No abstract

Comparing probabilities is a useful skill in life. Binary choice tasks are popular means in research on probabilistic reasoning. Falk, Yudilevich-Assouline, and Elstein (Educational Studies in Mathematics, 81 (2), 2012) noted that many of these tasks contain design flaws. We designed and evaluated an extended and improved binary choice item set. In each trial, children were shown two boxes containing desired and undesired elements and had to identify the best box to blindly draw from. We took into account four necessary item set characteristics:(un)desired elements in the correct box, total number of elements in the correct box, and difference between desired and undesired elements in the correct box. Furthermore, some extensions to Falk et al.'s design (2012) were implemented: items in which one box certainly provided a desired element, items with three competing colors of elements, and items with lower "countability" of the elements. Results showed that extensions added to the design did not imperil internal consistency and validity. Younger children were more likely to give correct answers when the correct box contained more desired elements and older children are better at comparing probabilities than younger ones overall.

The many studies with coin-tossing tasks in literature show that the concept of randomness is challenging for adults as well as children. Systematic errors observed in coin-tossing tasks are often related to the representativeness heuristic, which refers to a mental shortcut that is used to judge randomness by evaluating how well a set of random events represents the typical example for random events we hold in our mind. Representative thinking is explained by our tendency to seek for patterns in our surroundings. In the present study, predictions of coin-tosses of 302 third-graders were explored. Findings suggest that in third grade of elementary school, children make correct as well as different types of erroneous predictions and individual differences exist. Moreover, erroneous predictions that were in line with representative thinking were positively associated with an early spontaneous focus on regularities, which was assessed when they were in second year of preschool. We concluded that previous studies might have underestimated children’s reasoning about randomness in coin-tossing contexts and that representative thinking is indeed associated with pattern-based thinking tendencies.

reasoning abilities. In the present study, we examined children's numerical abilities in the second grade of preschool and their probabilistic reasoning abilities one and two years later. On the one hand, our results indicate that early numerical abilities assessed in the second grade of preschool predict the use of erroneous solving strategies to compare probabilities and create equal probabilities in the third grade of preschool. On the other hand, our results also indicate that the same early numerical abilities predict the use of more advanced or correct strategies when children are in the first grade of primary school. We discuss these seemingly contradicting findings in light of current research on probabilistic reasoning abilities.

Findings on children's proportional reasoning abilities strongly vary across studies. This might be due to the different contexts that can be used in proportional problems: fair-sharing, mixtures, and probability. A review of the scientific literature suggests that the context of proportional problems may not only impact the difficulty of the problem, but that it also plays an important role in how children approach the problems. In other words, different contexts might elicit different (erroneous) thinking strategies. The aim of the present study was to investigate the role of context in third graders' (n = 305) proportional reasoning abilities. Results showed that children performed significantly better in a fair-sharing context compared to a mixture and a probability context. No evidence was found for a difference in performance on the mixture and the probability context. However, the kind of erroneous answers that were given in the mixture and probability context differed slightly, with more additive answers in the mixture context and more one-dimensional answers in the probability context. These findings suggest that the type of answers elicited by proportional problems might depend on the specific context in which the problem is presented.