Q: When it comes to learning disorders, teachers have begun to recognize the signs of reading disabilities in the classroom, but what about math difficulties? What are teachers (or other professionals) to do when they see the telltale frustration of a student getting stumped by things that come more easily to other students?

A: There are many different reasons why a student may have difficulty with math. One major goal of our book is to help teachers identify the sources of their students' difficulties by suggesting useful questions to ask and things to look for and think about. We also offer guidelines to school psychologists, who should be invaluable sources of insight regarding possible underlying perceptual, cognitive, executive, or reasoning difficulties that may be contributing to classroom frustration.

Q: You also emphasize the importance of matching instruction to a student's way of thinking. Can you elaborate?

A: It may help to think in terms of an analogy to learning directions to a new location. Most people manage to learn directions in whatever form they're provided. But some people, by virtue of how they happen to think best, have a strong preference for using a map, others for verbal directions, and still others for landmarks. Likewise in math: some students can master equations but struggle to comprehend diagrams and graphs or to move easily between one and the other. Some students understand principles but fail to remember facts and algorithms (or vice versa). One-size-fits-all instruction will be less effective for these students than teaching that takes these discrepancies into account. Some students will need more help to absorb certain sorts of information or approach problems in different ways.

Q: Can you give one example of a conceivable math difficulty and explain what could be behind the difficulty as well as how understanding the basis for that difficulty can guide teachers' instruction?

A: Some young children, especially those from lower-income homes, enter kindergarten with a poor feel for quantity—for example, they have trouble saying which is bigger, 5 or 4. Beginning school without that fundamental skill can have long-lasting negative consequences for math learning, from which many students never recover. We now know that the brain processes quantity as if on an imaginary mental number line and that experience in the preschool years with certain kinds of number line activities, such as linear numerical board games (e.g., Candyland)—activities accessible to higher-income children—is critical to the early development of number sense. Fortunately, recent research has also shown that such difficulties can be remedied quite easily in kindergarten using simple number line activities. (There is no research yet on effective interventions for older students.)