Journal of Scholastic Inquiry: Education
Volume 1, Page 153
two different approaches would be the same. To this end, the idea of MR they learned from third grade was reinforced and connected to the geometry learning. Following the concrete modeling stage, a semiconcrete level of instruction with graph paper was used for students to create the unit squares for covering the area based on the given scale on the two dimensions (i.e., length and width) (see Figure 1-upper panel for reference). Building on their understanding from the concrete/semiconcrete modeling, students were then transitioned to the conceptual model equation for area problem solving (see Figure 1-lower panel) without relying on concrete models. Students engaged in a discussion about how the model equation tells the story about the multiplicative relationship between the length and the width. That is, the area is expressed as a multiple of the CU (the base strip). In a mathematical expression, A (length) × B (width) = Area. At this point, students were ready to advance to using the abstract model for area problem solving. For instance, to solve “How much carpet do you need to cover the floor of a room that is 18 feet long and 9 feet wide,” students would first identify from the problem, the number of units (of ones) for the length (i.e., the unit rate = 18) and the number of CUs for the width (9). Next, students would map the information of the two dimensions onto the model equation (see Figure 1-lower panel) and use the equation to solve for the area (18 × 9 = 162). To strengthen conceptual understanding, students were guided to verify the answer they obtained from the abstract model (the formula) through building a mini-sized concrete model and counting the actual unit squares that covered the area. Through this “two-way” connection (from concrete to abstract, and then from abstract back to concrete), the students realized that the symbolic model equation indeed represented the concrete model and could be used directly for solving area problems. The abstract model was no longer “abstract” or unfamiliar to them when it was attached to a concrete model, and, consequently, the students could make sense of the abstract model. After solving problems with the area as the unknown, students were also provided opportunities to solve problems with a missing factor (e.g., “What is the length of a rectangle that has an area of 54 square feet and a width of 18 feet?”). Students were engaged in representing the problem in the area model equation with a letter x to represent the unknown quantity. Then, they completed the problem through solving for the unknown quantity in the equation (Length × Width = Area). After students learned to use the model equation (Length × Width = Area) for solving relevant area problems, the model equation for solving the volume of rectangular prisms was introduced (see Figure 2). The same instructional sequence was applied to the teaching of solving problems that involve the volume of rectangular prisms. Again, the instructor (the second author) started the instruction with concrete manipulatives for modeling the concept. Students covered the prism’s base with the unit cubes, and they counted the number of unit cubes that made up the base. Next, students counted the base’s number of layers to complete the rectangular prism. The students were guided to discover that the number of iterations of the base unit would be the height (or thickness) of the prism. Finally, students could find out the volume