{"id":624,"date":"2013-06-01T01:11:00","date_gmt":"2013-06-01T01:11:00","guid":{"rendered":"http:\/\/localhost:8888\/cite\/2016\/02\/09\/using-online-error-analysis-items-to-support-preservice-teachers-pedagogical-content-knowledge-in-mathematics\/"},"modified":"2016-06-04T02:26:58","modified_gmt":"2016-06-04T02:26:58","slug":"using-online-error-analysis-items-to-support-preservice-teachers-pedagogical-content-knowledge-in-mathematics","status":"publish","type":"post","link":"https:\/\/citejournal.org\/volume-13\/issue-3-13\/mathematics\/using-online-error-analysis-items-to-support-preservice-teachers-pedagogical-content-knowledge-in-mathematics","title":{"rendered":"Using Online Error Analysis Items to Support Preservice Teachers\u2019 Pedagogical Content Knowledge in Mathematics"},"content":{"rendered":"
Errors are inevitable in the learning of mathematics, as the human brain is not genetically programmed to memorize multiplication facts, carry out multistep operations, or perform exact mathematical calculations (Sousa, 2008). The history of error analysis (also called error pattern analysis) in mathematics education dates back to the work of Radatz (1979). Since then, much of the literature has focused on two key areas of error analysis in mathematics: (a) the identification and interpretation of students\u2019 common error patterns as a result of misconceptions and (b) best practices for instructional remediation. This article addresses both areas and, specifically, highlights the importance of exposing preservice elementary and secondary mathematics teachers to common student error patterns that are the result of underlying mathematical misconceptions.<\/p>\n
The first component of error analysis, the ability to identify and interpret children\u2019s common errors, implies that teachers must not only possess strong mathematics content knowledge, but also the ability to focus on students\u2019 levels of understanding. Researchers have suggested that the ability to interpret students\u2019 understanding is a necessary skill for good teaching (Carpenter & Lehrer, 1999; Davis, 1989; Graeber, 1999). This ability to interpret student understanding helps teachers to become more acutely aware of the process of learning and the aspects of mathematics that are difficult to grasp (Shulman, 1987). Identifying and interpreting students\u2019 understanding also provides teachers with useful information about the underlying cognitive processes related to how students think and develop mathematical knowledge (Sousa, 2008), rather than simply focusing on the right or wrong answer, which does not necessarily provide a window into what students are learning (Ashlock, 2006).<\/p>\n
An increased focus on student thinking and the problem solving process also serves as a powerful mechanism that helps to connect pedagogy, mathematics, and student learning (Franke & Kazemi, 2001) and provides teachers with the first step in providing targeted remedial instruction (Ketterlin-Geller & Yovanoff, 2009). Despite the general consensus that error analysis is a basic and important task needed for mathematics teaching, a 2009 study conducted by Morris, Hiebert, and Spitzer suggested that many preservice mathematics teachers lacked a complete ability to plan effectively for and evaluate students\u2019 mathematical thinking. Additional research has suggested that preservice mathematics teachers share many of the same misconceptions as and make errors similar to those made by their students (Ryan & McCrae, 2005).<\/p>\n
The second component of error analysis, the ability to diagnose and remediate common errors with targeted instruction, is perhaps the most important skill for mathematics teachers to possess. Peng and Lou (2009) considered error analysis to be an \u201cinseparable part of the routine of mathematics teaching\u201d (p. 25), one that can be used as a tool in organizing instruction. The ability to remediate misconceptions with developmentally appropriate and efficient instructional techniques underlies Shulman\u2019s (1986) concept of pedagogical content knowledge (PCK). He described teachers who require knowledge of strategies that are most likely to reorganize the understanding of learners.<\/p>\n
More recent research has supported Shulman\u2019s idea and indicated that preservice teachers can demonstrate growth in PCK, especially with regard to the knowledge of student difficulties, by observing and discussing real classroom settings and remediation techniques (Akko\u00e7 & Yesildere, 2010). Finally, Hill, Ball, and Schilling (2008) cited the abilities not only to remediate errors, but rather to proactively anticipate student errors, interpret incomplete thinking, and predict how students will approach specific tasks, to be key components of an effective mathematics teacher.<\/p>\n
Despite the fact that all teachers encounter students who make mathematical errors, on a daily basis, many preservice teacher courses do not include authentic opportunities for teacher candidates to analyze and discuss common errors.\u00a0 This paper describes an online system that was used to give preservice teachers an opportunity to analyze and remediate student work.\u00a0 It includes a brief overview of two mathematics education courses where the online items were implemented. Next, error analysis problem structure, predicated upon the technology-enhanced formative assessment (TEFA) framework, which was developed by Beatty and Gerace (2009) to support and assess teachers\u2019 PCK, is discussed. This paper concludes with a discussion of the importance of engaging preservice teacher candidates in dialogical discourse and the implications of using similar online error analysis items in preservice teacher coursework.<\/p>\n
Description of Preservice Teachers and Math Education Courses<\/strong><\/p>\n Preservice teacher candidates from two separate mathematics courses enrolled at a mid-sized, suburban university in central Colorado completed the error analysis items described in this paper.\u00a0 Error analysis items were developed and administered across two separate mathematics education courses: (a) Secondary Mathematics Methods (fall 2010, 2011) and (b) Mathematics and Cognition (spring 2011, 2012).\u00a0 Secondary Mathematics Methods,<\/em> a fall-only class, is specifically designed for undergraduate preservice teacher candidates enrolled in a secondary mathematics teacher licensure program.\u00a0 The primary objectives of this course include developing preservice teachers\u2019 PCK in secondary mathematics.\u00a0 All students enrolled in the Secondary Mathematics Methods course were in their final semester of coursework prior to their student teaching experience (junior or senior class standing) and were secondary mathematics education majors in the College of Education.<\/p>\n Mathematics and Cognition is a graduate level class designed for preservice and in-service teacher candidates.\u00a0This course explores mathematical development from birth to adulthood and makes explicit connections between teaching, thinking, and learning in mathematics.\u00a0Unlike the Secondary Mathematics Methods course, Mathematics and Cognition spans a much wider age range and covers grades prekindergarten through high school.\u00a0 Despite the fact that this is a graduate level course, all students enrolled over the past 2 years have been preservice teacher candidates seeking certification in the areas of mathematics education or special education. \u00a0\u00a0\u00a0<\/strong><\/p>\n Error Analysis Problem Structure <\/strong><\/p>\n The error analysis problem structure described in this article was developed based on the theoretically and empirically grounded TEFA framework outlined by Beatty and Gerace (2009) and included three separate but related levels (see Table 1). Each error analysis item was intentionally designed to be diagnostic in nature, but also to enjoin the three basic principles of the TEFA framework in that they provided (a) question-driven instruction, (b) opportunities for formative assessment, and (c) dialogical discourse. Table 1 provides an overview of each level of the error analysis problems and how these instructional design principles supported the development of preservice teacher candidates\u2019 PCK in mathematics.<\/p>\n Table 1<\/strong>
\nOverview of the Error Analysis Three-Level Problem Structure<\/p>\n