Teachers’ Understandings of Content Influence the Tasks the Design

Teachers’ understandings of area, perimeter and volume content were highly, significantly predictive of the level of cognitive demand in the tasks they designed for students. While subject matter knowledge alone may be insufficient for proficient teaching (Ball et al., 2005), the results of this study found that teachers were unable to design engaging tasks if they were not proficient with the content (Baumert et al, 2010; Hill et al. 2008; Kilpatrick et al., 2001; Ma, 1999). Teachers who solved most, or all, of the problems in the Problem Solving Task were far more likely to design higher level tasks. The finding that 57.5% of the variation in the level of cognitive demand in tasks was explained by teachers’ proficiency with the content reinforces the importance of primary teachers possessing connected, coherent, structured understandings of mathematics (Ma, 1999). For teachers to provide students with increasing opportunities to solve more complex problems, teachers may first need opportunities to engage with problems that probe and deepen their own understandings of the content.

As teachers’ proficiency with the content increased, the levels of challenge in the tasks they designed also increased. Significant differences were identified between the Problem Solving Task means for responses in different Design Task categories. For example, the Problem Solving mean for responses in the Doing Mathematics Design Task category was significantly higher than the means for responses in all other categories with the exception of the Thorough category. While the mean Problem Solving score for teachers who designed Doing Mathematics tasks was eight out of nine, the mean Problem Solving score for teachers who designed Pre-Structural tasks was just one out of nine. The relationship between the two aspects of teacher knowledge showed that the variability identified in both aspects of teacher knowledge was interconnected. Teachers who designed tasks with higher levels of cognitive demand were those who demonstrated stronger subject matter knowledge. Conversely, teachers who designed tasks with lower levels of cognitive demand were those who demonstrated weaker subject matter knowledge. While high levels of content knowledge do not guarantee higher levels of

teacher effectiveness, low levels of content knowledge reduce teacher effectiveness (Ball et al., 2005) because they lower teachers’ expectations for student learning.

The results identified that increases in teachers’ Problem Solving scores would predict increases in the level of cognitive demand in the tasks they design. In practical terms this means that, if teachers increased their understandings of content sufficiently to support them in solving two or more additional problems this would be associated with an increase of one level of cognitive demand in the tasks they design for students. For the 19 teachers who designed Procedures without Connections tasks, the most common Design Task category, this would represent a shift from the lower levels of cognitive demand to the higher levels. Similar increases in the level of demand in tasks would be associated with increases in Problem Solving scores for teachers with responses in the Pre-structural and Memorisation categories. However, increases in content knowledge for teachers with responses in the Procedures with Connections category would not necessarily predict a shift to designing Doing Mathematics tasks.

The results reinforced the proposal that “teachers who do not themselves know a subject well are not likely to have the knowledge they need to help students learn this content” (Ball et al., 2008, p.404). In this respect, the results were reminiscent of research by Ball et al., (2005) where significantly lower student achievement was evident in the classrooms of teachers who performed in the lowest 20 – 30% of their assessment of Content Knowledge for Teaching Mathematics. In the present study, none of the teachers with scores in the lowest 23% on the Problem Solving Task designed tasks with higher levels of cognitive demand. It is possible that significantly lower levels of student achievement in the classrooms of teachers with lower levels of subject matter knowledge might be due to learning opportunities that are insufficient to prompt learning or identify misconceptions. By contrast, as no significant differences were observed between the Problem Solving scores of teachers who designed tasks in the two higher levels of cognitive demand, increases in subject matter knowledge beyond a certain threshold might not be associated with higher levels of teacher effectiveness (Ball et al., 2005; Hattie et al., 2012). Essentially, there was a strong positive relationship between teachers’ understandings of content, the clarity of learning goals, the communication of high expectations for learning, opportunities to develop deeper conceptual understandings of content and opportunities for students to become proficient in mathematics.

The positive association identified between teachers’ understandings of content and the levels of challenge in the tasks they designed builds upon the findings of Charalambous (2010). Charalambous’ research revealed notable differences between the unfolding of tasks in the classrooms of teachers with stronger and weaker mathematical knowledge. In Charalambous’ study, more than twice as many

tasks implemented by a teacher with stronger mathematical knowledge were presented with higher levels of cognitive demand when compared to the implementation of tasks by a teacher with weaker mathematical knowledge. The empirical findings of the present study may be connected to Charalambous’ observation that, while a teacher with stronger mathematical knowledge spent about half of each lesson on challenging aspects of learning, a teacher with weaker mathematical knowledge spent around 80% of lesson time on less demanding material. In this study, there was a strong association between stronger subject matter knowledge and designing tasks with higher levels of cognitive demand, representing teachers’ intended lessons.

The extent to which teachers’ understandings of area, perimeter and volume content accounted for variation in the levels of cognitive demand in the tasks they designed can be considered in light of positive correlations in previous studies (Baumert et al., 2010). Baumert and colleagues found correlations between increases in PCK and areas such as the cognitive activation of students in the classroom and teachers holding expectations for student learning that corresponded to the curriculum. They identified teachers’ PCK scores as accounting for 39% of the variance in student achievement. The present study builds on these findings by examining the extent to which primary teachers’ understandings of content influence a specific aspect application of their PCK: the design of mathematical tasks. The levels of cognitive demand in tasks and the cognitive activation of students are intertwined. Tasks provide insight into teachers’ interpretations of the intended curriculum through the expectations for student learning embedded within them. In the study of Baumert et al. PCK was highly predictive of teacher effectiveness. In this study teachers’ understandings of content were highly, significantly predictive of an aspect of PCK, which is in turn correlated with teacher effectiveness.

The results of this study supported Ma’s (1999) conceptualisation of PUFM. Increases in teachers’ understandings of the content were associated with increases in pedagogical knowledge, or the ability to act more effectively on existing pedagogical knowledge. Differences in understandings of basic mathematical principles seemed to impact on the schemas required to support the teaching and learning of area, perimeter and volume as specific mathematical topics. Two things need to be considered simultaneously to understand this relationship. First, the questions embedded within the tasks that teachers design provide insight into their interpretations of the curriculum and their expectations for student learning because tasks are the conduit for students’ attainment of the intended curriculum and reflect how the curriculum is implemented in classrooms. Second, the development of schemas or ‘knowledge packages’ impact on the coherence of teachers’ understandings of important mathematical ideas. In this study limitations in teachers’ understandings

of area, perimeter and volume limited their ability to design mathematical learning experiences. The findings have strong implications for achieving the goals of equity and excellence in Australian education. Hill et al. (2005) identified a comparability between teachers’ mathematical knowledge for teaching and the effects of socioeconomic status on students’ learning gains. The results of this study support the recommendation that deepening teachers’ understandings of the mathematics they teach should be a priority in overcoming educational inequity.