Maths and complex learning
Professor Lyn English has coordinated mathematics curriculum subjects in Queensland University of Technology's undergraduate BEd and graduate BEd and Diploma courses, and is internationally recognised for her extensive research and publications in mathematics education. This article is adapted from the author's paper Promoting student understanding through complex learning, which was presented at the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, October 2010.
The first recommendation of the US National Academies’ Rising above the Gathering Storm (2007) was to initiate a vast improvement in K-12 science and mathematics education. Likewise the Australian Academy of Science (2006) has indicated the need to address the ‘critical nature’ of the mathematical sciences in schools and universities.
Complexity refers here to the study of systems of interconnected components whose behaviour cannot be explained solely by the properties of their individual parts. Work with complex systems may involve constructing, describing, explaining and manipulating them, or predicting their behaviour over time. It may also mean working on multi-phase and multi-component projects, or adapting rapidly to ever-evolving conceptual tools, artefacts and resources (Gainsburg, 2006; Lesh & Doerr, 2003; Lesh & Zawojewski, 2007).
Complex problems tend to span disciplines and industries, and the mathematical sciences underpin many areas of society including financial services, the arts, humanities, and social sciences. The Australian Academy has emphasized the importance of such interdisciplinary research.
The rise of complex systems is also associated with new technologies, which have led to significant changes in mathematical thinking. For example, technology can ease the thinking needed in information storage, retrieval, representation, and transformation, but demands more complex thinking to interpret data and communicate results. (Hamilton, 2007; Lesh, 2007a; Lesh, Middleton, Caylor & Gupta, 2008). Digital technologies have changed the world of work, and the way mathematics is being used in workplace settings.
The new technologies demand new mathematical competencies while eliminating other mathematical skills that were once part of the worker's toolkit. Some of the core competencies needed at work include problem solving in collaborative environments; the ability to work with and transform complex data sets; critical thinking, including the ability to distinguish reliable from unreliable information sources; synthesising, engaging in research and effective dissemination of results; and flexibility in working across disciplines.
The employees who are most in demand in the mathematical sciences are those that can interpret and work effectively with complex systems; function efficiently and communicate meaningfully within diverse teams of specialists; plan, monitor, and assess progress within complex, multi-stage projects; and adapt quickly to continually developing technologies (Lesh, 2008). Research indicates that such employees draw effectively on interdisciplinary knowledge in solving problems and communicating their findings.
Although such employees draw upon their school learning, they often need to rework it in a flexible and creative manner. They have to generate or reconstruct mathematical knowledge to suit the problem situation in ways that they did not experience in school (Gainsburg 2006; Hamilton 2007; Zawojewski, Hjalmarson, Bowman & Lesh, 2008).
Today’s students are well equipped to deal with these challenges, thanks to their exposure to current technology and the ways of thinking that accompany it. However, schools do not always draw on it as well as they might. In fact, students’ mathematical curiosity and talent appear to wane as they progress through school (National Research Council, 2005; Curious Minds, 2008). We need to explore effective ways of fashioning learning environments and experiences that challenge and advance students’ mathematical reasoning.
The need to rethink students’ mathematical learning experiences that we provide to students applies particularly to the activities classified as ‘problem solving’. This has long been treated as an isolated topic, with problem-solving abilities assumed to develop through the initial learning of basic concepts and procedures that are then practised in solving word (‘story’) problems. In solving such word problems, students generally engage in a one- or two-step process of mapping problem information onto arithmetical quantities and operations. These traditional word problems tend to restrict problem solving to artificial and contrived contexts (Hamilton, 2007).
More opportunities are needed for students to generate important concepts and processes in their own mathematical learning as they solve thought-provoking, authentic problems. Recent research has argued for students to be exposed to learning situations in which they are not given all of the required mathematical tools, but rather are required to create their own versions of the tools as they determine what is needed (eg English & Sriraman, 2010; Hamilton, 2007; Lesh, Hamilton, & Kaput, 2007). Unfortunately, such opportunities appear scarce in many classrooms, despite repeated calls over the years to engage students in tasks that promote high-level mathematical thinking and reasoning (eg Henningsen & Stein, 1997; Silver et al., 2009; Stein & Lane, 1996).
