levels of thinking in conditional probability were identified, with few children being able to recognize that the probabilities changed in situations of selection without replacement. Following instruction, 51% were able to recognize that conditional probabilities changed in these situations.168 Children have difficulty determining the conditioning event and may be confused about the context of a conditional probability problem.

Independence

Students’ intuitive understanding of independence is measured by their ability to recognize and justify when the occurrence of one event has no influence on the occurrence of another. In one study, students in grades 4 through 8 were asked to determine which event was more likely: obtaining 3 heads by tossing one coin 3 times, or by tossing 3 coins simultaneously.169 Some 38% of fourth and fifth graders and 30% of seventh and eighth graders with no prior instruction in probability responded that the probabilities were not equal. Follow-up interviews revealed that these students harbored the pervasive misconception that the outcomes of a coin toss can be controlled. Similar misconceptions were evident in other studies of middle school students.170 Misconceptions of the kind illustrated above have been characterized more generally as representativeness—a belief that a sample or sequence of outcomes should reflect the whole population.171

From Arithmetic to Mathematics

As children move from number to other domains of mathematics, they both use their proficiency with number and develop it further. The school mathematics curriculum, although separated into domains for the purposes of this report, needs to be experienced by the learner as a unified whole.

In general, the arithmetic thinking of number-proficient students emerging from the typical elementary school mathematics program is different from the thinking that is central to algebra. Some of the conceptual understanding of the arithmetic thinker requires an adjustment when the student engages in the main types of activities in algebra. Whereas arithmetic focuses on number and numerical answers, school algebra focuses on relations. Algebra remains, however, a natural extension of arithmetic. Students’ numerical thinking can therefore continue to grow and develop into algebraic thinking, but their numerical thinking habits must be taken into account.

Just as current research has influenced conceptions of algebra in the early grades, the nature of school algebra in higher grades has likewise been evolv-



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