for students’ preconceptions on the subject, will leave many students with incorrect understanding (for a review of studies, see Mestre, 1994).

For young children, early concepts in mathematics guide students’ attention and thinking (Gelman, 1967; we discuss this more in Chapter 4). Most children bring to their school mathematics lessons the idea that numbers are grounded in the counting principles (and related rules of addition and subtraction). This knowledge works well during the early years of schooling. However, once students are introduced to rational numbers, their assumptions about mathematics can hurt their abilities to learn.

Consider learning about fractions. The mathematical principles underlying the numberhood of fractions are not consistent with the principles of counting and children’s ideas that numbers are sets of things that are counted and addition involves “putting together” two sets. One cannot count things to generate a fraction. Formally, a fraction is defined as the division of one cardinal number by another: this definition solves the problem that there is a lack of closure of the integers under division. To complicate matters, some number-counting principles do not apply to fractions. Rational numbers do not have unique successors; there is an infinite number of numbers between any two rational numbers. One cannot use counting-based algorithms for sequencing fractions: for example, 1/4 is not more than 1/2. Neither the nonverbal nor the verbal counting principle maps to a tripartite symbolic representations of fractions—two cardinal numbers X and Y separated by a line. Related mapping problems have been noted by others (e.g., Behr et al., 1992; Fishbein et al., 1985; Silver et al., 1993). Overall, early knowledge of numbers has the potential to serve as a barrier to learning about fractions— and for many learners it does.

The fact that learners construct new understandings based on their current knowledge highlights some of the dangers in “teaching by telling.” Lectures and other forms of direct instruction can sometimes be very useful, but only under the right conditions (Schwartz and Bransford, 1998). Often, students construct understandings like those noted above. To counteract these problems, teachers must strive to make students’ thinking visible and find ways to help them reconceptualize faulty conceptions. (Strategies for such teaching are discussed in more detail in Chapters 6 and 7.)

Prior knowledge is not simply the individual learning that students bring to the classroom, based on their personal and idiosyncratic experiences (e.g., some children will know many things because they have traveled widely or because their parents have particular kinds of jobs; some children may have suffered a traumatic experience). Prior knowledge is also not only a generic set of experiences attributable to developmental stages through which learners may have passed (i.e., believing that heaven is “up” or that milk comes