SCAFFOLDING NUMERACY IN THE MIDDLE YEARS A Linkage Research Project

PROJECT OVERVIEW The project will investigate the efficacy of a new assessment-guided approach to improving student numeracy outcomes in Years 4 to 8. In particular, it will identify and refine a Learning & Assessment Framework for the development of multiplicative thinking using authentic assessment tasks.

2. Identify learning needs in terms of LAF 3. Work in teams to develop/revise Learning Plans 4. Implement two Learning Plans complete an Implementation Report for each 6. Develop an appropriate authentic task to assess key ideas 1. Use initial assessment/authentic tasks to assess student learning against LAF Collect Ss work samples Collect Learning Plans Analyse data and revise LAF as appropriate Provide professional support as needed Collect authentic tasks, analyse, refine, distribute Observe lessons 5. Observe and be observed – discuss at Cluster Level Interview Students Visit Clusters & schools PROJECT DESIGN

WHY MULTIPLICATIVE THINKING? For more details, see the Final Report of the Middle Years Numeracy Research Project (Siemon, Virgona & Corneille, 2001) a significant number of students in Years 5 to 9 appear to experience difficulty in relation to some key aspects of numeracy; multiplicative thinking, particularly in relation to fractions, decimals and per cent, and the capacity to interpret, apply and communicate what was known in context, were among the most common sources of student difficulty. The Middle Years Numeracy Research Project has shown that:

WHY AUTHENTIC TASKS? For more details, see the Final Report of the Middle Years Numeracy Research Project (Siemon, Virgona & Corneille, 2001) these measures of numeracy performance are valid and reliable; they support the identification of an underlying framework distinguished by strategy usage, conceptual understanding and a capacity to monitor one’s thinking and actions; teaching informed by the results of these tasks leads to improvements in student numeracy. The Middle Years Numeracy Research Project has shown that:

WHY THE MIDDLE YEARS? For more details, see the Final Report of the Middle Years Numeracy Research Project (Siemon, Virgona & Corneille, 2001) the needs of many students, and ‘at risk’ students in particular, are not being met; there is a significant ‘dip’ in Year 7 and 8 performance relative to Years 6 and 9; there is as much difference within Year levels as between Year levels (spread); and there is considerable within school variation. The Middle Years Numeracy Research Project and Middle Years research more generally, eg, MYRAD, has shown that:

RESEARCH QUESTIONS: Essentially, the research is aimed at: identifying key points in the development of multiplicative thinking and rational number; investigating the efficacy of using authentic assessment tasks to identify student learning needs more specifically; and describing the teaching strategies and learning environments most conducive to scaffolding numeracy in the middle years. For more detail, see the Project Information handout distributed at the Cluster Leaders Meetings held in December 2003 (Tasmania) and February 2004 (Victoria)

THE LEARNING & ASSESSMENT FRAMEWORK: A learning assessment framework is generally understood to be a loose hierarchy of key ideas and strategies related to the development of an important aspect of the mathematics curriculum, that is expressed in way that is accessible and useful to teachers. To date, these have tended to be developed for the Early Years in fairly independent ‘domains’. See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004)

THE LEARNING & ASSESSMENT FRAMEWORK: In this case, the Learning and Assessment Framework for Multiplicative Thinking attempts to bring together, in a loose hierarchy, all the key ideas, strategies and representations of multiplication and division needed to work flexibly and confidently with whole numbers, fractions, decimals, and per cent across a wide and expanding range of contexts. See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004)

MULTIPLICATIVE THINKING: See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004) The development of multiplicative thinking requires that students construct and coordinate three aspects of multiplicative situations: groups of equal size; the number of groups; and a total amount. And that this is done “in such a way that one of the composite units is distributed over the elements of the other composite unit”. (Steffe, 1994, p.19)

MULTIPLICATIVE THINKING: See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004) Having accomplished this, students then need to move from the models and representations that work for whole number to more general ideas accommodating rational number and algebra. These ideas include ratio, proportion, multiplicative comparison, multiplication of measures, and the use of intensive quantities. This process is complex and may take many years to achieve.

MULTIPLICATIVE THINKING: Initially, multiplicative thinking develops as each aspect is abstracted and can be dealt with as a mental object MODELLEDABSTRACTED Equal group Makes all, counts all Trusts the count, sees equal group as a composite unit Number of groups Sees in terms of each group, counts all groups, eg, 1 group, 2 groups, 3 groups,... Can deal with number of groups in terms of part-part-whole understanding, eg, 6 groups is 3 groups and 3 groups, or 5 groups and 1 group Total Arrived at by counting all, skip counting Total seen as a composite of composites, eg, 18 is seen as 2 nines, 9 twos, 3 sixes, 6 threes

MULTIPLICATIVE THINKING: A critical step in this process appears to be the shift from counting groups: 1 three, 2 threes, 3 threes, 4 threes,... to seeing the number of groups as a factor 3 ones, 3 twos, 3 threes, 3 fours,... and generalising: 3 of anything is double the group and 1 more group. See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004)

MULTIPLICATIVE THINKING: Existing frameworks tend to move directly from modelling to abstracting. Given that a significant number of children experience difficulty in crossing the ‘abstracting barrier’ (Sullivan et al, 2001), this seems to suggest that there is a considerable gap between skip counting (arguably additive) and the flexible, non-rote application of number fact knowledge in a variety of situations. See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004)

MULTIPLICATIVE THINKING: Skip counting by anything other than 2s, 3s, 5s, 9s and 10s may in fact be a deterrent to multiplicative thinking While working with concrete models and representations is important in the early stages, a key aspect in crossing the ‘abstracting barrier’ appears to be the capacity to work with mental images and strategies based on doubling and known facts without physical objects. See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004)

THE LEARNING & ASSESSMENT FRAMEWORK: See the Draft Learning and Assessment Framework for Multiplicative Thinking (February, 2004) The Draft Learning Assessment Framework for Multiplicative Thinking is organised in 9 levels from initial explorations with concrete materials through to the confident use of a wide variety of multiplicative structures and symbolic forms. The framework has been developed on the basis of the research literature and the work of Callingham (eg, 2003) in relation to the growth of higher-order cognitive thinking.

RESEARCH TEAM: Associate Professor Dianne Siemon, Project Director, RMIT University Dr Shelley Dole, Research Associate, RMIT University Ms Jo Virgona, Senior Project Officer; RMIT University Ms Margarita Breed, Ph.D student, RMIT University; and Professor Peter Sullivan, Latrobe University and Adjunct Professors John Izard and Max Stephens, RMIT University - Consultants