# Standard 2: Reason Abstractly & Quantitatively

## 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.

## 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.

## Connections to Classroom Practices

### Kindergarten

*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.*

Tracy Sola asks the students in her transitional kindergarten class to represent five nuts using square tiles. Students are at the beginning of decontextualizing the problem and representing the quantity while considering the units involved.

**See this video in the context of an entire lesson.**

*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.*

A group of transitional kindergarten students struggle to represent the problem. Tracy Sola asks students from other groups to come and analyze the struggling group’s work. After repeating the problem—a squirrel had five nuts and shared one with his friend—she asks the students which student’s work represents the problem.The students are decontextualizing a problem situation and recontextualizing the representation back into the context of the problem. They analyze the struggling group’s work and use the context of the problem to find who represented the problem correctly. Tracy Sola supports this practice by asking the students in the class to represent another problem situation.

**See this video in the context of an entire lesson.**

### 3rd Grade

*Mathematically proficient students make sense of quantities and their relationships in problem situations.*

Mia Buljan circulates around her 3rd-grade classroom while her students work on a multiplication/division word problem. Buljan asks, “When you make two tens and four blocks, which number are you making?” The student responds, and Buljan continues to probe with questions to find out whether the student has identified the quantities and relationships in the problem situation.

**See this video in the context of an entire lesson.**

### 4th Grade

*Quantitative reasoning entails habits of creating a coherent representation of the problem at hand…*

Michelle Makinson's students create visual representations and area models, comparing them in a whole-group sharing. They explain their visual representations, clarifying their thinking and making corrections when necessary.

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### 5th Grade

*Mathematically proficient students make sense of quantities and their relationships in problem situations…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.*

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.

**See this video in the context of an entire lesson.**

**(Lesson Part 2B)**

*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 Lewis-Wolfsen leads a re-engagement 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. Lewis-Wolfsen 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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*Mathematically proficient students make sense of quantities and their relationships in problem situations.*

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, her students engage in a card sort, matching different representations of mixed numbers. She helps them name and make explicit the correspondences between the representations, e.g. “So you're going to say ‘Card C,’ and then say the number, “is equivalent to,” and then I want you to name the fraction. And then say, ‘Because it shows ...’ and explain to your partner what it shows. “

**See this video in the context of an entire lesson.**

**(Parts 3 - 6)**

### 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).

**See this video in the context of an entire lesson.**

### 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 two-week period. His students are asked to name familiar rates. He then presents them with interactive and real-life 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.

**See this video in the context of an entire lesson.**

**(Parts 1-6)**

*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 whole-group experiment of the teacher power walking a 5-meter 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?"

**See this video in the context of an entire lesson.**

### 8th Grade

*Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring …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…Antoinette Villarin transitions her students to a matching activity, having her students use a recording sheet to make sense of different quantities and representations, find models, match graphs, and explain their justifications for the match.*

She distributes a stack of twelve graphs that represent the flow of liquid between containers, and challenges students to match the two graphs that show movement for a given pair of containers. She notes that for some cards, information has intentionally been left out, and recommends that students begin with the cards that have complete information.

She reminds her students that they are trying to build a mathematical argument, so they must explain to their partner why a given pair of cards is a match, and the partner in turn needs to agree or disagree with each explanation. Antoinette explains how to use the cards, reminding students to refer back to the sentence frames for conversation if they don’t know what to say.

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

**See this video in the context of an entire lesson.**

**Lesson 2C**

*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.*

Antoinette Villarin asks her students to 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.

As she closes the day’s learning, Antoinette asks the students who returned from the gallery walk and sharing to describe their findings to their partners. The pairs work together, discussing and justifying modifications to their recording sheets.

**See this video in the context of an entire lesson.**

**Lesson Parts 3B and 3C**

### 9th-12th 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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**(Parts 8 and 10)**