Keywords: Technology
Ref: LukeB1
Author(s): Picciotto, Henri
Date: May 1996
Title: Make These Designs
Journal or Publisher: NCTM
Volume, Issue, Pages: 89, Mathematics Teacher, 4
Reviewer: LukeB
Date of Review: 1/31/00
This was an excellent article. It involved using graphing calculators. The whole idea is to show the students many different graphs and have them reproduce those graphs on their own graphing calculators. The students must enter functions on their graphing calculator in the form of y = mx + b. They could be given a bunch of parallel lines for one graph. The next graph would have them graph parallel lines with a different slope. They could also be given a graph that has all lines with different slopes that intersect at the origin. In any case the students have to figure out on their own the relationship between the graph of a function and the equation of a function. They will have to answer questions like, "How do you make lines steeper? Less steep? How do you make lines that go uphill? Downhill? How do you make lines horizontal? Vertical?" etc. This use of the graphing calculator helps students understand the parameter-graph connection. By showing the stude! nts the graph and then having them try to duplicate it, they learn to see what happens when you change the "m" or slope variable or if you change the "b" variable. Students learn these things by trial and error on their graphing calculators. This way students have more ownership to the things they learn in this manner.
Keywords: Technology, Geometry
Ref: LizA1
Author(s): Dwyer, Marlene C., Pfiefer, Richard E.
Date: 1999
Title: Exploring Hyperbolic Geometry with The Geometer' Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, Number 7, p 632-637
Reviewer: LizA
Date of Review: 2-1-2000
Exploring Hyperbolic Geometry with the Geometer's Sketchpad This article introduces additional tools that can be used with Geometer's sketchpad to explore hyperbolic geometry. The article explains that hyperbolic geometry includes all of Euclid's axioms except it replaces the parallel postulate with the hyperbolic postulate. The hyperbolic postulate states, through a given point P, not on a given line n, can be drawn more than one line that does not intersect the line n. The article continues to give a brief description that could be used as supplement to other knowledge about hyperbolic geometry.
The main part of the article is intended to be used along with the program. It gives different sets of instructions to use with geometer's sketchpad to explore the hyperbolic parallel postulate, hyperbolic triangle, hyperbolic circles, circumcenters, circumcircles, centroids, orthocenters, and incenters. The instructions are given in a way were the reader is exploring the concepts and sometimes finds properties similar to Euclidian geometry and sometimes finds properties very different from Euclidian geometry.
The article does not discuss how to use the software in the classroom but it presents it as a tool for students to explore their conjectures about Euclidian geometry. I would probably use this software along with other resources on hyperbolic geometry. If you are interested in using this extension to geometer's sketchpad, it is located at forum.Swarthmore.edu/sketchpad/gsp.gallery/poincare/poincare.html. The files need to be expanded with Stuffit Expander.
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, Technology
Ref: LeifN1
Author(s): Alan R Brown
Date: 1999
Title: "Geometry's Giant Leap"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(9), pp 816-819
Reviewer: LeifN
Date of Review: 1/28/2000
I thought this article was a good one, but it would have been better if there had been more examples given on how to use the technology instead of explaining the project the students did. The overall theme, that technology in the geometry classroom is not limited to expensive computer programs anymore, was the important information and that was conveyed well.
The article focuses on the capabilities of the TI 92 for exploring and demonstrating geometry. Students were instructed to complete a calculator geometry project that would be presented to the class. Most of the projects demonstrated geometric principles, conducted further investigations in to geometric topics or simply demonstrated the geometry learned in class. The students were able to use the different menus and capabilities of the TI 92 to complete these projects.
The article also talked about the effects that the TI 92 can have on the geometry Curriculum versus what can be done with the traditional tools of geometry: paper, pencil, straight edge and compass. The TI 92 allows students to do the same constructions but at the same time allows further discover by allowing them to investigate multiple what-if scenarios (i.e. What if I move this vertex does that statement still hold true?). Prior to the TI 92 the only technology that was available required computers and lab space, now with the TI 92 there is fairly affordable and mobile technology that expand the geometry curriculum.
This is all great if it can be implemented. The main problem I for see is making the technology available. Who is going to pay for it, the school or the parents? There is also the continuing criticism that calculators are becoming a crutch for today's students.
Keywords: Technology, Connections
Ref: TomD1
Author(s): Picciotto, Henri
Date: 1996
Title: Make The Designs
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89(5), p.424-427
Reviewer: TomD
Date of Review: 01/29/2000
This article is about using technology, in this article a basic graphing calculator, to make connections between the graphs of functions and that functions parameters. This article, particularly uses the function of the form y = mx + b and connections it has if its parameters (x and b) change. Students will be given a dozen descriptions of functions, all different, in which they must come up with an equation in the form y = mx + b.
After this, the students for each function are to change the parameters, x and b, and record how their new function differs from the original by the changes that they made. For maximum results and student understanding the students must work unguided from the teacher, allowing them to set their own path through the activity and arise questions among themselves on why a function behaved a certain way when the parameters changed. In order for maximal success, the teacher must interact with each group and assist them on pulling out some conclusions about the parameters and relationship with the graphs. For instance, if the students haven’t noticed yet that the b in the equation is the y-intercept, then the teacher must pose questions to the students that will help them see this correlation. Another important aspect of this activity is it works for all level of students. There are close to a dozen equations students are asked to come up with and examine, with a couple whose equations could be very difficult to find. This is great for all level of students because there are hard problems for the higher students, and plenty of others for the average level students.
I think the first sentence of the conclusion summed up my feelings, “Technology presents an excellent context for the reversal of standard tasks, which yields powerful educational benefits”. I like this activity because it makes use of technology that every high school age student has access to (a graphing calculator), and it would be a very popular activity among students because (a) they aren’t doing the every day thing out of the book, and (b) they are allowed to learn at their own pace and on their own path, they are allowed to do their own thinking, evaluations, and questioning of what is going on.
Keywords: Geometry, Technology, Activities
Ref: ChrisW1
Author(s): Dwyer, Marlene; Pfiefer, Richard
Date: 1999
Title: Exploring Hyperbolic Geometry with The Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), 632-637
Reviewer: ChrisW
Date of Review: 29 January 2000
Dwyer and Pfiefer present a set of seven investigations of hyperbolic geometry that one can complete on The Geometer’s Sketchpad. Using a set of special script tools available at forum.swarthmore.edu/sketchpad/gsp.gallery/poincare/poincare.html, one can turn The Geometer's Sketchpad into a tool to investigate the hyperbolic geometry of the Poincaré disk. Within this world within a circle on the Euclidean plane, the parallel postulate has been done away with and replaced with the hyperbolic postulate that "through a given point P, not on a given line n, can be drawn more than one line that does not intersect the line n." In the Poincaré disk, these lines are the arcs of circles that intersect the boundary circle (the circle that separates the hyperbolic geometry world from the Euclidean world) at right angles.
These investigations of the Poincaré disk seem easy to follow even for someone with minimal knowledge and experience with The Geometer’s Sketchpad; however, a fairly substantial understanding of Euclidean geometry is necessary for understanding most of the investigations. To the geometrically uninitiated, being introduced to circumcenters, circumcircles, centroids, orthocenters, and incenters in hyperbolic space might be intimidating. The investigations of the hyperbolic parallel postulate, hyperbolic triangles, and hyperbolic circles are much more appropriate for a high school level geometry course, and would provide students with an easily accessible look at a non-Euclidean space. These investigations provide a practical way to use technology in order to develop a deeper understanding of geometric material.
Keywords: Technology, Geometry, Activities
Ref: MiriamN1
Author(s): Dwyer, Marlene C.; Pfeifer, Richard E.
Date: 1999
Title: Exploring Hyperbolic Geometry with The Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, October, pp.632-637
Reviewer: MiriamN
Date of Review: 1/30/00
This article describes activities that can be done using The Geometer's Sketchpad to explore some principles of hyperbolic geometry. First the fundamental difference between hyperbolic and Euclidian geometry is explained. In hyperbolic geometry, all the axioms of Euclidean geometry are retained except for the parallel postulate, which is replaced by the hyperbolic postulate: "Through a given point P, not on a given line n, can be drawn more than one line that does not intersect the line n. " Brief references are made to the history of this field of geometry. One of the most popular models of hyperbolic geometry is the Poincare (accent above the "e") disk model, which is what this Sketchpad program is based on. In this model, you begin with a circle in the Euclidean plane, where all the points inside the circle are points in the model. The "lines" in the model are arcs of (Euclidean) circles that meet the boundaries of the circle at right angles, as well as straight line segments through the center of the circle. Inside the circle, most other geometric objects are defined as usual, e.g., triangles are defined as 3 hyperbolic line segments joining 3 (hyperbolically) noncollinear points, etc. After this brief explanation of some fundamental properties of the Poincare model, the remainder of the article is spent presenting several constructions and activities. It is recommended that the reader download the program from the website "forum.swarthmore.edu/sketchpad/gsp.gallery/poincare/poincare.html" in order to follow along with the described exercises. The activities involve exploring the hyperbolic parallel postulate, hyperbolic triangles and circles, exploring the circumcenter and circumcircle, and investigating properties of the centroid, orthocenter, and incenter. Several other topics are also mentioned, although details of those investigations are not provided. The authors feel that the purpose of these activities is to present a different type of geometry to students than what they are traditionally exposed to in the classroom and allowing them to compare and contrast the hyperbolic and Euclidean systems, and thus broaden and abstractualize their conception of geometry. This article aroused my curiosity, since I have never dealt much with hyperbolic geometry in any of my coursework. It is exciting to know that there is such accessible software out there to explore this area - it should be of interest to teachers and students alike. Since I have such limited experience with this subject myself, I find it difficult to critique the content of the exercises. As interesting as they seem, however, the topic strikes me as fairly abstract and advanced, and may be best suited for gifted and talented programs or even college geometry.
