# Willingham: We have math standards, but now what?

*University of Virginia cognitive scientist Daniel Willingham, author of "Why Don't Students Like School?" is my guest today.*

By Daniel Willingham

We’ve got good standards . . . now what?

The national standards for mathematics have been drafted by the Common Core State standards initiative and I think that the standards look quite good. But it is well to remember that standards are necessary for successful education, but they are probably not sufficient. We not only need to set appropriate standards, we need to know how to achieve them.

Implicit in the standards is the need for three types of knowledge.

Students must memorize some math facts (e.g., the times table). They must know certain procedures (e.g., how to multiply fractions). Both types of knowledge need to be well-learned—so well learned that the information can be retrieved rapidly and effortlessly. It’s hard to solve a complex long division problem if you need to stop what you’re doing to calculate simple products along the way.

Students also need conceptual knowledge. They need to understand why the procedures work, e.g., why “invert and multiply” yields the right answer when dividing fractions. Without conceptual knowledge, it is difficult to solve novel problems. The student can recognize that certain procedures apply to certain problem types, but if a problem is dressed up in a slightly different format, the student likely will be stumped.

American students generally have adequate (not terrific) factual and procedural knowledge. Their conceptual knowledge is, on average, terrible. One of the most startling data points I’ve seen on this topic is that about 70% of sixth graders don’t really understand what an equal sign is. They often think it means “put the answer here,” and don’t understand that it signifies equality.

It’s great that the Common Core standards acknowledge the importance of conceptual knowledge, but prior documents have done so—sometimes to the exclusion of factual and procedural knowledge. The problem is that this is the most difficult type of knowledge to teach and to learn.

Even the most basic mathematical thinking does not come naturally to the human brain. Humans (and many other animals) are born with an understanding of numerosity, and we can appreciate the difference in cardinal numbers up to about five. More than five is simply “many.”

In addition, we have a natural appreciation for a relationship between numbers and space. But the natural understanding of this relationship is not the linear relationship between number and space found on the number line. Our natural understanding is logarithmic: picture a number line with more room between 1 and 2 than between 9 and 10.

Formal mathematics must be bolted on to this framework that is provided by our genetic inheritance. The earliest grades are most crucial because if a student doesn’t understand the basic concepts, it becomes increasingly difficult to catch up. The student can still memorize formulas and solve math problems, but will have very little idea of what he or she is doing.

Singapore math solves this conceptual problem brilliantly by introducing the spatial-numeric relationship very early on. Children are introduced to two simple ways of manipulating spatial models. These two models are so flexible that small variations in them can be used to introduce all of the concepts students will learn, up to algebra. Thus, each new concept seems familiar because it’s based on a model that the student has seen many times.

American schools don’t necessarily need to adopt the Singapore method, but we need something that is equally effective.

Hung-Hsi Wu, a mathematics professor at the University of California at Berkeley, who has devoted his career to math education, has suggested that teaching mathematics is so difficult that schools should consider having specialized mathematics teachers beginning no later than the fourth grade.

Teaching math *is* that difficult, and that different than the teaching of other subject matter. I think he’s right, and I might amend that to the first grade.

By
Valerie Strauss
| November 16, 2009; 11:30 AM ET

Categories:
Daniel Willingham, Guest Bloggers, National Standards
| Tags:
Dan Willingham, math

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Posted by: harvard3 | November 17, 2009 11:36 AM | Report abuse

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Having struggled with math throughout my education, I only hope teachers will consider what has been proposed here.