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. 

Becoming a confident teacher of mathematics through inquiry pedagogies takes time. As teachers of mathematics we aim to promote intellectual quality in mathematics classrooms to promote engagement of our students in meaningful mathematics experiences. How can teachers’ pedagogies promote intellectual quality when teaching mathematics through inquiry?

The students decided to focus initially on constructing 3-sided, closed polygons (triangles) to measure the internal angles of.  Students completed this independently although each time a student constructed a triangle and measured to calculate the sum of the internal angles, they were required to have two other students validate this process. As students gathered mathematical evidence of the sum of the internal angles of triangles they had drawn, they shared their evidence with each other. Through classroom discussion, some students noticed how many calculations summed to 180°, or very close to it. The focus of conversations was between students as they considered the data they had collected as evidence. The students pondered why so many of their calculations for the sum of the internal angles of a triangle clustered around 180° degrees. Did they have enough evidence yet to form a conjecture? What may have caused variation in the data they collected? Students in groups negotiated what evidence they would need to convince others that the internal angles of triangles would always sum 180°.

Traditional approaches to mathematics which focus on reproduction of low-level, taught procedures point to low levels of intellectual quality. Mathematical inquiry has been argued to promote the intellectual demands desired in mathematics:

• Shifting responsibility to students to propose and defend mathematical ideas and conjectures (Goos, 2004)
• Expecting students to respond thoughtfully to mathematical arguments presented by their peers (Goos, 2004)
• Solving ill-structured problems containing ambiguities that require negotiation
  

The Productive Pedagogies framework (QSRLS, 2001) was an observation scheme developed in Queensland in 2001 which characterised classroom practices; intellectual quality being one of the clusters. This provided a useful framework for the author of this paper to use to identify classroom practices which promoted the development of engaging students in high quality work (QSRLS, 2001). A scale was provided for each dimension which was used as an indicator of pedagogical practice which reflected ideals of mathematical inquiry we valued.

Ongoing improvement may suggest that these are areas that teachers embrace and were possibly not initially very fluent with. It may speak to areas of regular mathematics lessons that we can improve. Most inquiry lessons by the third year were characterised as “Students are engaged in at least one major activity during the lesson in which they perform higher order thinking, and this activity occupies a substantial portion of the lesson and many students are engaged in this portion of the lesson” (QSRLS, 2001, p.6).

As teachers gain experience in teaching mathematical inquiry there is potential to affect their students’ understandings of mathematics as a contestable rather than fixed discipline, and to improve students’ mathematical reasoning through higher order thinking.

Record breaking is an inquiry unit you will find on the Resources page of this website. The inquiry can also be found in Book 3 of the Thinking through mathematics series, for students aged 10-13 years. An excellent way to mathematically consider records broken at the Olympic, Commonwealth, Paralympic, Pacific or Youth Olympics (depending on which is most relevant to your location and the year), in this inquiry students explore the notion that athletic ability has continued to improve over time. This is a popular classroom topic and such an inquiry could take place in many different year levels. The beauty of inquiry pedagogy is the ability to open tasks up so students of various abilities can successfully participate - low floor, high ceiling tasks.

Here we consideration, with a little imagination, ways in which you might adapt this unit for your own year level.

Foundation Year
Consider whether students can jump further from a standing jump or a frog jump. Direct comparison to determine which is longer.
Each student jumps. Records which jump was further (using markers to enable comparison) and then yes/no questions are asked to determine the most common response for the class.

• Use direct and indirect comparisons to decide which is longer, heavier or holds more, and explain reasoning in everyday language (ACMMG006)
• Answer yes/no questions to collect information and make simple inferences (ACMSP011)

Year One
Similar to above, but the students measure their jumps using informal objects. This provides an opportunity to discuss the need for uniform objects (imagine if you wanted to compare each other’s jumps). Count and record the jumps. How could we record the class data? What does the data mean? What would the data look like for other classes (inference).

• Measure and compare the lengths and capacities of pairs of objects using uniform informal units (ACMMG019)
• Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013)
• Choose simple questions and gather responses and make simple inferences (ACMSP262)
  Represent data with objects and drawings where one object or drawing represents one data value. Describe the displays (ACMSP263)

Year Two
Consider whether students get better at jumping over time. Have the student record a jump. Practise jumping for a short period each day and then record the jump distance and weekly intervals. After three jumps (say three consecutive Mondays), students compare their jump data (you could use lengths of string/wool – blue for first jump, red for second, green for third etc). Did students jump further with practice? How can they record this data? What inferences can they make? 
Measure the string lengths with informal objects, how much further/less did the student jump from one jump to the next? Show your working (evidence).

• Compare and order several shapes and objects based on length, area, volume and capacity using appropriate uniform informal units (ACMMG037)
• Collect, check and classify data (ACMSP049)
• Create displays of data using lists, table and picture graphs and interpret them (ACMSP050)
• Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

Year Three
As Year 2, however the measurement are now able to be made in centimetres.

• Measure, order and compare objects using familiar metric units of length, mass and capacity (ACMMG061)
• Collect data, organise into categories and create displays using lists, tables, picture graphs and simple column graphs, with and without the use of digital technologies (ACMSP069)
• Interpret and compare data displays (ACMSP070)

Year Four
As at Yr 3, with the additional connection between metric measures (metres and centimetres) and decimal place value notation. ie 123 cm is 1m 23cm. NB measurement should not be used to introduce decimal notation but only introduced once decimal PV is in place.
With this age group, consider the jumping events as these use length to two decimal places only (cm). Using timed events involves students with Base 60 and, if using hundredths (eg running or swimming) or thousandths of seconds (eg kayaking) – this can be quite difficult.

