**Dr. Tom Sallee, Davis, CA**

Most parents want their children to do well in school and they are usually willing to trust the teacher. But because CPM does not teach in a familiar way sometimes parents become nervous for their child. After talking to lots of parents, I have come to believe that much of the conflict comes from a mismatch of expectations about what mathematics is.

I first came to understand this mismatch while reading the book *The Psychology of Learning Mathematics* by Richard Skemp, especially chapter 12 (this chapter is available separately at https://www.skemp.org.uk). Skemp pointed out that for many (most?) people, to “know mathematics” means to be able to know when to apply a rule or to replicate an algorithm and to do it properly. Then you know this little piece of mathematics. (Skemp calls this “instrumental understanding.”) For example, if you know the fact that you divide fractions by “invert and multiply,” you know that part of math. In contrast, if all you know is the rule, but not **why** the rule is true (Skemp’s “relational understanding”) most college instructors (certainly including me) would say you know only the beginning.

Notice that if “knowing math” just means knowing the rules and procedures, then lecturing *seems* effective. If you want students to know a specific rule and be able to use it today, then explicitly tell them the rule, and let them practice many examples. Ask them to use the rule the next day and they can probably do so. Everyone is now happy: the teacher has taught the rule; students have learned it and the parents know that their child can use the rule. Success! This happy result will probably last a week but, too often, not a month.

To oversimplify a long history, the assumption has long been that if you know the rules, then in general these rules will get you by, and the brighter students will create the unified understanding necessary to advance for themselves. However, a German report about the 1997 TIMSS study found that students were *less likely* to learn either math or science if they had a lot of routine practice homework. Grappling with the ideas was more effective than lecture and practice in the long run.

CPM is about long-term learning—not regurgitating for tomorrow’s test. CPM starts from the assumption that understanding the big picture will help students develop the rules as an integral part of the interconnections they learn. In practice, there is a continual interplay between understanding rules and understanding the larger concepts and each reinforces the other.

To learn this way means serious intellectual effort. And serious thinking is harder than memorizing. Many centuries ago Ptolemy asked Euclid if learning geometry could not be done more easily and heard the reply, “There is no royal road to geometry.” The same is true today—not only for geometry but for algebra and every other part of mathematics.

It is understandable that parents want to make school easier for their children. It is too bad that easy ways are rarely effective.