A Reliance on Routine Problems and Procedural Knowledge

The influence of teachers’ solutions to area, perimeter and volume problems on their noticing of student thinking reinforced the finding that teachers’ own learning experiences may not allow them to transform knowledge of content into knowledge of how to teach it (D’Amore et al., 2006). Teachers’ understandings of content are largely the consequence of the quality of their own educational experiences and “many have difficulty clarifying mathematical ideas or solving problems that involve more than routine calculations” (Kilpatrick et al., 2001, p.394). The student work sample demonstrating Extensive understanding and reasoning in this study applied a novel approach to solve a problem that most teachers rated as unfamiliar, non-routine and moderate in complexity. Teachers’ interpretations of the student’s thinking might have been influenced by learning that the only way to find the area of a triangle is to use the correct formula and this may in turn influence what they look for when evaluating student work. A challenge for improving the quality of mathematics education is therefore: how teachers who are reliant on procedural knowledge develop the knowledge, understandings and confidence to teach students to communicate, problem solve, reason and

understand mathematics. From a teacher perspective, “if no-one has ever taught us these things, how can we possibly know them” (D’Amore et al., 2005, p.14).

A conceptual understanding, or “integrated and functional grasp of mathematical ideas” (Kilpatrick et al. 2001, p.118) beyond knowing isolated facts and skills to understanding why ideas are important and when to use them, is essential. In this study, conceptual understanding alone was insufficient for noticing higher levels of student thinking. Teachers needed to Know-to as well as Know-about the content (Mason et al., 1999). Mathematical proficiency (the intertwining strands of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition) was required to support teachers in developing their own solutions and responding flexibly to the range of solutions presented by students (Kilpatrick et al., 2001; Sullivan, 2011). Proficiency is not synonymous with getting all of the answers correct. Responses in the High category for Problem Solving did not ensure the identification of higher levels of student reasoning. For instance, one teacher with a response in the High Problem Solving Category did not interpret higher levels of understanding and reasoning as Extensive or Thorough because they ranked procedural thinking over understanding as evidence of achievement. The teacher’s feedback, “you managed to get the correct answer this time; however this strategy will not work for other examples. To improve your work please use the correct formula”, provides insight into why a teacher who solved all items on the Problem Solving Task ranked Amelia’s work as Sound, rather than Extensive. Higher levels of content knowledge predicted, but did not ensure, higher levels of PCK (Hill et al., 2008).

Teachers’ interpretations of procedural thinking as representing Extensive achievement may have been due to the prevalence of procedural knowledge among teachers in the study. While the strategies used by teachers were not recorded, one of the most common points of feedback to Amelia was to record and use the correct formula. Notably, almost half of the teachers in the study identified the work sample exhibiting procedural knowledge as representing Extensive achievement. The majority of these teachers solved only problems rated as familiar and routine. The association between teachers who ranked procedural knowledge as Extensive and understood the content sufficiently to solve familiar, routine problems, reflected Ma’s (1999) observation that “teachers who expected students merely to learn the procedure tended to have a procedural understanding” (p.3). The findings reinforce the observations of Simpson et al. (2017) that limitations in teachers’ content knowledge are associated with a lack of evidence when ascertaining students’ understandings. Simpson and colleagues noted that many teachers with lower levels of content knowledge fail to take students’ thinking into account. In response to students’ solutions, teachers with limited content

knowledge sometimes discuss any idea related to the initial problem, rather than ideas based on evidence of what students do or do not understand.

Emphasis on procedures in the teaching of measurement concepts has been long identified as impacting on students’ understandings and mathematical development (Bragg et al., 2000). The continued emphasis on procedural knowledge may result from limitations in teachers’ proficiency with measurement content (Sowder et al., 1998). Hirstein (1981) found that students’ errors were largely due to an absence of conceptual understanding of area rather than from difficulties in making computations. He proposed that this was the consequence of teachers’ urgency to introduce formulas for calculating the areas of triangles and rectangles. The high proportion of teachers interpreting procedural knowledge as representing the highest level of mathematical thinking in the present research indicates that the emphasis on procedural knowledge prevails. Yet it also highlights that changes in student learning cannot occur without changes in teacher knowledge. Knowing-what, knowing-why and knowing-how (Mason et al., 1999), can assist teachers in correctly solving problems and identifying which work samples present correct solutions. However, Knowing-about area measurement does not necessitate being able to discriminate between levels of thinking in student work samples with different yet correct solutions. Knowing-about mathematics is insufficient for assessing, providing feedback and reporting on student learning (AITSL, 2014). The proportion of teachers interpreting inefficient, procedural thinking as evidence of Extensive understanding and reasoning raises the challenge that Knowing-about has generally formed the core of mathematics education, but does not indicate being able to respond to the novel, sophisticated reasoning of a student (Mason et al., 1999). Only teachers with stronger subject matter knowledge in this study were able to demonstrate skilful perception of how, rather than whether, students responded to a question, made a calculation, or the reasons used to develop an approach when solving a problem. Teachers with weaker subject matter knowledge did not demonstrate the pedagogical knowledge needed to make comparable judgments of student work because they were unable to validate the mathematical soundness of students’ solutions (Ball et al, 2005).

In summary, teachers’ understandings of content influenced their noticing of higher levels of mathematical thinking in a number of ways. Teachers’ interpretations of student thinking relied to some extent on their overall proficiency with the selected content (Kilpatrick et al., 2001). At the same time, solving the problem that was the basis for student work, flexibly and in multiple ways, played a critical role in teacher noticing. Teachers are unlikely to “engage their students in productive conversations about multiple ways to solve a problem if they themselves can only solve it in a single way” (Kilpatrick et al., 2001, p.399). While knowledge of the content that teachers need to teach does

not encompass all of the knowledge needed to interpret and respond to student thinking, teachers must be able to understand concepts correctly, perform procedures accurately and understand the conceptual foundations of that knowledge (Kilpatrick et al., 2001). The results of this study illuminate the likelihood that novel, sophisticated mathematical thinking, based on conceptual rather than procedural understanding, is likely to go unnoticed in classrooms where teachers are not mathematically proficient.