Developing Student Thinking
Developing student thinking is at the core of Maths-for-Life lessons. First student thinking is revealed, and then support is provided to help move their thinking on. This support may be from a peer or teacher, or the students’ own self-reflection.
Students are encouraged to make their thinking not only audible but also visible. Several design strategies, including card matching activities, and the use of visual representatives, help clarify how students are thinking. The development of student thinking is supported through a number of carefully constructed tools, including:
Lead teachers from the Maths-for-Life programme discuss how elements of the Maths-for-Life materials contribute to developing student thinking and mathematical understanding.
Teachers are encouraged to foster a classroom culture in which mistakes are valued as a learning opportunity. This does not happen overnight, but requires persistence and resilience.
The Challenge of a task
Getting the challenge of a task right is key to developing student thinking. If the challenge is too little, then students are not learning. If the challenge is too great, there is a risk of non-engagement. This Goldilocks dilemma is further exacerbated by classes in which there is a wide range of prior student achievement. Designing tasks with a low threshold but high ceiling offers the opportunity for everyone to get started, and everyone get stuck.
This approach is grounded in the growth mindset philosophy that all students can succeed in mathematics, regardless of their prior attainment. By giving everyone the same task, no-one's achievement is limited before even beginning. It makes for a more equitable classroom.
Visual representations
The research indicates that students who use visual representations when solving a problem, are more likely to be successful compared to those that don’t. Studies also indicate when students work together, visual representations can serve as focal points for coordinating discussions and enhancing performance. The trials of the Maths-for-Life resources concurred with these findings.
Visual representations are used extensively throughout the programme. They take the form of diagrams, including the double number line, box diagrams for algebra, and graphs.
Designed student responses
In trials of the Maths-for-Life resources it was found that although many problems could be solved using various distinct strategies, most students, in any one class, preferred to use the same concrete method. A trial and improvement method, for example, was used by students, rather than more powerful methods. To address this issue, carefully designed, hand written and coherently organised, student responses to a problem were introduced in several lessons. These have multiple benefits:
- They expose students to more sophisticated methods
- They clarify assessment criteria – in particular how solutions could be best presented
- A student-led approach to learning is integral to their use
- They provide an opportunity for students to focus on understanding rather than performance, and answer-getting
Structure of each lesson
The structure of a lesson can determine the extent and depth of learning.
- Introduction
The Maths-for-Life lessons often begin with a whole class ‘setting the scene’. These real-life contexts provide concrete and imaginable footholds into the mathematics. As the lesson progresses, the scene can be a useful touchstone to refer back to.
- Paired work
To further support dialogic learning, students then work in pairs on a task. The intention is that students support each other’s learning, through explanation, clarification, questioning, and justification.
- Closure
Another key element of the lessons is the whole class closure. Closure can take place at any point in the lesson, but is particularly important towards the end where we want students to leave the classroom feeling that they have progressed in their understanding.
Some of the lessons include exam questions. These can motivate, and help students recognise how their mathematical understanding has developed.
The lesson plan
When students are working on problems that can be solved in multiple ways, teachers may encounter unexpected student approaches. Unsure how best to react, it can be tempting to simply tell students of their own preferred method. This can limit student agency, and deny them the opportunity for productive struggle.
To support teachers, included in the lesson plan, are examples of different student approaches, misconceptions that may arise as students solve a problem and ways they are likely to get stuck. Such student issues are accompanied by practical suggestions on how teachers can interpret and react in a constructive way.
The plan also outlines the different stages of the lesson, including clear learning goals and ideas on how achieve them. It provides, for example, suggestions on how to conduct whole class discussions – an activity that is known to be challenging for many teachers.