A framework for understanding what algebraic thinking is
In relation to the learning of mathematics, algebra occupies a very special place, both because it is in itself a powerful tool for solving problems and modelling situations, and also because it is essential to the learning of so many other parts of mathematics. On the other hand, the teaching of algebra has proven to be a difficult task to accomplish, to the extent of algebra itself being sometimes considered the border line which separates those who can from those who cannot learn mathematics. A review of the research literature shows that no clear characterisation of the algebraic activity has been available, and that for this reason research has produced only a local understanding of aspects of the learning of algebra. The research problem investigated in this dissertation is precisely to provide a clear characterisation of the algebraic activity. Our research has three parts: (i) a theoretical characterisation of algebraic thinking, which is shown to be distinct from algebra; in our framework we propose that algebraic thinking IS • thinking aritmnetically, • thinking internally, and • thinking analytically. and each of those characteristics are explained and analysed; (ii) a study of the historical development of algebra and of algebraic thinking; in this study it is shown that our characterisation of algebraic thinking provides an adequate framework for understanding the tensions involved in the production of an algebraic knowledge in different historically situated mathematical cultures, and also that the characteristics of the algebraic knowledge of each of those mathematical cultures can only be understood in the context of their broader assumptions, particularly in relation to the concept of number. (iii) an experimental study, in which we examine the models used by secondary school students, both from Brazil and from England, to solve "algebraic verbal problems" and "secret number problems"; it is shown that our characterisation of algebraic thinking provides an adequate framework for distinguishing different types of solutions, as well as for identifying the sources of errors and difficulties in those students' solutions. The key notions elicited by our research are those of: (a) intrasystemic and extrasystemic meaning; (b) different modes of thinking as operating within different Semantical Fields; (c) the development of an algebraic mode of thinking as a process of cultural immersion- both in history and for individual learners; (d) ontological and symbolical conceptions of number, and their relationship to algebraic thinking and other modes of manipulating arithmetical relationships; (e) the arithmetical articulation as a central aspect of algebraic thinking; and, (f) the place and role of algebraic notation in relation to algebraic thinking. The findings of our research show that although it can facilitate the learning of certain early aspects of algebra, the use of non-algebraic models-such as the scale balance or areas-to "explain" particular algebraic facts, contribute, in fact, to the constitution of obstacles to the development of an algebraic mode of thinking; not only because the sources of meaning in those models are completely distinct from those in algebraic thinking, but also because the direct manipulation of numbers as measures, by manipulating the objects measured by the numbers, is deeply conflicting with a symbolic understanding of number, which is a necessary aspect of algebraic thinking.