One approach to promoting complex learning through intellectually challenging tasks is mathematical modelling. Models can be understood as ‘systems of elements, operations, relationships, and rules that can be used to describe, explain, or predict the behaviour of some other familiar system’ (Doerr & English, 2003, p.112). From this perspective, modelling problems are realistically complex situations where the problem solver engages in mathematical thinking beyond the usual school experience and where the products to be generated often include complex artefacts or conceptual tools that are needed for some purpose or to accomplish some goal (Lesh & Zawojewski, 2007).
In one such activity, the Water Shortage Problem, two classes of 11-year-old students in Cyprus were presented with an interdisciplinary modelling activity that was set within an engineering context (English & Mousoulides, in press). Students are given background information on the water shortage in Cyprus and sent a letter from a client, the Ministry of Transportation, who needs a means of (model for) selecting a country that can supply Cyprus with water during summer. The letter asks students to develop such a model using the data given, as well as the web. The quantitative and qualitative data provided for each country include water supply per week, water price, tanker capacity, and port facilities. Students can also obtain data from the web about distance between countries, major ports in each country, and tanker oil consumption. After students have developed their model, they write a letter to the client detailing how their model selects the best country for supplying water. An extension of this problem gives students the opportunity to review their model and apply it to an expanded set of data. That is, students receive a second letter from the client which includes data for two more countries and are asked to test their model on the expanded data and improve their model, if needed.
Modelling problems of this nature provide students with repeated opportunities to express, test and refine or revise their current ways of thinking as they endeavour to create a structurally significant product—structural in the sense of generating powerful mathematical (and scientific) constructs. The problems are designed so that multiple solutions of varying mathematical and scientific sophistication are possible and students with a range of personal experiences and knowledge can participate. The products students create are documented, shareable, reusable and modifiable models that provide teachers with a window into their students’ conceptual understanding. Furthermore, these modelling problems build oral and written communication and teamwork skills.
We need to make the mathematical experiences we include for our students more challenging, authentic and meaningful. Developing students’ abilities to work creatively with, and generate, mathematical knowledge, as distinct from working creatively on tasks that simply provide the required knowledge, is especially important in preparing our students for success in a knowledge-based economy.
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English, L. D. & Mousoulides, N. (In press). 'Engineering-based modelling experiences in the elementary classroom'. In M. S. Khine, & I. M. Saleh (Eds.), Dynamic Modeling: Cognitive Tool for Scientific Enquiry. Springer.
English, L. D., & Sriraman, B. (2010). 'Problem solving for the 21st century'. In B. Sriraman & L. D. English (Eds.), Theories of Mathematics Education: Seeking New Frontiers (pp 263-285). Advances in Mathematics Education, Series: Springer.
Gainsburg, J. (2006). 'The mathematical modeling of structural engineers'. In Mathematical Thinking and Learning, 8(1), 3-36.
Henningsen, M., & Stein, M. K. (1997). 'Mathematical task and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning'. Journal for Research in Mathematics Education, 29, 514-549.
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Lesh, R. (2007b). 'What changes are occurring in the kind of elementary-but-powerfulmathematics concepts that provide new foundations for the future?' In R. Lesh, E.Hamilton, & J. Kaput (Eds.) Foundations for the Future in Mathematics Education (pp. 155-160). Mahwah, NJ: Lawrence Erlbaum.
Lesh, R., & Doerr, H. (2003). 'Foundation of a models and modeling perspective on mathematics teaching and learning'. In R. A. Lesh & H. Doerr (Eds.), Beyond Constructivism: A Models and Modeling Perspective on Mathematics Teaching, Learning, and Problem Solving (pp. 9-34). Mahwah, NJ: Erlbaum.
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Lesh, R. & Zawojewski, J. S. (2007). 'Problem solving and modeling'. In F. Lester (Ed.). The Second Handbook of Research on Mathematics Teaching and Learning (pp. 763-804). Charlotte, NC: Information Age Publishing.
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The National Academies (2007). 'Rising above the storm: Energizing and employing America for a brighter economic future'. In www.national-academies.org (accessed 2/24/2010).
Zawojewski, J., Hjalmarson, J. S., Bowman, K., & Lesh, R. (2008). 'A modeling perspective on learning and teaching in engineering education'. In J. Zawojewski, H. Diefes-Dux, & K. Bowman (Eds.), Models and Modeling in Engineering Education: Designing Experiences for All Students (pp. 1-16). Rotterdam: Sense Publications
Key Learning AreasMathematics
Subject HeadingsInterdisciplinary learning