MATHEMATICS AND ALGORITHMIC THINKING

Unit Overview: Understanding Behaviour through Simulation

Summary of Learning Goals As part of the Special Topic Mathematics and Algorithmic Thinking, this unit demonstrates simulation as an extension of, and complement to, traditional mathematical methods for the investigation of complex real world phenomena. Simulation can help us to understand real-world scenarios that seem to be beyond our mathematical grasp. This unit shows why and how simulation complements more traditional mathematical methods of investigation and investigates the comparative advantages and disadvantages of simulation-based approaches. The emphasis is on fundamental questions in biology and sociology, where simulation approaches have enabled significant scientific discoveries by shifting the perspective of the investigation. Students work hands-on with interactive simulations and in this way learn the role of virtual experiments, gaining exposure to computational methods as the third pillar of science. Along the way, they learn the basics of some fundamental simulation methods, including Monte-Carlo simulations, agent-based simulation, individual-based and population-based simulation. Another goal is to expose students to problems that incorporate randomness and unpredictability, i.e. to stochastic models. This is very important, since almost every real-world problem contains random influences, and often these have a crucial influence on the situation. Students learn how such problems can be addressed with Monte-Carlo simulations (randomised simulations) and through systematic investigations of the influence of changing parameters. They learn a number of basic principles, most importantly the law of large numbers, and gain hands- on experience with the implications. Australian Curriculum: Mathematics (Years 9 and 10) ACMSP225: List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (Year 9) ACMSP226: Calculate relative frequencies from given or collected data to estimate probabilities of events involving 'and' or 'or’. (Year 9) ACMSP246: Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence. (Year 10) ACMSP247: Use the language of ‘if .... then, ‘given’, ‘of’, ‘knowing that’ to investigate conditional statements and identify common mistakes in interpreting such language. (Year 10) ACMSP278: Calculate and interpret the mean and standard deviation of data and use these to compare data sets. (Year 10A) ACMSP251: Use scatter plots to investigate and comment on relationships between two numerical variables (Year 10) ACMSP252: Investigate and describe bivariate numerical data where the independent variable is time (Year 10)

Australian Curriculum: Digital Technologies (Years 9 and 10) ACTDIP037: Analyse and visualise data to create information and address complex problems and model processes, entities, and their relationships using structured data. ACTDIP041: Implement modular programs, applying selected algorithms and data structures including using an object-oriented programming language.

Summary of lessons

Who is this unit for? This unit is targeted mainly at students in Years 10 and above. No prior mathematical knowledge is assumed beyond the ability to construct and interpret line graphs and scatter plots and a basic understanding of probability and means. No prior coding knowledge is assumed. The student notebooks have been written to be entirely self- contained, allowing them to be used for targeted differentiation. Nearly all students will benefit from regular interaction with the teacher. The presentation and language used assumes a good level of literacy. There are many opportunities for extension, especially for students who have python coding skills. It is recommended to start with Lessons 1 and 2, then Lessons 3, 4, and 5 stand by themselves. Lessons 6 and 7 independently rely on Lesson 5. “Algorithmics Lesson X - Using Jupyter Notebooks” can be used if students have not used these notebooks before. Lesson 1: Why simulate? This short lesson introduces students to the concept of simulation and the motivations for running simulations, both physical and computational. The lesson draws on many examples, including flight simulators, wave tanks, an inventive physical model of the forces on the roof of Sagrada Familia cathedral in Spain, and computational modelling of molecules. Lesson 2: Monty Hall Puzzle Students investigate the solution to a classic probability puzzle, the Monty Hall problem. They run simulations to arrive at the counter-intuitive conclusion. Two pre-made programs are used for simulation: a spreadsheet and a Python program. Plotting the results shows that simulation findings are highly variable for a small number of games, so many games are required to get a reliable result. The mathematics of the problem is then explored.

Lesson 3: Schelling’s Model of Social Segregation Students explore Schelling’s model of segregation, which reveals how social segregation arises from individual behaviours, even when people want to live in diverse communities. They begin with hands-on simulation of Schelling’s model, moving coins on a grid to see how neighbourhoods change over time. Subsequently, students use an applet for the simulation, investigate the effect of parameters, and modify measures of segregation and of happiness.

Lesson 4: Optimal Stopping Students play a simple random number game that requires a strategy for when to stop, initially relying on their intuition to suggest a best approach. Subsequently, they simulate a similar ‘game’ relating to making weekend plans. They consider how the ideas that develop from the simulation can be applied to a wide range of real world contexts. Optionally, they simulate the famous Feynman restaurant problem to decide how to maximise deliciousness when making choices at a restaurant.

