Algebra; ratio; functions Nuffield Secondary School Mathematics BSRLM March 12 th 2011

Algebraic reasoning formulating, transforming and understanding unambiguous generalizations of numerical and spatial situations and relations; understanding and using symbolic mathematical models of situations to predict and explain; controlling as well as using and understanding and adapting spreadsheet, graphing, programming and database software (adapted from Mason and Sutherland)

What goes wrong emphasis on manipulations can lead to error- prone work and disaffection (e.g. conjoining) students need to understand if they are to progress in mathematics, control software, and apply algebraic reasoning algebra notation is ambiguous students often apply the last thing they learnt, or respond to visual appearance modern textbooks rarely draw attention to the structures, relations

Teaching methods Express relations among quantities Model phenomena Learn about expressions through substitution, ‘reading’, modelling Transform expressions to generate equivalent expressions Explore behaviour through CAS or virtual models

RPR The expert sense of ratio and proportion cannot be fully pinned down by definition

Division a component of multiplicative relationships the inverse of multiplication algorithms: chunking, multiplication facts, repeated subtraction, remainder, or decimals encapsulated in fraction notation (rational number) needs fractions and decimals partitioning (continuous or discrete) or unit quantities

What goes wrong four variables: a/b = c/d. an increase in a can be compensated by decreasing d, or increasing b, or increasing c as well as by decreasing a non-integers, co-prime denominators, unfamiliar wording or contexts -> additive strategies applying taught methods is error-prone Textbooks over-simplify to try and make RPR ‘do-able’

And … implicit throughout the curriculum learnt over time in contexts. concrete models can hide the abstract nature of RPR continuous contexts need imagination gradient and trigonometric ratios are expressed as single numbers students are more likely to use unitary methods with contextual problems where they design the unit many problems with ratio are due to lack of facility with multiplication and division facts and associated methods

Experiences learners need to have Use learners’ knowledge about contexts and relative quantities Build on learners’ capacity to design suitable units in contexts Represent and carry out division in many ways, maintaining the connection between division and ratio by using simplified fraction notations. Represent ratio in a variety of ways: a:b, a/b, and as a single number Use extendable models, images, metaphors, diagrams (e.g. ratio tables) Develop multiplication and division of non-integers Use problems that need proportional reasoning

Functions One-to-one or one-to-many mappings Input/output machines Input/output ‘black boxes’ such as trigonometric or exponential functions Relations between particular x-values and y-values Relations between a domain of x-values and a range of y- values Representations of relations between variables in ‘realistic’ situations Graphical objects which depict particular values, characteristics Graphical objects which can be transformed by scaling, translating and so on Structures of variables defined by parameters and relations

What goes wrong? pointwise correspondence focuses on quantity covariation of variables focuses on change what relations are functions? mappings between sets of discrete data can reinforce a view that a function is a set of discrete data points not all functions are continuous, smooth, invertible and calculable textbooks do not in general offer coherent, explicit, comments about how these relate to the overarching idea of relations between variables and often use strongly prototypical examples.

Experiences students need to have Identification of functions and their properties and definitions Tabulation and graphical tasks Calculate value of function at a point Construct functions with given properties Relate representations: pictorial, symbolic- algebraic and physical context Identify functions which are equal/equivalent Operate with functions (+, -,x, o) Use functions in proofs