Main Content

Lesson 1: Thinking About Mathematical Thinking

Relational and Instrumental Understanding

Have you ever heard the phrase "Ours is not to reason why, just invert and multiply"? If you're anything like me, your reaction to that is probably something like Ugh! None of us who care about students learning and truly understanding mathematics would ever want to teach using mnemonics, rhymes, or other "tricks." On the other hand, most of us can probably understand or be sympathetic to reasons teachers sometimes fall back on using shortcuts. Can you think of times when you've resorted to "teaching through tricks"?

From the Classroom

I have to admit that I used to teach my eighth graders to say, "IS over OF is equal to n over 100?" Many of the students were grateful to learn this trick, and they reported to me that it served them reliably on standardized tests over the years. For me as a teacher, it felt good to provide them a tool that they could turn to. But I was also unsettled on the larger messages I was sending them about mathematical understanding and—just as importantly—the opportunities for developing number sense and estimation skills that I was neglecting.

–Dr. Andrea McCloskey, Course Author

 

The next article talks more about the issue of teaching through tricks. The article, by Richard Skemp, was first written in 1976, but was reprinted in the magazine Mathematics Teaching in the Middle School in 2006. Skemp does a great job discussing the drawbacks of teaching through tricks, but he is also honest about some of the benefits. The terms relational understanding and instrumental understanding are no longer used very much, but Skemp's descriptions of these two contrasting types of understanding is still quite illustrative. (Consider adding the terms relational understanding and instrumental understanding somewhere in your concept map.)

One interesting site where mathematics teachers discuss the appropriate use of teaching tricks is Nix the Tricks. I encourage you to look around and want to point out that, in addition to publishing a book, this project maintains a publicly accessible, living document that you can read and contribute to. You are welcome to join any of the discussions if you care to engage, but please do note that this a Google document open to the public. Any information you provide, including personally identifiable or work-related information, will be available to the general public.

Reading

Skemp, R. R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95.


Top of page