Conclusión cruce padres, niños y profesora

The divergence in enacted practices were the most evident in the nature of mathematical discourse that was exemplified by each pair. Lillian and Tiffany exhibited a conceptually-oriented stance toward mathematics teaching while

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Rebecca and Stephanie exhibited a “computationally” oriented stance toward mathematics teaching (Thompson, Thompson, & Boyd, 1994). Specifically, Lillian and Tiffany focused “students’ attention away from thoughtless application of procedures and toward a rich conception of situations, ideas, and relationships among ideas” (Thompson, Thompson, & Boyd, 1994, p. 46). Additionally, Lillian and Tiffany’s expectations for student explanations reflected sociomathematical norms (Yackel & Cobb, 1996) that valued: (1) Explanations that consist of

mathematical arguments, not simply descriptions of procedures or summaries of steps. (2) Capitalizing on errors as valuable opportunities for discussion,

exploration, and reconceptualization. (3) Understanding the relationships among multiple strategies. (4) Collaborative work that involves individual accountability and consensus reached through mathematical argumentation. Students were expected to explain and justify their solutions and to use different representations to support those explanations. Additionally, Lillian and Tiffany capitalized on student errors as opportunities to clarify and deepen mathematical

understanding. Discourse played a major role in both Lillian and Tiffany’s

mathematics classrooms. The prominent role that mathematical discourse played in the classrooms is a reflection of the value that both teachers place on

verbalizing and discussing mathematical ideas. Lillian and Tiffany were

intentional about setting the expectation for student talk with their students. Lillian reminded her students, “You’re going to solve, share with your partner, and then we’re going to talk about it.” Tiffany instructed her students, “Be prepared to share your information. We will take a look at a couple different ways people

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solved it.” Both teachers also encouraged student-to-student sharing of ideas by employing strategies such as, “knee-to-knee, toe-to-toe.” Lillian frequently

encouraged her students to, “Turn to your partner and explain how you solved the problem.” Tiffany insisted, “In a classroom there should be many

opportunities for the children to voice their learning.” This occurs, she explained, through asking questions such as, “How did you figure that out?” Lillian shared this sentiment stating, “There’s value in talking.” Lillian explained the benefits of mathematical discourse in stating, “I think it’s helping them, like, firm up their understanding and it’s helping me see what they know.” Tiffany also described the benefits of student discourse, “I think definitely for different viewpoints to come into play and I think sometimes children are more responsive to their peers showing them a different way. They can put it in a language they understand, connect with it.”She continued, “I think they realize as we talk about

mathematics how to use math in the everyday world, that it’s not isolated.” Lillian suggested that student discourse is an essential component of mathematical proficiency.

She described evidence of mathematical understanding in the following way:

The ability to solve problems in diverse ways, justify those problems, listen to other people’s justifications for theirs, and think about yours. Like, making connections to other ways to solve it. Not thinking that there’s just one, like, one way. Um, basically, being able to solve problems and

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Lillian and Tiffany both encouraged multiple mathematical representations (Practice #3) as tools for developing mathematical understanding (Practice #6) and facilitating student discourse (Practice #4). Tiffany explained, “The children being able to represent their work in different ways and also, included with that, that they can verbalize what they’re thinking. I think if they can verbalize it then they really understand what they are doing.” Lillian described how she thinks students learn mathematics best in stating, “I think through doing and through talking about it and through representing problems.” Tiffany explained to her students that “a stranger should be able to look at your work and be able to figure out what you did.” In response to a student who said, “I did it on my fingers.” Lillian said, “Show me what you did on your fingers. How could you represent that using a drawing? I want you to practice drawing it so you can see what you are doing.” In response to a student’s explanation about making a ten and then adding five more, Lillian encouraged, “Use the ten-frame so everybody can see in their head what you’re talking about.”

Finally, both Lillian and Tiffany used student talk to capitalize on errors that occurred during problem solving tasks. Tiffany noticed two students who were leaving gaps and overlaps when completing a measurement task. She called all of the students to the carpet and said, “I have some measurement strategies I want you to be aware of.” She explained, “We’re going to observe, not judge, and then we are going to talk about it.” Each student then

demonstrated her measurement strategy. Once the demonstrations were complete, Tiffany said, “Sit like morning meeting so we can talk about our

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observations.” She continued, “Tell me what you noticed. Would you think that was an accurate way to measure?” She then allowed the students to share their observations and offer suggestions for how the measuring would be more

accurate. Tiffany summarized the students’ suggestions stating, “Remember, we talked about the importance of accuracy with measurement around the world. Make sure the ruler is at the very end. You have to mark it.”

