Word problem (mathematics education)

Example

Here is a mathematical problem in mathematical notation:

Solve for J: J = A − 20 J + 5 = (A + 5) / 2

The same problem might be presented in the form of 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 that J equals 15 (and A equals 35).

Structure

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

Word problems commonly include mathematical modelling questions, where data and information about a certain system is given and a student is required to develop a model. For example:

1. Jane has $5 and she uses $2 to buy something. How much does she have now?
2. 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, 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.