Download Use of Prior Knowledge In Geometry

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.