Students:
Year Six Students determine the number of pools that might run in a competition, by drawing on their knowledge of triangular numbers. A handball tournament can be one way to identify triangular numbers and represent them using a real life context.
Year Five Students decide on an appropriate inquiry question that could be answered if the class conducted a tournament. As a class, have students decide how pool participants will be determined (random, seeded, etc.). Pool members can construct a workable draw and data collection sheet which includes match results and durations.
The full version of Round Robin: Who is the best handball player in our class? is available in the Members section of this site. For further information regarding alignment with the Australian Curriculum: Mathematics, including how the inquiry supports student development in each of the Proficiencies, please download the alignment document we have created.
We hope you enjoy running a handball tournament in your classroom. 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 (FieldingWells, 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 evidencebased approach to mathematics. Seven experienced GMI teachers (between 1 and 10years’ 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.
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?
Authors:
FieldingWells, 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: FieldingWells, 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., PerretClermont, A.N., Tartas, V., & Iannaccone, A. (2009). Psychosocial processes in argumentation. In N. Muller Mirza & A.N. PerretClermont (Eds.), Argumentation and education: Theoretical foundations and practices (pp. 6790). New York: Springer. Zeidler, D.L. (1997). The central role of fallacious thinking in science education. Science Education, 81(4), 483496. 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 3sided, 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 lowlevel, taught procedures point to low levels of intellectual quality. Mathematical inquiry has been argued to promote the intellectual demands desired in mathematics:
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.
The intellectual quality of lessons continued to significantly increase as teachers gained experience teaching mathematics through inquiry. 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. Makar, K. (2016). Improving the Intellectual Quality of Pedagogy in Primary Classrooms through Mathematical Inquiry. Mathematics Education Research Group of Australasia. Queensland School Reform Longitudinal Study (QSRLS) (2001). Productive Pedagogies Classroom Observation Scheme. Brisbane: The University of Queensland.
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 1013 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.
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).
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).
Year Three As Year 2, however the measurement are now able to be made in centimetres.
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.
Year Five Intended year level of document. Be very careful of using events as cautioned in Year 4 notes).
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).
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. 
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