In my last post I suggested that public visions of what classrooms ought to look like need to change in order for more schools to enable students to develop the broad range of knowledge and skills they need. An example of an outdated vision is a Facebook post that has gone viral. In it, Jack Severt, the father of a second grader in Cary, North Carolina, expressed his exasperation at a mathematics problem his son brought home.

The problem stated that “Jack” had used a number line to solve a subtraction problem, 427-316, and asked students to explain what he did wrong. Severt wrote:

Dear Jack,

I have a Bachelor of Science degree in Electronics [sic] Engineering which included extensive study in differential equations and other higher math applications. Even I cannot explain the Common Core mathematics approach, nor get the answer correct. In the real world, simplification is valued over complication. Therefore,

427

-316

111

The answer is solved in under 5 seconds--111. The process used is ridiculous and would result in termination if used.

Sincerely,

Frustrated Parent

As Liana Heitin notes in her fine *Education Week* article on the changes the Common Core State Standards have brought about in mathematics, this problem is not a “Common Core mathematics” problem. In fact, the two lead authors of the Common Core State Standards in mathematics concluded that it was just badly written curriculum.

Yet the problem does illustrate a key aspect of the Common Core, and why the standards can help students learn mathematics more deeply. The “Frustrated Parent” might want to reconsider his complaint.

Traditionally, mathematics instruction has focused largely, if not exclusively, on procedural fluency. The idea was to ensure that students could perform basic calculations quickly and easily, perhaps in under five seconds, as Jack Severt did with the subtraction problem. To enable students to attain fluency, teachers assigned sets of problems so that students could increase their proficiency in solving them quickly. Word problems that presented real-world situations were included too, but these tended to be textual versions of number problems; fluent students could easily identify the structure of the problem and plug in the appropriate numbers.

The architects of the Common Core recognized that procedural fluency remains critically important. Students should not get bogged down in relatively simple calculations. That’s why the standards call for students to know the times table by the end of third grade, for example. But the Common Core drafters also recognized, based in part on the experiences of high-performing nations, that conceptual understanding is important as well. Without conceptual understanding, students can calculate well without knowing what the numbers mean. They would have no way of determining whether a solution is reasonable, or how to solve a problem that isn’t well-structured for them.

One way to teach for conceptual understanding is to show students that there are multiple ways to solve problems--for example, to show subtraction on a number line as well as a calculation. This approach is used commonly in Japan. There, a teacher poses a problem, and students write their solutions, one by one, on the board. The students then discuss the solutions, to explain what makes them reasonable or not. In that way, students understand what the problem represents, so that they can solve non-routine problems in the future.

Another way of teaching for conceptual understanding is to curtail some practices that are now common in schools. In another article in the Education Week series, Heitin says that some mathematics educators are questioning the need to reduce fractions to the lowest common denominator. She quotes Zachary Champagne of Florida State University: “Fifty-two one-hundredths is more than half. If you put that in lower terms, it becomes 26/50. If you go further, it’s 13/25. But that’s way harder to picture than 52/100.” Again, the goal is for students to understand what the numbers represent, rather than just go through the calculating motions.

To be sure, teaching for conceptual understanding can go too far, and students still need to become fluent and get the right answers. But without understanding, young people are unlikely to know how to go about solving a problem that isn’t tailor-made for them. And not being able to do that would “result in termination” as well.