Judy Bailey, University of Waikato
This is the first post in a three-part blog series highlighting two pedagogical approaches that are increasingly popular both in Aotearoa New Zealand and internationally.
- This post, by Judy Bailey, focuses on mathematics teaching that centres on rich tasks and problems. Judy shares an effective lesson structure plus findings from a NZ-based research project exploring the use of workshops within the problem-based approach.
- The second post, by Jessica Rubin, focuses on the “Writing Workshop” approach to teaching writing. In this approach, students spend time daily doing the real work of writers – making choices, exploring interests and problems, and using writing for a range of worthwhile purposes.
- The third post, by Katrina McChesney, brings Writing Workshop and problem-based mathematics teaching together. Katrina considers the connections between these powerful pedagogies and how both approaches align with important goals for education in Aotearoa New Zealand.
Teaching mathematics through problem solving (similar to using challenging, rich or worthwhile tasks) is known to support students’ learning and attitudes towards mathematics. It is also a very different way for most of us to teach mathematics, as students learn mathematics as they solve a problem or challenging task. Learning to teach mathematics in this way can be a challenge in itself.
Recently, a team of NZ primary school teachers I’ve been working with has been exploring how they can position problem solving at the heart of their mathematics curriculum. In this post, I share a lesson structure known to support teaching in this way, as well as these teachers’ adaptations of this structure to include workshops focussed on ‘just-in-time’ teaching.
A child excited about solving an algebra problem!
At the end of last year, while working with the team of teachers, I had the pleasure of talking with a nine year old boy, Rick (pseudonym), about his experiences participating in mathematical problem solving. Rick was known by his teachers to sometimes disengage during mathematics lessons. However, he genuinely and enthusiastically told me how he had solved an algebra problem (see this blog post for a copy of the problem) based on recognising a sequential pattern with a recurring rule of add 4. Interestingly, my discussion with Rick occurred approximately three weeks after his work on the problem. Rick’s enthusiasm was delightful, and his understanding of the problem was still clearly evident after this time.
Rick was part of an innovative learning environment of approximately 70 year 5-8 students engaged in a three-week mathematical problem solving unit as part of a Ministry of Education funded 2019-2020 Teacher-Led Innovation Fund (TLIF) project. Central to the project were two action research cycles including a professional development session (approximately two days each time) where the teachers spent time solving mathematical problems themselves, drew on these experiences to plan a three week unit, and finally reflected on the process. As part of the professional development sessions, we used a structure designed to support lessons based on challenging tasks and/or problems. The structure was drawn from the work of Peter Sullivan and colleagues engaged in the Encouraging Persistence, Maintaining Challenge (EPMC) Project. A recent NZ iteration of that project has found teachers and students were positive about their experiences, with highly significant learning occurring for students.
A structure for teaching mathematics through problem solving
The lesson structure for teaching mathematics through problem solving that was developed by Sullivan et al. comprises four phases: ‘Launch’, ‘Explore’, ‘Summary’ and a ‘Consolidation’ phase. While the term ‘lesson structure’ is used, the four phases can in fact be spread over several class sessions.
‘Launching’ the task is a critical part of structuring a problem-solving lesson. Two key aspects are establishing a common language so the task is interpreted appropriately, and deliberately maintaining the cognitive demands of the task by “inviting and clarifying without explaining and demonstrating“.
During the ‘Explore’ phase students work individually or in small groups. Teachers will have already thought about how different students might respond to the challenge by pre-planning enabling and extending prompts that differentiate the experience. Enabling prompts can involve reducing the number of steps and/or simplifying the complexity of the numbers. These prompts are offered with the explicit intention that students return to work on the initial task. Extending prompts are provided to students who finish the original task quickly; these prompts provoke students to think further than the original problem. For example, some of Rick’s peers were asked if they could write an equation that represented the answer to the algebra problem.
In the ‘Summary’ phase, student activity on a problem/task is reviewed as a whole class, including solutions and strategies.
The last phase, consolidating the learning, involves posing a task similar in structure and complexity to the original challenging task. Some elements of the original task remain the same, while other aspects change to help the learner avoid over generalisation from solutions to one example.
The role of struggle during problem solving
If learners immediately know what to do when presented with a task, then it is not ‘rich’, problematic or challenging. Thus, part of teaching mathematics through problem solving is the expectation that students will struggle. Researchers in Australia advise that struggle needs to be both facilitated and encouraged, suggesting that teachers be flexible in implementing appropriate pedagogies that are responsive to individual students. Teaching strategies that can be useful include:
- using enabling prompts
- recognising effort and persistence
- encouragement to give the problem a go
- suggesting that students collaborate and/or observe others
- suggesting that students trial of multiple ideas.
However, even despite all the above being in place, we know children can still disengage when struggle becomes unproductive. Therefore, a key query for the team of teachers engaged in the TLIF project became: What might be appropriate additional pedagogies that would simultaneously support the students’ struggle and maintain the integrity and demands of problem solving?
