Abstract
Research study on students’ conceptions of irrational numbers before entering university is of importance in order that the construction of the real number field introduced at the university level will have the desired effect. The aim of the present study is to analyze the way students approach the question of the existence of irrational numbers. 91 10th graders and 97 11th and 12th graders participated in the research study. The analysis focuses first on students’ conceptions of irrational numbers by using their representation as non-repeating infinite decimals. Then, the analysis focuses on students’ conceptions of irrational numbers on the number line. In spite of the fact that around 80% of the students claimed that they had learned about irrational numbers, only a small percentage of students (19%) showed awareness of the existence of irrational numbers as non-repeating infinite decimals. The same percentage of students (20%) showed awareness of the existence of irrational numbers on the real number line. I also observe some students’ answers that expressed their awareness of the existence of irrational numbers as non-repeating infinite decimals but the same students wrote that there are only rational numbers on the number line. Analyzing students’ conceptions of irrational numbers using both representations, as non-repeating infinite decimals and as points on the number line, I observe the incoherence of some of their conceptions.
Résumé
La recherche sur les conceptions que les étudiants ont des nombres irrationnels avant leur entrée à l’université est. importante afin que la construction du champ de nombres réels introduite au niveau universitaire soit faite de façon appropriée. Le but de la présente étude est. d’analyser comment les élèves abordent la question de l’existence de nombres irrationnels. 91 élèves de seconde et 97 élèves de premières et terminales ont participé à la recherche. Dans cet article, le focus de l’analyse est. d’abord sur les conceptions des étudiants des nombres irrationnels à l’aide de leur représentation décimale infinie sans répétition. Ensuite, le focus de l’analyse est. sur les conceptions des nombres irrationnels sur la droite réelle par les élèves. Bien qu’environ 80% des étudiants ont déclaré qu’ils avaient appris le concept de nombres irrationnels, seul un faible pourcentage d’étudiants (19%) ont démontré qu’ils étaient conscients de l’existence de nombres avec représentation décimale infinie sans répétition. Le même pourcentage d’étudiants (20%) sont conscients de l’existence des nombres irrationnels sur la droite réelle. J’ai également observé des réponses de certains étudiants qui reconnaissent l’existence de nombres irrationnels avec représentation décimale infinie sans répétition, mais les mêmes étudiants ont écrit qu’il existe seulement des nombres rationnels sur la droite réelle. En analysant les conceptions des nombres irrationnels par les élèves en utilisant les deux représentations: représentation décimale infinie sans répétition et des points de la droite réelle j’ai observé l’incohérence de certaines de leurs conceptions.
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Acknowledgments
I would like to thank Shlomo Vinner for his substantial contribution to this research. This research was supported by the Israel Science Foundation (grant number 1815/16).
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Kidron, I. Students’ Conceptions of Irrational Numbers. Int. J. Res. Undergrad. Math. Ed. 4, 94–118 (2018). https://doi.org/10.1007/s40753-018-0071-z
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DOI: https://doi.org/10.1007/s40753-018-0071-z
Keywords
- Extension of rational numbers to irrational numbers
- Existence of irrational numbers
- Intuition
- Non-repeating infinite decimal
- Real number line