Monday, May 4, 2009

Math in Everyday Life & Careers

Often, we hear that the poor performance of students on assessments will have a huge impact on America's ability to compete in the global economy.  However, there is little (if any) research to back such scare tactics.  In fact, the research tells a different story about mathematics in the real world and the traditional skill based computational curriculum that anti-reformists back.  Julie Gainsburg's research about mathematics in work and in life tells us a different story.  One of her studies, School mathematics in work and life: what we know and how we can learn more finds:

Ethnographic studies of child and adult math use in everyday life have generally all come to the same conclusion: there is a wide gap between school-taught mathematical methods and the math people use outside of school. Context plays a central role in everyday problem solving. Studying farmers, carpenters, and fishermen in Brazil, Nunes et al. showed that these adults relied on contextual aspects of the quantitative problems of their work to help them calculate in ways unlike school-taught algorithms. While school arithmetic is considered rule-based and divorced from meaning, these adults' ‘street arithmetic’ preserved situational meaning at each calculation step. In their study of grocery shoppers, Lave et al. demonstrated that the setting of the store (the layout of the aisles, the price labels, the appearance of products, etc.) was inextricably linked to the shoppers' arithmetic decision making, with the setting both constraining and supporting their solutions. The shoppers' arithmetic performance in shopping was far better than on a test of mathematically analogous, school-type operations. The shoppers' school backgrounds correlated only with their scores on the school-type test and not with their performance on, or willingness to use, arithmetic in shopping. This study also illuminated a key feature of real-life problem solving: gap closing. Typically, the shoppers would reshape their shopping calculation problems while simultaneously shaping the solutions, thereby moving problem and solution towards each other—an impossible strategy for most school math problems.

People avoid even well-known school-taught algorithms when solving everyday problems. Asked to find 3/4 of a 2/3-cup serving of cottage cheese, with the actual cheese and measuring cups available, adult dieters in de la Rocha's study ignored the standard algorithm for multiplying fractions and instead invented strategies that often involved manipulating the physical objects.

What about engineers, you ask?

Hutchins's (Cognition in the wild, The MIT Press, Cambridge, 1995) extensive ‘cognitive ethnography’ of the practice of navigation on a Navy ship illuminated several features of ‘cognition in the wild’. Navigation, Hutchins showed, is accomplished through a complicated, interconnected system of human tasks, methods of representation and communication, instruments, tools, charts, tables, codes, techniques, and traditions. The knowledge required to dock a huge ship is distributed among these system elements, too great for any one person or tool to possess entirely. The cognitive demands on any single crewmember are actually quite mundane and technological tools have supplanted most calculations. According to Hutchins, tools do not amplify the cognitive abilities of individuals in the system, as is often claimed, but replace their more difficult tasks (like calculating) with easier ones (like chart reading). Stressing the cultural diffusion of knowledge, Hutchins warned researchers not to infer the cognitive requirements of an individual from observations of the overall activity in which he participates. Hutchins's interest, in fact, was not the individual but the ‘environment for thinking’ that the culture of navigation has developed over its history. According to Hutchins:

The real power of human cognition lies in our ability to flexibly construct functional systems that accomplish our goals by bringing bits of structure into coordination… [A] proper understanding of human cognition must acknowledge the continual dynamic interconnectivity of functional elements inside with functional elements outside the boundary of the skin.

Hutchins's warning challenges the educational policy rhetoric that our highly mathematized world imposes high-level mathematical demands on each citizen. Today's technologies may indeed have replaced many of our mathematical tasks with button pushing and chart reading, thereby reducing our need to personally possess sophisticated calculation skills.

Analyzing the mathematical behavior of architects and structural engineers jointly engaged in a design session, Hall found that quantitative reasoning was prevalent but mostly involved only simple arithmetic with routine quantities. More interestingly, Hall showed how forms of mathematical representation are socially negotiated, both historically and on the spot. The diverse perspectives and accountabilities of the architects and engineers shaped their mathematical descriptions, models, and uses of quantity. Hall's portrayal of these workers' mathematical behavior—culturally and socially dependent, flexible, parochial, and open for reinvention and interpretation—clashes with the traditional image of mathematics as a static, objective, universal language that exists virtually independently of people, untainted by subjective perceptions and motivations.

Hoyles et al. investigated experienced nurses' use of proportional reasoning in calculating drug doses. Though trained to use a single computational routine for this purpose, in practice the nurses employed a wide range of effective proportional reasoning strategies. Hoyles et al. believe their findings confirm the existence of a large gap between school math and the math of the workplace, where abstract rules ‘fade into the background’, and culturally held and artifact-reliant procedures serve just as well or better. While rejecting the notion that the nurses used general, school-taught algorithms at work, Hoyles et al. proposed that the nurses had themselves ‘abstracted’ the theoretical concept of concentration (the proportion of drug to solvent) in the course of their years of practice.

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