Using and Understanding Algorithms to Solve Double and Multiple Proportionality Problems
- PDF / 874,030 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 21 Downloads / 171 Views
Using and Understanding Algorithms to Solve Double and Multiple Proportionality Problems Sintria Labres Lautert 1,2 & Analúcia Dias Schliemann 3 Received: 6 February 2020 / Accepted: 17 August 2020/ # Ministry of Science and Technology, Taiwan 2020
Abstract We analyzed the written answers and explanations by 26 Brazilian high-school students who attempted to solve double and multiple proportionality problems using the crossproduct algorithm. We focused on students’ awareness of the scalar and functional relations among the quantities described in verbal problems, an aspect that has been part of their school instruction. Our results show that, in their written work, the students included data tables or pairs of values, usually with their referents and connected by arrows indicating the scalar or the functional relations described in the problems. In most cases, during individual interviews, they explained their solutions in terms of these relations. Our findings suggest that students’ understanding of proportion relationships between quantities in verbal problems constitutes a basis for their understanding and correct use of algorithms. Keywords Cross-product algorithm . Double proportionality problems . Multiple
proportionality problems . Scalar and functional relations
Introduction This study explores how high-school students, while using and explaining their use of the cross-product algorithm, represent, solve, and understand the relationships between quantities described in double and multiple proportionality verbal problems. * Analúcia Dias Schliemann Sintria Labres Lautert [email protected]
1
Universidade Federal de Pernambuco, Recife, PE, Brazil
2
Departamento de Psicologia, UFPE–Centro de Filosofia e Ciências Humanas, Recife, Brazil
3
Tufts University, Paige Hall, Medford, MA 02155, USA
S. L. Lautert, A. D. Schliemann
Participants were enrolled in a Brazilian school where adopted textbooks and instructors placed emphasis, before the cross-product algorithm was introduced in the classrooms, upon the functional and scalar relationships (Vergnaud, 1983, 1988, 1994) among quantities described in proportionality problems. Proportionality, a core concept in cognitive development and in the characterization of adolescent reasoning (Inhelder & Piaget, 1958), is also a central topic in the middleschool curriculum (Lesh, Post, & Behr, 1988) and an entry point to algebraic reasoning and to algebra and function representations in middle and high school (Post, Behr, & Lesh, 1988). It is part of the conceptual field of multiplicative structures that includes fractions, ratios, rational numbers, linear and n-linear functions with dimensional analysis, and vector spaces (Vergnaud, 1983, 1988, 1994). As Vergnaud cautions, “The main problem for students is that rational numbers are numbers and that entities involved in multiplicative structures are not pure numbers but measures and relationships (p. 161). Therefore, teaching for understanding of concepts and procedures within the multiplicative structures re
Data Loading...
