Keywords: Geometry, Problem Solving, Teaching Strategies
Ref: MichaelR1
Author(s): Panasuk, Regina M.; Greenleaf, Yvonne
Date: 1998
Title: Using ROOTine Problems for Group Work in Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 91, Number 9, pp. 794-798
Reviewer: MichaelR
Date of Review: 2/1/00
In this article, Panasuk and Greenleaf introduce a technique for developing problems suitable for solution in a cooperative-learning small group environment. Students begin by solving example problems which have numerical solutions and then move inductively toward generalizations. After general tendencies have been identified, the teacher presents a set of follow-up problems to the group which stem from the general (or "root") problem. This establishes a problem-solving cycle of induction/deduction not unlike a crescendo/decrescendo in music or a warm-up/cool-down in a workout program.
The authors offer examples involving the angle sum of triangles and area of equivalent figures, noting the explorative usefulness of dynamin geometry software such as Sketchpad or Cabri in the former example. I was appreciative of the examples, as it made the intention of the technique much more clear than the relatively vague description given in the introductory paragraphs. However, one sticking point continued to be the claim that the approach is based on van Hiele's phases of learning. This is strongly contradicted by text in the angle-sum example: ". .. the teacher leads a class discussion in which groups share their results on each of the subproblems. .. Depending on the developmental level and the students' ability, the teacher may provide a rigorous proof of the theorem...or just state the theorem, highlighting that this theorem can be proved." We know from experience that most high school geometry courses are taught at van Hiele level 3, meaning that the ability for! *students* to independently develop proofs is groomed. How then, with the teacher doing all the hard (and therefore satsifying!) work, will students ever take ownership of the proof process?
In summary, though the idea of a "root" problem was a strong one, power should not be taken away from the students at a moment when the most crucial results of the education process are about to be achieved.
Keywords: Geometry, Planning, Problem Solving
Ref: MichaelR2
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: Volume 92, #2, pp. 114-116
Reviewer: MichaelR
Date of Review: 2/7/00
Manaster and Schlesinger guide the reader through a sequence of four problems that are intended to teach increasingly formal geometric reasoning prior to enrollment in an actual geometry course. The four problems all involve the relationship between the perimeter and area of a triangle, and use algebraic manipulation (or in one case, a very clever diagram) to prove conjectures about this relationship.
I was impressed with the authors' use of these problems to expand student experiences in geometry and proof beyond the normal boundary of the geometry classroom. The add-subtract argument used to prove that the largest rectangle of a given perimeter is a square is particularly commendable. This argument and accompanying diagram require only that a student understand the most basic tenets of inequality and the formula for area of a rectangle.
The algebraic techniques, however, are disappointing. Most of them utilize the method of completing the square for a quadratic expression, a technique that is not even taught in most traditional Algebra I curricula. Moreover, the decision to complete the square in the given example is rather contrived, and other very clever manipulations are necessary to reach an expression suitable for a ninth- grader's examination.
In summary, problems A, B, and D fulfill the author's purposes, and are worthy of inclusion and study in the Algebra I classroom. However, the algebraic proof of problem B and the entirety of problem C are at best teacher-centered items to be presented to an above-average Algebra II class.
Keywords: Inquiry, Problem Solving, Research
Ref: MichaelR3
Author(s): Miller, Catherine M.
Date: 2000
Title: Student-Researched Problem-Solving Strategies
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93 Number 2, pp. 136-138
Reviewer: MichaelR
Date of Review: 2/15/00
Inquiry learning derives its power from allowing students to develop their own questions about a topic and then seek methods for answering those questions. In this article, Miller takes the inquiry process a step further, by having students ask questions to discover what questions others have, then categorize the methods used by the interviewees in order to ask, "Which method would I use to solve this problem?"
Miller's process begins by distributing a problem set and asking students to find three individuals (family members, friends, coaches, etc.) who will attempt one of the problems. The students then collect field notes about the problem- solving strategies used, documenting associated behaviors and emotions as well. The researched data is examined as a large group and categorized into positive and negative problem-solving experiences. Finally, the students attempt the problems themselves, using their class-generated list of strategies as a toolbox.
This is an ingenious activity for two reasons: first, students discover the validity and existence of different solution strategies that are used by real people, without being force-fed these strategies from a text; and second, students work cooperatively to learn mathematics with individuals both in and out of the classroom. Not only is the inquiry process extended to an additional layer, but it is also begun in a non-threatening manner, where students are allowed to observe and learn from the development of others before engaging in the activity themselves.
Keywords: Connections, Geometry, Activities
Ref: MichaelR4
Author(s): Drost, John P.
Date: 1999
Title: The Vortex Tessellation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92 Number 4, pp. 286-290
Reviewer: MichaelR
Date of Review: 3/1/00
Tessellations have always offered a strong set of connections within geometry itself, involving concept blocks such as coordinate mapping, transformations, and 2D shape properties. Vortex tessellations, built on individual tiles that become smaller and spiral toward an interior vanishing point, provide further and more exterior connections to the areas of sequences and series, polar coordinates, and the theoretical underpinnings of fractals.
