Elaborating a model of learning to teach mathematical inquiry
School mathematics is often presented essentially as a set of fixed rules and relationships to be learned and applied. This view of mathematics is usually accompanied by a mindset that mathematical solutions are a priori true, and that there is one set answer to every mathematical question.
By contrast, an inquiry-based approach to maths rests on the concepts that mathematical knowledge is fallible; that mathematical knowledge is created through a non-linear process which often includes the generation of multiple hypotheses; and that the production of mathematical knowledge is a social process (Siegel & Borasi, 1994).
To help students manage a complex and uncertain world, teachers must develop experience in approaching mathematics with an inquiry-based perspective. However, this approach places new demands on teachers. The teaching of mathematical inquiry requires the ability to embrace uncertainty, and the capacity and skills to support student decision-making. It requires a balance between collaboration and independence. It also requires the experience to recognize opportunities for learning that arise from unexpected outcomes, a commitment to flexible thinking, a deep understanding of disciplinary content, and tolerance of periods of noise and disorganisation in the classroom (NRC, 2000).
Evidence indicates that teachers’ initial experiences in using inquiry-based learning are often very challenging and frustrating (Hancock et al, 1992; Confrey & Makar, 2002; Makar & Confrey, 2005; Makar, 2004, 2006, in press). These frustrations discourage many teachers from persisting with this approach beyond a first attempt.
Attempts to implement an inquiry-based approach may also get no further than the trappings of reform (eg the setting up of cooperative groups and activity-based learning, or the use of manipulatives), rather than changing the content of what is taught (Cohen & Hill, 2001; Stigler & Hiebert, 1999).
Teachers’ difficulties in applying inquiry to maths may be due in part to the high level of content knowledge required to effectively structure open-ended problems for inquiry (Voss & Post, 1988). Another set of challenges lies in the way that inquiry-based learning broadens the discipline of mathematics from a network of knowledge to a wider system of beliefs and practices. It demands an understanding of how ideas are communicated, of what counts as evidence, and of the ‘lenses through which we look at the world and interpret it’ (p.18, Mansilla, Miller, & Gardner, 2000). Creating an inquiry-based classroom environment requires teachers to help students develop a classroom identity as a community of learners (Cobb, 1999).
As Lehrer and Schauble point out, mathematical inquiry is not something that can be taught in a few lessons. Rather, it is ‘mastered only over an extended period and depends on thoughtful instructional support and repeated opportunities for practice and use’ (p.114, Lehrer & Schauble, 2000, emphasis added).
Cycles of inquiry
One promising means to developing teachers’ abilities in this area is cycles of inquiry involving repeated, cumulative phases of investigation, reporting, and refinement of the problems posed. Within this approach, an initial investigation of a topic generates new questions to explore, creating the potential for further investigations, beginning a new cycle of inquiry. Investigators must consider evidence to defend their claims sufficiently to convince their peers. Feedback and supporting evidence may lead to a refinement of the question, theories to explain outcomes, or discussion of ways to explore the relationships generated. Inquiry work may require the learner to return to the phase of investigation and seek deeper meaning through more powerful mathematical tools.
The cycle ends when the learner is satisfied that the results of their investigation sufficiently resolve or explain the problem. Within a single inquiry, learners undergo cycles that see them continually revise and refine their hypotheses, investigational approach, evidence-base and reporting.
The use of these cycles over an extended period is an innovative approach in mathematics.
The current project
A new study, conducted by the University of Queensland and funded by the Australian Research Council, aims to elaborate a model of how these cyclic processes evolve over multiple experiences with inquiry, and to provide a realistic vision of how expert teachers engage students in this form of learning.
The study involves 19 primary teachers at two Queensland public primary schools, one metropolitan, one rural. The participants, which include both expert and beginning maths teachers, will be required to develop, plan and implement multiple authentic curriculum units, each two weeks long, four times per year. The study will run from Term Three, 2007, to the end of Term Two, 2009. At each school, a senior staff member will act as a site-based coordinator to the project to assist in professional development seminars, liaise with research staff, and provide day-to-day support.
Both schools began initiatives in 2006 aimed at improving student outcomes in mathematics and orienting teachers to the demands of the new Queensland Year 1-10 Mathematics Syllabus, by implementing new curricular materials, increasing teachers’ professional development in mathematics, and implementing innovative programs to support teachers’ capacity to teach with mathematical investigations, as outlined in the new syllabus. The current researcher worked directly or indirectly with both schools in 2006 to pilot some aspects of the project.
The current project is grounded in an educational methodology identified as Design Research (Brown, 1992; Cobb, Confrey, diSessa, Lehrer & Schauble, 2003), in which the researcher simultaneously studies and works to improve the learning of subjects through an engineered learning environment refined through multiple cycles. This approach capitalizes on the practicality and complexity of authentic classroom contexts. The study puts forward an iterative design that refutes, revises, or refines evolving conjectures, the planned learning environment, and measures created.
