Effects of Visual Working Memory Training and Direct Instruction on Geometry Problem Solving in Students with Geometry Difficulties.

Side-Side-Side postulate states that if three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent. We integrated a CRA instructional sequence to teach each of the five postulates. The CRA instructional sequence of teaching the “Side-Side-Side (SSS)” postulate is illustrated in Figure 2. First, at the concrete level, the experimenters provided with parts (i.e., sticks) that can make triangles, then asked the participant to assemble a triangle in specific given conditions (e.g., a stick with a length of 4 units, one with a length of 3 units, and the other one with a length of 5 units on each side respectively). Then the experimenter instructed the participants to assemble a second triangle under the same conditions (i.e., using three sticks of the same lengths), and encouraged the participants to try as many approaches as possible to make different triangles under the given conditions. Afterwards, the instructor guided the participants to conclude that one cannot make two different triangles with the three given side lengths. Second, at the representational level, the instructor used visual diagrams to demonstrate with appropriate flipping and rotation procedures, two triangles with the congruent side lengths overlapping with each other. Lastly, at the abstract level, the instructor helped students to state the postulate verbally. Representational Learning Disabilities: A Contemporary Journal 15(1), 117-138, 2017 128 After introducing the postulate with the CRA sequence, the instructor adopted an explicit modeling guided practice independent practice sequence to help the participants to solve geometry problems using the postulate that was just taught. First, the instructor explicitly modeled three problems, including both triangle congruence proof problems, or explorative triangle problems that required a step of triangle congruence proving (e.g., to find out the degree of an angle or the length of a side of a triangle, students had to first prove this triangle was congruent with another triangle of which the angle/side information was given or could be calculated). Next, the instructor asked the participants to solve another three problems, and provided guidance or cues when they needed help. For example, the instructor may ask “how can we decide if two triangles are congruence” “Can you find any pairs of sides/angles that are the equivalent? ” Finally, we asked the participants to independently work on three problems. Two trained research assistants implemented the instruction to the four participants. One research assistant was a master student in a mathematics education program, with a bachelor degree in mathematics, whereas the other research assistant was a psychology major. Before implementing the interventions, both research assistants thoroughly reviewed the middle school geometry curriculum on triangle congruence and related properties, and worked on a variety of triangle problems on their own to make sure they had adequate content knowledge and problem solving skills to teach the lessons. The two instructors also practiced teaching the lessons with the CRA sequence before teaching it to the participants with GD. Treatment Fidelity The author and two research assistants developed a teaching script and a lesson plan for each of the six tutoring sessions on triangle congruence. The teaching script and the lesson plan were used to guide explicit instruction with a CRA instructional sequence during the intervention on triangle congruence postulates as shown in the appendix. The author observed 30% of treatment sessions to assess fidelity or quality of implementation of specific performance indicators. In addition, a checklist of the instructional steps was developed to assess the instructor’s adherence to the instructional components, which was judged on the presence or absence of the features listed on the fidelity checklist. These components were included if the instructor followed the CRA sequence, if the instructor explicitly modeled three problem-solving processes, if the instructor guided students to solve problems, if the instructor provided feedback appropriately, and so forth. Treatment fidelity was calculated as the percentage of steps correctly completed and was 100%. Inter-rater Reliability Inter-rater reliability was checked by an undergraduate research assistant who was unaware of the purpose of this study. Participants’ working memory level and accuracy in the n-back program was automatically recorded in the computer, so we only checked the reliability of data of the four generalization tests (i.e., the KeyMath Geometry Subtest, the PSVT: R, the Triangle Problem Solving Test, and the Triangle Congruence Identification Test). The research assistant randomly chose 30% of the tests, and re-scored all the items of the selected tests. The inter-rater reliability was 100%. Learning Disabilities: A Contemporary Journal 15(1), 117-138, 2017