# Standard 3: Construct Viable Arguments & Critique the Reasoning of Others

## Classroom Observations

Teachers who are developing students’ capacity to "construct viable arguments and critique the reasoning of others" require their students to engage in active mathematical discourse. This might involve having students explain and discuss their thinking processes aloud, or signaling agreement/disagreement with a hand signal. A middle childhood teacher might post multiple approaches to a problem and ask students to identify plausible rationales for each approach as well as any mistakes made by the mathematician. An early adolescence teacher might post a chart showing a cost-analysis comparison of multiple DVD rental plans and ask his students to formulate and defend a way of showing when each plan becomes most economical. A teacher of adolescents and young adults might actively engage her students in extended conjecture about conditions for proof in the construction of quadrilaterals, testing their assumptions and questioning their approaches. Visit the video excerpts below to view multiple examples of teachers engaging students in formulating, critiquing and defending arguments of mathematical reasoning.

## The Standard

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

## Connections to Classroom Practices

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures...They justify their conclusions, communicate them to others, and respond to the arguments of others. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Liz O'Neill works with her first grade students engaging them in composing and decomposing numbers within twenty. Using sentence frames, students shared with their partner responses to two warm-up activities engaging them with the content. A variety of solving strategies were discussed as a whole group after everyone had a chance to share with their partner. Her students then were given a bag with 10 cubes, a paper plate, and the "How Many Are Hiding Recording Sheet." In addition, sentence frames were posted on the board so students could produce academic language using structured student talk and convince their partners with oral justification. During the game, one partner takes some of the cubes and "hides" them under the plate. The remaining are placed on the top. The second partner uses sentence frames to answer the questions "What number do you see?", "How many are hiding?", "How do you know __ are hiding"? In addition, the answers are recorded. Roles are then reversed. The partner game gives students practice in composing and decomposing numbers within ten.

Students…can listen [to] or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Mia Buljan’s 3rd-grade students defend their approaches to a multiplication number talk and respectfully engage each other when a student makes an error in his or her reasoning.

Students…can listen [to] or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

As her 3rd-grade students work on a word problem, Buljan circulates around her classroom. She says to two students, “He did it slightly differently. Can you guys get together and talk about what you think the problem means?”

Students…can listen [to] or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

After working individually on various problems, each of Buljan’s 3rd-grade students has selected a card on the board that represents the problem they worked. She then invites her students to “find the other people who have your problem and see if they agree with you.”

Mathematically proficient students … build a logical progression of statements to explore the truth of their conjectures.

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. This clip also relates to standard 1 (make sense of problems & persevere in solving them)standard 5 (use appropriate tools strategically) and standard 6 (attend to precision).

Mathematically proficient students … build a logical progression of statements to explore the truth of their conjectures.

Michelle Makinson challenges her students to individually select one grouping of representations from their chart, and move those representations over to a poster without the justifications. Each student then writes responses to several questions, each of which has a sentence stem prompt:

• Why did you select this grouping? / I selected this grouping because...

• How do you know that all the cards represent the same quantity? / I know that all the cards represent the same quantity because...

• When sharing your work with other pairs, what did you learn from the discussion? / When I shared my chart with another partnership I learned... from the discussion.

This clip also relates to standard 8 (look for & express regularity in repeated reasoning).

Mathematically proficient students understand and use …definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases…They justify their conclusions, communicate them to others, and respond to the arguments of others…Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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, 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?”

See this video in the context of an entire lesson.
(Lesson Part 1)

Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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 them do not have visual representations; she invites her students to create drawings of those problems if it would be helpful to their process.

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(Lesson Part 2A)

Mathematically proficient students… justify their conclusions, communicate them to others, and respond to the arguments of others.

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

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(Lesson Part 2B)

They justify their conclusions, communicate them to others, and respond to the arguments of others...Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later...Students at all grades can listen or read the arguments of others, decide whether they make sense, and ... improve the arguments.

