As you contemplate your next inquiry, you might be pleased to know that a range of supports have been identified that enable teachers to develop their expertise, confidence, and commitment in taking on such innovative practices. These ‘levers’ connect teaching cycles of planning, teaching and reflection, as teachers apply learning from one teaching experience to subsequent ones. Makar’s (2007) paper reported on the beginning journey for four new-to-inquiry teachers as they embraced teaching mathematics with inquiry. Here are the connection levers she identified as supporting classroom teachers to develop capability with inquiry, and to be able to envision and embrace the approach.

Experience inquiry as a learner
Step into the shoes of learners in your class and experience the open-endedness of an inquiry-based problem. The four teachers in this paper had to work together on an ill-structured problem to design an ergonomic chair. The experience helped the teachers consider how they might scaffold the inquiry process for their own students. As a result of their experience as learners, they were inspired to give more control to their own students and for them to value the importance of struggle and ambiguity. The openness of an inquiry question presented opportunities for teachers to design mathematical inquiry lessons that incorporated these experiences for their students. To cope with struggle and  ambiguity, the teachers ​felt that it was important to scaffold and structure children’s initial experiences with inquiry. This would help to keep the experience positive, so students would be inclined next time to take a risk.

Validation
​Plan and teach inquiries with other teachers so that you can experience this together. It was useful for the teachers in this paper to have someone to compare experiences with; to validate that it was ‘normal’ when things did not go as anticipated. Sometimes it can be a relief to hear that others are also concerned with whether they are ‘doing it right’ too. Reflecting on your experiences with others can provide strong support to help you get through your first inquiry units, when things might not go as planned. 

Resources
The teachers reported on in this paper used good quality resources as an initial structure to follow, particularly before they had developed a vision of what inquiry would look like. There are teaching resources available to help. Take a look at mathsinquiry.com for instance…

Sustained support and feedback
The researcher was present in the classrooms reported on in this paper and the teachers highlighted the importance of receiving support and feedback, when the context seemed to invite input. In the spirit of collaborative inquiry, you can offer suggestions in a positive light in ways that might spark reflection for other teachers.

Collegiality
​The teachers in this project relied on one another as they spent time together, interacting, sharing ideas and concerns, to develop a community together. As a community of learners, the teachers shared what they were doing in their classroom with each other knowing that others were thinking through the unit with them as they were teaching it. If you are able to work closely with a colleague, listen to and learn from each other, ‘bounce’ ideas off each other, give and receive feedback and try new ideas in a supported, respectful and collegial partnership. 

Relevance
The international emphasis on 21st Century skills—creative and creative thinking, collaboration and communication—in  STEM classrooms  directly links to mathematics through inquiry. The teachers reported on in this paper saw their opportunity to participate in inquiry experiences during professional development as contributing to their growing expertise. There is a strong emphasis on 21st Century skills in the Australian Curriculum: Mathematics so feel assured that an inquiry approach to teaching mathematics is at the forefront of teaching mathematics and is promoted by state and local initiatives.

Accountability
With competing demands on your time as a teacher, some accountability can be helpful to keep you on track.  When you feel part of a community of learners, they can “push” you to continue when time is tight. Scheduling time for a colleague to observe your inquiry lesson, for example, makes sure that you teach one. 

Makar, K. (2007). 'Connection levers': Developing teachers expertise with mathematical inquiry. In Watson, J. and Beswick, K., Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA). Mathematics: Essential Research, Essential Practice. Hobart: MERGA.​

​How do experienced teachers set up a culture in their classroom that encourages their students to participate in inquiry? In inquiry, students are expected to address complex problems where solutions are not immediately obvious, and a single correct solution does not exist. This can be confronting for students who have - until finding themselves in your classroom - enjoyed the satisfaction of getting the right answer in maths by taking a specific taught approach which leads to the preferred answer. Learning through inquiry requires student exploration, obtaining mathematical evidence to make a claim and then defending the claim in a process of argumentation. When addressing such problems, children need to be supported in their learning and thus the term ‘Guided Mathematical Inquiry (GMI)’ is adopted.

One key requirement, which is the focus of this case study, is that of learners becoming accustomed to working with mathematical evidence.  Evidence needs to be appropriate and sufficient to support and justify the claims students make in GMI. Research suggests that students experience difficulties when working with evidence: students may make an assertion but tend not to see a need for evidence to support that assertion (Fielding-Wells, 2010; Muller Mirza et al., 2009); or may not recognise when they have too little or inaccurate evidence (Zeidler, 1997); or may be used to relying on the teacher to provide this detail. The researchers in this paper were interested in seeing how experienced GMI teachers focused their students on an evidence-based approach to mathematics. 

​Seven experienced GMI teachers (between 1- and 10-years’ experience each) were asked to consider the ways in which students engaged with mathematical evidence, based across all Inquiries they had undertaken. The teachers brainstormed all instances of student involvement with evidence during each phase. The comprehensive overview reflected how students need to Envisage Evidence in the Discover phase; Plan for Evidence in the Devise phase; Generate Evidence in the Develop phase and Conclude with Evidence in the Defend phase. 

Authors:
Fielding-Wells, J. & Fry, K. (in press). Introducing Guided Mathematical Inquiry in the Classroom: Complexities of Developing Norms of Evidence. Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia). Perth: MERGA. 

​References:
Fielding-Wells, J. (2010). Linking problems, conclusions and evidence: Primary students’ early experiences of planning statistical investigations. In C. Reading (Ed.), Proceedings of the Eighth International Conference on Teaching Statistics. Voorburg, The Netherlands: International Statistical Institute.

Muller Mirza, N., Perret-Clermont, A.-N., Tartas, V., & Iannaccone, A. (2009). Psychosocial processes in argumentation. In N. Muller Mirza & A.-N. Perret-Clermont (Eds.), Argumentation and education: Theoretical foundations and practices (pp. 67-90). New York: Springer.

Zeidler, D.L. (1997). The central role of fallacious thinking in science education. Science Education, 81(4), 483-496.