• Use scaled instruments to measure and compare lengths, masses, capacities and temperatures (ACMMG084)
• Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation (ACMNA079)
• Select and trial methods for data collection, including survey questions and recording sheets (ACMSP095)
• Construct suitable data displays, with and without the use of digital technologies, from given or collected data. Include tables, column graphs and picture graphs where one picture can represent many data values (ACMSP096)
• Evaluate the effectiveness of different displays in illustrating data features including variability (ACMSP097)

Year Five
Intended year level of document. Be very careful of using events as cautioned in Year 4 notes).

• Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108)
• Recognise that the place value system can be extended beyond hundredths (ACMNA104)
• Compare, order and represent decimals (ACMNA105)
• Pose questions and collect categorical or numerical data by observation or survey (ACMSP118)
• Construct displays, including column graphs, dot plots and tables, appropriate for data type, with and without the use of digital technologies (ACMSP119)
• Describe and interpret different data sets in context (ACMSP120)

Year Six
As Year 5 with the additional connection between metric measures (metres and centimetres) and decimal place value notation. ie 123 cm is 1m 23cm is 1.23m.
Opportunities to extend the maths for this age would include: average time, proportional reasoning (Is the 200m run in twice the time of the 100m etc).

• Connect decimal representations to the metric system (ACMMG135)
• Convert between common metric units of length, mass and capacity (ACMMG136)
  
• Solve problems involving the comparison of lengths and areas using appropriate units (ACMMG137)
• Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129)
• Interpret and compare a range of data displays, including side-by-side column graphs for two categorical variables (ACMSP147)

At all levels where students are constructing data representations (graphs, tables, tallies etc) – there are multiple opportunities to compare these representations and discuss the relative merits of, for example, a stem and leaf plot with a line graph.
We hope you are able to adapt the inquiry,  Record Breaking: Are athletes getting better over time? , to your own year level.

This post summarises the chapter The pedagogy of mathematical inquiry (Makar, 2012).

What does the pedagogy of mathematical inquiry mean specifically for teachers and learners? This chapter provides an in-depth overview of the key elements of mathematical inquiry pedagogy, what it looks like in a real year 6/7 classroom (ages 10-13) and provides theoretical connections to highlight important implications for teaching and teacher education.

The idea of exploring the question Which bubble gum is best? enthuses students, as they determine the qualities that they value in bubble gum. Benefits of posing mathematical problems through ill-structured questions include opening up ways for students to enter the solution process, providing opportunities for students to negotiate with peers how to address such problems, and presenting a need for students to justify their conclusions using the process they use to answer the question. The inquiry teacher in this chapter mathematises the context of exploring bubble gum to determine which is best and students create mathematical strategies to compare different brands of bubble gum. Four phases provide a framework for teaching and learning mathematics through inquiry: Discover, Devise, Develop and Defend. These phases offer teachers and their students, purposeful ways to navigate mathematically through an inquiry.

The chapter illustrates how one teacher, April Frizzle, conducts an inquiry in her multi-age classroom to find the best bubble gum. Although she initially refers to the experience as "the disaster of the bubble gum experiment!" April reflects on the challenges she has in trying to balance taking control of the lesson, and giving her students the opportunity to develop ways to collect and interpret the data they collect. The author of this chapter is able to summarise the positive experiences that April identifies and how these experiences benefit the learners in April’s classroom the next time they encounter a mathematical inquiry. What is the best orange? is a similar inquiry task that the students tackle the following term. It becomes clearer how the struggles from the first inquiry are now beneficial to the teachers and learners and April articulates some transfer between the two units. In particular, April comments on how the inquiry experiences have materialised her students’ beliefs about the nature of mathematics and how that changed the way they were now willing to approach problems.

The final sections of the chapter considers theoretically, how the pedagogy of mathematical inquiry aligns with research on teaching and learning mathematics. These links highlight the importance of engaging diverse learners in mathematics and the author refers particularly to a theoretical framework (Harel & Koichu, 2010) that analyses learning through inquiry in three ways: as struggle, as purposeful, and as providing students with repeated opportunities to reason in meaningful ways. Ambiguity in inquiry topics offers a space for students to construct understandings as they struggle to reach their mathematical conclusions. Teachers are required to balance letting students construct their understandings with the need to support and scaffold that learning. The relevance of mathematics becomes important to students when teachers successfully mathematise a problem situation. Meaningful contexts help students see a problem as important to solve and the openness of inquiry questions lets students see there can be more than one way to solve such problems. Over time, engaging with mathematical learning through inquiry presents repeated opportunities to apply and build understandings. Students and their teacher in the class described above transferred “met-before” (McGowena & Tall, 2010) mathematical struggles with the bubblegum unit of understanding the need for measurement, to quantitatively measure the qualitative characteristics of oranges.

Bubblegum and oranges surprisingly presented opportunities for Ms Frizzle to negotiate what a culture of inquiry meant in her classroom, with her students. This illustration may not reflect the habits and norms in your own classroom and this process requires guidance and explicit support. Consider some of the research on this site on developing a culture of inquiry in your classroom, to help you decide upon what you want your mathematical inquiry classroom to look like.

Summary by Kym Fry

Harel, G., & Koichu, B. (2010). An operational definition of learning. Journal of Mathematical Behavior, 29, 115–124.
McGowena, M. A., Tall, D. O. (2010). Metaphor or met-before? The effects of previous experience on practice and theory of learning mathematics. Journal of Mathematical Behavior, 29, 169-179.