Читать книгу Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong - Страница 11

In our work with teachers, we often see students being taught a list of “key words” that are linked to specific operations. Students are told, “Find the key word and you will know whether to add, subtract, multiply, or divide.” Charts of key words often hang on classroom walls, even in middle school. Key words are a strategy that works often enough that teachers continue to rely on them. They also seem to work well enough that students continue to rely on them. But as we saw in the history assignment problem, not only are key words not enough to solve a problem, they can also easily lead students to an incorrect operation or to an operation involving two numbers that aren’t related (Karp, Bush, & Dougherty, 2014). As the history assignment problem reveals, different operations could successfully be called upon, depending on how the student approaches the problem:

1 A student could use subtraction to determine that of the students in the class made graphic novels.

2 A student could use division to find the number of students in the class, dividing the 9 students doing graphic novels by of the class to get 27, the number of students in the whole class. This could even be modeled using an array solution strategy like the one in the student’s drawing seen earlier.

Let’s return to your imaginary classroom. Having seen firsthand the limitations of key words—a strategy you had considered using—where do you begin? What instructional approach would you use? One of the students mentioned a strategy she likes called CUBES. If she learned it from an elementary teacher and still uses it, you wonder if it has value. Your student explains that CUBES has these steps:

Circle the numbers

Underline important information

Box the question

Eliminate unnecessary information

Solve and check

She tells you that her teachers walked students through the CUBES protocol using a “think-aloud” for word problems, sharing how they used the process to figure out what is important in the problem. That evening, as you settle down to plan, you decide to walk through some problems like the history assignment problem using CUBES. Circling the numbers is easy enough. You circle (podcast) and 9 (students), wondering briefly what students might do with the question “How many?” Perhaps it’s too early to think of that for now.

Then you tackle “important information.” What is important here in this problem? Maybe the fact that there are two different assignments. Certainly it’s important to recognize that students do one of two kinds of history assignment. You box the question, but unfortunately the question doesn’t help students connect to 9 with a single operation.

If you think this procedure has promise as a way to guide students through an initial reading of the problem, but leaves out how to help students develop a genuine understanding of the problem, you would be correct.

What is missing from procedural strategies such as CUBES and strategies such as key words, is—in a word—mathematics and the understanding of where it lives within the situation the problem is presenting. Rather than helping students learn and practice quick ways to enter a problem, we need to focus our instruction on helping them develop a deep understanding of the mathematical principles behind the operations and how they are expressed in the problem. They need to learn to mathematize.