Ms. Snyder has just presented a challenging task to her fifth-grade students. They know something about subtracting fractions but have never solved problems with unlike denominators. Snyder gives her students time to complete the mathematical task. She waits for them to come up with their own solution strategies and then hopes to orchestrate a whole-class discussion around those strategies that will best help students understand the key concepts underlying subtracting fractions with unlike denominators. The students present a variety of strategies, so Snyder now faces the challenge of first choosing a few of the strategies to discuss and then sequencing them in a productive way. How does she make these decisions?

To ensure that their whole-class discussions are more than just a time to share different solution strategies (Stein et al. 2008), many teachers are using the Five Practices model to guide them in orchestrating whole-class mathematics discussions that focus on understanding:

1. Anticipate an array of possible student-generated solution strategies to a mathematical task before implementing the lesson.

2. Monitorstudents’ work as they grapple with the task.

3. Select a subset of the student-generated solution strategies to be shared and discussed during the whole-class discussion.

4. Sequence the selected student-generated solution strategies in a coherent way.

5. Connect the solution strategies in ways that will highlight important mathematical ideas.

These five practices are intended to make orchestrating whole-class discussions around student-generated solution strategies more manageable, but they are still difficult to enact.

Although the selecting and sequencing practices seem especially critical, they might be difficult to learn, so I studied them in more detail. I wanted to better understand the thinking behind the selecting and sequencing decisions that teachers make so that I could help prepare future teachers. During one semester of a mathematics methods course for elementary school teachers, I worked with twenty-three preservice teachers (PSTs) on a set of selecting and sequencing solution strategies activities; twenty-two of the PSTs agreed to let me use their written work.

Throughout the activities, PSTs were asked how they would select and sequence solution strategies in ways that promote the mathematical learning goal of the lesson and to explain why they would select and sequence the solution strategies in this way. The PSTs read and discussed the Five Practices article (Smith et al. 2009) to become familiar with the Five Practices model before (a) considering the mathematical task and learning goal (see fig. 1)

and then (b) thinking about how and why they would select and sequence the anticipated solution strategies to this task. Because PSTs generally have difficulty generating multiple solution strategies to mathematical tasks (Borko and Livingston 1989), I furnished the anticipated solution strategies (see fig. 2).

The PSTs’ rationales for the ways they selected and sequenced solution strategies could be grouped into three categories. The purpose of this article is to describe these three categories and to show how one of them has the potential to translate into a more productive whole-class discussion than the others. Before you read any further, take a moment to solve the task in figure 1 and carefully read the mathematical learning goal and its underlying concepts. Then, think about how you would select and sequence the solution strategies in figure 2 for a whole-class discussion. Why would you select and sequence them in this way?

Why did PSTs select particular solution strategies?

PSTs selected solution strategies for pedagogical reasons (pedagogical moves) and two mathematical reasons (mathematical procedures and underlying concepts). In this section, I will explain each category and present an example from the PSTs’ rationales for each category. These examples are from PSTs’ rationales for selecting solution strategies from figure 2. Each rationale could be classifiedinto more than one category because PSTs gave more than one reason for choosing a solution strategy.

The first category of rationales for their selections is called pedagogical moves. Considering pedagogical moves when planning discussions is always important for teachers to do, and in fact all the PSTs justified their selections that were related to pedagogical moves. However, some students provided rationales that primarily focused on pedagogical moves and failed to include deeper consideration of the mathematical learning goals. For example, consider part of Francine’s rationale:

I will show solution strategies b, c, f, and d. I want to make sure students are getting a visual representation of the problem as well as being able to see the problem in a systematic manner. . . .

By showing a misconception that they can just do the operation straight across without changing, they will understand that a change must be made to the fractions.

Francine’s rationale focuses mostly on superficial features of the solution strategies that she found attractive, such as a misconception in one strategy and a visual representation in another strategy. She appears to be selecting so she can implement particular pedagogical considerations, like moving from a category with a misconception to a correct strategy. But she is not using the mathematical learning goal to help determine the direction of the whole-class discussion. Surface-level features, such as visual representations, seem to be driving her selection decisions.

Category 2: Mathematical procedures

The second category for selecting rationales is mathematical procedures, which gets a little closer to analyzing solution strategies for the potential to promote the mathematical learning goal. Mathematical procedures rationales address important practices related to a learning goal but do not directly address the underlying concepts related to the mathematical learning goal. All the PSTs provided rationales that were related to mathematical procedures. For example, consider part of Eduardo’s rationale:

If I was teaching this class, I would choose solution strategies a, b, d, and f. Solution strategy a: They knew that the original denominators, 5 and 6, are both multiples of 60. Though they did not articulate how they found the equivalent fractions with 60 as the denominator, they seem to understand that the common denominator is essential. . . . Solution strategy b: I would want them to share with the class why common denominators are so important. Solution strategy d: This group was very similar to solution strategy a. I think it is important to contrast an answer with the denominator of 30 with a denominator of 60. . . . Solution strategy f: They found equivalent fractions by dividing their sixths by 5 and their fifths by 6.

