This study reports findings from a teaching experiment in which 4th grade children engaged in solving multiplication tasks with rectangular arrays in a computer environment. The environment provided flexible task-solving through dynamic virtual manipulatives (VMs) so children could use their existing knowledge of multiplication to complete array tasks in a variety of ways appropriate to their level of development. I hypothesized that children's multiplicative thinking, consisting of the counting scheme (Steffe, 1994) and splitting structure (Confrey, 1994), could advance through what I called insights, which arose when students hit an impasse which resulted in strategy revision. This study described the actions four children took to solve tasks. Findings suggested that the array tasks, the computer environment, array VMs, and verbal prompts helped children develop multiplicative thinking through engaging in multiple pathways to solution including either or both counting and splitting actions. Results showed the students were able to solve the tasks correctly and all four children experienced impasses and two resolved them with insights. Two children demonstrated advancements in multiplicative thinking and counting solution strategies were used by more children than splitting strategies. Successful task-solving with arrays appeared to include three important types of thinking: 1) Developing an awareness of the dimensions of the array; 2) Decomposing or splitting larger portions of the original array; and 3) Manipulating the numbers in the array. Findings suggest that children work in an intermediate space between repeated addition and memorized facts that includes the counting and splitting constructs. In this space, they can face impasses as they work with large factors in new computer environments but these environments can also provide helpful constraints, giving children insight opportunities to restructure their initial depictions of the problem to manipulate arrays to match the multiplication facts they know. Findings have implications on the Common Core State Standards for 4th grade mathematics, the use of visual representations and computer environments in the classroom, and current multiplication curriculum.This study reports findings from a teaching experiment in which 4th grade children engaged in solving multiplication tasks with rectangular arrays in a computer environment.
|Title||:||Developing Multiplicative Thinking with Rectangular Array Tasks in a Computer Environment|
|Author||:||Amy I-Yu Huang|