Problems of Cartesian Product Solved by Elementary School Students

Abstract: The study investigated the solution of direct (which requires multiplication for its resolution) and inverse (which requires division for its resolution) Cartesian product problems by elementary education students, examining the level of problem complexity and the children procedures according to the type of problem. A total of 269 8 and 10 year-old students attending from 3rd to 5th grade, were asked 8 and 10 years to solve direct and inverse Cartesian product problems. As expected, the inverse problem was the most difficult one. The strategies showed that levels of combinatorial reasoning vary according to the type of problem. It was also found a progression in the solution of direct Cartesian product problems, but not in relation to the solution of inverse problems.

摘要：本研究探究了小学阶段学生对笛卡尔积（Cartesian product）正问题（求解需运用乘法）与反问题（求解需运用除法）的解题情况，考察了问题的复杂程度以及儿童针对不同题型所采用的解题步骤。本研究共计招募269名就读于3至5年级的8岁与10岁学生，要求其完成笛卡尔积正问题与反问题的求解。正如预期，反问题是难度最高的题型。解题策略分析显示，组合推理（combinatorial reasoning）的水平因题型不同而存在差异。研究同时发现，学生在笛卡尔积正问题的求解中存在能力进阶，但反问题的求解未呈现类似的进阶趋势。