A patient with an aggressive cancer faces two options for treatment: With Option A, he has a 40% chance of surviving for a year, but if he makes it that long then his chance of surviving a second year is two out of three. With Option B, he has a 50-50 chance of surviving each of the first and second years. Survival rates beyond the second year are similar for each option. Which choice should he make?

Mathematics in Life and Work

• Reading Maps
• Ensuring Quality
• Using Spreadsheets
• Building Things
• Thinking Systemically
• Making Choices

            The diverse contexts of daily life and work provide many realistic views of functional mathematics--of the mathematical practice underlying routine events of daily life. These contexts offer episodic views, incomplete in scope and less systematic than a list of elements, but more suggestive of the way functional mathematics may be introduced in courses.

            Reading Maps. Road maps of cities and states provide crucial information about routes and locations. For those who know how to "read" them, maps also convey scale and direction, helping drivers know which way to turn at intersections, permitting quick estimates of driving time, and revealing compass directions that relate to highway signs at road intersections. Map scales are just ratios--an essential part of school mathematics. Different scales not only convey different detail, but also require different translations to represent distance.

            Reading maps is not just a matter of thinking of distances in different scales. In many cases, the geometry of maps represents other features such as temperature or soil content. Most common are weather maps with color-coded regions showing gradations in recorded or predicted temperatures. Similar maps sometimes display recorded or predicted precipitation, barometric pressure, vegetation features, or soil chemistry. Like topographic maps used by hikers, these maps represent some feature of the landscape that changes from place to place. The spacing between regions of similar temperature (or pressure, or elevation) conveys the steepness (or gradient) of change--what mathematicians call the "slope" of a line.

            Scale-drawings and blueprints are also widely used to illustrate details of homes, apartments, and office buildings. These drawings represent sizes of rooms, locations of windows and doors, and--if the scale permits--locations of electrical outlets and plumbing fixtures. Architects' rulers with different units representing one foot of real space make it possible to read real distances off scale drawings, taking advantage of the geometrical properties of similar figures. New geographic information systems (GIS) encode spatially oriented data in a form suitable for computer spreadsheets, thereby enabling other factors (e.g., costs, environmental factors) to be logically linked to the geometric structure of a map.

            Ensuring Quality. Statistical process control (SPC) and statistical quality control (SQC) are crucial components of high-performance manufacturing, where "zero defect" is the goal. Instead of checking and repairing products after manufacture, firms like Boeing, General Motors, Kodak, Motorola, and Siemens now insist that at every step in the manufacturing process, materials, parts, and final products be manufactured within tight tolerances. Moreover, workers on assembly lines are responsible for ensuring this consistent level of quality.

            The two tools that make this possible are based on statistics--the science of collecting and organizing data. The first, statistical process control, occurs during manufacture: assembly line workers chart key indicators of the process--perhaps the temperature of a mixture or the pitch of a grinding tool--on graph paper marked with curves representing the limits determined by the required (or contracted) tolerances. If the process strays outside these limits, or approaches them too often, workers may decide to shut down the assembly line to make adjustments in the manufacturing process rather than risk producing products that do not meet design specifications.

            Statistical quality control is like statistical process control, but takes place when components (e.g., computer chips) are completed. By sampling finished products and charting their performance characteristics, workers can identify potential problems before products exceed permitted tolerances--and then take action to prevent the shipping or further manufacture of defective (i.e., out-of-tolerance) products.

            Using Spreadsheets. Almost everyone who works with a computer uses a word processor for writing, whether for correspondence or business reports. Almost as popular are "number processors," commonly known as spreadsheets. Originally designed as a tool for accountants, spreadsheets are ubiquitous both in the office and at home--wherever anyone deals with budgets and expenses, taxes and investments. Spreadsheets are used to record business inventories and scientific data, to keep track of medical records and student grades, to organize crop records and airline schedules. Virtually any systematic information can be made more useful by being put in a properly organized spreadsheet.

            To a mathematician, a spreadsheet is just algebra playing on a popular stage. The basic operations of a spreadsheet--adding cells together, calculating percentages, projecting growth rates, determining present values--are entered as formulas into the appropriate cells. More complex formulas (e.g., exponential, financial, trigonometric) are available from a pull-down menu. Once the computations are completed, the results can be displayed in graphs of various sorts (lines, bars, pies), often in vivid color.

            Figuring out how to translate a task into a spreadsheet design is just like setting up a word problem in algebra: it involves identifying important variables and the relations among them. Preparing a spreadsheet requires equations which are suitably located in the cells. The spreadsheet does the arithmetic, and the designer does the algebra. Then, as in any mathematical exercise, the designer needs to check the results--typically by specifying independent computations to confirm key spots in the spreadsheet. (For example, adding all the entries in a grid can confirm the accuracy of the sum of the row totals, thus catching possible errors in the spreadsheet formulas.) Variables, equations, graphs, word problems--the ingredients of a good algebra course--are just the ticket for mastering spreadsheets.

