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by Annie and John Selden

Mathematics in Automobile Production

Proportional Reasoning by Nurses

Modeling the Mathematics of Banking

Mathematical Models as Seen by Biologists

How do Scientists Interpret Graphs?

How much mathematics is used in various occupations? What kind and in what ways? Are there any implications for teaching or learning? Answers to these questions will vary with the jobs -- auto workers use mathematics differently from biologists -- and with the perspectives of those who do the looking. In the past few years, researchers (mainly in mathematics education) have observed auto workers, nurses, bankers, biologists, ecologists, and others, as they go about their day-to-day activities.

While all such studies have gathered empirical data on the mathematics used in various workplaces, they have also investigated such things as the nature of modeling and abstraction, the role of representations, and various associated learning difficulties. Below is a description of several such studies conducted in the U.S., U.K., and Canada, progressing from mainly empirical to more theoretical. We do not here discuss any of the equally interesting studies of Brazilian street sellers or South African carpenters that are often classified as ethnomathematics.

To check claims that today's workers need higher levels of mathematical skills, at least for those automobile industry jobs open to high school graduates, John P. Smith, III of Michigan State University undertook a three-year observational study of sixteen diverse sites employing 7,500 automobile production workers. He made thirty-nine visits, totaling ninety hours of observation, to sites varying from two Japanese "transplant" suppliers to various U.S. and Canadian suppliers, final assembly plants, and "after market" suppliers. He found three kinds of mathematical domains embedded in workers' activities: measurement, numerical and quantitative reasoning, and spatial and geometric reasoning.

Ten sites involving high-volume assembly work required only minimal mathematics; most workers repeatedly did the same small set of actions, such as bolting on components using air-pressure wrenches, with manual dexterity, eye-hand coordination, and visual acuity being very important. The mathematical demands on the majority of these workers were limited to counting, measurement, arithmetic with whole numbers or decimals, and interpreting numerical information; only a small number of quality control workers did jobs with more mathematical content. At three sites having a "team" structure which included the two Japanese "transplants," there were higher mathematical demands -- more diverse numerical calculations including ratios and rates, translation among fractions, decimals, and percentages, and evaluation and interpretation of algebraic expressions such as

At the remaining sites where machining and quality control lab work was done, the mathematics included substantial spatial and geometric reasoning mediated by "smart" tools such as a Conturograph, composed of an input device, a personal computer, and a software package that measured two linear dimensions and two angles of anti-lock brake valves. The workers had to match the "lines of best fit" to the profile which affected the reliability of the final measurements and judge whether the critical dimensions fell outside the design tolerances.

Smith concludes that for such durable goods manufacturing jobs, which admittedly constitute only about 8% to 9% of all U.S. nonfarm wage and salary jobs, "The equivalent of an eighth-grade mathematics education is adequate." However, there are caveats. This mathematics needs to be taught within "situated, problem-based curricula" that narrow the gap between "abstract" mathematics and its uses in the world. There also needs to be more emphasis on visualizing, orienting, plotting, locating, and reasoning in two- and three-dimensional coordinate systems and an introduction to trigonometry as soon as students can conceptualize ratios. [Cf. "Tracking the Mathematics of Automobile Production: Are Schools Failing to Prepare Students for Work?", *American Educational Research Journal* 36:4, 835-878, 1999.]

How do nurses calculate drug dosages on the ward? As a part of a larger study, researchers watched twelve pediatric nurses in a specialist U.K. children's hospital for a total of two hundred and fifty episodes, with thirty related to intravenous drug administration (all error free). They observed a variety of situated proportional reasoning strategies; to take a hypothetical example, a nurse might have to administer 300 mg of a drug that comes as 120 mg diluted in 2 ml of fluid.

All nursing textbooks examined mentioned "the nursing rule" for calculating drug dosages, one version of which is,

Also, all the nurses could recite this as "the mantra,"

but only four of the thirty episodes showed a variant of this rule being used. Instead, in their mental calculations, the nurses used several correct contextually-based within-measure scalar strategies, similar to those described in previous studies of proportional reasoning with sixth to ninth graders, and an across-measure functional strategy.

