Keywords: Manipulatives, Geometry, Technology
Ref: TinaM1
Author(s): Perham, Arnold E.; Perham, Bernadette H.; Perham, Faustine
L.
Date: 1997
Title: Creating a Learning Environment for Geometric Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 90 Number 7
Reviewer: TinaM
Date of Review: February 17, 2000
This article is about the value of the use of manipulatives, computer software, and graphing calculators in geometry. The authors use a series of lessons about the centroid of a triangle. First, they allow the students to experience with hands-on manipulatives, such as a pencil and soda straw balance to show that the centroid is the balancing point of the triangle. This gets them to ask questions and heightens their interest about the topic at hand. Then, Geometer's Sketchpad is introduced and the students are able to further test the conjectures they have made. They are able to create new figures quickly and validate their ideas quickly, resulting in possible generalizations being made. Next, the students are introduced to Mathcad, which is another mathematics software package that works similar to a spreadsheet. The students are able to see the calculations that are being made behind the scenes in the Geometer's Sketchpad. The students can change their triangle and all of their calculations will change, as well. Lastly, they look at the usefulness of a graphing calculator, specifically the TI-82. The most important message of this article is that the use of such tools helps students better visualize aspects of geometry. They also help students come up with conjectures and prove them informally. These tools also can make the lessons more interesting and enjoyable for the students.
Keywords: Teaching Strategies, Geometry, Problem Solving
Ref: TinaM2
Author(s): Mark Fidler
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 7, pgs. 565-7
Reviewer: TinaM
Date of Review: February 6, 2000
Mark Fidler, a math teacher at Buckingham Browne and Nicholas School in Cambridge, MA, tried hard for the past several years to get his students engaged in working out longer and more difficult geometry problems. He felt his students gave up too soon. The content of this article deals with his attempts to change the working habits of his students when dealing with such problems. Fidler attempted several strategies to get his students more engaged in proofs. He tried assigning harder problems for classwork and homework. He also changed his grading practices. First, he gave students full credit for correct completed proofs, but deducting points for logical errors made by the students. He then began giving most of the credit for incomplete proofs that were accompanied by all work that lead the students to a dead in as well as for providing a goal statement for what they would do if all the information needed to complete the proof was given. Lastly, Fidler attempted cooperative learning. He assigned the students several problems to work on in their groups (which were based on ability level). The students split the problems up and turned in their work as a collective. He then gave them the opportunity to work in their cooperative learning groups again. This time their was more dialogue and the students really worked together, helping each other with the proofs they were given to solve. Fidler became more impressed each time the students worked together. He truly believed that he helped his students become better problem solvers as well as better able to handle more difficult proofs. In the article, Fidler suggests reasons for grouping the students the way that he did. He also included an example of a take home test that he gave his students. The article was really interesting and you could see the excitement that Fidler felt by conquering a challenge that he had for so long. He hoped to pass his experience onto his students so that they could see that by chipping away at a problem, they would eventually reach the solution that they desired.
