Word problems as a genre in Mathematics Education

Gerofsky quotes Hoyrup who says, “Babylonian problems look like real-world problems at first; but as soon as you analyze the structure of the known vs unknown quantities, the complete artificiality of the problem is revealed” (2004). This chapter of Gerofsky’s book A Man left Albuquerque Heading East provided an interesting analysis on Babylonian word problems and posed several ideas I did not notice when reading and solving modern word problems. The main argument was that some Babylonian problems appeared to be typical problems seen in everyday life with realistic quantities, while the context of other problems also took place in everyday life but had far-fetched numbers or dimensions to the problem making them appear impractical and unrealistic. This reminds me of a simple word problem I saw plenty of times in grade school: “Jim bought 200 apples, 300 bananas, and 400 coconuts for $1000. If apples cost twice as much as bananas and three times as much as coconuts and bananas are twice the cost of coconuts, how much is a single apple?” (I just made this problem up to illustrate a point, however, feel free to try and solve it!). As this problem is “relevant” in the real-world with a shopper buying fruits at a grocery store, the quantities are unrealistic and impractical. Whiles these problems are unrealistic, they can still be solved quickly using contemporary algebra. Similarly, word problems like this can be generalized quite easily by changing quantities and ratio of fruits.

Pure mathematics refers to the study of mathematical theory while applied mathematics focuses on application of mathematics to the world in a very direct sense. I believe that word problems can come from a pure mathematics point of view or an applied mathematics point of view. Consider the 2020 AMC 10A Problems booklet. I would categorize the first three problems as pure maths questions while problem 4 is an applied maths question. All four however, can be solved using contemporary algebra methods. Despite being viewed as a pure maths or applied maths problem, these are still mathematical exercises where background information is presented using ordinary language rather than mathematical notation (Vershaffel et. al, 2000).

Near the end of the chapter, Gerofsky poses several questions to the reader; “Are word problems used primarily to train students in the use of methods without necessarily providing understanding of those methods? Are problems chosen to illustrate the methods at hand?” I think word problems can be solved using multiple methods, so it is possible that students may not understand explicitly what method is being used besides arriving at the correct solution. However, word problems can be created to test a student’s ability to determine what the word problem is being asked and carry out the appropriate method to solve the problem. From Gerofsky’s paper, Hoyrup says that Babylonians trained students in methods available at hand rather than understanding of methods (2004). As historical evidence has shown, the Babylonians were very good at computations as their methods were very systematic. I think these ideas of Babylonian word problems illustrate the fact that the Babylonians were also great writers and thinkers and were among the first civilizations to incorporate ordinary language in mathematical exercises.

 

Sources:

Gerofsky, S. 2004. A Man left Albuquerque Heading East. Word Problems as Genre in Mathematics Education.

AoPS Online. 2020 AMC 10A Problems. Retrieved on October 4, 2020 from https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A.

Verschaffel, L., Greer, B., De Corte, E. 2000. Making Sense of Word Problems, Taylor & Francis.