Developing Pedagogical Tools for Intervention of low-attaining Grade 3 & 4 students 

Robert (Bob) Wright Southern Cross University David Ellemor-Collins Southern Cross University Gerard Lewis (2007)

This paper reports on a project aimed at developing pedagogical tools for intervention in the number learning of low-attaining 3rd- and 4th-graders. The experimental framework for instruction consists of five aspects: number words and numerals, structuring numbers to 20, conceptual place value, addition and subtraction to 100, and early multiplication and division.

How do we teach children to be numerate? 

Mike Askew & Margaret Brown, Kings College London

This is an overview of research conducted across Britain which covers a range of of general ideas which could be of interest to primary teachers and curriculum leaders.

**NEW** Building a strong conception of the numberline

John K. Lannin,  Delinda Van Garderen, and Jessica Kamuru (2020)

Students' views of the number sequence and their ability to recognise units are two important ideas for developing developmental understanding. 

*NEW** Mental computation in the Primary classroom

Angela Rogers

Explores the importance of teaching mental computation across the Primary school years.

Subitising: What is it? Why teach it? (This link will take you to the Semantic Scholar site.  Click on the pdf download link to access article)

Doug Clements State University of New York, Buffalo (1999)

Investigation of elementary students on immediate recognition of dots and how this affects counting. 

The Acquisition of Numeracy 

Jenny Young-Loveridge (1999)

This article outlines a framework that shows how children think about the number system. It explores the notion that the understanding of the number system is the key to teaching and learning mathematics. 

Understanding Assessing Children's Mathematical Thinking

D Clarke, J Cheeseman, P Sullivan (2001)

This reading unpacks the research on the Early Numeracy Research Project. The project involved three main components: the development and refinement of a set of research based “growth points” in mathematical understanding in various mathematical domains; the creation and use of a one-to-one, task-based assessment interview with all children twice a year; and a multi-level professional development program.

Insights About Children's Understanding of 2 and 3 Digit Numbers 

Ann Gervasoni & Linda Parish (2011)

Five interpretive place value tasks were added to the Early Numeracy Interview (ENI) to gain further insight about students’ construction of conceptual knowledge associated with 2-digit and 3-digit numbers. The new tasks assist teachers to identify students who need further instruction to fully understand 2-digit and 3-digit numbers. 

Structuring Numbers 1-20 

David Ellemor-Collins & Robert (Bob) Wright (2009)

The Numeracy Intervention Research Project (NIRP) aims to develop assessment and instructional tools for use with low-attaining 3rd- and 4th graders. The NIRP approach to instruction in addition and subtraction in the range 1 to 20 is described. The approach is based on a notion of structuring numbers, which draws on the work of Freudenthal and the Realistic Mathematics Education program. 

Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction 

Thomas P. Carpenter & Elizabeth Fennema, University of Wisconsin-Madison and Megan L. Franke The University of California, Los Angeles (1996)

This reading provides a framework for looking at addition and subtraction problems and the various structures teachers should consider when exploring these concepts.

Integration, Compensation and memory in mental addition & subtraction 

Ann Heirdsfield, Centre for Mathematics and Science Education, Queensland University of Technology, Brisbane, Australia (2001)

This paper reports on a study of Year 3 children's addition and subtraction mental computation abilities, and the complexity of interaction of cognitive and affective factors that support and diminish their ability to compute proficiently (accurately and flexibly).

A Longitudinal Study of Invention and Understanding in Children's Multi-digit Addition and Subtraction   (This link will take you to the jstor site.  On the right hand side is drop down menu 'Alternate access options for independent researchers'.  Click on the 'Read online - read 100 articles/ month free'. You will need to join to access the free articles).

Thomas P. Carpenter, University of Wisconsin-Madison Megan L. Franke, University of California at Los Angeles Victoria R. Jacobs, California State University-San Marcos Elizabeth Fennema, University of Wisconsin-Madison Susan B. Empson, University of Texas at Austin (1998)

This 3-year longitudinal study investigated the development of 82 children's understanding of multi-digit number concepts and operations in Grades 1-3.

