Impact
  • Getting started
  • Free Resources
    • Member request
    • Members
  • Research
    • Links
    • Further reading
  • IMPACT
    • reSolve Inquiries
  • About
    • Contact

Connection levers: Support mechanisms for teachers teaching mathematics through inquiry

27/6/2019

 
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.
Repeated opportunities
(or Multiple iterations) ​Accept that you are likely to encounter difficulties in the first inquiry you conduct in your classroom as this is unfamiliar territory. All of the teachers in this paper saw improvement over one year of teaching mathematics with inquiry, although it was harder to recognise success in their first classroom inquiry. It’s important to persist through the unfamiliar so that your skills and confidence can improve. Don’t feel you have to undertake a full inquiry every time. You might emphasise or focus on one particular part of an inquiry. For instance, you may wish to provide the data for students to focus on drawing conclusions. Multiple attempts at inquiry were central to these teachers and their students, building their expertise.  
Picture
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. 
Development of
​deep disciplinary knowledge

There are benefits also for you as a learner while you consider the mathematics in your own classroom more deeply. Push your students, and yourself, to think deeply about the mathematical connections that arise in your inquiry.
​Time and support
​for reflection

Set aside time to reflect. Appreciate the learning curve that inquiry may present. Stand back and consider how to apply shared experiences that will improve your practice.
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. 
So find a fellow teacher (or a group of you) and take the plunge to teach mathematics with inquiry! With an emphasis on developing children’s mathematical conceptions and proficiency at applying mathematics to new situations through inquiry, teachers must develop their capability with this approach. The connection levers listed here can sustain your ability to persist beyond the challenges encountered during your initial teaching experiences, as well as continue to support you towards building your expertise, confidence and commitment to the approach. We hope you find these connection levers helpful in supporting you in your own inquiry teaching and learning journey.
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.​

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

7/6/2019

 
Picture
​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. 
​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. 
Picture
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?

Picture
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 
Picture
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
Picture
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
Picture
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. 

    Archives

    September 2019
    August 2019
    June 2019
    April 2018
    July 2017

    Authors

    Katie Makar
    Jill Fielding-Wells
    Sue Allmond
    Kym Fry
    Judith Hillman
    Karen Huntly
    Debra McPhee

    Categories

    All
    Alignment
    Argumentation
    Australian Curriculum
    Checkpoints
    Culture
    Evidence
    Ill-structured Problems
    Intellectual Quality
    Measurement And Geometry
    Number And Algebra
    Proficiencies
    Research
    Statistics And Probability
    STEM

    RSS Feed

Picture
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. 
To view a copy of this license, visit Creative Commons.
Picture
  • Getting started
  • Free Resources
    • Member request
    • Members
  • Research
    • Links
    • Further reading
  • IMPACT
    • reSolve Inquiries
  • About
    • Contact