Commentary 2: knowledge integration in mathematics learning: the case of inversion

Abstract

In their most common form, inversion problems have the form of a+b−b0?. The correct answer in this case is a. It can be derived without computation because addition and subtraction of the same number cancel each other out. This solution method, that is, finding the answer without computation, is called the inversion-shortcut strategy. Many researchers use inversion problems, related principles, and related problem types to investigate understanding by the learners of the inverse relation between mathematical operations. In the following, I use the terms "inversion" and "area of inversion" to refer to the principles, strategies, problem types etc. that are related to the concept of inversion. Researchers frequently emphasize the importance of learning about inversion. However, they rarely explain why inversion is important, even though this is hardly self-evident.There are at least three reasons to doubt the importance of learning about inversion. First, empirical studies show that it is possible to be good at arithmetic problem solving without having a good understanding of inversion and vice versa (Bryant, Christie, & Rendu, 1999;Gilmore & Papadatou-Pastou, 2009). To my knowledge, there is not a single experiment that shows a direct positive causal effect of an understanding of inversion on arithmetic or other mathematical skills (cf. Schneider & Stern, 2009). Second, inversion problems rarely occur in everyday life. The reader of this article might try to remember when he or she last encountered a problem of the form a+b−b (with arbitrary numbers for a and b) in their life. Most people have troubles coming up with even just one or two examples. Third, learning about the inversion shortcut might be an inefficient use of learning time. Based on the existing studies, it can be estimated that using the inversion shortcut instead of left-to-right computation saves roughly one second, at least if relatively small and easy numbers are involved. Consequently, if a teacher used only 90 min on inversion instruction, a learner would have to solve 90×6005,400 inversion problems by using the shortcut strategy before he gained more time by using the shortcut than he invested by learning about inversion.

Exploring the Shift in Children’s Incline Motion Predictions: Fragmentation and Integration of Knowledge as Possible Contributors

“…This would appear to fit in well with ideas concerning knowledge integration (Linn, 2006;Schneider, 2012;Schneider & Hardy, 2013;Schneider & Stern, 2009). Fragmented ideas are interlinked and coordinated within larger configurations of knowledge, and reinforcing such connections over time should result in more systematic forms of reasoning (Thaden-Koch, Dufresne, & Mestre, 2006;Wagner, 2010).…”

“…In relation to knowledge integration (Linn, 2006;Schneider, 2012;Schneider & Hardy, 2013;Schneider & Stern, 2009), it would seem that as children are making less use of the horizontal motion dimension they are equally aligning their incline motion predictions with their fall predictions. However, these predictions never reach the same disparity level as horizontal or incline alone, suggesting that it is unlikely children are only relying on one source of knowledge alone.…”

Applying the design-based learning model to foster undergraduates’ web design skills: the role of knowledge integration

“…In order to avoid the traditional teaching model being too rigid, some scholars (Lin et al, 2020;Linn & Eylon, 2006;Luo & Chea, 2020;Schneider, 2012) believe that teachers may assist students in integrating the knowledge of science and other disciplines into their design thinking. The research of Linn and Eylon (2006) revealed that knowledge integration strengthened the breadth and depth of students' thinking in exploring science-related issues.…”

“…If learners can induce knowledge integration behavior when learning, it may be helpful for their learning. In addition, knowledge integration can reduce the fragmentation of learners' knowledge, and form long-term comprehension by integrating knowledge to enhance learning performance (Schneider, 2012).…”

“…If students combine knowledge with related concepts in the process of designing a project, they may expand their experience in different subjects (So et al, 2018). By integrating learned skills and ideas, students may apply knowledge to problem solving practice and gain meaningful understanding (Elder & Paul, 2008, 2012. The above discussion shows that knowledge integration may play a certain role in DBL.…”