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Mathematical Modelling and Mathematical Education – What, why and how? Dr Max Stephens Graduate School of Education THE UNIVERSITY OF MELBOURNE [email protected] The world of work in the 21st century In her plenary at ICTMA 15, Lyn English identified competencies that are now seen as important for productive and innovative work practices (English, Jones, Bartolini, Bussi, Lesh, Tirosh, & Sriraman, 2008). Her list included: Problem solving, including working collaboratively on complex problems where planning, overseeing, moderating, and communicating are essential elements for success; Applying numerical and algebraic reasoning in an efficient, flexible, and creative manner; Generating, analysing, operating on, and transforming complex data sets; Applying an understanding of core ideas from ratio and proportion, probability, rate, change, accumulation, continuity, and limit; The world of work in the 21st century Constructing, describing, explaining, manipulating, and predicting complex systems; Thinking critically and being able to make sound judgments, including being able to distinguish reliable from unreliable information sources; Synthesizing, where an extended argument is followed across multiple modalities; Engaging in research activity involving the investigation, discovery, and dissemination of pertinent information in a credible manner; Flexibility in working across disciplines to generate innovative and effective solutions; Techno-mathematical literacy (“where the mathematics is expressed through technological artefacts” Hoyles, Wolf, Molyneux-Hodgson, & Kent, 2010, p. 14). Implications for schools and schooling These changes to the world beyond school cause us to reconsider what we ask children to learn in school Human resource development requires learning to become more future oriented, interdisciplinary, involving problem solving and modelling that mirror similar experiences beyond school More powerful links are needed between classrooms and the real world where students can apply their mathematics to solve authentic problems Students for the st 21 century “I think the next century will be the century of complexity.”----- Stephen Hawking (2000) We need to develop students who are: Knowledge builders Complex, multifaceted and flexible thinkers Creative and innovative problem solvers Effective collaborators and communicators Optimistic and committed learners Mathematical modelling Modelling is a powerful vehicle for not only promoting students’ understanding of a wide range of key mathematical and scientific concepts, but also for helping them appreciate the potential of the mathematical sciences as a critical tool for analysing important issues in their lives, communities, and society in general (Greer, Verschaffel, & Mukhopadhyay, 2007) Importantly, modelling needs to be integrated within the primary school curriculum and not reserved for the secondary school years and beyond as it has been traditionally. Research has shown that primary school children are indeed capable of engaging in modelling (English & Watters, 2005) English, ICTMA 15, 2011 Mathematical modelling The terms, models and modelling, have been used variously in the literature, including … solving word problems, conducting mathematical simulations, creating representations of problem situations (including constructing explanations of natural phenomena), and creating internal, psychological representations while solving a particular problem (English & Halford, 1995; Gravemeijer, 1999; Lesh & Doerr, 2003) English, ICTMA 15, 2011 Mathematical modelling One perspective on models … is that of conceptual systems or tools comprising operations, rules and relationships that can describe, explain, construct, or modify … a complex series of experiences Modelling involves the crossing of disciplinary boundaries, with an emphasis on the structure of ideas, connected forms of knowledge, and the adaptation of complex ideas to new contexts (Hamilton, Lesh, Lester, & Brilleslyper, 2008) English, ICTMA 15, 2011 Mathematical modelling Modelling activities provide students with opportunities to repeatedly express, test, and refine or revise their current ways of thinking. Modelling problems need to be designed so that multiple solutions of varying mathematical sophistication are possible and such that students with a range of personal experiences and knowledge can participate In this way, the mathematical experiences of students become more challenging, authentic and meaningful English, ICTMA 15, 2011 Teacher’s role How does a teacher cultivate students’ thinking of modelling? Outside School School mathematics Modelling Modelling By scientists/experts Contrast By students Problem is familiar to them and they have clear reasons to solve a problem. They can observe a situation/phenomena for a long time. Abstraction is relative easy. These three points are quite different for students. They know the modelling process and have good modelling skills. Ikeda, ICTMA 15, 2011 Pedagogical aims of modelling As an objective Modelling for its own sake Teacher’s role How does the teacher cultivate students’ thinking about modelling? As a means to an end Mathematical knowledge construction Relation between modelling and