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The aim of the study was to identify the understanding of elementary school students of numerical concepts, and the beliefs of their teachers about this understanding. To achieve this goal, the content of the mathematics sixth grade textbook was analyzed to identify the numerical concepts. Then, a conceptual diagnostic test consisting of 24 multiple-choice questions to measure the understanding of numerical concepts was designed. Another questionnaire was built to measure the teachers' beliefs about their students understanding of these concepts. The teachers' questionnaire included 24 concepts as well. The sample size was 1411 male and female students; and the sample size of teachers was 528 mathematics teachers. Both were drawn randomly from three cities in the Kingdom of Saudi Arabia. The test of students showed that about 4.2% of the numerical concepts were highly understandable, whereas the questionnaire showed that teachers believed their students understood 17% of the concepts. Furthermore, students moderately understood about 37.5% of the concepts, while the questionnaire showed that teachers believed their students moderately undersood about 83% of the concepts. Moreover, the test showed that about 58.3% of the concepts were poorly understood, whereas the questionnaire showed that there were no concepts that would be poorly understood. The study found that there were statistically significant differences due to gender, credential, and experience; however, there was no statistically significant difference attributed to training on teachers' beliefs about students' understanding of mathematics.

Numerical Concepts between the Reality of Elementary Students' Understanding and their Teachers' Beliefs towards this Understanding in Saudi Arabia