What is intellectual quality in a mathematics classroom?

Becoming a confident teacher of mathematics through inquiry pedagogies takes time. As teachers of mathematics we aim to promote intellectual quality in mathematics classrooms to promote engagement of our students in meaningful mathematics experiences. How can teachers’ pedagogies promote intellectual quality when teaching mathematics through inquiry?

What does intellectual quality look like in a mathematics classroom? In one Year 5 classroom in this study, students were required to estimate, measure and compare angles using degrees. To engage students in a task of high intellectual quality, the classroom teacher posed the inquiry question to students, How can we accurately estimate the sum of the internal angles of a polygon?

The students decided to focus initially on constructing 3-sided, closed polygons (triangles) to measure the internal angles of.  Students completed this independently although each time a student constructed a triangle and measured to calculate the sum of the internal angles, they were required to have two other students validate this process. As students gathered mathematical evidence of the sum of the internal angles of triangles they had drawn, they shared their evidence with each other. Through classroom discussion, some students noticed how many calculations summed to 180°, or very close to it. The focus of conversations was between students as they considered the data they had collected as evidence. The students pondered why so many of their calculations for the sum of the internal angles of a triangle clustered around 180° degrees. Did they have enough evidence yet to form a conjecture? What may have caused variation in the data they collected? Students in groups negotiated what evidence they would need to convince others that the internal angles of triangles would always sum 180°.

Traditional approaches to mathematics which focus on reproduction of low-level, taught procedures point to low levels of intellectual quality. Mathematical inquiry has been argued to promote the intellectual demands desired in mathematics:

• Shifting responsibility to students to propose and defend mathematical ideas and conjectures (Goos, 2004)
• Expecting students to respond thoughtfully to mathematical arguments presented by their peers (Goos, 2004)
• Solving ill-structured problems containing ambiguities that require negotiation
  

The author of this paper (Makar, 2016) was interested in understanding how primary teachers’ experiences and pedagogies evolved as they taught mathematics though inquiry. The study presented in this paper observed aspects of teachers’ pedagogical practices that showed evidence of intellectual quality. The author compared data from regular mathematics lessons and initial inquiry lessons from 41 primary teachers and continued to follow 19 of these teachers over three years.

Intellectual Quality

• Knowledge presented as problematic
  
• Higher order thinking
• Depth of knowledge
• Depth of understanding
• Substantive conversation
• Meta-language

Productive Pedagogies (QSRLS, 2001): Intellectual Quality Cluster and Dimensions

The Productive Pedagogies framework (QSRLS, 2001) was an observation scheme developed in Queensland in 2001 which characterised classroom practices; intellectual quality being one of the clusters. This provided a useful framework for the author of this paper to use to identify classroom practices which promoted the development of engaging students in high quality work (QSRLS, 2001). A scale was provided for each dimension which was used as an indicator of pedagogical practice which reflected ideals of mathematical inquiry we valued.

The intellectual quality of 41 teachers’ regular mathematics lessons and their first term of (in)experience in teaching mathematics through inquiry were compared. The results overall were interesting:

The intellectual quality in teachers’ initial inquiry lessons was significantly higher than in their regular mathematics lessons.

Higher order thinking had a high effect size in the initial inquiry lessons.

The greatest difference was in how mathematical knowledge was presented. The nature of mathematical inquiry is that it is ambiguous and requires negotiation.

The gains in Intellectual Quality which were the greatest were in terms of Higher order thinking and Knowledge as problematic. This suggests that the aspects of intellectual quality highlighted here potentially align with the nature of mathematical inquiry.

How did the intellectual quality of teachers’ pedagogical practices change as they gained experience teaching mathematics through inquiry? The author compared the teachers’ lessons at four junctures over three years:

The intellectual quality of lessons continued to significantly increase as teachers gained experience teaching mathematics through inquiry.

Ongoing improvement may suggest that these are areas that teachers embrace and were possibly not initially very fluent with. It may speak to areas of regular mathematics lessons that we can improve. Most inquiry lessons by the third year were characterised as “Students are engaged in at least one major activity during the lesson in which they perform higher order thinking, and this activity occupies a substantial portion of the lesson and many students are engaged in this portion of the lesson” (QSRLS, 2001, p.6).

As teachers gain experience in teaching mathematical inquiry there is potential to affect their students’ understandings of mathematics as a contestable rather than fixed discipline, and to improve students’ mathematical reasoning through higher order thinking.

Makar, K. (2016). Improving the Intellectual Quality of Pedagogy in Primary Classrooms through Mathematical Inquiry. Mathematics Education Research Group of Australasia.

Queensland School Reform Longitudinal Study (QSRLS) (2001). Productive Pedagogies Classroom Observation Scheme. Brisbane: The University of Queensland.