Understanding MP 2: Reason abstractly and quantitatively

The Common Core’s Mathematical Practice 2, Reason abstractly and quantitatively, calls for students to make sense of quantities and their relationships in problem situations. Students de-contextualize – translate the actions and events described in a problem into models or numbers and symbols that they can manipulate. They also contextualize – interpret the meaning of the numbers, symbols, and units of measure in their work in terms of what they represent in the original problem. Students are able to imagine a problem context for an abstract problem, enabling them to make sense of the mathematics and judge the reasonableness of their answer. They understand the properties of operations, numbers, and geometric objects and use them flexibly. MP 2 is closely related to several other Practices, especially MP 4, Model with Mathematics.

Why is reasoning important?

Reasoning is the foundation of mathematics. It is one of the principal vehicles for developing understanding, exploring new ideas, and constructing new knowledge. Through reasoning we are able to analyze and solve problems, assess our results, and generally make sense of mathematical situations. Students who can reason are often able to find their own way through problems, even when their memory of procedures and rules fails them.

How should we develop reasoning?

Like other Mathematical Practices, MP 2 is best addressed in the context of doing worthwhile mathematics. Rich problems and investigations provide authentic opportunities for students to develop the number sense and operation sense that support understanding. Teachers can help students develop reasoning by asking guiding questions; by encouraging students to visualize problems and search for connections with previous work; by modeling a variety of tools and representations and having children compare them; and by fostering the development of math language and providing numerous opportunities for students to share their thinking.

How can these resources help?

The resources in this collection are intended to help teachers and math specialists understand the intent and scope of MP 2 and plan for successful attainment in the classroom. They include: documents and websites that help us interpret the practice and appreciate its importance; a webcast and videos that illustrate teachers fostering reasoning in a problem context; and articles suggesting effective instructional strategies. A companion Classroom Collection includes resources to help teachers foster this practice with their students.

Collection Discussion

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Last Post:09-02-2014 by bethb

Resource Title/Description

This 26-minute Flash presentation helps educators understand the intent of CCSS Mathematical Practice 2 and its implications for the K-5 classroom. It suggests strategies for helping children develop the following proficiencies: strong number sense, the ability to decontextualize problems from words to symbols and re-contextualize results, the ability to make sense of quantities and relationships in problem contexts, and operational fluency. A transcript (pdf) of the video is available for download.
This 19-page draft document (pdf) offers guidance to K-5 educators in interpreting the CCSS Mathematical Practices for the elementary grades. The first section provides annotations to the original language. The second section integrates those annotations into coherent descriptions of how each practice standard applies in the K–5 classroom.
This series of nine articles interprets and illustrates each of the eight Mathematical Practices of the Common Core State Standards as they might be exemplified in grades K-5. The final article sheds light on how curriculum needs to connect the Practices with the Content Standards.
In this 1-hr webcast Dr. Cathy Fosnot and a group of math educators discuss the significance of problem contexts for students' mathematics learning. They solve a problem involving fractions and proportional reasoning, share multiple solution strategies, and discuss the importance of solving problems before assigning them to children in order to anticipate the range of knowledge and skills that students might demonstrate. Video clips of a classroom foster a discussion of strategies for designing lesson problems and managing a congress.
This 19-page article (pdf) by Deborah Loewenberg Ball and Hyman Bass is a chapter from A Research Companion to Principals and Standards for School Mathematics, published by NCTM. The authors discuss the foundations of mathematical reasoning and proof and support their position that it is central to understanding and using mathematics. Two episodes from a third grade class help illustrate how children can develop the ability to reason. Ball and Bass suggest teaching practices that foster this development.
This article states that research shows 93% of teacher questions are focusing on recall of facts and that type of questioning does not stimulate the mathematical thinking of students who are engaged in open ended investigations. To improve the quality of teacher questioning techniques, the author suggests dividing questions into four main categories: starter questions, questions to stimulate mathematical thinking, assessment questions and final discussion questions and then to follow with a rigorous question. A link to an addendum to the article provides a table of generic questions that can be used by teachers to guide children through a mathematical investigation, and at the same time prompt higher order thinking, as espoused by Bloom and others.
This 24-min video includes 16 classroom vignettes that illustrate the central role of reasoning in mathematics. Students use a variety of models and strategies; they explain and justify their thinking and solutions. Teachers emphasize the importance of reasoning in the problem solving process.
This professional development video clip shows kindergarten students engaged in the Common Core Practice Standard #2 Reasoning abstractly and quantitatively. In this video learners use three known quantities of beans for reference, they use abstract and quantitative reasoning to make sense of the number of beans in their bags individually and in a group and then groups share and justify their results. Additional resources include a video transcript, teaching tips, and a link to a professional development reflection activity based upon the video.
This professional development video clip shows students engaged in Common Core Practice Standard #2—Reason abstractly and quantitatively. In this video clip students using various problem-solving strategies to understand the cost of household items, as well as the partitive model of division. Additional resources include a video transcript, teaching tips, and a link to a professional development reflection activity based upon the video.
In this 13-minute video Teacher Suney Park's students apply their knowledge of area and perimeter of rectangles in the context of a dinner table. The resource includes a transcript of the video, extension activities, and reflection questions for teachers.
In this blog entry Michelle Flaming describes what the implementation of MP 2 looks like in the classroom. She lists student actions, teacher actions, and open-ended questions that teachers can ask students to elicit the kind of reasoning called for in MP 2.