A simple within-measure scalar strategy was observed as two nurses prepared a morphine prescription. They needed to administer 1.5 mg of morphine packaged in 20-mg ampules diluted in 10 ml of fluid. The mental calculations of one nurse at the time were verbalized as, "Ten in five; five in two point five; one in point five, . . . . Zero point seven five." When interviewed later, she explained that she had made the following parallel computations: Given 20 mg in 10 ml; that's 10 mg in 5 ml; so that's 5 mg in 2.5 ml; 1 in 0.5; and 0.5 in 0.25; and then 1.5 in 0.75. The usual explanation for employing such seemingly awkward calculations is that they tend to preserve the proportion in the (situated) quantities, rather than having to work abstractly to calculate *(1.5/20) x 10*.

A variant of an across-measure functional strategy occurred when another nurse needed to give 120 mg of the antibiotic, amakacine, prepared in 100 mg per 2 ml vials. When interviewed, she explained, "With amakacine, whatever the dose is, if you just double the dose, it's what the mil is." In effect, this amounts to a transformation of dose mass (120 mg) to dose volume (2.4 ml) by doubling and moving the decimal point. Since the relationship of mass to volume is fixed for a given concentration of drug, one can calculate *(120 x 2)/100* instead of *(120/100) x 2*, obtaining an ml answer by beginning with the number of mg.

The researchers concluded that the nurses had a "culturally shared set of calculational strategies that served as well as, if not better than, the abstract rule they were taught," that "they had abstracted a concept of concentration" which could be seen as an example of situated abstraction. [Cf. Celia Hoyles, Richard Noss, and Stefano Pozzi, "Proportional Reasoning in Nursing Practice," *Journal for Research in Mathematics Education* 32:1, 4-27,2001.]

When a major U.K. investment bank handling billions of pounds called in Richard Noss and Celia Hoyles, two mathematics education researchers, to remedy what its management saw as a widespread reluctance on the part of its employees "to think mathematically about transactions," they began by trying to understand the essence of the problem. What did employees do that was mathematical? What would be a reasonable way to simplify and mathematize the banking situation? Prior to designing a specialized course, *Time is Money*, which included a small number of modifiable programs modeling future-value and present-value, Noss and Hoyles interviewed bank employees. These were not clerical personnel or janitors, but rather ranged from administrators to one fellow in charge of computer equipment support, whose budget ran to some million or two pounds a year. They used spreadsheets or the bank's theoretical models, often involving functions from *R*^{n} to *R*, designed by the bank's "rocket scientists," the term used for the mathematics Ph.D.s responsible for them.

The bank employees spoke their own specialized language; they saw dozens of distinct financial instruments whereas Noss and Hoyles saw them all as more or less the same. For example, treasury bills were considered distinct from certificates of deposit because of the way interest was calculated. Indeed, when asked the question, *Suppose you want $100 in one year. You have the chance of buying a simple instrument (say a CD) paying 8% or an instrument (like a Treasury bill) which offers a discount over the year again of 8%. Which would you choose and why?*, many employees gave situated, rather than abstract mathematical, answers. For instance, one person rejected the artificial world of simple interest and answered in terms of the bank's practice of compounding interest, *I would choose the Treasury bill. The discount will take account of the compound interest and will make the Treasury bill cheaper to buy than the simple interest instrument.*

While graphs were a part of the language of communication in the bank, they were not viewed holistically. For example, employees were asked to indicate which of four graphs (without explicit scales) showed the following combined scenarios most realistically: *An agent received commission for each transaction he makes as follows: (a) for transactions less than $30,000 . . . $750 plus 2 1/2 % of the transaction; (b) for transactions more than $30,000 . . . 5% of the transaction.*

To answer this, many added scales to the graphs and worked out commissions at specific points. Only when a graph represented real data, as if in an oddly displayed table, could they proceed. For them, "Graphs were just pictures of numbers, not graphical representations of functional relationships."

In the conventional view of modeling, one translates the situation into (abstract) mathematical terms, finds a mathematical solution, and translates that back to the "real world." Noss and Hoyles see this as simplistic. Instead, they had bank employees explore various interest rate scenarios by modifying specialized programs, thereby encouraging them to generalize and abstract while staying close to the banking setting. As the employees edited the programs, switching variables and parameters to model various financial situations, the mathematical and banking structures and their interconnections became more visible and meaningful to them. In this way, the specialized software was not a black box. One pedagogical challenge Noss and Hoyles see is "to employ technology which has contributed so much to the invisibility of mathematics, in order to make these meanings visible." [Cf. "The Visibility of Meanings: Modelling the Mathematics of Banking," *International Journal of Computers for Mathematical Learning* 1:1, 1996, 3-31.]

