them into parts, and it seriously changes their nature. Thus, joining rods will support an extension of arithmetic into fractional quantities much more easily than counting cats will.

Similarly, multiplication has multiple interpretations. We introduced it as adding the same number many times. The set-combination interpretation of multiplication would be to combine several essentially identical collections, such as the packages of cookies mentioned above. If you think of addition in terms of joining rods, then multiplication would amount to joining several rods of the same length end to end. Thus, 4×6 can be visualized by laying four rods of length six end to end, where you can think of each rod as a little row of boxes. A more compact way to arrange the rods would be to lay them side by side rather than end to end. This arrangement produces an array of four rows of boxes with six boxes in each row, which may be called a rectangular array interpretation of multiplication. When the rods have height one, there is an added benefit: The array looks like a rectangle of boxes, and the area of the rectangle (measured in box areas) is just 4×6. This is the area interpretation of multiplication.

The multiple interpretations of the basic operations is symptomatic of a general feature of mathematics—the tension between abstract and concrete.3 This tension is a fundamental and unavoidable challenge for school mathematics. On the one hand, as we indicated above, the abstractness of math-



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