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 ½ 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: Technology, Geometry
Ref: StaceyS1
Author(s): Finzer, W.F.; Bennett, D.S.
Date: 1995
Title: From Drawing to Construction with the Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88(5), pp. 428-431
Reviewer: StaceyS
Date of Review: 1/30/00
The theme of this article is to point out the importance of a construction versus a drawing in geometry using the Geometer's Sketchpad. Many students will try to "eyeball" the correct lengths or angles of a figure they are trying to produce making the dimensions almost perfect, but not exact. A figure may look exactly what the teacher wants, but when the object is dragged, the elements of the sketch might vary. The article gives tips on emphasizing construction: make sure students know the difference between construction and simply drawing and emphasize dragging the figure to make sure the constraints of the sketch remain the same. By learning how to construct rather than draw, the students will gain a deeper understanding of what they are investigating; including definitions and properties.
I enjoyed this article because I even try to draw a figure instead of constructing it with the appropriate constraints. I think that the Geometer's Sketchpad will further a student's knowledge of many different shapes and figures along with their properties since it is much quicker than using a paper, pencil, ruler, and compass. This article gave me a brand new perspective on the importance of construction and how to convey it to the students.
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
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, Technology
Ref: LoriLu1
Author(s): Brown, Alan R.
Date: 1999
Title: Geometry's Giant Leap
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(9), pp. 816-819
Reviewer: LoriLu
Date of Review: 01-30-99
Brown describes his experience using a new technology, Cabris software on the TI-92, in the geometry classroom. Dynamic geometry software programs, such as The Geometer's Sketchpad, have existed for several years; however, they require expensive hardware and dedicated space and have limited transportability. The TI-92 is the first handheld graphing calculator with dynamic software. Student groups completed calculator projects in which they investigated a geometric topic. Written reports were submitted and peer demonstrations were presented. According to Brown, this project was the best one that students completed that year.
This article demonstrates how interactive technology can enhance the geometry curriculum and better meet the needs of visual and tactile learners. The TI-92's Cabris software allows for repeated and consistent student involvement while helping the student learn geometry through visualizing, problem-solving, exploring, and conjecturing. I was particularly inspired by Brown's initiative in borrowing loaners through the TI Internet site after attending summer workshops on using the TI-92.
Keywords: Technology, Calculus, Connections
Ref: LoriLa1
Author(s): Dubinsky, Ed
Date: 1995
Title: Is Calculus Obsolete?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 88, No. 2, pgs. 146-148
Reviewer: LoriLa
Date of Review: Jan. 30, 2000
In this article, the use of technology in the classroom is addressed. Technology use, in this case, is discussed through a calculus problem of analyzing a curve. The author's students want to know why they should be working out problems that can be done easily through technology. Through the illustration of this problem, the reader sees that one cannot simply plug a function into a graphing calculator for some problems without loosing some pertinent information. The author conveys the need for studying calculus, or mathematics in general, in the long-hand form. That is to say, students will benefit in seeing the "power, beauty, and subtlety" of mathematimcs by doing the somewhat grueling calculations in order to see the real methods and nature of mathematics. By the end of the article, the reader discovers that the calculus problem the author has illustrated for us, which could not be accurately described through current technology, is a problem from a chemistry tex! tbook describing the chemical reaction of a certain substance. The reaction was not described accurately through technology, and this is a problem that we would find in the real world.
The author tries to make the point that technology may not always give the best, most accurate answer. He also tries to get the point across that students need to appreciate mathematics, which is enhanced by the careful study and calculations of problems. I tend to agree with the author that technology, while being helpful classroom aide, should not be looked at as the best approach to mathematics. Students need to understand the underworking of mathematics before they accept what technology computes for them. As the author illustrates, we cannot always rely on technology because it is not always accurate!
Keywords: Problem Solving, Technology
Ref: AndreaA1
Author(s): Jones, Graham A,; Thornton, Carol A.; McGehe, Carol A.; Colba, David
Date: 1995
Title: Rich Problems - Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7, pp. 520 - 525.
Reviewer: AndreaA
Date of Review: January 30, 2000
Keywords: Problem Solving, Technology
Ref: AndreaA2
Author(s): Jones, Graham A,; Thornton, Carol A.; McGehe, Carol A.; Colba, David
Date: 1995
Title: Rich Problems - Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7, pp. 520 - 525.
Reviewer: AndreaA
Date of Review: January 30, 2000
This article describes a problem that a middle school math teacher got from an architecture friend of his. This friend had a job to design a hotel with a brass railing around an atrium. They wanted the atrium view to be as large as possible so that the most people could enjoy it as possible. The only restriction was that they could only use 650 feet of brass because the price of brass was high and they had a set budget. The students’ jobs were to find how they could best solve this problem.
Some great extensions were provided for those who had an easy time with the problem. Other possibilities for the project were shown that the students could work with. Also the article gives detailed instructions for how to work with this problem on a graphing calculator. This problem is an excellent way for students to work with a specific problem and then transfer what they learn about maximum area to the general case.
I thought this article showed a variety of activities to do with a class. There were things that could be done with different levels of students and it was a problem that came from real life. It also had detailed instructions for a calculator activity, which might be helpful for those who aren’t that familiar with a grapher.
Keywords: Communication
Ref: TracyA1
Author(s): Doerr, Helen; Hecht, Caroline
Date: 1995
Title: Navigating the Web
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 88, Issue 8, pages 716-719
Reviewer: TracyA
Date of Review: January 31, 2000
The internet has become a major resource in the field of mathematics. Students can use it as a tutor; teachers can use it as a resource and as a learning opportunity. The World Wide Web can be a wonderful opportunity for collaboration, communication and exploration; if you know how to use it and access the information you are looking for. Unfortunately, not everyone is internet experienced and knowledgable. Personally, I would rather have another root canal, than have to do any work on the internet. For those of you who feel the same way, this article could be a big help in getting started.
This article explains how the internet works, what a "internet address" means, and how to generally get around. It offers a variety of web site to try out and addresses for discussion groups. There were two sites mentioned that got my attention. First, The Geometry Forum. It is "built around a set of seven newsgroups for people interested in teaching and learning geometry at all levels from high school through college and graduate school" (page 717). The other site was AskERIC. This is a question-and-answer service about any aspect of teaching, learning and information about technology. This could be a great resource for a new teacher.
Personally, I found the article interesting becuase I have basically no internet experience. It offers good starting points and had almost step-by-step instructions on how to use the different search resources available on the web. For me the article was helpful, but if you have internet experience this would probably be a waste of time. I wasn't kidding about another root canal versus using the internet!
Keywords: Geometry, Teaching Strategies, Technology
Ref: EoinO1
Author(s): Litchfield, Dan; Goldenheim, Dave; Dietrich, Charles H.
Date: 1997
Title: Euclid, Fibbonaci, Sketchpad
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 1, 8-12
Reviewer: EoinO
Date of Review: 1/20/00
This is an article by and about two high school geometry students who used Sketchpad to create a new construction of partitioning lines into equal segments.
During a summer session class, (in which it is feasible to set aside large blocks of time for an exploratory exercise such as this) the teacher had challenged these two students to find such a construction. After they asked, he allowed them to use Sketchpad to help them with their constructions. Sketchpad helped them in this endeavor in several ways. First of all, it allowed them to construct exactly, not having to wonder "did I place that line exactly between those two dots. " Secondly, It allowed them to try many ideas from one basis without having to redo work. Thirdly (but I am sure not lastly) the measurement feature of sketchpad allowed them to say whether or not the pieces were even fractions of the whole easily, without guess work.
After working with the problem for a few hours the sudents not only found a construction, but it was apparently a novel and elegant construction. Which helps demonstrate strength and flexibility of Sketchpad. The students went on to prove that their constrcution produced what it was heralded to produce and find yet another constuction from one of their discarded ideas related to the fibonacci sequence.
In addition to the intertesting mathematics and the demonstration of the strength of Sketchpad, this article shows that students can and will exceed expectations at times. It is important that teachers recognise this fact, and to ecourage successes such as these by challenging their students by setting high standards and giving encouragement (though not always in the manner of the teacher in the article, he said to the students at the beginning of this endeavor that "you don't have a prayer of figuring this out" and while this motivation worked with these students, it will not work for all).
Keywords: Communication, Technology
Ref: AndreaB1
Author(s): Dick,Thomas; Kubiak, Evelyn
Date: 1997
Title: Issues and Aids for Teaching Mathematics to the Blind
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 5, P. 344-349
Reviewer: AndreaB
Date of Review: 01/31/00
The purpose of this article is to help mathematics teachers learn how to communicate with visually impaired students. An example used is, "if the expression is meant to be (x+3)^2, then the teacher should say, 'the quantity x plus 3 [pause] squared' rather than 'x plus 3 squared,' which sounds like x+3^2." The article explains some common learning challenges for visually impared students. Because the much of the language of mathematics relies on visual cues, it will take a blind student much longer to grasp a concept. The article gives suggestions for teachers. One suggestion that surprised me was to have a conversation with the student or parents to find out the visual limitations of the student. Another suggestion is to reduce the ammount of homework. The homework should be carefully selected to insure the student will keep up with the class. This is done because the time it requires for a visually impaired student to complete an assignment can be 1.5 to 2 times longer than their classmates. Helpful resources are listed, such as textbooks in braille, taped versions of textbooks, calculators with large displays or talking calculators and graphing and drawing aids.
I thought that the article had a lot of useful information. Learning the sight experience of a student can give the teacher insight in to how to explain a concept. The talking calculator is a great idea. I hope that, the problem of a useful graphing calculator for the blind has been resolved. There is a list of suppliers that sell the graphing and drawing kits. If you have a visually impaired student in your class I would recommend reading this article.