From Additive to Multiplicative Thinking  The Big Challenge of the Middle Years 

Dianne Siemon, Margarita Breed & Jo Virgona, RMIT University, School of Education (2010)

The transition from additive to multiplicative thinking is one of the major barriers to learning mathematics in the middle years. This workshop will explore some of the tasks from a current research project that are being used to identify steps in the development of multiplicative thinking from Years 4 to 8 in a number of Victorian and Tasmanian schools.

Links Between Children’s Understanding of Multiplication and Solution Strategies For Division 

Ann Downton, Australian Catholic University (2008)

The focus here is on the strategies used by case study students to solve equivalent groups and times as many division tasks. Results suggest that young children are capable of solving complex division problems given experience with a range of semantic structures for multiplication and division.

Challenging Multiplicative Problems Can Elicit Sophisticated Strategies 

Ann Downton, Australian Catholic University

This paper reports on 13 Grade 3 students’ approaches to Equivalent groups and Times as many multiplicative word problems. The findings are part of a larger study relating to children’s development of multiplicative thinking. Of particular interest was the extent to which task level of difficulty influenced students’ strategy choice. The results suggest a relationship between the level of difficulty and strategy choice: the more difficult the task the more sophisticated strategy choice.

A motivation for developing more flexible meanings and strategies for division in Primary School 

Dr Karen Newstead: Mathematics Learning and Teaching Initiative, South Africa Dr Julia Anghileri, Dr David Whitebread: Homerton College, Cambridge, U.K(1995). 

This study provides a motivation for exposing young pupils to a wider variety of division problems than is currently the case. Year 5 and 6 English pupils' strategies for solving six written symbolic division problems were classified and analysed in terms of success. 

It Seems to Matters Not Whether it is Partitive or Quotitive Division When Solving One Step Division Problems 

Ann Downton, Australian Catholic University (2015)

This paper reports on strategies 26 Grade 3 students used to solve a range of division word problems in a one-to-one interview. The focus is on the strategies used by the students to solve partitive and quotitive division problems pertaining to four different semantic structures. Of particular interest was the range of strategies used for each form of division. Results suggest that there was little difference between the strategies used for partitive and quotitive division.

Proportional Reasoning

Associate Professor Shelley Dole, The University of Queensland, presents her research on Proportional Reasoning.

Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction 

Doug M. Clarke & Anne Roche, Australian Catholic University (2009) - This link will take you to the research gate website where you can download a PDF.

This research was based on tasks were expected to be undertaken mentally by primary school students. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.

10 Practical Tips for Making Fractions Come Alive and Make Sense 

Clarke, D; Roche, A; Mitchell A (2008)

Sharing advice and research on what is important and less important, and practical classroom approaches and activities

A Canadian effort to address fractions teaching and learning challenges 

Shelley Yearley and Catherine Bruce (2014)

This article explore the teaching and learning of fraction concepts and common misconceptions. In this article, the authors describe a fractions-based research project conducted in Ontario, Canada.

Decimats: Helping Students To Make Sense Of Decimal Place Value 

Anne Roche, Australian Catholic University (2010)

This article introduces "decimats" and describes how they can be used to make sense of decimal size and decimal place value.

Detection and Remediation of Decimal Misconceptions 

Vicki Steinle (2004)

Research and suggestions for teaching to overcome decimal misconceptions

Longer is Larger - Or is it? 

Anne Roche, Australian Catholic University (2005)

The author explores research about students' misconceptions of decimal notation that indicates that many students treat decimals as another whole number to the right of the decimal point and considers the implications for classroom teachers. 

Decipipes: Helping Students to "Get the Point" 

Bruce Moody (2011)

Decipipes are a representational model that can be used to help students develop conceptual understanding of decimal place value. They provide a non-standard tool for representing length, which in turn can be represented using conventional decimal notation. This article reviews theory around why student intuitions concerning decimals are resistant to change and presents the use of the Decipipes equipment as one way of helping students "get the point".

**NEW** Unhelpful, false or misleading analogies - Decimals 

University of Melbourne

"Some children misinterpret decimals by making false analogies with other ideas. Common sources of confusion are with money, sport and reminders in division. " This webpage explores the impact of these analogies further.