mathematical knowledge construction Where to locate modelling in the teaching of mathematics? Ikeda, ICTMA 15, 2011 Teacher’s role How does teacher cultivate students’ thinking of modelling? Scientists and other experts Problem is familiar with them. They have clear reasons to solve a problem. In school, How about for students? Why do students solve a problem? Selecting Material, Setting a situation Ikeda, ICTMA 15, 2011 What is an appropriate modelling task? Galbraith (2007) Introducing real world modelling tasks (a) the importance of using models based on experience Further Questions Does the problem situation concern the surroundings of students at present, in the past or in the future? Is it relevant to most students or to a few students? Future Is it concerned with situations they will confront as citizens, as individuals or in their profession/vocation? Ikeda, ICTMA 15, 2011 Compare with PISA context categories What is an appropriate modelling task? Galbraith (2007) Introducing real world modelling tasks (b) motivation Two points To clarify the reason why someone had to solve the problem To set the appropriate situation so that students can accept the problem posed by someone else as their own problem Ikeda, ICTMA 15, 2011 (b) motivation Clarify why someone had to solve the problem in the first place Set an appropriate situation so that students can accept a problem as their own problem Observing or analyzing the phenomenon or action How can we win in a relay in school sports? (Osawa,2004) It is important to consider the order of runners, how to pass the baton, etc One of the issues: Focusing on the baton pass When does the next runner begin to run to get the baton from the previous runner, for the shortest baton pass time? Ikeda, ICTMA 15, 2011 Distilling essential mathematical structure in complex situations Abstract content can be only understood by connecting it with its concrete contents. Concrete activity is essential! A real world Mathematical world Students have limited experience to observe a real world situation/phenomena. Distilling essential structure is difficult for students Observation/Manipulation by using Concrete Model Ikeda, ICTMA 15, 2011 How can teacher make students realize how to control many variables to solve a real world problem? Generating relating variables Checking whether or not generated variables affect problem solving Is it possible to solve by using my acquired mathematics knowledge? Conflicting Situations Meaningful conflicting situations so that students can derive key ideas. Setting up assumptions as simple as possible at the beginning, after then modifying them into more general situation gradually. Ikeda, ICTMA 15, 2011 Communication on mirror problem Formulating a real problem What minimum size of mirror do you need in order to see all your face? (Shimada,1990; Matsumoto, 2000; Ikeda,2004) Is the following sentence true or false? Half size of mirror is needed at least in order to see my whole face It might be true because it seems to be half by drawing a figure. It might be false because if the mirror is far from my face, it is sufficient to use small mirror. Let’s draw a figure to check their answer. Ikeda, ICTMA 15, 2011 Communication on mirror problem Please draw a figure on the blackboard. How can we treat these variables? Setting Assumptions How about the width of the face? Are three points, namely the point of the eye, the point of head and the point of chin, on a same line? Are the two planes, namely face and mirror, parallel or not? Is the eye located at the midpoint between the point of head and the point of chin? Ikeda, ICTMA 15, 2011 Communication on mirror problem Are the two planes, namely face and mirror, parallel or not? It seems easy to solve the problem if the relation of the two planes is parallel. However, the relation of two planes is not always parallel in a real situation. Conflicting Situations If the relation of two planes is not parallel, it is too difficult to solve the problem. Let’s set up an assumption that the relation of two planes is parallel at first. Regarding the case of not parallel, let’s consider that later. Ikeda, ICTMA 15, 2011 side width When we see one ear with two eyes left ear left eye right eye right ear Mirror Size: Width between left eye and right ear Ikeda, ICTMA 15, 2011 Side width When we see one ear with one eye Mirror Size: Width between left eye and left ear Error elimination Is it OK in any situation? left ear left eye right eye right ear Invisible Assumption Width between two eyes is shorter than double of width between left eye and left ear Ikeda, ICTMA 15, 2011 Pedagogical aims of modelling As an objective Modelling for its own sake Teacher’s role How does the teacher cultivate students’ thinking about modelling? As a means to an end Mathematical knowledge construction Relation between modelling and mathematical knowledge construction Where to locate modelling in the teaching of mathematics? Ikeda, ICTMA 15, 2011 Thinking about the balance between modelling and constructing math knowledge Role 1 Build up the model to mathematize in order to solve real world problems Real world Mathematical world Role 2 Build up the model to test the validity of mathematical concepts Real world Mathematical world Clarifying From which