Investigating a case of hidden misinterpretations of an additive word problem: structural substitution

Abstract

According to numerous studies (Barrouillet & Camos 2002; Brousseau 1988; Chevallard 1988; Riley et al. 1984; Schubauer-Leoni & Ntamakiliro, Revue Des Sciences de L’éducation, 20(1): 87–113, 1994; Vergnaud 1982; Xin, The Journal of Educational Research, 100(6):347–360, 2007), a combination of many factors, including curriculum, didactic contract and task design, can potentially lead to students experiencing difficulties in developing of a full understanding of addition and subtraction and their relationship in problem solving. Few studies (Conne, Recheche En Didactique Des Mathématiques, 5, 269–332, 1985; DeBlois, Éducation et Francophonie, 25(1), 102–120, 1997; Giroux & Ste-Marie, European Jornal of Psychology of Education, 16(2), 141–161, 2001) describe the misinterpretations of problems as a factor related to learning difficulties. We have studied how and why elementary school students misinterpret the mathematical structure of a simple additive word problem and what kind of possible (hidden) misinterpretation may occur. We analysed possible mechanisms of misinterpretations in word problem solving, discussing various examples of correct and incorrect solutions resulting from the misinterpretation of a problem. We gave the elementary school students a word problem, which could potentially be misinterpreted, and observed their solving strategies. Our results show how the particular form of mathematical misinterpretation—structure substitution—may help students obtain a correct answer and thereby hinder the development of their mathematical reasoning. We further discuss different ways of addressing this phenomenon in teaching practice.

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Notes

  1. 1.

    The authors (Giroux and Ste-Marie 2001) use the expression shift of meaning.

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Acknowledgements

We want to acknowledge the support grant from the Ministère de l'Éducation, Enseignement supérieur et Recherche du Québec, Programme de soutien à la formation continue du personnel scolaire.

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Correspondence to Elena Polotskaia.

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Elena Polotskaia. Département des sciences de l’éducation, Université du Québec en Outaouais, Pavillon Alexandre-Taché, 283, boulevard Alexandre-Taché, bureau C-2332 C.P. 1250, Succursale Hull, Gatineau, Québec, Canada J8X 3X7. E-mail: elena.polotskaia@uqo.ca, http://www.uqo.ca

Current themes of research:

Elena Polotskaia, Ph.D. and Assistant Professor in the Department of Educational Sciences at the Université du Québec en Outaouais. Her research interests focus on mathematical reasoning development in school students and learning difficulties related to mathematics.

Most relevant publications in the field of Psychology of Education:

Arkhipova (Polotskaia), E. (2014). How elementary students learn to mathematically analyze word problems: The case of addition and subtraction. Ph.D. thesis, McGill University. Available at: http://mcgill.worldcat.org/title/how-elementary-studentslearn-to-mathematically-analyze-word-problems-the-case-of-addition-andsubtraction/oclc/908962593&referer=brief_results.

Polotskaia, E., & Savard, A. (2014). Reasoning duality in solving additive word problems: how to mesure students’ performance. In P. Liljedahl et al., (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 6, pp. 382–383.). Vancouver, Canada: PME.

Polotskaia, E., Savard, A., & Freiman, V. (2015). Duality of mathematical thinking when making sense of simple word problems: theoretical essay. Eurasia Journal of Mathematics, Science & Technology Education, 11(2), 251–261. Available at: http://www.ejmste.com/arsivAyrinti.aspx?kim=35.

Savard, A., & Polotskaia, E. (2014). Tasks to promote holistic flexible reasoning about simple additive structures. In The 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education. Vancouver, Canada.

Annie Savard. Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, Quebec, Canada, H3A 1Y2. E-mail: annie.savard@mcgill.ca, http://www.mcgill.ca/dise

Current themes of research:

Annie Savard is an associate professor in Mathematics Education in the department of Integrated Studies in Education at McGill University. Her research interests concern the contribution of mathematics in primary school to the development of citizenship skills such as decision making and critical thinking in relation to financial literacy, according to a Ethnomathematics perspective. Her interests include the conceptual field of probabilities in the teaching and learning of mathematics, as well as inquiry based-learning. She is also interested in the use of robotics for the development of scientific and mathematical skills in a multidisciplinary context. She studied professional development of teachers in initial training and through professional learning communities (PLCs). She is a member of the Center for the Study of Learning and Performance (CSLP / CEAP).

