Standard 2: Reason abstractly and quantitatively
The Standard:
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Classroom Observations:
Teachers who are developing students’ capacity to "reason abstractly and quantitatively" help their learners understand the relationships between problem scenarios and mathematical representation, as well as how the symbols represent strategies for solution. A middle childhood teacher might ask her students to reflect on what each number in a fraction represents as parts of a whole. A different middle childhood teacher might ask his students to discuss different sample operational strategies for a patterning problem, evaluating which is the most efficient and accurate means of finding a solution. Visit the video excerpts below to view these teachers engaging their students in abstract and quantitative reasoning.
Connections to Classroom Practices
5th Grade 
[Mathematically proficient students bring] the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
Hillary LewisWolfsen leads a reengagement lesson on the proportions and ratios, helping students to recognize what a visual representation of a simplified fraction looks like. In this clip, she gives her students “think time” to jot down their ideas, then responds to a student with the correct answer by asking another student to explain her answer. LewisWolfsen comments that her “students often understand more if they hear the explanation in a variety of ways, and not just from the teacher.”

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5th/6th Grade 
Mathematically proficient students make sense of quantities and their relationships in problem situations... Quantitative reasoning entails... attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Fran Dickinson leads a lesson on numeric patterning, helping students to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson describes the importance of individual think time before he asks his students to discuss the relative strength of two different approaches to a patterning task. One pair discusses the numbers within the sample strategy, and Dickinson repeats back their conversation to the whole group, telling his students, “I’ve heard two really good questions about Learner B’s strategy. One was, what are all these 3’s? and Kelcey’s question was, what about this 4? Where’s the 4 coming from?” This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).

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6th Grade 
Mathematically proficient students make sense of quantities and their relationships in problem situations... Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Joe Condon works with his 6th grade students to identify strategies for comparing unit rates. In these clips, he begins with a definition of rate developed and expanded by the class over the twoweek period. His students are asked to name familiar rates. He then presents them with interactive and reallife scenarios and asks them to calculate the fastest unit rates and justify their answers. Condon refers his students to strategies for comparing rates, asking the students to use these strategies with new data. The students then discuss their responses with the whole group, and conclude that the best rates depend on the goal of the calculation.

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Mathematically proficient students make sense of quantities and their relationships in problem situations... Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Joe Condon presents a lesson study for 6th grade students and observing teachers to identify strategies for comparing unit rates. In these clips, he starts by establishing classroom norms for active listening. Then he engages the class in a wholegroup experiment of the teacher power walking a 5meter strip. The members of the class are given jobs for collecting data or monitoring time. Then Condon uses a ratio table to talk about rate and find equivalent rates and unit rates. 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.
Students conduct 3 rate experiments: stringing beads on a shoelace, picking up cubes with chopsticks, and counting rice. After the experiments the teacher debriefs the results as a whole class. "Who won?" "Why can't we tell from the raw data?"

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9th12th Grade 
Mathematically proficient students make sense of quantities and their relationships in problem situations...Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Carlos Cabana works with his high school algebra for English language learning students on algebraic reasoning and multiple representations around parabolas. In these clips, Cabana analyzes groups' work in terms of logical flow and carefulness of organization, trying to make a guess about what they really understand. Students explain their thinking to each other as they look to find the vertex of the parabola. Cabana challenges them to move between their representations.

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