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

Abstract

IntroductionElementary teachers need a solid understanding of so that they can teach it as a coherent, reasoned activity and communicate its elegance and power (Conference Board of Mathematical Sciences [CBMS], 2001, p. xi}. However, research studies on prospective teachers' knowledge have shown that many possess a limited knowledge of in key content areas as number (e.g.. Ball, 1990a; Thanheiser, 2009; Tobias, 2013}. This is particularly true in case of which, along with ratio and proportion, Lamon (2007} calls, the most protracted in terms of development, most difficult to teach, most mathematically complex, most cognitively challenging, most essential to success in higher and science, and one of most compelling research sites (p. 629}.The National Mathematics Advisory Panel (2008} affirmed that with fractions is a major goal for K-8 education because such proficiency is foundational for algebra and, at present time, seems to be severely underdeveloped (p. xvii}. Therefore, developing proficiency in prospective elementary teachers (PTs} is a critical task for educators. As authors of 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 courses for prospective teachers that will help them to develop solid understanding of mathematics called for by Conference Board of Mathematical Sciences (2001}, including a deep understanding of and with fractions, we must begin by determining what it is that PTs know. In this paper, we discuss 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 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 discipline are organized to incorporate its facts and truth or falsehood, validity or invalidity, are established (p. 9]. Pedagogical content knowledge refers to 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 conceptions and preconceptions students bring to learning processes. third type of knowledge, curricular knowledge, includes knowledge of a full range of programs designed for teaching of particular subjects and topics at a given level, variety of instructional materials available in relation to those programs, and set of characteristics that serve as both indications and contraindications for 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 (e.g., Ball et al., 2008; Davis & Simmt, 2006; Hiebert, 1986; Ma, 1999]. Deborah Ball and her colleagues introduced term mathematical knowledge for teaching (MKT) (e.g., Ball & Bass, 2002), which focused on 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. …