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Multiplicative Reasoning - The teaching units

Created on 01 February 2016 by ncetm_administrator
Updated on 04 April 2016 by ncetm_administrator

The teaching units

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.

The research background informing the lesson design

  • The lessons have been written by 3 designers and are arranged in sets of 3 within each unit (A, B & C) reflecting the particular teaching approach in which the designer has expertise. The 3 particular approaches are described by the designers in the materials and run through the units allowing this to be an aspect of the PD. This is shown in the table below.

Choosing the lessons to focus on in the workshop

  • 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 (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 PD lead will need to decide which one or two lessons from the unit to focus on in depth at the workshop. 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.

Research approaches

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.

Structure of the teaching units

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)

Unit 0
School Task

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
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
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



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