The evaluation of 4th grade students’ Errors based on the Newman's Error Analysis model in problem solving

Word problems, despite their difficulties, are of great importance in mathematics. Many students' ability to solve word problems is lower than their ability to solve computational problems (Burns, 2000; Miller et al., 2017). It can be argued that understanding these kinds of problems requires an active interaction between linguistic knowledge and mathematical knowledge. If these difficulties are not solved in the early stages of learning math, these difficulties will become more widespread over the years and reach the academic level. Therefore, initial action is essential for students' success in such problems in elementary school (Clements and Sarama, 2011). Most elementary school students, in the face of these problems, after reading the problem without realizing it, simply try to find the keywords and as soon as they find them, they start solving the problem, unaware that this method does not always lead them to the correct answer. Therefore, since elementary schools are the keystone of learning mathematics for future levels, conducting research to find students' difficulties in solving word problems as well as examining and recognizing their errors in solving such problems and eliminating them. They are very important in all grades of in elementary schools, especially the fourth grade; because this is the first grade to be addressed in the TIMSS international Study.
There are several ways to analyze student errors. One of these methods is the Newman error analysis method, which includes a systematic method for analyzing the errors that students make while answering word mathematical problems. This method is hierarchical and based on the assumption that students follow orderly mental patterns in problem solving that can be divided into specific stages. Other models ignore students' specific and individual methods, which are quite vital (Clarkson, 1991). According to Newman, in order to reach the correct answer in solving word problems, one must go through five hierarchies: read the problem; understand what he/she has read; to be able to choose an appropriate mathematical strategy, make a mental transformation of the words used in the problem; depending on the strategy chosen, use the required process skills and finally present the answer in an acceptable written form. These five hierarchies are abbreviated as: 1) reading, 2) comprehension, 3) transformation, 4) process skills, and 5) coding. Newman used the term "hierarchy" because failure at each level prevents the problem solver from advancing properly in the problem-solving process. In solving word problems, students go through five stages, in each of which, an error may occur that prevents the student from reaching the correct answer (Clements and Ellerton, 1996).
Newman's hierarchical model of error analysis enables teachers to identify five obvious difficulties students have when solving word problems, to recognize students' errors before solving problems, and to use appropriate strategies (Watson, 1980; White, 2005). Therefore, conducting research in this field will be useful. In this regard, the present study examines the types of students' errors in solving word problems using Newman analysis.
The study aimed to investigate the ability to solve word mathematical problems of 4th-grade students based on the Newman error analysis model using descriptive-survey method of cross-sectional type. Among the all 4th-grade primary school students from district 12th of of Tehran, 399 students were selected by using cluster random sampling. To collect data, a researcher-made test which composed of four questions as well as Newman questions was applied. To examine the validity of the research instrument, face and content validity was used and Cronbach's alpha (α=0.82) was used for reliability. The paper and pencil tests were corrected by two raters, and the inter-rater reliability was computed. Data analysis was done using descriptive and inferential statistics (MANOVA).
Based on the results, among the errors of reading, comprehension, transformation, process skills and coding, students in the process of achieving the correct answer to the word problems have the most error in the transformation (24.9%) and the least in coding (0.5%).of the Newman hierarchy stages. The results of Chi -square test at the level of 0.05 showed that there is a significant relationship between the rate of reading error and students' performance in solving math word problems (P <0.05, ²χ = 71.637). The results of Chi-square test showed that there is a significant relationship between the rate of comprehension error and students' performance in solving math word problems (P <0.05, ²χ = 61/984). The results of Chi -square test showed that there is a significant relationship between the rate of transformation error and students' performance in solving math word problems (P <0.05, ²χ = 33.543). The results of Chi-square test showed that there is a significant relationship between the rate of error of process skills and students' performance in solving math word problems (P <0.05, ²χ = 63.262). The results of Chi-square test showed that there is a significant relationship between the rate of coding error and students' performance in solving mathematical word problems (P <0.05, ²χ = 44.645).
 Also, the results showed a significant difference in the performance of the two groups of boys and girls, while boys generally made fewer errors than girls. In fact, the results of Wilks' Lambda statistic showed that there was a significant difference between the total problem solving error in the Newman analysis model in comparison between the two groups of girls and boys (ƞ² = 0.135, P = 0.002, 4.365, F (4, 394)= 4.394). The findings of this research can be used to review the word problem solving teaching process and to set up educational materials for mathematical textbooks.