Keywords: Technology
Ref: JennieN1
Author(s): Eds: Cuoco, Albert; Goldenberg, E. Paul; Mark, June; et al
Date: October 1994
Title: Technology Tips: A Potpouri
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 87, Num 7, pp 566-569
Reviewer: JennieN
Date of Review: January 30, 2000
This is a helpful selection of tips on classroom technology compiled by various teachers and the magazine's editors.
Grace Kelemanik, from EDC (Educational Development Center) writes, "Texas Instrument's Workshop Loan Program lends out free classroom sets of calculators for two weeks." She follows with additional details and e-mail info (woskhop-loan@lobby.ti.com). The magazine's editors have developed free software (anarres.cs.berkeley.edu in directory pub/ucblogo) which has geometric capabilities as well as high-powered algebra abilities. Michelle Manes of EDC gives tips on using Geometric Golfer in the classroom. Midian Kurland, also of EDC, writes that publishers will often give teachers "sizable discounts," but that teachers have to write directly to the publishers to inquire for more information. She also advises that joining a local computer society is a great way to learn more about computers and classroom technology. Barry Kort has found an underground n! etwork (conect@musenet.bbn.com) of "computer-savvy professionals who spend volunteer time helping schools get up to speed on computers and computer networking."
These are just a few of the many tips given in this article. The various contributors gave detailed tips as well as where to write for more information, but often do not tell how the students react to the technology. Overall, though, the article is well-written and valuable to teachers.
Keywords: Geometry
Ref: JennieN2
Author(s): McGivney, Jean M.; DeFranco, Thomas C.
Date: October 1995
Title: Geometry Proof Writing: A Problem-Solving Approach a la
Polya
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 88, Num 87, pp 552-555
Reviewer: JennieN
Date of Review: February 06, 2000
The NCTM's C&E suggest that teachers should view mathematics as "a process involving problem solving, reasoning, and communication." McGivney and DeFranco write that geometry in general, and geometric proofs in particular, are facilitators in achieving this goal. Furthermore, they believe that teacher-student and student-student dialogues "create a classroom environment in which students justify, clarify, support, argue, and defend their thoughts throughout the solution process." In the article, the authors elaborate on successfully incorporating these dialogues into a classroom. The authors deal with two specific types of problems-- "problems to find" and "problems to prove." These problems have several common characteristics: they contain given information; there exist a set of allowable operations associated with the problem; and the goal of the problem is defined. Working backwards, establishing subgoals, means-ends analysis, and recognizing relationships are helpful when trying to solve these types of problems. Geometry proofs are often "problems to prove." Information is given and the assumptions are built upon using axioms and theorems all so that the end goal (the completed proof) can be reached. By employing the above strategies, students learn to attempt proofs from various angles, explore different methods, and eliminate superfluous approaches. Through teacher-student and student- student dialogues, students are challenged to effectively communicate their ideas, defend their reasoning, and clarify when needed. Teachers can subtly nudge students in the right direction through appropriate questions, or stimulate debates by questioning responses. McGivney and DeFranco devote much space to an example of teacher-student dialogue in which the teacher skillfully guides the students through a proof through careful questions and responses using the above strategies. I found this part of the article particularly enlightening as it provided a concrete example of what to do in the classroom. As a whole though, the article was somewhat confusing-I had to read it several times before I made sense of it. The authors include many (somewhat miscellaneous) quotes from the 1930's through the 1970's whose relevance to the article was at times dubious. After wading through theories on 1950's Information Processing analogous to artificial intelligence and computer simulation, the article took a turn for the better and was interesting, straight-forward, and very applicable to today's classroom.
Keywords: Technology, Geometry,
Ref: JennieN3
Author(s): Cuoco, Albert A.
