A dialogue between Socrates and a researcher who claims to know what constitutes effective teaching is used to critique the implicit assumptions held by researchers on teaching, highlighting their differences with construcrivist researchers. Teacher effectiveness research is criticized for its empiricism; its disregard of the interaction between content and instruction; and its conceptions of mathematics, teaching, and learning. Potential bridges for communication are identified from the two communities' shared interest in how students perceive different examples and explanations and how they might be intrinsically motivated to learn mathematics. In Socratic style, the dialogue ends by emphasizing that the question of what constitutes effective teaching remains unresolved.
Jere Confrey and Erick Smith
Exponential and logarithmic functions are typically presented as formulas with which students learn to associate the rules for exponents/logarithms, a particular algebraic form, and routine algorithms. We present a theoretical argument for an approach to exponentials more closely related to students' constructions. This approach is based on a primitive multiplicative operation labeled “splitting” that is not repeated addition. Whereas educators traditionally rely on counting structures to build a number system, we suggest that students need the opportunity to build a number system from splitting structures and their geometric forms. We advocate a “covariation” approach to functions that supports a construction of the exponential function based on an isomorphism between splitting and counting structures.
P. Holt Wilson, Marrielle Myers, Cyndi Edgington and Jere Confrey
Teaching young children to create equal-size groups is your treasure map for building students' flexible, connected understanding of and reasoning about ratios, fractions, and multiplicative operations.
P. Holt Wilson, Cynthia P. Edgington, Kenny H. Nguyen, Ryan C. Pescosolido and Jere Confrey
Develop and strengthen students' rational number sense with problems that emphasize equipartitioning.
Timothy Boerst, Jere Confrey, Daniel Heck, Eric Knuth, Diana V. Lambdin, Dorothy White, Patricia C. Baltzley and Judith Reed Quander
The National Council of Teachers of Mathematics (NCTM) is committed to strengthening relations between research and practice and to the development of a coherent knowledge base that is usable in practice. The fifth of NCTM's strategic priorities states, “Bring existing research into the classroom, and identify and encourage research that addresses the needs of classroom practice” (NCTM, 2008). The need to work toward connection and coherence is not unique to the field of mathematics education. Fields such as medicine (e.g., Clancy, 2007), software engineering (e.g., Gorschek, Garre, Larsson, & Wohlin, 2006), and social work (e.g., Hess & Mullen, 1995) routinely attend to these issues. Researchers in many fields strive to find new ways or to engage more effectively through existing means to enhance coherence and connection. In a sense, this is not a goal that can be achieved definitively, but one that requires persistent engagement. In education, the constant flux of variables in the system, such as curriculum, goals for student learning, and school contexts, requires that new connections between research and practice be investigated and that old connections be reexamined. Changes in educational contexts open new territory in need of study and also challenge the coherence of explanations grounded in previous research. In this way, attention of the field to connection and coherence is neither unique to mathematics education nor an effort due solely to inadequacies of research efforts in the past.
NCTM Research Committee
Michael Battista, Timothy Boerst, Jere Confrey, Eric Knuth, Margaret S. Smith, John Sutton, Dorothy White and Judith (Reed) Quander
I do not think any thoughtful researcher today believes that experiments or randomized field trials are the “gold standard” for addressing all the important questions in educational research. Yet, because these designs are now required by the 2001 No Child Left Behind Act (NCLB) and are being strongly encouraged in other federal legislation and funding initiatives, scholars, practitioners, parents, and researchers must devote time and energy to fighting these designs when they are inappropriate or irrelevant, which is often the case. Despite long-standing objections from prominent methodologists and reservations expressed by national groups and committees, key policymakers in the federal government are encouraging the pursuit of experimental designs primarily or exclusively (Eisenhart, 2005, p. 246).
NCTM Research Committee
Jere Confrey, Marilyn E. Strutchens, Michael T. Battista, Margaret Schwan Smith, Karen D. King, John T. Sutton, Timothy A. Boerst and Judith Reed
Research on curricular choices has attracted widespread attention and merits increased investment by the research community. Multiple studies, publications, conferences, and a multicampus center (The Show-Me Project, n.d.) speak to the need to discuss what is taught in our classrooms and to whom, how, and when. For example, the January 2007 issue of JRME focused on research related to the development, implementation, and evaluation of curricula (NCTM, 2007). Williams (2007) commented that one of the challenges we face as a field is determining what questions of practice are worth asking and how research on those questions can help advance the field. Since the 1990s, with the creation of Standards-based curricula, practitioners have had to choose between new and traditional curricula. In addition, the No Child Left Behind Act of 2001 has put pressure on schools to use curriculum materials that are research based and approved by the U.S. Department of Education (n.d.). These two milestones related to mathematics education justify making curricula issues an area that warrants the attention of both the research community and practitioners. The purpose of this article is to highlight advances related to curricular research; pose questions that require further investigation; and describe related, emerging subfields.