Comparison shopping

Summary: Both simple and complex algorithms are used to compare consumer prices and contextualize mathematics instruction.

The globalization of the marketplace has resulted in a plethora of choices for any given item at both the local store and via the Internet. People comparison shop for both very expensive items like a car or a plane ticket and fairly inexpensive purchases like a box of cereal. Comparison shopping is perhaps one of the most widely used applied mathematics lessons, both in K–12 and lower-level college courses.

Mathematics is at the forefront of comparison shopping through unit pricing, which makes use of division and fractions. Geometric methods can be used to compare volume or weight. Notions from pre-algebra and algebra model financial decisions such as purchasing a cell phone plan or taking out a car loan. Students explore parameters in order to make balanced and informed choices. Mathematics educators not only use these examples in classrooms, but they also study their effectiveness. Researchers and online shopping agents take advantage of mathematical methods to extract, compare, and mine huge amounts of data. Comparison techniques also include data envelopment analysis and multiple regression.

Unit Pricing

One method of comparing differently priced items in different sized containers is through unit pricing. Dividing the price by the quantity or amount of items, such as how many ounces, will yield a cost per unit term that can be used for comparison purposes. For example, an 11.5 oz box of cereal might cost $4.49, while a 24 oz box of cereal costs $4.99. The unit price of the first box is $4.49/11.5≂$0.39 per ounce, while the second box is $4.99/24≂$0.21 per ounce. Some items are already priced by their weight, like meats, fruits, vegetables, or coffee, and others are priced according to their volume, that is, by the container size. For those items that are not priced by weight or volume, unit pricing is listed on the shelf tag in many stores. However, the unit price is not the only important feature in comparison shopping. Personal preferences and other important factors must also be taken into consideration, like whether one will be able to use up a larger quantity before the expiration date. Unit pricing examples proliferate in lessons on fractions and in classes like pre-algebra and developmental mathematics. Students also compare scenarios in which sales occur or other discounts are applied.

Debt and Interest

Another common classroom scenario is found in comparing house and car purchases in financial mathematics segments. For instance, students can use the loan payment formula to calculate the monthly payment R in terms of the monthly interest rate r, the loan amount P, and the number of months, n

Then they can calculate the total interest by multiplying the monthly payment and the number of months and subtracting the loan amount. One comparison scenario is determining how the monthly payment and total interest change as the price of the car or house changes or the interest rate fluctuates. Another is determining whether one should take out a smaller loan versus paying loan points to buy down the interest rate. Students also compare car prices to income level using the debt-to-income ratio. The debt-to-income ratio is the debt divided by the income, which is the percentage of debt. Banks use the debt-to-income ratio in making decisions about mortgage or car loans. From the Great Depression in the 1930s until the deregulation of banking restrictions in the 1970s, an upper limit of 25% was typical. However, that level rose after deregulation and with the increase in consumer credit card debt. In the twenty-first century, it is common for an upper limit to range between 33% and 36%. Given a monthly car payment, house payment, and other monthly debts, students can add up the total debt and solve for the necessary income level in order to stay below 36%. They can also compare the way that debt and the needed income change as the interest rates vary.

Contextualizing Instruction

Mathematics educators use purchasing scenarios in the classroom and study and debate their effectiveness. Some studies have found that the contextualization of mathematics using examples from shopping helps students. Terezinha Nunes, Analucia Schliemann, and David Carraher compared the mathematical abilities of children who were selling items in Brazilian street markets to questions in school. They found that the closer to the real-life situation, the more successful the student. Other studies have also found that there can be a disconnect between performance in the supermarket and performance in school. Some researchers assert that the contextualization may disguise the mathematics and be problematic in elucidating the underlying mathematical processes.

Mathematical Models for Comparison Shopping

Businesses and researchers employ a variety of mathematical techniques in order to compare large shopping data sets. Online shopping agents use mathematical methods in situations such as a Web search for airplane ticket prices or hotel rooms. Historically, dating back to at least the nineteenth century, travel agents sold vacations to consumers on behalf of suppliers. Travel agencies grew in popularity with the increase in commercial aviation after World War I. At the end of the twentieth century, the Internet vastly changed the way in which consumers compared and purchased vacation travel. Airlines, hotels, and other vacation companies offered online services directly to consumers, bypassing travel agents. In response, some travel agencies created travel Web sites that would compare options. Their computer programs extracted comparative price data from Web sites in order to build comparison shopping engines. Researchers continue to develop advanced comparison shopping techniques including methods in data mining, data envelopment analysis, and multiple regression. They create sophisticated algorithms to analyze data and find patterns. In data envelopment analysis, networks can be viewed as decision-making units, and efficient configurations are selected. In multiple regression, several variables are combined in an attempt to create a meaningful predictor or measure. Mathematical methods are also important in predicting shopping preferences and consumer behavior.

Bibliography

Berry, Michael, and Linoff, Gordon. Data Mining Techniques for Marketing, Sales and Customer Support. Hoboken, NJ: Wiley, 1997.

Boaler, Jo. “The Role of Contexts in the Mathematics Classroom: Do They Make Mathematics More ‘Real’?” For the Learning of Mathematics 13, no. 2 (1993).

Devlin, Keith. The Math Instinct: Why You’re a Mathematical Genius (Along With Lobsters, Birds, Cats, and Dogs). New York: Basic Books, 2005.

Herzog, David. Math You Can Really Use—Every Day. Hoboken, NJ: Wiley, 2007.

McKay, Lucia, and Maggie Guscott. Practical Math in Context: Smart Shopping Math. Costa Mesa, CA: Saddleback Educational Publishing, 2005.

Nunes, Terezinha, Analucia Schliemann, and David Carraher. Street Mathematics and School Mathematics. New York: Cambridge University Press, 1993.