Lillian used a similar strategy when two students presented a solution in which the model, the equations, and the student explanations did not match. Lillan instructed the students to, “Turn toward your partner and figure out what happened. How does this strategy compare to what Laney shared?” She then facilitated a discussion that helped the students clarify their thinking.

The sociomathematical norms (Yackel & Cobb, 1996) exemplified in Lillian and Tiffany’s teaching practices are reflected in the Mathematics Teaching

Practices (NCTM, 2014) in which teachers implement tasks that promote

reasoning and problem solving (Practice #2), use and connect mathematical representations (Practice #3), facilitate meaningful mathematical discourse (Practice #4), pose purposeful questions (Practice #5), build procedural fluency from conceptual understanding (Practice #6), and elicit and use evidence of student thinking (Practice #8). The sociomathematical norms and Mathematics

Teaching Practices share commonalities with the STEAM instructional

approaches. Namely, STEAM instructional approaches prioritize problem solving, authentic tasks, inquiry, process skills, student choice, and integration.

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Conversely, the nature of the mathematical discourse that occurred in Rebecca and Stephanie’s classrooms was teacher-directed. They demonstrated a “calculational” orientation toward mathematics teaching by focusing on the problem to be solved, prioritizing the answer, and maintaining expectations for student explanations that were shallow and incomplete (Thompson, Thompson, & Boyd, 1994). The teachers led the conversations and explicitly modeled mathematical representations with no explanation of how the representations were related to each other or to the mathematics. They quickly corrected student errors with no explanation on the part of the teacher or the student of where the flaw in thinking occurred.

Rebecca and Stephanie did express the belief that student talk should play a major role in mathematics teaching and learning.

Rebecca explained:

Student talk is huge. For example, this morning with that worksheet there would be a child who didn’t get it and so the rest of the table, they were explaining it to them how they needed to complete the worksheet for morning work. You can just sit back and listen and it’s very interesting to see how the way I explain it to them may not necessarily be how their peers explain it. Their peers might explain it better than I can, just because they’re on the same level. Um, so the kids, I mean, in every center for math they’re constantly talking. They’re talking it out. So, for instance, at the light table they’re building structures of MagnaTiles. They have to communicate and talk about what they want to build and who’s going to

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use what pieces because we don’t have a thousand pieces. So they have to share, they have to talk it out. They have to, you know, use math talk to build a structure that’s going to stand and not fall over. Um, same thing with Legos, same thing with if they’re doing a puzzle, you know, “Who has this piece?” They’re looking at different flat sides and curved sides on puzzles. They’re constantly talking about math. They don’t realize it, but as a teacher if I sit back and listen, they really are talking a lot and a lot of it is problem solving. “How can we work together to create a structure? How can we work together to finish a puzzle?” or “How can I tell you how to complete this sheet because you’re just not getting it?” So, I think it’s huge, I mean especially in kindergarten and especially because we do centers. They have that opportunity to talk, where in past times, I’ve taught math whole group and there’s no talk. They just, it’s me talking and them answering questions and I feel like having math centers in kindergarten has really helped open up the talking and problem solving among peers. Contrary to the beliefs expressed by Rebecca and Stephanie, the nature of the mathematical discourse that was observed in each of the classrooms was teacher-directed and teacher-centered. For example, while guiding students through applying both the number line strategy and the counting on strategy, Stephanie asked very rote and predictable questions such as “Did he start in the right spot? How many hops? What’s the sum?” and ”What do I put in my head? Count up how many times?” Rebecca instructed the students, “Write it on your

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board, don’t tell me.” The following exchange between Rebecca and her students exemplifies the teacher-centered nature of the discourse in these two classroom. Student: “I know what six plus six is!”

Rebecca: “You do? What is it?” Student: “Twelve!”

Rebecca: “Good.” Hushes other students who are trying to join in the

conversation reminding them, “Make sure you have a bubble in your mouth.” While Rebecca and Stephanie utilized different representations for

mathematics problems such as the ten-frame and the number line, the focus was on the representation itself, not on the use of the representation as a tool for facilitating mathematical discourse or understanding mathematics. Both teachers modeled explicitly how to “do” the strategy. Rebecca modeled the use of the ten- frame stating, “We have an empty ten-frame. How many dots are missing? Remember, where do you start when you’re filling in your ten-frame? Do you start at the bottom? No, you start at the top.” The students then copied the ten- frame in their number books. Similarly, Stephanie modeled the use of the number line for addition repeatedly instructing the students to “start with the biggest number and count up.”

Finally, Rebecca and Stephanie treated students’ errors much differently than Lillian and Tiffany. While the nature of Lillian and Tiffany’s mathematical discourse reflected sociomathematical norms (Yackel & Cobb, 1996) by