Keywords: Problem Solving, Geometry
Ref: JenM1
Author(s): Manaster, Alfred B., Schlesinger, Beth M.
Date: 1999
Title: Geometry Problems Promoting Reasoning and Understanding
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 2, Pages 114-116
Reviewer: JenM
Date of Review: 1/30/00
The authors of this article take a new approach to explaining the students thought process behind problem solving and reasoning. They present four problems that would be appropriate for geometry students and there solutions. However, they provide much more than only the solutions. They describe many thoughts that the students may have when given these problems. If the teacher knows the potential problems they can guide students through solutions in a way that will be beneficial to the students understanding of geometric properties and how it relates to other branches of mathematics and the real world. It is not just important for the problems to be completed, it is most important for the students to reason their way through a problem and really get an understanding for it.
This would be a great article for teachers to read. After reading the authors insight to these selected problems one can apply the same line of reasoning to any problems they might already be using or have got from other sources and would like to use in the classroom.
Keywords: Technology, Problem Solving
Ref: JenM2
Author(s): McGehee, Jean J.
Date: 1998
Title: Interactive Technology and Classic Geometry Problems
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol.91,No.3, Pages 204-208
Reviewer: JenM
Date of Review: 2-5-00
This was a very interesting article about how using technology in the classroom can change how students learn. The main focus is on using The Geometer's Sketchpad to do a traditional constuction problem like the circle of Appolonius. By using something like this instead of traditional methods, teachers can prevent turning off students to geometry. The article begins by walking through the traditional construction and telling how students will react and what they are likely to get out of it. It then moves on to describe how it might look using interactive software and telling of the advantages of this approach. With the latter method students will begin to demonstrate a real understanding of geometry.
The article stresses the importance of ownership for the students and the importance of mathematics as a process. Mathematicians spend so much time making conjectures and proofs before coming up with a final product and it is importance for students to realize this. The article includes what a finished set of steps might look like after a student finished with this construction, as well as the drawing that would be included. It shows that the new steps will be much like the old but since the student found them they will be much more likely to understand and remember them. The article ends by providing a list of other classic problems would be great for interactive computer activities like, applications of Menelaus and Ceva's theorem, Steiner's theorem, and problems with the nine-point cirlce. This would be a great resource for any geometry teacher.
Keywords: Proof, Teaching Strategies
Ref: JenM3
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs: A cooperative approach
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 7
Reviewer: JenM
Date of Review: 2-13-00
This is a really interesting article written by a math teacher about his struggle to get students to do well at contructing proofs and to enjoy them. It begins with his frustrations about his students giving up too soon on difficult problems. They were bright students but they had an all or nothing approach that did not apply well to proofs. He first tried to assign harder problems and changed his grading criteria. A proof with a logical error received ery little credit but an incomplete proof with many attempts and much information received quite a bit of credit. This did not work though becasue his students still felt compelled to get it rights so they would fudge steps to get it right. He wanted to prevent this so he tried cooperative learning. This did not work right away either. He still rewarded on correctness, even if it was incomplete but gave large penalties to logical errors. Students did not check over each others work and many did poorly on assignmen! ts because of logical errors. He then began having the studetns work cooperatively on in-class quizzes. Things began to turn around. Once the students began to discuss what they were doing and checking one anothers work their scores began to improve and they began to enjoy mathematics and constructing proofs.
Along the way the author shares some tips that turned his classroom around. He created his groups by ability. He found that mixing stron students with weak students can stunt real discussion and he wanted everyone to feel comfortable in their groups. No one asked about how he formed the groups so this system worked well for him. He intersperses doable proofs with more challenging ones so to avoid complete frustration with the students. He always encourages students to document everything they do which includes all dead ends they might have come acrosses. In his class an incomplete proof will include much more work then a complete proof. Students are gr! aded on effort and persaverence, as well as mathematical ability. He includes four or five group quizzes proceeding the test. Students get a chance to work together and refine their skills. The article also includes a sample group take-home test. This is great to get some ideas for the geometry classroom,
Keywords: Probability, Activities, Geometry
Ref: JenM4
Author(s): Florence, Hope
Date: 2000
Title: Free Pizza? Slim Chance!