Keywords: Problem Solving, Connections, Teaching Strategies
Ref: TinaM3
Author(s): Kahan, Jeremy
Date: 1999
Title: Ten Lessons from the proof of Fermat's Last Theorem
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 6, pgs. 530-1
Reviewer: TinaM
Date of Review: February 12, 2000
This is a pretty short article by our own Jeremy Kahan. He uses the process of the proof of Fermat's last theorem, by Andrew Wiles, to bring out several important points about the `human process of constructing the proof.'(The Mathematics Teacher, 1999, 530). First, Jeremy points out that sometimes math is hard and it takes time. He illustrates this point by siting the fact that it took hundreds of years before Fermat's last theorem was proved. Next he discusses how mathematics can be both an individual and collaborative process. Although Wiles worked alone for several years attempting to prove this theorem, he also relied on other mathematicians' contributions to mathematics in order to complete this proof. Jeremy points out here that it is important to give students the opportunity to work individually as well as cooperatively. Jeremy then discusses the social aspect of mathematics. He explains that it is necessary to create communities in our classrooms where students can present results and have their peers them. Wiles presented the proof of Fermat's theorem to a community that, at first, rejected it. He also points out that it is important that the students check the work of the instructor because even he or she can omit something substantial when constructing a proof. Evaluating other's work and thought processes is something that is stressed in the NCTM Reasoning Standards. During problem solving, sometimes we must take a step back in order to go forward. This is what Wiles had to do before he could complete the proof of Fermat's theorem, Jeremy explains. Another point that is brought out in this article is encouraging students to relate a problem to one that they know how to solve. This is a problem solving strategy that gets students out of the habit of giving up too soon on a problem, which is a big problem when it comes to proof construction. It is then pointed out the relationship between geometry and algebra. Although many high school curricula treat the two subjects as they were separate entities, Jeremy points out that NCTM encourages teachers to make connections between the two because sometimes algebra can be used to solve geometric problems, and vice versa. Diversity within mathematics is also brought out in this article. The mathematicians who helped with the construction of the proof of Fermat's last theorem represent a variety of nationalities and were both women and men. The history of this proof, Jeremy notes, `furnishes diverse role models for students' (The Mathematics Teacher, 1999, 531). Lastly, Jeremy notes that teachers often present mathematics as a `fait accompli' (The Mathematics Teacher, 1999, 531). Yet, the proof of Fermat's last theorem shows us that very difficult mathematics can be solved with contributions of modern mathematicians. All of the points made in this article should be considered as we begin our journey into the world of mathematics teacher's.
Keywords: Problem Solving, Connections, Teaching Strategies
Ref: TinaM3
Author(s): Kahan, Jeremy
Date: 1999
Title: Ten Lessons from the proof of Fermat's Last Theorem
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 6, pgs. 530-1
Reviewer: TinaM
Date of Review: February 12, 2000
This is a pretty short article by our own Jeremy Kahan. He uses the process of the proof of Fermat's last theorem, by Andrew Wiles, to bring out several important points about the `human process of constructing the proof.'(The Mathematics Teacher, 1999, 530). First, Jeremy points out that sometimes math is hard and it takes time. He illustrates this point by siting the fact that it took hundreds of years before Fermat's last theorem was proved. Next he discusses how mathematics can be both an individual and collaborative process. Although Wiles worked alone for several years attempting to prove this theorem, he also relied on other mathematicians' contributions to mathematics in order to complete this proof. Jeremy points out here that it is important to give students the opportunity to work individually as well as cooperatively. Jeremy then discusses the social aspect of mathematics. He explains that it is necessary to create communities in our classrooms where students can present results and have their peers them. Wiles presented the proof of Fermat's theorem to a community that, at first, rejected it. He also points out that it is important that the students check the work of the instructor because even he or she can omit something substantial when constructing a proof. Evaluating other's work and thought processes is something that is stressed in the NCTM Reasoning Standards. During problem solving, sometimes we must take a step back in order to go forward. This is what Wiles had to do before he could complete the proof of Fermat's theorem, Jeremy explains. Another point that is brought out in this article is encouraging students to relate a problem to one that they know how to solve. This is a problem solving strategy that gets students out of the habit of giving up too soon on a problem, which is a big problem when it comes to proof construction. It is then pointed out the relationship between geometry and algebra. Although many high school curricula treat the two subjects as they were separate entities, Jeremy points out that NCTM encourages teachers to make connections between the two because sometimes algebra can be used to solve geometric problems, and vice versa. Diversity within mathematics is also brought out in this article. The mathematicians who helped with the construction of the proof of Fermat's last theorem represent a variety of nationalities and were both women and men. The history of this proof, Jeremy notes, `furnishes diverse role models for students' (The Mathematics Teacher, 1999, 531). Lastly, Jeremy notes that teachers often present mathematics as a `fait accompli' (The Mathematics Teacher, 1999, 531). Yet, the proof of Fermat's last theorem shows us that very difficult mathematics can be solved with contributions of modern mathematicians. All of the points made in this article should be considered as we begin our journey into the world of mathematics teacher's.
