The Noticing Task results captured enormous variability in teachers’ interpretations of higher levels of students’ mathematical thinking. This was evident from the perspective of where teachers ranked a work sample exhibiting sophisticated mathematical thinking. Less than one quarter of the teachers in the study ranked the work sample evidencing a correct solution, using sophisticated understanding and efficient reasoning, in the Extensive position. From the five levels available, teachers were equally as likely to rank the same work sample as Thorough or Sound as they were to rank it as Extensive and even more likely to rank it as Basic. Notably, a small number of teachers ranked this work sample as Limited, suggesting that they interpreted it as demonstrating no relevant evidence of achievement. The almost equal distribution of teachers’ rankings for the same student work sample across four of the five reporting descriptors reinforced that teachers’ understandings of achievement vary greatly (Hattie et al., 2012).
Noticing relies upon skilful perception of how, as well as whether, a student responds to a question, makes a calculation, or reasons to develop an approach when solving a problem (Jacobs et al., 2010). Yet, the results of this study illuminated that many teachers were not able to discriminate between different levels of understanding and reasoning used to achieve correct answers. Teachers were more likely to rank Amelia’s work sample in the Basic position than in any other position. This means that the most popular ranking for a work sample with a correct answer, achieved using sophisticated reasoning and understanding, was lower than at least one of the two work samples with incorrect solutions. The proportion of responses ranking Amelia’s work sample as Basic or Limited indicated that
approximately one-third of the teachers in the study did not notice whether or how Amelia achieved a correct answer. As teachers’ judgements of student work have a significant influence on student learning, it was concerning that a small number of teachers identified Amelia’s thinking as Limited. While Amelia used a novel approach to solve an unfamiliar, non-routine problem, using her knowledge of geometrical properties and reasoning, the proportion of teachers ranking her work in the Basic or Limited positions suggests that she is likely to be disadvantaged at some point on her journey through schooling. Notably, none of the teachers who ranked Amelia’s work in the two lowest positions had correctly solved this item in the Problem Solving Task. This highlighted the importance of opportunities for teachers to successfully complete tasks prior to providing them for students. However, the relationship between teachers’ understandings of content and their noticing of student thinking will be discussed in detail in Section 5.3.
Teachers’ feedback responses were used to confirm their interpretations of a student work sample exhibiting Extensive achievement when solving an area problem. Feedback illuminated that teachers’ uncertainty about whether a response was correct or incorrect, in combination with the student’s novel solution approach, influenced teacher’s rankings of higher levels of student thinking. Responses suggested that many teachers held deeply-rooted misconceptions about area relationships (D’Amore et al., 2006). The results reinforced the importance of teachers experiencing “mathematical problem solving from the perspective of the problem solver before they can adequately deal with its teaching” (Thompson, 1985, p.292). Further, teachers’ feedback supported the notion that teachers whose knowledge of mathematics is founded on procedures and facts tend to focus their attention on students’ acquisition of procedural knowledge (Ma, 1999), such as knowing the formula for finding the area of a triangle.
Feedback from teachers in the two lowest Noticing Task Position categories confirmed that a reliance on standard figures may be an obstacle to improving students’ understandings of measurement relationships (D’Amore et al., 2006). A number of feedback responses that referred to finding the area of three triangles suggested that some teachers did not identify the importance of the perpendicular heights in the figure. For example, the suggestion that “the shape is actually made up of three triangles. You need to calculate the area of each triangle first and then add three of them together”, provided evidence of misconceptions. It was not possible to calculate the areas of three separate triangles based on the dimensions given. The reference to three triangles suggested an inclination to use formulas to find the areas of right angled triangles, rather than the area of any triangle. Feedback offered by teachers with responses in the two lowest Noticing Task Position Categories reinforced the
finding that teachers who are low in subject matter knowledge may overlook students’ misconceptions, or sometimes even reinforce them (Baumert et a., 2010; Gess-Newsome, 2002). Around one-fifth of the teachers in this study interpreted the sophisticated, efficient understanding and reasoning evident in a work sample as representing Extensive mathematical thinking. This may be because these teachers possess the type of Profound Understandings of Fundamental Mathematics (PUFM) described by Ma (1999). An important characteristic of PUFM is the ability to recognise a variety of strategies for solving the same problem so as to engage in mathematics as a dynamic process that draws upon properties and relationships. For example, the ability to recognise that the student had calculated the area of a quadrilateral by averaging the perpendicular heights of two triangles that were formed by one of the quadrilateral’s diagonals. Ma also emphasised the importance of understandings of mathematics that support longitudinal coherence. An understanding of how ideas in the syllabus from one stage to the next may be central to interpreting higher levels of student thinking. For example, students in Stage 2 in NSW (years 3 and 4 combined) learn to describe the relationship between the dimensions of a rectangle and its area. In Stage 3 (the final two years of primary school) students build on this knowledge and describe the relationship between the areas of rectangles and triangles. Understanding these relationships lays an essential foundation for students to establish algebraic representations of measurement relationships in Stage 4 (the first two years of high school in Australia). The results of this study suggest that around one in five teachers noticed the dynamic process used by Amelia and the way that she drew upon properties and relationships to solve a problem. To notice the higher levels of understanding and reasoning in Amelia’s work, teachers needed a profound understanding of area and its connection to the geometrical properties of shapes. The important yet basic principles of proportion, congruence and equivalence needed to be identified. To make comparable judgements between solutions achieved using procedural thinking and those demonstrating understandings of relationships, demanded longitudinal coherence. Teachers required an awareness of why averaging the perpendicular height of a shape without parallel sides could be used to make an efficient calculation. They needed to understand why Amelia’s approach embraced a powerful idea in the structure of measurement units that went beyond learning a rule and beyond the particular shape in question.
The proportion of teachers who identified Amelia’s work sample as representing Extensive understanding and reasoning raises a challenge when teachers are increasingly called to personalise learning through ongoing formative assessment and feedback (Dinham, 2016; Hattie et al., 2012; Hattie, 2015). At the proficient standard, teachers make comparable judgements of student learning and provide quality feedback, regardless of the novelty of the approach taken by the student. They
need to validate the mathematical soundness of all students’ solutions and assist students to generalise their findings (AITSL, 2014; Ball et al, 2005; Baumert et al., 2010). If teachers are not able to identify and interpret higher levels of student thinking, then they will not be able to explicitly respond in ways that improve the learning of their most able students (Dyer et al., 2016). Educational equity cannot be achieved by offering challenging tasks that are open to all learners if teachers can only interpret and extend the thinking of students with routine responses (Smith et al., 2011). The results exemplify the extent to which students’ thinking may go unnoticed when teachers do not recognise the opportunities offered by a problem (Chick et al., 2006).