Keywords: Geometry, Connections, Algebra
Ref: JeffD1
Author(s): Forringer, Richard
Date: 2000
Title: (A + B + C)^3
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93,1,6-8
Reviewer: JeffD
Date of Review: 1/23/00
Mr. Forringer describes an exercise in which he tells his students is "advanced-placement" blocks. Using wooden building blocks, students solve the square and cube of a trinomial.
The exercise makes a nice connection between geometry and algebra and provides great motivation for students faced with polynomial expansion.
Keywords: Geometry, Problem Solving, Assessment
Ref: JeffD2
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92,7,565-67
Reviewer: JeffD
Date of Review: 02-06-00
This article presents a nice assessment strategy for motivating students in high school geometry courses to do rigorous proofs. The author uses a group quiz format and rewards points for correct proofs as well as "leads" that are logically correct. In this way students avoid giving up so easily since guessing or the all-or-nothing approach does not result in the most points. Past experiences that have not been so successful are also shared. The author has come to the conclusion that cooperative homogenous groups arranged by ability work best for this type of assessment. Group members are more likely to participate and are more content with group arrangements. A nice sample quiz is provided at the end of the article.
I like the way the author motivates his students to enjoy the challenge of cracking difficult proofs. Judging from the sample quiz, the difficulty level for high school is nothing short of spectacular. These students are benefiting tremendously from doing this type of problem solving! I am a little concerned with the grading scheme though since "low" students are grouped together. Perhaps "improvement" could be factored in the grading criteria to compensate?
Keywords: Curriculum, History, Standards
Ref: JeffD3
Author(s): Usiskin, Zalman
Date: 1999
Title: Educating the Public about School Mathematics
Journal or Publisher: UCSMP Newsletter
Volume, Issue, Pages: Winter 1999-2000, p.4-12
Reviewer: JeffD
Date of Review: 2-13-00
This article is the written version of a talk given by UCSMP Director Zalman Usiskin at the Fifteenth Annual UCSMP Secondary Conference, November 6-7, 1999. Usiskin covers the development of math curriculum since the 1950's and makes some striking comparisons between the current NCTM Standards Era and the "New Math" Era that began shortly after Sputnik in the early 50's. Usiskin cites statistical data including test scores that accurately describe how curriculum reform under these movements faired as compared to traditional mathematics instruction. He blames falling SAT scores in the mid 70's as a direct consequence of a backlash movement away from these reform efforts. Usiskin says that the current backlash against the new NSF curriculum is the result of misinterpretation of findings from the TIMSS study and concludes that it is difficult to find any value in this "back-to-the-basics" backlash.
This is a very scholarly report. Zalman is careful to cite findings accurately and points out where evidence isn't conclusive on both sides of the argument. He presents a very balanced prospective on curriculum developments and suggests that knowing the history and politics behind these developments will help teachers better educate the public about school mathematics. I couldn't agree more. What I found most intriguing was some of the statistical data on math and computer science degrees in the U.S.-there have been some rather huge swings in these numbers over the past decade or two! Could the current shortage of math degrees and teachers be because of poor math instruction? You decide. This is a must read article. < BP>
Keywords: Geometry, Connections,
Ref: JeffD4
Author(s): Peterson, Blake E.
Date: 2000
Title: From Tessellations to Polyhedra: Big Polhedra
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Feb 2000, p.348-57
Reviewer: JeffD
Date of Review: 3-14-00
Keywords: Geometry, Technology,
Ref: JeffD5
Author(s): Scher, Daniel P
Date: 1996
Title: Theorems in Motion: Using Dynamic Geometry to Gain Fresh
Insights
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol.89, No.4, April 1996, p.330-32.
Reviewer: JeffD
Date of Review: 3/14/00
Software programs such as Geometer's Sketchpad allow students to take static figures from their textbook and move and manipulate them around. Constant-Perimeter and Constant-Area BP>rectangles are two such figures that foster fresh insights into traditional geometry theorems. For instance, by constructing a rectangle with a constant perimeter, students can make a connection to the< "pivoting chord" theorem. Similarly, a constant area rectangle relates to geometric mean. The author points out that by setting these theorems in motion, students are able to generalize and uncover relationships that the static textbook counterparts could not.
