Graphs before algebra: an exercise in imaginary curriculum

The technology that we use to present a topic to our students, and the technology that we expect our students to use in class, can have profound effects on what is taught and the way in which it is taught. Sometimes this does not happen for a number of reasons, including concerns as to how much the students will 'understand' using the new technology, and the development of new curriculum and new methods can be slowed or stopped altogether. One way to work through these difficulties is to present blocks of 'imaginary curriculum' as a basis for discussion among teachers and authorities, working through the sorts of understanding we would expect students to develop and identifying points where something might be gained or lost. A specimen topic is developed in this paper. In Australia in the middle years of secondary school students typically learn to work algebraically with quadratic functions, first multiplying out products of terms and then reversing the process to factorize quadratics. Later this is extended into completing the square and solving quadratic equations by factorizing, by completing the square or by use of the standard formula. Later again students use these algebraic skills in constructing graphs of quadratic functions. In this paper the author demonstrates how the process can be completely reversed, using modern hand-held graphing calculators, by studying graphs first and then deducing all the standard algebraic processes. Examination of the underlying processes suggests that this method is at least as easy to understand, if not clearer; that students will 'understand' at least as much as they do by learning the traditional processes; and that they will be more able to extend their skills into more advanced problems. In an ideal curriculum students would probably learn both approaches and the connections between them, to enable them to check their work and reinforce their understanding in multiple ways.