Academic journal article The Mathematics Enthusiast

A Cognitive Framework for Normative Reasoning under Uncertainty, and Reasoning about Risk, and Implications for Educational Practice

Academic journal article The Mathematics Enthusiast

A Cognitive Framework for Normative Reasoning under Uncertainty, and Reasoning about Risk, and Implications for Educational Practice

Article excerpt

Introduction

Clarifying what is normative or appropriate reasoning under various circumstances provides a valuable reference for guiding what should be taught, and, in contrast, what should not be. This paper proposes a cognitive framework for viewing normative reasoning and behavior under uncertainty, including the applying of knowledge of probability and statistics in real world situations; and identifies implications for educational practice. In sections below, factors relevant to normative reasoning under uncertainty are identified, illustrated with examples, and related to the research literature on reasoning under uncertainty. In particular, examples involving reasoning about risk are addressed, including reasoning reflected in industry standards for risk management. In the final sections, the factors are integrated into a cognitive framework; and implications for educational practice are identified.

In real world situations, we are often in the position that the outcome of a situation, which is subject to uncertainty, matters to us. To reason and behave normatively at such times is important, since doing so helps to bring on potential benefits and/or to stave off potential difficulties. Such real world situations draw interest and are engaging, and typically call for action, because the results matter. Such real world situations also are aptly described as involving risk. Not only is the outcome of the situation uncertain, with alternative possible outcomes, but the possible outcomes have positive or negative impact, so that there is risk that a positive outcome will not occur, and/or risk that a negative outcome will occur. By addressing what is normative reasoning and behavior in such situations, the cognitive framework presented here applies in general to reasoning about risk.

As a simple example, consider observing the rolling of a pair of six-sided dice. There is uncertainty in the outcomes, but unless the rolling occurs in the context of a game or other real world consequences, the outcomes don't really matter. Now, consider that the rolling of the dice is occurring in the context of gambling, and that you are about to place a large sum of your money as a bet on the outcome of the roll. Now the outcome is more important, the situation is more engaging, you are interested in your options for action that may make a difference in the situation, and there is risk. As another example, consider that you are a young person and occasionally contemplate your own mortality, but realize that, due to your generally safe environment and healthy habits, your odds are good that you will live a long life; and so the issue of your possible early death is not of real concern to you. Now consider that you and your spouse are just starting a family. Although the probability that you will die relatively young is still low, now the possibility of your early death is of concern to you, since it would have a great financial and otherwise life-impacting effect on your remaining family. The situation regarding your mortality is now more important, more engaging, involves risk, and has led you to consider possible actions, including buying life insurance.

The focus of the framework presented here is not just on people's judgments of probability of outcomes in situations involving uncertainty, but a broader sense of reasoning under uncertainty that includes consideration of risk, perceived consequences of outcomes, and human actions/ behavior in that context.

Mathematical and Non-mathematical Reasoning

Historically, the research area of "reasoning under uncertainty" in cognitive psychology and decision science has been closely identified with the mathematics of probability and statistics. In a seminal paper in the field, Tversky and Kahneman (1974) reported their research in which adults had been posed written problems calling for them to reason and make judgments under uncertainty; and the authors concluded that their subjects showed "biases" and "errors" in their judgments, using reasoning heuristics, such as representativeness and availability, while not being influenced by relevant mathematical information provided in the problems, such as prior probabilities or base rates, and sample sizes. …

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