What is Practice?

Hello!

Today’s blog is devoted to the idea of practice. When I say practice, I am referring to the idea of “drill and practice”, that is, doing many similar problems in a row to develop procedural fluency. My placement has recently switched over to the Interactive Mathematics Program curriculum, which is a heavily problem-based mathematics curriculum. While this curriculum has been great for fostering student’s problem solving and critical thinking skills, many people (students, parents, and faculty) are concerned the curriculum does not have enough practice built into it. Being so, the idea of practice was a hot topic at our most recent department meeting.

There is no doubt that practice makes perfect. I don’t think there is a single educator who believes there is no need for practice in the classroom. That being said, doing 20 isolated, repetitive problems for 30 minutes is not particularly cognitively simulating, and this time and energy could be better spent. Another idea that is built into this whole practice debate is the difference between conceptual understanding and procedural fluency.

Conceptual understanding is the knowledge of the content in a deep, connected, and meaningful way. On the other hand, procedural fluency is the ability or skill of using algorithms and formulas. For example, knowing the area of a right triangle is 1/2(bh) and being able to substitute values and carry out the calculation would show procedural fluency, while knowing the formula is derived by cutting the area of a rectangle in half would show conceptual understanding. Now, when many people talk about practice, they are referring to honing procedural fluency. However, if students have the conceptual understand and problem solving skills from IBL, they can derive the formula themselves on a test if they forget it, while a student with nothing but procedural fluency would be stuck if they ever forgot the formula. While this is a fairly simplistic example, I believe it illustrates the overall distinction between the two domains. That being said, I do want to make it clear that I believe having both conceptual understanding and procedural fluency is very important in mathematics. Therefore, how can we foster procedural fluency in a meaningful way, while still developing a conceptual understanding?

My MTF made an excellent point at our latest faculty meeting.

“Teachers love to make sports metaphors. ‘If you want to be better at baseball, you go out in the back yard and hit 100 baseballs’. This works because when you hit 100 baseballs, you get immediate feedback. You see the ball slam into the dirt or fly over the fence. When students do 30 or 40 problems on their own they do not get that feedback”. (I am paraphrasing here).

I thought this was an important distinction, and it also supports our standards-based grading system that has promoted constant and immediate feedback. With that being said, it is clear that the traditional form of drill and practice is not the most productive exercise, so, what does practice look like in IMP, or in an IBL classroom in general?

At our department meeting, our principal laid out his idea of practice as he understands it.

“Practice is not a long list of repetitive problems that will never be reviewed or revisited. Practice is a small set of carefully chosen problems that support the curriculum and will most likely be review”. (Once again, paraphrasing). In essence, this sounds like the direction we want to move in. However, like with all things, it is much easier said than done.

In our classroom, my MTF and I have been using our standards-based Learning Target Assessments (LTAs) as practice. One or two times a week we give the students a short two or three question quiz, grade it on the spot, and give the students immediate feedback. This lets the students know if they hit the ball over the fence or into the dirt. Also, if students continue to not demonstrate understanding on these LTAs, they can come after school or during a study hall for more practice. This model has been working well for us in the sense that every students knows where they are at and where they need to be, as well as letting us know where every student is at. Another upside of this model is that if students are proficient and don’t need the practice, they don’t have to waste their time on a 40 question worksheet.

All that being said, I am really interested in how others work practice into their classrooms. Is your model more drill and practice, more feedback intensive, or something completely different? Let me know in the comments, and as always, I welcome any suggestions or ideas!

Thanks for reading 🙂