Hillary Lewis-Wolfsen invites students to examine a problem about proportions and ratios with a strategy used by a student to organize the information in the problem. In these sequential clips, mathematics coach Linda Fisher probes a pair to elaborate their thinking. Lewis-Wolfsen and Fisher had hoped that students would be able to see the logic in this answer and be able point out where the answer could have been found. In responding to one another’s arguments, Lewis-Wolfsen’s appreciates the “supportive language this class uses, i.e. ‘logical,’ ‘heading in the right direction.’ The discussion between the kids is respectful of one another too.”

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(Parts C & D)

Students…can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, she asks students to review the approaches of two example students to representing a mixed number. Her students engage in a pair-share, stating which approach they agree with and explaining why.

Students…can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. Throughout the lesson, her students listen attentively to each other (in a whole group, in pairs, and in small groups), making use of sentence frames modeling academic vocabulary. While this mathematical practice is evident throughout the lesson, in the second half of these clips pairs of students work to develop and challenge each other's thinking.

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(Parts 3 & 5)

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.

Erika Isomura engages her 4th and 5th graders in a mental math talk, inviting multiple iterations of dividing by 10, e.g., 8,000/10, 800/10, 80/10, 8/10, (8/10)/10. She asks students to justify and revise their answers if needed. She reviews norms for whole-class participation, and asks students to evaluate the reasonableness of their results. Erika then presents multiple iterations of dividing 8 by multiples of 10 (e.g., 8/10, 8/100, 8/1,000, 8/10,000), noting similarities and patterns.

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(Day 1 Math Talk)

Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.

Erika Isomura works with her 5th graders to develop their understanding of how decimals work. She begins her lesson having her students work in pairs to do a “calculate and check” mental math decimal task, using calculators to check, then evaluating their work with symbols and writing (e.g. “We disagree because,” “I have a question because”).

She reminds her students to do the mental math, then first discuss what they got “in their brains,” and then check with their calculators. As the pairs work, Erika circulates around the classroom and probes their thinking (e.g. “Did you agree? Did that make sense?”) This clip also relates to standard 5 (use appropriate tools strategically).

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(Day 1 Lesson Part A)

Mathematically proficient students…justify their conclusions, communicate them to others, and respond to the arguments of 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.

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(Day 1 Lesson Part C)

Mathematically proficient students…justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Erika Isomura’s 5th-grade students work in pairs to classify, sort, and glue down decimal representations in numerical order. She challenges her students to explain and defend their thinking. This clip also relates to standard 7 (look for and make use of structure).

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(Day 2 Lesson Part B)

Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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 6 (attend to precision)and standard 7 (look for and make use of structure).

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(Day 2 Lesson Part D)

Mathematically proficient students….justify their conclusions, communicate them to others, and respond to the arguments of others.

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 pairs’ work to check and compare their work. This clip also relates to standard 1 (make sense of problems and persevere in solving them)standard 6 (attend to precision), and standard 7 (look for and make use of structure).

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(Day 2 Lesson Part E)

Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

As her students finish their card sorts and defend their thinking, Erika Isomura increases the rigor, inviting pairs to go beyond just solving the problem: “Make something challenging, make something that the other kids are going to be like ... then after they think about it like ‘oh yes, I got it.’ Give them a little bit of a push, okay?” She connects their thinking back to the previous lesson with fractional lengths of strings. Erika closes the lesson by praising students’ perseverance and the value of mistakes —“We like mistakes because they give us something to learn from…think of this as your prototype, this is our test run, we're trying it out, once we learn a little bit more, once we become more experts we can always go back and make some changes.” This clip also relates to standard 1 (make sense of problems and persevere in solving them).

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(Day 2 Lesson Part F)

Mathematically proficient students….are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Erika Isomura connects her students’ thinking back to the previous lesson with fractional lengths of strings. Erika closes the lesson by praising students’ perseverance and the value of mistakes —“We like mistakes because they give us something to learn from…think of this as your prototype, this is our test run, we're trying it out, once we learn a little bit more, once we become more experts we can always go back and make some changes.” This clip also relates to standard 1 (make sense of problems and persevere in solving them).

