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For example, Jackie worked with a crop and soil scientist. She was intrigued by the way in which measurement of weight is used to count seeds. First, her sponsor would weigh a test batch of seeds to generate a benchmark weight. Then, instead of counting a large number of seeds, the scientist would weigh an amount of seeds and compute the number of seeds such a weight would contain.
Rebecca worked with a carpeting contractor who, in estimating costs, read the dimensions of rectangular rooms off an architect's blueprint, multiplied to find the area of the room in square feet doing conversions where necessary , then multiplied by a cost per square foot which depended on the type of carpet to compute the cost of the carpet. The purpose of these estimates was to prepare a bid for the architect where the bid had to be as low as possible without making the job unprofitable.
Rebecca used a chart Table to explain this procedure to the class. Joe and Mick, also working in construction, found out that in laying pipes, there is a "one by one" rule of thumb. When digging a trench for the placement of the pipe, the non-parallel sides of the trapezoidal cross section must have a slope of 1 foot down for every one foot across. This ratio guarantees that the dirt in the hole will not slide down on itself. Thus, if at the bottom of the hole, the trapezoid must have a certain width in order to fit the pipe, then on ground level the hole must be this width plus twice the depth of the hole.
Knowing in advance how wide the hole must be avoids lengthy and costly trial and error. Other students found that functions were often embedded in cultural artifacts found in the workplace. For example, a student who visited a doctor's office brought in an instrument for predicting the due dates of pregnant women, as well as providing information about average fetal weight and length Figure While the complexities of organizing this sort of project should not be minimized—arranging sponsors, securing parental permission, and meeting administrators and parent concerns about the requirement of off-campus, after-school work—we remain intrigued by the potential of such projects for helping students see mathematics in the world around them.
The notions of identifying central mathematical objects for a course and then developing ways of identifying those objects in students' experience seems like an important alternative to the use of application-based materials written by developers whose lives and social worlds may be quite different from those of students. Chazen, D. Algebra for all students?
Journal of Mathematical Behavior , 15 4 , Eckert, P. Jocks and burnouts: Social categories and identity in the high school. New York: Teachers College Press. Fey, J.
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Concepts in algebra: A technological approach. Dedham, MA: Janson Publications. Kieran, C. Introducing algebra by mean of a technology-supported, functional approach. Bednarz et al. Kincaid, J. The autobiography of my mother. New York: Farrar, Straus, Giroux. Nemirovsky, R. Mathematical narratives, modeling and algebra. Approaches to algebra , pp. Schwartz, J.
Getting students to function in and with algebra. Dubinsky Eds. Serres, M. Mathematics and philosophy: What Thales saw … In J. Bell Eds. Baltimore, MD: Johns Hopkins. Thompson, P. Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics , 25 , Yerushalmy, M. Seizing the opportunity to make algebra mathematically and pedagogically interesting. Romberg, E. Carpenter Eds. To assist his research in mathematics teaching and learning, he has taught algebra at the high school level. His interests include teaching mathematics by examining student ideas, using computers to support student exploration, and the potential for the history and philosophy of mathematics to inform teaching.
She has interest in mathematics reform, particularly in meeting the needs of diverse learners in algebra courses. A city is served by two different ambulance companies. City logs record the date, the time of the call, the ambulance company, and the response time for each call Table 1. Analyze these data and write a report to the City Council with supporting charts and graphs advising it on which ambulance company the operators should choose to dispatch for calls from this region. This problem confronts the student with a realistic situation and a body of data regarding two ambulance companies' response times to emergency calls.
The data the student is provided are typically "messy"—just a log of calls and response times, ordered chronologically. The question is how to make sense of them. Finding patterns in data such as these requires a productive mixture of mathematics common sense, and intellectual detective work. It's the kind of reasoning that students should be able to do—the kind of reasoning that will pay off in the real world. In this case, a numerical analysis is not especially informative. On average, the companies are about the same: Arrow has a mean response time of The spread of the data is also not very helpful.
The ranges of their distributions are exactly the same: from 6 minutes to 19 minutes. The standard deviation of Arrow's response time is a little longer—4. Graphs of the response times Figures 1 and 2 reveal interesting features. Both companies, especially Arrow, seem to have bimodal distributions, which is to say that there are two clusters of data without much data in between.
“What if this is what we should be doing? What if it’s that simple?”
The distributions for both companies suggest that there are some other factors at work. Might a particular driver be the problem? Might the slow response times for either company be on particular days of the week or at particular times of day? Graphs of the response time versus the time of day Figures 3 and 4 shed some light on these questions. These graphs show that Arrow's response times were fast except between AM and AM, when they were about 9 minutes slower on average. Similarly, Metro's response times were fast except between about PM and PM, when they were about 5 minutes slower.
Perhaps the locations of the companies make Arrow more susceptible to the morning rush hour and Metro more susceptible to the afternoon rush hour. On the other hand, the employees on Arrow's morning shift or Metro's afternoon shift may not be efficient. To avoid slow responses, one could recommend to the City Council that Metro be called during the morning and that Arrow be called during the afternoon.
A little detective work into the sources of the differences between the companies may yield a better recommendation. Comparisons may be drawn between two samples in various contexts—response times for various services taxis, computer-help desks, hour hot lines at automobile manufacturers being one class among many. Depending upon the circumstances, the data may tell very different stories.
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Even in the situation above, if the second pair of graphs hadn't offered such clear explanations, one might have argued that although the response times for Arrow were better on average the spread was larger, thus making their "extremes" more risky. The fundamental idea is using various analysis and representation techniques to make sense of data when the important factors are not necessarily known ahead of time. Practice "back-of-the-envelope" estimates based on rough approximations that can be derived from common sense or everyday observations.
Some 50 years ago, physicist Enrico Fermi asked his students at the University of Chicago, "How many piano tuners are there in Chicago? Thus, many people today call these kinds of questions "Fermi questions. Scientists and mathematicians use the idea of order of magnitude , usually expressed as the closest power of ten, to give a rough sense of the size of a quantity. In everyday conversation, people use a similar idea when they talk about "being in the right ballpark.
To say that these salaries differ by a factor of or 10 3 , one can say that they differ by three orders of magnitude.
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Such a lack of precision might seem unscientific or unmathematical, but such approximations are quite useful in determining whether a more precise measurement is feasible or necessary, what sort of action might be required, or whether the result of a calculation is "in the right ballpark. On the other hand, determining whether 5, or 6, is a better estimate is not necessary, as the strategies will probably be the same. Careful reasoning with everyday observations can usually produce Fermi estimates that are within an order of magnitude of the exact answer if there is one.
Fermi estimates encourage students to reason creatively with approximate quantities and uncertain information. Experiences with such a process can help an individual function in daily life to determine the reasonableness of numerical calculations, of situations or ideas in the workplace, or of a proposed tax cut.