Keywords: Technology, Geometry
Ref: StaceyS1
Author(s): Finzer, W.F.; Bennett, D.S.
Date: 1995
Title: From Drawing to Construction with the Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88(5), pp. 428-431
Reviewer: StaceyS
Date of Review: 1/30/00
The theme of this article is to point out the importance of a construction versus a drawing in geometry using the Geometer's Sketchpad. Many students will try to "eyeball" the correct lengths or angles of a figure they are trying to produce making the dimensions almost perfect, but not exact. A figure may look exactly what the teacher wants, but when the object is dragged, the elements of the sketch might vary. The article gives tips on emphasizing construction: make sure students know the difference between construction and simply drawing and emphasize dragging the figure to make sure the constraints of the sketch remain the same. By learning how to construct rather than draw, the students will gain a deeper understanding of what they are investigating; including definitions and properties.
I enjoyed this article because I even try to draw a figure instead of constructing it with the appropriate constraints. I think that the Geometer's Sketchpad will further a student's knowledge of many different shapes and figures along with their properties since it is much quicker than using a paper, pencil, ruler, and compass. This article gave me a brand new perspective on the importance of construction and how to convey it to the students.
Ref: StaceyS2
Author(s): Galbraith, Peter
Date: 1995
Title: Mathematics as Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88(5), pp. 412-417
Reviewer: StaceyS
Date of Review: 2/6/00
This article explains why standard 3 of the Curriculum and Evaluation Standards, mathematics as reasoning, is important. First of all, examples of students' reasoning is shown through three different problems. The research was conducted on thirteen- to fifteen-year old students enrolled in years 8 to 10 of British or Australian secondary schools. The main misconceptions and misunderstandings that students have involving proof and logic include: the implications of a counterexample, the need for only one case that disproves a claim instead of many examples, and students little appreciation for the usefulness of proofs. The author discussed how many students adopt a wholly empirical approach to proofs by testing different cases. They convince themselves that a claim must be true if it works for x number of different trials. This is a great way to get students geared toward thinking analyitically, but does not constitute a complete process of proving a claim. To get s! tudents to think more deductively and generalize more will lead to a higher proficiency with proofs. Finally, the article discusses ways for teachers to apply this research in their teaching methods. The teacher and student need to understand the vocabulary used and the meanings attached to words and techniques. A common appreciation of what is being achieved needs to be shared by all. The teacher cannot just "tell" his or her students that a claim is true, they need to actively figure it out for themselves using a more constructivist approach. Otherwise, the meaning behind "proof" is thrown out the window. If the students believe what the teacher says as true without seeing it for themselves, then they cannot fully appreciate the meaning of truth and proof.
I liked this article in the sense that I need all the guidance I can get in preparing to teach students about proof. However, I found the article hard to read and follow. I am not sure that I picked out the important concepts because they seemed deeply embedded. Many of the articles on effective ways to teach reasoning and proof are backed up by research, but they fail to include ways to excite students about developing good proofs. They find no purpose in it and therefore fail to put forth effort. If you cannot convince the student of the importance of proofs pertaining to the "real world", they fail to take interest. This article does not succeed in explaining how to captivate students interest level, it focuses on the proper process of reasoning.
Keywords: Proof, Teaching Strategies
Ref: StaceyS3
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), pp. 565-567
Reviewer: StaceyS
Date of Review: 2/10/00
This article was short, but sweet in the sense that it reflected a positive tone of students learning and liking proofs. The author began by explaining how he would change his way of teaching proofs each year until he finally got a model that he felt worked well. Fidler first changed his grading system to one that gave credit to proofs that were incorrect, but the students had written down everything they could deduce. He thought this would get rid of the "all- or-nothing" attitude. He went on to cooperative learning whcih worked better, but something was missing. The following year he broke the class up into groups where the students were at similar levels and gave them small quizzes to work on together. Afterwards, he gave them a 10 proof take home test which they were excited to work on. Proofs had become fun to work on!
I enjoyed this article because it opens up a way of teaching proofs that could allow the students to enjoy the rigor of mathematics. I would like to try this approach in my own classroom even in other mathematical topics especially problem solving!
