Standard 3: Construct viable arguments and critique the reasoning of others

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.

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.

## 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.

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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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See this video in the context of an entire lesson (Parts C and D).

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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See this video in the context of an entire lesson (Parts B and 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).

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See this video in the context of an entire lesson.

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.

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See this video in the context of an entire lesson.(Parts 1 - 6)

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).

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See this video in the context of an entire lesson (Parts B and C).

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?"

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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.

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See this video in the context of an entire lesson.

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.

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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).

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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.

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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. 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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See this video in the context of an entire lesson (Parts A-D).

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 and make use of structure), and standard 8 (look for and express regularity in repeated reasoning).

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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).

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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 and persevere in solving them) and standard 6 (attend to precision).

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