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, Activities
Ref: RyanV1
Author(s): Peterson, Blake E.
Date: 1997
Title: A New Angle on Stars
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 90, Number 8, pp. 634-638
Reviewer: RyanV
Date of Review: 2/1/00
This is an article with a great application of Geometer's Sketchpad in the classroom. It analyzes the interior angle measures of a 5, 6,…, n pointed star (interior meaning the angle measures at the vertices of the given star). It also gives a very detailed description of how you could form a lesson around this material. Along with the lesson plan ideas and application of Geometer's Sketchpad, this article also gives a formal proof for the following theorem: The sum of the measures of the point angles (interior angles at the vertices) of a five-pointed star is 180 degrees.
This article also includes some generalizations and extensions that arose from teaching the lesson. As students first examined five-pointed stars, they were able to make a conjecture and (most of them) were able to complete the proof of the above theorem. Then they were led into a discussion about 6, 7,…, n pointed stars. Each group examined a different number of points on a star and then the class collaborated their data to come up with the formula that the sum of the point angles of an n-pointed star is (n-4)180 degrees. From there some other useful extensions were given that could be examined by the whole class (although might take some time) or for the more advanced students . These extensions involved looking at stars that were constructed by connecting every 3, 4,…,n points (the first extension involved only every other or every second point). In conclusion, I found this article very interesting and I hope to use it in my classroom someday.
Keywords: Teaching Strategies
Ref: MiriamN2
Author(s): Samide, A.J., Warfield, A.M.
Date: 1996
Title: A Mean Solution to an Old Circle Standard
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89, May, 411-413
Reviewer: MiriamN
Date of Review: 2/5/00
This article is a case study of the phenomenon of students proposing unique solutions to standard problems, and the interesting exercises in conjecturing, inquiry, and proof that can arise from their solutions. The author begins by discussing the general phenomenon and giving several reasons why such solutions deserve some investigatory class time, including the fact that they tend to interest the entire class and promote mathematical connections and reasoning. The author then launches into a specific case that occurred during a geometry class. Students were supposed to solve for the length of a line segment in a figure with two externally tangent circles. The traditional solution that appears in most textbooks involves the construction of an auxiliary line segment and use of the Pythagorean theorem. However, one student came up with a completely different relationship between the length of the segment in question and the radii of the two circles. Students immediately became interested in the question of whether this result was purely coincidental or whether it would hold true in every case. The instructor had groups of students choose different values for measures of objects the figure and explore whether the student’s result worked in each case. They were excited to discover that it always did! Additional searches for counterexamples were unsuccessful. Then students from an enriched geometry class joined this class in the investigation. Ultimately the students came up with two proofs of the result, one algebraic and one geometric. From here, students began to make further conjectures, such as what would happen if the circles were not externally tangent. These questions were explored by applying algebraic and geometric methods similar to those they had used to do the initial proof, and they ended up discovering interesting new relationships and generalizations. In all, several days were devoted to this unanticipated project. To me, this case study is a great example of how unanticipated conjectures that arise during class can be extended into exciting and meaningful investigations in which skills of creativity, mathematical connections, reasoning, and proof can be developed. It is also possible that original discoveries may be made, and this is a great way to illustrate that real mathematics is an ongoing academic pursuit rather than a “dead” subject for which everything is already documented in textbooks. The only reservation I might have about applying a similar model in my classroom is simply the length of time that was spent on this spur-of-the-moment conjecture. I think it would be difficult to be that flexible with the limited time that teachers have to fit in all the required topics during a school year, but it would be great if it can be done.
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: Geometry
Ref: TracyA2
Author(s): Scher, Daniel
Date: 1996
Title: Folded Paper, Dynamic Geometry, and Proof: A Three-Tier Approach to the Conics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volumn 89, Issue #3, pages 188-193
Reviewer: TracyA
Date of Review: February 5, 2000
There are many things a teacher needs to consider when planning a lesson. How much time is allotted for this lesson? How much knowledge do the student already have? How can this lesson meet the needs for everyone in the class, at the same time? Everyone has different learning styles; some students need hands-on experince, some comprehend through listening or reading, yet others need to see the material before they reach a level of understanding. This article explains how to teach ellipses, hyperbolas, and parabolas by using patty paper and Geometer's Sketchpad to generate proofs. The author stressed the importance of using patty paper and having hands-on experience. He said, "It is tempting to abandon the folding process and move directly to the computer. Yet Sketchpad, or any other geometric computer program cannot replace the experience of hands-on construction. .. the folding process also possesses a simplicity that stays with us after we have forgotten the speci! fics of the computer modeling" (page 192). The computer is great for moving figures around and for probing more in-depth investigations, but personal experience is still vital in the learning process.
I found this article to be very worth while and informative. I really liked ths way the author described how to use patty paper to make discoveries and conjectures about ellipses, hyperbolas and parabolas. Then he described how to verify those conjectures on Geometer's Sketchpad, followed by a BRIEF discussion about how to use this information to generate a proof. I enjoyed reading this article.
Keywords: Geometry, Curriculum
Ref: TomD2
Author(s): Gregg, Jeff
Date: 1996
Title: The Perils of Conditional Statements and the Notion of Logical Equivalences
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90 (7), 48 - 54
Reviewer: TomD
Date of Review: 02/06/2000
In this article, Jeff Gregg, talks about how the traditional two column proofs in no way promotes any mathematical thought processes. He believes that before attempting a proof students need to make conjectures, and justify their arguments through constructing mathematical arguments (in particular conditional statements). For example, the author mentions the use of sketchpad and doing activities very similar to what we did. He suggests that the students should be given a problem and attempt drawing it out. From this they need to make some conjectures about their figure and then use conditional (If …, then …) statements that they are trying to validify or will show are untrue. The students will then use the sketchpad to support or deny their conditions. At this point, the students haven’t even started a proof yet. However, by conjecturing, making conditional statements, and supporting or denying those statements these students are going to have a better understandi! ng for this problem now than if they were just doing a two column proof. The author’s strong feelings on conjecturing and using conditional statements are summed by this following quote, “Conditional statements and the notion of logical equivalences should not be introduced as individual and isolated topics with no apparent connection to proving but rather should be taught to arise the students efforts to construct valid arguments that will justify their conjectures and communicate their reasoning to others”.
Reading this article, I could relate to the author on how he feels that the traditional geometry courses and two column proofs never call for “real” thought to go on by the students. For my practicum this fall, I observed two geometry classes that still taught from the traditional textbooks and were doing two column proofs. Like the author of this author states, the students learned virtually nothing. They were simply regurgitating information, and really had no reasoning or conjecturing about what they were doing. As I would help the students, I would try to pose questions that would make them think about what the problem asked and what would help them. And just about every student said, “We don’t need to know that much information, all we need to know is a definition or theorem as a reason for that statement. ” It’s crazy to think that this is what the students are taking away from their geometry class.
Keywords: Problem Solving, Geometry, Technology
Ref: RyanV2
Author(s): Watanabe, Tad; Hanson, Robert; Nowosielski, Frank D.
Date: 1996
Title: Morgan's Theorem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 5, pp. 420-423
Reviewer: RyanV
Date of Review: 2/6/00
This is a short article dealing with a Theorem about triangles, student discovery, and the application of technology in the classroom (Geometer's Sketchpad). The article begins by discussing how a 9th grade student presented his findings on a Theorem (Walter's Theorem) to a special mathematics colloquium. The student, Ryan Morgan, took the already known Walter's Theorem and extended it to dimensions not thought of before by other mathematicians. Walter's Theorem states: If the trisection points of the sides of any triangle are connected to the opposite vertices, the resulting hexagon has area one-tenth the area of the original triangle. Ryan extended this theorem beyond simply trisecting a side, into n-secting a side and he found some great patterns.
The article proceeds to discuss Ryan's findings and also how he came about them. First, it was plainly noted that Ryan's findings would not have come about if it weren't for Geometer's Sketchpad (and if they had they would have been much more difficult to obtain). The article then gives a detailed proof of Morgan's Theorem (which Ryan had not quite been able to do at that time) and also some great extensions of it for classroom use with Geometer's Sketchpad. Finally, the article talked about how what had occurred was very much in line with the NCTM's standards using student discovery to heighten motivation and interest in math.
In conclusion, this article was very interesting to see an application of Geometer's Sketchpad as well as how it affected one student's mathematical development. It also gives great ideas to use in the classroom.
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: Technology
Ref: AndreaB2
Author(s): Watanabe, Tad; Hanson, Robert; Nowosielski, Frank D.
Date: 1996
Title: Morgan’s Theorem
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 5, p.420 - 423
Reviewer: AndreaB
Date of Review: 02/06/00
This Article is a wonderful example of what can happen while using technology. It began with an assignment to rediscover Walter’s theorem. Walter’s Theorem states: If the trisection points of the sides of any triangle are connected to the opposite vertices, the resulting hexagon has an area one-tenth the area of the original triangle. The class was using the software GeoExplorer. One ambitious ninth-grade student wanted to know what happened when the sides of the triangle were broken up into more than three pieces, perhaps n. We will call this “n-secting”. Using The Geometers Sketchpad, this student discovered that when n was odd, a pattern emerged. Using regression on a calculator, he discovered that the ratio of the inner hexagon to the original triangle was (9n^2 – 1) / 8 to 1. The teacher was unfamiliar with this conjecture and took it to the mathematics department of Towson State University. The teacher and student were invited to present their findings at a colloquium. There are a couple reasons that they were invited. First, the students work involved significant mathematics. Second, they were unable to find an equivalent theorem in the existing resources. Third, technology made this discovery possible. Without the use of Geometers Sketchpad, this exploration would have been difficult, if not impossible. The icing on the cake was that when scholars offered the student proofs to this theorem, the student declined them because he wanted to prove it without any help.
