Promoting structural thinking in the early mathematics curriculum
Joanne Mulligan is Associate Professor of Mathematics Education at Macquarie University. She is Associate Director of the Centre for Research in Mathematics and Science Education (CRiMSE) and lectures in teacher education programs.
A number of major studies have suggested that, given appropriate opportunities, young children can develop algebraic thinking from the early years of schooling (Carraher, Schliemann, Brizuela & Earnest, 2006). The recognition of pattern and structure underlying algebraic thinking has also been shown to be important in studies of young children's mathematical reasoning and problem solving (English, 2004). This surge of new research has raised awareness among educators that young students can develop complex mathematical ideas much earlier than previously expected.
Pattern and structure lie at the very heart of school mathematics. A pattern may be seen as a numerical or spatial regularity, and the relationship between the various components of a pattern as its structure. The generalisations on which algebra is based are one type of pattern that children encounter in school mathematics. The underlying mathematical structures that children need to recognise include the base ten structure which underpins the number system. The early development of 'pattern and structure' in children's mathematical thinking is critical to ongoing development of abstract mathematical ideas and relationships.
Recent mathematics curriculum reform, both in Australia and internationally, reflects these new findings. The NSW mathematics K–6 syllabus, for example, has recently incorporated a 'Patterns and Algebra' strand from the first years of schooling, and a 'Structure' strand has also been developed in the Victorian mathematics syllabus. These advances in curriculum and research enable us to gain insight into the link between the development of early algebraic thinking and students' difficulties in learning mathematics, as well as the potential early algebraic learning may have for improving mathematics achievement generally.
The Macquarie Pattern and Structure Project
Over the past decade, Macquarie University researchers have been investigating early causes of difficulty in mathematics learning. A 2-year longitudinal study of students in Grades 2 to 3 showed that low achievers were more likely to produce poorly organised representations (drawings, diagrams, symbols) compared with others. These children also lacked flexibility in their thinking and were unable to replicate structured models such as groups, arrays or patterns. A follow-up study indicated that these difficulties persisted right up to Grade 5. High achievers, on the other hand, used abstract representations with highly developed structures that were already well in place in Grade 2 (Thomas, Mulligan & Goldin, 2002).
A subsequent study of 103 first-graders provided clear evidence that early school mathematics achievement was directly related to children’s underlying development of both mathematical pattern and structure (Mulligan, Mitchelmore & Prescott, 2006). Low-achievers’ mathematical ideas tended to be idiosyncratic, focused on particular features of special interest to them and often varied over time without showing any progress towards use of structure. These students did not seem to look for, or did not recognise, the underlying mathematical similarities between superficially different situations – a crucial step in effective mathematics learning.
On the basis of this evidence, we formulated a series of classroom-based initiatives to assist us in developing a 'pattern and structure' teaching program. A numeracy initiative conducted in one NSW metropolitan primary school trialled an early version of this program (Mulligan, Prescott, Papic & Mitchelmore, 2006). The 9-month project involved 683 low-achieving students aged from 5 to 12 years, 27 teachers, and three researchers, and comprised a Pattern and Structure Assessment (PASA) interview, and a teaching program.
Project activities focused on improving students' visual memory, ability to identify and apply patterns and ability to seek structure in mathematical ideas. Activities were regularly repeated in varied form to encourage generalisation. For example, some students learned that, in a 2 x 3 rectangular grid of squares, the squares are of equal size, they touch each other along their sides, there is the same number in each row and in each column, and the total number can be counted. The next lesson repeated the exercise with different sizes and shapes. Essentially, activities highlighted the 'spatial structure' of the numerical and measurement ideas.
After two school terms, there was a marked improvement in PASA scores, particularly in the early grades. Substantial improvements were also found in school-based and system-wide measures of mathematical achievement. Many students who had previously experienced learning difficulties made rapid, effective improvements in their perceptions of mathematical situations. For example, they had a better understanding of what to look for in mathematical representations, and why mathematics usually involved seeing some similarity or regularity.
