## Background

The difficulty that algebraic thinking poses to children has been thoroughly studied in the field of mathematics education. Generalisation problems are not causing difficulties to students but the way they are presented to the. Often, the constraints of the teaching approaches used are the source of confusion (Moss and Beatty, 2006). These difficulties have to be investigated in the context of the curriculum, the nature of the tasks posed and the tools available for their solution (Noss, Healy and Hoyles, 1997). The general tension in schools is towards pattern spotting. Most instructions emphasise the numeric aspect of a pattern (Noss and Hoyles, 1996). These unfortunately lead to the variables becoming obscured and students ability to conceptualise relationships between variables, justify the rules and use them in a meaningful way is limited (Moss and Beatty, 2006). “Generalising problems are usually presented as numeric or geometric sequences, and typically ask students to predict the number of elements in any position in the sequence and to articulate that as a rule” (ibid, p.443). Teachers, then, tend to teach “the abstracted techniques isolated from all context” or alternatively “the technique as a set of rules to be followed in specific contexts” (Sutherland and Mason, 2005) to help their students find the rule. This could result in students own powers atrophying due to lack of use (Mason, 2002).

Another difficulty secondary school students face is their inexperience with the use of letters. They struggle to grasp the idea of letters representing any value (e.g. Duke and Graham, 2007) and lack some of the mathematical vocabulary needed to express generality at this age. Even though, it is a reasonable idea to introduce algebra early, there is still the issue of how to introduce it so that students can make the transition from simple arithmetic to algebra smoothly. Students could succeed in expressing generality at an early age, but even if they do so, it is in natural language and as expected, their written responses lack precision. The right design of tasks though could actually encourage students to write expressions in a general form rather than give a description in words. This articulation process needs to be addressed so that students could learn to express their thinking using algebraic notation. The use of ICT could help students see different representations, like symbolic, iconic, numeric or even verbal ones, and realise the relationships and the equivalence of different representations. Showing students creative as well as symbolic representations reinforces connections between them (Warren and Cooper, 2008).

It is important, though, to introduce different approaches to students and allow them to explore. This can be further enhanced by having students construct their own mathematical models (Noss, Healy and Hoyles, 1997). This modelling approach is inspired by Papert’s notion of constructionism (e.g. Papert, 1990) which supports the idea that learning happens most effectively when learners are actively making things in the real world or, in other words, “the idea that learners build knowledge structures particularly well in situations where they are engaged in constructing public entities” (Noss and Hoyles, 1996, p.61). Such a constructionist pedagogical approach, allows students to explore and construct their own models and could be adopted when building an appropriate system, as the microworld we are building, and the relevant tasks. The aim of the tasks we are designing is to generat understanding with the help of the system rather than inducing repetitive behaviour, as are many of the tasks in current school mathematics books.

### References

1. Duke, R., Graham, A. (2007): Inside the letter. Mathematics Teaching Incoporating Micromath, 200, pp. 42-45.
2. Mason, J. (2002): Generalisation and algebra: Exploiting children’s powers. In Haggerty, L., ed.: Aspects of Teaching Secondary Mathematics: Perspectives on practice. Routledge Falmer and the Open University, pp. 105-120.
3. Moss, J., Beatty, R. (2006): Knowledge building in mathematics: Supporting collaborative learning in pattern problems. International Journal of Computer-Supported Collaborative Learning 1, pp. 441-465.
4. Noss, R., Healy, L., Hoyles, C. (1997): The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33(2), pp. 203-233.
5. Noss, R., Hoyles, C. (1996): Windows on mathematical meanings: Learning cultures and computers. Dordrecht: Kluwer.
6. Papert, S. (1990): An Introduction to the 5th anniversary collection. In I. Harel (Ed.), Constructionist Learning: A 5th anniversary collection of papers. Cambridge, MA: MIT Media Laboratory.
7. Sutherland, R., Mason, J. (2005): Key aspects of teaching algebra in schools. QCA, London.
8. Warren, E., Cooper, T. (2008): The effect of different representations on year 3 to 5 students’ ability to generalise. ZDM Mathematics Education, 40, pp. 23-37.