Teachers’ understandings of area, perimeter and volume were significantly predictive of their interpretations of students’ mathematical thinking. Teachers who did not know and understand the concepts and structures of the content sufficiently to solve a variety of problems did not identify higher levels of students’ mathematical thinking in work samples based on the same content (Baumert et al, 2010; Hill et al. 2008; Jacobs et al., 2010; Kilpatrick et al., 2001; Ma, 1999). As teachers’ proficiency with the content increased, they were increasingly likely to notice higher levels of student thinking and identify them as representing Extensive or Thorough achievement. By contrast, lower levels of proficiency with content were associated with misinterpreting higher levels of student thinking. The positive correlation between teachers’ responses to the Problem Solving Task and the Noticing Task confirmed that disciplinary knowledge plays an important role in teacher noticing. The results reinforced Ma’s (1999) conviction that strong beliefs about teaching mathematics for understanding cannot overcome limitations in subject matter knowledge. Even teachers who believe steadfastly in teaching for understanding cannot attend to, interpret and respond to students’ understandings if they do not have the depth and type of subject matter knowledge to support them in doing so.
Stronger subject matter knowledge was associated with noticing higher levels of student thinking. The results suggested that increasing teachers’ understandings of content by two or more categories would be associated with increases in noticing higher levels of student thinking. Based upon these results, increasing teachers’ understandings of the content they teach should be prioritised. The 22 teachers with responses in the two lowest Noticing Task Position categories, representing just over one-third of the teachers participating in the study, all ranked one or both of the incorrect student solutions ahead of one or more correct solutions. None of the teachers in the two lowest Noticing Task Position categories demonstrated strong subject matter knowledge. By comparison, among the 14 teachers who interpreted the sophisticated thinking in Amelia’s work sample as Extensive, 11 had responses in the two highest Problem Solving categories. Differences in teachers’ understandings of basic mathematical principles in the Problem Solving Task seemed to impact on the networks of
conceptual and procedural knowledge available to them when identifying the mathematical thinking of students in response to a problem (Ma, 1999). None of the teachers with responses in the highest Problem Solving Task category ranked a work sample with an incorrect student solution higher than a correct solution. The results reinforced the extent to which teachers’ Knowledge of Content and Students relies on their understandings of the content they teach and that minimum levels of proficiency with the content are required for proficient teaching. Subject Matter Knowledge matters. In this study, differences in Subject Matter Knowledge were strongly associated with differences in teachers’ judgements of student learning. The result that many teachers with weaker understandings of content ranked incorrect solutions higher than correct solutions reflects the observation that, in areas lower in subject matter knowledge, misconceptions are often reinforced by teachers (Gess- Newsome, 2002).
The results confirmed the finding that there is a minimum threshold of content knowledge required for effective teaching (Ball et al., 2005; Hattie, 2012). Significant differences were evident between the noticing of teachers with Low and Low-Moderate levels of proficiency with content in comparison to teachers with Moderate-High and High levels of proficiency. Teachers in the lowest 40% of proficiency with the selected content demonstrated a significantly lower ability to “scrutinize, interpret, correct, and extend” (Ball et al., 2005, p.17) student thinking. Only one of the teachers who solved less than half of the items in the Problem Solving Task ranked Amelia’s work as Thorough or Extensive. However, increases in teachers’ proficiency with area, perimeter and volume content beyond the lowest 40% were not associated with significant increases in teacher noticing. Hence, there may be a minimum aptitude for teaching mathematics that influences teachers’ abilities to ask questions, use models to support explanations and identify and use non-standard methods for solving problems (Ball et al., 2005).
Differences in teachers’ understandings of content accounted for almost half of the variation in noticing higher levels of student thinking. Teachers needed to draw upon their own knowledge as well as students’ current knowledge, the ideas that supported it, relied on it and how they could build upon them (Ma, 1999) in order to rank student work samples. The results reinforce the complex relationship that exists between knowledge of students and knowledge in the subject matter domain and the challenge of discerning the strength of this relationship (Hill et al., 2008). While the correlational analysis in this study identified a strong, positive relationship between teachers’ understandings of the content and their noticing of student thinking, teachers’ subject matter knowledge was not as significantly predictive of teacher noticing as it was of the level of cognitive demand in the tasks they designed. Analysis of the influence of teachers’ responses to Item 9 as the discriminator variable
suggested that this may be due in part to the finely grained nature of teacher knowledge. Possessing knowledge of content overall and possessing knowledge regarding a particular problem provided different insights into the relationship between teachers’ understandings of the content and their noticing of student thinking.