Mathematics Teacher TPACK Standards and Development Model

What knowledge is needed to teach mathematics with digital technologies? The overarching construct, called technology, pedagogy, and content knowledge (TPACK), has been proposed as the interconnection and intersection of technology, pedagogy, and content knowledge. Mathematics Teacher TPACK Standards offer guidelines for thinking about this construct. A Mathematics Teacher Development Model describes the development of TPACK toward meeting these standards. The standards and model provide structured detail to further the work of various groups. The proposals may guide teachers, researchers, teacher educators, professional development consultants, and school administrators in the development and evaluation of professional development activities, mathematics education programs, and school mathematics programs. Contemporary Issues in Technology and Teacher Education, 9(1) 5 In 1986 Lee Shulman launched a new way of thinking about the knowledge teachers need for teaching with a construct that he called pedagogical content knowledge (PCK). This new way of thinking about the knowledge teachers need for teaching called for the integration of content knowledge (the knowledge previously considered the primary knowledge for teachers) and pedagogical knowledge (the knowledge about teaching and learning). The intersection of these two knowledge bases, PCK, was described as the way of representing and formulating subject matter knowledge, the knowledge that makes the subject matter comprehensible to learners (Shulman, 1986, 1987; Wilson, Shulman, & Richert, 1987). More specifically, Shulman (1986) characterized a teacher’s PCK as knowledge of the most regularly taught topics in one’s subject area, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations . including an understanding of what makes the learning of specific concepts easy or difficult: the concepts and preconceptions that students of different ages and backgrounds bring with them to the learning. ( p. 9) During those early discussions of the construction of knowledge growth in teaching, teacher preparation programs were challenged to determine how they might guide the development of this teacher knowledge. Some programs honed in on the development of six primary domains of knowledge essential for effective instruction: subject matter knowledge, pedagogical knowledge, knowledge of schools, knowledge of learners, and curricular knowledge, with PCK as the essence of the intersection of these five domains of knowledge (Niess, 2001). The relationship was viewed as a complex and integrated structure where no domain was totally distinct or separate from the other, with the relative amount of overlap and interaction among the domains constantly changing as preservice teachers made sense of and prioritized the multiple factors affecting student learning. Attention to PCK through research studies provided insight into the preparation of preservice mathematics teachers’ development of PCK (Ball, 1988; Civil, 1992; Grossman, 1991; McDiarmid, 1990; Simon & Brobeck, 1993; Simon & Mazza, 1993; Wilcox et al., 1990). Grossman’s (1989, 1990) research identified four central components of PCK to focus the description and understanding of the knowledge needing development in the preparation programs: (a) an overarching conception of what it means to teach a particular subject; (b) knowledge of instructional strategies and representations for teaching particular subject matter topics; (c) knowledge of students’ understandings, thinking, and learning in the subject area; (d) knowledge of curriculum and curriculum materials with learning subject matter (Borko & Purtnam, 1996). As this understanding of PCK evolved, modern digital technologies also began to be recognized as useful for teaching and learning. During the late 1970s and 1980s, the focus in mathematics education was on identifying places in mathematics instruction for inserting digital technology applications. A myriad of software programs afforded drill and practice in a variety of environments that were more entertaining than traditional paper-and-pencil worksheets for providing practice with computational skills. Graphing calculators offered capabilities for efficiently generating visuals of graphs useful for demonstrating mathematical ideas such as slope and y-intercept for linear functions and points of intersection for multiple functions. The primary vision for employing mathematical digital technologies was for demonstration and verification of ideas previously developed in the classroom. Calculators – from limited four-function calculators to scientific calculators – were Contemporary Issues in Technology and Teacher Education, 9(1) 6 restricted with the belief that these tools trivialized the mathematics rather than engaging students in learning mathematics. The lack of an in-depth integration of these technologies prompted Kaput’s (1992) lament that the “major limitations of computer use in the coming decades are likely to be less a result of technological limitations than a result of limited human imagination and the constraints of old habits and social structures” (p. 515). An examination of mathematics teachers’ PCK in the late 1980s and early 1990s revealed an overarching conception that teachers’ beliefs about how to teach mathematics generally were aligned with how they learned mathematics. Although a few teachers embraced the use of graphing calculators, spreadsheets, and software like Logo and Geometric Supposer, many did not. Mathematics teachers’ knowledge of instructional strategies and representations for teaching particular mathematical topics relegated the application of such digital technologies to demonstration, verification, and drill and practice. Their knowledge of students’ understandings, thinking, and learning in mathematics held to the importance of mastery of skills with paper and pencil prior to using modern digital technologies (Kastberg & Leatham, 2005; Walen, Williams, & Garner, 2003; Yoder, 2000). Furthermore, access to technology without necessary knowledge of related curriculum materials did not encourage teachers to incorporate the technology in their classroom instruction (Kastberg & Leatham, 2005). “In the absence of professional development on instructional technology and curriculum materials that integrate technology use into the lesson content, teachers are not particularly likely to embed technology-based or technology-rich activities into their courses” (Ferrini-Mundy & Breaux, 2008, p. 437438). Fast forward to 2008 to see that many mathematics teachers’ PCK lacks a solid and consistent integration of modern digital technologies in mathematics curriculum and instruction. Technologies, such as dynamic geometry tools or advanced graphing calculators with computer algebra systems (CAS), are primarily used for modeling and providing examples, where students imitate the actions and use the technologies for verification, demonstration, and drill and practice. In essence then, while digital technologies have evolved, strategies for their effective integration into the learning of mathematics have not evolved as rapidly. Mathematics TPACK: The Total Package for Teaching Mathematics As time shifted and digital technologies became more accessible and incorporated into citizens’ work and play, the International Society for Technology and Education (ISTE) challenged teachers to think about the technology skills and knowledge students would need in an increasingly technology savvy society. By the turn of the 21st century, the National Education Technology Standards for Students (NETS-S; ISTE, 2000) were released with the goal of supporting the evolution of effective use of appropriate technologies in school settings. ISTE recognized that these new standards called for different teacher knowledge than was currently operating in the schools. Within the following 2 years, the National Educational Technology Standards for Teachers (NETS-T; ISTE, 2002) were also released. Although embedded in a rapidly changing digital society, little real instructional change filtered into classrooms. Therefore, ISTE moved to shift the focus of the NETS-S from basic skills and knowledge needed to operate the technology to learning how to effectively use the technology; the NETS-S were updated in 2007. Subsequently, to assist Contemporary Issues in Technology and Teacher Education, 9(1) 7 teachers in responding to the call of learning environments supported by multiple technologies, a revision of the teacher standards was released in 2008. These standards effectively shifted the focus on digital technologies toward a concern about the curriculum and instructional uses of the digital tools and resources. Earle (2002) framed this shift most clearly: Integrating technology is not about technology – it is primarily about content and effective instructional practices. Technology involves the tools with which we deliver content and implement practices in better ways. Its focus must be on curriculum and learning. Integration is defined not by the amount or type of technology used, but by how and why it is used. (p. 8) Numerous researchers focused on the integration of technology, content, and pedagogy in much the same way that Shulman described PCK, to gain a broader perspective on the knowledge teachers need for teaching with technology. In essence, they defined technological pedagogical content knowledge (TPCK) as that body of knowledge teachers needed for teaching with and about technology in their assigned subject areas and grade levels. TPCK was presented as the interconnection and intersection of content, pedagogy (teaching and student learning), and technology (Margerum-Leys & Marx, 2002; Mishra & Koehler, 2006; Niess, 2005; Pierson, 2001). The idea of TPCK developed to the point that the American Association of Colleges of