Keywords: Activities, Technology,
Ref: StaceyS4
Author(s): Reinstein,D.; Sally,P.;Camp,D.R.
Date: 2000
Title: Generating Fractals through Self-Replication
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90(1); 34-38,43-45
Reviewer: StaceyS
Date of Review: 3/10/00
This article discusses a hands-on activity of fractal geometry, "geometry of nature." The basis of this geometry is the limiting result of an infinite number of self-replications. Students are given 3 activity sheets and a link to a TI-82 program. Through hands-on experience, technology, and geometric visualization, the students can explore fractal geometry in a cooperative classroom. In order that this activity run smoothly, students must have prior knowledge of series and sequences. Otherwise, the teacher can give some examples before the investigation takes place.
I liked this article because it provides the worksheets and a clear insight to an investigation of fractals. This would be a great break from the traditional geometry book or even an integrated curriculum to explore a topic that most students do not explore until college. This activity is a wonderful way to incorporate discovery, creativity, technology, and the desire to find mathematical connections.
Keywords: Resoning, Connections,
Ref: StaceyS5
Author(s): Perham,A.E.; Perham, B.A.; Perham,F.L.
Date: 1997
Title: Creating a Learning Environment for Geometric Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90(7); 521-526
Reviewer: StaceyS
Date of Review: 3/10/00
This article describes how students in a 10th grade geometry class discovered relationships that led to the development of conjectures, theorems, and directions of proofs regarding the centroid of a triangle. This is accomplished through 3 different means: manipulative experiences, software, and the graphing calculator. The author explains the benefits of each method and what can be accomplished.
I really enjoyed this article because of the clear analysis of two related theorems and how students can investigate its properties through a paper-pencil method, Geometer's Sketchpad and Mathcad, and finally, the graphing calculator. Of course, a teacher would not have enough class time to use all of these learning methods for each topic, but I think that by varying the investigation and conjecturing of theorems and postulates should be done with each at one time or another.
Keywords: Technology, Problem Solving,
Ref: StaceyS6
Author(s): Purdy,D.C.
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume
Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(3); 224-228
Reviewer: StaceyS
Date of Review: 3/10/00
This article discusses integrating various areas of mathematics into traditional courses. Technology including graphing calculators and the Geometer's Sketchpad can further a student's investigation and allows them to start working with problems involving advanced algebra, pre-calculus, and calculus. The problem posed by the author is the "Maximum-Volume Box Problem." The objective is to find the largest possible volume of a box constructed from a 10 in-square piece of flat metal. The stipulation is that the equal squares must be cut out of the 4 corners of the original metal. This can be explored through paper models, organized charts, graphing calculator, and the Geometer's Sketchpad in order to discover a pattern for solving the problem. Once solved, students can generalize this problem to a rectangular sheet.
I think this article has some great ideas for exploring patterns and utilizing technology. However, I am trying to imagine this in one of the classrooms I have done a practicum at and I cannot see where one could find the time to complete an activity such as this one. Even though the author thought this would be appropriate for students in grades 9-12, I think it would work best in a pre-calculus or calculus class.
Keywords: Problem Solving
Ref: StaceyS7
Author(s): Gannon,G.E.; Martelli,M.U.
Date: 2000
Title: The Prisoner Problem-A Generalization
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(3); 192-193
Reviewer: StaceyS
Date of Review: 3/10/00
This article gives a great approach to the discovery of the "Prisoner Problem" and then how to generalize the problem. In the initial problem there are 3 prisoners and each wears one of 3 white hats or 2 black hats. The prisoners are in a line facing the wall and if they know (not guess) the color of their hat, they can go free. The prisoner furthest from the wall removes his blindfold and can see the two prisoners in front of him. The middles prisoner can only see the prisoner closest to the wall and the last prisoner cannot see either of the other two. Once the student solves this problem, they can formulate a problem involving 4 prisoners. This can lead to a generalization of n prisoners. The authors recommend this problem as a way to emphasize to students the final step in a problem solver's kit - considering possible generalizations when a particular problem has been solved.
I loved this article! This is a fun way to investigate patterns and could work at any level in high school getting more involved as the student gets older.