Rethinking Probability Education: Perceptual Judgment as Epistemic Resource

Abstract

The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct outcomes, e.g., HT and TH in the case of flipping a pair of coins, and thus infer erroneous predictions. Educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Yet whereas empirical activities per se are crucial for any probability design, because they introduce the pivotal contents of randomness, variance, sample size, and relations among them, empirical activities may not be the unique or best means for students to accept the logic of combinatorial analysis. Instead, learners may avail of their own pre-analytic perceptual judgments of the random generator itself so as to arrive at predictions that agree rather than conflict with mathematical analysis. I support this view first by detailing its philosophical, theoretical, and pedagogical foundations and then presenting empirical findings from a design-based research project. Twenty-eight students aged 9-11 participated in tutorial, task-based clinical interviews that utilized an innovative random generator. Their predictions were mathematically correct even though initially they did not discern variations. Students were then led to recognize the formal event space as a semiotic means of objectifying these presymbolic notions. I elaborate on the thesis via micro-ethnographic analysis of key episodes from a paradigmatic case study. Along the way, I explain the design-based research methodology, highlighting how it enables researchers to spin thwarted predictions into new theory of learning.

How do students reason about statistical sampling with computer simulations? An integrative review from a grounded cognition perspective

“…Consistent with the concreteness fading approach, another route for the grounded cognition approach in pedagogical simulations is to employ tangible objects before students transition to computer simulations. Examples of such objects have included marbles for drawing samples from a box (Abrahamson, 2014, van Dijke-Drookers et al, 2021b, different-colored candies to demonstrate sampling distributions (Hancock & Rummerfield, 2020), a video demonstration of an instructor manually shuffling data on paper to teach the randomization test function in R (Zhang et al, 2022), and data cards in a population bag (Arnold et al, 2017). Although many studies did not evaluate the particular impact of these tangibles on learning, Hancock and Rummerhield's quasi-experiment revealed that the group that engaged with tangible activities before computer simulations had significantly higher improvement in their exam scores compared to the group that engaged with only computer simulations.…”

Professional competency: Pre-service mathematics teachers’ understanding toward probability concept

“…Probability is a challenging lesson for students especially in compound events. They tend to make incorrect predictions of events http://journals.ums.ac.id/index.php/jramathedu (Abrahamson, 2014) so that many students have difficulty learning probability. They find it tricky to connect theoretical and empirical contexts (Nilsson, 2014).…”