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Use of Prior Knowledge In Geometry Why Is This Strategy Useful? Researchers have suggested that schemas or mental models drive the way students approach and solve word problems in geometry. These mental models contain knowledge about concepts, the relation among these concepts, and how to use these concepts. For successful problem-solving process, these mental models need to be organized and accessed in students’ memory in appropriate contexts. To help students organize and access mental models during problem solving exercises, teachers should provide ongoing assessment and feedback about the mental models that students have used and should have used. Description of Strategy Teachers should use explicit instruction to ensure that all students have mastered that mental models required for problem solving. Teachers should also ask students to describe the prior knowledge (the schemas) they have used in solving a problem in order to identify cases in which students fail to use relevant schemas. Then, teachers should model students how to use knowledge of concepts and how to clearly describe the relation between this knowledge and the solution of a variety of word problems. The emphasis in this instructional strategy is on ongoing assessment of the schemas that struggling students fail to use and prescriptive feedback that highlights these concepts until students master them. Here are some examples of schemas: • Isosceles triangle • Right-angled triangle • Equilateral triangle • Cosine ratio for a right-angled triangle • Sine ratio for a right-angles triangle • Tangent ratio for a right-angled triangle • Tangent-radius theorem • Pythagoras rule • Sum of angles in a triangle theorem • Perpendicularity rule • Difficulty between segments • Radii for a given circle are equal in length • Ratio of sides of triangles • Congruency of triangles • Exterior angle theorem • Complementarity rule • Supplementarity of angles INQUIRE summaries available at schools.nyc.gov/inquire Study abstract and sample activity reproduced with permission from Springer, copyright © 1998. All rights reserved. Research Evidence One case study of thirty 10th grade students in a suburban high school serving middle-class population provides support for this approach. The students were classified into two groups: high-achieving and low-achieving. Analysis of students’ interviews showed that high-achievers activate a larger number of mental models than low-achievers when attempting to address a word problem. Sample Studies Supporting this Strategy Chinnappan, M. (1998). Schemas and mental models in geometry problem solving. Educational Studies in Mathematics, 36(3), 201-217. Recent investigations of mathematical problem solving have focused on an issue that concerns students‘ ability at accessing and making flexible use of previously learnt knowledge. I report here a study that takes up this issue by examining potential links between mental models constructed by students, the organizational quality of students‘ prior geometric knowledge, and the use of that knowledge during problem solving. Structural analysis of the results suggest that the quality of geometric knowledge that students develop could have a powerful effect on their mental models and subsequent use of that knowledge. Sample Activity Source: Chinnappan (1998) Linking Concepts to Solution Options In the word problem below, the student needs to recognize that a component may serve more than one function in order to solve the problem. For example, the student needs to recognize that AE is (a) a straight line, (b) a tangent to the circle, (c) the hypotenuse of the right-angled triangle ACE. This is essential for linking the appropriate concepts and theorems. For example, recognizing that AE is a tangent is needed to know that CDA is a right angle (radius-tangent theorem). Students may use different approaches to solve this problem depending on the schemas activated. Trigonometric solution: determining the angle ACD can lead to the application of trigonometric ratios in the triangle ADC to determine the length of AC. Trigonometric / Pythagoras solution: understanding that AE is a hypotenuse could result in setting up an equation that uses the Pythagoras’ formulate. Equilateral / isosceles triangles: Understanding that CDA is a right angle and drawing the segment BD could result in a solution that uses prior knowledge about the angles of equilateral triangle and isosceles triangles. By using a series of deductive reasoning, students can compare the length of segments of triangles to arrive at the length of AB, BC, and then AC. Similar triangles / trigonometry: Students may solve this problem by setting up equations that show the ratios between the corresponding sides of triangles. INQUIRE summaries available at schools.nyc.gov/inquire Study abstract and sample activity reproduced with permission from Springer, copyright © 1998. All rights reserved. INQUIRE summaries available at schools.nyc.gov/inquire Study abstract and sample activity reproduced with permission from Springer, copyright © 1998. All rights reserved.