Case study 4: Towards an enquiry and discussion based approach to teaching and learning mathematics

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Case study 4: Towards using an enquiry and discussion based approach to teaching and learning mathematics – changing practice and beliefs
We have chosen this case study as it shows how teachers at a selective 11-18 girls’ grammar school who were initially nervous about introducing a new way of working, changed their practice and beliefs as a result of introducing enquiry and discussion based ways of teaching and learning.

Background to the study
Four Year 7 classes and their teachers were involved, with the assistance of a consultant. Generally mathematics had been taught in a traditional manner, based on exposition and practice and there was concern that the Year 7 curriculum did not provide sufficient challenge or progression for many of the students, nor did it allow links to be made across topics.

The school completely re-wrote the Year 7 scheme of work for mathematics with an emphasis on enquiry-based approaches. A collaborative planning approach was adopted with teachers working in pairs. Two of the teachers planned and delivered a unit of work over a number of weeks. The pairs then discussed effectiveness of the new approach before swapping and teaching the new units again.  Finally, at the end of the year, the Year 7 team met to run through the entire scheme of work, discussing which approaches had been effective and amending the written scheme of work for future use.

What new approaches were used?
In moving from a textbook-based exposition and practice model to an enquiry-based approach, the teachers involved developed open-ended tasks, providing students with opportunities to consolidate previous knowledge whilst also offering a potential challenge for more confident students.

For example when working on long multiplication:

Use the digits 1, 2, 3 and 4 once each to make two numbers. Multiply your numbers together. Which numbers give you the largest and smallest possible products?

This task enabled the teacher to identify students who had difficulty with the basic skill required while allowing others to make generalisations and to justify their reasoning.

Typically students would work individually for a short period but then be asked to discuss their work in pairs, groups or as a class. Solutions and methods could then be justified and compared.

Students were also given additional resources to stimulate learning, such as geoboards, multi-link cubes or paper-folding.

How did the project challenge teachers’ practice and beliefs?
Staff involved initially had mixed feelings about moving from a familiar way of working to what was viewed as an untested approach. In reality the teachers involved found it to be an inspiring experience, benefiting both themselves and their students. Comments included:

What was surprising was the richness of the content of the work and how one piece of work could open the doors to many areas of mathematics.

I was inspired to try a new way of teaching Year 7 by having [consultant] in my Year 8 lesson last summer term, when he showed how an open-ended approach to a topic can yield results which involve students doing higher level Maths than I thought possible.

Trying an enquiry approach with some of my classes.… I found that I could very quickly assess what students did or didn’t already understand, and I have been consistently surprised by the insights students have shared when asked to find their own method to solve a problem.

Wider impacts on pedagogical approach

Staff commented:

The project has helped teachers to explore new ideas of delivery and has made us reflect on our teaching styles.

I am looking for more ‘experimental’ ways and open-ended ideas for other year groups for example: more brainstorming, ‘what if’, ‘what do you notice’, try this with a friend’, ‘make up your own example’.

What impact did the project have on student learning and attitudes?
Students who experienced the new curriculum with a focus on open-ended tasks and problems became:

• more likely to work at a problem that might initially appear difficult and less likely to be discouraged by being stuck;
• more inquisitive and critical in their thinking and more likely to pose their own mathematical questions;
• more likely to produce creative and varied solutions and happier to justify them to peers; and
• more enthusiastic about learning mathematics.
  
  

My class are not fazed when faced with an open-ended question or when asked to brainstorm or explain their solutions… they feel that they can ‘do’ maths as they aren’t limited by one particular method.

As a result of the project the enquiry approach was rolled out across the whole school. A similar complete overhaul of the Year 8 and 9 schemes of work followed.

Reference
Richards, M. (2008) Teaching and learning mathematics using an enquiry-based approach.
Summary of the National Teacher Research Panel conference, 2008