DANIEL CHAZAN ON TEACHERS MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION A PERSONAL STORY ABOUT TEACHING A TECHNOLOGICALLY SUPPORTED APPROACH TO SCHOOL ALGEBRA In the United States examples of elementary school mathematics teaching Ball 1993b Lampert 1990 Peterson Fennema and Carpenter 1991 Wood Cobb Yackel and Dillon 1993 have been celebrated as exemplars of broader trends to reform students experience of schooling and re focus their attention and efforts on understanding core ideas of traditional academic disciplines 1 For example Magdalene Lampert s 1990 When the problem is not the question and the solution is not the answer lays out one influential image of teaching aimed at student understanding If traditional mathematics instruction links mathematics problems and their solutions to teacher taught solution methods Lampert severs that connection In the described episode students are not solving the problem by applying a method taught previously in class Instead they are posed a problem for which they have not been taught an algorithm However they have methods for checking the validity of proposed solutions Using conclusions developed from previous classroom work they then seek both to find the answer to the problem and to find a more general method of solution to problems of this type In other contexts teaching of this sort goes by other names inquiry teaching a problem solving orientation child centered instruction mathematical investigations or mathematical activity Cuban 1993 Jaworski 1994 Lester 1994 Love 1988 Morgan 1998 What does such teaching require of the teacher Is it simply a matter of withholding of not teaching solution methods Is it simply the will to interact differently with students to listen to their mathematical ideas Or do attempts to create childcentered mathematics classrooms in which students explore and conjecture require anything special of the teachers knowledge of mathematics 2 Are there qualities of the teachers knowledge which are crucial to this sort of teaching If so do advances in calculator and computer technology have any role to play International Journal of Computers for Mathematical Learning 4 121 149 1999 1999 Kluwer Academic Publishers Printed in the Netherlands 122 DANIEL CHAZAN To approach these questions I will reflect on my own experiences teaching beginning algebra and the sorts of mathematical understandings I could call upon as a teacher In particular I will compare and contrast two three year long experiences with different pedagogical approaches to beginning algebra I will compare teaching I did between 1982 and 1985 with Dolciani and Wooton s Algebra One text 1970 73 with teaching of a functions based approach to school algebra between 1990 and 1993 Each of these two approaches is a conceptualization of algebra designed for use in teaching Each seeks to provide teachers with a perspective on the content which will aid in providing students access to a set of mathematical ideas 3 In this paper I explore ramifications of these two curricular approaches for teaching In particular I focus on differences in the nature of the subject matter understandings which each provided me as a teacher I explore whether these differences had an impact on my capacity to support student classroom exploration of mathematical ideas I argue that the functionsbased approach provided a type of knowledge of the subject matter which supported involvement of students in the exploration of the subject and that it provided opportunities for engagement with students questions about the purposes of studying mathematics ON THE ROLE OF TECHNOLOGY IN ALGEBRA REFORM Technology plays a pivotal but often indirect role in this story Much of what changes between 1982 and 1990 is related to technology available for the teaching of high school mathematics For neo Vygotskian activity theorists like Tikhomirov 1972 1981 it is not surprising that technology would have such an impact They suggest that technology by virtue of the way it restructures activity has the potential to play a critical role in the shaping of thought Crucial to the appreciation of such arguments is the recognition that such shaping does not necessarily require the actual presence of the technology in the context of particular interactions the opening up of new technological possibilities even in the absence of the relevant technology in a particular interaction can have profound impact This sort of analysis suggests that in mathematics education technology by virtue of the ways in which it restructures activity and provides new opportunities for analysis and solution of problems can shape the thinking of teachers and curriculum developers in addition to supporting the learning of students For me an eloquent example of this sort of impact of technology is found in recent developments in the teaching and learning of x in school TEACHERS MATHEMATICAL KNOWLEDGE AND STUDENT EXPLORATION 123 algebra In the past school algebra has been dominated by a conceptualization of x as unknown solving equations to find the unknown number has been a central activity Yet in the past fifteen years for example since the early eighties van Barneveld and Krabbendam 1982 and since the entrance of microcomputers and graphing calculators into schools particularly in North America school algebra curriculum and research projects have begun to explore a more central role for the conceptualization of x as a variable 4 Arguably see critics of this development for example Pimm 1995 much of this change has been influenced by computer technology Specifically the capacity of graphing calculators and computers to carry out many calculations rapidly supports the transition from examination of single cases towards the examination of groups of cases at once This possibility is supported in technological environments by the possibilities of graphing the output of an expression for all real values of x in an interval of creating a table of values of the output of an expression involving x of linking such graphical and tabular representations or even the possibility of having a spreadsheet recalculate a series of expressions as a particular cell is varied Thus arguably technology has helped it become easier to conceptualize x as taking on a series of values To my view these technological capabilities have had an impact even on projects which claim to continue earlier traditions of examination of x as unknown even projects which claim to