Eduardo highlights mathematical procedures related to subtraction of fractions, such as finding a common denominator and finding equivalent fractions. But he does not analyze the solution strategies in terms of their potential to reveal the three underlying concepts of the mathematical learning goal in figure 1.

Category 3: Underlying concepts

The third category for selecting rationales is underlying concepts, which directly address the underlying concepts related to the mathematical learning goal. Of the PSTs, 54.5 percent provided selecting rationales that were related to underlying concepts. Dominic’s rationale seems to have the most potential to highlight the key mathematics concepts of the task. Consider part of Dominic’s rationale:

I would select b because . . . it is important to make sure students know that the denominator represents the piece size and the numerator represents the number of pieces. I would select d because it addresses the fact 5/6 and 3/5 do not have the same-size pieces . . . The students subtracted the number of pieces they got from 18/30 from the number of pieces they got from 25/30 to get 7/30. This strategy is correct and addresses finding a common denominator and the fact that you are subtracting the number of pieces you have and not the size of the group, like strategy b did. I would show strategy f, because . . . this would really help them see that the denominator represents the size of the pieces and the numerator represents the number of pieces. These students represented 5/6 by cutting a whole into 6 pieces and shading 5 pieces in and 3/5 by cutting a whole into 5 pieces and shading 3 of them in. When they realized the piece sizes were different, they cut the whole of sizes 1/5 pieces into 6 parts for each piece so that both wholes had pieces of size 1/30.

Dominic’s rationale attends to the underlying concepts of the mathematical learning goal. First, he notes how each solution strategy selected has the potential to promote the first two underlying concepts: the meaning of the numerator and the meaning of the denominator. Second, he mentions the third underlying concept (the necessity of same-size pieces for subtracting two fractions) when he explains why he selected solution strategies d and f. In particular, he explains how the visual representation in solution strategy f could help illustrate that the piece sizes are different. Based on their rationales, it seems like a whole-class discussion led by Dominic has the potential to be more productive than the others because the underlying concepts seem to be driving Dominic’s selecting decisions. Because Dominic has clearly analyzed the solution strategies for the underlying concepts, he might be more likely to orchestrate a whole-class discussion during which the underlying concepts of the mathematical learning goal are made public to the entire class.

Practical suggestions for selecting and sequencing

Francine, Eduardo, and Dominic had similar selections of solution strategies—they all selected solution strategies b, d, and f from figure 2. But their rationales for their selected solution strategies varied to the extent that they were driven by the mathematical learning goal. Smith and Stein (2011) recommend that the first step to engaging in the Five Practices is identifying a mathematical learning goal and a mathematical task that aligns with this goal. Then, anticipate an array of solution strategies that students might generate to the task. Once the mathematical learning goal, mathematical task, and anticipated solution strategies are in place, planning for selecting and sequencing solution strategies can begin. The remainder of this article presents two suggestions to develop a plan and rationale for selecting and sequencing solution strategies in ways that will direct the whole-class discussion toward the mathematical learning goal.

Keep the mathematical learning goal front and center when selecting

A mathematical learning goal should describe what a teacher would like students to understand by the end of a lesson. “Describing learning goals precisely requires unpacking them into component goals or subgoals” (Hiebert et al. 2007). Table 1 shows an example of an unpacked mathematical learning goal and one that is not.

One way of uncovering the underlying concepts that lie behind a mathematical learning goal that is related to a mathematical procedure would be to work out the procedure in detail and think at each step about what knowledge is needed to understand why that step works the way it does. For example, underlying concept c of the unpacked mathematical learning goal in table 1 could be uncovered by performing the standard algorithm for addition and thinking about why the “little one” is written above the tens column.

After the underlying concepts of the mathematical learning goal have been identified, analyze the solution strategies first and foremost for their ability to reveal the underlying concepts of the mathematical learning goal. Analyzing the solution strategies might be easier if the underlying concepts have been clearly specified ahead of time so that what one needs to look for in the solution strategies becomes evident. If a solution strategy does not seem to have the potential to promote any of the underlying concepts of the mathematical learning goal, then do not select this solution strategy for the whole-class discussion. Keeping the mathematical learning goal front and center will help focus a teacher’s attention on selecting solution strategies for their mathematical potential and not only to fulfill a pedagogically based sequencing strategy (such as starting with a misconception and building up to a correct solution strategy). Make sure to double check that the compilation of selected solution strategies has the potential to attend to all the underlying concepts of the mathematical learning goal and not just some of the underlying concepts.

