Introducing Guided Mathematical Inquiry in the classroom: A focus on evidence

​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.

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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. 

​To test this framework, the researchers applied it to the first (introductory) full GMI unit taught by an experienced project teacher, in a school year. It is important to mention that the classroom teacher moved between phases of the inquiry as the students did not find easy the need for evidence to be accurate and sufficient. 

However, through the inquiry (described below) the students were able to develop a more robust conceptual understanding of aspects of geometry and measurement: they developed a referent benchmark for a litre; made links between 3D shapes and their nets; and made connections between volume and capacity. These are key conceptual understandings children require to continue to more complex concepts.

Inquiry question: Can you make a one litre container out of paper?

Teaching and learning:
What is the nature of inquiry?
Inquiry addresses a question mathematically
What mathematics will we need to answer the question?
Write down your (initial) ideas about how you will make the container?
Will your (initial) plan work? Test by making a prototype

Focus on evidence: Envisaging and Planning Evidence

• Unpacking the mathematics
• Identifying a need for mathematical evidence
• Establishing a need to plan
• Building and/or trialling a representation 

Teaching and learning:
Establish the need for proof and evidence
Consider ways of testing the container
Consider quality of testing (accuracy)
​Record plans for testing

Focus on evidence: Envisaging and Planning Evidence

• Establishing the need for mathematical evidence
• Considering ways to obtain evidence
• Considering evidence quality
• Establishing the need to plan

Teaching and learning:
Revisit previous learning
Highlight how construction, measurements and testing needs to be accurate
Share containers (prototypes and solutions) and describe measuring processes
Introduce the need for a conclusion to be drawn, based on evidence
Demonstrate deconstruction of boxes to show their nets
Refine the question: “Can you make a container which holds half a litre? (using a net)”
Continue planning the container (nets) using labels for measurements
​Build designs from plans

Focus on evidence:
​Planning and Generating Evidence

• Addressing/evaluating evidence
• Considering evidence quality
• Making a claim from evidence
• Unpacking the mathematics
• Obtaining feedback on decisions/processes
• Refining evidence
• Establishing the need to plan
• Considering the mathematics
• Considering evidence quality
• Representing their evidence
• Evaluating representations

Teaching and learning:
​Groups share solutions including: containers, results of repeated iterations of testing, and proposals for ways to improve their next iteration if they were to continue

Focus on evidence:
Concluding Evidence

• Providing evidence to support a claim
  
• Evaluating evidence

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.