Mathematical Practice Standard #5

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet…. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. —CCSS

The standard in elementary school

Counters, base-10 blocks, Cuisenaire® Rods, Pattern Blocks, measuring tapes or spoons or cups, and other physical devices are all, if used strategically, of great potential value in the elementary school classroom. They are the “obvious” tools. But this standard also includes “pencil and paper” as a tool, and Mathematical Practice Standard #4 augments “pencil and paper” to distinguish within it “such tools as diagrams, two-way tables, graphs.” The number line and area model of multiplication are two more tools—both diagrammatic representations of mathematical structure—that the CCSS Content Standards explicitly require. So, in the context of elementary mathematics, “use appropriate tools strategically” must be interpreted broadly and sensibly to include many choice options for students.

Essential, and easily overlooked, is the call for students to develop the ability “to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” This certainly requires that students gain sufficient competence with the tools to recognize the differential power they offer; it also requires that their learning include opportunities to decide for themselves which tool serves them best. It also requires curricula and teaching to include the kinds of problems that genuinely favor different tools. It may also require that, from time to time, a particular tool is prescribed—or proscribed—until students develop a competency that would allow them to make “sound decisions” about which tool to use.

The number line is sometimes regarded just as a visual aid for children. It is, in fact, a sophisticated image used even by mathematicians. For young children, it helps develop early mental images of addition and subtraction that connect arithmetic with measurement. Rulers are just number lines built to spec! This number line image shows “the distance from 5 to 9” or “how much greater 9 is than 5.” Children who see subtraction that way can use this model to see “the distance between 28 and 63” as 35 , and to do so without crossing out digits and borrowing and following a rule they may only barely understand. In fact, many can learn to see this model in their heads, too, and do this subtraction mentally. This is essentially how clerks used to “count up” to make change. The number line model also extends naturally to decimals and fractions by “zooming in” to get a more detailed view of that line between the whole numbers. And it extends equally naturally to negative numbers. It thereby unifies arithmetic, making sense of what is otherwise often seen as a collection of independent and hard-to-remember rules. We can see that the distance from -2 to 5 is the number we must add to -2 to get 5: . And we can see why 42 – (-36) can also be written as 42 + 36: the “distance from -36 to 42 is . The number line remains useful as students study data, graphing, and algebra: two number lines, at right angles to each other, label the addresses of points on the coordinate plane.

The area model of multiplication is another powerful tool that lasts from early grades through college mathematics. Images like along with appropriate questions like “how many columns, how many rows, how many little squares” help establish the small multiplication facts. So might pure drill, of course, but this array image goes much further. Seeing the same array held in different positions like and makes clear that we can label any of these 3 × 4 or 4 × 3 and the number of little squares is always 12. In grades 3 through 5, array pictures like help clarify the distributive property of multiplication. This is the property that makes multi-digit multiplication possible, makes sense of the standard multiplication and division algorithms, and underlies the multiplication that students will encounter in algebra. In this picture, we see that “two 7s plus three 7s is five 7s.” The conventional notation of the idea, 2×7 + 3×7 = (2 + 3) × 7, can be a useful, even informative, summary after students already understand: a conclusion, rather than a starting place.

A schematic version of this image—the area model of multiplication—organizes students’ thinking as they learn multi-digit multiplication. In the 3×4 array, counting the squares was not impractical, though remembering the fact was certainly more convenient. But in a 65×24 array ,

neither a memorized fact nor counting are practical. Instead, by partitioning the array, we get a set of steps for which a combination of memorized facts and an understanding of place value help.

This image, combined with a spreadsheet-like summary, models the conventional algorithm exactly, making total sense of what can otherwise feel like an arbitrary set of steps.

The same image also allows students to acquire and understand the algorithm for division as a process of “undoing” multiplication, greatly simplifying the learning of a part of arithmetic that has a long history of being difficult. What makes this a powerful tool is that it serves the immediate goals of elementary school arithmetic in a way that prepares students for algebra.

Algebraic multiplication of (x + 5)(b + 4) which has no “carry” step, is

modeled perfectly by exactly the same tool. And for 147 × 46 or for (y + s – 5)(q – 2), no new image is needed. We just extend the “area” model to include the extra terms.

These versatile tools build mental models that last. What makes a tool like the number line or area model truly powerful is that it is not just a special-purpose trick or temporary crutch, but is faithful to the mathematics and is extensible and applicable to many domains. These tools help students make sense of the mathematics; that’s why they last. And that is also why the CCSS mandates them.