Modeling the Real-World Context with Concreteness Fading

Worked examples that are situated in real-word contexts (e.g., word problems) should be modeled with concreteness fading. In this book, the concreteness fading method refers to a representational sequence that shifts from concrete real-world contexts to semi-concrete models (e.g., manipulatives, diagram) and eventually to abstract numerical solutions. I would like to point out two aspects of this structure in particular. First, this representational sequence, from concrete to abstract, refers to an overarching developmental goal with abstract thinking being the destination. This is because real-world contexts contain lots of distracting information that could hinder students’ ability to grasp the structural information. As such, it is important to set up the ultimate learning goal to go beyond the concrete. In a prior study (Cai, 2005), U.S. teachers were observed accepting student solutions indicated by manipulatives or drawings while their Chinese counterparts valued students’ numerical solutions more.

This reported cultural difference was consistently observed in my project. Given that early algebraic concepts like inverse relations and basic properties are fundamentally abstract in nature, I believe that students should strive to obtain mastery over these concepts using abstract mathematical thinking. Therefore, the targeted approach in this book encourages using manipulatives and drawings only as a path or tool that leads to numerical solutions. Of course, as observed in many Chinese videos, concreteness fading is not a one-way street. There are many opportunities to fold back from abstract to concrete in ways that ensure sense-making.

The second aspect of this representational sequence is that concreteness fading can be initiated more gradually through the intermediate use of semi-concrete representations. In my project, I noticed that Chinese lessons sometimes shifted directly from word problems to numerical solutions while U.S. lessons often directly started with semi-concrete models. In this sense, a combination of the U.S. and Chinese lesson insights may provide excellent examples of how elementary mathematics lessons can be designed to apply concreteness fading both gradually and completely. For instance, when teaching additive inverse relations, the swimming pool context in Chinese lessons could be modeled using the connecting cubes that occurred frequently in the U.S. lessons. The connecting cubes could also be used to model the story problems about the CP and AP of addition in Chinese lessons. Similarly, when teaching the CP of multiplication, the Chinese lessons tended to directly solve the story problems with numerical solutions. In this case, the array model, which was unique to U.S. lessons, could be added to visualize the story problems and aid students’ meaning- making. The array/area model is also an effective tool to illustrate the DP, which may be considered by Chinese teachers to model relevant story problems.

In addition to the cubes and arrays, linear quantity models such as number lines and tape diagrams (Ding, Chen, & Hassler, 2019) play an important role in facilitating concreteness fading (Pashler ct al., 2007). During my project, I observed instances of the linear quantity model occurring in both U.S. and Chinese lessons. Chinese teachers often spent more time guiding students to discuss the diagrams and then reason about the story situations. This can be seen in the G2 lesson where the teacher taught students to draw tape diagrams to model a pair of additive comparison tasks (see Section 2.4). The G3 lesson that informally integrated the DP also contained a sophisticated discussion about the number lines that modeled the story context of shopping for clothes (see Section 5.5). The linear quantity models began in East Asian countries and have been drawing increased attention from the U.S. (Ding & Li, 2014; Murata, 2008; Ng & Lee, 2009). In fact, Murata (2008) articulated a couple merits of tape diagrams such as illustrating the structures of a problem and offering a platform for class discussions.

However, simply presenting semi-concrete representations (e.g., connected cubes, array/area models, tape diagrams, number lines) does not guarantee a successful bridge from concrete to abstract mathematical ideas. In other words, following the recommended representational sequence of concreteness fading (concrete, semi-concrete, and abstract) does not guarantee a successful transition between these representations. Teachers need to ask deep questions that help students draw meaning from these representations and that promote the connections between the varied representations. Below I elaborate on the features of teachers’ deep questioning.

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