Journal or Publisher: Mathematics, Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No.5
Reviewer: JenM
Date of Review: 2-20-00
This article was based on a problem that first appeared in the journal in March 1999. The problem combines geometry and probability in a scenario at Mario's Pizza Parlor. In the pizza parlor there is a round dart board with three rings with radii of 2, 4, and 6. If the dart hits the bulls eye the customer wins a large pizza, if a dart hits the middle ring a medium, and if it hits the outer ring a small. The question is, if a dart is thrown at random, what is the probability of winning a large, medium, or small pizza respectively?
The article contains some answers that students at Nipher Middle School in Kirkwood, MO supplied. Students used their knowledge of area and probability to solve this problem. It is good to follow students thought process in problem solving. This article is also good because it provides a interesting, real-life problem where math is applied to something students would find interesting. Areas of mathematics are not isolated from each ot! her, just as mathematics is not isolated from the real world.
Keywords: Activities, Manipulatives,
Ref: JenM7
Author(s): Eggleton, Patrick
Date: 1999
Title: Experiencing Radians
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 6
Reviewer: JenM
Date of Review: 3/1/00
This is a great article about the unit circle and radians. The author provides a great activity for anyone who is learning about the circle and various measurements that can be taken from it. Patrick Eggleton expresses that as a student, he too had difficulty with the concept of expressing angle measures with radians. He shares with us a relatively simple way to teach students about radians.
The only materials that are needed for this activity are paper plates, adding machine tape, and scissors. By wrappiing the tape around the plate, measuring the radius, and marking off that distance over the length of the adding machine tape students see the length of tape (which equals circumference of circle) equals just over 3 diameters. This leads to understanding pi. Later by labeling the plate from 0 to 2pi (which equals the length of the tape) they see where the other angle measures are. Students could then create a table with radian and degree measures. After examining the table they could be led to develop the proportion used to convert degrees to radians.
This was a very informative article with a very practical, well- planned activity that could be implemented into almost any classroom. I would definately suggest that any math teacher read this.
Keywords: Geometry, Connections,
Ref: JenM8
Author(s): Lornell, Randi, Westerberg, Judy
Date: 1999
Title: Fractals in High School: Exploring a New Geometry
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 3
Reviewer: JenM
Date of Review: 3-6-00
Before I read this article I did not know that in the most simple terms, fractal geometry is the geometry of nature. I came in knowing very little and upon completion have a feel for fractal geometry. The authors did a good complete overview including these three sections: What is Fractal Geometry and How can it be studied in the classroom, Where did Fractal Geometry come from, and Why should Fractals be included in the Math Curriculum. In addition to such broad background information, they included four activities for various levels of high school geometry. The well-roundedness of this make it applicaple for anyone interested in teaching geometry.
The authors also included the actual activity sheets that one might use for the four activities. It is helpful to hear how it would be utilized and then to see exactly how a teacher would set it up. This could be very useful for a novice in this area. The descriptions really walk someone through the activity, saying things like here ask to students to do this or now discuss this property. It also encourages students to observe what is around them and to see how these things may be very different from definate Euclidean shapes to use things that they are familiar with, like cauliflower, to observe the meaning of self-similiarity by breaking off chunks and comparing it to the whole. The article stresses the need for additional ways to describe objects found in the natural world that are not Euclidean.
Keywords: Geometry, Activities,
Ref: JenM9
Author(s): Morgan, Frank, Melnick, Edward R., and Nicholson Ramona
Date: 1997
Title: The Soap Bubbles Geometry Contest
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 9, pages 746-749
Reviewer: JenM
Date of Review: 3-13-00
This sentence written by the authors, sums up the article, "The following soap-bubbles-geometry contest allows students to mesh observation and mathematical reasoning." Apparently simple things like bubbles can lead to complex geometric concepts. Bubbles are researched by many mathematicians, young and old and the activities described in this article are appropriate for grade levels 8-12. The required materials are a bucket of cold water, JOY liquid dishwashing detergent, a bottle of bubble liquid, wire cutters, and pliable wire.
I think this is a great article for high school math teachers to read. This would be a really interesting mathematical activity. It is hands on and also exposes some great geometry. The article provides some diagrams and some answers and explanations to questions that are answered in the contest. Students need no prior knowledge to participate which makes it great for all levels and abilities. The authors also provide good directions on how to structure the activity and to get the students thinking and problem-solving.