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

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

References

  • Barrouillet, P., & Camos, V. (2002). Savoirs, savoir-faire arithmétiques, et leurs déficiences (version longue). Ministère de la Recherche, programme cognitique, école et sciences cognitives.

  • Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams : a method demonstrated in grade 4–6 texts used in Singapore. Mathematics Educator, 14(1), 42–46.

    Google Scholar 

  • Brousseau, G. (1988). Théorie des situations didactiques. Grenoble: La Pensée Sauvage.

    Google Scholar 

  • Brousseau, G., & Gibel, P. (2005). Didactical handling of students’reasoning process in problem solving situations. In C. Laborde, M.-J. Perrin-Glorian, & A. Sierpinska (Eds.), Beyond the apparent banality of the mathematics classroom (pp. 13–58). Springer.

  • Brun, J. (1990). La resolution de problernes arithrnetiqucs: Bilan et perspectives. Math Ecole, 141, 3–14.

    Google Scholar 

  • Checkley, K. (2006). The essentials of mathematics K-6: effective curriculum, instruction, and assessment. Alexandria: VI: Association for supervision and curriculum development.

    Google Scholar 

  • Chevallard, Y. (1988). Sur l’analyse didactique : deux études sur les notions de contrat et de situation. Marseille: Institut de recherche sur l’enseignement des mathématiques d’Aix-Marseille.

    Google Scholar 

  • Christou, C., & Philippou, G. (1999). Role of schemas in one-step word problems. Educational Research and Evaluation, 5(3), 269–289. doi:10.1076/edre.5.3.269.3884.

    Article  Google Scholar 

  • Conne, F. (1985). Calculs nurneriques et calculs relationnels dans la resolution de problernes d’arithmetique. Recheche En Didactique Des Mathématiques, 5, 269–332.

    Google Scholar 

  • Davydov, V. V. (1982). Psychological characteristics of the formation of mathematical operations in children. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: cognitive perspective (pp. 225–238). Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Davydov, V. V. (2008). Problems of developmental instruction: a theoretical and experimental psychological study. Hauppauge: Nova Science Publishers.

    Google Scholar 

  • DeBlois, L. (1997). Quand additionner ou soustraire implique comparer. Éducation et Francophonie, 25(1), 102–120. Retrieved from http://www.acelf.ca/revue/XXV1/articles/rxxv1-08.html#SEC20.

  • DeBlois, L. (2003). Interpréter explicitement les productions des élèves: une piste. Éducation et Francophonie, 31(2), 21. Retrieved from http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:Interpréter+explicitement+les+productions+des+élèves+:+une+piste+…#0.

  • Elia, I., Gagatsis, A., & Demetriou, A. (2007). The effects of different modes of representation on the solution of one-step additive problems. Learning and Instruction, 17, 658–672. doi:10.1016/j.learninstruc.2007.09.011.

    Article  Google Scholar 

  • Fagnant, A. (2005). The use of mathematical symholism in problem solving: An empirical study carried out in grade one in the French community of Belgium. European Journal of Psychology of Education, XX(4), 355–367.

  • Ferreira, F., Christianson, K., & Hollingworth, a. (2001). Misinterpretations of garden-path sentences: implications for models of sentence processing and reanalysis. Journal of Psycholinguistic Research, 30(1), 3–20. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/11291182

  • Gervais, C., Savard, A., & Polotskaia, E. (2013). Les structures additives et le développement du raisonnement. Bulletin AMQ, 53(3), 58–66.

    Google Scholar 

  • Gibson, E. J., & Levin, H. (1975). The psychology of reading. Cambridge: The MIT Press.

    Google Scholar 

  • Ginsburg, H. P. (1977). Children’s arithmetic. New York: D Van Nostrand.

    Google Scholar 

  • Giroux, J., & Ste-Marie, A. (2001). The solution of compare problems among first-grade students. European Jornal of Psychology of Education, 16(2), 141–161.

    Article  Google Scholar 

  • Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: a comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87(1), 18–32. doi:10.1037/0022-0663.87.1.18.

    Article  Google Scholar 

  • Helmholtz, L. F. V. (1860). Handbuch der physiologischen Optik [Handbook of physiological optics]. Leipzig: Voss.

  • Houdement, C. (2011). Connaissances cachées en résolution de problèmes arithmétiques ordinaires à l’école. Annales de Didactique et de Sciences Cognitives, 16, 67–96.

    Google Scholar 

  • Iannece, D., Mellone, M., & Tortora, R. (2009). Counting vs. measuring: Reflections on number roots between epistemology and neuroscience. In M. Tzekaki & M. Kaldrimidou (Eds.), Proceedings of the 33rd Conference of the International group for the Psychology of Mathematics Education (Vol. 3, pp. 209–216). Thessaloniki, Greece.

  • Jackson, K. J., & Cobb, P. A. (2010). Refining a vision of ambitious mathematics instruction to address issues of equity. Denver: Annual meeting of the American Educational Research Association.

