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VISIBLE THINKING IN MATH: NAMING, SEQUENCING, AND CONNECTING Math StrategiesOne “Visible Thinking” strategy in mathematics that we encouraged among the lessons was the idea of eliciting multiple representations, naming strategies, and evaluating efficient uses of representations to develop conceptual understanding, and make connections among representations (see Figure 6.1). In the thirdgrade lesson, preplanning discussions focused on how teachers might expose students to multiple strategies (use of concrete manipulatives: coins, pictures, table, graph) to solve the problem while giving students choice. Evidence from the jointly planned research lessons indicated that teachers were interested in having students compare their strategies and used questioning as a way to elicit students to look for patterns. One useful tool that teachers used was our problemsolving strategy cards. Teachers laminated the cards and attached magnets and placed them for students to pull down when sharing strategies. This allowed for students to name their strategies. When there was a strategy that students could not name within the strategy cards, they would pull down the student’s strategy card and name it with the student’s name. This allowed that student with a unique yet worthwhile strategy to have his name to fame and allow for others to refer to his/her strategy as they connected the strategies shared. The naming process using the strategy cards also helped the class community learn efficient strategies and keep a collective record of the strategies that could be used as a future reference related problems. Along with the strategy cards, we encouraged teachers to use Stein and Smith’s (2011, p. 8) framework called the Five Practices, which breaks down the important process in orchestrating thoughtful mathematical discourse. The Five Practices include: (1) anticipating likely student responses to challenging mathematical tasks; (2) monitoring students’ actual responses to the tasks (while students work on the task in pairs or small groups); (3) selecting particular students to present their mathematical work during the wholeclass discussion; (4) sequencing the student responses that will be displayed in a specific order; and (5) connecting different students’ responses and connecting the responses to key mathematical ideas (see Figure 6.1). Figure 6.1 Nameable problemsolving strategies. In their study, Stein, Engle, Smith, and Hughes (2008) stated that when teachers used the Five Practices to anticipate student strategies, they were more adept at identifying multiple strategies during the lesson, and thus better at selecting the student artifacts, and sequencing the order of presentation. This purposeful and thoughtful planning allowed for mathematics discourse to be rich with meaningful mathematics (see Figure 6.2). Figure 6.2 Five practices for orchestrating math talk (Smith and Stein, 2011, 8). Through these intentional teaching moves, the lesson study team for the “Saving for the Present” focused on how students demonstrated their understanding using different strategies and representations. The teachers did not have a prescribed method for the students; instead, students were given the choice of using an approach that made most sense to them. This also allowed for teachers to analyze the most common approaches among students and how they interacted with the mathematics. The following is an excerpt from the thirdgrade teacher’s reflection that described students’ interaction with the problem and their problemsolving strategies, I had two boys come up to the front of the room and pretend they were the brothers who were saving their money to buy their mom a birthday present. My students were very interested to watch their peers “act it out.” One of my students, a very articulate little girl, created her own table. Her table was very close to what I was hoping to find so I had her present it to our class. Working off of her table, I had my class look for patterns on the white board where she was using subtraction. Together we established that the amount of money in Nick and Dan’s piggy banks were decreasing. When we “discovered the pattern” I observed that several light bulbs were steadily going off in my students’ heads. Having students work with the “five representations” allowed the class to name the different strategies: using numbers and symbols, manipulatives, words, graphs, and tables. The teacher and the observers shared their noticings that students had access to the problem with the physical manipulatives of the coins but when they started putting the coins in their “piggy bank,” it was apparent that they had difficulty keeping track. They counted the coins that went in the bank but could not keep track of the amount each of the characters had put in without the help of a recording sheet or a table, which a girl in the class started generating. By working in collaborative groups, others in the group observed and started to mimic her method. The students in these groups instinctively wanted to draw a picture or create a table, which a lot of them did, but then they could not translate that into words or a number sentence. We also felt that the use of dry erase boards presented a problem because students would erase their work and start over rather than build on their previous mistakes.—Teacher collaborator and observer at the thirdgrade research lesson. The idea of accessibility through multiple representations came up in the sixthgrade and the eighthgrade lessons. The sixthgrade teacher noted how the students took the problem and acted it out while one student kept track of the mathematics: Another group of students used a concrete model along with a table and tally marks to record their thinking. The three boys each took on a designated role in the problem. Shequeem was Aleah’s brother and Raul played Aleah while Daniel was in charge of keeping track of how much money had been spent with tally marks. This team of boys really worked together to both understand and solve the problem by using many represen tations—verbal, concrete, and with a table.—Sixthgrade teacher. Focusing on multiple approaches allowed students greater access to the problem and in turn, more success with the problem. Additionally, teachers promoted the idea of organizing information and keeping track of the changes as an important part of solving the algebraic problems and analyzing the recursive pattern. The lesson study team intentionally planned for the students to solve the problem in their most natural way, and then to give them opportunities to sit sidebyside with Figure 6.3 Two sixgrade students worked individually then compared strategies. Source: Authors. a partner with a Venn diagram to compare their approaches. This builtin activity allowed for rich discussion of comparison and connection between the representations. For example, in figure 6.3, one student compared the quantities by crossing off the same number of coins for each of the brothers, while another student kept track of the remaining balance using a table. One of the focal points of our lesson was to have students look at different strategies to solve the problem and also think about the similarities and differences among their solution paths. We had students use many of the multiple representations that we focused on in our class this summer—concrete, verbal, pictorial, graphically, and symbolic.Fifth grade teacher special educator. In order to connect the strategies, the teacher posed important questions for students. Pivotal questions for students: What are some similarities and differences among our strategies? How can you describe your strategy in one sentence? What is happening each day? How will you organize your coins? How will you keep track of how much money Nick and Dan have? What tool could you use to help you solve the problem?—Excerpt from 3rd grade research lesson. As teachers codesigned, observed, and revised their lessons, they discussed the vertical algebraic connections stating where in the continuum they would find most of their students in their algebraic reasoning with this particular problem. For example, teachers observed how most of the students in the earliest stages of the learning progression represented their thinking through a repeated subtraction method or using a comparing quantity approach to show the decreasing values. They also had difficulty fully understanding the word problem and setting up mathematical statements for the relationship that was being expressed. The lesson study team discussed how the majority of students used the most common two methods we had seen in younger classes—a chart and a drawing. This idea became a critical discussion point. Can students become so fixated by a pictorial or concrete representation that it prevents them from looking for more efficient strategies? Or do most learners need entry into a problem using a concrete, pictorial, or tabular approach to make sense of the problem before advancing to an algebraic equation? What is the teachers’ role and how do teachers use questioning to advance students’ thinking when they are “stuck”? How can we use classroom discourse and representations as a means of moving student from inefficient to sophisticated strategies? The host teacher also made some “horizontal connections” as she related this problem to other algebraic problems with linear equations called the cell phone plan problem where students find the best plan using similar strategies as this problem. 
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