Problem Solving

Problems provide important examples of tasks that create cognitive conflict and set the scene for future learning (Chick et al., 2006). The ability to solve problems related to, and beyond, the content that teachers are responsible for teaching (Ball et al., 2005; Baumert et al., 2010; Ma, 1999) is essential, because teaching opportunities may be lost if teachers overlook the mathematics offered by an example (Chick, 2009). Recognising the learning opportunities in a problem, including the variety of ways of representing it and leading discussion of the mathematics embedded within it, requires a deep understanding of mathematics that is ultimately reliant on being able to successfully solve the problem. Problem solving offers substantial insight into teachers’ conceptual understandings of mathematics because it requires them to mobilise existing knowledge and make connections between concepts, information and embedded mathematical properties (Lampert, 2001; Silver & Cai, 2005; Thompson, 1985). Consequently, problem solving is an integral component of teachers’ subject matter knowledge.

Problem solving lies at the heart of mathematics education (Polya, 1973; Schoenfeld, 2011; Silver et al., 2005). It offers the answer to why, as well as how, people learn and experience mathematics and reflects the aim of developing quantitatively literate citizens who can apply knowledge and skills to make informed decisions (Mevarch et al., 2014). Students learn to analyse, reason and communicate by interpreting and solving a variety of mathematical problems (OECD, 2012). A challenge in providing mathematics education for the 21st _{century may be the extent to which teachers’ knowledge supports} them in engaging students in analysing, reasoning and solving problems.

Anderson, White and Sullivan (2005) studied links between primary teachers’ use of problem solving in classrooms and the role of problem solving for learning mathematics in the syllabus. They considered the issue that, while most teachers endorsed the importance of problem solving, limited opportunities for problem solving were evident in classrooms. Their study identified factors related to differences between teachers’ beliefs and practices in using problem solving. Through the collection and analysis of survey data the research team reported that some teachers held traditional views, while others held more contemporary views that influenced their intentions regarding problem solving. Factors included the grade level, school culture and the pressure of time. Beliefs about how formal mathematics should be taught was identified as a major constraint impacting on problem solving in classrooms. The research concluded that, as problem solving was central to learning mathematics, professional learning should include opportunities for teachers to engage in wondering about mathematics through problem solving. While teachers’ beliefs are a critical factor influencing teaching practice (Davis, 2006), teachers’ abilities to act on their beliefs may be enabled or restricted by their understandings of the content, the syllabus and their own experiences as learners of mathematics (Putnam & Borko, 2000; Schoenfeld, 2011).

The ability to solve problems that are applications of the content they teach is integral to the role of teachers in a context where problem solving is an outcome and component of the syllabus. Standard 2 in the Australian Professional Standards for Teachers (AITSL, 2014) makes clear the expectation for teachers to know the content and how to teach it. To achieve this standard, teachers must know and understand the concepts, structures and processes relevant to the syllabuses and programs they teach. This includes being able to teach fundamental concepts, such as area, perimeter and volume, as well as the working mathematically outcomes of communicating, problem solving and reasoning. Knowledge of content and how to teach it encompasses understanding developmentally appropriate strategies for student learning and the ability to make the content meaningful to students. Teachers need to know and be able to apply content knowledge. While subject matter knowledge alone is insufficient for proficient teaching, teachers cannot apply knowledge of content and teaching to

develop engaging tasks if they do not know and understand the concepts and structures of the content they teach (Baumert et al, 2010; Hill et al. 2008; Ma, 1999).

Global research supports the role of problem solving in developing and extending students’ thinking (City, Elmore, Fiarman & Teitel, 2009). While the emphasis on problem solving in mathematics education is not new, the nature of problem solving has evolved because the types of problems that need to be solved have changed. The extent to which something is perceived as a problem is relative to the individual (Schoenfeld, 1985), and the resources available to them. With ready access to technology, the meaning of problem solving has shifted from applying formulas to known variables to analysing, reasoning and communicating about problems where all of the information is not given (English & Sriraman, 2010; Lesh & Zawojewski, 2007). For the purposes of this research, mathematical problems refer to tasks for which a solution method is not known in advance by the problem solver (NCTM, 2000).

