Kristen Robinson

EMAT 4680

10 October 2000

Midterm Exam Part A

The selected reading reflects different aspects of the Geometry Standard, the Technology Principle, the Reasoning and Proof Standard, and "The Role and Function of Proof with Sketchpad." The first sentence of the excerpt states that "one of the most important challenges in mathematics teaching has to do with the roles of evidence and justification, especially in increasingly technological environments" (311). "Evidence" represents the Geometry Standard. "Increasingly technological environments" represents the Technology Principle. "Justification" represents the Reasoning and Proof Standard. The entire sentence represents the article on "The Role and Function of Proof with Sketchpad." The entire excerpt elaborates on each of the previous representations in further detail.

The paragraph applies to different sections of the Geometry Standard. The reading represents the expectation that "in grades 9-12 all students should . . . analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships" (308). The reading states that "students can quickly generate and explore a range of geometric examples" (311). By creating numerous two- and three-dimensional figures, students can make conjectures about relationships between different objects. The Geometry Standard includes understanding the concepts of similar triangles, parallelograms, regular polygons, and many other geometric relationships. Students are able to construct multiple examples of these figures and to justify why these figures fit the requirements of their definitions using the Geometry Standard.

A further expectation of the Geometry Standard is evident in the selected reading. The standard states that "in grades 9-12 all students should . . . use visualization, spatial reasoning, and geometric modeling to solve problems" (308). By constructing several figures, students can visualize problem situations which are difficult to envision in the mind alone. Additionally, students can manipulate these figures to make conjectures and reasons for problem solutions. Thus, this last expectation of the Geometry standard gives students the ability to develop evidence fitting the bounds of the math problem and to justify the final solution to the problem.

The chosen paragraph also directly relates to the Technology Principle. The reading states that "being able to generate and explore many examples can result in deeper and more-extended mathematical investigations than might otherwise be possible" (311). Students with technology "can examine more examples or representational forms than are feasible by hand, so they can make and explore conjectures easily" (25). These two previous statements are redundant, but the prove that the excerpt from the Geometry Standard and the Technology Principle correlate quite closely. Technology gives students abilities which would be much more difficult to access without technology. Students can analyze various aspects of a situation, be it a simple figure or an entire geometric proof, more quickly and more accurately than with ruler, compass, and paper.

The Reasoning and Proof Standard is very evident in the assigned reading. The reading states that "if they have not learned the appropriate uses of proof and mathematical argumentation, [students] might argue that a conjecture must be valid simply because it worked in all the examples they tried" (311). The Reasoning and Proof Standard demands that curriculums "enable all students to . . . [both] make and investigate mathematical conjectures [and] develop and evaluate mathematical arguments and proofs" (342). Therefore, the Reasoning and Proof Standard should be addressed in the classroom before the given aspect of the Geometry Standard is brought into use. Students should be able to reason repeated observations into conjectures which can be proved or disproved using geometric knowledge. In this sense, the Geometry Standard and the Reasoning and Proof Standard go hand in hand in the high school classroom.

Furthermore, "The Role of Function of Proof with Sketchpad" from Rethinking Proof with The Geometer’s Sketchpad by Michael D. de Villiers collaborates the concepts from the Technology Principle and the Reasoning and Proof Standard into one clear argument which also can be applied to the excerpt from the Geometry Standard. de Villiers describes six different types of proof which should all be used in specific situations in order to prove a conjecture. Honestly, these six means overlap and blur together so often that no one can be positive when one choice of proof is better than another. de Villiers agrees with the reading from the Geometry Standard that technology instigates conjectures and reinforces proofs. In most high school geometry classrooms these days, teachers find it difficult to teach geometry without elaborating on the subject through the use of technology. This is why computer software such as The Geometer’s Sketchpad has become such staples in high school math computer labs.

In the technology-dependent world of today, teachers are forced to address technology’s application to every subject they teach. Technology’s primary purpose in the geometry classroom is to reinforce the rest of the subject’s curriculum. Students can explore different figures, solve problems, and prove their solutions all through the use of technology. However, without first using the definitions of these figures, understanding the problems, and knowing the means of proving a solution students would not be able to utilize technology’s full power. The chosen excerpt merely touches on the necessary elements of the geometry curriculum, and the other mentioned readings expand on the ideas set forth in the excerpt.