Solving Familiar, Routine Problems and Tasks with Lower Levels of Demand

There was a strong connection between designing tasks with lower levels of cognitive demand and Problem Solving scores that were achieved by solving familiar, routine problems. This association raised a central issue: teachers were once students and students who learned the content through instrumental instruction may experience difficulties in solving unfamiliar, non-routine problems (Pesek et al., 2000). If teachers learned area, perimeter and volume content as sets of fixed rules and formulas during their own education, then it is not surprising that they may experience difficulty in thinking flexibly in response to non-routine problems. Nor is it surprising that teachers who understand the content as sets of rules and formulas design tasks that focus on Procedures without Connections - for procedures would be central to their schemas for the selected concepts. Teachers who solved a variety of unfamiliar, non-routine problems also represented, connected and highlighted powerful mathematical ideas in the tasks they designed, while teachers who only answered problems that were familiar, routine and low in complexity focused on procedural knowledge. As noted by Ma (1999), the structure of teacher knowledge may influence teaching more than the quantity. How teachers understood the content, rather than whether, may have influenced the types of learning experiences they designed. In considering the question of what teacher knowledge matters more and why (Bobis et al., 2012), the results of this research suggested that teachers’ subject matter knowledge matters greatly. When founded on deep, understandings of content, subject matter knowledge is the type of powerful, flexible, adaptable knowledge that can support teachers’ pedagogical knowledge. The strength of the relationship between Problem Solving and Design Task results reflected the origins of the TAG (Stein et al., 1998). It reinforced the important, yet additional demands that developing deep, interconnected understandings of mathematical concepts places on teacher knowledge in comparison to memorising and applying formulas. To design higher level tasks teachers needed to understand the concepts in the selected content to the extent that they could apply their knowledge to solve unfamiliar, non-routine, more complex problems. Complete understanding of mathematics involves “the capacity to engage in the processes of mathematical thinking, in essence doing what makers and users of mathematics do: framing and solving problems” (Stein et al. 1996, p.456). The results of this research suggested that framing and solving problems related to the same content are

interconnected. Teachers who designed tasks that were open to a variety of solution approaches were those who were able to use knowledge flexibly to solve a variety of problems.

Teachers who demonstrated Low levels of content knowledge mostly designed Pre-structural tasks that did not reflect a learning goal related to the content they selected. Low levels of content knowledge influenced the design of tasks by making the content unclear – to the teacher and the students. Low levels of content are particularly concerning because it is difficult to determine what counts as evidence of student learning or how student learning is linked to the learning task without clear learning goals (Hiebert et al., 2007). Most of the teachers who solved less than two of the problems involving measurement relationships designed tasks that did not focus on the selected content. The results reflected Steele’s (2013) finding that teachers who performed better on a task “were better able to write more specific goals” (Steele, 2013, p.265) for using that task with students. In this study, Low levels of understanding area, perimeter and volume content when solving problems were associated with not knowing what, why or how to design learning tasks for students.

Teachers who demonstrated Moderate levels of content knowledge when solving area, perimeter and volume problems tended to design Procedures without Connections tasks. These were the types of tasks that required students to engage in more than just memorisation of isolated facts, but did not require an understanding of how knowledge and skills were connected to the concepts being learned (Smith et al., 1998). Teachers whose scores relied on solving problems rated as familiar, routine and/or low in complexity mostly designed tasks that limited opportunities for conceptual understanding because the content, the tasks and the concepts were not integrated sufficiently to build connections. Teachers demonstrating Moderate levels of content knowledge predominantly emphasised applying rules and learning the types of fixed plans that would support students in knowing what to do and how to get answers when responding to familiar, routine problems. The results reinforced the extent to which teachers may demonstrate a dependence on rule-bound knowledge as well as shortcomings in their conceptual understandings (Hourigan et al., 2013) and how a dependence on rule bound knowledge can influence their implementation of the curriculum. With the exception of two teachers, Moderate levels of proficiency did not support teachers in designing the types of learning experiences that would enhance students’ awareness, reasoning and understanding (Chick et al., 2006) of area, perimeter or volume. Teachers whose understandings of content supported them in solving about half of the problems tended to focus on students Knowing-how to use procedures (Mason et al., 1999), but not why.

Teachers who demonstrated High levels of content knowledge were most likely to design tasks with higher levels of cognitive demand. Ten of the eleven teachers demonstrating the highest level of

proficiency with the selected content designed tasks with higher levels of demand and the majority of these tasks focused on Doing Mathematics. They engaged students in exploring and understanding the nature of the concepts, processes and relationships in the content in ways that required students to self-monitor and regulate cognitive processes in order to reach solutions. High levels of content knowledge appeared to influence the design of tasks that embedded principles of effective mathematics teaching (Sullivan, 2011) and provided substantial opportunities for developing the five desirable mathematical actions (Kilpatrick et al., 2001). These tasks required students to plan their own approach, sequence multiple steps, process multiple pieces of information, make connections between ideas, see concepts in new ways, engage with important mathematical ideas, choose their level of access to the task, explain their strategies and justify their thinking (Sullivan et al., 2011). Teachers who solved all of the problems presented, regardless of the degree of familiarity, routineness or complexity, designed tasks through which students might experience Knowing-to, rather than Knowing-about, the content (Mason et al., 1999). They demonstrated the ability to think about the content and structure learning situations “in ways that [would] enable learners to extend understandings” (Mason et al., 2013, p.193).

Variations in teachers’ knowledge of the specific subject matter accounted for more than half of the variation in the levels of challenge available for students. While knowledge of the content did not encompass all of the knowledge needed to design tasks, teachers needed to understand the conceptual foundations of the content in order to design tasks with higher levels of cognitive demand. Analysis of the relationship between aspects of teacher knowledge in the subject matter and pedagogical knowledge domains illuminated that teachers’ designs for learning were influenced by both whether and how they understood the content. The strength of the relationship identified teachers’ understandings of the content as a cornerstone of designing learning for proficiency (Kilpatrick et al., 2001). Analysis of the relationship between teachers’ subject matter knowledge and the tasks they designed for students confirmed that,

attention to teachers’ mathematical knowledge and its central role in practice is crucial to ensure that their study of mathematics provides teachers with mathematical knowledge useful to teaching well (Kilpatrick et al., 2001, p.395).