Word problem (mathematics education)
This article is about a type of exercise in mathematics education. For other uses, see Word problem (disambiguation).
In science education, a word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation. As most word problems involve a narrative of some sort, they are sometimes referred to as story problems and may vary in the amount of technical language used.
A typical word problem:
Tess paints two boards of a fence every four minutes, but Allie can paint 14 boards every two minutes. If there 160 boards total, how many hours will it take them to paint the fence, working together?
To solve this by algebra, one first translates the words into mathematical variables, operations, and equations:
- Write the rate, which is how long it will take Tess and Allie to paint the whole fence
The problem thus becomes:
Solve the system of equations 2 ÷ 16 × 150 = 300 ÷ 16
The answer is 18.75, or in ordinary language: It will take Tess and Allie 18.75 minutes to paint the whole fence.
Word problems such as the above can be examined on three levels:
- A. The verbal formulation;
- B. The underlying mathematical relations;
- C. The symbolic mathematical expression.
Linguistic properties can include such metrics as the number of words in the problem or the mean sentence length. One scheme to analyze the logico-mathematical properties is to classify the numerical quantities in the problem into known quantities (values given in the text), wanted quantities (values to be found) and auxiliary quantities (values found as intermediate stages of the problem).
Purpose and use
- Jane had $5.00, then spent $2.00. How much does she have now?
- In a cylindrical barrel with radius 2 m, the water is rising at a rate of 3 cm/s. What is the rate of increase of the volume of water?
These examples are intended to lead students to develop mathematical models on their own, and to promote mathematical interest and understanding by analyzing real-life situations. The relevance of these problems to students varies. The first example is known even to primary school students, and is often used to teach the concept of subtraction. The second example presents a high school student with the challenge of translating the problem into symbolic form:
- Given and , find
Word problems are a common way to train and test understanding of underlying concepts using a descriptive problem, instead of solely exercising 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, when a lengthy word problem ("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...") trails off with a schoolboy character instead imagining that he is 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. JSTOR 748989
- ^ 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. JSTOR 3482158
- ^ Jump up to:a b 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., JSTOR 3482220
- ^ 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
- Word problems that lead to simple linear equations at cut-the-knot