Procedural and Conceptual Understanding

The terms relational and instrumental are no longer commonly used to describe types of understanding. More common in educational research, curriculum materials, and standards documents are the terms procedural and conceptual understanding. This is not to say that there is widespread agreement about the definitions of these terms and whether/how much they overlap—some people consider procedural and conceptual understanding to be very distinct (with conceptual considered more important than procedural), while other people consider the two to overlap quite a bit.

All this is to say that the debate over what constitutes mathematical understanding is alive and well!

For the final reading, we’re going to go back in history a little further with a 1973 article by a researcher named Stanley Erlwanger. He studied the mathematical understanding of a sixth grader named Benny. Benny’s mathematics instruction consisted of Individually Prescribed Instruction (IPI), a type of program that was popular at the time, in which students worked individually through cards or booklets that presented mathematics lessons and then took mastery tests to confirm whether or not they could progress. Some things about this program may seem old-fashioned to us now, but in fact this model is similar to many computer-based self-guided instructional programs used today. Benny’s understanding of mathematics topics is quite interesting and revealing. As you are reading, you should again be noting terms, descriptions, or ideas that you might want to incorporate into your concept map.