The Pattern and Structure Mathematics Awareness Project (PASMAP) arising from this pilot is now being expanded and adapted to new contexts. A revised form of PASMAP, with increased attention to early algebraic reasoning, is currently being implemented with a Year 1 class in a one-year study. A revision of PASMAP assessment and teaching tasks for kindergarten students experiencing difficulties in mathematics learning is also being evaluated.
Implications for teaching, curriculum and assessment
A significant implication of this research is that, if low achievers have a poor awareness of pattern and structure, then their achievement may be improved if they are explicitly taught to recognise a variety of mathematical patterns and structures across mathematical strands. Another important implication for curriculum is that the development of spatial and measurement concepts may provide a complementary pathway to number learning in early mathematical development.
A number of new classroom-based studies are looking at ways that teachers can promote the development of structure and generalisation in children’s early mathematics learning. The issue of effective professional development for teachers must be integral to this work. One of the questions the research raises is how teachers can develop a deeper understanding of why pattern and structure is critical to early mathematical thinking, and overcome traditional perceptions that algebra is the exclusive domain of secondary school mathematics.
Recent Commonwealth and State government numeracy initiatives have successfully encouraged teachers to increase their professional knowledge about how children develop mathematical concepts and strategies. Programs such as Count Me In Too (NSW Department of Education and Training, 2002) and the Early Numeracy Research Project (Cheeseman & Clarke, 2005) have demonstrated that interview-based assessment linked to a framework of mathematical strategies enables teachers to scaffold learning based on an individual's level. Our research complements and extends these programs, and supports the direction of the NSW State Numeracy Plan 2006–2008 (Tebbutt, 2006).
Our research aims to provide reliable evidence that can readily inform teaching and learning programs, to better understand why there is still a gap between those students who develop mathematical knowledge and skills easily and those who do not. We approach the problem of 'closing the gap' between achievers and non-achievers by describing not merely a 'gap', but mathematics' learning of a different dimension. At the very least we have discovered that we need to focus on something quite different from traditional approaches.
This discussion highlights the importance of focusing on the general mathematical processes that underpin effective learning. For many reasons, some children appear to focus on aspects that may be meaningful to them in their own way, but are not conducive to formal mathematical development. As educators we may not always be aware of this, or aware that individuals are interpreting school-based mathematical ideas differently to what we might expect. Other students easily "see" mathematical structure and seek to use it effectively.
Attention needs to be refocused on essential elements of mathematical pattern and structure, such as counting in patterns or drawing a grid from memory. In this way, we can assist students to make better sense of what may seem a confusing and artificial system of symbols and language. Once better communication is established about what is important, other strategies to build accuracy and further mathematical knowledge can be more readily established, improving access to effective mathematical development for all students.
Board of Studies New South Wales 2002, Mathematics K–6 Syllabus 2002, Board of Studies, Sydney, NSW.
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English, LD 2004, 'Promoting the development of young children’s mathematical and analogical reasoning' in LD English (Ed), Mathematical and Analogical Reasoning of Young Learners (pp 210–215), Lawrence Erlbaum, Mahwah, NJ, USA.
Mulligan, JT, Mitchelmore, MC & Prescott, A 2006, 'Integrating concepts and processes in early mathematics: The Australian Pattern and Structure Mathematics Awareness Project (PASMAP)', Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education, Prague, July 2006.
Mulligan, JT, Prescott, A, Papic, M & Mitchelmore, MC 2006, 'Improving early numeracy through a pattern and structure mathematics awareness program (PASMAP)', Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia, Canberra, MERGA.
NSW Department of Education and Training 2002, Count Me in Too: A professional development package, NSW Department of Education and Training Curriculum Directorate, Sydney.
Tebbutt, C 2006, Numeracy Plan 2006–2008, New South Wales, Curriculum Leadership article, volume 4, issue 8. http://www.curriculum.edu.au/leader/default.asp?id=13489 (Retrieved 16/7/2006)
Thomas, N, Mulligan, JT & Goldin, GA 2002, 'Children's representations and cognitive structural development of the counting sequence 1–100', Journal of Mathematical Behavior, 21, 117–133
Key Learning AreasMathematics
Subject HeadingsThought and thinking