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BSRLM discussion March 12th 2011 Institute of Education What’s the same and what’s different? A=1/2bh v2- u2 = 2as y = 2x+5 3x - 5=9 - 2x One variable Two variables More variables Interval data One-to-one or one-to-many mappings between sets Input/output machines with algebraic workings Input/output ‘black boxes’ such as trigonometric or exponential functions Expressions to calculate y-values from given x-values Relations between particular x-values and y-values Relations between a domain of x-values and a range of yvalues Representations of relations between variables in ‘realistic’ situations Graphical objects which depict particular values Graphical objects which have particular characteristics Graphical objects which can be transformed by scaling, translating and so on Structures of variables defined by parameters and relations unknown situations use numbers in number sentences like x + y = 6, or x – y = 4 of formulae: areas of simple shapes; conversions of units. pictorial representations bar charts trend lines experimental data against time discussed in terms of steepness, higher/lower distinctions, and zeroes scaling the vertical axis scaling the horizontal axis. term-to-term rules for simple sequences one-to-one, many-to-one, and one-tomany mappings qualitative relations between variables in realistic phenomena Discuss ... f as a label f(y) and f(b) as different functions f(x) as the formula for a function f(3) means the function has value 3 f(x+y) = f(x) + f(y) f(y) is the ordinate f(x) = g(x) is an instruction to find an unknown f(x) is a graph f(x) means ‘f times x’ Car hire with a $100 cost per day and $5 cost per mile f(n)= n. 100 + x * 5 f(n) +f(x) = n * 100 + x. 5 f(n, b) = (n . 100) + (b. 5) (n .100) + (k.5) =f(n, k) f(d, k) = n(d) . 100 + n(k) * 5 Methods which retain the notion of equality of expressions seem particularly powerful, and relate well to understanding the properties of functions. Graphs and graphing both realistic and algebraic data: problems of conceptual understanding rather than technical problems. Swan and colleagues: cognitive conflict Ainley, Pratt and colleagues: purpose, utility, meaning. Understanding, both in pure mathematics and as tools for modelling, takes many years to develop. • continuous and discrete; • with and without time in the x-axis; • smooth and non-smooth • calculable and non-calculable • do/do not depict the underlying relations in obvious ways Consistent use of multiple representation software over time can, with appropriate tasks and pedagogy, enable students to understand the concepts and properties associated with functions, and be able to use functions to model real and algebraic data. anne.watson@education.ox.ac.uk