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(Day 2 Lesson Part G)

They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is... Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he continues a class conversation about input and output numbers. Dickinson notes that “It was interesting to hear all of the different opinions of how to state the rule. I think this illustrates where we were as a group as far as our familiarity with algebraic expression goes.” For example, the students discuss “3 groups of x versus x groups of 3.” Dickinson also models whole-group strategies for consensus and disagreement, which he explains as “silent signals.” This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).

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(Parts B & C)

...Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades....Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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 6 (attend to precision).

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Joe Condon works with his 6th grade students to identify strategies for comparing unit rates. In these clips, the students actively engage with the concepts individually, in small groups, and in large group discussions. His students are asked to write a rate for each student and justify who is fastest at counting beans. Students use different approaches to calculate the rate, and one student observes that if the rate is seconds per bean it needs to be the smallest rate to win, but if it is beans per second it needs to be the largest rate.

Students first think on their own, then share in groups before class discussion. Another student shares a conjecture with the class: If the rate is seconds per step Alex wins, but if the rate is steps per second Joe wins. Students talk in groups to try to figure out how she got this conjecture and whether it makes sense. A final student concludes that the goals are different for each rate. During whole class discussion, one group interprets a simpler strategy for simplifying unit ratio and shares with the class.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, [and] decide whether they make sense...

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 6 (attend to precision).

Mathematically proficient students understand and use stated assumptions, ... and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, [and] decide whether they make sense...

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?"

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.

Disston leads his students to a deeper level of noticing. The lesson keeps recycling or layering the ideas, forcing students to new levels of describing and deciphering the symbol strings, sorting and classifying. The students are now much more talkative than at the beginning of the lesson. They seem more invested in reaching an understanding and questioning each other.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, [and] decide whether they make sense...

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. This is an opportunity to make all the knowledge or thinking from the groups public, and honored the idea that there could be different categories.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, [and] decide whether they make sense...

Cecilio Dimas leads a lesson on constructing, communicating, and evaluating student-generated tables while making comparisons between three different financial plans, helping students use multiple representations of mathematical problems: verbal, tabular, graphical, and algebraic generalization. In this clip, Dimas asks his students to examine a table comparing DVD rental plans, and ask themselves, “Does this make mathematical sense? Why or why not?” His goal is for students to make all three representations for a new and different cost analysis situation and discuss the merit of each representation in that particular situation. This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments...They justify their conclusions, communicate them to others, and respond to the arguments of others.

Jake Disston and Jesse Ragent work with their middle school students to deepen their understanding of the connections between different representations of functions—graphs, t-charts, and equations. In this clip, he challenges his students to work at their tables to group the cards they have in front of them, creating as many different groupings as they can and devising language to describe each group. As the first student reports it becomes clear that while there is some understanding of the information, there still remains plenty of misunderstanding and confusion. Disston models the need for teachers to honor all responses in class discussion. He brings the incorrect notion of the student's "linear" grouping of the quadratics into proper focus and clarity in a way that does not denigrate the student's thinking. Ragent reviews group work protocols for turn-taking and talking, asking students to "think out loud, giving mathematical reasons for the selections" they make using language generated by the class.

See this video in the context of an entire lesson.
(Parts 1 & 2)

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.

Antoinette Villarin begins her lesson on graphing constant rates 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 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… justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose...Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Antoinette Villarin asks her students to engage in a “turn and talk” with a fellow student, with the purpose of describing how they would know how many centimeters of liquid will be in the bottom container, given the number of centimeters in the top container.

In the “turn and talk,” Antoinette’s students use the vocabulary and sentence frames she provided to make sense of the problem. Antoinette gathers the students back together and asks the pairs to share their findings with the whole group.

Antoinette describes the constraints for the problem. She then asks students to look at a graphical representation of the problem and respond to the prompt "I think this graph represents ..." on their whiteboards.

Her students then share their statements with their partners. After the students share with each other, Antoinette asks pairs to report out on their conversations.

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

Mathematically proficient students…make conjectures and build a logical progression of statements to explore the truth of their conjectures…. They justify their conclusions, communicate them to others, and respond to the arguments of others…. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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 relate to standard 6 (attend to precision).

Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Antoinette Villarin’s students return to their groups to discuss and share their thinking about the relationship between graphs showing rate of change. She challenges them to try to find the two graphs out of four with the correct combination to describe the same container. As the student partners work, Antoinette circulates around the room engaging pairs in questioning.