world is the problem derived ? Ikeda, ICTMA 15, 2011 Spread Infectious Diseases – modelling a natural disaster for senior high school students Dr Max Stephens Graduate School of Education THE UNIVERSITY OF MELBOURNE [email protected] Infectious disease Movie Contagion (2011) – “Don’t speak to anyone. Don’t touch anyone!” reflects the media frenzy attaching to the perceived threat. Infectious disease media images Some have great potential to scare Emerging Infectious Diseases (EIDs) A more careful study of the web gives a less panicked view, and causes us to us some important questions Since 1940 more than 300 Emerging Infectious Diseases have been identified. However, most do not take off So we have to ask why some do and some don’t Emerging diseases go global Mark Woolhouse (2008) Centre for Infectious Diseases at the University of Edinburgh: Novel human infections continue to appear all over the world, but the risk is higher in some regions than others. Identification of emerging-disease 'hotspots' will help target surveillance work Nature 451, 898-899 (21 February 2008) Global trends in emerging infectious diseases Jones et al. Nature 451, 990-993 (21 February 2008) Modelling Spread of disease One Sunday evening, five people with infectious influenza arrive by plane in a large city of about 2 million people They then go to different parts of the city and so the disease begins to spread At first when a person becomes infected, the disease is latent/incubating and he/she shows no sign of the disease and cannot spread it Modelling Spread of disease About one week after first catching the disease the person becomes infectious and can spread the disease to other people The infectious phase also lasts for about one week. After this time the person is free from influenza, although he/she may catch it again at some later time Modelling Spread of disease Scientists are trying to model the spread of influenza. They make a simplifying assumption that the infection progresses in one week units That is, they assume that everyone who becomes infected does so on a Sunday evening, has a one week latent period, and then becomes infectious one week later, and is free of infection exactly one week after that Modelling Spread of disease People who are free of the disease are called “susceptibles” (= capable of catching it) The scientists also assume that the city population is large and so can be assumed to be constant for the duration of the disease. That is, they ignore births, deaths and any movements into or out of the city Modelling Spread of disease It’s very hard to follow these descriptions. A picture (Becker, 2009) shows the key stages: Modelling Spread of disease The scientists assume that each infectious person infects a fixed fraction f of the number of susceptibles, so that the number of infectious people at week n + 1 is: f × (number of susceptibles at week n) × (number of infectious at week n) and the number of susceptibles at week n+1 is: (number of susceptibles at week n) + (number of infectious at week n) – (the number of infectious at week n + 1) Modelling Spread of disease The modelling uses the variable ‘weeks’. This simplification ensures that at any time there are only susceptible people and infectious people. The model excludes people who are in a “latent” stage – i.e. infected but not infectious This allows the model to be investigated easily Modelling Spread of disease The number of infectious people and the number of susceptible people will be constant from week to week. Choosing values of f between 10-6 and 2 × 10-6 we can make a model showing how the number of infectious people changes from week to week Modelling Spread of disease Three equations connect In the number of infectious people in each week n and Sn the number of susceptible people at week n: In+1 = f × Sn × In Sn+1 = Sn + In f × Sn × In In + Sn = 2 × 106, eliminating Sn to give In+1 = f × [2 × 106 In ] × In Modelling spread of disease f = 10-6 means that each infectious person spreads the disease to 2 people in a week. It will be important to show how any limiting values are connected to the size of f and to the size of the population. For what values of f will there be a situation where the number of infectious people eventually oscillates between two values? Modelling Spread of disease How can simple technology help us to investigate In+1 = f × [2 × 106 In ] × In One accessible way for senior students is to use EXCEL to plot graphs for different values of f. The recursion relation cannot be investigated easily without technology. Graphs for different values of f Remember that f = 10-6 means that each infectious person spreads the disease to 2 people in a week The following four graphs show what happens when f = 0.1 × 10-6 , f = 0.5 × 10-6 , f = 0.8 × 10-6 , f = 1 × 10-6 The first two show low rates of infection: f = 0.5 × 10-6 means that only one person is infected by each infectious person in a week, this rate of infection is too low to spread the disease Graphs for different values of f y=0.1*10^-6*(2*10^6-x)*x 1 1 1 1 0 0 0 y=0.5*10^-6*(2*10^6-x)*x 5 5 y=0.1*10^-6*(2*10^6x)*x 5 y=0.5*10^-6*(2*10^6x)*x 5 5 0 5 10 15 5 20 0 10 20 30 f= y=10^-6*(2*10^6-x)*x y=0.8*10^-6*(2*10^6-x)*x 800000 600000 400000 y=0.8*10^-6*(2*10^6x)*x 200000 0 0 20 