I read an article titled Calculational and Conceptual Orientations in Teaching Mathematics by Thompson et. al. It blew my mind when I was a young teacher. For many of you this concept may not be new but for me it was career changing. When I visited classrooms, it was obvious which orientation the teacher practiced and it helped me relate to teachers from their perspective.
In a nutshell, calculational orientation means you may not know why the math is the way it is, but you can do it. There is a heavy focus on knowing the math and the process steps. Conceptual orientation means you understand why and how the math works and is applied. There is a heavy focus on the why and how. Below is a table to compare and contrast the two:
In a nutshell, calculational orientation means you may not know why the math is the way it is, but you can do it. There is a heavy focus on knowing the math and the process steps. Conceptual orientation means you understand why and how the math works and is applied. There is a heavy focus on the why and how. Below is a table to compare and contrast the two:
In a classroom setting with a limited mathematics block it might seem easier to take the calculational route. When I taught using calculational orientation, my lesson was very much like this: Here is the math. We will apply it to these 20 problems that you will do in class and what you don’t finish will become homework. I will meet with a small group for intervention while the rest of you work on these problems individually.
In a conceptual orientation lesson, a whole mathematics block might be used to build the concept without students solving any additional problems. For the teacher, it might appear or feel like you are behind. I hope that even with the pressure of teaching and getting so much math done in one class period, you still consider the conceptual approach, for the sake of how much it benefits your students. I spoke with my 17-year-old son, who at the time of our conversation was a junior in high school taking a college statistics course. On the topic of conceptual learning he said, “My world feels so much more open when teachers allow me to do things, like being told a question and told to figure it out. Quite plainly, I had no clue what a variable was until my ninth-grade teacher, who when she taught variables, asked us what a variable is. Before that moment, I always just solved for an equation. I never thought about what it meant. When a teacher skips steps, I skip those same steps. I have no idea what I missed because I was never taught it.”
Conceptual orientation empowers students with discovery and understanding that calculational orientation skips. It is time consuming on the front end but embeds conceptual foundations that do not have to be "retaught" throughout the year. In this sense, teaching with a conceptual orientation is actually the better investment.
In a conceptual orientation lesson, a whole mathematics block might be used to build the concept without students solving any additional problems. For the teacher, it might appear or feel like you are behind. I hope that even with the pressure of teaching and getting so much math done in one class period, you still consider the conceptual approach, for the sake of how much it benefits your students. I spoke with my 17-year-old son, who at the time of our conversation was a junior in high school taking a college statistics course. On the topic of conceptual learning he said, “My world feels so much more open when teachers allow me to do things, like being told a question and told to figure it out. Quite plainly, I had no clue what a variable was until my ninth-grade teacher, who when she taught variables, asked us what a variable is. Before that moment, I always just solved for an equation. I never thought about what it meant. When a teacher skips steps, I skip those same steps. I have no idea what I missed because I was never taught it.”
Conceptual orientation empowers students with discovery and understanding that calculational orientation skips. It is time consuming on the front end but embeds conceptual foundations that do not have to be "retaught" throughout the year. In this sense, teaching with a conceptual orientation is actually the better investment.
