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Students’ Problem Solving and Transitioning from Numerical to Relational Thinking

Abstract

The transition from arithmetic to algebra in middle school can be critical for students having persistent difficulties in mathematics. The leading perception in our project is that holistic relational thinking is an effective problem-solving tool in both arithmetic and algebra. We worked in collaboration with two teachers of special classes to implement elements of this approach with Secondary I students. We implemented several problem-solving-related activities with students. We collected data through individual interviews with students before and after the intervention. Our data shows the positive effects of the intervention in students changing their strategies as well as overall success in solving word problems requiring relational analysis.

Résumé

Le passage de l’arithmétique à l’algèbre à l’école intermédiaire peut s’avérer déterminant pour les élèves qui éprouvent des difficultés persistantes en mathématiques. Le principe qui oriente notre projet suppose que la pensée relationnelle holistique est un outil de résolution de problème efficace à la fois en arithmétique et en algèbre. Nous avons travaillé en collaboration avec deux enseignants de classes d’éducation spécialisée de première année de l’enseignement secondaire afin de mettre en œuvre des éléments de l’approche relationnelle holistique. Durant cette intervention, nous avons présenté aux élèves plusieurs activités liées à la résolution de problèmes puis nous avons recueilli des données par le biais d’entrevues individuelles avant et après l’intervention. Nos résultats indiquent que l’intervention a eu, dans l’ensemble des effets positifs sur la façon dont les élèves composent avec les problèmes écrits. Nous avons relevé des changements dans les stratégies adoptées par les élèves ainsi que dans leur taux global de réussite pour résoudre les problèmes écrits qui requièrent de l’analyse relationnelle.

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Notes

  1. In Quebec, students start secondary school at the age of 12, immediately after completion of Grade 6.

  2. Students proposed that this is about baby feet and a doll room.

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Appendix. Problems used for interviews (pre- and post-test). Translated from French by authors

Appendix. Problems used for interviews (pre- and post-test). Translated from French by authors

Problem Text
P1: Lynne Lynne is 8 years old. Lyne has ___oranges and ___bananas. How many fruits does she have?
P2: Crayons-I Iren bought a box of ___ coloured crayons for $ ___. Jacob bought ___ of the same boxes. How many coloured crayonss did Jacob buy?
P3: Biking Sarah went biking today. She made some distance before noon and again ____km in the afternoon. She made the total of ___km. What distance did Sarah make in the morning?
P4: Crayons-J Jacob bought ___ identical boxes of coloured crayons for $ ____ and now has ____ crayons. How many crayons are in each box?
P5: Ropes The red rope is ___ cm shorter than the green rope. How long is the green rope if the red rope is ____cm?
P6: Rain Last week it rained ____ times more than this week, ____ millimeters of rain to be exact. They also announced a rain forecast of ____ millimeters for next week. How much rain fell this week?
P7: Flowers The mayors of Laval and Sherbrooke decide to plant _____flowers in each of their cities to beautify the landscape. The two areas reserved for planting flowers are rectangular and have the same area. The dimensions of the Laval land are ___m by _____m. The one in Sherbrooke is _____m long. How wide is the land in Sherbrooke?
P8: Book Nicolas read a few pages of his book on the first day and three times as many pages on the second day. He read ____pages less on the first day than on the second. He still has ___pages to read. How many pages did Nicolas read on the second day?

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Polotskaia, E., Fellus, O.O., Cavalcante, A. et al. Students’ Problem Solving and Transitioning from Numerical to Relational Thinking. Can. J. Sci. Math. Techn. Educ. 22, 341–364 (2022). https://doi.org/10.1007/s42330-022-00218-1

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Keywords

  • Relational thinking
  • Problem solving
  • Transition to algebra
  • Students with difficulties
  • Secondary school mathematics