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Word problem

The term "word problem" in abstract algebra applies to groupss in a different sense than below: see word problem for groups.

In mathematics education, a word problem is a mathematical question written without relying heavily on mathematics notation.

The idea is to present mathematics to the students in a less abstract way and to give the students a sense of "usefulness" of mathematics. Word problems are supposed to be interesting problems that can motivate students to learn mathematics.

It should be noted that "word problem" is not a well-defined term in mathematics. In fact, all mathematical problems are expressed primarily in words, and the notation is merely a method of facilitating brevity and conciseness in writing these words. Therefore some mathematicians dislike the use of the term "word problem".

Table of contents
1 Modelling questions
2 Other problems
3 Misunderstandings

Modelling questions

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 cyclinder of radius 2m is rising in a rate of 3m per second, what is the rate of increase of the volume of water?

It is believed that the first example is really useful in helping
primary school students to understand the concept of subtraction. The second example, however, might not be so interesting or so "real-life" to a high school student. A high school student may find that it is easier to handle the following problem:
Given r=2, dh/dt=3. Find d/dt (π r2× h).

This type of problems is the so-called "problem in equations", the term "equation" is a popular misnomer referring to mathematical notation.

Indeed, in senior high school level or higher, this type of problems is often used solely to test understanding of underlying concepts within a descriptive problem, instead of testing the student's capability to perform algebraic manipulation or other "mechanical" skills. As a result, a word problem may even harder then the so-called "problems in equations" and indeed, it may inhibit a student's desire to learn mathematics.

Other problems

Some word problems aren't modelling questions, but merely express an exercise for a student to perform. For example:

  1. Prove that if the sum of two numbers is odd then the product of them is even.
  2. Differentiate x2+3xy+2y with respect to x.

These problems seem to fail completely in motivating students to learn. In the second example, half the sentence is notations and it is called a "word problem" just because it is not written as Find d/dx (x2+3xy+2y) !

Recall that every mathematical problem is expressed primarily in words. No doubt the term "word problem" is sometimes regarded as meaningless.


A commonplace misunderstanding of mathematics is that some mathematical problems are expressed primarily in words and others are "problems in equations". Why people fails to notice that all mathematical problems are expressed primarily in words may result from the way textbooks are written.

A student sees this:

and thinks this is a problem posed in "equations" (a misnomer in this case -- what is displayed is a polynomial expression, not an equation). Such a student has failed to notice that somewhere above this "equation" there were some words that said "Factor the following polynomials" or "Find the derivative of each of the following functions of x", or otherwise state what is to be done.

Worse still, students may consider

  1. Differentiate x2+3xy+2y with respect to x.
  2. Find d/dx (x2+3xy+2y)
to be two different problems. The difference between the two is just the words.