How do biologists use mathematical models? Are there implications for how we might teach modeling? To find out, a Cornell University advanced modeling class consisting of biology, ecology, agronomy, applied math, and statistics graduate students was studied. Two groups, each containing biologists and mathematicians, worked separately to formulate specific aphid-wasp-fungus models of population dynamics. Sara, one of the two instructors, contributed a theoretical biologist's perspective, as well as her field data on a species of pea aphid, its host plants (clover and alfalfa), a predatory wasp, and a fungal pathogen found in the fields of upstate New York.

As the work proceeded, it became clear to everyone just how differently biologists and mathematicians/statisticians view the task of modeling. The mathematicians wanted to quickly turn a jointly agreed upon schematic conceptualization of the problem (a kind of flow-chart of observed aphid-wasp-fungus behavior) into an interesting mathematical project not directly related to the data on aphid populations. For the mathematicians in the first group, this meant a simplified set of differential equations which they could analyze. For the mathematicians in the second group, this meant investigating the effect of aphids' genetic resistance to wasps and fungus on their birth and death rates, despite the fact that the experimental data showed no significant variation. As a result, the biologists in both groups became frustrated -- although they learned a lot of mathematics, they "learned nothing or almost nothing about the system."

At the root of the difficulty lay not only the differing perspectives coming from the two disciplines, but also their respective views of what models are for. The mathematicians thought in terms of mimicking the situation, i.e., providing a model that is both descriptive and predictive, often of large scale phenomena. The biologists, on the other hand, felt that biological systems are far too complex for this -- they wanted models to address specific biological questions, e.g., the population dynamics of pea aphids. They saw the role of models as stimulating conjectures, ruling out possibilities, and serving as 'experiments' for theoretical claims. As Carlos, the other instructor explained, Darwin's theory of natural selection was a model, which though now questioned by biologists, ruled out Lamarckian and other previous theories and allowed a first approximation explanation for evolution. Rather than striving for a good 'fit' between model output and empirical data, one seeks a good 'fit' between one's understanding of biological processes and the model. Models are a way of understanding, much like metaphors.

It was suggested that perhaps many of our assumptions about how models are constructed and used in science should be reevaluated. The idea that one "gathers all the relevant information, creates the appropriate mathematical relationships, enters the data, runs the model, and then learns from the results" seems too simplistic for many situations. [Cf. Erick Smith, Shawn Haarer, and Jere Confrey, "Seeking Diversity in Mathematics Education: Mathematical Modeling in the Practice of Biologists and Mathematicians," *Science and Education* 6:5, 441-472, 1997.]

By analyzing the use of graphs by 45 students in a second-year university ecology course, 10 fifth-year post-baccalaureate elementary science education students, and 15 practicing scientists from theoretical and field ecology, forest engineering, and physics, Wolff-Michael Roth of University of Victoria not only noted that graphs are often not transparent, but also developed a semiotic model for activities involved in learning to interpret graphs. Roth who was trained as a physicist now investigates various aspects of mathematics and science learning.

One population graph, typical of that found in introductory ecology textbooks, displayed birth and death rates versus population density.

It showed a concave down parabolic birth rate superimposed on an increasing linear death rate. When Roth asked his subjects about this graph, not only did he find that students incorrectly attended to the height of the graph instead of its slope and vice versa, he also found some scientists interpreted *b-d, where **b*=birth rate and *d*=death rate, incorrectly as a situation where the population goes extinct. Rather than saying what this graph meant 'en bloc,' the scientists moved back-and-forth between individual features of the graph and various natural phenomena, trying to relate individual aspects of the graph to particular phenomena. For example, they asked whether the maximum of *b* or the maximum of *b-d* was relevant. Meaning for the graph was slowly constructed and emerged only after considerable interpretive activity.

Using the language of semiotics (sign-referent-interpretant), Roth observes that, unlike words where recognition is often instant and meaning clear (as with the word 'graph'), graphs themselves are neither unequivocal nor complete signs pointing to unique 'natural objects.' He conjectures that individuals move from viewing graphs as *things* to considering graphs as *signs* which come to stand for '*natural objects*,' and only subsequently, as was the case for his physicists and theoretical ecologists, do graphs become '*natural objects*' in their own right.

