rather a system for which mathematics is constructed, a way of understanding the world rather. I would want my students to develop a habit of thinking about ways in which to interpret ideas mathematically. Because when it comes down to it, most students can do the procedures but fail when given a word problem that expresses the same thing. This tells me that they didn't quite understood the math in the first place, and that they need to develop a deeper understanding of what mathematics, specifically algebra, really means. The biggest problem comes into play when we talk about how to teach our students to think algebraically. Of course it is important to give students multiple representations of the same material. Kieran does an excellent job of laying out a plethora of ways in which students can think about equations and variables, including balancing models, geometric models, arithmetic models, as well as a couple more. Things like algebra tiles, scales, or virtual manipulatives can all aid in acquiring an understanding of what exactly equations are and what it means to manipulate quantities. The other key part of this is that enough time should be allowed where students can really wrap their head around complex ideas like variables and functions. Students really just need plenty of time to master the skills AND understandings that surround algebra. Of course high expectations and emphasis on the process of problem solving, not the product can also help students get on the right track, especially when standardized testing and the state of mathematics education has driven them so far off the path to mathematical competency. Lastly I think it's important to value student backgrounds when teaching algebra, because it is likely they have developed ways of understanding the world that can align with algebraic thinking (like the notebook model Kieran suggests). Not only does involving student's lives show them that you respect who they are and where they come from, but it also helps to scaffold their learning in a really genuine and relevant way.

Memo #5 Generalizations

Generalizations are a key component to the acquisition of algebraic thinking. Driscoll suggests that it is important to not only generalize arithmetic, but also functions and relations between numbers. For example, observing that powers of 2s (ie 1, 2, 4, 8, 16, 32...), 3s (1, 3, 9, 27, 81...) and so on all grow larger, but numbers like 1/2 to positive powers grow smaller and smaller, approaching 0, helps students grasp the concept of the function that is "repeated multiplication." In a way, if we think about fostering algebraic thinking, rather than procedural fluency that is warned about in Kilpatrick and Swafford, then we will students building rules for themselves, and understanding the functions they are applying, or tasks they are following through with.

One of the large hurdles for students is systems of equations, a topic that Driscoll touches on in Chapter four. As in many mathematical situations, sometimes all of the computation is not necessary, and the skillful student will see patterns and develop strategies to take shortcuts while still arriving at the same answer. The hardest part about all of this is inspiring students to pick up on the knowledge for themselves, since problem solving and algebraic thinking is not the easiest to teach.

I would suggest that instead of modeling problems, teaching procedural strategies, and then letting them go to fend for themselves on the homework, that instead we foster a general curiosity and a genuine interest in math by allowing them to come to terms with new ideas on their own. By this I mean that students come into the classroom with so much information already, and (I would like to think) an innate desire to learn. So instead of talking at them for 45 minutes, or an hour, or even two, we should rather entrust them with problems and puzzles that will instead lead to learning. Too often I see students who become frustrated and give up on problems that are not that different from what they know, but they don't see themselves as competent problems solvers. The truth is that they ARE capable of extending their knowledge, like Driscoll suggests. We just need to give our students plenty of time and support.