Encouraging Math Students to 'Make Sense of Problems'
As I've written before, the Standards for Mathematical Practice are a unique feature of the Common Core State Standards. They describe the habits and methods demonstrated by proficient math students.
In illustrating how to "make sense of problems," the first practice standard, Ginsburg offers this example of a problem he was given as a new teacher. (If you were a fan of Cheryl's Birthday, you may like this one as wellthough I found it a lot less brain-twisting.)
Place the numbers 1 - 8 in the grid, using each number once, such that no consecutive numbers are in boxes that touch vertically, horizontally, or diagonally.
Ginsburg explained that the problem takes most people about 20 minutes to solve. But, he writes:
[A] handful of teachers and students, most notably a 4th grader who was an average math student, have nailed this problem in a minute or two. What did they do that the rest of us didn't do? Simple: They thought about the problem before trying to solve it. They made sense of the problem, and the solution became obvious to them. The rest of us, on the other hand, just grabbed a pen or pencil and plugged in numbers.
The common core practice standard puts it this way: Proficient students "plan a solution pathway rather than simply jumping into a solution attempt."
The first educator to introduce me to the idea of "sense-making" in mathematics was Dan Meyer, a Stanford University doctoral candidate and former math teacher, best known for his viral TED Talk.
Meyer told me in 2011 that teachers need to be "delegating the sense-making of math to students." The way to do that? By being "less helpful." That is, by throwing a problem like the one above up on the board, and not saying anythingletting the class try it on their own or talk it out if they want to.
Too often, textbooks (and teachers) hand-hold students through a solution, Meyer says. But wouldn't the value of "planning a solution pathway" be better learned by having students see some of their peers answer the puzzle in under a minuteand then asking how they did it?
Would be great to hear from math educators (and anyone who tried the problem above) in the comments section below.