We analyzed the written answers and explanations by 26 Brazilian high-school students who attempted to solve double and multiple proportionality problems using the cross-product 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.

我们分析了 26 名巴西高中生的书面答案和解释，他们试图使用叉积算法解决双重和多重比例问题。我们专注于学生对语言问题中描述的数量之间的标量和函数关系的认识，这是他们学校教学的一部分。我们的结果表明，在他们的书面作业中，学生包括数据表或数值对，通常与他们的参照物相连，并用箭头连接，指示问题中描述的标量或函数关系。在大多数情况下，在个别访谈中，他们根据这些关系解释了他们的解决方案。