These are really nice exercises if you are new to Geometers sketchpad. The author provides step-by-step illustrations and explains clearly how each figure relates to the theorems. I constructed the figures in under a half hour as I read the article. This is really nice math content and fun! Go for it.
Keywords: Geometry, Technology,
Ref: JeffD5
Author(s): Scher, Daniel P.
Date: 1996
Title: Theorems in Motion: Using Dynamic Geometry to Gain Fresh
Insights
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol.89, No.4, April 1996, p.330-32
Reviewer: JeffD
Date of Review: 3-14-00
In 1994, the World Cup soccer championship was held indoors at the Silver-dome in Detroit. But since soccer is played on natural grass, pallets of grass needed to be crated into the stadium. Soil experts from Michigan State University chose a hexagon tessellation pattern for the grass pallets, which were first grown outdoors and then shipped in right before the tournament. The question of why this pattern was chosen by the soil scientists provides the motivation for this article.
The author provides a series of investigations to explore this question starting with an activity that determines which regular polygons can completely cover (or tessellate) a flat surface. The next activity explores angle measure relationships between the polygons that tessellate and those that do not. This leads to the discovery that polygons must fit around a single point in order to tessellate. The key is finding polygons or combinations of polygons that can be put together so that the their angular measures around a single point add up to 360 degrees. Next, students are given time to construct various tessellations with regular polygons that all have sides of 1 inch in length. Students can soon discover that there are only 8 semi-regular tessellation patterns possible with regular polygons. Finally, the author provides activities for constructing the 5 Platonic and 13 Archimedean solids.
The investigations presented in this article provide an excellent way to do constructions in geometry while making important mathematical connections. I often hear teachers say such activities are fun but waste too much time. This article shows just how valuable the math content can be in doing such investigations. A warning to the wise: once your students start, they will not want to put them down. I'm hooked and I am sure you will be too.
Keywords: Geometry, Manipulatives,
Ref: JeffD6
Author(s): Johnson, Donavan A.
Date: 1957
Title: Paper Folding For the Mathematics Class
Journal or Publisher: NCTM
Volume, Issue, Pages: 1957, p.1-32
Reviewer: JeffD
Date of Review: 03-24-00
This is a classic paper-folding booklet that seeks to familiarize students with the basic constructions in Geometry. Everything from lines, angles, and triangle properties to the most sophisticated polygon constructions are treated. There is a little bit of everything. How about tying paper knots or trying your hand at a hexaflexagon? Donovan Johnson helps you investigate geometric concepts through folding paper. This 1957 classic is once again available through Key Curriculum Press.
This is the best geometry related paper folding resource I have seen. I have several origami resources, but none as fitting for use in the mathematics classroom. Students should find these construction activities as helpful conceptually as they are fun.
Keywords: Geometry, Technology, Problem Solving
Ref: JeffD7
Author(s): Purdy, David C.
Date: 2000
Title: Using Geometer's Sketchpad to Visualize Maximum-Volume
Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol.93,No.3,March 2000,p.224-28
Reviewer: JeffD
Date of Review: 03-24-00
The author explains that reform efforts have broken traditional barriers of what mathematics is reserved for higher-level courses. Consequently, classic Calculus problems such as the maximum-volume-box problem are now accessible to algebra and geometry students with the help of graphing calculators and Geometer's Sketchpad. This results in even more ways to solve the same problem. So a one-dimensional problem becomes rich and full of numerous conncections that make it a very valuable teaching tool. The author reasons that solving this problem with Geometer's Sketchpad as a stand-alone activity would be ineffective and weak. Combining Sketchpad with some preliminary discrete experiences and other representations, on the other hand, transforms the problem into a rich source of interconnected mathematics. He points out that this is what NCTM reform efforts have aimed to accomplish all along, that is; to teach mathematics as a logically interconnected body of thought.
This was a very scholarly and thoughtfully written article. The article really illustrates how valuable it is to solve problems in many different ways, using multiple representations. The Sketchpad instructions are easy to follow and doing this activity lends itself to discovering all the idiosyncrasies of this classic problem. It's worth the effort.