I absolutely loved this article. It shows that with the use of technology there are going to be many more discoveries in the field of Mathematics. Opportunities are out there for anyone, even a ninth-grader in high school, to discover something new.
Keywords: Geometry
Ref: LoriLu2
Author(s): Izen, Stanley P.
Date: 1998
Title: Proof in Modern Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91(8), pp. 718-720
Reviewer: LoriLu
Date of Review: 02/06/00
In this article, Izen explores the role of deductive proof in modern geometry given the recent availability of such excellent geometry software as The Geometer's Sketchpad. For the past two years, he has been writing his own geometry curriculum. His goal is to give students both an inductive and a deductive look at some geometry theorems. Izen first writes computer exercises that encourage students to discover geometric relationships, to deter- mine the likelihood of the truth of their conjectures, and to write their conclusions as theorems. Thus, inductive reasoning is used to demonstrate the likelihood that a theorem is true. Then, later, either the same theorem is presented and proved deductively or the student is asked to prove it. Izen provides a very good example of a computer exercise, theorem, and proof actually used in his classroom. Izen makes a convincing case that, whether or not one studies geometry with a computer, deductive proof is still crucial. Geometry learned by inductive development alone is surface learn- ing, whereas deductive proof by itself can be complicated and inaccessible. Izen has done a good job illustrating how both inductive and deductive reasoning can be used together as complementary processes to give students the fullest under- standing and deepest insight. I especially enjoyed a line from the article as Izen is explaining the importance of a student learning why a theorem is true--"A student who solves problems but does not understand why the methods work is a technician, not a mathematician."
Keywords: Geometry, Activities, Manipulatives
Ref: ChrisW2
Author(s): Peterson, Blake E.
Date: 2000
Title: From Tessellations to Polyhedra:Big Polyhedra
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 6, pp348-357
Reviewer: ChrisW
Date of Review: 6 February 2000
Peterson presents an investigation complete with sample worksheets, stencils, possible "hip pocket" questions, and extensions for a teacher to help their students learn about tessellations and the important role that angle measure plays in tiling planes and constructing polyhedra. Conjecture plays an important role in this investigation. Students start by testing various polygons in order to see if they tessellate by themselves, and commenting why they think the polygons do not tessellate if they do not tessellate by themselves. After this, students work to learn about the various polyhedra they have been working with by calculating the measure of the interior angles for each polyhedra. While this skill might prove difficult for some students, Peterson has provided questions that a teacher might use to help elicit a method for determining these angle measures. Once students have calculated these angle measures, Peterson suggests that students revisit their initial tessellation discoveries to see if students can find a pattern between a polygon's angle measure and its ability to tessellate itself. Once students have discovered or been shown that the sum of the angles around any given point needs to be 360 degrees, teachers can have students come up with different combinations of regular polyhedra that fit together around one point and consider the different ways that they can be arranged.
Ref: StaceyS2
Author(s): Galbraith, Peter
Date: 1995
Title: Mathematics as Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88(5), pp. 412-417
Reviewer: StaceyS
Date of Review: 2/6/00
This article explains why standard 3 of the Curriculum and Evaluation Standards, mathematics as reasoning, is important. First of all, examples of students' reasoning is shown through three different problems. The research was conducted on thirteen- to fifteen-year old students enrolled in years 8 to 10 of British or Australian secondary schools. The main misconceptions and misunderstandings that students have involving proof and logic include: the implications of a counterexample, the need for only one case that disproves a claim instead of many examples, and students little appreciation for the usefulness of proofs. The author discussed how many students adopt a wholly empirical approach to proofs by testing different cases. They convince themselves that a claim must be true if it works for x number of different trials. This is a great way to get students geared toward thinking analyitically, but does not constitute a complete process of proving a claim. To get s! tudents to think more deductively and generalize more will lead to a higher proficiency with proofs. Finally, the article discusses ways for teachers to apply this research in their teaching methods. The teacher and student need to understand the vocabulary used and the meanings attached to words and techniques. A common appreciation of what is being achieved needs to be shared by all. The teacher cannot just "tell" his or her students that a claim is true, they need to actively figure it out for themselves using a more constructivist approach. Otherwise, the meaning behind "proof" is thrown out the window. If the students believe what the teacher says as true without seeing it for themselves, then they cannot fully appreciate the meaning of truth and proof.
I liked this article in the sense that I need all the guidance I can get in preparing to teach students about proof. However, I found the article hard to read and follow. I am not sure that I picked out the important concepts because they seemed deeply embedded. Many of the articles on effective ways to teach reasoning and proof are backed up by research, but they fail to include ways to excite students about developing good proofs. They find no purpose in it and therefore fail to put forth effort. If you cannot convince the student of the importance of proofs pertaining to the "real world", they fail to take interest. This article does not succeed in explaining how to captivate students interest level, it focuses on the proper process of reasoning.
Keywords: Technology
Ref: LoriLa2
Author(s): Hirschhorn, Daniel B.; Thompson, Denise R.
Date: 1996
Title: Technology and Reasoning in Algebra and Geometry
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 2, pgs. 138-142
Reviewer: LoriLa
Date of Review: February 7, 200
Many mathematicians fear that students do not leave high school with an adequate understanding of the nature of proofs and their importance. The authors of the article "Technology and Reasoning in Algebra and Geometry," invite teachers to incorporate technology into their classrooms in order for students to discover the importance of reasoning and proof to the world of mathematics. With the help of technology students will be able to move to higher-level thinking and use sophisticated reasoning. They will be able to test and explore conjectures more easily. They will learn to hypothesize, or make a conjecture, and discover whether or not a counterexample exists or whether their conjecture seems suitable, urging them to pursue a formal proof of their conjecture. The authors illustrate in this article how the use of technology will speed up the testing of conjectures in algebra and geometry by processing information or tedious calculations. The authors claim that with the use of technology, students will be more engaged with the problem and more willing to discuss a proof. Technology will also enable them to easily visualize the problem. With a more hands-on aspect to mathematics, many more students may be able to understand and use the logic and techniques of reasoning and proof. I feel that the authors of this article are making a valid assertion that technology will provide an enriching avenue to mathematics. If technology is used correctly in the classroom, I believe it can prove to be invaluable to the learning and visualization process. If students are able to test conjectures with the help of technology, they are given ownership of their learning, and will be likely to use the skills learned by proving or disproving conjectures both inside and outside the mathematics classroom.
Keywords: History, Problem Solving
Ref: LeifN2
Author(s): Wilkens, Jesse
Date: 2000
Title: Why Is the Year 2000 a Leap Year?
Journal or Publisher: Mathenatics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 6, Febuary 2000, pg 360-362
Reviewer: LeifN
Date of Review: 2/6/2000
I found the article very interesting and well written. The article is about Leap Years; what makes a certain year a leap year, why we have leap years, how did they figure out what years are leap years and why those specific years are leap years. The article gives a lot of the history behind Leap Years, their purpose, and the math that is used in calculating them. It also gives the reader good references on where to find out more about Leap Years so that extensions of the problem can easily be done. In my opinion this topic is not appropriate for most middle school classes, it could possibly be used in an 8th grade classroom, but I think that middle school students would miss a lot of what is involved in this problem. It is better suited for older students that can better understand and appreciate the complexity of what is involved in this problem; the societal and cultural influences on this problem, and the mathematics involved. The teacher also has to be very careful when dealing with religious influences, because of the separation of church and state. Given the proper amount of time this would be a great problem to use in the classroom, as a group or individual project. Leap Years are interesting phenomenon and many students are curious about them, therefore interest and motivation in doing the problem would be high. It requires the student to use a variety of problem solving skills, and forces them to use inquiry and conjecturing about why certain decisions were made. This problem also gives the students that opportunity to see the role that math has had throughout the history of the world. The problem is an easy one to link to science class, because of the astronomy involved. With the proper planning it could be used as an inter-disciplinary project, with science, as well.
Keywords: Activities, Problem Solving, Teaching Strategies
Ref: LukeB2
Author(s): Gonzales, Nancy A.; Fernandez, Albert; Knecht, Corine
Date: May 1996
Title: Active Participation in the Classroom Through Creative Problem Generation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89 Number 5, 3 pages
Reviewer: LukeB
Date of Review: 2/6/00
The authors of this article came up with a good alternative idea to get students actively involved in the creation of mathematics problems. The class is divided into several groups. The first group decides on a statement or phrase and writes it on the board. The next group then adds a phrase, and passes this task on until the last group finishes the question for the whole class to solve. Such questions could be, "How large is the bed of a pickup truck? How many pints of blood are in the human body?" This activity gets the students to communicate mathematical ideas and to get the students to be involved in generating real world problems and then solving these problems. The authors want the students to evaluate four major things. They are; "Understanding the Problem, Devising a Plan, Carrying Out the Plan, and Looking Back. " I think that these authors have a good alternative idea here and there are several versions of the "pass it along" activity.
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: Problem Solving, Teaching Strategies
Ref: LizA2
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs:A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, Number 7, pp 565-567
Reviewer: LizA
Date of Review: 2-6-2000
In this article a teacher shares his ideas on how to encourage students to keep working on a proof when they cannot solve the problem with a single stroke of inspiration. Over a few years this teacher developed several strategies. First he has the class regularly work in groups on quizzes that include two difficult proofs. The rules of how the quiz will be graded state that less than half the credit will be given for logically incorrect proofs or if little is written. But, students can earn credit if they record clearly all paths of logic they explored and these dead-end paths listed should be much longer than the actual proof would be. The teacher noted that on the first quiz students used incorrect logic but after this they quickly learned to check each other's work and to monitor all their own steps carefully. This teacher had great success using these strategies. Students found working on proofs to be exciting and fun, they worked cooperatively in groups and learned to brainstorm ideas for proofs, and finally they learned, "that sometime they can solve a problem by just chapping away in the hopes that something will happen" (566).