    Google Scholar 

  • Jackson, K. J., Shahan, E. C., Gibbons, L. K., & Cobb, P. A. (2012). Lunching complex tasks. Consider four important elements of setting up challenging mathematics problems to support all students’ learning. Mathematics Teaching in the Middle School, 18(1), 24–29.

    Article  Google Scholar 

  • Julo, J. (1995). Représentation des problèmes et réussite en mathématiques (Presses Un.). Rennes: Presses Universitaires de Rennes.

    Google Scholar 

  • Julo, J. (2002). Des apprentissages spécifiques pour la résolution de problèmes? Grand N, 69, 31–52.

    Google Scholar 

  • Kintsch, W. (1988). The role of knowledge in discourse comprehension: a construction-integration model. Psychological Review, 95(2), 163–82. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/3375398.

  • Kintsch, W. (1994). Text comprehension, memory, and learning. The American Psychologist, 49(4), 294–303. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/8203801.

  • Kintsch, W. (2005). An overview of top-down and bottom-up effects in comprehension: The CI perspective. Discourse Processes, 39(2), 125–128. doi:10.1207/s15326950dp3902&3_2.

    Article  Google Scholar 

  • Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92(1), 109–29. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/3983303.

  • Masson, S., Potvin, P., Riopel, M., Foisy, L. B., & Lafortune, S. (2012). Using fMRI to study conceptual change : why and how ? International Journal of Environmental & Science Education, 7(1), 19–35.

    Google Scholar 

  • Mellone, M., Verschaffel, L., & Van Dooren, W. (2014). Making sense of word problems: The effect of rewording and dyadic interaction. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38 th Conference of the International Group for the Psychology of Mathematics Education and the 36 th. Conference of the North American Chapter of the Psychology of Mathematics Education. Mathematical tasks and the student (pp. 4 – 201–208). Vancouver, Canada: PME.

  • Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics, 13, 373–394. doi:373-394. 0013-1954/82/0134-03735.

  • Piaget, J. (1964). PART I: cognitive development in children: Piaget: development and learning. Journal of Research in Science Teaching, 2(3), 176–186.

    Article  Google Scholar 

  • Polotskaia, E., Freiman, V., & Savard, A. (2013). Developing reasoning about simple additive structures: one task for elementary students. In A. M. Lindmeier & A. Heinze (Eds.), Mathematics learning across the life span. The 37th conference of the international group for the psychology of mathematics education (Vol. 5, p. 253). Kiel, Germany.

  • Riley, M. S., & Greeno, J. G. (1988). Developmental analysis of understanding language about quantities and of solving problems. Cognition and Instruction, 5(1), 49–101.

    Article  Google Scholar 

  • Riley, M. S., Greeno, J. G., & Heller, J. L. (1984). Development of children’s problem-solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). Orlando, FL: Academic Press, Inc. Retrieved from http://www.eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED252410&ERICExtSearch_SearchType_0=no&accno=ED252410.

  • Salmina, N. G., & Sohina, V. P. (1981). Obuchenie obcshemu podhodu k resheniu zadach [Teaching general method of problem solving]. Voprosy Psikhologii, (I), 151–157.

  • Sarrazy, B. (2002). Effects of variability of teaching on responsiveness to the didactic contract in arithmetic problem-solving among pupils of 9–10 years. European Journal of Psychology of Education, 17(4), 321–341. doi:10.1007/BF03173589.

    Article  Google Scholar 

  • Savard, A., Polotskaia, E., Freiman, V., & Gervais, C. (2013). Tasks to promote holistic flexible reasoning about simple additive structures. In C. Margolinas (Ed.), Proceedings of ICMI Study 22 Task Design in Mathematics Education (Vol. 1, pp. 271–280). Oxford, England.

  • Schubauer-Leoni, L. M., & Ntamakiliro, L. (1994). La construction de réponses à des problèmes impossibles. Revue Des Sciences de L’éducation, 20(1), 87–113.

    Article  Google Scholar 

  • Stavy, R., & Babai, R. (2010). Overcoming intuitive interference in mathematics: insights from behavioral, brain imaging and intervention studies. ZDM Mathematics Education, 42(6), 621–633.

    Article  Google Scholar 

  • Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 39–59). Hillsdale, New Jersey: Lawrence Erlbaum Associates. Retrieved from http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:A+classification+of+cognitive+tasks+and+operations+of+thought+involved+in+addition+and+subtraction+problems#0.

  • Vygotsky, L. S. (1984). Detskaia psihologia, Sobranie sochinenii, T.4 [Child Psychology]. Moscow: Pedagogika.

    Google Scholar 

  • Xin, Y. P. (2007). Word problem solving tasks in textbooks and their relation to student performance. The Journal of Educational Research, 100(6), 347–360. doi:10.3200/JOER.100.6.347-360.

    Article  Google Scholar 

  • Xin, Y. P., Wiles, B., & Lin, Y.-Y. (2008). Teaching conceptual model-based word problem story grammar to enhance mathematics problem solving. The Journal of Special Education, 42(3), 163–178. doi:10.1177/0022466907312895.

    Article  Google Scholar 

Download references

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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