The author spends the majority of the article leading us through the construction technique for these intriguing plane tilings. Though the underlying mathematics is tremendously complex for even the most advanced high school students, Drost's explanations are as elementary as could be expected. He does a commendable job of establishing step-by-step instructions for creating one's own vortex tessellation. This is certainly of primary value for the high school-environed reader.
At the same time, specific attention is drawn to the geometric sequences involved in the radii and areas of circles around the vanishing point, as well as an alternate way of thinking about the tessellations that revolves (no pun intended) around polar coordinates and rotational mapping. This latter connection should be no surprise, given the relationships between complex numbers, polar coordinatization, and self-similar spiraling images (a.k.a. fractals). Drost calls the creations "truly an interdisciplinary project," and despite their conceptual difficulties, I would strongly agree.
Keywords: Geometry, Connections, Research
Ref: MichaelR5
Author(s): Foletta, Gina M.; Leep, David B.
Date: 2000
Title: Isoperimetric Quadrilaterals: Mathematical Reasoning with
Technology
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93 Number 2, pp. 144-147
Reviewer: MichaelR
Date of Review: 3/7/00
As stated in its opening, "This article evolved as an extension of a lesson created in 1995 as part of the Kentucky Partnership for Reform Initiatives in Science and Mathematics (PRISM)." The lesson was performed and analyzed by in- service secondary mathematics teachers, which makes it particularly reviewable in this forum. The authors "thought that [work with general quadrilaterals] might be too difficult for many students," and thus limited discussion within the article to that portion of the lesson dealing with parallelograms.
Unfortunately, the lesson is still beyond most students' understanding. It begins with the simpler proof of the fact that among isoperimetric parallelograms, the square has maximal area. This proof involves aspects of geometry, trigonometry, inequality, quadratic manipulation (specifically, completing the square) and conic sections. However, clearly the mathematical requirements for this proof demand a highly capable student who has advanced at least as far as late precalculus. The last three-fourths of the article deals with the "too difficult" quadrilateral proof, concluding with a terribly complicated n-gon generalization! I spent quite some time examining the proof, and still determined that I would need to spend significant additional time with my own pencil, paper, and Geometer's Sketchpad to grasp the big picture.
The article makes a valiant attempt to connect geometry to other areas of mathematics, particularly those of algebra, calculus, and dynamic technology, but focuses on a proof that is simply outside the realistic scope of the high school classroom.
Keywords: Activities, Communication, Research
Ref: MichaelR6
Author(s): Williams, Nancy B.; Wynne, Brian D.
Date: 2000
Title: Journal Writing in the Mathematics Classroom: A Beginner's
Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93 #2, pp. 132-135
Reviewer: MichaelR
Date of Review: 3/11/00
As math teachers, most of us have found ourselves, at one time or another, weighing the benefits of journal writing against the detriments. The question is never whether journal writing would be productive or not, but rather if it would be sufficiently productive to offset the student complaints, extra grading time, "goofball" journals, and so on. This article presents an informal case study of two Georgia teachers who took the plunge into journal assignments and lived to tell the tale.
Williams and Wynne's account is extremely useful to those teachers who are considering integrating journal writing into their classroom procedure. The two authors are very careful to point out how they viewed the situation during the planning stages, and, more importantly, how that view changed as the school year commenced. What makes their case study so appealing is that they were teaching at different schools at the time, giving them an opportunity to exchange papers and compare grading practices, collect data from two different environments, and compare procedures.
Though the recommendations and advice at the end of the article involve personal tastes and are therefore of questionable general value (for example, using a color other than red for grading, with no substantiating reason given), the article is nevertheless a valuable first look into mathematical journal writing.
Keywords: Resoning, Activities, Teaching Strategies
Ref: MichaelR7
Author(s): Gannon, Gerald E.; Martelli, Mario U.
Date: 2000
Title: The Prisoner Problem -- A Generalization
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93 #3, pp. 192-193
Reviewer: MichaelR
Date of Review: 3/12/00
"During an ancient war three prisoners were brought into a room. In the room was a large box containing three white hats and two black hats. Each man was blindfolded, and one of the hats was placed on his head. The men were lined up, one behind the other..."
The Prisoner Problem, as introduced above, is familiar to many in mathematics as an enjoyable excursion into deductive reasoning. The key to solution lies in the fact that the number of black hats is one less than the number of prisoners. The article uses the key to extend the problem to a generalization for (n) prisoners, (n) white hats, and (n-1) black hats. Special attention is given to formulation of a four-prisoner problem, using tables to organize the deductive data clearly and logically.
The authors conclude, and correctly so, that not only do "...most students enjoy thinking about and solving [the Prisoner Problem]...", but also that "...the generalization is still within the reach of most students."