Standards for Mathematical Practice 2

The second standard of practice is Reason abstractly and quantitatively. 


The explanation reads: Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complimentary 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. 


The last phrase explains the what standard two is really all about. Students need strategies of how to look at problems different ways, to be able to isolate and solve a part of a problem, and to be able to look at and see connections and relationships within a problem and how to manipulate that information to help them solve the problem. 


Again, across grade levels and abilities this can take on many shapes.. for instance, a first grader who understands that 4 + 3 = 7 and can demonstrate this by showing OOOO +  OOO = OOOOOOO is able to represent the equation symbolically. If they can then relate the symbols to the number and show how 4+3 and 3+4 are interchangeable then they are using quantitative reasoning about number and structure. 


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 a problem make sense. So, for example, if fourth graders  are asked to solve a problem that asks how many busses are needed for 99 students if each bus seats 44, they might decontextualize a problem  and 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.

Our goal throughout all of these NCTM standards is to promote deeper and richer mathematical understandings.  Think about this standard as you become more familiar with the Common Core and how you might  design lessons, investigations or activities that will enable your students to have a richer experience and a deeper understanding of number.  

Not to sound like a broken record, but I feel that students will need inquiry experiences working  with manipulatives, investigating in groups, drawing or modeling their problems and spending time discussing, questioning and proving their thoughts. Accountable Talk.... Inquiry Circles....