40 60 1200000 1000000 800000 600000 400000 200000 0 y=10^-6*(2*10^6-x)*x 0 10 20 30 Graphs for different values of f y=1.1*10^-6*(2*10^6-x)*x 1200000 1000000 800000 600000 400000 200000 0 1500000 y=1.1*10^-6*(2*10^6x)*x 1000000 y=1.4*10^-6*(2*10^6x)*x 500000 0 0 10 20 30 40 0 y=1.5*10^-6*(2*10^6-x)*x 1500000 1000000 -y=1.5*10^-6*(2*10^6 x)*x 500000 0 0 y=1.4*10^-6*(2*10^6-x)*x 20 40 60 20 40 60 An interesting feature appears for f = 1.5 × 10-6 , where the graph begins to oscillate. This occurs when the value for y at any week is equal to the value of y two weeks later. Graphs for different values of f y=1.9*10^-6*(2*10^6-x)*x y=1.6*10^-6*(2*10^6-x)*x 2000000 2000000 1500000 1500000 1000000 y=1.6*10^-6*(2*10^6x)*x 500000 1000000 -y=1.9*10^-6*(2*10^6 x)*x 500000 0 0 10 20 30 40 0 0 20 40 The oscillating feature which appears for f = 1.5 × 10-6 , appears to continue for f = 1.6 × 10-6 , and possibly (?) for f = 1.9 × 10-6. But do we know if it starts at f = 1.5 × 10-6 ? We need other technology to decide this. TI-Nspire CAS TE worksheet can answer this question. 60 Utilising CAS to investigate further Only after looking at the different graphs and the effect of different values of f does it make sense to use CAS technology to explore the mathematical relationships. This cannot be done by hand. And should not be. Yet a CAS solution to the equation provides a powerful finding that students can anticipate from their exploratory work using EXCEL Where p = 2 × 106 , f > 3/p = 1.5 × 10-6 Implications for teaching Modelling the spread of disease requires much more than traditional textbook resources To explore the mathematical relationships students need access to programs such as EXCEL CAS capacity is highly useful Web-based information is important for students to understand the context E-book formats integrate these different resources in ways that students and teachers can easily use. Concluding: What principles of curriculum design are important when considering modelling activities? How do they help us to think about the balance between mathematical modelling and mathematical education? Principles of curriculum design How will a modelling investigation help develop: Underpinning mathematical concepts and skills from across the discipline (numerical, spatial, graphic, statistical and algebraic) Mathematical thinking and strategies Appreciation of context Communicating to a wider audience Principles of curriculum design What tasks are suitable for modelling activities? Tasks that require information and resources that are not easily available in textbooks or single printed source Tasks that are extended in time Tasks that are interdisciplinary, crossing over and integrating several curriculum areas Tasks that link mathematics to the real world Principles of curriculum design A modelling investigation changes teacher’s roles: Students moving in different directions Technological fluency is not the same as mathematical fluency Ensuring that students understand and communicate the key mathematical ideas Clear criteria on mathematical performance, reasoning, and communication are needed Principles of curriculum design Teachers as designers: More than just using new technology Deciding what technology is mathematically appropriate for students and whether students are mathematically ready to use the technology Getting students to ask: what are we looking for; framing and repeatedly testing conjectures; justifying and communicating conclusions. Principles of curriculum design Teachers as designers: Writing tasks: investigative and problem-solving tasks are very different from textbook tasks Developing new mathematical skills (especially in graphing and data); representing and interpreting graphs and data displays Exploring data sets; cleaning up large data sets; sampling and Exploratory Data Analysis Principles of curriculum design Why spend time on a modelling investigation? Only if the opportunities and time invested in designing and using a modelling investigation: advance students’ mathematical knowledge build their mathematical capacity in ways that will inform their other school subjects, and build habits of inquiry that they can carry forward into their future study, life and work Real world <=>Mathematical world In this activity students use mathematical relationships that are partly new to them These relationships can be manipulated using technology using mathematical understanding and understanding of the phenomena Different mathematical behaviours can be produced by careful variation of key terms These variations are powerful because they can help explain real world phenomena Real world <=>Mathematical world These mathematical variations require careful selection and analysis by students This analysis depends on students being able to connect mathematical behaviours to the phenomena that they are trying to model These mathematical variations help to explain why some diseases take off while others gradually die out What is the balance between mathematical modelling and mathematical education? Role 1 Build up the model to mathematize in order to solve real world problems Real world Mathematical world Role 2 Build up the model to test the validity of mathematical concepts Real world Mathematical world Ikeda, ICTMA 15, 2011