General overview of the project PD lessons

The aims of the lessons:

The lessons are in the form of detailed booklets with accompanying commentary on the different stages of the lesson together with other information such as the research background that informed the design of the lesson. They have been designed to be used as part of a professional development programme rather than as a coherent set of lessons for pupils, which will alone improve their learning and standards. The aim is as part of a PD programme, teachers and departments will improve their understanding, knowledge, skills and teaching approaches to enable them to develop their teaching and schemes of learning at KS3. However many teachers report that the approaches have very quickly enabled them to adapt these materials and design their own lessons to make an impact on pupil learning across KS3 and KS4.

How they are organised:

The 3 sets of lessons in each unit either comprise a pair of lessons (i), (ii) in units 1 & 2 or single lessons in unit 3. Where the lessons are in pairs then the first lesson (the lead lesson (i) should be taught before lesson (ii) though it is not necessary that all the lessons are taught as part of the planned PD, but priority should be given to the lead lessons.

The research background informing their design

The 3 pairs of lessons in each unit have been written by 3 authors (A, B, & C). The lessons reflect the particular research approach that the author has expertise in. Hence the author and more importantly the particular research background informing the design of the lesson can be identified by the lesson reference used eg; lessons 1A(i) & (ii), 2A(i) & (ii) and 3A(i) & (ii) have all been written by the same author (A) and follow an RME approach in their design and span the three units. The three main research approaches are described below. The table shows how the lessons and their particular research approach progress across the units. This allows the research background to also be an aspect of the PD including the option of choosing a particular research approach as the focus of the PD programme if desired.

Choosing which lessons to use

PD lead will need to decide which one or two lessons from the unit to focus on in depth at the workshop or in departments. This might be the lead lesson from one of the pairs (lesson (i)) chosen to emphasise a particular approach. The lesson chosen for the lesson study gap task should be one of the lessons studied. Familiarity with the aims of the PD materials and the delivery model used to maximise impact need to gained in order to decide how best to use these materials to meet the identified need of the target audience.

Unit 1 lessons:
Reasoning and making sense of fractions / Unit 2 lessons:
Understanding and identifying proportional contexts / Unit 3 lessons:
Application to a range of proportional problems.
Author/design approach / A
Realistic mathematics education / 1A(i) Fair shares
1A(ii) Our survey / 2A(i) Working with contexts that lead to the bar model
2A(ii) Percentages on the bar model / 3A The ratio table
B
Constructivist / 1B(i) Parts of a shape
1B(ii) Pieces of a cake / 2B(i) Using the double number line to explore relations
2B(ii) Using the Double number line to solve ratio (and non-ratio) tasks / 3B Using stories and diagrams to model division and multiplication
C
Enquiry/cognitive conflict / 1C(i) Ordering and equivalence
1C(ii) milkshakes / 2C(i) Identifying proportional scenarios
2C(ii) Directly or inversely proportional / 3C Exploring multiplicative structures

The research approaches

Below are brief descriptions of the 3 main broad research approaches that have influenced the design of the lessons.

Approach A

Realistic Mathematics Education (RME) is an approach to mathematics education developed in The Netherlands by the math 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 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

Approach B

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.

Approach C

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.