The table below details the four units of school-based activity, corresponding to each of the first four workshop phases in this professional development model. Unit 0 contains questions and tasks to set in class, the responses to which will help prepare the ground for the lessons contained in Units 1, 2 and 3.

## A - Realistic mathematics education

Realistic Mathematics Education (RME) is an approach to mathematics education developed in The Netherlands by the maths educators of the Freudenthal Institute. It is proposed here that the teaching and learning of mathematics should be connected to reality, stay close to children’s experience and be relevant to society, in order to be of human value. Mathematics lessons according to Freudenthal should give students the ‘guided’ opportunity to ‘re-invent’ mathematics by doing it; the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization.

The features of RME include the following.

- Use of realistic situations to develop mathematics
- Well researched activities that encourage pupils to move from informal to formal representations
- Less emphasis on algorithms, more on making sense
- Use of 'guided reinvention'
- Progress towards formal ideas seen as a long-term process

## B - Constructivist

The lessons here can, in places, appear very structured and teacher-led. However, they are designed to help students explore and contemplate mathematical ideas. The intention is that students are given the opportunity to reveal their thinking, to value their ideas, and to explore, critique and develop them through discussion and activities guided by the teacher. In the process, it is hoped that the teacher will get a richer understanding of their students’ thinking and of the complex nature of multiplicative reasoning. The lessons are predicated on the belief that multiplicative reasoning is not learnt in a ‘linear’, step by step, level by level way, but that it comprises a complex network of ideas that is constructed, strengthened, extended and modified over a long period of time. These few lessons can only provide snapshots of some of these ideas but it is hoped they will stimulate the teacher to revisit and take them further with their students.

## C - Enquiry/cognitive conflict

A collaborative inquiry based approach to learning where students are engaged in cognitive conflict has been shown to promote long-term learning (see for example Birks (1987) *‘Reflections: a Diagnostic Teaching Experiment’,* Cobb (1988) *‘The tension between theories of learning and instruction in Mathematics Education’,* Onslow (1986) *‘Overcoming conceptual obstacles concerning rates: Design and Implementation of a diagnostic Teaching Unit’,* Swan (1983) *‘Teaching Decimal Place Value – a comparative study of ‘conflict’ and ‘positive only’ approaches’*). Students become aware of the inconsistencies in their own conceptions and this awakens a curiosity and desire to seek resolution through discussion. A final whole class discussion allows students to share their different understandings and provides an opportunity for generalisation and extending what has been learned.

## Lesson commentary:

Each part of the actual lesson is accompanied by a detailed commentary written by the designers giving reasons behind the activities, possible pupil thinking, misconceptions and suggestions for how the teacher might respond. Other features important to subject and pedagogical knowledge are also highlighted.

## Adapting the lesson:

General notes on how the work might be pitched to challenge and meet the needs of different groups of pupils across KS3. More specific advice might also be given in the lesson commentary.

## Lesson study suggestions

The lessons also include suggestions for possible research questions for use in any lesson study.

## Research background:

The lessons are written by researchers who have particular expertise in how children learn mathematics. The lessons are written in sets of 3, each set reflecting one of 3 particular research approaches. These particular research approaches can be broadly described under the headings: Realistic mathematics education, socio- constructivist and enquiry (cognitive conflict). Details of the particular research background that has informed the design of the lesson is given in each lesson.

## Other supporting documents

**Recording Common issues:** A key function of the lesson commentary is to anticipate what pupils may think including any difficulties/misconceptions and suggest possible responses. However not all responses can be anticipated or particular teacher responses be successful. Hence many of the teachers following the teaching of the lessons found that the brief listing of any unexpected learning issues that occurred during the lesson alongside responses that proved successful or otherwise greatly assisted the next teacher planning to use the lesson. A simple sheet to accompany a lesson/unit was designed to record this as brief bullet points which accumulate as the lesson is taught across the department over time. This was considered a simple but effective way to improve the learning. *(***Click here** to see)