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### Reasoning Abstractly and Quantitatively

##### Description of the Practice

A central aspect of sense making is being able to reason abstractly and quantitatively. Some of the important components on this habit include the following abilities:

• representing the relationships in a problem context mathematically (decontextualizing)
• connecting and/or interpreting mathematical representations, operations and results within the context of the problem situation (contextualizing)
• treating quantities in the problem as representing objects, or some measurable characteristic (e.g., feet per second per second, ft/sec2) (quantitative reasoning)

Elaboration of these aspects of this habit can be found in various NCTM and CCSS documents.

In describing what it means to reason abstractly and quantitatively, Common Core State Standards for Mathematics states the following:

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. (http://www.corestandards.org/Math/Practice/MP2)

The habit of reasoning abstractly and quantitatively is further examined in Connecting the NCTM Process Standards and the Common Core Standards for Mathematical Practice (2013) when the authors make the point that this habit requires an understanding deeper than simply providing a numerical answer to a question:

Notice that the emphasis in this practice is on “making sense of quantities and their relationships in problem situations” (CCSSI 2010, p. 6; italics added), not blindly computing with numbers or algebraic symbols. It is important to understand that quantity should be seen as a measurable attribute of an object and thus different from numbers. (p 19)

Focus on High School Reasoning and Sense Making (2009) suggests that students who reason abstractly and quantitatively must engage in “seeking and using connections across different mathematical domains, different contexts, and different representations” and “generalizing a solution to a broader class of problems and looking for connections with other problems” (p. 10).

A Mathematician's View of the Practice (Wade Ellis)

When exploring real world problems, mathematicians consider previously existing mathematical concepts, objects and theorems or develop new ones. They often resort to new ways of thinking to see the relationship between known mathematics and the real world situation. For example, it is a giant imaginative leap to see group theory as a way to understand chemical structures. In order to think about ways to represent a problem mathematically, mathematicians and theoretical physicists create and explore toy problems where assumptions and constraints are more easily controlled. The abstractions (definitions, theorems, properties) that they develop in these toy problems are then used in comparing their abstract reasoning against the real world quantitative measurements determined from the problem situation.

Reasoning abstractly is then followed by quantitative reasoning that is followed by more abstraction and more quantitative reasoning. This cycle is repeated until a satisfactory solution to the problem (approximate or exact) arises. Quantitative reasoning about real world problems can create the need for new mathematical objects and theorems. For example, Copernicus used his observations of the skies to determine that the Earth moved around the sun. Kepler used this idea, precise measurements, and the mathematics of conic sections to determine that the motion of any planet around the sun was an ellipse with the sun at one focus. Newton created the physical notions of gravity (force), mass, the concept that Force = Mass times Acceleration, and the mathematical relationship between derivatives (rates of change) and integrals (accumulations) to formulate the concept of differential equations. Then he used these physical and mathematical ideas to construct differential equations he solved to understand why the planets moved as they do, a major triumph of human ingenuity. In the following four centuries, mathematicians and theoretical physicists sharpened Newton’s results and use of differential equations in a  variety of real world situations, creating new abstract mathematical objects and theories (Control Theory, Quantum Theory and String Theory, for example). They used quantitative reasoning to validate these theories and to continue to work on abstract reasoning to enhance their understanding of the real world and to develop new mathematical improvements to their models.

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