Discover To start the inquiry, investigate the use of integers in everyday contexts. A game that includes positive and negative numbers on a number line (vertical or horizontal) can create an authentic context for modelling positive and negative numbers. You might like to consider the Elevator Challenge developed by re(Solve): Maths by Inquiry. Introduce the inquiry question and reflect on the challenge of the inquiry. 
Devise In this phase, have students collaborate to design their own games to model positive and negative integers. Guide the coconstruction of a class set of criteria to define ‘best’ in this context. Use content from the Australian Curriculum and the Proficiencies described for Year 6 to inform this criteria e.g. the ‘best’ game needs to give players lots of chances to represent positive and negative numbers on a number line as well as opportunities to formulate and solve authentic problems involving positive and negative integers. Encourage students to share their ideas and provide feedback on planning ideas. 
Develop Provide time for students to make their games based on their planning and ideas in the previous phases. Have students play games other students have made to generate feedback on challenges to game designs, and on improvements and innovations that can be made. In this phase, have students prepare the mathematical evidence they will need to Defend their solution to the inquiry question. 
Defend Students now get to play the games designed by others and reflect to Defend how a game ‘best’ models positive and negative integers. Have students consider the mathematical evidence generated in games they play and focus learning on the many opportunities to problem solve as they play – locating and representing positive and negative integers on a number line. 
John: As you can see from my graph – and I did gold medals in men’s long jump – in 1968, George Beaman jumped 8.9 metres and that was the world record at the time and the reason for that was that it was in Mexico City and it has a high altitude and the oxygen’s thinner so you can jump further so from there it’s obviously gone down a lot [points to graph]. In 1972, it started to increase again [points to graph] and from then on it went a little bit up and down … in conclusion out of our four events, three are getting better at jumping (women and men’s pole vault, women’s triple jump) but not in the men’s long jump because it has to do with the places where the Olympics are being held. (Muir & Wells, 2019) 
Year Four Devise an everyday context that would necessitate the school librarian visiting the class to ask them to make decisions about favourite books. For example, a scenario that says Year Four have won the ‘Best Borrowers’ competition, which entitles the class to choose books for a library display. Invite the school librarian to share the context with the class and ask students: What are the best books to display, to represent our class? In Year Four, students can select and trial methods for data collection. Remind students that graphs and tables are a way to help them understand and analyse the data as well as tools to communicate the answer.

Round Robin: Who is the best handball player in our class? is a great inquiry that you can find in book 2 of the Thinking Through Mathematics series. Designed initially for students aged 810 years, we have now aligned the mathematical content with the intent of the Australian Curriculum: Mathematics. What happens in the inquiry? Handball is a popular, easytolearn schoolyard game which can be adapted for tournament use. This unit provides opportunities for students to explore triangular numbers and to apply their understanding to create an appropriate draw for their tournament. 
Year Four Students investigate how long a game is. Timekeeping (including timeouts) using a stopwatch or timer will be useful for students to get a feel for the duration of a game (e.g. 1:15 is one minute and 15 seconds or 75 seconds). Pools play and collect data for several (about six) matches following constructed draws. Students will need to organise and interpret data discussing frequency and range. 
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. 
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. 
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.
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. 
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

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

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

What does intellectual quality look like in a mathematics classroom? In one Year 5 classroom in this study, students were required to estimate, measure and compare angles using degrees. To engage students in a task of high intellectual quality, the classroom teacher posed the inquiry question to students, How can we accurately estimate the sum of the internal angles of a polygon? 
The author of this paper (Makar, 2016) was interested in understanding how primary teachers’ experiences and pedagogies evolved as they taught mathematics though inquiry. The study presented in this paper observed aspects of teachers’ pedagogical practices that showed evidence of intellectual quality. The author compared data from regular mathematics lessons and initial inquiry lessons from 41 primary teachers and continued to follow 19 of these teachers over three years.  Intellectual Quality
Productive Pedagogies (QSRLS, 2001): Intellectual Quality Cluster and Dimensions 
The intellectual quality of 41 teachers’ regular mathematics lessons and their first term of (in)experience in teaching mathematics through inquiry were compared. The results overall were interesting: The intellectual quality in teachers’ initial inquiry lessons was significantly higher than in their regular mathematics lessons. Higher order thinking had a high effect size in the initial inquiry lessons. The greatest difference was in how mathematical knowledge was presented. The nature of mathematical inquiry is that it is ambiguous and requires negotiation. The gains in Intellectual Quality which were the greatest were in terms of Higher order thinking and Knowledge as problematic. This suggests that the aspects of intellectual quality highlighted here potentially align with the nature of mathematical inquiry. How did the intellectual quality of teachers’ pedagogical practices change as they gained experience teaching mathematics through inquiry? The author compared the teachers’ lessons at four junctures over three years: 
The intellectual quality of lessons continued to significantly increase as teachers gained experience teaching mathematics through inquiry.