Having listed the five strands of mathematical proficiency in Chapter 6, I’ll now look in turn at how teachers can take advantage of a suitable video game (when they become available) to help students acquire mastery of each strand, and what game developers can do to facilitate that learning. I start with conceptual understanding. How Important Is Conceptual Understanding?First, I’ll note that full conceptual understanding, while desirable, is not strictly necessary in order to be able to apply mathematics successfully. Differential calculus was first developed in the middle of the seventeenth century, and has been used extensively and successfully ever since. Yet it was not until the very end of the nineteenth century, some 250 years later, that mathematicians achieved full conceptual understanding, and then only after considerable effort had been exerted to develop the mathematical machinery necessary to ground such understanding. Even today, the experience most students have when they learn calculus is that it’s possible to ace all the calculus exams without understanding what calculus is or how it works. That was definitely how I learned calculus. And I know from hav-ing included on my calculus exams for many years at least one question that tries to elicit from my students their degree of understanding, that almost all of them don’t understand it any more than I did at that stage. So much for the much-touted myth that doing mathematics requires conceptual understanding.What mathematical thinking does require, I suggest, is functional understanding. Let me explain what I mean by that term. I’ll start out with a question:

what exactly does it mean to understand an abstract mathematical concept? Take the most familiar example of all: positive whole numbers. What does it mean to say that a child (or adult for that matter) has mastered the concept of a positive whole number? This question is not as clear-cut as might first be assumed. Moreover, we can appreciate just what a major cognitive leap it is to grasp this purely abstract concept by recalling that it took many generations to develop; early humans were able to count collections long before numbers came on the scene. The Number ConceptNumbers,1 specifically whole numbers, arise from the recognition of patterns in the world around us; the pattern of oneness, the pattern of twoness, the pattern of threeness, and so on. To recognize the pattern that we call threeness is to recognize what it is that a collection of three apples, three children, three footballs, and three rocks have in common. “Can you see a pattern?” a parent might ask a small child, showing her various collections of objects-three apples, three shoes, three gloves, and three toy trucks. The counting numbers 1, 2, 3, and so on are a way of capturing and describing those patterns. The patterns captured by numbers are abstract, and so are the numbers used to describe them.