Date: October 1995
Title: Visualizing the Behavior of Functions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 88; Number 7; pages 604-607
Reviewer: JennieN
Date of Review: February 13, 2000
Albert A. Cuoco estimates that about 50% of students struggle a great deal with "thought experiments." By "thought experiments," Cuoco refers to visualizing problems in one's mind, such as imagining a long skinny rectangle gradually changing into a tall skinny one. For the students who struggle, chalk is of little help and television programs turn children into "passive spectators." Thus, Cuoco recommends using geometry software packages in the classroom to aid in inquiry, conjecturing and proof. In this article, he details two "thought experiments" which he tackles using Cabri Geometry II from Texas Instruments. The first experiment is the classic "burning tent" problem: You are on a camping trip, returning from a walk. Standing at A, you notice your tent, located at B, on fire. You need to run to the river, fill your bucket with water, and get to the tent. To what point P should you run to minimize the total distance of the trip. Though this type of problem is traditionally reserved for a calculus class, it can be solved quite easily using geometry. By reflecting A over the river to from a point A' and joining A' to B, P can be found as the point where line A'B intersects the river. However, as a teacher we can use geometry software packages to broaden the scope of the problem. Students should think of the sum of the distances AP + PB as a function of P, and to realize that the sum changes continuously as P moves back and forth. The purple parabola is a graph of the function that assigns AP + PB to P. This graph will allow students to see the continuity of the graph as well as see the graph develop "dynamically right from the geometry of the situation." Cuoco develops this problem extensively while giving many helpful hints on helping students to visualize problems. I found this article extremely insightful, especially since the author addresses a common math obstacle. Many of his ways of "seeing" a problem were new to me, but the article was clear enough that I could follow all of his steps. Indeed, he gives you concise advise on leading your students through the inquiry and conjecturing phase of several geometry problems. I give this article two thumbs up.
Keywords: Connections, Geometry,
Ref: JennieN4
Author(s): McIntosh, Margaret E.
Date: October 1994
Title: Word Roots in Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 87; Number 7; pages 510-515
Reviewer: JennieN
Date of Review: February 20, 2000
This is an interesting article in which the author, Margaret E. McIntosh, relates geometry to vocabulary and word study. During the first week of school, which is traditionally hectic and somewhat fruitless due to students' changing schedules, lack of books, etc., she devotes her class time to establishing a firm geometric base via word power. Hence, the article details five days of lessons, with the goal of enabling students to move through the following four phases of development: Defining the concept Recognizing the concept Producing the concept Appreciating the concept Day 1, defining the concept, begins by having groups of three students match given note cards with shapes in the classroom. For example, a picture of a house is made up of triangles, rectangles, quadrilaterals, parallel lines, an octagonal stop sign, and various other shapes. Students are given dictionaries and geometry books as aids. Students conclude the lesson by writing journal entries beginning, "Well, after today's learning activity, I'll tell you what I notice.." On days 2 and 3, recognizing the concept, McIntosh begins by promptly returning the journals, in which she had written reinforcing and enlightening comments. Students, again in groups of three, are given another set of note cards on which Greek or Latin roots are written. Students try to generate as many words as possible containing the roots. Later in the lesson, the teacher asks students to highlight any words which pertain to geometry. The next day, students are given note cards on which geometry terms are written and asked to determine the roots, and the meaning of the words. McIntosh made a game out of the activity by dividing the students into groups and giving points for each correct activity. At the end of each day, journals were again distributed and students were allowed class time to write their entries. Days 4 and 5, producing the concept and appreciating the concept, involve similar inquiry-based activities. Students are again divided into groups and asked to do certain activities building on Greek and Latin root words. I would have to say I liked this lesson best of the four article reviews I have done. It was very practical, well-written, and not in the least confusing. McIntosh gave great ideas on how to fill that first hectic week of school that can easily be applied to any mathematics or science class. In addition to this, she effectively connected mathematics to grammar and vocabulary. Nicely done.
