# Standard 6: Attend to Precision

## standard 6: attend to precision

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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.

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.

## connections to classroom practices

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Mathematically proficient students try to communicate precisely to others.

Mia Buljan’s 3rd-grade students defend their approaches to solving various problems in a multiplication number talk. Two students respectfully engage each other when one of the students makes an error in her reasoning. She recognizes her error and corrects it in her problem-solving approach.

In the elementary grades, students give carefully formulated explanations to each other.

After her 3rd-grade students work individually on various problems, Buljan invites them to “find the other people who have your problem and see if they agree with you.” In their conversations, the students explain their thinking clearly to each other, striving to reach consensus about the best approach and solution to a problem.

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

On the first day of the learning segment, Michelle Makinson engages her learners in a math talk focused on unit fractions, combining into wholes, “parts of,” and the idea of equivalence, using manipulatives to create and explain a visual representation of a contextualized representation / word problem. Her students then share with their partners, explaining their approach.

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

Michelle Makinson's students work in pairs to match cards showing different representations of fractional quantities, matching pictures with verbal descriptions, using sentence stems to follow a discussion process. They then create a justification card to describe their reasons for making a match. The sentence stems support students’ agreement, disagreement, requests for clarification, and restatements of each other’s ideas. Michelle circulates among the pairs, probing partners’ work and reminding them about the discussion process. In their discussions, they “try to communicate precisely to others.”

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

On the second day of the learning segment, Michelle Makinson's students return to their sentence starters and discussion sheets. They examine their work, looking for a set of cards that they matched with a particularly strong justification that they can “save-the-planet-level defend.” Pairs then share their justifications with the whole class, and Michelle challenges her students to communicate precisely and to evaluate their own justifications. She uses a “turn and talk” strategy to generate suggestions to improve the clarity of a justification. She models academic language: “Does anyone have a different strategy for enhancing the justification?” Students consider their own justification cards to make improvements: “Is all the evidence you need there? If not, add it.” This clip also relates to standard 7 (look for and make use of structure).

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

Students continue to create sets of cards that match each other (Green: Set or Area Model, White: Verbal Representation, Dark Blue: Contextual / Word Problem, Yellow: Justification). Michelle Makinson asks pairs to explain the connections between the cards. She challenges students to persevere in grappling with a scenario working with 22/12. The class works with incomplete sets of 12 and incomplete wholes, recognizing that students seek to make connections to real life and become confused about how an improper fraction works in reality. The students create connections between the representations based on their observations of patterns and structures. This clip also relates to standard 7 (look for and make use of structure).

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

On the third day of the learning segment, Michelle Makinson challenges her students to work with various representations of fractional wholes. She reminds students of the connections between set / area models, verbal representations, contextual representations / word problems, and number lines. Her students do a quick-write about which representation(s) made the most sense to them and why. Students then create multiple representations for different fractions and share them in various ways—with partners, by showing their whiteboards to the class, and by mingling with other students.

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

Michelle Makinson and her students work with a number line representation for a new fraction: 1 & 1/5. She identifies that a mixed-number fraction is a new concept for some students. She challenges them to create, consider, explain, and defend their number line representations to other students. With the fifth representation of a fraction, she gives students the choice of what representation makes the most sense to them. Most students select an area / set model representation.

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

Erika Isomura begins her lesson with engaging her 4th- and 5th-grade students in a conversation about the terms whole and part, activating their prior knowledge of work with mentor “string” problems, and asking the students to identify the parts and wholes in each scenario.

Her students then work with new problems, sorting and describing differences and similarities between the new problems and the ones they’d done before:

“Are any of these problems a lot like Jesus's problem, where he already knows his pieces or his parts, but he needs the whole amount? And are any of these problems like Camila’s, where she needs the pieces because she already has the whole?”

| In the elementary grades, students give carefully formulated explanations to each other.

As her 4th- and 5th-grade students work to describe and classify their problems, identifying which are more like Jesus’s problem (multiplying fractional quantities) and which are more like Camila’s problem (finding a fraction of a whole), Erika Isomura circulates around the classroom, questioning them and probing their understanding. Two of the problems she has given her students do not have visual representations; she invites her students to create drawings of those problems if it would be helpful to their process.

Mathematically proficient students try to communicate precisely to others…In the elementary grades, students give carefully formulated explanations to each other.

After her 4th- and 5th-grade students’ initial sorting process, Erika Isomura reminds them that they’ve been working with bar model representations. She asks them to make sure the pictures match the stories.

Mathematically proficient students try to communicate precisely to others.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, she models expectations for how her students are to communicate agreement and disagreement with each other’s reasononing, for example, “Make sure you're really clear with your explanation. You may want to use some of the fraction vocabulary when you're explaining to your partner. “ and “You can ask them to repeat or you can ask them what they meant by that, have them explain a little bit more, push them to explain so that you can understand their thinking.”

In the elementary grades, students give carefully formulated explanations to each other.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, students warm up by engaging in evaluating examples of students’ solutions. In their sharing, she models and expects precision of language.

Mathematically proficient students try to communicate precisely to others.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In these clips, students engage in pair work and then share their conversations with other pairs, communicating their rationales for a card sort matching mixed numbers to fractional representations.

