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The empirical law of large numbers and the hospital problem: systematic investigation of the impact of multiple task and person characteristics

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Abstract

The empirical law of large numbers is an important content in secondary school mathematics. Tasks used to analyze students’ understanding of this law are often based on the hospital problem, but vary in various features, leading to mixed and conflicting empirical results. To identify task features that support students when approaching this type of task, we systematically investigated the impact of multiple task and person characteristics on the accuracy of students’ responses in a cross-sectional study with N = 242 mathematics teacher education students. Students answered several variants of the hospital problem in different sequences. Our assumption was that differences in performance between tasks could be traced back to the salience of relevant task features and the sequence of the tasks. Results of GLMM analyses of our data support that in particular larger deviations from the expected relative frequency and a bigger ratio between the large and small sample size increase solution rates. Moreover, a verbal presentation of a 100% frequency in the case of maximal deviation increased solution rates. A within-subject analysis revealed that effects of task characteristics were more pronounced for the first task and weakened substantially for subsequent tasks. Finally, we found that 100% frequency tasks have a positive cueing effect, supporting students to solve subsequent tasks, even if the relevant features are less salient there. These tasks thus seem to be a promising starting point to connect the empirical law of large numbers with students’ prior intuitions.

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Notes

  1. Note that the translation from the original language (German) into English may cause subtle differences between tasks. Words might have different connotations in different languages or evoke different associations or even different reasoning processes (e.g., Lem et al., 2011).

  2. A GLMM analysis containing only the intercept and no predictors indicated that the solution rate over all tasks was 53.7% (31.8% for first tasks). Thus, solution rates were substantially different from 33.3% over the whole study, which would be expected if participants were choosing one option at random for each task. Moreover, in line with previous studies (e.g., Fischbein & Schnarch, 1997; see also Lem et al., 2011), solution rates varied systematically by task variant, even when considering only the first task in each booklet. This indicates that students did not apply a general guessing strategy.

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Correspondence to Simon Weixler.

Appendix

Appendix

1.1 Exemplary explanations provided by students on the final, open question

Fig. 4
figure 4

Example of a student’s response (original resp. translated illustration) with a conceptual explanation that addresses the relation between sample size and variability of the observed relative frequency. This student answered all closed questions correctly

Fig. 5
figure 5

Example of a student’s response without a conceptual explanation. This student answered all closed questions correctly

Fig. 6
figure 6

Example of a student’s response without a conceptual explanation, but with a reference to his/her “intuition”. This student answered all but one of the closed questions correctly

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Weixler, S., Sommerhoff, D. & Ufer, S. The empirical law of large numbers and the hospital problem: systematic investigation of the impact of multiple task and person characteristics. Educ Stud Math 100, 61–82 (2019). https://doi.org/10.1007/s10649-018-9856-x

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