Abrahamson, D. (2011). Towards instructional design for grounded mathematics learning: the case of the binomial.

In N. L. Stein & S. Raudenbush (Eds.), Developmental cognitive science goes to school (pp. 267-281). New York: Taylor & Francis / Routledge.


The Problem

Consider a penny. It is flipped four times. Now consider two possible outcomes of this experiment:

(a) Heads, Heads, Heads, Heads

(b) Heads, Heads, Tails, Tails

Is one of these two outcomes more likely than the other, or are they equally likely?

This item targets basic knowledge of probability. Namely, it aims to evoke the phenomenon of a random generator (e.g., coins, dice, spinners, etc.) as a context for eliciting and gauging an understanding of randomness, independence, and distribution. Solving this item does not demand any numerical reasoning or arithmetical calculation—one need only grasp the logic of the situation so as to determine the appropriate response. We might thus hope that graduates of the U.S. school system, who have studied at least basic probability concepts, fare well on this simple item. But do they?

According to probability theory, this penny-flipping situation describes a compound-event problem, because each trial is defined as a conjunction of two or more singleton events. Here, each trial is a conjunction of four singleton coin flips. Also, this is a binomial situation, because each singleton-event variable—that is, each coin flip—yields one of exactly two possible values (Heads or Tails). In this particular compound-event binomial problem, the two hypothetical experimental outcomes, HHHH and HHTT, are equiprobable, because the conjunction consists of a sequence of independent events. That is, the coin has no memory, so each of the four singleton flips is not affected by the result of a preceding flip nor does it affect the result of a subsequent flip. And yet an overwhelming proportion of the adult population chooses option (b), HHTT, as more likely, arguing that it appears to better capture the structure and function of the random generator, that is, its equal numbers of Heads and Tails better represent the essence of a two-sided coin (Tversky & Kahneman, 1974). These findings have been robustly replicated in numerous studies (Jones, Langrall, & Mooney, 2007). Moreover, there is reason to believe that the findings reflect early or even innate reasoning mechanisms. For example, Xu and Vashti (2008) showed eight-month-olds a tub full of balls of two colors from which there issued into a narrow tube a sample of several balls that were thus arranged sequentially. The infants’ reactions to different types of samples suggested that they found more interesting those samples whose ratio composition was less representative of the population, for example, a sample of four green balls was more interesting than a sample of two green balls and two blue balls, irrespective of their order.1

What are the pedagogical implications of these empirical findings of flawed probabilistic reasoning across the ages? Should we conclude that the subject matter of probability is inherently counterintuitive and that therefore students can at best develop basic fluency in the rote execution of probability solution procedures?