Chatper 4


Different tasks lend themselves to different styles of representation (Sloman 1985). As a rule of thumb it makes sense to use the most appropriate technology to model the phenomena of interest. An example would be to use connectionism for low-level processes (e.g., motor control), but switch to symbolic systems for higher-level tasks, such as planning (Clary 1989). Such hybrid views of cognition are attractive (Thornton 1991, 1992a; Rose 1991; Hendler 1989), but how do we know which technology is most appropriate for a given task?

This chapter reviews the symbolic models that have been built to capture the way children learn multicolumn arithmetic. Solving problems like 32-27 or 49x12 involves a host of skills: not only do you need to know the arithmetic facts, you also need to know how to borrow and carry, which column to process next, what to do when a number does not have a number below it (as in 12+5), and so on.

It appears that students are following rules when solving multicolumn problems. Perhaps the main evidence for this comes from the observation that children discover faulty rules (malrules) when learning arithmetic. Hence, it seems appropriate to use something like a production system to model multicolumn arithmetic. Indeed, to date the only systems used to investigate arithmetic have been rule-based systems (Brown & Burton 1978; Young & O'Shea 1981; Brown & VanLehn 1980; VanLehn 1983, 1990).

This chapter first looks at the kinds of mistakes children make when solving multiplication problems. The style of symbolic modelling is then described by briefly discussing a production system for multiplication. Section 4.2.2 considers the Young & O'Shea (1981) approach to modelling children's errors, and section 4.2.3 looks at the way VanLehn (1990) has modelled the errors. The conclusion is that the symbolic models offer a very plausible interpretation of children's arithmetic. Nevertheless, section 4.3 argues that there are good reasons for looking into a connectionist account of the phenomena. In particular, I aim to show that connectionism may be an "appropriate technology" for this domain, despite the phenomena's rule-like appearance. Many of the assumptions and ideas from the VanLehn and Young & O'Shea accounts are taken on board in chapter 5 which describes a connectionist model of multicolumn arithmetic.

  • Bug phenomena
  • Models
  • A production system for multiplication
  • Modelling bugs: the Young & O'Shea way
  • Modelling bugs: the VanLehn way
  • Summary
  • Why connectionism
  • Development
  • Implementation and theory
  • What is a symbol, anyway?
  • Other symbolic machine learning approaches
  • Comments
  • table of contents