In response to these questions, the TLIF team decided to trial planning and providing ‘workshops’ for selected groups focussed on just-in-time teaching and learning. It was envisaged the workshops would occur during the “explore” phase after considerable engagement and trialling of ideas had occurred, and when it was observed that learning a key mathematical idea would be useful to keep ‘struggle’ productive.
A sample lesson
Many teachers will be familiar with an area-perimeter problem known by the team as “Holly and the Lamb” (shown below). It is important that problems are set in relevant cultural contexts, and given the rural setting of this school, the teachers thought it likely students would understand and be engaged by the context.
In essence, this task engages learners in determining that a set perimeter can link to shapes with different areas – a mathematical idea known to surprise many learners. For example, possible shapes for Holly’s pen include a square with each side 6m long (area 36m2) or a series of rectangles (e.g., 2×10 rectangle – area 20m2 or 3×9 rectangle – area of 27m2, etc).
As the team planned how to enact this problem using the lesson structure, possible points of struggle were identified. From this, brief activities were devised for ‘workshops’ that would be offered to groups of students as and when the need arose. Many of these workshop plans drew on the teacher’s existing repertoire of activities. Given the nature of this particularly diverse syndicate (with children from Years 5-8), it was anticipated that points of struggle might involve quite a range of ideas including:
- clarifying the terms ‘perimeter’ and ‘area’ (we opted not to do this within the ‘launch’, deciding it would be more beneficial for children to realise for themselves that they needed to understand these terms);
- devising the formulae for the area of a rectangle, and for some students, a triangle and/or a circle;
- noticing the relationship between the diameter and radius of a circle
- realising that the diagonal of a square is longer than its sides
- … and so on.
In the spirit of problem solving, these workshops focussed on students being engaged with drawing, cutting, measuring, thinking and responding to questions, rather than being taught didactically or given exercises and/or formulae.
Three class sessions of approximately 75 minutes were set aside for students to solve ‘Holly and the Lamb’. At the end of each session, students were given an opportunity to reflect on and write about their thinking, including possible solutions. Judicious summarising as a whole group of 70 students also occurred at the end of each session. The first session was primarily spent in the “launching” and “exploring” phases of Sullivan et. al.’s lesson structure, with an early opportunity for some students to clarify the terms ‘perimeter’ and ‘area’. After this first session, students’ initial written recordings were collected and analysed. All students were then allocated to a ‘workshop’ for approximately 30-40 minutes at the beginning of the second session, after which students then continued their work on the original problem. Critical pieces of information (e.g., that the formulae for finding the area of a rectangle is length x width) were able to be learned ‘just in time ’and then applied within the original task. By the end of the third class session, all students were expected to have a photo/video and/or written explanation of their solution with a justification and explanation of why they believe their pen would give Holly’s lamb the most grass or more grass than another shape. Some workshops (in smaller groups) were also offered during the third session, particularly for those still ‘struggling’. Consolidation tasks were completed in the second and third weeks of the unit.
What we learned
The teachers in this project found that teaching mathematics through problem solving engaged students and supported learning. They appreciated the ‘workshops’ as a useful addition to the lesson structure. The provision of ‘just-in-time’ learning which students could then apply to the problem, sometimes in unexpected ways (but that’s another story!) were helpful in maintaining an appropriate level of struggle.
Trudy (teacher; pseudonym) said:
“I think having pre-thought-out workshops worked really well”
and commented on
“how the concept is relevant to the learning process … that is already happening”.
Emma (teacher; pseudonym) also appreciated being able to support students with just-in-time teaching. In her final reflection, she wrote:
“I believe that there is a need for workshops within PS [problem solving]. While some students made connections to didactic teaching that was given previously, a ‘just in time’ model for workshops seems to me to be more pertinent. Don’t we all remember things we have learnt so much better if we then use them soon after?”
The teachers attributed part of the success of these workshops to having solved the mathematics problems themselves and to their careful planning. These two key factors enabled the teachers to identify possible solutions to the problems and to develop enabling and extending prompts, as well as to anticipate likely ‘points’ where support might be needed. At the end of one of the PD sessions where potential workshops had been identified and planned for, one of the teachers, Pearl (pseudonym), commented:
“Having clear workshops set out is good. I felt that during the last round of teaching this wasn’t planned with as much depth / detail and therefore became difficult. I [now] feel much more confident with the potential learning opportunities for the kids and how to approach this”.
Like researchers in Australia, the team involved in this TLIF project in New Zealand acknowledges that there is no “one way” to support students with productive struggle when teaching mathematics through problem solving. Results from our project suggest that planning and providing ‘workshops’, in addition to the use of enabling and extending prompts, supported teachers’ use of challenging tasks or problems.
Judy Bailey is a primary mathematics teacher educator and researcher. She enjoys working alongside pre-service and in-service teachers as they investigate what the teaching and learning of mathematics can look like. Most recently she has been collaborating with beginning teachers exploring what supports are needed for beginning teachers to embed mathematical problem solving within their teaching programmes.