Completed and submitted to the Working Group on Early Algebra, PME 25, on Monday, March 5 th 2001. Reflections on Early Algebra David Tall Mathematics Education Research Centre University of Warwick, UK e-mail: David.Tall@warwick.ac.uk This paper is a contribution to a discussion on early algebra at the 25 th Annual Conference of PME. It is predicated on a perceived need to frame early algebra within a wider theory of symbol development. I shall use an existing theory (Tall et al, 2000) to place the study of Carraher Schliemann and Brizuela (2001) within a broader framework. It will reveal the study to be situated in a preliminary (but vital) stage between arithmetic and algebra. Furthermore it will suggest theoretical and practical links to earlier arithmetic that need to be considered and reveal the reported activity as a small step for many of the children involved along the path to what I shall term ‘evaluation algebra’. After this point, however, there is still much to do and the traditional discontinuities that occur in the transition to full ‘manipulation algebra’ still remain to be faced at a later stage. Introduction At this conference we celebrate 25 years of research meetings organised by the International Group for the Psychology of Mathematics Education. It is therefore fitting to place this response within this context. Carraher, Schliemann and Brizuela (2001) refer broadly to a range of previous research, partly to report observed difficulties and partly to respond to suggestions to ‘bring out the algebraic character of arithmetic’. This means that, apart from using the advice to ‘algebrafy’ arithmetic, the 25 years of previous research is not used in any foundational way. My analysis of their paper therefore uses a global theory of developing symbolism to place the research in context. Analysis Carraher et al (2001) base their research in a ‘typical’ class of 9 year-olds and is implicitly an approach to teach ‘algebra for all’. I begin, therefore, by looking at their data to see if they are actually reaching every child in the class and also to analyse precisely what kind of algebra the children appear to be learning. The class is presented with a story in which two children start with the same unspecified amount of money on Sunday and spend and receive specific amounts on successive days. When the researcher Bárbara asks the class if they know how much money they have, ‘the children state a unison “no”’, but ‘a few utter “N” and Talik states “N, it’s for anything.”’ Thus we have some children who have already met the idea of using a letter to stand for a number and some who presumably have not. Our first piece of evidence is that, faced with adding 3 dollars to the initial unspecified amount, we are told that ‘only three children do not write N+3 as a representation for the amounts on Monday.’ What is missing is an analysis of what the children individually bring to the – 2 – class from their previous development and why some children are more adept at algebraic thinking than others. This in turn requires a theory of longer-term development that is consonant with empirical evidence. Tall et al (2000) present a theory of symbolic development arising from earlier work of Gray & Tall (1994) and many others. This reveals a bifurcation in performance in arithmetic between those who become entrenched in a procedural mode of counting to do arithmetic and those who develop proceptual thinking involving the flexible use of symbols as both process and concept. (This is not to be interpreted as a naïve prescription that the successful always get better and the less successful get worse. The case of Emily (Gray and Pitta, 1997) reveals a child growing from counting procedures to flexible number concepts by being given support using a calculator that carries out the procedures for her so that she can concentrate on the conceptual relationships.) However, the theory does intimate that what children bring to a given situation—depending on their preceding development—radically affects how and what they learn. It can have a profound effect on early algebra. For instance, the English National Curriculum in England intended to use arithmetic problems such as the following as a precursor of algebra: (1): 3+4 = , (2): 3+ = 7, (3): + 3 = 7. Although these look like algebra, they are certainly not. Children perform them using their repertoire of methods of counting and deriving or knowing facts. Question (1) can be done by any counting method, (2) can be done by ‘count-on’ from 3 to find how many are counted to get to 7. Equation (3) is more subtle. If the child senses that the order of addition does not matter, the problem is essentially the same as (2); and can be solved by count-on from 3. If not, the child who counts has a far more difficult task to find out ‘at what number do I start to count-on 3 to get 7?’ This involves trying various starting points to count-up using a ‘guess-and-test’ strategy. Foster (1994) used these three types of question in a study of ‘typical’ children in the first three years of an English Primary School. He found a significant spectrum of performance in the first year where the lower third were almost totally unable to respond to questions of types (2) and (3). By the third year the top two-thirds of the class obtained almost 100% correct responses but the lower third obtained 93% correct on type (1), 73% correct on type (2) and 53% on type (3). Seen in the light of procept theory, this suggests that the lower third operate more in a procedural than a flexible proceptual level. This would be consistent with the lower third of a class in Grade 3 in the USA including children who are more procedural than proceptual, which, in turn is consistent with difficulties with algebraic qualities of arithmetic exhibited by some children in this study. I would counsel, therefore, that in carrying this activity out in a classroom context, some children are already struggling and need special individual care. Even those who succeed in writing down the symbolism ‘N+3’ are likely to be using it in a manner different from that observed by an expert.