Keywords: Proof, Connections,
Ref: TinaM4
Author(s): Dodge, Walter; Goto, Kathleen; and Mallinson, Philip
Date: 1998
Title: "I Would Consider the Following to Be a Proof..." -
www.nctm.org/mt/1998/11/soundoff.html
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 91, Num. 8
Reviewer: TinaM
Date of Review: February 20, 2000
This was a really short but interesting article about proof. The authors present several proofs presented in different mathematics classes, including pre-algebra, first year algebra, geometry, second year algebra, trigonometry, and calculus. Some of the proofs are technology based including the use of the TI-83 and TI-92 calculators, and the Geometer's Sketchpad software package. Some of the proofs were solved using diagrams and others were solved using algebraic manipulations. Two points should be made from this article. First, although proof has generally been associated with geometry coursework, it is definitely useful in other math courses. The purpose of proof must be determined and whether it is the same in different classes. Second, there are many ways proofs can be presented. The instructor must decide what is acceptable as a proof as well as evaluate what the students find acceptable.
Keywords: Activities, Games, Geometry
Ref: TinaM5
Author(s): Foshay, John D.; Wells, Wendy L.
Date: 1997
Title: "Table Tennis Anyone?" Using Ping-Pong to Teach the
Coordinate Plane
Journal or Publisher: Mathematics Teacher online
www.nctm.org/mt/1997/12/90.09.tabletennis.htm
Volume, Issue, Pages: Vol. 90, Num. 9
Reviewer: TinaM
Date of Review: March 5, 2000
John Foshay and Wendy Wells used the game ping-pong to teach their students about the coordinate plane. Foshay teaches first- year algebra and Wells teaches students with moderate mental retardation and helps them with various skills such as hand-eye coordination. They recognized that the Ping-Pong table could make the coordinate plane, the net could represent the x-axis and the center line could represent the y-axis. The rules of the game, as well as basic definitions of the coordinate plane, including the concept of ordered pairs were introduced to the students. The students were split into two teams. Although only four students could actually play the game at any given time, the other students (academic players) were matched up with an active player. Each time a point was scored an academic player placed a red dot on the approximate spot where the ball last hit. The teacher would then ask a question regarding the ordered pair. At first, if the team that scored the point answered incorrectly, the teacher would correct the response. The students, however, encouraged the change in the game so that if an incorrect response was given, the other team would have the opportunity to answer the question correctly and `steal' the points from the other team. In this respect, the students had the opportunity to participate in the game as well as have input regarding rules of the game. At the end of the unit, students reflected on their experience. Many said that the method was enjoyable and that they could "see" the concept of coordinate geometry in the ping-pong table. I believe this was a very interesting lesson/activity for the students as well as the teachers. It allowed them to learn and have fun at the same time. It also appealed to the needs of the students with different learning styles.
Keywords: Connections, Teaching Strategies, Technology
Ref: TinaM6
Author(s): Purdy, David
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume
Problems
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93 No. 3
Reviewer: TinaM
Date of Review: March 22, 2000
Maximum-volume problems are common in advanced algebra, precalculus and calculus classes, however, recently they have found themselves in high school geometry texts such as Discovering Geometry: An Inductive Approach. This article highlights this problem as well as several approaches to solving it. The students start out by actually cutting out paper boxes to model the problem. They created charts to organize and display the data. They recognized patterns in their data and were able to come up with a volume function. They finally graphed the function and came up with the maximum size for their cutouts. This problem can also be looked at using The Geometer's Sketchpad. This tool can either be used by the teacher to demonstrate the construction or the students can use the tool to perform the constructions themselves. The advantage of using such a tool is that the students can vary the dimensions of the construction and see the affects of these changes on the maximum volume. They can then test these results using their graphing calculators. "One hallmark of a good problem is that it defies instant and complete solution yet yilds parts of its solution over time." (Purdy, March 2000, p. 228) Purdy points out that this problem can be extended over many years, each time tackling a different aspect of the problem. For instance, algebra geometry students can tackle the problem as described in the article; calculus students can use the derivative to solve this same problem. Such a problem, if approached by integration over time, will allow students to see strong connections among mathematics topics including algebra, geometry, and calculus. The use of technology only enhances their learning process.