Problem solving is the means through which students learn to think (Polya, 1962). Polya’s distinction between teaching students how to think and teaching students what to think heralded a move beyond routine problems that emphasise procedural knowledge and are therefore inconsistent with genuine problem solving. Solutions to familiar, routine problems can be misleading. While the solutions might be precise, routine problems do not convey the nature of understanding and reasoning used in arriving at a solution (Mevarech et al., 2014). Schoenfeld’s (1992) example of giving an elementary geometry problem to university students majoring in mathematics, and finding that most were unable to solve it, typifies the observation that high levels of mathematical coursework do not necessitate the ability to solve unfamiliar problems. This reinforces the claim that primary teachers require PUFM (Ma, 1999), rather than higher levels of mathematics, to teach effectively. For this reason, the selection of problems in this study to evaluate teachers’ understandings of content takes into account the degree to which problems are experienced by participants as familiar, routine and complex.

The strands of mathematical proficiency proposed by Kilpatrick et al. (2001) describe desirable mathematical actions for students embedded within the Australian Curriculum (Sullivan, 2011). Problem solving draws upon these desirable actions in varying combinations. For example, strategic competence, is characterised by the ability to formulate, represent and solve mathematical problems (Kilpatrick et al., 2001; Sullivan, 2011). Yet, to solve a problem where the solution method is not known in advance, problem solvers need more than strategic competence. They need to identify relevant concepts, select and apply appropriate procedures, make sense of their actions in relation to the problem and have the willingness to engage and persist in planning and sequencing the steps involved.

Problem solving offers a valuable measure of teachers’ content knowledge reflective of knowing–to, rather than knowing-about, mathematics (Mason et al., 1999). It evaluates connections between ideas as well as the consolidation of mathematical knowledge and thinking and therefore provides a window to understandings of concepts and processes (Arcavi & Friedlander, 2007; English et al., 2010). In order to teach and assess through problem solving, teachers need to be able to solve problems that provide appropriate experiences of the selected content. As recognised by Chick et al. (2013), teachers need to be mathematically confident to lead mathematical discussions, and students are more likely to learn to become good problem solvers if teachers can show how students’ ideas can lead to a solution. Without confidence and skill in solving mathematical problems, teachers may experience difficulty in designing cognitively challenging learning tasks (Sullivan et al., 2009) and evaluating students’ solutions. The specialised knowledge required for teaching mathematics is inclusive of the ability to “carry out and understand multi-step problems” (Ball et al., 2005, p.21).

While familiar, routine problems are necessary for providing practise in mathematics, they do not probe understanding. This is particularly the case for area, perimeter and volume content where the learning of formulas prior to the development of conceptual understanding is associated with the phenomenon of rules without reasons (Skemp, 1987). Problems that are unfamiliar, non-routine and complex are needed to evaluate “flexibility in exploring mathematical ideas and trying alternative solution paths” (NCTM, 2000, p. 21). While research identifies problem solving and the teacher’s role in developing it as paramount, little research is identifies how teachers’ learn to teach through problem solving. For example, little is known about how to support more frequent use of alternative solutions by teachers in classroom instruction (Silver et al., 2005).

As problem solving lies at the core of mathematical experience, measures of teachers’ knowledge for teaching mathematics should include responses to problems. The OECD framework for evaluating students’ mathematical achievement (2012) identified the importance of blending together assessments of content with competency in problem solving and the ability to identify mathematics within a context. Given the broad recognition of problem solving as a process, instructional goal, instructional method and means through which students’ abilities, beliefs, attitudes and performance develop, problem solving offers a valid measure of teachers’ understandings of mathematics. To understand how teachers can support their students to become better problem solvers, teachers need to experience “mathematical problem solving from the perspective of the problem solver before they can adequately deal with its teaching” (Thompson, 1985, p.292).