Abstract
Ensuring students correctly understand the notion of equality is a fundamental problem in teaching mathematics. Is it possible to teach arithmetic in such a way that students do not misinterpret the equal sign "=" as indicating the result of an arithmetic operation and, consequently, viewing the arithmetic operation symbols (+, -, or x) as systematically meaning perform a computation? The present paper describes the implementation of an experimental arithmetic teaching program (called ACE) drawn up in conjunction with teachers and focusing on these issues. We assessed the program's impact via an experiment involving 1140 experimental group students and 1155 control group students, and using a pretest/posttest design. The experimental group students achieved higher composite arithmetic scores, combining all four subdomains (arithmetic writing, mental computation, word-problem solving, and estimation), than control group students. The effect size, computed using individuals' posttest scores adjusted for pretest scores was d = 0.559. The ACE program was effective in all four subdomains tested; however, it was particularly successful in the arithmetic writing subdomain, especially in writing equalities. The program's effect not only held one year later, it was cumulative, as the benefits produced by following the program in both first and second grade were almost double those of following the program in just one grade.
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Acknowledgments
We thank all the children and teachers who have taken part in the experimentation. We are also grateful to the experimenters, coders, and administrative officials who have facilitated its performing.
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Jean-Paul Fischer. Lorraine Laboratory of Psychology and Neuroscience (2LPN), University of Lorraine, 54015 Nancy, France. E-mail: jean-paul.fischer@univ-lorraine.fr
Current themes of research:
Interactive development of procedural and declarative knowledge in the field of number and arithmetic. Mirror writing in typically developing children.
Most relevant publications in the field of Psychology of Education:
Fischer, J. P., (2017, on line). Studies on the written characters orientation and its influence on digit reversal by children. Educational Psychology. https://doi.org/10.1080/01443410.2017.1359239
Fischer J. P., 2017. Does finger sense really have a delayed effect on children’s addition performance? Cognitive Processing, 18(1), 105-106. https://doi.org/10.1007/s10339-016-0782-5
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Fischer, J. P. (2015). Commentary: Development of magnitude processing in children with developmental dyscalculia: space, time, and number. Frontiers in Psychology (Developmental Psychology), 6:804.https://doi.org/10.3389/fpsyg.2015.00804
Fischer, J. P., & Tazouti, Y. (2012). Unraveling the mystery of mirror writing in typically developing children. Journal of Educational Psychology, 104(1), 193-205. https://doi.org/10.1037/a0025735
Emmanuel Sander. Developmental psychology and educational sciences, IDEA Lab, University of Geneva. Email: emmanuel.sander@unige.ch
Current themes of research:
Arithmetical word problem-solving. Analogical reasoning.
Most relevant publications in the field of Psychology of Education:
Brissiaud, R., & Sander, E., (2010). Arithmetic word problem solving: a situation strategy first framework. Developmental Science, 13(1), 92-107. https://doi.org/10.1111/j.1467-7687.2009.00866.x
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Gamo, S., Sander, E., & Richard, J. F. (2010). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction, 20(5), 400-410. https://doi.org/10.1016/j.learninstruc.2009.04.001
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Gérard Sensevy. Educational Sciences, CREAD Laboratory, University Bretagne Occidentale. E-mail: gerard.sensevy@espe-bretagne.fr
Current themes of research:
Didactics, teaching and learning, and their relationships.
Most relevant publications in the field of Psychology of Education:
Sensevy, G., Forest, D., Quilio, S. & Morales, G. (2013). Cooperative engineering as a specific design-based research. ZDM - The International Journal on Mathematics Education, 45(7), 1031-1043. https://doi.org/10.1007/s11858-013-0532-4
Sensevy, G., Mercier, A., Schubauer-Leoni, M.L., Ligozat, F. & Perrot, G. (2005). An attempt to model the teacher’s action in mathematics. Educational Studies in Mathematics, 59(1), 153-181. https://doi.org/10.1007/s10649-005-5887-1
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Bruno Vilette. Developmental Psychology, PSITEC Laboratory, University Lille. E-mail: bruno.vilette@univ-lille3.f
Current themes of research:
Numerical cognition and development.
Most relevant publications in the field of Psychology of Education:
De Clercq-Quaegebeur, M., Rassel, A., Casalis, S., Vilette, B., Lemaitre, M.-P. & Vallée, L. (2017). Arithmetic abilities in children with developmental dyslexia : Performances on French ZAREKI-R Test. Journal of Learning Disabilities.https://doi.org/10.1177/0022219417690355
Vilette, B. (2016). Effets d'entraînements basés sur l'estimation numérique auprès d'enfants avec une dyscalculie ou des troubles du calcul. Développements, 20-21, 57-77.
Jean-François Richard. Paragraphe Laboratory, University Paris 8. E-mail: jf.richard.paris8@gmail.com
Current themes of research:
Modelling problem-solving behavior. Role of problem-solving activity in learning mathematics. Analysis of errors as a tool for cognitive diagnosis.
Most relevant publications in the field of Psychology of Education:
Richard, J. F., & Sander, E. (2000). Activités d’interprétation et de recherche de solution dans la résolution de problème. In J. N. Foulin & C. Ponce (Eds) Lire, écrire, compter, apprendre : les apports de la psychologie des apprentissages. Bordeaux : CRDP d’Aquitaine.
Gamo, S., Sander, E., & Richard, J. F. (2010). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction, 20(5), 400-410. https://doi.org/10.1016/j.learninstruc.2009.04.001
Mengue-Metoule, E., Richard, J. F., & Sander, E. (2015). Classification des problèmes additifs : statut intermédiaire de la transformation relativement au complément et à la comparaison. L’année psychologique, 115(4), 497-531.
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Fischer, JP., Sander, E., Sensevy, G. et al. Can young students understand the mathematical concept of equality? A whole-year arithmetic teaching experiment in second grade. Eur J Psychol Educ 34, 439–456 (2019). https://doi.org/10.1007/s10212-018-0384-y
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DOI: https://doi.org/10.1007/s10212-018-0384-y