Three Balloons for Two Dollars: Developing Proportional Reasoning

Abstract

Ellen, Jim, and Steve bought three helium-filled balloons and paid $2 for all three. They decided to go back to the store and buy enough balloons for everyone in the class.

Prospective Primary School Teachers’ Competence for the Cognitive Analysis of Students’ Solutions to Proportionality Tasks

“…Proportional reasoning is understood as the ability to establish multiplicative relationships between two quantities and to extend this relationship to another pair of quantities (Lamon 2007). This places proportional reasoning as a precursor to algebraic thinking (Langrall and Swafford 2000) and functional thinking (Lichti and Roth 2019).…”

“…The strategies used are affected by whether the problem context is familiar to the student or not, the placement of the unknown value, and the nature of the numerical relationships, that is, the presence or absence of integer ratios, the size of the ratios or the numbers involved (Tourniaire and Pulos 1985;Misailidou and Williams 2003). Langrall and Swafford (2000) or Misailidou and Williams (2003) proposed different levels of students' proportional reasoning in terms of the tasks in which students may or may not succeed. However, authors such as Modestou and Gagatsis (2010) consider that proportional reasoning does not only imply the success in solving a range of proportional problems, but it also involves handling verbal and arithmetical analogies, as well as the awareness of discerning non-proportional situations from other situations.…”

An analysis of proportional reasoning ability of junior high school students

“…Research on students' proportional reasoning abilities has been done a lot. Dooley (2006, p. 6) said that in the last 20 years, there had been several studies on strategies used by students in solving proportional reasoning problems (e.g., Ben-Chaim et al, 2012;Carney et al, 2015;Lamon, 1993;Langrall & Swafford, 2000;Prayitno et al, 2019;Sumarto, 2013). Several previous studies have exhibited theories about proportional reasoning problem-solving strategies at the elementary, junior high, and senior high school levels.…”

“…The difference in students' proportional reasoning ability can be reflected in the different strategies they use to solve a problem. The study of Langrall and Swafford (2000) described the characteristics of the strategies in solving proportional reasoning problems that later grouped into levels of proportional reasoning ability. Guided by the student proportional reasoning level described by Langrall and Swafford (2000), this study will report the analysis of students' proportional reasoning ability based on student learning achievement refers to the average score of the national examination in 2016 to acquire new knowledge about the proportional reasoning ability based on the learning achievement of junior high school students.…”

مستویات التفکیر التناسبی لدى طلاب الصف الثانی المتوسط فی منطقة القصیم The levels of proportional reasoning among grade 8 students in Qassim schools

“…‫آزشػ‬ ‫ٗ٠ب‬ Langrall and Swafford ( 7000 & Izsak, 2015;Byerley & Thompson, 2017;Langrall & Swafford, 2000;Orrill & Brown, 2012;Son, 2013 )…”