Lesson 5: Competitive Behaviour This lesson introduces the basic aspects of game theory, an important mathematical field with applications in many contexts. Students devise and specify a consistent strategy for paper-scissors-rock and try it against an opponent. They play Morra, another simple two-player game, and learn to write down the payoff matrix for such games. Students use payoff matrices to find good and bad strategies, and then use simulation to find optimal solutions.

Lesson 6: Evolutionary Game Theory This lesson extends the previous lesson by applying game theory to situations where behaviour evolves rather than the consequence of the choices of a few rational players. Students use payoff matrices to describe a competition between two animals for a single resource, and through simulation analyse how the payoffs influence the development of traits in an animal population. Students perform simulations demonstrating the development of animal and human populations, and investigate how changing assumptions influences the results.

Lesson 7: Mixed Strategy This lesson goes ‘behind the scenes’ to show how to make an Excel simulation related to Lesson 5. Students complete a partially prepared Excel sheet (e.g. by making a cell to choose a strategy randomly according to a given probability) and then use the model to investigate two-player, zero-sum games. Students can change the payoff matrix to test various mixed strategies. An optional extension shows how to do this and more with Python.

2 Reflection on this sequence

Rationale These activities set out a “journey of discovery”. They start by introducing a problem that despite its simple statement seems difficult or even impossible to solve. In some cases, this is because the relevant mathematics is too hard for the target students, and in some cases does not exist. In other cases, arguably the most interesting ones, the problem seems has an intuitive solution, but this turns out to be wrong. The lessons then introduce simulation models that allow students to approach the problem as a virtual experiment and solve it with an experimental approach. Where possible, the mathematical solution is discussed and contrasted with the computational approach. The topic emphasizes the complementarity of computational and mathematical approaches. Students learn that many problems that are not mathematically tractable can be solved with simulation models, but they also learn about the limitations of simulation models compared to a full mathematical treatment. Simulation is a very broad field. The lessons focus on four independent case studies which are technically accessible, yet its topic had a major scientific impact. In a few cases, the scientists who developed the corresponding field have received Nobel Prizes for their work. This should assist in conveying the importance of the topics. To this end, the lessons also introduce the central characters behind this work. Despite the conceptual complexity of the topic, the lessons should be engaging and entertaining, since they are built on an active learning approach, including exploration, discussion, role-play, and experiments and often involve an exciting element of surprise. reSolve Mathematics is Purposeful This unit is driven by the study of important real-world problems. Computational techniques are studied as tools to solve fundamental, challenging questions in biology and sociology. Students begin to comprehend that computation is not just what enables the technologies around us, but that computational thinking provides a different conceptual framework to interpret and explain our world. This approach can lead to a deeper understanding across a surprising range of subjects, including Biology, Medicine, History, Economics, Sociology, Politics and others. reSolve Tasks are Inclusive and Challenging The premise of this unit is that simulation provides an accessible way to study challenging problems that are mathematically complex. Simulation is used as an entry point to the study of problems that are out of the mathematical reach of Year 10 students or indeed entirely out of the reach of current mathematics. The units introduce computation methods in mathematics and scientific investigation requiring no previous exposure to coding. Students use computational tools, but they do not need to construct them. This is achieved by providing custom functions for students to use, removing any need to memorise (or even understand) syntax. For interested students, the code for all custom functions is available and editable through the Jupyter notebooks, allowing significant extension and enrichment opportunities. As the unit progresses, students are allowed increasing freedom in their inquiries and the tools and methods covered begin to stretch beyond the standard Australian Mathematics Curriculum. reSolve Classrooms Have a Knowledge Building Culture The unit follows an inquiry-based approach by students that leads students to actively explore authentic and meaningful real world data using simulation and visualisation. Making mistakes and learning from these is positively encouraged. Students are invited to form and investigate their own hypotheses about real world dependencies. These will often be incorrect and students learn that making mistakes is an integral and necessary part of any exploratory approach. They will experience this first hand whenever their initial hypotheses are incorrect or incomplete, in which case they will learn to invalidate their own assumptions and to revise and refine their thinking incrementally from evidence.

Further Information The Teacher’s Guide to ST6 Mathematics and Algorithmic Thinking contains further information, further reading and acknowledgements.

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