# Model with mathematics

## Mathematical Practice Standard #4

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life…. In early grades, this might be as simple as writing an addition equation to describe a situation.… Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation…. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs…. They…reflect on whether the results make sense…. —CCSS

## The standard in elementary school

The intent of this standard is not to pretend that “problems arising in [the] everyday life” of an adult would be of educational value, let alone interest, to a child. The *serious* everyday-life problems that children face must be solved by the adults who take care of them. Children, themselves, pay closest attention to learning how the world around them works. What we call “play” is their work: they experiment, tinker, push buttons (including ours), say and ask whatever comes to mind, all in an attempt to see what happens. And their curiosity naturally includes ideas we call mathematics: thinking about size, shape and fit, quantity, number. Figuring out where to be to catch a ball is not a paper-and-pencil calculation, but it certainly does involve attention to and rough quantification of the speed, direction, and position of both the ball and oneself, as well as other mathematical ideas. Doing a KenKen® puzzle is also an application of raw arithmetic skills to an everyday life problem, in the honest sense that puzzling is very much an important part of the life of a child (and important enough even in the lives of adults to assure that even supermarkets stock puzzle books).

One intent of this standard is to ensure that children see, even at the earliest ages, that mathematics is not just a collection of skills whose only use is to demonstrate that one has them. Even puzzles suffice for that goal.

Another intent is to ensure that the mathematics students engage in helps them see and interpret the world—the physical world, the mathematical world, and the world of their imagination—through a mathematical lens. One way, mentioned in the standard, is through the use of simplifying assumptions and approximations. Children typically find “estimation” pointless, and even confusing, when they can get exact answers, but many mathematical situations do not provide the information needed for an exact calculation. The following problems suggest two ways children might encounter situations like these in elementary school.

About how many children are in our school? 50? 200? 1000? To figure that out, we could count, but that’s a lot of work. Besides, we don’t need to know exactly. How can we come reasonably close, just sitting here in our classroom?

Sam Houston Elementary School has nearly 1,000 children from kindergarten through 5th grade, with about the same number of students in each grade. No class has more than 25 students, but most classes are close to that. What can you figure out from this information? (Adapted from

Think Math!Grade 5.)

Both of these examples require “assumptions and approximation to simplify” and also the essential step of “reflect[ing] on whether the results make sense.”

What’s important here is not the context that’s used, but the kind of thinking it requires. Using “approximations to simplify a complicated situation” can be valuable even within mathematics and even when exact answers are required. For example, students in 4th grade who are beginning to learn that there are many fractions equivalent to ½ can quickly become competent, and inventive, at contributing entries to a table, on the board, with three columns: fractions between 0 and ½, fractions equal to ½, and fractions between ½ and 1. They can then apply what they know to simplify, cleverly, such “naked arithmetic” problems such as “Arrange the fractions 4/9, 5/8, and 7/12 in order from least to greatest.” The comparison can be performed entirely without calculating by noting, first, that 4/9 is less than ½ (because 4 is less than half of 9) and it is the only one that is less than ½, so it is the smallest. The other two are both greater than ½, but 5/8 is one eighth greater than ½ whereas 7/12 is only one twelfth greater, so 5/8 is the largest. The point, of course, is not to replace one technique with another. The point is that mathematical thinking simplifies the work.

A completely different kind of modeling involves spatial location: a map or diagram models the real thing. Even children in K and 1 can lay out strips of paper in a grid on the floor, name the streets and avenues, place houses and schools and libraries at various locations, and describe the distances and directions to get from one to another.