Mathematize It! [Grades 6-8]. Kimberly Morrow-Leong

      Linda M. Gojakworked as an elementary mathematics specialist and classroom teacher for 28 years. She directed the Center for Mathematics and Science Education, Teaching, and Technology at John Carroll University for 16 years, providing support for districts and more than 10,000 teachers. Linda continues to work with K–8 mathematics teachers and coaches nationally and internationally. She is a recipient of the PAEMST from Ohio. She served as the president of NCTM, NCSM, and the Ohio Council of Teachers of Mathematics. Linda is the coauthor of three other books for Corwin Mathematics—The Common Core Math Companion, K–2; The Common Core Math Companion, 6–8; and Visible Learning for Mathematics, Grades K–12. Linda also wrote Path to Problem Solving for Grades 3–6 (ETA hand2mind, 2008) and What’s Your Math Problem? (Teacher Created Materials, 2011).

      CHAPTER ONE Introduction Why You Need to Teach Students to Mathematize

      Imagine you are a new teacher. You are teaching eighth grade at a new school and are eager to get to know your students—their interests, skills, and how prepared they are to meet the challenges of eighth grade. You have just emerged from your teacher education program knowing various approaches you have seen modeled in classrooms and described in the literature, some of which you have tried with varying degrees of success. You aren’t sure what approaches you want to use but are excited about challenging your students, introducing the rigor you have read so much about. But first, you need to know what your students can and can’t do.

      You decide to start with a couple of word problems, ones that involve relatively simple mathematical operations:

      Mrs. King wanted her American history students to do a project about the Emancipation Proclamation. of the class chose to make podcasts. The other 9 students chose to create graphic novels. How many students are in Mrs. King’s American history class?

       Armando started his descent into the cave. He was 10 feet down before he realized that he had forgotten to bring a flashlight. He climbed back up to the 2-foot mark to take the flashlight his friend handed to him. How many feet did he have to climb to get the flashlight?

      You circulate around the room, noting who draws pictures, who writes equations, and who uses the manipulatives you have put at the center of the table groups. While some students take their time, quite a few move quickly. Their hands go up, indicating they have solved the problems. As you check their work, one by one, you notice most of them got the first problem wrong, writing the equation . Some even include a sentence saying, “6 students will do a podcast.” Only one student in this group draws a picture. It looks like this:

      Even though the second problem demands an understanding of integers, a potentially complicating feature, most of these same students arrive at the correct answer, despite the fact that they do not write a correct equation to go with it. They write the incorrect equation 10 − 2 = 8 and are generally able to find the correct answer of +8, representing an 8-foot climb toward the cave opening. Some write K C C above their equation. You notice that other students make a drawing to help them solve this problem. Their work looks something like this:

      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:

      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

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