One interesting piece in the article is that the author suggests using ability groups within the class. He puts students who usually get A's in groups of two, students who usually get B's in groups of three, and other students in groups of four. He claims the students do not question why they are grouped this way. Although I do not necessarily agree with this method, I appreciate this teacher's honesty in presenting a less trendy method of accommodating different ability levels.
I believe this article could be useful for teachers, both when teaching proofs or any challenging problem solving. He also ends the article with examples of challenging proofs a teacher might wish to use.
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: Technology, Activities, Geometry
Ref: AndreaA7
Author(s): Scher, Daniel P.
Date: 1996
Title: Theorems in Motion: Using Dynamic Geometry to Gain Fresh Insights
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 4, pp. 330-332
Reviewer: AndreaA
Date of Review: 2/6/00
This article distinguishes between drawing a rhombus with Sketchpad and constructing one. When a rhombus is drawn in sketchpad and a vertex is dragged, it becomes an arbitrary quadrilateral. In contrast a constructed rhombus maintains four equal sides when dragged. The article then gives an exercise to do in sketchpad to do and questions to think about. It also gives suggestions to use in figuring out how to construct a rectangle of given perimeter with the greatest area. The next example it shows is in constructing a rectangle whose perimeter can change but the area remains fixed. Another thing the article shows is something I haven’t seen before. It shows the construction of a rectangle using one circle inside of another. The closer the circles are to one another, the more square the rectangle. The farther apart they are, the more oblong the rectangle. The vertices of the rectangles are on the two circles are in the center. This looks like something that could be used in the classroom to show different methods of construction and playing around with sketchpad.
Keywords: Teaching Strategies, Geometry, Problem Solving
Ref: TinaM2
Author(s): Mark Fidler
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 7, pgs. 565-7
Reviewer: TinaM
Date of Review: February 6, 2000
Mark Fidler, a math teacher at Buckingham Browne and Nicholas School in Cambridge, MA, tried hard for the past several years to get his students engaged in working out longer and more difficult geometry problems. He felt his students gave up too soon. The content of this article deals with his attempts to change the working habits of his students when dealing with such problems. Fidler attempted several strategies to get his students more engaged in proofs. He tried assigning harder problems for classwork and homework. He also changed his grading practices. First, he gave students full credit for correct completed proofs, but deducting points for logical errors made by the students. He then began giving most of the credit for incomplete proofs that were accompanied by all work that lead the students to a dead in as well as for providing a goal statement for what they would do if all the information needed to complete the proof was given. Lastly, Fidler attempted cooperative learning. He assigned the students several problems to work on in their groups (which were based on ability level). The students split the problems up and turned in their work as a collective. He then gave them the opportunity to work in their cooperative learning groups again. This time their was more dialogue and the students really worked together, helping each other with the proofs they were given to solve. Fidler became more impressed each time the students worked together. He truly believed that he helped his students become better problem solvers as well as better able to handle more difficult proofs. In the article, Fidler suggests reasons for grouping the students the way that he did. He also included an example of a take home test that he gave his students. The article was really interesting and you could see the excitement that Fidler felt by conquering a challenge that he had for so long. He hoped to pass his experience onto his students so that they could see that by chipping away at a problem, they would eventually reach the solution that they desired.
Keywords: Geometry, Activities, Problem Solving
Ref: EoinO2
Author(s): Casey, James
Date: 1998
Title: Perfect and Not so Perfect Rollers
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Vol 91, No 1, 12-20
Reviewer: EoinO
Date of Review: 2/6/00
This article provides an activity for the geometry classroom that provides a real-world hands-on activity that is very broad in scope but is reasonable and adjustable in its expectations.
The problem is how rate rollers. Obviously a perfectly round roller is a perfect choice, but what about when your roller isn't round (the example in the text is rolling your book on pencils with hexagonal cross-sections). The problem has several stages. One is to look at a variety of rollers. Try to determine what characteristics are the most desired in these rollers. Then, if possible, quantify these characteristics to give you a way of rating various rollers. Along the way a wide variety of skills are going to be needed, from problem solving skills to figure out the best way to take measurements to analytical skills to do the final analysis.
The article walks you through this experiment. providing a wide variety of example, motivations, and even two quantifiable characteristics (though these are not the only way to caracterize rollers). Overall, the article is very good, though I think that it might be hard to find a class in which this project would engage the whole class (I feel that many students would not be particularly motivated by this problem). I see it more as a project appropriate to give a small group (three or four) when all of the class is involved in different projects, and then they could present their findings to the class.
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: Conjecturing, Problem Solving, Games
Ref: RyanV3
Author(s): Quinn, Anne Larson; Koca Jr., Robert M.; Weening, Frederick
Date: 1999
Title: Developing Mathematical Reasoning Using Attribute Games
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, Num. 9, 768-775
Reviewer: RyanV
Date of Review: 2/9/00
This is a great article for any math educator teaching the principles of proof, patterns, combinatorics, logic, problem solving, or just plain old fun. The article begins with the authors discussing their mathematics club meetings and a game called “Set” that had become very popular at these meetings. From there it describes how the game is played and how it is set up in a rather lengthy discussion. Basically, there are 81 different cards each with a different symbol (diamond, oval, or squiggle), different shading (open, solid, or striped), different number of shapes (1,2, or 3), and different color (red, green, or purple). Then to play the game you place 12 random cards in front of the players and each player looks for a set of 3 cards (a set must have the characteristics on the cards either all the same or all different). The set is then removed and 3 more cards are added to the table. The game continues until all cards are dealt and no more sets are found. The player with the most sets at the end wins.
After the article describes the game and its’ rules, it continues by examining eight mathematical questions that were posed to different aged students (from high school freshman to college sophomores). They pose the questions, one at a time, and then discuss both the answers and the ways that students tried to solve them. The questions themselves are very interesting (and somewhat complicated) but the discussions of how different aged students tried solving them was even more interesting. For example , I thought the fact some college students took longer to solve the problems than high school students was rather comical. Finally, the article sums up their findings and pedagogical concerns, along with some other extensions of this game that were not discussed in the article.
In conclusion, this is a must read article for any math teacher . It gives a great way of exploring the concepts of sets, logic, probability, combinatorics, abstract thinking, deductive reasoning , problem solving, and proof through hands on activities that students will definitely be interested in.
Keywords: Activities, Problem Solving
Ref: TomD3
Author(s): Gonzales, Nancy; Fernandez, Albert; Knecht, Corine
Date: 1996
Title: Active Participation in the Classroom through Creative Problem Generation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89 (5), 383 - 385
Reviewer: TomD
Date of Review: 02/09/2000
This article focuses on how a mathematical classroom must change the role of students from being passive recipients of mathematical problems to becoming active participants. However, in order for such a task to occur then we as teachers must structure activities within the classroom that will involve all students, provide opportunity for mathematical communication, and link in creative thinking with mathematical content. In this article they describe one such activity, kind of like the game “pass it along”. What happens is a designated group makes a mathematical phrase and puts it on the board. The next group must then add an additional phrase to the original, and so on until the final group who then brings closure by summing up the entire phrase. Now, where the inquiry and conjecturing comes in is that while the problem generation is going the other students are commenting on whether or no the phrases fit into the whole overall idea or problem. Then in each of their own groups they follow the four step method of problem solving: (1) Understanding the problem - here the students ask there group “What exactly is the problem?”, or “Is this a problem at all?”; (2) Deciding a plan - they inquire with each other some ideas on how the problems could be solved and then the methods that they might use to solve them; (3) Carrying out the plan - at this point they must verify their solution or no solution either by a formal or informal proof, a diagram, a theorem or a postulate; and finally (4) Looking back - each group must evaluate what they did and discuss any changes they would make, and then the entire class will reflect on the problem and discuss maybe how it could be presented differently.
I really like this idea because it incorporates more than one area of mathematics to accomplish. The students are first of all working with word problems, and trying to understand what the problem is. Then each group must decide themselves how they would answer the question, and venture on their own to either proof or disproof their conjectures. Then they must reflect on what they did, listen to what other groups did, and then make an overall evaluation of what they did. Students will learn so much about communication, inquiry, conjecturing, and reasoning that hopefully they will see how useful it is and use it when they work alone on other projects in the future.
Keywords: Proof, Teaching Strategies
Ref: StaceyS3
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), pp. 565-567
Reviewer: StaceyS
Date of Review: 2/10/00
This article was short, but sweet in the sense that it reflected a positive tone of students learning and liking proofs. The author began by explaining how he would change his way of teaching proofs each year until he finally got a model that he felt worked well. Fidler first changed his grading system to one that gave credit to proofs that were incorrect, but the students had written down everything they could deduce. He thought this would get rid of the "all-or-nothing" attitude. He went on to cooperative learning whcih worked better, but something was missing. The following year he broke the class up into groups where the students were at similar levels and gave them small quizzes to work on together. Afterwards, he gave them a 10 proof take home test which they were excited to work on. Proofs had become fun to work on!
I enjoyed this article because it opens up a way of teaching proofs that could allow the students to enjoy the rigor of mathematics. I would like to try this approach in my own classroom even in other mathematical topics especially problem solving!
