Dual-Process Theory

Just some quotes and notes on a very intriguing group of ideas. I'll have to get around to commenting on them in depth at another time.

A large part of mathematics education research is concerned (explicitly or implicitly) with the relationship between intuitive and analytical modes of thinking and behavior (e.g., Fischbein, 1987; Stavy and Tirosh, 2000). Empirical findings on misconceptions are often explained by the mismatch between students' intuitions and the requirements of the formal system of mathematics. As it turns out, the distinction between intuitive and analytical modes of thinking is rigorously and solidly expressed in what cognitive psychologists call dual-process theory. . . .

According to this theory, our cognition and behavior operate in parallel in two quite different modes, called System 1 (S1) and System 2 (S2), roughly corresponding to our common sense notions of intuitive and analytical thinking. These modes operate in different ways, are activated by different parts of the brain, and have different evolutionary origins (S2 being evolutionarily more recent and, in fact, largely reflecting cultural evolution). . . .

S1 processes are characterized as being fast, automatic, effortless, unconscious and inflexible (hard to change or overcome) . . . In contrast, S2 processes are slow, conscious, effortful and relatively flexible. The two systems differ mainly on the dimension of accessibility: how fast and how easily things come to mind. In many situations, S1 and S2 work in concert, but there are situations (such as the ones concocted in the H&B [heuristics and biases] research) in which S1 produces quick automatic non-normative [incorrect] responses, while S2 may or may not intervene in its role as monitor and critic.¹

I would not presume to imagine that any education reforms were ever introduced with the explicit intention of harnessing the ideas presented in dual-process theories, but it is not difficult to imagine, given that S1 and S2 correspond respectively to the more accessible notions of intuitive and analytical thinking, that academic reformers—called to respond to a math education "crisis"—would find fault with the "fast, automatic, effortless, unconscious and inflexible" S1 thinking and favor the "slow, conscious, effortful" S2 thinking.

Stanovich and West² describe S1 thinking as being similar to Levinson's (1995) idea of "interactional intelligence":

[Levinson] speculated that evolutionary pressures were focused more on negotiating cooperative mutual intersubjectivity than on understanding the natural world . . . Having as its goals the ability to model other minds in order to read intention and to make rapid interactional moves based on those modeled intentions.³

Our S1 thinking, in adapting to the world we live in, is designed to be fast and contextual, operating on a limited number of inputs, and is constrained by the features of situations that are immediately accessible. Thus, it often fails to find optimal (or normative or correct) solutions. This can be seen in a classic probability contingency experiment:

The subject sits in front of two lights (one red and one blue) and is told that she or he is to predict which of the lights will be flashed on each trial and that there will be several dozen of such trials (subjects are often paid money for correct predictions). The experimenter has actually programmed the lights to flash randomly, with the provision that the red light will flash 70 percent of the time and the blue light 30 percent of the time. Subjects do quickly pick up the fact that the red light is flashing more, and they predict that it will flash on more trials than they predict that the blue light will flash. Most often, they switch back and forth, predicting the red light roughly 70 percent of the time and the blue light roughly 30 percent of the time.

This strategy of probability matching is suboptimal because it insures that, in this example, the subject will correctly predict only 58% of the time . . . compared to the 70% hit rate achieved by predicting the more likely color on each trial. In fact, much experimentation has indicated that animals and humans often fail to maximize expected utility in the probabilistic contingency experiment.⁴

There is, apparently, a reason behind this madness (or I should say, irrationality), however:

Turning to more complex organisms, it is a well-established fact that many higher organisms, including humans, exhibit a behavior known as probability matching, which involves making a kind of randomized foraging decision. The behavior naturally arises in a situation where several resources (usually food sources) of varying abundance are available to a group of animals. The animals spread themselves out so that the concentration of creatures at any one source matches the proportion of the total available resource accounted for by that source. Twice as many bread crumbs, for example, attract twice as many birds.

Experiments show that the mechanism primarily responsible for this phenomenon is a probabilistic foraging rule: choose between resources probabilistically, assigning a probability of choosing any source equal to the proportion of food delivered by that source. (Link, p. 325)

And this unlikely source tells us that we are not just like birds, but like bees as well:

A notable strategy by which bumblebees (and other animals) optimize choice in such situations is probability matching. When faced with flowers offering similar rewards but with different probabilities, bees match their choice behavior to the reward probabilities of the flowers. This seemingly "irrational" behavior with respect to optimization of reward intake is explained as an Evolutionary Stable Strategy (ESS) for the individual forager when faced with competitors, as it produces an Ideal Free Distribution (IFD) in which the average intake of food is the same at all food sources. (253)

There are several examples of reforms that move math education away from the primitive, fragile, and fallacious S1 thinking and into the apparently more stable S2 thinking. Explaining answers, alternative approaches, virtually all of NCTM's process skills, etc.

1 Leron, U. & Hazzan, O. "The Rationality Debate: Application of Cognitive Psychology to Mathematics Education". Educational Studies in Mathematics (2006) 62: 105-126.

2 Stanovich, K. & West, R. "Individual Differences in Reasoning: Implications for the Rationality Debate?". Behavioral and Brain Sciences (2000) 22(5).

3 Stanovich, K. (1999). Who Is Rational? Studies of Individual Differences in Reasoning. Lawrence Erlbaum Associates.

4 Stanovich, K. & West, R. "Evolutionary versus instrumental goals: How evolutionary psychology misconceives human rationality." Evolution and the psychology of thinking: The debate. Psychological Press.