## ABSTRACT

The United States ranks much lower than most of our major economic com-petitors in the mathematics performance of high-school students. That na-tional slide begins in middle school. Many attempts have been made to improve US middle-school mathematics education, but most have failed to achieve the de-sired results. I think the reason is clear. They have focused on improving basic math skills. In contrast, I, and a great many of my colleagues, believe the emphasis should be elsewhere. Mathematics is a way of thinking about problems and issues in the world. Get the thinking right and the skills come largely for free. This simple phrase, while correct as far as it goes, requires some elaboration. First, what exactly is a basic skill? As a rough rule of thumb, the basic skills are the computations that can become so routine that they can be performed in an almost subconscious fashion. For example, most people would say the basic skills include knowledge of the multiplication tables and an ability to perform addition, subtraction, and multiplication of whole numbers and fractions. But what else? Handling parentheses in manipulating arithmetic or algebraic expressions is another skill that I think most educators would say is important. What about long division? In the pre-calculator era it was a very important skill to master. But today many mathematics educators would probably say this is not an essential basic skill, though the process of acquiring mastery does lead to an important understanding of place-value arithmetic and experience with a computational algorithm.Second, my claim that the basic skills follow “largely for free” from understand-ing does not mean a student does not have to practice those skills. I’ll come to this issue in due course. The complexity of the relationship between the development of

mathematical thinking and the mastery of basic skills becomes clear when you recog-nize that the former requires the latter.There are two reasons why the focus of mathematics education video games has until now been largely on skills. First, many people-even those in positions of power and influence-don’t understand what mathematics is and how it works. All they see are the skills, and they think-wrongly-that mathematics is about those skills. Given that most people’s last close encounter with mathematics was a skills-based school math class, it is not hard to see how this misconception arises. But to confuse mathematical thinking with mastery of basic skills is akin to con-fusing architecture with bricklaying, or playing a musical instrument with being able to play the musical scale. Of course you need the basic skills. You can’t build a brick house unless you know how to lay bricks, and you can’t play an instrument if you don’t know how to sound the different notes. Similarly, you can’t think math-ematically if you have not mastered the basic skills. But mathematical thinking is far more than merely having the basic skills at your fingertips, just as architecture is more than laying bricks and music more than playing notes. This comparison should not be taken as a suggestion that students should first master basic skills and then develop mathematical thinking. The former is in-deed required for the latter, but the skills are much more easily acquired when en-countered as a part of mathematical thinking. In my opinion, viewing these issues simplistically and separately is a large part of what is wrong with much math-ematics instruction in the United States. The two are each major components of a symbiotic whole. The same is true for learning to play a musical instrument. Doing so requires an ability to play the right notes, but that basic musical skill is acquired far more easily and effectively by trying to play tunes than by endlessly practicing the scales.The other reason for the continued focus on skills is more substantial. For over 2000 years, the only way to provide mathematics education to everyone was through the written word-textbooks. But in order to learn mathematical thinking from a textbook, you have to approach it through learning the skills. That means you have to master the skills first. And for reasons I just indicated, that is not an effective educational approach.So mathematics is not about acquiring basic skills or learning formulas. It’s a way of thinking about problems in the world. The skills are merely the tools you need in order to do that thinking. Math is not a body of knowledge, it’s something you do. And the printed word can be a terribly inefficient way to learn how to do something. The best way for an individual to learn how to do something is, as the Nike slogan says, “Just do it!” For example, if you want to learn to play chess, you can learn the rules from a book, but you won’t learn to play chess until you start playing games. The same is true for learning to ride a bicycle, learning to swim, to ski, to play tennis, or to play the guitar.