Joanne Mulligan

Young Children's Intuitive Models of Multiplication and Division

In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children’s intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it.

Awareness of Pattern and Structure in Early Mathematical Development

Recent educational research has turned increasing attention to the structural development of young students’ mathematical thinking. Early algebra, multiplicative reasoning, and spatial structuring are three areas central to this research. There is increasing evidence that an awareness of mathematical structure is crucial to mathematical competence among young children. The purpose of this paper is to propose a new construct, Awareness of Mathematical Pattern and Structure (AMPS), which generalises across mathematical concepts, can be reliably measured, and is correlated with general mathematical understanding. We provide supporting evidence drawn from a study of 103 Grade 1 students.

Children’s representation and structural development of the counting sequence 1–100

Abstract In an exploratory study, we interviewed 172 children from Grades K to 6, and an additional 92 high ability children from Grades 3 to 6, seeking to infer aspects of their internal imagistic representations from their drawings and explanations of the numbers 1–100. We interpret our observations with respect to developing theoretical models for mathematical learning and problem solving, based on characteristics of internal systems of representation. Focusing on children’s understandings of the conventional base ten system of numeration, we explore how internal representational systems for numbers may change through a period of structural development, to become eventually powerful, autonomous systems.

Reconceptualizing early mathematics learning

Introduction: Lyn English and Joanne Mulligan: Perspectives on Reconceptualising Early Mathematics Learning.- Chapter 1: Kristie J. Newton and Patricia A. Alexander: Early Mathematics Learning in Perspective: Eras and Forces of Change.- Chapter 2: Joanne Mulligan and Michael Mitchelmore: Early Awareness of Mathematical Pattern and Structure.- Chapter 3: Joanne Mulligan, Lyn English, Michael Mitchelmore, and Nathan Crevensten: Reconceptualising Early Mathematics Learning: The Fundamental Role of Pattern and Structure.- Chapter 4: Lyn English: Reconceptualising Statistical Learning in the Early Years.- Chapter 5: Herb Ginsburg: Cognitive guidelines for the design and evaluation of early mathematics software:The example of MathemAntics.- Chapter 6: Doug Clements and Julie Sarama: Rethinking Early mathematics: What Is Research-Based Curriculum for Young Children?.- Chapter 7: Bob Perry and Sue Dockett: Reflecting on Young Childrens Mathematics Learning.- Chapter 8: Anita Wager: Practices that Support Mathematics Learning in A Play-based Classroom.- Chapter 9: Bert van Oers: Communicating about Number: Fostering Young Childrens Mathematical Orientation in the World.- Chapter 10: Kristy Goodwin and Kate Highfield: A Framework for examining technologies and early mathematics learning.- Chapter 11: Marja van den Heuvel-Panhuizen and Iliada Elia: The role of picture books in young childrens mathematical learning.- Chapter 12: Marina Papic: Improving numeracy outcomes for young Australian Indigenous children.- Chapter 13: Elizabeth Warren and Jodie Miller: Enhancing teacher professional development for early years mathematics teachers working in disadvantaged contexts.- Chapter 14: Heidi A. Diefes-Dux, Lindsay Whittenberg, and Roxanne McKee Mathematical modeling at the intersection of elementary mathematics, art, and engineering education.

Children’s solutions to multiplication and division word problems: A longitudinal study

AbstractChildren’s solution strategies to a variety of multiplication and division word problems were analysed at four interview stages in a 2-year longitudinal study. The study followed 70 children from Year 2 into Year 3, from the time where they had received no formal instruction in multiplication and division to the stage where they were being taught basic multiplication facts. Ten problem structures, five for multiplication and five for division, were classified on the basis of differences in semantic structure. The relationship between problem condition (i.e. small or large number combinations and use of physical objects or pictures), on performance and strategy use was also examined.The results indicated that 75% of the children were able to solve the problems using a wide variety of strategies even though they had not received formal instruction in multiplication or division for most of the 2 year period. Performance level generally increased for each interview stage, but few differences were found between multiplication and division problems except for Cartesian and Factor problems.Solution strategies were classified for both multiplication and division problems at three levels:(i)direct modelling with counting;(ii)no direct modelling, with counting, additive or subtractive strategies;(iii)use of known or derived facts (addition, multiplication). A wide range of counting strategies were classified as counting-all, skip counting and double counting. Analysis of intuitive models revealed preference for a repeated addition model for multiplication, and a ‘building-up’ model for division.

