"Shutting the Box": Fostering Collaboration Among Early Grades and Secondary Preservice Teachers Through Authentic Problem Solving

In this paper are discussed recent efforts to provide preservice mathematics teachers with opportunities to connect elementary teaching methods and content with the content and methods of secondary school mathematics. Through an indepth exploration of the game, Shut the Box, preservice elementary and secondary mathematics teachers thoughtfully analyzed and manipulated computer-generated output, developed and tested their own conjectures, and collaboratively answered questions involving theoretical probabilities across courses and content levels. Through their collaboration, the preservice teachers gained a better appreciation of mathematics content and pedagogical strategies that lie beyond the grades they will likely teach, as they reconsidered the importance of content and pedagogical knowledge at every level of mathematics instruction. These interactions are considered in this document through a discussion of the mathematical underpinnings of the popular board game. Contemporary Issues in Technology and Teacher Education, 6(4) 375 Results of curricular and classroom investigations of the Third International Mathematics and Science Study (TIMSS, 1995) reported that time in the American mathematics classroom is more likely spent learning specific concepts and skills—at a somewhat shallow level—without providing the deep connections that experts believe will “improve students’ ability to learn and understand a subject in an integrated way” (National Center for Educational Statistics, 1997). This state of affairs lies in direct contrast to recommendations set forth in the Curriculum Principle and the Connections Standard of the Principles and Standards for School Mathematics (National Council Teachers of Mathematics [NCTM], 2000). Two major goals for American mathematics classrooms detail the need for a coherent curriculum and classroom experiences that help students develop a connected, integrated, holistic view of mathematics, rather than experience it from year to year, or grade to grade, as a disconnected discipline (NCTM, 2000). Against this backdrop of the K-12 mathematics classrooms, I sought a preventive approach that might build on a specific recommendation for teacher preparation. It highlights the need for prospective teachers to make connections extending beyond their specific grade-level boundaries (Conference Board of the Mathematical Sciences [CBMS], 2001). Perhaps successful classroom teachers—particularly those with significant, varied classroom teaching experience—ably and meaningfully communicate connections among various mathematical concepts and levels of mathematics. Unfortunately, prospective mathematics teachers (hereafter referred to as preservice teachers or PSTs) often struggle to make these same connections for their students. As a result many PSTs do not have a good sense of where their students "came from" or where they “are going" mathematically. They may, therefore, benefit from experiences in pedagogy and content courses that prepare them to make mathematical connections beyond their particular level of certification (or licensure). In this article, is described an approach and the resulting benefits in which various technology-based mathematics and communication tools (e.g., programming and data analysis utilities, computer algebra systems, flowcharting software, and Internet-based research applications) were used to explore connections between elementaryand secondary-level mathematics. While building new mathematical knowledge related to probability elementary and secondary mathematics, PSTs collaboratively engaged in authentic mathematical activity: generating original conjectures, testing hypotheses, and taking "wrong turns.” The elementary mathematics PSTs need college mathematics experiences in which their ideas for solving problems are elicited and taken seriously, their sound reasoning affirmed, and their missteps challenged in ways that help them make sense of their errors (CBMS, 2001, p. 17). To further enrich their mathematical understanding they also need to explore and develop connections to the mathematical concepts that extend beyond the grades for which they will be licensed (CBMS, 2001). Similarly, secondary mathematics PSTs need knowledge of the mathematical concepts and skills that students acquire before they reach the high school years (CBMS, 2001). When provided with opportunities to collaborate—particularly when engaged in authentic, long-term problem-solving projects—both groups (elementary and secondary mathematics PSTs) may begin to break the "cycle of disconnect" that exists between their respective groups and become better prepared for teaching mathematics to their prospective students. Contemporary Issues in Technology and Teacher Education, 6(4) 376 Recent efforts to provide mathematics PSTs with opportunities to connect elementary teaching methods and content with the methods and content of secondary school mathematics are shared in this article. Through an in-depth exploration of the popular board game, Shut the Box (Red Fern Enterprises, 2006), elementary and secondary PSTs thoughtfully analyzed and manipulated computer-generated output, developed and tested their own conjectures, and collaboratively answered questions involving theoretical probabilities across courses and content levels. Through their collaboration, the PSTs gained a better appreciation of mathematics content and pedagogical strategies that lie beyond the grades they will likely teach. They reconsidered the importance of content and pedagogical knowledge at many levels of mathematics instruction and, as hoped, they experienced “their own capacity for mathematical thought” (in the words of CBMS, 2001, p. 24). These interactions are considered in the context of discussions of the mathematical underpinnings of Shut the Box.