Conference Agenda

Algebra includes the relationships between quantities, the use of symbols, the modeling of phenomena, and the mathematical expression of change (Carraher, Martinez & Schliemann, 2008). In order to learn algebra by understanding its content, it is necessary to have algebraic thinking skills, which is one of the types of mathematical reasoning. Driscoll (1999, 2001) interpreted algebraic thinking as thinking about quantitative situations that support clarifying relationships between variables, based on Cuoco, Goldenberg, and Mark (1996)'s useful ways of thinking about mathematical content which they defined as habits of mind. Driscoll (1999) put forward a theoretical framework for the habits that students should acquire in order to develop algebraic thinking skills, by claiming that when the student learns symbols, they will take an important step in expressing generalizations, revealing algebraic structures, forming relationships, and formulating mathematical situations. Driscoll (1999) conceptualized habit of algebraic thought as Building Rules to Represent Functions and Abstracting from Computation as habits of mind, which are taken place the umbrella term of the Doing-Undoing.

Doing-Undoing: This algebraic habit of mind is an umbrella term for the other two habits. Students should be able to both conclude an operation related to algebra and reach the starting point by working backwards from the result of an operation which they found the result. Thanks to this mental habit, students not only focus on reaching the result, but also think about the process.

Building Rules to Represent Functions: This mental habit includes recognizing and analyzing patterns; investigating and representing relationships; making generalizations beyond specific examples; analyzing how processes or relationships have changed; and looking for evidence of how and why rules and procedures work (Magiera, van den Kieboom & Moyer, 2013). The sub-themes of this habit are; organizing information, predicting patterns, chucking the information, different representations, describing a rule, describing change, justifying a rule.

Abstracting from Computation: It is the capacity to think about calculations regardless of the numbers used. Abstraction is important for this habit of mind. Abstraction is the process of extracting mathematical objects and relations based on generalization (Lew, 2004). The sub-themes of this habit are; computational shortcuts, calculating without computing, generalizing beyond examples, equivalent expressions, symbolic expressions, justifying shortcuts.

Magiera et. al. (2013) investigated algebraic habits of mind 18 elementary school pre-service teacher with problems. Magiera et. al. (2017) examined pre-service teachers' habits of building rules to represent functions in the scope of the algebra problem. Strand & Mills (2017) stated that the studies in the literature examined pre-service teachers' algebraic thoughts within the scope of problem solving. Therefore, in the studies in the literature, it is seen that problems are used as a tool to examine the pre-service teachers' algebraic thoughts. Also, Kieran et al. (2016) mentioned the importance of problems in developing algebraic thinking. In this context, unlike other studies, this study will examine how pre-service teachers' algebraic habits of mind differ in well-structured and ill-structured problems. Simon (1973) stated that students' solutions to ill-structured problems differ from their solutions to well-structured problems. In addition, Webb & Mastergeorge (2003) examined the differences in the solution strategies of student groups solving ill-structured problems and well-structured problems. Kim & Cho (2016) examined how pre-service teachers' motivations affect their problem solving processes in ill-structured problems. In this study, it is aimed to compare the algebraic thinking styles used by pre-service elementary mathematics teachers in the process of solving a well-structured and ill-structured algebra problem. For this purpose, "What is the difference between the algebraic habits of mind that teacher candidates use in solving a well-structured and ill-structured algebra problem?" an answer to the research question was sought.

Given the ongoing advancement of technology across many areas, the use of technology has become not only a significant tool, but an inevitable component of education. National Council of Teachers of Mathematics (NCTM, 2000) underlines the importance of technology in mathematics education as follows: “technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning” (p. 11). Although there is an increasing access to technology in classrooms, mathematics teachers have difficulty in integrating it into their teaching practices (Erduran & Ince, 2018). Teachers’ knowledge determines how effectively technology is used during instruction (Guerrero, 2010). Therefore, mathematics teachers should be equipped with the necessary knowledge and the positive stance to arrange an effective utilization of technology in supporting students’ learning.

What teachers’ knowledge should encompass has been the subject of many studies. Shulman (1987) proposed that approaching to content and pedagogy as separate entities in education is inadequate, and instead, integration and balance between the two must be achieved. Thus, Shulman (1987) offered pedagogical content knowledge (PCK). Subsequent studies, building upon Shulman’s conceptualization of PCK, have associated technology with the discussion of teacher knowledge. For example, Niess (2005) abbreviated technology-enhanced PCK as TPCK. Similarly, Koehler and Mishra (2005) used TPCK to refer to the term “technological pedagogical content knowledge” which presents how teachers’ understanding of technologies and pedagogical content knowledge interact with one another to produce effective teaching with technology. Then, in 2007, TPCK was changed to TPACK, to make it a more easily pronounced and remembered term (Angeli &Valanides, 2015).

