Prospective middle school mathematics teachers’ knowledge of linear graphs in context of problem-posing

IntroductionContent knowledge is one type of knowledge that teachers of mathematics need to possess to ensure student achievement (Aslan-Tutak & Adams, 2015; Ball, Thames & Phelps, 2008; Goos, 2013; Mewborn, 2001). This is because mathematics content knowledge is a crucial factor influencing the quality of mathematics teaching (Ball, Lubienski & Mewborn, 2001). According to National Council of Teachers of Mathematics [NCTM] (2000), teachers should have an in-depth understanding of the mathematical concepts they teach. Similarly, Ma (2010) stated that teachers should have a profound understanding of the mathematical concepts they use in their teaching. One of the assessment tools used to examine teachers' mathematical knowledge and to identify their errors and conceptual misunderstandings is problem-posing (Kilic, 2013; Rizvi, 2004; Ticha & Hospesova, 2009). Stoyanova (1998) emphasizes the common agreement among researchers that the problems posed by students provide important clues about their mathematical skills. The current study therefore examined prospective middle school mathematics teachers' mathematics knowledge, using linear graph problem-posing activities that emphasize the skill for translating between representations.Students' ability to understand and use representations is influenced by their teachers' knowledge of representation (Hjalmarson, 2007; Stylianou, 2010). Teachers must possess fluent knowledge about representation types and the transition among these types in order to establish a conceptual learning environment (Ball, Hill & Bass, 2005; McAllister & Beaver, 2012). Problem-posing is an important assessment tool for determining the ability to transition among representation types. Friedlander and Tabach (2001) argue that problemposing is a frequently used tool when translating from different types of representation to daily life situations. Walkington, Sherman and Howell (2014) found that personalized problems related to students' out-of-school interests are more effective at improving achievement with regard to linear functions. Moreover, these researchers argue that personalization can be accomplished through simple mathematics story problems and that problem-posing can serve as an important tool in this context. In addition, many studies confirm that problem-posing can serve as a means of associating mathematical concepts with daily life situations and can thus contribute to mathematics learning (Abu-Elwan, 2002; Dickerson, 1999; English, 1998). Therefore, the investigation of content knowledge through problem-posing should provide us with an accurate assessment of whether prospective teachers have the skills that their students are expected to acquire.A fundamental mathematical domain in middle school mathematics involves the concept of functions, particularly functional relations of the form y=mx+b (Brenner et al., 1997; Ministry of National Education (MONE), 2013; NCTM, 2000). According to Wilkie (2014), many real-world applications are modeled as functions and significant emphasis is placed on functional thinking in mathematics courses during the later years of schooling. In addition, functional relationships play a key role in building algebraic thinking. The lines of the graphs used for problem-posing in this study have the form y=ax+b. Therefore, problems involving such graphs may provide significant evidence for understanding how prospective teachers perceive visually presented functional relations.Theoretical FrameworkProblem-posing and the Classification of Problem Posing ActivitiesProblem-posing, also referred to as problem generation or problem finding, is defined as the process of generating new problems or reformulating existing ones (Akay, 2006; Leung, 1993). In his classification of problem types, Pehkonen (1995) includes problem-posing in the category of open problems. Different theoretical frameworks are offered in the literature for classifying problem-posing activities, each using different criteria (e. …