Section 2: Case 1 Developing communities of mathematical inquiry

Introduction

'Effect size' is a statistical measure of the impact of an intervention on an outcome. Hattie¹ shows that the average yearly effect of teaching in New Zealand in reading, mathematics, and writing from year 4 to year 13 is d = 0.35. Effect sizes above 0.40 represent an improvement on business-as-usual and effect sizes of d = 0.60 are considered large.

This case describes how two teachers worked to develop classroom learning communities in which students learned to engage with the teacher and each other in mathematical inquiry, reasoning, and argumentation². It traces significant changes in teacher knowledge and pedagogy and in student behaviour and mathematical practices through a collaborative, school-based, professional learning process. That process was led by the researcher over one school year. The effect sizes for the gains in both classes were very large: d = 2.39 for Ava's class and d = 2.53 for Moana's class. This is extraordinary progress, representing the equivalent of several years' progress (compared with business-as-usual teaching) in just one year.

The case dramatically illustrates the benefits of developing a genuine learning community within the peer culture. Many studies across the curriculum report high gains when using co-operative learning approaches (for example, Hattie³ found an effect size of d = 0.59 for co-operative learning approaches). On the other hand, studies also reveal the problems that arise when students are not effectively trained to work collaboratively or when they simply work in seating or social groupings. The source study for this case reveals how two teachers took action to transform their teaching, thereby accelerating student development in terms of a wide range of valued outcomes, including cognitive, metacognitive, and social outcomes.

Both teachers had previously engaged in the Numeracy Development Project (NDP) but had adapted what they learned to fit with their traditional practices. Through inquiry into their own practice and the use of a smart tool, the Mathematics Communication and Participation Framework⁴ (see Appendix 1), they were able to build on what they had learned in the NDP and transform the ways in which their students interacted and participated. These changes, in turn, accelerated the students' progress in terms of academic achievement and self-management. The case describes the wide range of instructional strategies, scaffolds, and prompts used by the two teachers and explains how these influenced student learning and behaviour.

Case 7 describes the 2006 TIMSS⁵ findings that New Zealand students in their middle primary school years experience a high rate of bullying from their peers, the second highest on an international index comparing rates of student safety in their peer cultures in thirty-five countries. Māori boys and Pasifika girls and boys experience the highest rates of bullying. Given these findings, the magnitude of the changes achieved through teacher actions in Case 1 have important implications for policy and practice nationally. This case highlights the connections between the collaborative inquiry that the teachers engaged in and the shifts for students. It was the professional learning process and support that the teachers experienced that enabled them to achieve the changes.

Learners and learning context

The learners in this study were two teachers and their students. Ava, a teacher of Māori and New Zealand European descent, was in her ninth year of teaching. Ava's class included year 4 and 5 students. Moana, a Māori teacher, was in her fifth year of teaching. Moana's class included year 4, 5, and 6 students.

The setting was a small, suburban, multicultural decile 3 primary school. The students were mostly of Māori and Pasifika ethnicity. Tumeke School⁶ had a transient population, with around 30% of the students in each class leaving or arriving during the school year.

The professional development and research project described in this case was led by the study's author, doctoral student Roberta Hunter. The project was funded from the larger Numeracy Practices and Change Project. This project was a Teaching and Learning Research Initiative (TLRI) project, co-directed by Professor Glenda Anthony and Associate Professor Margaret Walshaw.

Outcomes

The students in these classrooms experienced a variety of ways of engaging in mathematics. Working through a collaborative process of mathematical inquiry supported the students' well-being, enhanced their understanding of mathematics, and developed their mathematical proficiency.

The qualitative data reported in this case shows that the collaborative inquiry process supported the students in developing productive dispositions and mathematical identities. The students grew in their confidence as mathematicians. They could see the value of mathematics, wanted to learn mathematics, and believed that they could succeed if they applied themselves. These dispositions are not just a desirable end product of mathematics education; they are the means by which students learn and do mathematics.

The following graphs show levels of student achievement in mathematics on the number framework, as revealed by the NumPA diagnostic interview. It compares the students' achievement at the start of the year, before the process began, to their achievement at the end of the year, after the development of communities of mathematical inquiry. The teachers and the researcher checked the data independently.

Figure 1: Shifts in mathematics achievement in Ava's class after collaborative professional development underpinned by research and development

Figure 2: Shifts in mathematics achievement in Moana's class after collaborative professional development underpinned by research and development

As discussed, the effect sizes for the gains in both classes were very large: d = 2.39 for Ava's class and d = 2.53 for Moana's class. The confidence intervals about these values accord with these effect sizes. The effect sizes are also in accord (in terms of order of magnitude) with the values of Cramer's V, a measure of the strength of association of the Chi-Square tests (given categorical data). For Ava's class, V= 0.841 (out of a possible 1) was calculated for the achievement gain. The achievement gain for Moana's class was V = 0.907.

Curriculum relevance

Studying mathematics helps students develop the ability to think creatively, critically, strategically, and logically. They learn to create models, conjecture, justify and verify, and seek patterns and generalisations. In this case, both teachers developed units that included the study of fractional numbers, statistics and measurement, and adding, subtracting, multiplying and dividing. A further extension unit focused on algebraic reasoning was developed within the project. The students actively built capability in all five key competencies: thinking, using language, symbols, and texts, managing self, relating to others, and participating and contributing.

Footnotes

1. Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. London: Routledge.
2. 'Mathematical argumentation' refers to the process of resolving mathematical disagreement by examining the premises of the different positions to establish which outcome is correct. There are constructive ways of doing this, and students need teacher guidance and modelling.
3. Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. London: Routledge.
4. The case explains how the teachers used the Mathematics Communication and Participation Framework to scaffold the development of mathematical communities of inquiry amongst the students.
5. Mullis, I., Martin, M. O., & Foy, P. (2007). Findings from IEA's trends in mathematics and science study at the fourth and eighth grades. In collaboration with J. F. Olson, C. Preschoff, E. Erberber, A. Arora, & J. Galia (Eds.), Index of students' perception of being safe in school (SPBSS) and trends, (p. 367). United States: TIMSS & PIRLS International Study Center.
6. This is not the school's actual name.
7. This section draws from The New Zealand curriculum (Ministry of Education (2007). The New Zealand curriculum. Wellington: Learning Media.) and, in some cases, the parallel.