Standard 6: Attend to precision

The Standard:
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Classroom Observations:
Teachers who are developing students’ capacity to "attend to precision" focus on clarity and accuracy of process and outcome in problem solving. A middle childhood teacher might engage his students in a "number talk" in which students use an in/out table and a plotted graph to "guess [the teacher’s] number." An early adolescence teacher might distribute cards with different symbol strings to his students, asking them to mingle to group and categorize their symbol strings, explaining and defending their groupings. A teacher of adolescents and young adults might continually probe her students to defend whether their requirements for a particular quadrilateral will always be the case, or whether there are some flaws in their group’s thinking that they need to refine and correct. Visit the video excerpts below to view multiple examples of teachers engaging students in attending to precision.

## Connections to Classroom Practices

5th - 6th Grade

Mathematically proficient students try to communicate precisely to others... They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.

Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he wraps up the number talk, and the learners mention many different ways of representing the rule: x3 – 3, times 3 minus 3, 3x – 3. Dickinson notes that “So we’re doing a lot of talking about this rule. What is the rule? Can we write a rule here?” As the students respond, Dickinson notes some disagreement among the student responses and asks his students to explain their thinking to each other. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others), standard 7 (look for and make use of structure), and standard 8 (look for and express regularity in repeated reasoning).

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Mathematically proficient students try to communicate precisely to others.... They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately... In the elementary grades, students give carefully formulated explanations to each other.

In the closure section of his numerical patterning lesson, Dickinson chooses student pairs to present their thinking. His selection shows a progression of presenters that increases with sophistication and accuracy. He notes that “This ramping up allows learners the best chance to wrap their minds around the conversation that ensues… Note how we end with some clear disequilibrium in the room, yet we do have a bit of closure. I purposefully do not ‘give an answer,’ especially since the nature of this investigation was dissecting two different solutions.” This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others).

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6th Grade

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... They are careful about specifying units of measure.. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Joe Condon works with his 6th grade students to identify strategies for comparing unit rates. In these clips, he reads a definition of rate developed and expanded by the class over the two-week period, then engages the students in a lesson testing out strategies for calculating unit rate.

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... They are careful about specifying units of measure.. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context... By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Joe Condon presents a lesson study for 6th grade students and observing teachers to identify strategies for comparing unit rates. In these clips, students are asked to name some familiar or common rates. Students then try to give their own definition of rate after looking at these examples. Finally students are given a ratio with no words and asked what it could mean.

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7th Grade

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

Jacob Disston leads a lesson on connections between ideas about equations, inequalities, and expressions, helping students to use mathematical vocabulary for a purpose to describe, discuss, and work with these symbol strings. In this clip, his students have grouped themselves by the common features of the symbol strings. Disston then asks the students to explain their groupings, saying “Oliver, you had another whole category. Who are you standing with? … Why are you guys saying you guys are similar? You want to tell us?” He involves the whole class in evaluating the groups’ justifications. This clip is also indicative of standard 1 (make sense of problems and persevere in solving them) and standard 3 (construct viable arguments and critique the reasoning of others).

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

Disston asks his students to focus on a category of symbol strings (expression, equation, inequality), separate them, and discuss how they know how to categorize them. Students are asked to make statements like "I know this is an equation because..." or "I think ... what do you think?" The companion commentary by math coach Linda Fisher notes that “While everyone is using the same language of equations, students seem to have a variety of definitions for what that means. Some students name it an equation because there’s multiplication. For others the variable signifies equations.” Disston’s subsequent work in the lesson asks students to develop their thinking about the differences between symbol strings.

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

Disston creates opportunities for his students to use mathematical vocabulary for a purpose: “How do we pull the inequalities all together?” Students seem to puzzle over words for describing signs: there’s addition, subtraction, and other signs like equal. They understand that the difference is significant, but don’t know how to classify it. Their writing promotes further discussion and negotiation of the definitions.

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

Disston engages his students in the second task of describing and categorizing symbol strings, saying “I want you to now sort the equations into different types—pull them into which equations are like other equations. What subgroups are there?” The students discuss and defend their thinking to each other and to the teacher.