Early Awareness of Mathematical Pattern and Structure

This chapter provides an overview of the Australian Pattern and Structure Project, which aims to provide new insights into how young students can abstract and generalize mathematical ideas much earlier, and in more complex ways, than previously considered. A suite of studies with 4- to 8-year old students has shown that an awareness of mathematical pattern and structure is both critical and salient to mathematical development among young students. We provide a rationale for the construct, Awareness of Mathematical Pattern and Structure (AMPS), which our studies have shown generalizes across early mathematical concepts, can be reliably measured, and is correlated with mathematical understanding. A study of Grade 1 students and follow up case studies enabled us to reliably classify structural development in terms of a five structural levels. Using a Pattern and Structure Assessment (PASA) interview involving 39 tasks, students identified, visualized, represented, or replicated elements of pattern and structure. Students with high AMPS are likely to have a better understanding of Big Ideas in mathematics than those with low AMPS. They are likely to look for, remember and apply spatial and numerical generalizations and in particular are likely to grasp the multiplicative relationships that underlie the majority of the concepts in the elementary mathematics curriculum.

A Developmental Multimodal Model for Multiplication and Division.

This paper presents an analysis of young students’ development of multiplication and division concepts based on a multimodal SOLO model. The analysis is drawn from two sources of data: a two-year longitudinal study of 70 Grade 2 to 3 students’ solutions to 24 multiplicative word problems, and examples from a problem-centred teaching project with Grade 3 students. An increasingly complex range of counting, additive, and multiplicative strategies based on an equal-grouping structure demonstrated conceptual growth through ikonic and concrete symbolic modes. The solutions employed by students to solve any particular problem reflected the mathematical structure they imposed on it. A SOLO developmental model for multiplication and division is described in terms of developing structure and associated counting and calculation strategies.

Reconceptualizing Early Mathematics Learning: The Fundamental Role of Pattern and Structure

The Pattern and Structure Mathematics Awareness Program (PASMAP) was developed concurrently with the studies of AMPS and the development of the Pattern and Structure Assessment (PASA) interview. We summarize some early classroom-based teaching studies and describe the PASMAP that resulted. A large-scale two-year longitudinal study, Reconceptualizing Early Mathematics Learning (REML) resulted. We provide an overview of the REML study and discuss the consequences for our view of early mathematics learning.

Factors influencing Filipino children’s solutions to addition and subtraction word problems

Young Filipino children are expected to solve mathematical word problems in English, which is not their mother tongue. Because of this, it is often assumed that Filipino children have difficulties in solving problems because they cannot read or comprehend what they have read. This study tested this assumption by determining whether presenting word problems in Filipino or reading them aloud to children in either language facilitated solution accuracy. Contrary to the initial hypothesis, reading word problems aloud did not seem to improve student performance (p > 0.10). In contrast, presenting word problems in Filipino significantly improved solution accuracy (p < 0.0001) and led to differences in error patterns – children were less likely to use an inappropriate arithmetic operation when problems were presented in Filipino. However, the language of the problem had minimal effects on the more difficult Compare problem type. Finally, the benefits of using Filipino were more pronounced for low‐achieving students who may have lower proficiency in English than their high‐achieving peers (p < 0.01).

Dynamic imagery in children’s representations of number

An exploratory study of77 high ability Grade 5 and 6 children investigated links between their understanding of the numeration system and their representations of the counting sequence 1–100. Analysis of children’s explanations, and pictorial and notational recordings of the numbers 1–100 revealed three dimensions of external representation—pictorial, ikonic, or notational characteristics—thus providing evidence of creative structural development of the number system, and evidence for the static or dynamic nature of the internal representation. Our observations indicated that children used a wide variety of internal images of which about 30% were dynamic internal representations. Children with a high level of understanding of the numeration system showed evidence of both structure and dynamic imagery in their representations.