Pedagogy is where the nitty gritty of teaching really comes into play. Pedagogy refers to the method of instruction a teachers uses in their classroom. There are dozens of different strategies a teacher can use to deliver content to students, so decisions on pedagogy are done on a daily basis. One of the problems that I struggle with, in regards to my pedagogy choices, is the conflict between pedgogies that deliver content, and those that are designed for practicing content.
I like to use scavenger hunt activities in my classroom. This pedagogical strategy involves a set of problems that students solve, using their answer to move on to the next problem. Students continue this process until they have solved all the available problems and circle back around to their original problem. This is a great way for students to self-check, ensuring instrumental understanding (Skemp, 1978), and is often used to get students up and moving around the classroom. However, there are some practical problems with this process. Firstly, the larger your class is, the harder it is to space things so students can access the problems as necessary. Secondly, this is a pedagogy for practice, not initial delivery of content. Students need to be able to solve the problems in the first place, before they can participate in the scavenger hunt.
There are fewer pedagogies aimed at initial delivery of content, and then you have to choose carefully. Because I believe relational understanding (Skemp, 1978) is important, I like to give students the opportunity to explore new concepts when possible. I am considering trying out a combination of inquiry and the socratic questioning method. For example, I like to give students an inquiry activity when I introduce exponent rules. We begin with the definition of exponents, and then I let students consider and explore what happens when we multiply powers with the same base. Eventually, students discover the rule themselves, and we add it to our notes. This could be a good time to have a socratic questioning session, so students can process the parameters based on what they observed. By putting details under scrutiny, students would later be able to avoid common operational mistakes, due to their deeper understanding. This strategy would have its drawbacks as well. Allowing students to explore takes time, and some students who struggle with the base concepts, may be unable to see the connections. Nor does all content work well from an inquiry basis.
The truth is, it can be difficult to align pedagogy with in-depth learning goals, and the constraints of a demanding curriculum. I am torn between a desire to give my students a deeper understanding of math, and the necessity of getting through content. I end up doing a lot of explicit teaching and modeling in my classroom when introducing new content. Sure, I sprinkle in other pedagogy methods, like questioning and graphic organizers, but it still feels like a lot of direct instruction happens in my classroom. Is this a failure on my part, or an unavoidable part of high school math teaching? I still struggle with this question, even after 8 years.
Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15.