            Building Things. One in every four American workers builds things--automobiles or airplanes, bicycles or buildings, containers or chips. These products are three-dimensional, created by casting and cutting, by folding and fastening, by molding and machining. Designing things to be built (the work of engineers and architects) and building objects as designed (the work of carpenters and machinists) require impressive feats of indirect measurement, three-dimensional geometry, and visual imagination.

            In a typical aluminum airplane part, for example, some measurements are specified by the designers, while others must be calculated in order to program the cutting tool that will actually create the part. In three dimensions, things are even more complicated. Planning how to drill holes at specified angles in a block of aluminum whose base is not square and whose sides are tilted in odd directions would tax the skills of most mathematics teachers. But machinists are expected to perform these calculations routinely to determine settings on a "sine plate," a device whose surface can tilt in two different dimensions in order to compensate for odd angles on the part that is to be drilled.

            Both designers and builders now use computer-assisted design (CAD) and computer-assisted manufacturing (CAM) to ensure the exacting tolerances required for high-performance manufacturing. To use these tools effectively, workers need to have mastered the basic skills of drawing geometric objects, measuring distances, and calculating angles, distances, areas, and volumes. The basic principles of geometry in three dimensions are the same as those in two dimensions, but the experience of working in three dimensions is startlingly more sophisticated. A good command of geometry and trigonometry is essential for anyone building things in today's manufacturing industries.

            Thinking Systemically. Systems surround us--in commerce, science, technology, and society. In complex systems, many factors influence performance, thus making the task of solving problems inherently multidimensional. Indeed, the interaction of different factors is often difficult to predict, sometimes even counterintuitive. Complex systems defy simplistic single answers. Thus, the first step in mathematical analysis is often to prepare an inventory of all possible factors that might need to be considered.

            For example, the rise of efficient package delivery services and instantaneous computer communication have enabled many manufacturing companies to operate with minimum inventories, thus saving warehousing costs but risking a shutdown if any part of the network of suppliers fails. Understanding how a system of suppliers, communication, and transportation works requires analysis of capacity, redundancy, single-point failures, and time of delivery--all involving quantitative or logical analyses.

            Other system problems arise within the everyday work of a typical small business. For example, the stockroom of a shoe store holds several thousand boxes labeled by manufacturer, style, color, and size and arranged on floor-to-ceiling shelves. Deciding how to arrange these boxes can have a significant impact on the profit margin of the store. Obvious options are by manufacturer, by style, by size, by frequency of demand, or by date of arrival. Clerks need to be able to find and reshelve shoes quickly as they serve customers. But they also need to be able to make room easily for new styles when they arrive, to compare regularly the stockroom inventory with sales and receipt of new shoes, and to locate misshelved shoes. Mathematical thinking helps greatly in exploring the advantages and disadvantages of the many possible systems for arranging the stockroom.

            Making Choices. Life is full of choices--to rent an apartment or purchase a home; to lease or buy a car; to pay off credit card debt or use the money instead to increase the down payment on a house. All such choices involve mathematical calculations to compare costs and evaluate risks. For example:

The rent on your present apartment is $1,200 per month and is likely to increase 5% each year. You have enough saved to put a 25% down payment on a $180,000 townhouse with 50% more space, but those funds are invested in an aggressive mutual fund that has averaged 22% return for the last several years, most of which has been in long-term capital gains (which now have a lower tax rate). Current rates for a 30-year mortgage with 20% down are about 6.75%, with 2 points charged up front; with a 10% down payment the rate increases to 7.00%. The interest on a mortgage is tax deductible on both state and federal returns; in your income bracket, that will provide a 36% tax savings. You expect to stay at your current job for at least 5-7 years, but then may want to leave the area. What should you do?

            This sounds like a problem for a financial planner, and many people make a good living advising people about just such decisions. But anyone who has learned high school mathematics and who knows how to program a spreadsheet can easily work out the financial implications of this situation. Moreover, by doing it on a spreadsheet, it is quite easy to examine "what if" scenarios: What if the interest rate goes up to 7% or 7.25%? What if the stock market goes down to its traditional 10-12% rate of return? What if a job change forces a move after three years?

            In contrast to many problems of school mathematics which are routine for anyone who knows the right definitions (e.g., what is cos ([pi]/2)?) but mystifying otherwise, this common financial dilemma is mathematically simple (it involves only arithmetic and percentages) but logically and conceptually complex. There are many variables, some of which need to be estimated; there are many relationships that interact with each other (e.g., interest rates and tax deductions); and the financial picture changes each year (actually, each month) as payments are made.

            The complex sequence of reasoning involved in this analysis is typical of mathematics, which depends on carefully crafted chains of inferences to justify conclusions based on given premises. Students who can confidently reason their way through a lengthy proof or calculation should have no problem being their own financial advisors. And students who learn to deal with long chains of reasoning inherent in realistic dilemmas will be well prepared to use that same logic and careful reasoning if they pursue the study of mathematics in college.