Compare and contrast to help formulate a sequence

To construct a rationale for sequencing solution strategies that would be grouped under the underlying concepts category, a suggestion is to analyze the selected solution strategies for similarities and differences that are related to the underlying concepts of the mathematical learning goal. Dominic makes an underlying concepts connection when he says in reference to solution strategy d,

This solution is correct and addresses finding a common denominator and the fact that you are subtracting the number of pieces you have and not the size of the group like solution strategy b did.

He is connecting solution strategies b and d by contrasting how solution strategy d attends to the meaning of the numerator and denominator concepts and solution strategy b does not. It would make sense for Dominic to place solution strategies b and d next to each other in the sequence so that this connection can be highlighted to the class. Consequently, after analyzing solution strategies for underlying concepts connections, it makes sense to cluster solution strategies with connections together in the sequence. Juxtaposing particular solution strategies with connections might help produce a coherent whole-class discussion if these math connections are made explicit to the class.

Pedagogical considerations are not irrelevant when considering how to sequence students’ solution strategies. They can be useful but are best considered after decisions have been made on the basis of the mathematical potential of the strategies. A teacher can finalize the sequence by using such pedagogical moves as (a) starting with a misconception and building up to a correct solution strategy or (b) starting with a visual representation and building up to more abstract representations (Stein et al. 2008). These pedagogical moves should be used only after the solution strategies have been selected based on the underlying concepts of the mathematical learning goal and have been analyzed for similarities or differences related to the underlying concepts.

To select and sequence solution strategies successfully during a whole-class discussion, teachers need to plan ahead. It is difficult to (a) select solution strategies that will promote the learning goal, (b) compare and contrast solution strategies, and (c) use pedagogical considerations to formulate an impromptu sequence in this order during a lesson. If teachers plan ahead, it makes sense to follow these suggestions in this order to ensure that no parts of the mathematical learning goal are missed and so that connections can be made easily between or among solution strategies. If you are teaching a lesson to a mathematical task for the first time and do not know anyone else who has taught the lesson, then it might be difficult to anticipate students’ solution strategies, and you might have to manage these suggestions on the spot. Teachers should not expect to select and sequence solution strategies perfectly the first time they teach a lesson. To improve for the next time they teach the lesson, teachers can keep a careful record of students’ solution strategies that come to the surface during the lesson and then use these to develop a plan for selecting and sequencing solution strategies the next time they teach the lesson.

Helping students achieve ambitious learning goals

The underlying concepts of the mathematical learning goal must be clear to the teacher for the teacher to help make these underlying concepts clear to the students. These underlying concepts must guide all decision making for a whole-class discussion—selecting, sequencing, and connecting. If teachers consider only pedagogical and mathematical procedures when selecting solution strategies, then they might fail to make explicit the main mathematical learning goal during the whole-class discussion. Consequently, keeping the mathematical learning goals front and center is important to guide selecting, sequencing, and connecting decisions.

Unpacking learning goals for underlying concepts is difficult work, as are then selecting and sequencing solution strategies in ways that align with a mathematical learning goal. But this kind of work is crucial in creating whole-class discussions, which help students achieve ambitious learning goals. As teachers move toward meeting the Common Core State Standards for Mathematics (CCSSI 2010), adopting practices like these could give students richer opportunities to meet these more ambitious learning goals.

Borko, Hilda, and Carol Livingston. 1989. “Cognition and Improvisation: Differences in Mathematics Instruction by Expert and Novice Teachers.” American Educational Research Journal 26 (4): 473–98.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Hiebert, James, Anne Morris, Dawn Berk, and Amanda Jansen. 2007. “Preparing Teachers to Learn from Teaching.” Journal of Teacher Education 58 (1): 47–61.

Smith, Margaret S., Elizabeth K. Hughes, Randi A. Engle, and Mary Kay Stein. 2009. “Orchestrating Discussions.” Mathematics Teaching in the Middle School 14 (May): 548–56.

Smith, Margaret S., and Mary Kay Stein. 2011. 5 Practices for Orchestrating Productive Mathematical Discussions. Reston, VA: National Council of Teachers of Mathematics.

Stein, Mary Kay, Randi A. Engle, Margaret S. Smith, and Elizabeth K. Hughes. 2008. “Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move beyond Show and Tell.” Mathematical Thinking and Learning, An International Journal 10 (4): 313–40.

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Chris Kalmbach- 11/18/2016 1:00:48 PMWhile reading this I realized that I need to place a greater emphasis on the "Anticipating" stage of the 5 Practices, as the developing of a clear learning target helps with selecting student work samples that reflect the intended learning goals. I think my selection and sequencing have relied more on what the author calls "Pedagogical moves" rather than an initial focus on samples that demonstrate the underlying concepts intended by the lesson. Great article that gave me lots to think about and provided some actionable next steps to depen my use of the 5 Practices! Thanks!

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