# Reason abstractly and quantitatively

## Mathematical Practice Standard #2

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. —CCSS

## The standard in elementary school

This wording sounds very high-schoolish, but the same mathematical practice can be developed in elementary school. Second graders who are learning how to write numerical expressions may be given the challenge of writing numerical expressions that describe the number of tiles in this figure

in different ways. Given experience with similar problems so that they know what is being asked of them, students might write 1+2+3+4+3+2+1 (the heights of the stairsteps from left to right) or 1+3+5+7 (the width of the layers from top to bottom) or 10+6 (the number of each color) or various other expressions that capture what they see. These are all decontextualizations—representations that preserve some of the original structure of the display, but just in number and not in shape or other features of the picture. Not any expression that totals 16 makes sense—for example, it would seem hard to justify 2+14—but a child who writes, for example, 8+8 and *explains* it as “a sandwich”—the number of blocks in the middle two layers plus the number of blocks in the top and bottom—has taken an abstract idea and added contextual meaning to it.

More generally, Mathematical Practice #2 asks students to be able to translate a problem situation into a number sentence (with or without blanks) and, after they solve the *arithmetic* part (any way), to be able to recognize the connection between all the elements of the sentence and the original problem. It involves making sure that the units (objects!) in problems make sense. So, for example, in *decontextualizing* a problem that asks how many busses are needed for 99 children if each bus seats 44, a child might write 99÷44. But after calculating 2r11 or 2¼ or 2.25, the student must *recontextualize*: the context requires a whole number answer, and not, in this case, just the nearest whole number. Successful recontextualization also means that the student knows that the answer is 3 busses, not 3 children or just 3.