As an example of a P R mathematics task consider the following question in

# As an example of a p r mathematics task consider the

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As an example of a P-R mathematics task, consider the following question in relation to the classroom scenario described earlier: What would be a mathematically appropri- ate way in which the teacher could respond to Jane’s question about whether the class could use Mark’s method every time they had to find a fraction between two given, nonequivalent positive fractions? March 2014 Notices of the AMS 269 Figure 2. An algebraic proof of a general method for finding a fraction between two given positive and nonequivalent fractions. We will henceforth refer to this task as the Fractions Task . A solution to the Fractions Task would build on the “course of action” that we discussed earlier under possibility 2, which is the desirable possibility. According to this course of action, the teacher would engage the class in the discussion of a proof that would not only be valid but also accessible to the group of seventh-graders. Feature 1: A mathematical focus P-R mathematics tasks have a mathematical fo- cus that relates to one or more mathematical ideas that theory, research, or practice suggested are important for teachers to know. The mathematical focus is intended to engage prospective teachers in mathematical activity. In the Fractions Task, the mathematical focus is the mathematical evalua- tion of Mark’s method, which can be expressed algebraically as follows: Given two fractions a b and c d where a, b, c, d > 0 and a b < c d , a b < a + c b + d < c d . Feature 2: A substantial pedagogical context In addition to the mathematical focus, a P-R mathematics task has a substantial pedagogical context that is an integral part of the task and essential for its solution. The pedagogical con- text situates prospective teachers’ mathematical activity in a particular teaching scenario and helps prospective teachers engage with the mathematics from the perspective of a teacher. In the Fractions Task the pedagogical context describes the teacher’s need to formulate a re- sponse to Jane’s question about whether the class could use Mark’s method when asked to find a fraction between two positive and nonequivalent fractions. According to this context, the event happened in a seventh-grade class, which allows the solvers of the task (prospective teachers) to make certain assumptions about what the students in the class might know or be able to understand. Thus a solution to the task must not only satisfy mathematical considerations but also needs to take into account pedagogical considerations. Next we discuss four points related to feature 2 of P-R mathematics tasks. First, the pedagogical context in which a P-R mathematics task is situated determines to a great extent what counts as an acceptable/appropriate solution to the task, because it provides (or suggests) a set of conditions a possible solution to the task needs to satisfy. In the Fractions Task, for example, an algebraic proof of Mark’s method like the one in Figure 2, though mathematically valid, would likely not be within the conceptual reach of students in a seventh-grade class. A proof can  • • • 