Word problem

In abstract algebra one has the unrelated term word problem for groups.

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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, as well as to teach them how to apply their math knowledge to real world problems they may encounter.

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. Therefore some mathematicians dislike the use of the term "word problem". Here is an example of how this misunderstanding arises. In an algebra textbook, it says,

Factor the following polynomials into linear or quadratic factors with real coefficients.

Then a numbered list follows, going from 1 through 50. Within it we find that #28 says:

<math>2x^3-3x^2-10x-40.<math>

The student may then think that this problem was expressed not primarily in words, but in mathematical notation. But the without the words "Factor the following polynomials into linear or quadratic factors with real coefficients", the problem is incomprehensible.

Contents

1 Modelling questions 2 Other problems 3 Misunderstandings 4 External links

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 cylinder 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 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 problem is called a "problem in equations" by some students; however the use of "equation" in this sense is sometimes misused to refer 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 be even harder than the so-called "problems in equations" and indeed, it may inhibit a student's desire to learn mathematics.

Other problems

Some word problems are not 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 their product 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) !

Misunderstandings

A commonplace misunderstanding of mathematics is that some mathematical problems are expressed primarily in words and others are "problems in equations". People sometimes fail to notice that all mathematical problems are expressed primarily in words. This confusion may result from the way textbooks are written.

A student sees this:

<math>x^2+3x+2<math>

and thinks this is a problem posed in "equations". Such a student has failed to notice that somewhere above this expression 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 answer is the same in each case. The difference between the two is just in the words that pose the question. Recall that every mathematical problem is expressed primarily in words. No doubt the term "word problem" is sometimes regarded as meaningless.

External links

• Interactive algebra word problem solvers that teach problem solving (http://www.algebra.com/algebra/homework/word/)
• Word problems that lead to simple linear equations (http://www.cut-the-knot.org/arithmetic/WProblem.shtml) (requires Java)