Keywords: Geometry
Ref: AndreaA3
Author(s): Libow, Herb
Date: 1997
Title: Explorations in Geometry: The "art" of Mathematics
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 5, pp. 340-342
Reviewer: AndreaA
Date of Review: 2/10/00
The author explores in this article how he recaptured the art and discovery in math. While teaching geometry, Herb Libow recognized that two theorems were related and might be able to be unified. The theorems involve chords in a circle being rotated about a point. As he was playing around with these chords, he made some fascinating and unexpected discoveries. As a chord is rotated around a point and the distances from the point to each side of the circle, the two distances multiplied together remains a constant. He kept going with it and asked himself what that constant was. As he let his imagination continue to take him through these thoughts, he was able to combine the two theorems to form the chord-line theorem. This was then furthered until he came upon a totally unexpected yet exhilarating result – the Pythagorean theorem! This is the process the author tries to share with his students. He tries to demonstrate the artistic experience in math, what makes math interesting. If we don’t have this, math becomes boring for everyone. Students need to “see that math is more than axioms, definitions, and cold logic. It embodies feelings, intuition, excitement, exploration and artistry.”
Keywords: Geometry, Technology,
Ref: TracyA3
Author(s): Izen, Stanley
Date: 1998
Title: Proofs in Modern Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 91, Issue 8, Pages 718-720
Reviewer: TracyA
Date of Review: February 12, 2000
Proofs are a very crucial element in learning and understanding mathematical concepts. Proofs can teach more than just math, they can enrich logical thinking and argumentation skills. The article, "Proofs in Modern Geometry", starts addressing the issue of computer generated "proofs". Is the information that is generated on the computer screen a mathematical proof, or is it only compelling evidence for a mathematical theorem? Are deductive proofs a think of the past? In the age of technology, what is the role of inductive and deductive proofs? Will the mathematical software available enrich the student's knowledge or will the student's "lose the math" in the graphics and computer capabilities?
If you are a math teacher who is planning on using computers in your classroom, this is the article for you. The author really explains the role of the computer and why students still need the traditional "paper and pencil" method of learning. Put your students in awe with the computer programs, but give them the insight and understadning they can only get through learning proofs.
Keywords: Inquiry, Geometry, Research
Ref: MiriamN3
Author(s): Malloy, Carol E.
Date: 1999
Title: Perimeter and Area Through the van Hiele Model
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 5, October, 87-90
Reviewer: MiriamN
Date of Review: 2/12/00
The author believes that mathematics instruction based on the research of the van Hieles can help students in the middle grades succeed in high school geometry. The van Hieles proposed that there are five levels of mathematical thinking: 0) Concrete, in which the student identifies and operates on concrete geometric figures; 1) Analysis, in which the student analyzes properties of figures through observation; 2) Informal deduction, in which the student formulates generalizations and develops informal supporting arguments; 3) Deduction, in which the student proves theorems deductively and understands the geometric system; and 4) Rigor, in which different postulational systems are studied and compared. The van Hieles argue that students within a given classroom may be at a variety of levels in their thinking, and that progress in learning is hindered when the level of the student is not aligned with the level of the material being taught. Although geometry in the middle grades is traditionally taught at the level of informal deduction, many students of this age may still be in the analytical, or even concrete stages of thinking.
Students in the middle grades often have a weak understanding of the concepts of perimeter and area, and consequently confuse these two terms and apply them incorrectly in formulas when solving problems. The author describes a van Hiele-based lesson designed to clarify understanding of these concepts and help students at different levels of thinking progress to the next higher level. Students conduct an investigation in which they are provided with an arrangement of square tiles and asked to add tiles to achieve a given perimeter. This experiment was performed by three sixth-grade students at three different levels of geometric thinking during a summer enrichment program. Students were first asked to explore the problem individually. Then through a group discussion facilitated by the teacher, students shared their thinking with one another, worked together to solve related extension problems, and gained new insights into different properties and relationships between area and perimeter. Through careful questioning and suggestions on the part of the teacher, by the end of the lesson, each student had begun to demonstrate some characteristics of the next level of geometric thinking. The author's belief is that use of the van Hiele method, both in terms of the sequence of instruction (information-gathering, guided discovery, explication, free orientation, integration) and knowledge of different levels of thought, were critical to the progress these students made on this topic.
I found little to argue with in this article. I liked the activity very much, since it not only solidifies concepts of perimeter and area, but is interesting and beneficial for students at different levels of thinking and can lead to good conjectures and further extensions in many different directions. I agree that knowledge and application of the van Hiele levels of thinking and phases of investigation can enrich instruction of geometry and other mathematical subjects. My only comment, and the author would probably agree, is that moving students from one level of thinking to the next likely requires numerous investigations of this kind. Teachers reading this type of account should not be overly optimistic and assume that the entire transformation can take place in a single lesson such as this.
Keywords: Geometry
Ref: LoriLu3
Author(s): Brandell, Joseph L.
Date: 1994
Title: Helping Students Write Paragraph Proofs in Geometry
Journal or Publisher: Mathematics Journal
Volume, Issue, Pages: 87(7), pp. 498-502
Reviewer: LoriLu
Date of Review: 02/12/00
In this article, Brandell demonstrates how paragraph proof can be implemented and developed in the classroom. He likes the use of a flowchart as one way to help students organize their thoughts in a logical fashion. Once students demonstrate an ability to outline proofs correctly, they progress to writing proofs in paragraphs, writing the "statement" and the "reason" in one sentence. They can still use the flowchart as the basis for the proof if needed. As the geometry course develops, Brandell advocates allowing students to omit preselected steps for which mastery has been demonstrated. He also provides a five-point rubric that he uses to evaluate student proofs. Many times, the most difficult parts of solving a proof involve analyzing the given information and knowing what additional infor- mation is needed. Brandell suggests two group activities that address these problems separately. In the first activity, each group is supplied with the "Given" information, but no "Prove" information. The objective is to analyze the "Given" information by creating a flowchart. In the second activity, each group is supplied with a "Prove" statement only. The objective is to analyze the "Prove" statement by creating a flowchart, that is, work the proof backward. Brandell describes another activity whereby students are given a series of previously written and evaluated proofs. The students evaluate each proof and "grade" it, discussing the main points. The NCTM Curriculum and Evaluation Standards suggest that students should learn to express deductive arguments orally and in sentence form. We must teach students to think and communicate effectively. Brandell has designed some excellent activities that can help students to accomplish these objectives. I liked the idea that after some practice evaluating proofs, students are better able to write proofs. Brandell provides a useful framework for introducing a subject that is often difficult for students to grasp and just as difficult for teachers to convey. I especially appreciated the fact that Brandell included several examples illustrating his activities and ideas.
Keywords: Problem Solving, Connections, Teaching Strategies
Ref: TinaM3
Author(s): Kahan, Jeremy
Date: 1999
Title: Ten Lessons from the proof of Fermat's Last Theorem
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 6, pgs. 530-1
Reviewer: TinaM
Date of Review: February 12, 2000
This is a pretty short article by our own Jeremy Kahan. He uses the process of the proof of Fermat’s last theorem, by Andrew Wiles, to bring out several important points about the ‘human process of constructing the proof.’(The Mathematics Teacher, 1999, 530). First, Jeremy points out that sometimes math is hard and it takes time. He illustrates this point by siting the fact that it took hundreds of years before Fermat’s last theorem was proved. Next he discusses how mathematics can be both an individual and collaborative process. Although Wiles worked alone for several years attempting to prove this theorem, he also relied on other mathematicians’ contributions to mathematics in order to complete this proof. Jeremy points out here that it is important to give students the opportunity to work individually as well as cooperatively. Jeremy then discusses the social aspect of mathematics. He explains that it is necessary to create communities in our classrooms where students can present results and have their peers them. Wiles presented the proof of Fermat’s theorem to a community that, at first, rejected it. He also points out that it is important that the students check the work of the instructor because even he or she can omit something substantial when constructing a proof. Evaluating other’s work and thought processes is something that is stressed in the NCTM Reasoning Standards. During problem solving, sometimes we must take a step back in order to go forward. This is what Wiles had to do before he could complete the proof of Fermat’s theorem, Jeremy explains. Another point that is brought out in this article is encouraging students to relate a problem to one that they know how to solve. This is a problem solving strategy that gets students out of the habit of giving up too soon on a problem, which is a big problem when it comes to proof construction. It is then pointed out the relationship between geometry and algebra. Although many high school curricula treat the two subjects as they were separate entities, Jeremy points out that NCTM encourages teachers to make connections between the two because sometimes algebra can be used to solve geometric problems, and vice versa. Diversity within mathematics is also brought out in this article. The mathematicians who helped with the construction of the proof of Fermat’s last theorem represent a variety of nationalities and were both women and men. The history of this proof, Jeremy notes, ‘furnishes diverse role models for students’ (The Mathematics Teacher, 1999, 531). Lastly, Jeremy notes that teachers often present mathematics as a ‘fait accompli’ (The Mathematics Teacher, 1999, 531). Yet, the proof of Fermat’s last theorem shows us that very difficult mathematics can be solved with contributions of modern mathematicians. All of the points made in this article should be considered as we begin our journey into the world of mathematics teacher’s.
