Analogy and Arithmetics: An HDTP-Based Model of the Calculation Circular Staircase

Analogy and Arithmetic: An HDTP-Based Model of the Calculation Circular Staircase Tarek R. Besold (tbesold@uos.de) and Martin Schmidt (martisch@uos.de) Institute of Cognitive Science, University of Osnabr¨uck, 49069 Osnabr¨uck, Germany Alison Pease (a.pease@ed.ac.uk) School of Informatics, University of Edinburgh, Edinburgh, EH8 9AB, UK Abstract Analogical reasoning and its applications are gaining attention not only in cognitive science but also in the context of educa- tion and teaching. In this paper we provide a short analysis and a detailed formal model (based on the Heuristic-Driven The- ory Projection framework for computational analogy-making) of the Calculation Circular Staircase, a tool for teaching basic arithmetic and insights based on the ordinal number concep- tion of the natural numbers to children in their first years of primary school. We argue that such formal methods and com- putational accounts of analogy-making can be used to gain ad- ditional insights in the inner workings of analogy-based edu- cational methods and tools. Keywords: Analogy, Education, Teaching, Arithmetic, For- mal Model, Computational Analogy-Making, HDTP. primary school (Schwank, 2003; Schwank, Aring, & Blocks- dorf, 2005), before showing how a computational analogy- making framework as Heuristic-Driven Theory Projection (HDTP) (Schwering, Krumnack, K¨uhnberger, & Gust, 2009) can be used to provide a formal computational reconstruc- tion of the staircase as a prototypical example of analogy-use taken from a real-life teaching situation. We thereby also con- tinue the work started in (Besold, 2013) with a far more com- plex and deep-rooted case study. By doing so, we aim to show one way (amongst several) of how analogy-engines and their corresponding background theories can fruitfully be applied to modeling and analysis tasks from the field of psychology of learning, education, and didactics. Introduction Heuristic-Driven Theory Projection (HDTP) Analogical reasoning is the ability to perceive, and operate on, dissimilar domains as similar with respect to certain as- pects based on shared commonalities in relational structure or appearance. This has been proposed as an essential aspect of the ability to learn abstract concepts or procedures (Gentner, Holyoak, & Kokinov, 2001), and is recognised as ubiquitous in human reasoning and problem solving (Gentner, 1983), representational transfer (Novick, 1988), and adaptation to novel contexts (Holyoak & Thagard, 1995). Inherent in the structure of analogical reasoning is its role in education and learning: new ideas can be constructed and explored in relation to familiar concepts. While substantial research has been carried out into the role of analogical rea- soning and science education (see, for instance, (Duit, 1991; Arnold & Millar, 1996; Guerra-Ramos, 2011)), its role in mathematics education has been somewhat less explored – although notable exceptions include (Pimm, 1981; English, 1997). These studies support our assumption that analogies can be used for facilitating the understanding of concepts and procedures in abstract and formal domains, such as mathe- matics, physics or science. The pedagogical use of analogies as a means of triggering, framing and guiding creative insight processes still needs to be widely recognised as part of teach- ing expertise and incorporated into innovative teacher educa- tion schemes (Akgul, 2006). In this paper, we want to contribute to a deeper under- standing of the role and the mode of operation of analogy in an educational context by first providing a description and short analysis of the analogy-based Calculation Circu- lar Staircase used for teaching basic arithmetic to children attending their initial mathematics classes at the beginning of There is much work on both theoretical and computational models of analogy-making. Heuristic-Driven Theory Projec- tion (HDTP) (Schwering et al., 2009) is one such perspective: this is a formal theory and corresponding software implemen- tation, conceived as a mathematically sound framework for analogy-making. HDTP has been created for computing ana- logical relations and inferences for domains which are given in the form of a many-sorted first-order logic representation. Source and target of the analogy-making process are defined in terms of axiomatizations, i.e., given by a finite set of for- mulae. HDTP tries to produce a generalization of both do- mains by aligning pairs of formulae from the two domains by means of anti-unification: Anti-unification tries to solve the problem of generalizing terms in a meaningful way, yield- ing for each term an anti-instance, in which distinct subterms have been replaced by variables (which in turn would allow for a retrieval of the original terms by a substitution of the variables by appropriate subterms). HDTP in its present version uses a restricted form of higher-order anti-unification. In higher-order anti-unification, classical first-order terms are extended by the introduction of variables which may take arguments (where classical first- order variables correspond to variables with arity 0), making a term either a first-order or a higher-order term. Then, anti- unification can be applied analogously to the original first- order case, yielding a generalization subsuming the specific terms. The class of substitutions which are applicable in HDTP is restricted to (compositions of) the following four cases: renamings (replacing a variable by another variable of the same argument structure), fixations (replacing a variable by a function symbol of the same argument structure), ar-