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      To learn more about how your students went wrong with the history assignment problem, you call them to your desk one by one and ask about their thinking. A pattern emerges quickly. All the students you talk to zeroed in on two key elements of the problem: (1) the portion of students who did a podcast () and (2) the word “of”. One student tells you, “Of always means to multiply. I learned that a long time ago.” Clearly, she wasn’t the only student who read the word of and assumed she had to multiply by the only other number given in the problem. While a key word strategy led students astray in the first problem, visualizing the problem situation in the second problem led students to a correct answer, even if they were not able to write an accurate equation for the problem situation.

      Problem-Solving Strategies Gone Wrong

      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:

      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.

      What Is Mathematizing? Why Is It Important?

      Mathematizing is the uniquely human process of constructing meaning in mathematics (from Freudenthal, as cited in Fosnot & Dolk, 2002). Meaning is constructed and expressed by a process of noticing, exploring, explaining, modeling, and convincing others of a mathematical argument. When we teach students to mathematize, we are essentially teaching them to take their initial focus off specific numbers and computations and put their focus squarely on the actions and relationships expressed in the problem, what we will refer to throughout this book as the problem situation. At the same time, we are helping students see how these various actions and relationships can be described mathematically and the different operations that can be used to express them. If students understand, for example, that equal-groups multiplication problems, as in the history assignment problem, may include knowing the whole or figuring out the whole from a portion, then they can learn where and how to apply an operator to numbers in the problem, in order to develop an appropriate equation and understand the context. If we look at problems this way, then finding a solution involves connecting the problem’s context to its general kind of problem situation and to the operations that go with it. The rest of the road to the answer is computation.

      Mathematizing: The uniquely human act of modeling reality with the use of mathematical tools and representations.

      Problem situation: The underlying mathematical action or relationship found in a variety of contexts. Often called “problem type” for short.

      Solution:

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