Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Fractions

By Dana E. Olanoff, Jane-Jane Lo, Jennifer M. Tobias. The Mathematics Enthusiast, 2014

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IntroductionElementary teachers need a "solid understanding of mathematics so that they can teach it as a coherent, reasoned activity and communicate its elegance and power" (Conference Board of the Mathematical Sciences [CBMS], 2001, p. xi}. However, research studies on prospective teachers' mathematics knowledge have shown that many possess a limited knowledge of mathematics in key content areas such as number (e.g.. Ball, 1990a; Thanheiser, 2009; Tobias, 2013}. This is particularly true in the case of fractions, which, along with ratio and proportion, Lamon (2007} calls, "the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, the most essential to success in higher mathematics and science, and one of the most compelling research sites" (p. 629}.The National Mathematics Advisory Panel (2008} affirmed that "proficiency with fractions" is a major goal for K-8 mathematics education because "such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped" (p. xvii}. Therefore, developing such proficiency in prospective elementary teachers (PTs} is a critical task for mathematics educators. As the authors of The Mathematical Education of Teachers, Part 1 suggest, "The key to turning even poorly prepared prospective elementary teachers into mathematical thinkers is to work from what they do know" (CBMS, 2001, p. 17}. Thus, in order to design mathematics courses for prospective teachers that will help them to develop the "solid understanding of mathematics" called for by the Conference Board of Mathematical Sciences (2001}, including a deep understanding of and "proficiency with fractions," we must begin by determining what it is that PTs know. In this paper, we discuss the main findings from a research summary of existing studies on prospective elementary teachers' fraction knowledge to identify directions for future research.Theoretical FrameworkIn looking at teacher knowledge, we begin by examining the work of Shulman (1986], who proposed three categories of content knowledge for teachers: (a] subject matter content knowledge, (b] pedagogical content knowledge, and (c] curricular knowledge. For Shulman, subject matter content knowledge includes knowing a variety of ways in which "the basic concepts and principles of the discipline are organized to incorporate its facts" and "truth or falsehood, validity or invalidity, are established" (p. 9]. Pedagogical content knowledge refers to the knowledge of useful forms of representations (e.g., analogies, illustrations, explanations] of subject-matter ideas that make it understandable to others, as well as an understanding of the conceptions and preconceptions students bring to the learning processes. The third type of knowledge, curricular knowledge, includes knowledge of a "full range of programs designed for the teaching of particular subjects and topics at a given level, the variety of instructional materials available in relation to those programs, and the set of characteristics that serve as both the indications and contraindications for the use of particular curriculum or program materials in particular circumstances" (p. 10].Shulman's ideas on pedagogical content knowledge sparked a huge interest in knowledge for teaching, eliciting over a thousand studies (Ball, Thames, & Phelps, 2008] throughout a number of content areas, with a large number of these studies focusing on teachers' knowledge of mathematics (e.g., Ball et al., 2008; Davis & Simmt, 2006; Hiebert, 1986; Ma, 1999]. Deborah Ball and her colleagues introduced the term mathematical knowledge for teaching (MKT) (e.g., Ball & Bass, 2002), which focused on the work that teachers do when teaching mathematics.Building on Shulman's (1986) categories of knowledge, Ball, Thames, and Phelps (2008) introduced a framework for mathematical knowledge for teaching. …

Subfields: Education, Statistics and Probability

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