Problem solving as an important skill is, beside arithmetic, measure and algebra, included in standards of school mathematics (National Council of Teachers of Mathematics) (NCTM, 2000) and needed as a necessary skill for successfulness in science, technology, engineering and mathematics (STEM) (National Mathematics Advisory Panel, 2008). Since solving of human problems is connected to the real life, the arithmetic word problems (in short AWP) are an important kind of mathematics tasks in school and connect mathematics knowledge with real-life situations. When solving AWP a student needs to understand the language and numeric information in the problem in order to translate them into an adequate mental representation, create a solution plan and execute suitable procedural calculations. Foreign and domestic results (Geary, 1993; Montague, 1997; Jitendra and Hoff, 1999; Fuchs and Fuchs, 2007; Kavkler et al., 2011) indicate that students often experience difficulties with AWP solving. Empirical researches of numerous authors (Montague and Applegate, 1993; Bryant, Bryant and Hammill, 2000; Verschaffel, Greer and De Corte, 2000) mention numerous deficits in students with mathematics difficulties in (fields of) arithmetic, language, working memory and attention when solving AWP. Improvements in solving AWP have been emphasized with respect to cognitive, metacognitive, motivation and emotional aspects.
The central aim of the research was to investigate latent characteristics of students with mathematical difficulties and students without mathematical difficulties in solving AWP. The sample included 140 students from the fifth grade from Upper Carniola and Ljubljana, 70 students with mathematical difficulties and 70 students without them. In the research heterogeneous measurement instruments were used to investigate calculating, perceptio-motor, linguistic and nonverbal intellectual skills and executive function. Two questionnaires for teachers and students were used to gather information about the teacher's estimation of the students’ reading, calculating and strategies in solving AWP and information about the student's strategies of solving AWP reported by students. Qualitative and quantitative processing of gathered data was carried out in accordance with the purpose of research and hypothesis. For all included variables a descriptive analysis was made, all remainder statistical data was handled with the following statistical methods: t-test for independent samples, correlation coefficient, chi-square, analysis of variance, principal factor analysis and discriminative analysis.
The results showed significant differences among two groups of students in all included manifest variables of used measuring instruments. There were also differences in the latent structure of factors. So the latent structure of math difficulties consists of eight factors, among them three are very important: automatization of arithmetic facts and algorithmic calculation, working memory – executive functions and decoding speed. The latent structure of students without math difficulties consists of nine factors, among them three play an important role: judgment of capability of solving AWP, automatization of reading and calculation and verbal understanding. There are three common factors, i. e. automatization of reading and calculation, linguistic skills and judgment of capability of solving AWP. Upon the factor's structure of math difficulties in solving AWP it can be established that the factor of automatization of arithmetic facts with part of variance of 20,94 % is of essential importance. The analysis of variance and discriminant analysis showed that the groups of students are statistically different based on the results of the used instrument. The students with math difficulties are identified with the questionnaire for teachers (information about arithmetic knowledge and strategies for solving AWP), Test of AWP, 10-minute test for assessment of automatization of arithmetic facts and calculation, The questionnaire of solving AWP for students and Test of difficulties in writing and reading. Cognitive skills, such as the automatization of basic arithmetic facts and linguistic skills, are statistically significantly connected with solving AWP by students with math difficulties. In addition, among cognitive skills, the decoding speed, linguistic skills, nonverbal reasoning, selective attention and the ability to inhibit irrelevant information, are statistically significantly connected with solving AWP by students without math difficulties.
It was established that there was no difference among the groups in basic four steps in solving AWP (read the problem, underline the key words, calculate, write down the answer), but there is a statistically significant difference between groups in other steps in solving AWP. In short, students with math difficulties read the word problem just once, understand it less clearly after just one reading, do not paraphrase and do not use visual representation, rarely verify the accuracy of their solution, use fingers, are less motivated to solve the AWP, they find it difficult and they are not confident that they will be able to solve the AWP. On the other hand, students without math difficulties read the word problem many times and use more metacognitive processes and strategies because they control the whole process of solving. In accordance with the findings, we can conclude that students with math difficulties in comparison to students without math difficulties generally have the cognitive and metacognitive deficits that suggest a different approach and instructions. It is necessary to develop a treatment with direct and explicit instructions for solving AWP and training with the focus on automatization of basic arithmetic facts, language understanding and visual representation. Finally, the research has important implications for teaching AWP, treating students with math difficulties and further research.
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