I have taught algebra every year since gaining my teaching certification nine years ago. I have been working slowly on my master’s degree for the last several years, and at every chance I get, I try to apply what I learn to my teaching: teaching algebra. Every year, I have thought about what I teach, and how I teach it. The algebra I am teaching in my class right now, does not look like the algebra I was teaching my first year. It has evolved as my expertise and understanding have improved.
I have spent a great deal of time trying make my lessons more interconnected. Algebra flows from one topic to another, building on previously understood ideas, while maintaining certain themes and ideas throughout. These connections have become apparent to me as my knowledge of content and teaching (KCT)(Hill & Ball, 2009) improves, and I share as much as possible with my students. I have specifically designed my lessons to build upon one another so that no idea feels like it is “coming out of nowhere.” For example, I take the time to show how multiplying and factoring are opposites (like other inverse operations used in algebra), so students understand why they use particular techniques for factoring. These connections are important because I want my students to be able to reach relational understanding of algebra, not just an instrumental understanding (Skemp, 1978).
The more I teach algebra, the better I know where students will struggle. Although I could never anticipate every difficulty that every student will have, I do learn where there is most likely to be confusion. My classroom has a very mixed group of abilities, from students with math learning disabilities, to students set on the advanced track. It is my responsibility to teach them to the best of my ability, no matter who they are. My knowledge of the content and my students (KCS) (Hill & Ball, 2009) has led me to adjust my teaching. For example, I have greatly simplified the way I teach factoring, so that students are not given 4 different ways to factor. While my advanced students could handle 4 different ways to factor, my students who struggle with math found it difficult to recognize when to use each different strategy. Now, I teach one main way to factor, that will factor every quadratic expression students will encounter (assuming it is factorable). I then try to bring attention to various patterns and ‘shortcuts’ that students can use, if they see them. My advanced students love the shortcuts, and my students that struggle with math are able to continue with their main way of factoring if they are more comfortable with that.
Even after nine years I am not done learning and adjusting my teaching. I have more ideas I want to incorporate, about data and social justice, about intentionally building a cooperative learning community in my classroom. My evolution as an algebra teacher is still ongoing.
Hill, H., & Ball, D. L. (2009). The curious – and crucial – case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71.
Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15.