Each term during the two years of the project, teachers will participate in full-day professional learning seminars at the university. The seminars will allow time for participants to discuss their experiences and beliefs about inquiry. The sessions will also provide learning activities involving an inquiry-based problem, drawing on successful experiences of such work during the pilot study last year. For example, during the pilot participants were asked to mathematise the task of designing ergonomic chairs for the students in their classrooms. This process was initially frustrating and even overwhelming for the teachers, but with persistence, support, and validation of their efforts, they were able to refine the problem into a practical investigation.
Finally, time will be provided for teachers to work in teams to plan teaching units. Follow-up sessions will provide additional support during the planning phase.
A mixture of selected and randomly chosen units will be videotaped during classroom implementation. During videotaped lessons, observation notes will also be taken to provide summaries of specific teachers’ lessons.
The expert teachers will form one group for the purposes of professional development and data collection. Other teachers will typically be either kept together or separate, depending on the type of professional development planned. Those with the most basic concerns about logistics of teaching mathematical inquiry will be given the most structured support. Teachers will be specifically asked about the kind of support and scaffolding they have found to be most beneficial to their professional growth and capacity-building.
Mathematical and statistical reasoning is of utmost importance in a data-driven global economy and foundational to industrial research, ground-breaking technology, and the frontier sciences. Unfortunately, the teaching of mathematics and statistics is plagued by rule-based pedagogies which inhibit creativity and curiosity. Learning experiences which include open-ended inquiry into authentic contexts can foster persistence and generative thinking, critical skills in the uncertainty of the real world. In order for teachers to create these kinds of learning experiences for their students, they must develop capability with this approach themselves. This first-hand experience with the potential and challenges of inquiry-based learning has the additional benefit of providing them with an insight into the importance of creating an environment which will develop creativity, talent, and generative thinking.
ReferencesBrown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2(2), 141-178
Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5-43
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.
Cohen, D. & Hill, H. (2001). Learning policy: When state education reform works. New Haven: Yale University Press.
Confrey, J. & Makar, K. (2002). Developing secondary teachers’ statistical inquiry through immersion in high-stakes accountability data. Paper presented at the Twenty-fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Athens, GA.
Hancock, C., Kaput, J. & Goldsmith, L. (1992). Authentic inquiry into data: Critical barriers to classroom implementation. Educational Psychologist 27(3), 337-364.
Lehrer, R. & Schauble, L. (2000). Modeling in mathematics and science. In Glaser, R (Ed), Advances in instructional psychology: Educational design and cognitive science (Vol 5, pp. 101-159). Mahwah NJ: Lawrence Erlbaum Associates.
Makar, K. (2004). Developing statistical inquiry: Prospective secondary mathematics and science teachers' investigations of equity and fairness through analysis of accountability data. Doctoral disseratation, University of Texas at Austin. Online at http://www.stat.auckland.ac.nz/~iase/publications/dissertations/dissertations.php
Makar, K. (2006). Innovative inquiry-based mathematics teaching in the middle years. Paper presented at Excitement in the Classroom: First ACEL/Microsoft online conference on innovative teaching and learning. 15-21 May.
Makar, K. (in press, accepted October 2006). Beyond the bar graph: Primary teachers’ uses of informal inference to teach statistical inquiry. Paper to be presented at the Fifth International Research Forum for Statistical Reasoning, Thinking and Literacy. University of Warwick, UK, 11-17 August 2007.
Makar, K. & Confrey, J. (2005). Understanding distributions in ill-structured and well-structured problems. In Makar, K. (Ed), Proceedings of the Fourth International Research Forum for Statistical Reasoning, Thinking and Literacy, University of Auckland (NZ) [CD-ROM]. Brisbane: University of Queensland.
Makar, K & Confrey, J (2007). Moving the context of modeling to the forefront: Preservice teachers’ investigations of equity in testing. In Blum, W., Galbraith, P., Henn, H-W. & Niss, M. (Eds), Modelling and Applications in Mathematics Education. New York: Springer.
Mansilla, V., Miller, W. & Gardner, H. (2000). On disciplinary lenses and interdisciplinary work. In S Wineburg & P Grossman (Eds), Interdisciplinary curriculum: Challenges to implementation. New York: Teachers College Press.
National Research Council (2000). Inquiry and the National Science Education Standards: A guide for teaching and learning. Washington DC: National Academy Press.
Siegel, M. & Borasi, R. (1994). Demystifying mathematics education through inquiry. In Ernest, P. (Ed), Constructing mathematical knowledge: Epistemology and mathematics education (Vol 4, p 201-214). Washington, DC: Falmer.
Stigler, J. & Hiebert, J. (1999). The teaching gap: best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.
Voss, J. & Post, T. (1988). On the solving of ill-structured problems. In Chi, M., Glaser, R. & Farr, M. (Eds), The nature of expertise. Hillsdale, NJ: Lawrence Erlbaum Associates.
Key Learning AreasMathematics
Subject HeadingsProfessional development
Project based learning