{'fraction_non_alphanumeric': 0.0308760172331259, 'fraction_numerical': 0.020344662517951174, 'mean_word_length': 5.021613832853026, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Modeling the Logarithmic-To-Linear Shift In Representations Of Numerical Magnitudes

Modeling the Logarithmic-To-Linear Shift In Representations Of Numerical Magnitudes Leif K. Lende (leif.k.lende@diakonhjemmet.no) Department of Social Education and Occupational Therapy, Diakonhjemmet University College Rogaland Vaagsgaten 40, N-4306 Sandnes, Norway Niels Taatgen (n.a.taatgen@rug.nl) Department of Artificial Intelligence, University of Groningen Nijenborgh 9, 9747 AG Groningen, Netherlands Abstract We present findings from an experiment with 6-year-old children in Norway who are getting started with the process of learning the numbers from 1 to 10. The findings provide information on number line estimation - that is, translating a number to a spatial position on a number line. The results show different categories of representation of the magnitudes on the number line, which may represent different stages in a learning sequence. On this basis, we show a proposal for a cognitive model of the learning process towards a linear representation of magnitudes. Parallel Changes Keywords: Learning; numerical magnitudes; number line; dynamic decision making; memory; cognitive architectures; ACT-R. Introduction In this paper we will present our first findings from the pre-test of a larger experiment where the subjects are Norwegian 6 year old children just started on pre-school education learning numbers from 1 to 10. One important learning process that is involved when dealing with number magnitude is the estimation of what position a number value has on a number line. The learning sequence involved is the one that Siegler calls the logarithmic-to-linear shift in representations of numerical magnitude (Siegler, Thompson, & Opfer, 2009). In order to make a cognitive model of this learning process we want to investigate what cognitive processes are involved in learning the positioning of number magnitudes on a number line. Siegler et al (2009) show that children undergo parallel changes from logarithmic to linear representation on numerosity estimation tasks. The example we have reused from their article in figure 1 shows long-term changes in estimation of whole number magnitudes. (A) On 0–100 number lines, kindergartners’ estimates were better fit by the logarithmic function than by the linear, whereas second-graders’ estimates were better fit by the linear function than by the logarithmic; (B) On 0–1000 number lines, second-graders’ estimates were better fit by the logarithmic function than by the linear, whereas fourth- graders’ estimates were better fit by the linear function than by the logarithmic. Figure 1. The logarithmic to linear shift. From Siegler, Thompson, & Opfer, (2009), Copyright 2009 Wiley. Reprinted with permission. From Siegler et al’s research and from others (Dehaene, Izard, Spelke, & Pica, 2008) we know that a logarithmic representation is likely to be the starting point of this learning sequence. This representation is usually attributed to Weber’s law. However, in some of the Ramani and Siegler’s work (2008) the initial performance is not logarithmic, but on average linear but incorrect. Maybe this has something to do with the scale: 1..10 as opposed to 1..100 compared to the size of the task sheets. But then the simple Weber explanation will not hold universally. We also know from the research of Siegler and colleagues (2009) that this learning sequence happens several times at different stages during lifetime. That means children can have multiple representations of numerical magnitudes. As an example they can have a logarithmic representation of the numerical magnitudes on a number line from 0 to 100 while they have a linear representation on the number line from 0 to 10. As the children get older and learn a linear representation on the number line from 0 to 100 too, they may still have a logarithmic representation on the number line from 0 to 1000, and so on. A possible account for the transition from logarithmic to linear is that children learn the location of particular points on the number line. Schneider et al. (2008) have investigated the role of anchor points in an eye-movement