Keywords: Geometry, Problem Solving, Standards
Ref: KipK1
Author(s): Taylor, Lydotta; King, Joann
Date: 1997
Title: A Popcorn Project for All Students
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90, 3, 194 - 200
Reviewer: KipK
Date of Review: 1.30.00
This paper details a two-week classroom activity with strong allegiance to the 1989 NCTM Standards. The Popcorn Project takes students on a different course of learning, breaking up a typical classroom setting by allowing students in a prealgebra class to work alongside honors precalculus students. Two related activities are presented by each group of students (two prealgebra and one precalculus student).
The first of such exercises consists of finding the brand of popcorn that yields the greatest amount of popped kernels. This part of the assignment acted as a review of basic concepts, but also as a builder of team unity. Each of the three students in a group were assigned specific duties, and wisely the precalculus student was only to compile information. Prealgebra students were in charge of either constructing the bar graphs or presenting the team's data to the entire class. This activity is organized in a way that allows equal participation while adhering to standards by `summarizing data from real-world situations.' The article does a good job of depicting the standards involved with various parts of the assignment.
Students then worked towards finding the largest possible popcorn box by using folded sheets of 8 « inch-by-11 inch paper. On the last day of the project the students presented their findings and predictions. This final presentation provided students with the opportunity to express mathematical concepts orally and visually.
Results of the activity were positive. Students worked well together with those in other classes, either for reasons of peer pressure or simply the change of pace from the typical math class setting. Students' reactions were in favor of the project. The variance in level of mathematical ability was blurred in some examples from this project. In one instance the teachers (the precalc and prealgebra teachers team-taught) noticed that many of the advanced students had forgotten the specific meanings of mean, median and mode. Fortunately the prealgebra students had just spend time reviewing these concepts, and aided their groups with calculating these figures.
The article provides strong support for the project, and rightly so. This is made evident by detailing students' testimony of what they had learned and the enjoyment of the experience itself. Following the article are the `Activity Sheets' used by the students throughout the project.
Keywords: Number Theory, Teaching Strategies
Ref: KipK2
Author(s): Leonard, Bill
Date: 1997
Title: Proof: The Power of Persuasion
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: V. 90, iss. 3, 202 - 5
Reviewer: KipK
Date of Review: 2.6.00
In The Power of Persuasion the author raises points regarding the presentation of the concept of proof to high school students. Three exercises are presented as catalysts for ways to connect with students while working proofs. An interesting suggestion is to emphasize the idea of `convincing' in discussions. Leonard uses discrepancies in trisecting angles to show that disproving conjectures by using extreme counterexamples is a convincing means of relaying information. Another example using number theory - what is the longest string of non-prime numbers - depicts how sometimes teachers can convey a proof's general properties.
Teachers must also get students to want to think about mathematical concepts. The act of convincing will play a larger part to a student who has put some thought into problems. Leonard uses the phrase `create a thirst' in the student. In addition, focusing on logical concepts allows the students to reason to themselves more so than simply using symbols on the board.
I had a good time reading what Bill Leonard had to say. Especially beneficial to the reader is how Leonard brought in some specific examples of how to provide students with confidence in understanding the concept of proof.
Keywords: Proof
Ref: KipK3
Author(s): Epp, Susanna
Date: 1998
Title: A Unified Framework for Proof and Disproof
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v. 91, no. 8
Reviewer: KipK
Date of Review: 3.5.2000
A common complaint of college-level mathematics instructors is that students out of high school arrive ill-prepared to read and write proofs without instruction or special guidance. Epp presents examples for introducing students to different concepts of proof. The theorem for the `square of any odd integer being odd' is used as a guide to first extract the typical questions that novice students have regarding proofs. The author then address each notion, providing concepts for the students that are not complex in nature (for instance, defining truth or falsity for the classroom - everyone in the class is younger than twenty years old. Why not? Because one member of the class knows she is not). This section of the article provides keen insight as to presenting concepts to students early in their career of mathematical proofs.
Later Epp presents a framework for examination of proof, by means of disproof by counterexample and proof by contradiction. The claim of `today is Thanksgiving' is easily retorted by a student simply claiming that today is not Thursday - and this is enough for students to see the the basic of explanations is enough to provide a contradiction. A various mix of means to teach proofs is included, among them incomplete proofs requiring fill-in-the-blank work.
This article is indeed a benefit for any teacher looking for ideas to help their students understand proofs.
Keywords: Technology
Ref: KipK4
Author(s): Quinn, Anne
Date: 1997
Title: Using Dynamic Geometry Software to Teach Graph Theory
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v. 90, no. 4, p. 328-332
Reviewer: KipK
Date of Review: 3.5.2000
Implementing geometry software in the classroom can provide students with a different, and usually more malleable, point of view. The author Anne Quinn details how the geometry software package Geometer's Sketchpad 3 can plainly present students with various notions of graph theory. Any software that draws, labels, and drags figures will work. Validation of student conjectures, and subsequently development of proof are just two of the ways geometry software help students. Objects such as a 4-cube, which would be difficult to conceive and draw on paper are easy to draw.