Keywords: Problem Solving, Connections, Teaching Strategies
Ref: TinaM3
Author(s): Kahan, Jeremy
Date: 1999
Title: Ten Lessons from the proof of Fermat's Last Theorem
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 6, pgs. 530-1
Reviewer: TinaM
Date of Review: February 12, 2000
This is a pretty short article by our own Jeremy Kahan. He uses the process of the proof of Fermat’s last theorem, by Andrew Wiles, to bring out several important points about the ‘human process of constructing the proof.’(The Mathematics Teacher, 1999, 530). First, Jeremy points out that sometimes math is hard and it takes time. He illustrates this point by siting the fact that it took hundreds of years before Fermat’s last theorem was proved. Next he discusses how mathematics can be both an individual and collaborative process. Although Wiles worked alone for several years attempting to prove this theorem, he also relied on other mathematicians’ contributions to mathematics in order to complete this proof. Jeremy points out here that it is important to give students the opportunity to work individually as well as cooperatively. Jeremy then discusses the social aspect of mathematics. He explains that it is necessary to create communities in our classrooms where students can present results and have their peers them. Wiles presented the proof of Fermat’s theorem to a community that, at first, rejected it. He also points out that it is important that the students check the work of the instructor because even he or she can omit something substantial when constructing a proof. Evaluating other’s work and thought processes is something that is stressed in the NCTM Reasoning Standards. During problem solving, sometimes we must take a step back in order to go forward. This is what Wiles had to do before he could complete the proof of Fermat’s theorem, Jeremy explains. Another point that is brought out in this article is encouraging students to relate a problem to one that they know how to solve. This is a problem solving strategy that gets students out of the habit of giving up too soon on a problem, which is a big problem when it comes to proof construction. It is then pointed out the relationship between geometry and algebra. Although many high school curricula treat the two subjects as they were separate entities, Jeremy points out that NCTM encourages teachers to make connections between the two because sometimes algebra can be used to solve geometric problems, and vice versa. Diversity within mathematics is also brought out in this article. The mathematicians who helped with the construction of the proof of Fermat’s last theorem represent a variety of nationalities and were both women and men. The history of this proof, Jeremy notes, ‘furnishes diverse role models for students’ (The Mathematics Teacher, 1999, 531). Lastly, Jeremy notes that teachers often present mathematics as a ‘fait accompli’ (The Mathematics Teacher, 1999, 531). Yet, the proof of Fermat’s last theorem shows us that very difficult mathematics can be solved with contributions of modern mathematicians. All of the points made in this article should be considered as we begin our journey into the world of mathematics teacher’s.
Keywords: Geometry
Ref: LizA3
Author(s): Manaster, Alfred B.; Schlesinger, Beth M.
Date: 1999
Title: Geometry Problems Promoting Reasoning and Understanding
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Volume 92, No 2, P114-116
Reviewer: LizA
Date of Review: 2-14-2000
This article gives four related problems concerning perimeter and area of rectangles and circles. These problems are presented as ways to introduce the idea of proof before students enter a geometry class. The authors suggest they can be introduced at different times while students are studying quadratics and that students should be asked to justify their solutions. The problems are well developed in that each problem asks students to apply what they learned in the problem before and take it one step further. The author also gives possible justifications students might develop.
This article seems to be aimed at moving a traditional classroom towards being more integrated. The authors suggest introducing some aspects of geometry into an algebra class in order to start students justifying their answers. Although I think the questions given are good questions, I do not think that they are necessary to introduce the idea of justifying answers. In almost every aspect of mathematics students can be asked to find why their answers work. As students move to higher grades, they can be asked to do this with more and more rigor. Not only will this prepare students to write proofs, but it will also help them understand and remember the material they are learning.
Keywords: Proof
Ref: ChrisW3
Author(s): Driscoll, Mark
Date: 1982
Title: The Path to Proof
Journal or Publisher: NCTM
Volume, Issue, Pages: in Research Within Reach: Secondary School Mathematics
Reviewer: ChrisW
Date of Review: 13 February 2000
"The Path to Proof" starts out so promising. At first glance, one hopes that it might provide a strong method for presenting proof to high school students so that proof can become meaningful to them. The introductory statement and question "Constructing proofs is a very difficult task for many of my students. They can’t even get started on most proofs. How can I help them to analyze a question or problem well enough to discover a starting point?" is one that seems relevant to many teachers. Sadly, this question is not particularly well answered in the rest of the article. The author points out the importance that cognitive development and prerequisite skills play in the success of developing a student’s understanding of proof. In a discussion of van Hiele levels, the author cites research that suggests that van Hiele levels can be a good predictor of success in a geometry class which includes formal proof. (160 The Path to Proof) The geometry classes studied, however, did not do a particularly effective job at moving students up to higher van Hiele levels: after a year long geometry course, only about half of the students were able to solve moderately complex proofs. (160 The Path to Proof) The evidence provided in this article seems to suggest that the question asked at the beginning was a valid question; however, the evidence provides little in way of an answer to that question. The three kernels of pedagogic wisdom for teachers that one finds in this article are: 1. Model the type of reasoning that one wants one’s students to use. 2. Think aloud while attacking problems and constructing proofs. 3. Make student involvement in mathematical discussions a key part of the classroom atmosphere. While these are certainly good ideas, they hardly seem like they would be all that amazing to a teacher who is concerned about making proof a more important part of their students’ lives. This is why the article is a disappointment.
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: Proof, Planning,
Ref: LeifN3
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, Number 7, October 1999, pg 565-567
Reviewer: LeifN
Date of Review: 2/13/2000
This article was not like what I thought it was going to be, but after reading it found it kind of informative. The article is written by a teacher, Mark Fidler, and is about his methods for helping students understand and perform better at doing proofs. He shares the techniques he used in getting his classroom relating to proof, student performance, student interest and student understanding. He had been frustrated by his students’ attitudes about proof and was trying to figure out how to change those attitudes, particularly their performance. His students were not doing well when had the working alone, after changing his grading system he decide to try cooperative learning. He put the students in groups of ten and gave them a week to work on some very hard proofs, but this still did not produce the desired result. I found it curious that he seemed to forming groups the exact opposite way as to what we learned from Dave Johnson (Mr. Fidler was using large groups with homogenous ability levels, he later explains his reasoning for homogenous ability levels). I wonder if that is the reason the desired goals were not reached. I don’t group of ten high school kids could agree on anything let alone work cooperatively on a proof. Mr. Fidler got is desired results, the next year, when he used smaller groups. He had the students work on in class group quizzes together, the quizzes were worth enough that it made that students realize the need to work cooperatively to get the desired grade. He still used homogenous skill level groups though. “Assigning groups for graded work can be tricky. I find that mixing the strongest students with the weakest can stunt real discussion. And so I assign people to work with others who are earning similar grades on individual tests. Students work in groups of two, three or four. Mostly my A students work in groups of two. I try to but my B students in groups of three and my weaker groups usually have four students. This homogenous grouping tends to make students feel needed for the success of the groups and encourages involvement by all group members.” Mr. Fidler’s reasoning sounds good for using homogenous groups in this situation, but I wonder if it would work in other situations. I think that group work on proofs is a great idea, because often times what one students doesn’t see another student well and thus they can help each other through the proof. Another thing that Mr. Fidler does that I found interesting, was his grading system for proofs. He stressed documenting all dead ends that the students run into and when you get stuck finish the proof with a goal statement, i.e. if I could only prove this thing then I would be able to finish the proof. I found this article to be very informative and full of good ideas that I would like to try implementing in the 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: Geometry, Activities, Technology
Ref: EoinO3
Author(s): Reinstein, David; Sally, Paul; Camp, Dane R.
Date: 1997
Title: Generating Fractals Through Self-Replication
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Vol. 90 No. 1 January 1997
Reviewer: EoinO
Date of Review: 2/13/00
This article gives a series of activities that can be used to introduce the idea af fractals to a geometry class. These activities include both hands on constuctions and the use of programable graphing calculators.
Fractals are an area of geometry that has really come into its own in the last half century. The idea was present before, but much of the exploration of fractals has been very computer dependent. Because of their dependence on technology and the sublime esthetic of the fractal form, students usually find this topic very intriguing and are very interested and motivated. This activity can be used to introduce them to some of the prime characteristics about fractals and let the explore them on their own.
The activities were written for 9th to 12th graders, but could be accessable to some younger students. It is fairly straight forward. In addition to telling you that the program is available for download at the NCTM site (though whether that is still true three years later I haven't checked yet, but they give the code in the paper) they provide a very useful bibliography for students and teachers who want to explore this area further. On the bibliography are a number of books and articles that deal with using fractals in the classroom. So even if you find the activities in this article to be to basic or repetitious (which some might) there are resources that are availible which will challenge you.
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.
Keywords: Technology
Ref: LoriLa3
Author(s): Dion, Gloria A.
Date: 1995
Title: Fibonacci Meets the TI-82
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 88, No. 2, pgs. 101-105
Reviewer: LoriLa
Date of Review: Feb. 15, 2000
A study of the Fibonacci sequence, through the use of the TI 82, is illustrated in this article. The author believes that technology is a wonderful addition to the classroom for helping to solve problems. She says, however, because of the addition of such calculators, we have to "reexamine what we teach and how we teach." She also adds that while calculators are responsible for doing the "dirty work" of mathematics, such as number crunching, teachers are also required to be "experts" with the functions of the graphing calculator and take time in their classrooms to teach these function. With the help of the graphing calculator, the author adds that "the graphing calculator affords teachers and students an opportunity to generate data and then formulate and test conjectures, giving them a chance to do mathematics as mathematicians do." Following this statement the article illustrates her argument by providing a calculator problem concerning the Fibonacci sequence. The students are given the a variation of Fibonacci's original question of a male and female pair of adult rabbits who, placed in an enclosed pen, breed and multiply. The example brings us through various functions of the graphing calculator, using such things as the table function, the graphing function, and the sequence mode. The problem allows students to do the over-all, big picture thinking in this problem, without having to do all the foot work of making small calculations. Like the author, I think this kind of problem is invaluable to students in their mathematical career. In the real-world, students will need to be able to use and manipulate resources in order to solve problems, and this will give them good practice in generating an over-all thought process in order to solve a fairly complex problem. It will also give them exposure to the graphing calculator for possibly making their own conjectures and testing them.
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, Trigonometry
Ref: MiriamN4
Author(s): Pagni, David L.; Shultz, Harris S.
