To do math, or to understand math? That is the question. Well, why not both? My math colleagues and I have had this discussion before. Where we debate on whether to just teach the procedure, or why the procedure works first. We are not the only ones, math teachers address this question a lot (Gauthier, 2021). It is an important question, given the demands of fitting our curriculum into the school schedule without missing anything. I teach Algebra 1, which means my students need to be ready for Algebra 2 by the time they leave my classroom. So it is important that I set them up for success, but what does that mean?
I am a firm believer in what Skemp (1978) calls relational understanding. This means a deep understanding that fits ideas within a larger schema that explains and connects ideas across content and contexts. It is not enough to me that I know how math works, but that I know why it works. This is in stark contrast to how I was taught math growing up, which was largely instrumental, or procedural. I learned a rote procedure to get the answer to math problems, and then was expected to memorize it and use it going forward. Now, I was never bad at math, I just could not see how the pieces fit together. Math felt like a messy jumble of rules and procedures with very little rhyme or reason. It was really once I started digging into math and moved from learning it to teaching it that my understanding of math moved to a deeper level. Now I can see math as this beautiful continuum of ideas and patterns, and that is what I want to share with my students.
One of the first things I do with my algebra classes is give them a live problem-solving activity. I use both my knowledge of how math fits together across grades and content, horizon content knowledge (HCK) and my knowledge of content and students (KCS) to design this lesson for more impact. I want students to make connections across math classes (HCK), so I give them no solving directions just the problem prompt. I know (KCS) students come in with knowledge of how to work backwards to solve problems; a process most of them have internalized so much, they find it difficult to explain it. In my first lesson activity, students have to work their way backwards to get to the answer, and explain how and why they think they are correct. I take this process they know and draw a direct line to how solving equations works in algebra. Suddenly solving goes from separate, difficult to decide steps to a simple reversal of the operations they see in the problem. If they see addition, they reverse it with subtraction. If they see multiplication, they reverse it with division, and vice versa. I stress the pattern, how it relates to the operations, and how to work backwards to get to the answer. Students can see why the process works, and this makes them more flexible problem-solvers and thinkers.
It is my hope that students leave my classroom not just able to ‘do’ algebra, but seeing that math is connected and designed to make sense.
References:
Gauthier, J. (2021, January 8). Let’s Talk about High-Quality mathematics Instruction. Michigan Council of Teachers of Mathematics – Let’s talk about high-quality mathematics instruction – part 1. Retrieved January 23, 2022, from https://www.mictm.org/page-1075226/9663708
Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15.