The article brings to light the ways Sketchpad can represent isomorphic, bipartite, and planar graphs. The article itself is a good review for what these concepts of graph theory mean. Included in sections for each of the three are graphics depicting the manipulation of figures to determine if they are in fact isomorphic (a one-to-one correspondence between their vertex sets that preserve edges), bipartite (the vertices of the graph are divided into two sets, each edge of the graph connecting a vertex from one set with a vertex of another), or planar (can be drawn in the plane without any edges crossing). Drawing numerous figures on Sketchpad is many times more efficient than continuously redrawing figures on a chalkboard.
I thought the article was insightful, not only in that I was able to see yet another example of how the Sketchpad enhances the classroom environment, but also as a review of certain aspects of graph theory.
Keywords: Geometry
Ref: KipK5
Author(s): Brown, Alan
Date: 1999
Title: Geometry's Giant Leap
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v. 92, no. 9
Reviewer: KipK
Date of Review: 3.19.2000
This is a worth-while article depicting the advantages of today's graphing calculators with classroom geometry. Geometry computer software is great when available, but their cost and the need for an available computer can make them hindering. With handheld calculators being most applicable to algebra and other advanced subjects, the new TI-92 is the first calculator to contain dynamic geometry software and offer portability.
Brown describes the process of introducing the calculator to ninth-grade students: a fifty minute demo presentation, where each student had their own `loaner' TI-92, allowed for many of the calculator's functions to be previewed. Later, the students had a two-week period to create a project and demonstrate in front of the class. With the borrowed calculators returned, the students worked with two of the faculty's TI-92s. Projects ranged from what was already learned in the year to topics included later in the curriculum. Students were able to apply technology to geometry, and even make some connections - one student found that a circle inscribed within a triangle inscribed within another circle had a ratio of 4:1 between the two circles. This applied to all sizes of triangles used. This is an example the technology offering visual representations that would be difficult to recreate when drawing by hand.
The author makes note that by applying these technologies to the curriculum allows teachers and students to venture far beyond traditional figure construction - there are now possibilities to create multiple situations, giving students the ability to explore in a more thorough manner. Students take a more active part in learning when using the new technologies, as they have become a part of the learning process. Geometry's giant leap is supported by the availability of the new technologies, as the TI-92 is the first in what will be more calculators offering portability and affordability to geometric technology.
Keywords: Geometry
Ref: KipK6
Author(s): Purdy, David
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume
Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v. 93, no. 3
Reviewer: KipK
Date of Review: 3.19.2000
This fine article relays the importance of technology when visualization of a problem offers insight to a problem. The maximum-volume of a box problem can be applied to the geometer's sketchpad, not only to help the students but also to apply the basis of the problem to geometry, from the typical presentation in an algebra format.
The author is a teacher who has run the problem by the class, then having them construct the problem with paper boxes. After student conjectures have been tested, patterns were graphed, and even a graphing calculator was used to help identify patterns. The graphing calculator left some students frustrated when trying to interpret the pixeled image into the paper boxes they once had.
The author then presents instructions to construct the scenario in the Geometer's Sketchpad. The instructions are straight-forward for a user of the software, and apparently the students were able to design the box on the Sketchpad as well. Manipulation with the design was possible, and led to student experimentation with conjectures. The article claims that as a stand- alone exercise the lesson may be ineffective, and then offers some extensions of the lesson. This could be an excellent way for students to utilize the software while solving classroom problems in geometry.
Keywords: Geometry
Ref: KipK7
Author(s): Ryden, Robert
Date: 1999
Title: Astronomical Math
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v. 92, no. 9
Reviewer: KipK
Date of Review: 3.19.2000
This article is one that may be of interest to the historical mathematician, which is the focus of the first five pages. It depicts how early mathematicians (Eratosthenes of Egypt, Aristarchus of Samos, Copernicus and Kepler) used methods of measuring shadows, ratios of positions of heavenly bodies, and timed intervals of movement to discern the positions of planets and their distances from the sun.
Students would not be able to collect the data for most of the experiments listed in the article, but they are of interest as to how they were done - the author does a good job of making these methods of recording understandable. What the author's students were able to do was to create a device to measure the parallax. This concept is best described as the phenomenon that occurs when, if held in front of your face, your finger appears to shift when alternately closing eyes. If objects are far away (the face and the finger) the angle from one to another becomes smaller (an eye to the center). Ryden describes a tool that the students can construct out of Styrofoam to act as a parallax-measuring device. This may be a nice activity to relate to the historical aspect of mathematics, but one that might be difficult to relate to the immediate concerns of the geometry class. Perhaps this activity would be best fit in an astronomy class.