Defining Connections
 Introduction | Varieties of Connections | Other Contexts | The Teacher's Role | Your Journal
 Applying Mathematics NCTM contends, "In grades 9-12 students should be confidently using mathematics to explain complex applications in the outside world" (NCTM, 2000, p. 66). This kind of exposure to connections adds interest and value to the study of mathematics and also broadens a student's preparation for future academic work. As well, it helps provide a foundation for mathematical competence in adult life, in the workplace, and in other spheres. High school students are sometimes perplexed at first when they discover that many mathematical skills and concepts appear as important components of science courses, such as chemistry and physics. For example, the study of acceleration in a science course involves the use of equations and graphs, analysis of data, and consideration of a number of parameters. Solving chemistry problems can require solving simultaneous equations and a true understanding of percentages. In biology and environmental studies classes, students encounter population studies, such as predator-prey relationships, that can be modeled through the use of differential equations. Reports of survey data on predicted voting habits in a social science class may be analyzed using knowledge of probability to predict possible outcomes. Geometric ideas are frequently used in fine arts courses. Mathematics is also connected to physical education and sports. On a simple level, it is used to measure running times or to compare data to judge how one's performance has improved. Major league sports, of course, provides a popular context for demonstrating the use of statistics, equations, and precision of measurement and calculation. For example, in baseball, batting averages, on-base percentages, and slugging percentages are of great interest to fans, to team management, and to many students. Here are the formulas for calculating two important statistics: Slugging Average (aka SA or SLG) = (Singles + [2 x Doubles] + [3 x Triples] + [4 x Home Runs]) ÷ At Bats On Base Percentage (aka OBP or OBA) = (Hits + Walks + Hit-by-Pitch) ÷ (At Bats + Walks + Hit-by-Pitch + Sac Flys) Many other real-world concerns that are of interest to high school students can be set up as mathematics problems. Predicting future college costs, comparing the gas mileage and initial cost of several cars, analyzing car loans, making health-related decisions -- all of these can be modeled and analyzed mathematically. The ability to apply mathematics to such situations expands students' view of the utility of mathematics and better prepares them to make informed decisions. There are also connections between mathematics and life after high school. Work, participation in civic life, and managing household expenses all require fluency and proficiency in mathematics. In business and finance, decisions are made by the use of complex calculations that project profitability. For the consumer, it is critical to be able to analyze the terms and implications of various loan offers through the use of common formulas. For the citizen, it is important to be able to analyze data about such issues as environmental controversies, electoral politics, and medical risks, whether the data appear in tabular form or in a graphical presentation. Connections via Technology Technology provides another tool whereby both students and teachers can explore connections and see them more easily than might otherwise be possible. Calculators and computers are now common tools for living and working in modern society, and connections to basic mathematics are key to working effectively with technology. For example, when interpreting the reasonableness of data displays on a calculator or computer, basic estimation skills, number sense, and understanding of the magnitude of numbers and scientific notation are necessary. At the same time, technology makes it possible to work with more elaborate data sets and more complex problems. As we saw in Part A, technology not only provides quick access to graphs for equations, but it can also become a forum for further mathematical inquiry. The students' misconceptions related to significant digits, graph scales, and interpreting data screens all came out while using a graphing calculator. In Part D, we will see how technology can help develop students' understanding of the relationship between an acceleration situation and a parabola. Spreadsheets make possible the efficient analysis of a large quantity of data related to everything from the value of a stock portfolio to life insurance cost projections to seasonal rainfall. Data analyzed using spreadsheets can be displayed in a variety of ways through the use of technological applications, which increases opportunities for communication and further analysis. Technology provides a link between mathematics and applications, in both the classroom and the work world. For example, computer-based laboratory experiments use technology to gather and analyze data, as do devices used by technicians in many fields. These devices shorten the time needed to do basic data collection and representation, potentially saving time for more in-depth analysis of the data and for discovery of connections, both the obvious and the unexpected.
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