On Cooperative Learning, Routines, and Practices
The implementation of discussion protocols and the construction and inclusion of written reflections and explanations in each student’s touchstone atlas is tantamount. Cooperative learning is equally essential to the progression of this unit and is embedded in every activity through Math Language Routines (MLRs) and Standards for Mathematical Practice (MPs). “In a systematic review of 15 studies of talk in mathematics classrooms, Kyriacou and Issitt (2008) found that good learning outcomes result when teachers use questions not just to seek right answers, but also to elicit reasons and explanations.”21 As the mathematical lexicon is vast, students need support in developing their productive and receptive math language.
Math Language Routines, developed by the Stanford Center for Assessment, Learning, and Equity, were created to support sense-making, optimize output, cultivate conversation, and maximize meta-awareness as students acquire mathematical language. 22 The table below is not exhaustive but displays some of the ways the Touchstone Atlas unit implements the MLRs.
TABLE 1. Implementation of Stanford’s Math Language Routines in the Touchstone Atlas unit
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Math Language Routine |
How the Touchstone Atlas promotes the routine |
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MLR 1 Stronger and Clearer Each Time |
Through Pair Shares and switching partners after receiving feedback, students will improve their ability to defend their choices and explain their thinking. |
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MLR 2 Collect and Display |
Teacher will display student maps and a collection of student ideas she observes from listening in and will ask students to compare methods, as well as point out benefits and drawbacks of a strategy. |
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MLR 3 Critique, Correct, and Clarify |
The teacher will present students a model of a writing sample with incomplete explanation and students will be tasked with offering suggestions for refining the explanation, to promote more complete reflections in the written sections of their respective atlases. |
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MLR 4 Information Gap |
Students will reflect on information gaps in their mapping and articulate them to one another. In a more traditional use of the routine, students can play an information gap game such as “guess my scale factor.” |
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MLR 5 Co-Craft Questions and Problems |
Students will consider what kinds of questions can be answered from the maps they create. In turn, they will see the limitations as well. |
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MLR 7 Compare and Connect |
Students will determine “which one doesn’t belong” as they consider whether shapes are scaled copies of one another and provide evidence. |
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MLR 8 Discussion Supports |
Teacher revoices student responses using mathematical language and provides sentence stems as needed. |
A 2022 study on the neuroscience behind cooperative learning has found that students working in groups of three experience temporary synchronization of neural patterns when they worked to reason, explain, and elaborate on solving a problem together. Further the synchronization became enhanced on brain scans when the group reached consensus. 23 Students learn through their interaction with one another. As students implement mathematical practices such as constructing and critiquing the reasoning of others, they will work toward reaching consensus. The dissonance they hold en route to consensus is equally as important as the consensus. Disagreements create the need for clarifying a point and constructing a stronger argument that makes use of mathematical structure. Triads make it easier to ensure that all group members are contributing to the work, offering suggestions, asking questions of one another, and reasoning together. Consistent implementation of MPs helps to develop mathematically proficient students, and collaborative learning encourages them to meet proficiency interdependently.
TABLE 2. Demonstrating how the Touchstone Atlas unit promotes the standards for mathematical practice.
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Mathematical Practice |
How the Touchstone Atlas promotes the practice |
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CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. |
Students are presented with mapping tasks without being explicitly taught how to create them. Students are expected to make mistakes and include their mistakes in their atlases to discover how their work evolves. |
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CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. |
Students will create coherent representations of problems and make sense of the relationship between quantities and units as they sure up their models. |
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CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. |
The teacher will collect and display maps and ask students to record what they notice and wonder. Teacher will build skill by displaying her own maps, modeling how to engage in conversation about them, and progress to having students display their maps while the class discusses observations. |
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CCSS.Math.Practice.MP4 Model with mathematics. |
By virtue of consistently creating maps, students “routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.”24 |
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CCSS.Math.Practice.MP5 Use appropriate tools strategically |
Students will consider the need for various types of technology to help them communicate ideas and confirm their legitimacy. Technology includes mapmaking software such Open Street Maps as well as rulers, tracing paper, varied colors, pattern blocks, etc. |
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CCSS.Math.Practice.MP6 Attend to precision. |
Making an accurate scale drawing necessitates careful measurement to maintain the proportional relationship. |
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CCSS.Math.Practice.MP7 Look for and make use of structure. |
As students move to working with proportional relationships in unit 2, they will recall how they found the scale factor when presented with a figure or figures by dividing the dimensions of the new image by the dimensions of the old image new⁄old, and determine that they can find the constant of proportionality, k by dividing y⁄x. |
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CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning. |
Students will analyze what is revealed by their modeling and derive calculation shortcuts to implement when the model is not available. |
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