Difficulties in mathematical problem solving: What might be the cause(s)?



Children at primary schools experience different kinds of problems in their subjects. One of the subjects where difficulties are being experienced is mathematics. During math lessons, teachers try to improve the mathematical skills of the pupils and also teach how the pupils can solve arithmetic problems. There are different types of arithmetic word problems. One of those is the compare word problem which contains a relational statement that compares the value of two variables (Hegarty, Mayer & Green, 1992). A compare word problem consists of three kinds of sentences and can be described as assignments, relations or questions. The assignment sentence gives a single numerical value for a variable (e.g., At a supermarket, a piece of cheese costs 4 euros). Relational sentences define a variable in terms of another and these sentences therefore give a single numerical relationship between two variables (e.g., At a cheese shop, a piece of cheese costs 3 euros more than at the supermarket). The question in a word problem asks what the value is of a particular variable (e.g., How much do 3 pieces of cheese cost at the cheese shop?) (Mayer, 1982; Mayer, Larkin & Kadane, 1984).
The compare word problems cause more errors than other mathematical problems. Research has shown that pupils have more difficulties with solving those kinds of problems than with other kinds of mathematical problems (Kintsch & Greeno, 1985; Riley, Greeno & Heller, 1983). The difficulties children experience when solving problems might be the cause of low mathematical skills of the pupils. To be able to solve a word problem, a child has to know which operation is required and how to apply a certain mathematical operation. Nevertheless, there is another skill that might play an important role when solving arithmetic word problems. It is reading comprehension ability that may also have an influence on the solving process. According to research of Van der Schoot, Bakker Arkema, Horsley & Van Lieshout (2009) pupils who were less successful problem solvers had a lower reading comprehension ability than the more successful problem solvers. Less successful problem solvers may perform the correct mathematical operations but have wrongly interpreted the situation given in a word problem (Lewis & Mayer, 1987).
The incorrect representation of a problem might be caused by inconsistent relational terms in this problem (Lewis and Mayer, 1987; Verschaffel, De Corte & Pauwels, 1992). Compare word problems consist of a relational term which can be consistent or inconsistent with the required operation to solve the problem. A consistent language problem is a word problem which, for example, includes the word 'more' and also needs an additional operation to solve the problem. It is called inconsistent when the word problem includes the word 'more' but needs a subtraction to solve the problem. Previous research has demonstrated that children make more errors when solving word problems with inconsistent language than problems with consistent language (Riley et al., 1983; Hegarty et al., 1992; Lewis & Mayer, 1987). The difficulties solving those problems may be found in the comprehension phase and not in the solution phase of problem solving. In other words, pupils make correct computations on incorrect representation of the word problems. The error rate can be explained by the consistency effect (Lewis & Mayer, 1987). This effect states that children prefer a certain order in which the problem information will be presented. The way information is presented in a consistent problem, corresponds with the preferred order. The order of information presented in an inconsistent problem, on the other hand, is in conflict with the preference. The preference for the order of information was found in a study of Huttenlocher and Strauss (1968) which concluded that comprehension of a relational statement is easier when consistent language is used.
As well as consistency, markedness has been found to play an important role when solving a word problem. Word problems may contain a marked term like 'less than' or an unmarked term like 'more than'. Several studies have shown that markedness may also have an influence on the solving process (Hegarty et al, 1992; Lewis & Mayer, 1987; Verschaffel et al., 1992). Clark (1969) has called this effect the lexical marking principle. According to this principle a marked term (e.g., short) of an antonymous adjective pair is more difficult to process than an unmarked term (e.g., tall). The storage in memory is more complex for meanings of negative adjectives than it is for meanings of the positive adjectives of a pair. The effect of this principle has been found to influence the process of problem solving of problems which contain a marked or unmarked term. Difficulties are not only seen when solving problems that contain a marked term (Hegarty et al., 1992), but also when markedness and consistency interact with each other (Lewis & Mayer, 1987; Pape, 2003). 
Future research should focus on the possible factors that have an influence on the process of solving compare word problems. It is important to make a distinction between the different abilities of children, which are the mathematical and the reading comprehension ability. Research should try to find out which of those two skills is more important when solving a problem. It might turn out that either the mathematical or reading comprehension ability has more influence on problem solving, but it is also possible that both of them are equally important. This can be examined by using known scores of the mathematical ability and reading comprehension ability of pupils. The compare word problems that can be used in such research will contain marked and unmarked terms, and have consistent and inconsistent relational terms. By using the scores and the different kinds of compare word problems, it can become clearer which ability has more influence on solving a word problem successfully. When this research is done, attention can be paid to the development of new teaching methods to help children with their mathematical difficulties and to improve their mathematical performance. 





References
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Hegarty, M., Mayer, R.E. & Green, C.E. (1992). Comprehension of arithmetic word problems:   Evidence from student's eye fixations. Journal of Educational Psychology, 84(1), 76-84.

Huttenlocher, J. & Strauss, S. (1968). Comprehension and a statement's relation to the situation it  describes. Journal of Verbal Learning and Verbal Behavior, 7, 300-304.

Kintsch, W. & Greeno, J.G. (1985). Understanding and solving word arithmetic problems.   Psychological Review, 92, 109-129. 

Lewis, A.B. & Mayer, R.E. (1987). Students' miscomprehension of relational statements in arithmetic  word problems. Journal of Educational Psychology, 79(4), 363-371. 

Mayer, R.E. (1982). Memory for algebra story problems. Journal of Educational Psychology, 74(2),  199-216.

Mayer, R.E., Larkin, J.J. & Kadane, J.B. (1984). A cognitive analysis of mathematical problem-solving  ability. In R.J. Sternberg (Ed.), Advances in the psychology of human intelligence: Vol. 2 (pp.  231-273). Hillsdale, NJ: Erlbaum. 

Pape, S.J. (2003). Compare word problems: Consistency hypothesis revised. Contemporary  Educational Psychology, 28, 396-421. 

Riley, M.S., Greeno, J.G. & Heller, J.I. (1983). Development of children's problem-solving ability.  Cognition and Instruction, 2, 41-57. 

Van der Schoot, M., Bakker Arkema, A.H., Horsley, T.M & Van Lieshout, E.C.D.M. (2009). The   consistency effect depends on markedness in less successful but not successful problem  solvers: An eye movement study in primary school children. Contemporary Educational  Psychology, 34, 58-66.

Verschaffel, L., De Corte, E. & Pauwels, A. (1992). Solving compare problems: An eye movement test  of Lewis and Mayer's consistency hypothesis. Journal of Educational Psychology 84(1), 85-94.