There is a seemingly paradoxical situation with regard to mathematics learning. Students often have informal knowledge of mathematical concepts, but have difficulty acquiring the same concepts in their formal symbolic form. For example, very young children can equally divide a collection of objects among a group of people, but they can have difficulty acquiring formal knowledge of division and fractions. Similarly, older learners sometimes use informal strategies to solve algebraic word problems, but may struggle to solve analogous algebra problems in symbolic form. Why does mathematical knowledge instantiated in familiar settings often fail to transfer to standard symbolic mathematics? How should mathematical concepts be represented to facilitate both initial learning and subsequent transfer? Can individual differences in general cognitive abilities predict the effectiveness of different instructional material? I examine questions such as these through well-designed experiments and quantitative analyses of results. Participants in my research are school-aged students and undergraduate students. My current research has been funded by the Cognition and Student Learning Program of the U.S. Department of Education, Institute of Education Sciences.