Keywords: Geometry, Technology,
Ref: JennieN5
Author(s): Reinstein, David; Sally, Paul; Camp, Dane
Date: January 1997
Title: Generating Fractals through Self-Replication
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Unknown (Excerpt from class); pp34-36
Reviewer: JennieN
Date of Review: March 5, 2000
This is a succint article outlining a lesson plan on fractal geometry. Students can explore fractals through iterative functions, or algorithms. The objective of the lesson is to give students an interactive experience with fractal geometry via geometric visualization and technology. In the case, the technology is a program written on a Texas Instrument calculator (which is given in the article), though other programs are available. In addition to some type of software, activity sheets (which are also provided at the end of the lesson) are needed.
Before beginning this lesson, a review of angle measure may be needed. Knowledge of series and sequences would be helpful but the lesson can be edited for students without this prerequisite. The authors encourage students to work in groups to help facilitate creativity, discovery, and connections.
I thought this seemed like a fun activity and a great learning experience for students. The questions seemed manageabl! e and the calculator component would provide variety from the status quo. I thought the authors' "objective" paragraph somewhat weak and would have spent some time refining this aspect. I think there are a lot of connections that could have been made concerning iterative functions as well as the importance of fractals in mathematics which the authors never broached. Overall, though, if you are looking for a good lesson with a different approach, this one is definitely worth trying.
Keywords: Geometry, Connections, Standards
Ref: JennieN6
Author(s): Pacyga, Robert
Date: January 1994
Title: Making Connections by Using Molecular Models in Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 87, Num 1, pp. 43-47
Reviewer: JennieN
Date of Review: March 19, 2000
The author, Pacyga, has designed twelve activities with molecular models to help visualize crystalline structures, to relate their mathematical skills to other areas of math and science, and to learn to express their ideas in orally and in words. The lessons have also been successfully adapted to algebra, basic geometry, geometry, honors geometry, and chemistry classes. This article is devoted to two of the twelve activities.
In the first activity, students build a "simple cube," which forms the most basic building block for several crystalline structures, such as sodium chloride. The learner then uses geometric principles to describe interesting properties of a simple cube. For instance, students can compare the volume of the original sphere to the volume of the interior of the constructed simple cube. In a basic geometry class, students were asked to write some of their own questions as well.
The methane molecule, composed of a central carbon atom surrounded by! four equally spaced hydrogen atoms, lends itself to a mathematically rich exploration. This molecule can be modeled using an open tetrahedral form (similar to the ones we made in class out of straws) and a glycerin solution. Students can then see the six planes of the molecule and the angles formed by the line segments from the center sphere to each vertex can be measured. Sample discussion questions are given, as well as suggestions as to how to adapt this lesson into your classroom.
The goal of these lessons is to offer students concrete hands-on experiences to help visualize abstract geometric principles. The activities are consistent with van Heile's model of learning. In addition to this, the activities address an important Standard--connecting geometry to other areas of math and science.
Keywords: Activities, Standards,
Ref: JennieN7
Author(s): Showalter, Millard E.
Date: January 1994
Title: Using Problems to Implement the NCTM's Professional
Teaching Standards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 87, Num 1, pp. 5-7
Reviewer: JennieN
Date of Review: March 19, 2000
The Professional Standards for Teaching Mathematics (NCTM 1991) states that a primary goal of teaching and learning mathematics is the development of mathematical power for all students. For teachers, this means changing the emphasis of problem solving from, "`Here's a problem, solve it'" to "`Here's a situation, let's explore it!'" Thus, the focus of this article it to provide teachers, who are often stuck in the doldrums by inadequate textbooks or constricting curriculums, with a sequence of interesting problems. These problems illustrate how multiple goals can be accomplished through a careful selection of activities.
The article details four activities that can be easily incorporated into an existing curriculum or used as opening activities. Each activity can be expanded into full lessons as well. The first two activities uses paper folding to lead into exponents. Conversions from inches to miles are also used. The third and fourth activity lead into geometric series.
To fully meet the NCTM's standards, teachers should always be on the lookout for new activities. The problems should be interesting, allow for extended exploration, employ various problem-solving strategies, and permit multiple solutions. I thought this article was well-written, easy to understand, and full of great ideas. The four activities were easy to follow and aptly illustrated the author's points.