This research will focus on the acquisition of knowledge underlying understanding of mathematics and science. One set of studies will develop further Gelman's commitment to achieving a good description of the kinds of understandings that children have when they start their school math and science lessons. A second set of studies will pursue the fact that these early understandings sometimes do and sometimes do not facilitate further learning about mathematics and science. The research will be concerned especially with the conditions under which early knowledge acts more like a barrier to, as opposed to a support for, further learning. To illustrate, many elementary school children read fractions as if these kinds of symbols are novel ways of writing whole numbers; for example when asked to read 1/2, they say "one and two" or "one plus two, that is three", or even "12." Similarly, they will say that 1/4 is more than 1/2 because "four is more than two" or 2.09 is more than 29 because 209 is more than 29. The research will investigate ways to build understanding of why rational numbers are different from count numbers, that is that fractions are governed by different rules of mathematics than are the positive count numbers. There are several reasons to look forward to these efforts. First, understanding of fractions is related to understanding of a range of important ideas, including division, measurement, and the notion of equal units, and so on, just the kind of concepts one would expect to be included in a definition of mathematical literacy. Second, it is well known that fractions are a watershed for students; those who do master them mathematically go on to learn more mathematics, while those who do not have special trouble even with algebra. Additionally, these studies will offer insight into the more general need to understand how conceptual change can take place. This research's focus on the contrast between initial learnings that lead readily to further concept acquisition and those that take special efforts to surpass the boundaries of what is already known will offer important insights on the kind of environments that are needed to encourage and support a learner's advance to scientific and mathematical literacy, a knowledge state that reflects the ability to understand and use the relevant concepts as well as their related symbols and terms. Already, Gelman's progress with these research questions has led to the development by others of a curriculum that embeds science instruction into a High School English as a Second Language program. As a result, students are acquiring understanding of scientific terms they will encounter when they start science classes.
