A word problem is a verbal description of a problem situation wherein one or more questions are posed, the answers to which can be obtained by the application of mathematical operations to information (usually numerical data) available in the text. In its most typical form, a word problem describes the essentials of some situation assumed to be familiar to the solver. Within the text, certain quantities are explicitly given, while others are not. The student is required to give a numerical answer to a stated question by making exclusive use of the quantities given–and of the mathematical relationships between these quantities. Simple examples include: "Pete wins 3 marbles in a game and now has 8 marbles. How many marbles did he have before the game?" and "One kilogram of coffee costs 12 euros. Susan buys 0.75 kilogram of coffee. How much does she have to pay?"
Despite its label, a word problem need not constitute a problem in the cognitive-psychological sense of the word–higher-order thinking going beyond the application of a familiar routine procedure is not necessarily required. Indeed, in typical elementary mathematics instruction, many word problems provide thinly disguised practice in adding, subtracting, multiplying, or dividing.
Structural Dimensions of Word Problems
Several structural dimensions can be distinguished in word problems that affect their difficulty and how they are solved:
- Mathematical structure, which includes the nature of the given and unknown quantities of the problem, and the mathematical operations by which the unknowns can be derived from the givens.
- Semantic structure, which includes the ways in which an interpretation of the text points to particular mathematical relationships. For example, addition or subtraction is indicated when the text implies a combination of disjoint subsets into a superset, a change from an initial quantity to a subsequent quantity by addition or subtraction, or the additive comparison between two collections.
- Context, meaning the nature of the situation described. For example, an additive problem involving combination of disjoint sets might deal with physically combining collections of objects or with conceptually combining collections of people in two locations.
- The format, meaning how the problem is formulated and presented. Format involves such factors as the placement of the question, the complexity of the lexical and grammatical structures, the presence of superfluous information, and so on.
Over several decades, numerous studies have analyzed the role of these task variables on the difficulty of problems, on the kind of strategies students use to solve these problems, and on the nature of their errors, particularly for simple word problems involving addition and subtraction or multiplication and division.
Roles of Word Problems
Why does school mathematics include word problems? Perhaps simply because they are there, and have been for many centuries. Indeed, their role in mathematics education dates back to antiquity; the oldest known being in Egyptian papyri dating from 2000 B. C.E., with strikingly similar examples in ancient Chinese and Indian manuscripts. The following example is from the first printed mathematical textbook, a Treviso arithmetic of 1478: "If 17 men build 2 houses in 9 days, how many days will it take 20 men to build 5 houses?"
Despite this striking continuity across time and cultures, until recently there was little explicit discussion of why word problems should be such a prominent part of the curriculum, or of the variety of purposes behind their inclusion. Some have a puzzle-like nature and act as "mental manipulatives" (Toom, p. 36) to guide thinking within mathematical structures. Such problems are intended to train students to think creatively and develop problem-solving abilities. By contrast, the type mainly used educationally consists of a text representing (at least putatively) a real-world situation in which the derived answer would "work." Ostensible goals for the use of this type include offering practice for the situations of everyday life in which the mathematics learned will be needed, thereby showing students that the mathematics they are learning will be useful.
Apparent Suspension of Sense-Making
In recent years, the characteristics, use, and rationale of word problems have been critically analyzed from multiple perspectives, including linguistic, cultural, and sociological perspectives. In particular, it has been argued by many mathematics educators that the stereotyped and artificial nature of word problems typically represented in mathematics textbooks, and the discourse and activity around these problems in traditional mathematics lessons, have detrimental effects. Many observations have led to the conclusion that children answer word problems without taking into account realistic considerations about the situations described in the text, or even whether the question and the answer make sense. The most dramatic example comes from French researchers who posed children nonsensical questions such as: "There are 26 sheep and 10 goats on a ship. How old is the captain?" It was found that the majority of students were prepared to offer an answer to such questions. In another study, thirteen-year-old students in the United States were asked the following question: "An army bus holds 36 soldiers. If 1,128 soldiers are being bussed to their training site, how many buses are needed?" The division was correctly computed by 70 percent of the students to get a quotient of 31 and remainder 12–but only 23 percent gave the appropriate answer, "32 buses." Nineteen percent gave the answer as "31 buses" and 29 percent gave the answer as "31, remainder 12."
To explain the abundant observations of this "suspension of sense-making" when doing word problems, it has been suggested by Erik De Corte and Lieven Verschaffel that the practice surrounding word problems is controlled by a set of (largely implicit) rules that constitute the "word-problem game." These rules including the following assumptions: (1) every problem presented by the teacher or in a textbook is solvable and makes sense; (2) there is only one exact numerical correct answer to every word problem; and (3) the answer must be obtained by performing basic arithmetical operations on all numbers stated in the problem.
Reconceptualizing Word Problems as Modeling Exercises
One reaction to criticisms of traditional practice surrounding word problems in schools is to undermine the approach that allows students to succeed using superficial strategies based on the "rules." This is done by breaking up the stereotypical nature of the problems posed. For example, by including problems that do not make sense or contain superfluous or insufficient data, students can be guided to interpret word problems critically.
A more radical suggestion is to treat word problems as exercises in mathematical modeling. The application of mathematics to solve problem situations in the real world, termed mathematical modeling, is a complex process involving several phases, including understanding the situation described; constructing a mathematical model that describes the essence of the relevant elements embedded in the situation; working through the mathematical model to identify what follows from it; interpreting the computational work to arrive at a solution to the problem; evaluating that interpreted outcome in relation to the original situation; and communicating the interpreted results.
This schema can be used to describe the process of solving mathematical word problems as application problems. In the simplest cases, situations may be directly modeled by addition, subtraction, multiplication, or division, and children need to learn the variety of prototypical situations that fit unproblematically onto these operations. In other cases, the modeling is not so straightforward if serious attention is given to the reality of the situation described. In the example from the Treviso arithmetic, attention would be drawn to the assumptions that under-pin an answer based on direct proportionality–and to the fact that the answer thus derived would at best provide a rough approximation in the real situation. In the bus problem, the "raw" result of the computation has to be appropriately refined in the context of the situation described.
Reforming the Teaching of Word Problems
In line with the above criticisms and recommendations with respect to the traditional practice surrounding word problems in schools, researchers have set up design studies to develop, implement, and evaluate experimental programs aimed at the enhancement of strategies and attitudes about solving mathematical word problems. In these studies, positive outcomes have been obtained in terms of both outcomes (test scores) and underlying processes (beliefs, strategies, attitudes). Characteristics common to such experimental programs include:
- The use of more realistic and challenging tasks than traditional textbook problems.
- A variety of teaching methods and learner activities, including expert modeling of the strategic aspects of the competent solution process, small-group work, and whole-class discussions.
- The creation of a classroom climate that is conducive to the development in pupils of an elaborated view of mathematical modeling, and of the accompanying beliefs and attitudes.
To some extent, these characteristics of a new approach to word-problem solving are beginning to be implemented in mathematical frameworks, curricula, textbooks, and tests in many countries. Much remains to be done, however, to align the teaching of word problems with widely accepted principles that children should make connections between mathematics and their lived experience–and that mathematics should make sense to them.
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ERIK DE CORTE