Word problem (mathematics education)
In mathematics education, the term word problem is often used to refer to any mathematical exercise where significant background information on the problem is presented as text rather than in mathematical notation. As word problems often involve a narrative of some sort, they are occasionally also referred to as story problems and may vary in the amount of language used.
A mathematical problem in mathematical notation:
- Solve for J:
- J = A − 20
- J + 5 = (A + 5)/2
might be presented in a word problem as follows:
- John is twenty years younger than Amy, and in five years' time he will be half her age. What is John's age now?
The answer to the word problem is that John is 15 years old, while the answer to the mathematical problem is J = 15 (and A =35).
Word problems can be examined on three levels:
- Level a: the verbal formulation;
- Level b: the underlying mathematical relations;
- Level c: the symbolic mathematical expression.
Linguistic properties can include such variables as the number of words in the problem or the mean sentence length. The logico-mathematical properties can be classified in numerous ways, but one such scheme is to classify the quantities in the problem (assuming the word problem is primarily numerical) into known quantities (the values given in the text of the problem), wanted quantities (the values that need to be found) and auxiliary quantities (values that may need to be found as intermediate stages of the problem).
The most common types of word problems are distance problems, age problems, work problems, percentage problems, mixtures problems and numbers problems.
Purpose and use
- Jane has $5 and she uses $2 to buy something. How much does she have now?
- If the water level in a cylinder of radius 2 m is rising at a rate of 3 m per second, what is the rate of increase of the volume of water?
These examples are not only intended to force the students into developing mathematical models on their own, but may also be used to promote mathematical interest and understanding by relating the subject to real-life situations. The relevance of these situations to the students is varying. The situation in the first example is well-known to most people and may be useful in helping primary school students to understand the concept of subtraction. The second example, however, does not necessarily have to be "real-life" to a high school student, who may find that it is easier to handle the following problem:
- Given r = 2 and dh/dt = 3, find d/dt (π r 2× h).
Word problems are a common way to train and test understanding of underlying concepts within a descriptive problem, instead of solely testing the student's capability to perform algebraic manipulation or other "mechanical" skills.
History and Culture
The modern notation that enables mathematical ideas to be expressed symbolically was developed in Europe from the sixteenth century onwards. Prior to this, all mathematical problems and solutions were written out in words; the more complicated the problem, the more laborious and convoluted the verbal explanation.
Examples of word problems can be found dating back to Babylonian times. Apart from a few procedure texts for finding things like square roots, most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. Students had to find lengths of canals dug, weights of stones, lengths of broken reeds, areas of fields, numbers of bricks used in a construction, and so on.
Ancient Egyptian mathematics also has examples of word problems. The Rhind Mathematical Papyrus includes a problem that can be translated as:
There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all the enumerated things.
In more modern times the sometimes confusing and arbitrary nature of word problems has been the subject of satire. Gustave Flaubert wrote this nonsensical problem, now known as the Age of the captain:
Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain? 
Word problems have also been satirised in The Simpsons:
Bart: 7:30am an express train traveling 60 miles per hour leaves Santa Fe bound for Phoenix, 520 miles away. At the same time, a local train traveling 30 miles an hour carrying 40 passengers leaves Phoenix bound for Santa Fe. It's 8 cars long and always carries the same number of passengers in each car. An hour later, the number of passengers equal to half the number of minutes past the hour get off, but three times as many plus six get on. At the second stop, half the passengers plus two get off but twice as many get on as got on at the first stop.
(at this point Bart's imagination takes over and he imagines he's actually on the train)
- L Verschaffel, B Greer, E De Corte (2000) Making Sense of Word Problems, Taylor & Francis
- John C. Moyer; Margaret B. Moyer; Larry Sowder; Judith Threadgill-Sowder (1984) Story Problem Formats: Verbal versus Telegraphic Journal for Research in Mathematics Education, Vol. 15, No. 1. (Jan., 1984), pp. 64-68. http://links.jstor.org/sici?sici=0021-8251%28198401%2915%3A1%3C64%3ASPFVVT%3E2.0.CO%3B2-V
- Perla Nesher Eva Teubal (1975)Verbal Cues as an Interfering Factor in Verbal Problem Solving Educational Studies in Mathematics, Vol. 6, No. 1. (Mar., 1975), pp. 41-51. http://links.jstor.org/sici?sici=0013-1954%28197503%296%3A1%3C41%3AVCAAIF%3E2.0.CO%3B2-H
- Madis Lepik (1990) Algebraic Word Problems: Role of Linguistic and Structural Variables, Educational Studies in Mathematics, Vol. 21, No. 1. (Feb., 1990), pp. 83-90., http://links.jstor.org/sici?sici=0013-1954%28199002%2921%3A1%3C83%3AAWPROL%3E2.0.CO%3B2-8.
- Duncan J Melville (1999) Old Babylonian Mathematics http://it.stlawu.edu/%7Edmelvill/mesomath/obsummary.html
- Egyptian Algebra - Mathematicians of the African Diaspora
- Mathematical Quotations - F
- Andrew Nestler's Guide to Mathematics and Mathematicians on The Simpsons