Mathematically proficient students try to communicate precisely to others.

Erika Isomura’s students continue pair work on the problems, defending their thinking to each other. They make use of the sentence stems Erika gave them (e.g., “I think ____ because,” “This way is easier on our brains because______,” “What do you think?” “How did you feel about this problem?” “I knew it was this answer because I thought about the 0s”).

Erika engages pairs in explaining their thinking (e.g., “Show me in your answer,” “Do you think that will work every time?”). She reminds her students to note moments in their work that suggest that they might need to have a discussion about it. This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others) and standard 7 (look for and make use of structure).

Mathematically proficient students….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.

Erika Isomura’s 5th-grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Students discuss the reasonableness of their results with their partners prior to gluing down their cards. Erika connects pairs with each other to have them share their thinking, discussion, and results (“Why did you put this card here?” “Are both of these answers possible?” “What do you think of their idea? Does it make sense?”)

Mathematically proficient students try to communicate precisely to others….In the elementary grades, students give carefully formulated explanations to each other.

Erika Isomura’s 5th-grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Erika circulates around the classroom, engaging students in sharing their thinking. She says to one pair “This could be our prototype. This is our test run. We're kind of working on it, thinking. After we have some new ideas and maybe we have some better understanding of what we're doing, we can always come back into it. Okay?” This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others) and standard 7 (look for and make use of structure).

Mathematically proficient students.…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.

Erika Isomura’s 5th-grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Erika invites her students to compare their work to their work for prior investigations and other problems. She reminds each partner to contribute equally to the pair work. She asks “How did you get that? Can you show it to me in a picture or with numbers?” She challenges partners to contribute to their pair’s mutual work: “Make sure he's proving it to you. Don't just let him talk it out.” As partners finish their work, Erika invites them to do a “gallery walk” of other partners’ work to check and compare their work. This clip also relates to standard 1 (make sense of problems and persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others), standard 6 (attend to precision) and standard 7 (look for and make use of structure).

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 & critique the reasoning of others), standard 7 (look for & make use of structure), and standard 8 (look for & express regularity in repeated reasoning).

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 & critique the reasoning of others).

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.

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.

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 & persevere in solving them) and standard 3 (construct viable arguments & critique the reasoning of others).

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.

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.

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.

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.

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.

Antoinette Villarin begins her lesson on graphing constant rate of change by reviewing the learning goals and mathematical practices, naming Standards for Mathematical Practice 1, 3, 6, and 7. She notes that it is important that her students understand how to build a mathematical argument, and she shares sentence frames and key vocabulary that the students will use as they build their arguments.

Antoinette presents a model of two bottles attached to each other so that fluid can flow between them, and she asks her students to make sense of the problem by describing what they see happening.

Students share that as the amount of fluid in the top container/prism decreases, the amount in the bottom container/prism increases.

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

Antoinette Villarin returns to the academic language of the lesson: starting situation, initial value, rate of change. She asks students to use the sentence frames in pair-share conversations.

Antoinette shares a slide with new graphs showing rate of change and asks her students to use the sentence frames with their partners to describe what they see, using precise mathematical language as they do so.

This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others).

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

Antoinette Villarin asks her student pairs to share their discussions with the whole group. She models academic language — constraints, rate of change, initial value, starting situation — that she expects her students to use.

Once a student pair has shared, Antoinette asks the larger group to add on additional details that helped them identify a matched pair of graphs that show the flow of liquid between a given pair of containers.

Antoinette refers back to her anchor chart of the lesson vocabulary and sentence frames that she expects the students to use, and she names and reinforces students’ use of academic language.

This clip also relates to standard 1 (Make sense of problems and persevere in solving them), and standard 7 (look for and make use of structure).

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

Antoinette Villarin reminds her student pairs that — once they are confident about a match between graphs — they should record their matches and explain their justification, ensuring that both partners in each student pair understand and can explain their reasoning.

Student pairs discuss and record their thinking on their recording sheets. Antoinette circulates among the pairs and engages them in explaining their thinking — for example, reminding them to identify the starting situation to help guide them.

This clip also relates to standard 1 (make sense of problems and persevere in solving them).

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

As they close the day’s learning, Antoinette Villarin asks the students who returned from the gallery walk and sharing to describe their findings to their partners.

The student pairs work together, discussing and justifying modifications to their recording sheets. Antoinette collects the graph slides, cards, and other materials.

She asks students to engage in a brief reflection/exit ticket at the end of the lesson, responding to the prompts “Today I learned… One question I still have about interpreting parts of a graph is… My partner and I left off on… Tomorrow we will continue with….”

Antoinette adjourns the lesson by telling them that their exit ticket responses will help them remember where they will begin the next day’s continued work.

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

Patty Ferrant describes the role of homework in her classroom community to deepen understanding, to connect to high school expectations, and to jumpstart classroom conversations. Her students share their homework, clarify each other’s thinking and extend each other’s understanding. Students ask questions, listen attentively, and model academic language. This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others).

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 & persevere in solving them).

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 & persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others), and standard 7 (look for & make use of structure).

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 & critique the reasoning of others), standard 7 (look for & make use of structure), and standard 8 (look for & express regularity in repeated reasoning).

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).

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 & critique the reasoning of others).

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).