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• Author or Editor: Jeremy A. Kahan
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## The Spot Problem: Connecting Points, Connecting Mathematics

In this article, we discuss the spot problem, which we have used many times in our work with preservice and in-service teachers of secondary mathematics. This problem appeals to us because of its multiple connections. First, it illustrates that patterns and proof are both necessary in mathematics and are related to each other. Moreover, in working on this problem, students connect geometry, algebra, graph theory, and combinatorics. Finally, the problem connects with another well-known problem, the pizza problem, which we describe subsequently.

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## My Calculator Is Broken; It Says the Log of (−1) Is …

Recently, a student in the mathematical methods for secondary teachers class that I (Richgels) teach stopped me in the hall and asked a question: “What does it mean when a calculator gives you a number for the log of negative one?” Immediately, smiling inside because I thought that I had caught a rookie, I asked the student to show me the calculator. I was impressed because the student had a new TI-86. The student entered–1. When he pressed the key, I expected an error message. Instead the calculator displayed (0, 1.36437635384). I did a double take and asked for the calculator. I had seen the student enter the correct expression but had to verify the result for myself. For almost thirty years, I had taught that the logarithm of a negative number is not defined.

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## Problem Solving Can Generate New Approaches to Mathematics: The Case of Probability

In this article, we discuss the World Series problem, which we have used many times in our work with preservice and in-service teachers of secondary mathematics. This problem yields to multiple approaches. Furthermore, the history of the underlying problem illustrates how mathematics sometimes develops in response to the need to solve a problem, a process that we want to let our students share. In fact, we believe that this problem is a strong illustration of Stanic and Kilpatrick's (1989) metaphor that the problem is a vehicle. Those who work on (ride) the problem arrive at significant mathematics (a destination). We begin with a presentation of the vehicle, then explore several routes that students might take and notice how these routes all lead to the same destination.

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## Visions of Problems and Problems of Vision: Embracing the Messiness of Mathematics in the World

Problems and problem solving have a long history in mathematics education (Dewey, 1910; National Council of Teachers of Mathematics [NCTM], 1980; Pólya, 1945; Schoenfeld, 1992; Stanic & Kilpatrick, 1988). The Curriculum and Evaluation Standards for School Mathematics asserted, “Problem solving should be the central focus of the mathematics curriculum” and placed it as Standard 1 (NCTM, 1989, p. 23). The 1990s saw the development of school mathematics curricula based on various interpretations of these Standards. In most of these curricula, the mathematics emerges from the solution of problems, and there is a growing body of research evidence supporting the effectiveness of these curricula (Senk & Thompson, 2003). Teaching mathematics through problem solving also continues to be a focus of mathematics educators independent of the curriculum that is used (Schoen & Charles, in press).

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## Standards based Mathematics Curricula and Secondary Students' Performance on Standardized Achievement Tests

The current study examined the mathematical achievement of high school students enrolled for 3 years in one of three NSF funded Standards-based curricula (IMP, CMIC, MMOW). The focus was on traditional topics in mathematics as measured by subtests of a standardized achievement test and a criterion-referenced test of mathematics achievement. Students generally scored at or above the national mean on the achievement subtests. Hierarchical linear modeling results showed that prior mathematics knowledge was a significant but modest predictor of achievement, student SES had a moderate effect, and increasing concentrations of African American students in a classroom were associated with a stronger effect of attendance on achievement. No differences on the standardized achievement subtests emerged among the Standards-based curricula studied once background variables were taken into account. The two suburban districts providing data for the criterion-referenced test achieved well above the national norm.

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## Standards-based Mathematics Curricula and Middle-Grades Students' Performance on Standardized Achievement Tests

Approximately 1400 middle-grades students who had used either the Connected Mathematics Project (CMP) or the MATHThematics (STEM or MT) program for at least 3 years were assessed on two widely used tests, the Stanford Achievement Test, Ninth Edition (Stanford 9) and the New Standards Reference Exam in Mathematics (NSRE). Hierarchical Linear Modeling (HLM) was used to analyze subtest results following methods described by Raudenbush and Bryk (2002). When Standards-based students' achievement patterns are analyzed, traditional topics were learned. Students' achievement levels on the Open Ended and Problem Solving subtests were greater than those on the Procedures subtest. This finding is consistent with results documented in many of the studies reported in Senk and Thompson (2003), and other sources.