Mathematically proficient students…justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose …. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Antoinette Villarin’s students continue their partner work, matching graphical representations of rates of change with each other.

In this clip, two different student pairs engage in mathematical discourse, comparing graphs to each other to discern which graph represents the top prism out of which the fluid is draining, and which represents the corresponding receptacle prism. Students offer ideas and defend their thinking to each other.

This clip also relates to standard 5 (use appropriate tools strategically).

Mathematically proficient students…justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Antoinette Villarin asks her students to gather together the cards that they determined were matches for each other and to record a justification for each match on their recording sheet.

She explains that they will now engage in a gallery walk, in which one of the student partners will travel with a copy of their recording sheet of matches and justifications, comparing their thinking to that of other pairs, and the other partner will stay at their table and engage in conversation with visitors from other pairs.

Antoinette charges her students to be alert for differences in thinking and to make modifications to their recording sheets if necessary. She also reminds students to be especially attentive to other groups’ approaches to matching the three cards that had partial or missing information.

Mathematically proficient students… justify their conclusions, communicate them to others, and respond to the arguments of others.

Patty introduces her lesson by charging students to identify the “big ideas” they should be thinking about when they work with right triangles. Students pair-share their ideas, and Patty notes when they are making reference to available tools and supports, such as anchor charts, around the room. In her commentary, Patty notes that this lesson is intended to develop students’ capacity to engage in modeling mathematical situations. Students identify the Pythagorean Theorem, and Patty prompts them to attend to precision and communicate precisely. In a whole-group sharing, she engages all students to add on to, critique, extend, and clarify each other’s thinking. Students deepen their capacity to make sense of the problem or situation. Patty presents student work from a previous assessment and asks students to critique the person’s strategies and precision, giving advice to each exemplar learner about how to improve their approach. This clip also relates to standard 1 (make sense of problems & persevere in solving them)standard 4 (model with mathematics), and standard 5 (use appropriate tools strategically).

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lesson part 1

Mathematically proficient students… justify their conclusions, communicate them to others, and respond to the arguments of others.

Patty’s students give advice to their peers about perseverance and strategies they can use to help themselves in an upcoming MARS performance assessment task. She asks “What do you do when you’re stuck? What strategies should you try?” Students think, write, then share their strategies with each other. They identify strategies using anchor charts and calculators, drawing pictures, consulting a peer, taking their time, double-checking their work, and ensuring that it makes sense. This clip also relates to standard 1 (make sense of problems & persevere in solving them)standard 4 (model with mathematics), and standard 5 (use appropriate tools strategically).

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lesson part 2

Mathematically proficient students… justify their conclusions, communicate them to others, and respond to the arguments of others.

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 6 (attend to precision).

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students discuss homework

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.... Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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 6 (attend to precision), and standard 7 (look for and make use of structure).

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(Parts A-D)

See how SEL Competencies and Mathematical Practices work together in this classroom.
(Describing an Ideal Classroom, Appendix)

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.... They justify their conclusions, communicate them to others, and respond to the arguments of others.... Students at all grades can listen or read the arguments of others, decide whether they make sense..

In the closing of the group work, 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 6 (attend to precision)standard 7 (look for & make use of structure), and standard 8 (look for & express regularity in repeated reasoning).

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.... They justify their conclusions, communicate them to others, and respond to the arguments of others.... Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

On the second day of Humphreys’ exploration of the properties of quadrilaterals, 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 6 (attend to precision).

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.... They justify their conclusions, communicate them to others, and respond to the arguments of others... Mathematically proficient students are also able to ... distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Once each group has worked through the reasoning of the proof, Humphreys checks in with the group and instructs them to begin designing a poster that will display the proof they had created. Groups are instructed to design a poster that contains a drawing of the figure, the conjecture of what is to be proved, a list of the given from the conjecture, and what needs to be proved. The students can use a two column or a flow chart format of the proof. This clip is also indicative of standard 1 (make sense of problems & persevere in solving them) and standard 6 (attend to precision).