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

Disston and his students share their insights into the day's lesson and reflect on what they learned. He spends time discussing various ways that groups sorted equations: “We’re going to see—did we all come up with the same subcategories? What did different people see as important here?” This provides students with an opportunity to make all the knowledge or thinking from their groups public, and honored the idea that there could be different categories.

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9th-10th Grade

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... By the time they reach high school they have learned to examine claims and make ... use of definitions.

Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, she orients students to the task and explains how they are to communicate their ideas to one another in their group work. This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In these clips, students engage in the first of two block-length explorations of their proofs. The students are in groups of four. Each group has chosen a group member to perform these roles: a team captain, resource manager, recorder, and facilitator. The groups have access to the problem (one page per group), two packets of manipulatives, and other resources that they can retrieve, including a page of definitions of quadrilaterals. In some of the groups, students work individually for a while. In other groups, students work in pairs, and in still others, the entire group of four is collaborating. Humphreys’ commentary notes subtleties in the students’ discourse that either advance or impede the development of their thinking. This clip is also indicative of standard 1 (make sense of problems and persevere in solving them), standard 3 (construct viable arguments and critique the reasoning of others), and standard 7 (look for and make use of structure).

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... By the time they reach high school they ... make explicit use of definitions.

In the closing of the group work on the first day, Humphreys refers her students to the idea of “mathematical friends.” This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others), standard 7 (look for and make use of structure), and standard 8 (look for and express regularity in repeated reasoning).

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

In the opening of the second day, Humphreys shares students’ work sheets to illustrate how the students were thinking while investigating the quadrilateral. After sharing work, she has students write to a prompt about how well they keep track of their thinking. The students then share with the class their individual reflections. The class is still learning how to prove a conjecture. Each group begins to settle on the quadrilateral they will formally prove. The students use their prior knowledge of parallel lines and congruent triangles to approach the proofs of the quadrilaterals. This clip is also indicative of standard 5 (use appropriate tools strategically).

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose... By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Continuing their explorations of the properties of quadrilaterals, Humphreys’ students work through their understandings of congruent triangles, the triangle postulates, parallel lines, transversals, and other geometric properties to apply those to create proofs for the quadrilaterals. The students move between group work and whole class interaction throughout the lesson. In small groups, the students discuss and debate proof arguments. At selected times, the teacher pulls the class together to share findings, ideas, or sample justifications. After sharing ideas or arguments with the entire class, students then return to working in their small groups. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others).

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... By the time they reach high school they have learned to examine claims and make explicit use of definitions.

In this clip, Humphreys’ students continue their group based work to develop their thinking in proving the properties of quadrilaterals. One student asks her group, “How would we prove it?” and discuss which quadrilateral to focus in on. Her seat mate proposes, “If the two diagonals are not congruent, are perpendicular, and bisect each other, then the figure is a rhombus?” A third student responds, “What’s the difference between a rhombus and a (gestures to a drawing)?” The first two students respond, “All sides are congruent.” This clip is also indicative of standard 5 (use appropriate tools strategically).

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9th - 12th Grade

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context...By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In this clip, he pushes a group of students for a higher order of thinking that can advance the conversation. He observes, "I want the answer to model what mathematical thinking should look and sound like. It should have reasons, it should maybe point the way towards the generalization or trajectory or a strategy to be in the service of some mathematics that's a little bit bigger."

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning...they have learned to examine claims and make explicit use of definitions.

In this clip, Carlos Cabana's students work in groups with confidence; in his commentary he observes that the students are evidencing strengths that they had developed over time. Students advance each others' thinking and give each other encouragement to back up their reasoning.

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Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In the first clip, Cabana analyzes a group's work in terms of logical flow and carefulness of organization, trying to make a guess about what they really understand. In the second clip, a student group demonstrates that they have mastered the conceptual pieces and are engaged in moving to the next step; Cabana works with them, trying to assess the extent to which they see the conceptual whole.

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