Overview

The team lead had expertise in this area, both through research and professional practice. It was a great opportunity to explore this approach and how it might be put at the heart of the hub’s mathematics programme.

I was pleasantly surprised by how engaged nearly all the students were with this unit. I expected more off-task [behaviour] or reluctance. It was interesting to observe the Year 5 students who had not experienced maths like this before (they were not part of our 2019 unit). They were initially quite overwhelmed and struggled with the concept of there not being ‘one right answer’. They showed persistence and could talk to others about their learning. As the unit went on, they were approaching the problems with more of a growth mindset and seemed to be engaged with the tasks.

Teacher reflection following second inquiry cycle

The teachers began by exploring mathematical problems themselves and then looking at how the same problem might catalyse learning at different levels of the curriculum. They adopted and adapted a research-based model for designing lessons that shifted through a repetitive set of phases around three different, related problems. The original structure incorporated time for exploration and for sharing, comparing, and summarising learning. The inquiry team introduced workshops to help fill gaps in knowledge and skills when it was needed, and to whom it was needed. They sought to hand responsibility for learning to students, so entered each lesson prepared with prompts to help them get through difficulties or extend their ideas after they had found a solution. They reflected on impact, both on students who were initially disengaged from mathematics and on themselves.

Integrating a problem-solving approach with knowledge-building led to students becoming more engaged in mathematics and achieving better results. Giving teachers time to explore the problems as learners themselves and to break them down in relationships with curriculum achievement objectives left teachers feeling empowered and enthused.

The inquiry story

This inquiry involved the three teachers and the approximately 70 students who are part of the years 5–8 innovative learning environment (ILE) at Netherton School. It took place in two cycles, over 2019 and 2020. As in many other schools, the second cycle was stretched due to Covid 19.

What was the focus?

For several years, Netherton School had been engaged in whole school inquiry into the use of a problem-solving pedagogy for teaching mathematics. The school has also worked on embedding a growth mindset into teaching across the curriculum, along with the idea of the ‘learning pit’. (This is the idea that meaningful learning often involves challenge and struggle and that successful learners develop strategies for getting back out of the pit.) Staff see mathematical problem solving as a natural fit with these two pedagogical frameworks.

The inquiry lead has engaged in extensive research into problem-solving and rich tasks, discovering that while it has many benefits for mathematical learning, teachers often struggle with implementation. In their own school, the team had seen some learners disengage from mathematical learning. They seemed hesitant to embrace a growth mindset and reluctant to make mistakes.

The inquiry team believed students need to be supported to develop the mathematical knowledge and skills set out in the curriculum, but also need problem solving opportunities where they could apply, practise, and extend these capabilities. Thus, the purpose of this inquiry was for the team to redesign the school’s years 5 to 8 mathematics curriculum to put problem-solving at its heart while ensuring students would learn the mathematical content described in the curriculum.

The team developed the following innovation statement:

We want to know whether planning and teaching a unit in mathematics that has problem-solving at the heart of the unit and is aligned to New Zealand Curriculum achievement objectives and key competencies will positively impact on children's learning and attitudes towards mathematics, with a particular focus on students who have previously presented as mathematically disengaged.

What did the teachers try?

The teachers commenced their inquiry with workshops where they surfaced their own beliefs about mathematics and problem-solving and experienced mathematical problem solving for themselves. The problems had a ‘low floor’ and a ‘high ceiling’, meaning that the same problem could be attempted by students at a range of levels, but using different strategies. They explored the different strategies and mathematical content students are expected to deploy at different curriculum levels and how the same problem could enable learning at these different levels. These experiences provided deeper insight into mathematical progression and into the feelings students might experience when attempting challenging problems.

In each cycle of inquiry, the teachers collaboratively developed, planned, and taught a three-week unit built around the problems the teachers had explored. Each unit was implemented in nine lessons, and each lesson followed a set structure that had been adapted from work by Sullivan et al. (2015):

• Launch: Introduce and clarify the problem with the whole group. Allow independent thinking time.
• Explore: Students work in self-selected groups to solve the problem, drawing on mathematical equipment and supported by teacher prompts. They are expected to record their thinking.
• Summarise: Students move back into the whole group. Strategies are shared and connections made between them. Students have ten minutes silent writing time for reflection.
• Consolidate: Work through three similar, related problems.

During the inquiry, the teachers adapted the model to include workshops addressing the specific knowledge and strategies required to solve the problems and achieve the related achievement objectives. The workshops were pre-planned and offered to students on an “as needed” and “just in time” basis and offered it to them at that moment.

The lesson design also included pre-planned prompts for enabling and extending students without “rescuing” them from difficulty or reducing their cognitive load (for example, “What is the question asking you?”, “Are you certain?”) The prompts were laminated, and the teachers wore them on lanyards.

What happened as a result of this innovation?

The teachers observed the impact on students, and their critical friend interviewed them, including asking whether they preferred the problem-solving approach or more traditional methods of mathematics teaching. While the student survey data was ambiguous, the observations showed increased engagement. In the interviews, the students said that they:

• learned mathematics better or more ‘deeply’ when solving problems
• like being allowed to work with their friends
• like working on the same problem as a hub
• enjoying constructing a poster to display their learning at the end of the unit
• found this way of learning mathematics, exciting, new, or fun.
• preferred the choice that is a feature of the usual mathematics programme.

Student self-perception of their progress did not always align with what the teachers saw – some students’ assessment of their enjoyment and achievement in mathematics had declined when in fact, their achievement outcomes had improved. This may have been associated with the additional cognitive load created by the problem-solving approach.

In the first cycle of inquiry, e-asTTLe was used to monitor shifts through the three-week units. This did not prove to be very helpful, as the tool wasn’t sufficiently attuned to the specific learning intentions the teachers and students were addressing. In response, the teachers developed and implemented alternative forms of assessment, including using Seesaw to share samples of student work and having students create summative posters. Here, the record of progress is much more refined and enables deeper insight into student thinking.

The survey of teachers regarding their beliefs about mathematics teaching threw up some puzzles to explore. However, all teachers reported growing confidence in their content and pedagogical content knowledge and ability to teach mathematics through problem solving. These are important findings, as research shows teacher knowledge and self-confidence are strongly related to the ability to teach in this way.

What did they learn?

Learning from this inquiry included:

• Time spent exploring problems before teaching them provided invaluable insight in the learners’ experience.
• Offering workshops to address specific needs during problem-solving rather than before or after enables greater responsiveness to student need.
• Keeping the prompts on lanyards was a useful reminder for teachers not to slip back into explaining strategies to the students.
• Other forms of assessment (such as posts on Seesaw and summative posters) can be better attuned to specific learning intentions than standardised testing
• Taking time to unpack how problems link to the achievement objectives and competencies described in the curriculum can contribute to improved teacher knowledge and confidence.
• Student surveys can throw up puzzles that require digging into – the issue may be just as they say but there may also be something in the design and delivery of the survey that leads to unexpected results.

Inquiry team

The project was led by Hilary Rodley and involved two other teachers: Tany Simpson and Venessa Silvester.

The project’s critical friend was Judy Bailey (University of Waikato.)

For further information

If you would like to learn more about this project, please contact the project leader Hilary Rodley at hilary.rodley@netherton.school.nz

Reference list

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