Most relevant publications in the field of Psychology of Education:

Savard, A. (2014). Transition between university students to teachers: practice in the middle. Canadian Journal of Science, Mathematics and Technology Education, 14(4), 359–370.

Savard, A., Manuel, D., & Lin, T. W. J. (2014). Incorporating culture in the curriculum: the concept of probability in the Inuit culture. In Education Special Issue, Part 2: [Indigenous Education] 19(3): 152–171. Available online at http://ineducation.ca/ineducation/article/view/125/640.

Savard, A., Freiman, V., Larose, F., & Theis, L. (2013). Discussing virtual tools that simulate probabilities: what are the middle school teachers’ concerns? McGill Journal of Education, 48(2), 403–424. Available online at http://mje.mcgill.ca/article/view/9019/6882.

Savard, A., & DeBlois, L. (2013). Enumerating all possible outcomes: an analysis of students’ work. Scientia in Educatione, 4(1), 49–62. Available online at http://www.scied.cz/Default.aspx?PorZobr=1&PolozkaID=134&ClanekID=359.

Chichekian, T., Savard, A., & Shore, B. M. (2011). The languages of inquiry: an english-french lexicon of inquiry terminology in education. Learning Landscape, 4(2), 93–109. Available on-line at http://www.learninglandscapes.ca.

Viktor Freiman. Département d’enseignement au primaire et de psychopédagogie, Université de Moncton,Campus de Moncton, Pavillon Léopold-Taillon, 18, avenue Antonine-Maillet, Moncton, NB, Canada E1A 3E9. E-mail: viktor.freiman@umoncton.ca, http://www.umoncton.ca

Current themes of research:

Mathematics education: problem solving, challenging tasks, enrichment, giftedness, creativity and use of technologyDigital competences: life-long continuum of transferable digtal skills Interdisciplinary research: mathematics and its connection with the arts and science, robotics-based learning, problem-based learningHistory of mathematics education: use of computing procedures and devices in teching and learning mathematics, history of problems and problem-solving.

Most relevant publications in the field of Psychology of Education:

Freiman, V., & Manuel, D. (2014). Une ressource virtuelle de résolution de problèmes mathématiques : les perceptions d’utilisateurs et les traces d’usage. Sciences et Technologies de l´Information et de la Communication pour l´Éducation et la Formation, 20, Consulté le 10 janvier, 2015 à l’adresse, http://sticef.univ-lemans.fr/num/vol2013/18-freiman-reiah/sticef_2013_NS_freiman_18.htm.

Freiman, V., & Applebaum, M. (2014). Engaging elementary students in mathematical reasoning using investigations: example of a bachet game engaging elementary students in mathematical reasoning using investigations: example of a bachet game. In R. Leikin (Ed.) Proceedings of at the 8th International Conference on Mathematical Creativity and Giftedness, Denver, CO, July, 27–31, 2014.

Martinovic, D., Freiman, V., & Karadag, Z. (2012). Visual mathematics and cyberlearning in view of affordance and activity theories. In D. Martinovic, V. Freiman & Z. Karadag (Eds.), Visual Mathematics and Cyberlearning (pp. 209–238). Springer.

Freiman, V. (2011). Mathematically gifted students in inclusive settings: the example of New Brunswick, Canada. In B. Sriraman & K. Lee (Eds.). The Elements of Creativity and Giftedness in Mathematics (pp. 161–172). Sense Publishers.

Freiman, V., & Applebaum, M. (2011). Online Mathematical competition: using virtual marathon to challenge promising students and to develop their persistence. Canadian Journal of Science, Mathematics and Technology Education, 11(1), 55–66.

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Polotskaia, E., Savard, A. & Freiman, V. Investigating a case of hidden misinterpretations of an additive word problem: structural substitution. Eur J Psychol Educ 31, 135–153 (2016). https://doi.org/10.1007/s10212-015-0257-6

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Keywords

  • Problem solving
  • Elementary mathematics
  • Word problems
  • Reasoning development