Roth sees implications for science professors and teaching assistants who try to explain the meaning of graphs. For the instructors, the graphs are largely transparent so they talk about the phenomena without elaborating the correspondence between individual aspects of graphs and particular phenomena. For the students, there is then a double problem -- they neither know the phenomena nor have they constructed the graph as a sign object. [Cf. "Unspecified Things, Signs, and 'Natural Objects': Towards a Phenomenological Hermeneutic of Graphing," *Proceedings of PME-NA 20* (Vol. I), 291-297, 1998.]

In another study, Roth asks whether scientists are competent readers of graphs generally or whether they mainly have intimate knowledge of their fields and of particular graphs. He notes that Cartesian graphs are central to scientists' representations of the world. Scientists use graphs to construct phenomena, prove the existence of phenomena, and for rhetorical purposes in publications. He found more than 420 such graphs in 2,500 pages in five top-ranked ecology journals. To better understand how professionals read familiar and unfamiliar graphs, sixteen practicing scientists who had an M.Sc. or Ph.D. and at least five years experience doing independent research were asked to interpret the same set of three graphs (including the above population graph) and to explain one or more graphs from their own publications.

For an individual, Roth sees two fundamental difficulties in learning to read graphs of natural phenomena: (1) Structuring -- learning the arbitrary, but conventional, relations that exist between aspects of a graph and the phenomena represented. (2) Grounding -- interpreting graphs which generally contain little contextual information, that is, seeing graphs as describing specific phenomena. Two individuals, Ted and Karen, are used to illustrate the different roles played by structuring and grounding as individuals read unfamiliar and familiar graphs. When Ted, a physicist, interpreted the above population graph he struggled (as did 6 others of the 16) to make sense of it. He first made general observations such as, "rate would be a number differentiated by time, so this would be a measure of change," and later drew on resources from everyday life, "The birth rate increases to a maximum . . . probably because of limits in the environment or competition or disease or overcrowding or social problems within the population" to help him make sense of the graph. He repeatedly went back-and-forth between structuring and grounding.

Karen, a water technician, routinely reads graphs produced by a recorder monitoring the water level of a creek flowing through a local watershed. For her, these graphs are transparent -- she is able to leap directly to the things in her world. She can explain to visitors the meaning of the graphs on-the-fly, "In a really, really incredible rain event, we get right to the top here it's 5,000 liters per second . . . because we're getting run-off from all the pavements, rooftops, roadways . . . If that low in the summer gets below the first square, we dry up, farming cannot occur." Karen shows that, based on her four-year experience working in the valley, she can go effortlessly back-and-forth from a peak in the graph to an "incredible rain event" and from the natural phenomenon of "summer" to the graph being low (i.e., from sign to referent and back again). She is familiar with the geographical area, the measuring device, and the farming practices in the area. As a result, she can point to a "blip" in the graph and say, "this is a non-natural event, a natural event has a duration, kind of has a roundness . . . that's an obvious clogged pipe." For Karen, reading the graph is as transparent as reading a newspaper. Roth found a similar transparency and intimacy when scientists discussed graphs from their own research, but chose to describe Karen to avoid having to provide arcane technical background.

Roth sees the competent interpretation of graphs as requiring more time than traditional instruction has allowed. Students need experiences that develop competency with both graphs (the expressive domain) and the world (the referent domain), as well as translations between the two. [Cf. "Professionals Read Graphs (Imperfectly?)," *Proceedings of PME-NA 21* (Vol. 1), 385-391,1999 and the accompanying paper, "Professionals Read Graphs: A Semiotic Analysis."]

What the above studies have in common is a willingness on the part of the researchers to examine how non-mathematicians use or view mathematics in their jobs/disciplines, together with an openness to possible implications for teaching. The workers, whether nurses or scientists, were seen as having an embodied and situated knowledge of mathematics, but not one that is incapable of generalization and abstraction. This is reminiscent of the Freudenthal Institute's Realistic Mathematics Education (RME) perspective in which students are encouraged to investigate situations which are 'real' for them and to use what they know about these situations to help them mathematize the situations and gradually generalize and abstract from them. [For a description of RME, see Martin van Reeuwijk, "Students' Knowledge of Algebra," *Proceedings of PME-19*, 1995 or Hans Freudenthal, *Revisiting Mathematics Education*, Kluwer, 1991.]