Fast and Slow Strategies in Multiplication

Abstract

In solving multiplication problems, children use both fast, retrieval-based, processes, and, slower computational processes. In the current study, we explore the possibility of disentangling these strategies using information contained in the observed response latencies using a method that is applicable in large data sets.We used a tree-based item response-modeling framework to investigate whether the proposed qualitative distinctions in fast and slow strategies can be detected. We analyzed responses to two sets of multiplication items, totalling more than 500.000 responses, collected with an online computer-adaptive training environment for mathematics.Results showed qualitative differences between the fast and the slow strategies. Building on these results, both item and person characteristics were differently related to fast and slow processes. These characteristics, resulting from substantive models of multiplication, allowed us to further describe the fast and slow strategies. Results emphasize the quantitative and qualitative differences between strategies used for solving multiplication problems, and provide possibilities for tailored feedback on learning multiplication.

The dynamics of the development of mathematics skills: A comparison of theories of developing intelligence.

“…For more details on the Math Garden and its psychometric properties, see Appendix A. In this study we focus on two basic skills: counting (Jansen et al, 2014) and addition, and two more advanced skills: multiplication (van der Ven et al, 2015;Hofman et al, 2017) and division. Each skill is measured by a separate game.…”

Hidden Markov Models of Evidence Accumulation in Speeded Decision Tasks

“…The current model provides a proof of principle of a combination of an EAM with an HMM, and as such can lead to further interesting applications and extensions, as it opens new possibilities regarding modeling continuous and discontinuous patterns of response times and accuracy in a single modeling framework. Although the current article focused solely on speeded decision tasks, questions about the continuous and discontinuous relations between response times and accuracy is ubiquitous in higher cognitive applications as well, including study of more complex cognitive tasks and development of strategies used to solve these tasks (van der Maas & Jansen, 2003;Raijmakers et al, 2014;Hofman et al, 2018). An interesting feature of higher level cognitive tasks that might be relevant to explore using the current framework is the emergence of more efficient strategies, that lead to qualitatively better response accuracy as well as shorter response times.…”

“…An interesting feature of higher level cognitive tasks that might be relevant to explore using the current framework is the emergence of more efficient strategies, that lead to qualitatively better response accuracy as well as shorter response times. Such strategies have been described in many applications, such as multiplication tasks (Hofman et al, 2018), Mastermind game (Gierasimczuk et al, 2013;Kucharský et al, 2020), or Progressive matrices tasks (Vigneau et al, 2006;Laurence et al, 2018). Combination of HMM with EAM in this context would enable uncovering different relations between response times and accuracy depending on whether we look within or between strategies -it is possible to imagine that an efficient strategy would be faster and more accurate than less efficient strategy, but within those strategies separately, we will see the traditional speedaccuracy trade-off whereby increasing response caution increases accuracy at the cost of speed, which would be captured by the EAM part of the model.…”

Error detection through mouse movement in an online adaptive learning environment

“…This rich source of information has served as the ideal basis for numerous substantive and methodological papers. These topics range from replicating effects predicted by theoretical models about mathematics (van der Ven, Straatemeier, Jansen, Klinkenberg, & van der Maas, 2015), and cognitive strategies and models (Buwalda et al, 2016; Hofman, Visser, Jansen, Marsman, & van der Maas, 2018; Jansen & van der Maas, 2002), to longitudinal studies showing mutualism in mathematical abilities (Hofman, Kievit, Stevenson, Molenaar, & Van Der Maas, 2018; Ou et al, 2019) and the ideographic approach (Hofman, Jansen, de Mooij, Stevenson, & van der Maas, 2018). In Math Garden, every student has his/her virtual garden, where each plant represents a game from a different domain, such as addition, multiplication or percentages.…”