If a teacher is competent in using technology but lacks the ability to transfer this knowledge effectively during teaching or to integrate technology with the content, there is an issue related to TPACK (Hew & Brush, 2007; Mishra & Koehler, 2006). From the perspective of mathematics, teachers should have necessary TPACK in all areas of mathematics, such as numbers and geometry. According to Mathematics Education in Europe report (2011), numbers, algebra, data and chance, and geometry are the areas of mathematics which are widely presented in the curricula of European countries. Compared to these areas, probability is stated as a less frequent one. Similarly, NCTM (2000) stated that number and operations, algebra, geometry, measurement, and data analysis and probability are the main contents of mathematics. In this respect, mathematics teachers should not see technology as “a way to keep kids busy” (Hew & Brush, 2007, p. 229), but have high self-efficacy related to their TPACK in different areas of mathematics. Since self-efficacy is a domain-specific construct (Pajares, 1996), the purpose of this study is to investigate whether mathematics teachers’ self-efficacy beliefs toward TPACK differ across the areas of mathematics.

According to Bandura (1997), self-efficacy, which is a central aspect of social cognitive theory, is a concept related to perceived capability. In more detail, self-efficacy is defined as “beliefs in one’s capabilities to organize and execute the courses of action required to produce given attainments” (Bandura, 1997, p.3). Self-efficacy covers some distinctive features. One of the features is that self-efficacy includes judgments of capabilities to perform activities rather than personal characteristics. Another feature is that self-efficacy measures are not only domain-specific but also context-specific. For example, a student may have lower self-efficacy about learning in a competitive classroom than in a cooperative classroom. In addition, self-efficacy beliefs are multidimensional so that they vary across specific tasks within a particular domain (Zimmerman & Cleary, 2006).

By considering these points, the research question is given below.

Do self-efficacy beliefs of mathematics teachers regarding their technological pedagogical content knowledge (TPACK) vary across the areas of mathematics?

Problem-solving has been a part of school mathematics for a long time (Stanic & Kilpatrick, 1988), but problem-posing research (Cai & Hwang, 2002; Kilpatrick, 1987; Silver, 1994; Silver & Cai, 2005) is fairly new. Since educational research is done to help students learn better, research on problem-posing is no different. Researchers and curricula have mentioned how important it is for elementary school students to be able to pose mathematical problems. Researchers have suggested that problem-posing activities are good for students' creativity (Silver, 1997) and help them get better at solving problems (Brown & Walter, 2005).

On the other hand, reading comprehension research has a longer history than problem-posing research. Reading comprehension research has shown that getting students to pose problems can help them understand their reading much better. Rosenshine, Meister, and Chapman (1996) found gains in students’ reading comprehension when the students were engaged in problem-posing, with a 0.36 effect size measured by standardized tests and 0.86 when using researcher-designed tests. Yu (2011) mentioned that posing problems could be a good way to get students to do higher-order thinking instead of just trying to memorize and understand the learning content. This could help students figure out what the key ideas are while they are learning. Other researchers have also mentioned that when teachers help students with problem-solving activities, it can strengthen their understanding of the course material, improve their learning outcomes, and improve their ability to understand what they are reading (Sung, Hwang, and Chang, 2013). Studies have shown that those who are very good at reading comprehension point out that they are the most effective readers in constant contact with the text (Duke and Pearson, 2002; Cartwright, 2009; Brassell and Rasinski, 2008). Despite this interest in integrating mathematical problem-posing into classroom practice, little is known about the cognitive processes involved when students generate their problems and how problem-posing relates to other cognitive processes in students, like their reading ability.

Even though the research is limited, these quantitative results show that problem-posing should be used in math classrooms because it helps students understand what they are reading and solve problems. Even though it makes sense, in theory, to give students problem-solving tasks to help them understand and improve their learning, more research is needed to show the link between these two ideas. The research in reading comprehension can be used as a model for a systematic study of how mathematical exploration and problem-posing activities affect how well students learn math. This study addresses some of these questions by investigating students' problem-solving abilities and reading comprehension skills. Therefore, this study aimed to examine the relationship between problem-posing and reading comprehension skills.