Date: 1999
Title: Extending a (TIMSS) Japanese Lesson Using Trigonometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, March, 189-191
Reviewer: MiriamN
Date of Review: 2/15/00
A problem from the Japanese lesson on areas of triangles in the TIMSS study was extended to include the use of trigonometry. The problem in question was the one in which two non-intersecting line segments are drawn, one below the other, and the left-hand portion of the space between the segments is said to represent “Eda’s land”, while the right-hand portion is “Azusa’s land”. The border line between the two sides is bent, and the problem in the lesson is to straighten the line, while preserving the areas of both parties’ properties. The class’ solution was to form a triangle by drawing a “base” connecting the ends of the bent border, then draw a line through the point in the border that is parallel to the base, and to move the point down along the parallel line until it touches the bottom border. The concept used in the solution was the property that all triangles with the same base length and height have the same area. The authors have observed this exercise u! sed by teachers in a professional development setting, and have noted that preliminary ideas for solutions usually involve trying to find a straight border line that is parallel to the base of the triangle. This gave the authors the idea to extend this lesson to involve trigonometry, and pose the two problems: 1) find the segment parallel to the base of the triangle that will preserve the areas of the properties; and 2) find the line at some given angle to the bottom border that will preserve the areas of the properties. Trigonometric solutions were applied to answer both of these questions.
Although the authors did not discuss the use of these exercises in a trigonometry class, I think they would be good extension problems that would help connect the students’ trigonometry instruction to their prior knowledge of geometry. The exercises require a synthesis of a range of knowledge and skills and are set in an interesting real-world context. My concern is that the authors’ trigonometric solutions appeared to be fairly complex and involved. Perhaps these problems should only be tackled by classes at more advanced stages of trignonometry instruction, and perhaps they should be only presented as long-term, or even extra-credit projects.
Keywords: Manipulatives, Geometry, Technology
Ref: TinaM1
Author(s): Perham, Arnold E.; Perham, Bernadette H.; Perham, Faustine L.
Date: 1997
Title: Creating a Learning Environment for Geometric Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 90 Number 7
Reviewer: TinaM
Date of Review: February 17, 2000
This article is about the value of the use of manipulatives, computer software, and graphing calculators in geometry. The authors use a series of lessons about the centroid of a triangle. First, they allow the students to experience with hands-on manipulatives, such as a pencil and soda straw balance to show that the centroid is the balancing point of the triangle. This gets them to ask questions and heightens their interest about the topic at hand. Then, Geometer's Sketchpad is introduced and the students are able to further test the conjectures they have made. They are able to create new figures quickly and validate their ideas quickly, resulting in possible generalizations being made. Next, the students are introduced to Mathcad, which is another mathematics software package that works similar to a spreadsheet. The students are able to see the calculations that are being made behind the scenes in the Geometer's Sketchpad. The students can change their triangle and all of their calculations will change, as well. Lastly, they look at the usefulness of a graphing calculator, specifically the TI-82. The most important message of this article is that the use of such tools helps students better visualize aspects of geometry. They also help students come up with conjectures and prove them informally. These tools also can make the lessons more interesting and enjoyable for the students.
Keywords: Connections, Geometry,
Ref: TomD4
Author(s): Dodge, Walter; Goto, Kathleen
Date: 1998
Title: "I would consider the following to be a proof . . . "
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91 (8), Soundoff
Reviewer: TomD
Date of Review: 02/16/2000
This article appeared in the Soundoff section of that Mathematics Teacher in 1998, and it pertains to how important proofs are to geometry but can also but equally important in the other areas of mathematics. It talked about how in prealgebra and algebra classrooms students are first introduced to power series such as 1+2+3+4+....+n=n(n+1)/2. However, how often do you see an algebra or prealgebra teacher proving this. Better yet, how often do you see a teacher asking students to think about this equation and determine if in fact it is a valid statement. Another example from an algebra classroom is the binomial (a + b)2 = a2 + 2ab + b2. How many students in algebra actually know why that is true? Or, how many think very little about it and accept it as fact. This article talks about how geometry (and the point of proofs) attempts to make students look at mathematics and the things they are doing in a new and different manner. Geometry allows students to open their minds, and begin a new reasoning process in mathematics. However, why does this reasoning process be restricted to geometry? We need to have our students reason and rethink about things in all areas of mathematics because that is how our students will learn and be able to apply.
I definitely had some mixed reviews about this article. Primarily I supported the point that too often we as teachers make a statement and without making our students think about it, let our students automatically think it is true. I remember this from my algebra class and never really knew why many laws and rules were true. I was told it, and it worked so I took it as fact. There was no conjecturing, reasoning, or thought process about it. However, one aspect of this article I disagreed with (so I didn’t mention above) was the emphasis they put on technology. They talked about how geometry students should be able to use Sketchpad to do proofs. How does that show any thought process or knowledge? They might be able to use a program by fooling around with it, but do they know or understand what justifies what the computer is doing? This is an example where I believe that they are trying to use technology as “the answer”, rather than as “a tool”.
Keywords: Teaching Strategies, Assessment,
Ref: TomD5
Author(s): Gronseth, Phillip
Date: 1999
Title: Course Diary: A Valuable Information Source
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (6), 496 - 497
Reviewer: TomD
Date of Review: 02/16/2000
This article talks about the importance of using a diary, as a teacher, to write down thoughts, feelings and possibly new ideas. The diary is especially useful for new teachers to write down what they did, what worked, and what miserably failed that was thought initially as a brilliant idea. For maximum benefits you should have one journal for each subject that you teach. For example, during my student teaching I know that I have an algebra and a prealgebra for sure. So right now I am planning on having two exclusively separate journals. The diaries should also contain any handouts, quizzes, midterms, chapter tests, or worksheets. Anything that you did with the students; whether they needed turn it in or not you should keep a record of it. This article gave a few good reasons why, as a teacher, this would be an exceptional tool. First, it’s an excellent tool for teachers to assess the job they did in the classroom. To many times the only assessment we ever hear about is for the students, well isn’t just as important that the teachers are assessed on the job that they do! Secondly, this allows teachers to document on student performances; whether it is used for assessment of those students or to make the course better for the students of the future. Finally, it can be used a useful tool to assess the effectiveness of the course itself or the version of text that the school was using.
Until this class this quarter, I never had to keep a journal on anything. For me, I like the journaling that we are doing because it allows me to reflect on what I’m doing and to make changes or modifications to my approaches in my practicum. This is one reason that this article jumped out at me when I was reading it. I was considering to keep a journal during my student teaching to help me assess my teaching as a tool. Reading this article there is no doubt I will be journaling, in addition putting in anything that I did with the class. It will not only help me as a teacher, but would also be an eye opener if I walked in with that for a job interview !!!
Keywords: Games, Connections, Teaching Strategies
Ref: TomD6
Author(s): Johnson, Carl
Date: 2000
Title: Human Coordinates and Floor Tiles
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93 (1), 13
Reviewer: TomD
Date of Review: 02/16/2000
In this article the author, Carl Johnson, talked about how he used his classroom floor as a coordinate system and the students as points in that system. What he did was assign certain students as coordinates in each of the four quadrants, representing the four vertices of a square. Now, the other students no apart of the square had to answer questions about the length of each side and the length of its diagonals, and the area of the figure. Then they had to verify there work by actually using tape measures and measuring the distances. The students then had to make transformations of the “student figures”. For instance, they could do a reflection over any given line. Or, rotate a certain number degrees about a vertice (one of the students representing that point), and even translate the figure in a particular direction. From this he had the students break up into groups (I believe he called them teams) and make up two or three transformations they could do, and then they physically had to do the transformations themselves on the floor (coordinate system).
The main reason I enjoyed this article is that I like to read about how teachers have done different things in the classroom to keep student awareness high. This activity allows students to see the mathematical connections, and inspires the students to have fun and that they can be more creative in an ugly mathematics classroom.
Keywords: Activities, Geometry,
Ref: AndreaB3
Author(s): Van Dyke, Frances
Date: 1995
Title: A Visual Approach to Deductive Reasoning
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 88, Issue: 6, p. 481-484, 492
Reviewer: AndreaB
Date of Review: 2/14/00
This article describes activities that can be used to introduce students to three patterns of reasoning in inferential logic. Using Ven Diagrams and p-q symbolization, students explore direct reasoning, indirect reasoning, and transitive reasoning. Students find valid and invalid conclusions for each type of reasoning. There are five activity sheets for use in classroom. Group activities are built into the lesson plan.
Proving conclusions from a visual perspective can be helpful in today’s modern classroom. I really found this article helpful because when I studied logic and deductive reasoning there was no attention was paid to students who learned visually.
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: Geometry
Ref: LoriLu4
Author(s): Lornell, Randi and Westerberg, Judy
Date: 1999
Title: Fractals in High School: Exploring a New Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(3), pp. 260-265
Reviewer: LoriLu
Date of Review: 02/20/00
The authors of this article have included a unit about fractals in their traditional geometry classes. This article begins with a brief description of fractals and their characteristics. For example, fractals typically have the property of self-similarity, meaning that a part of the whole closely resembles the whole. The authors describe how fractal feometry can be studied in the classroom by sharing some activities from their classroom unit on the subject. A good way to begin the study of fractals is to compare familiar objects that have a definite Euclidean shape with other familiar objects that do not. The authors provide a compare/contrast activity that drives home the need for additional ways to describe objects found in the natural world.
The authors then describe a set of activities using two classic fractals, the Cantor set and the Koch snowflake. These activities are designed to help students to understand how iteration can create a fractal. For example, students construct the seed and first two iterations of the Koch snowflake with pattern blocks. They then use their structures to make conjectures about the growth patterns of the area and perimeter under iteration and to deveop recursive equations describing them. They can use the graphing calculator to verify their thinking and simulate the growth patterns in further iterations. Students discover the counterintuitive result that the Koch snowflake has an infinite perimeter but a bounded area.
Finally, the authors provide several reasons for including fractals in the mathematics curriculum. Students have the opportunity to investigate traditional mathematics topics from a new approach and to explore mathenatics in nonanalytic ways. Students can also make connections both within mathematics and between mathematics and the natural and human worlds. Many good examples were given to support such claims.
I learned a lot about fractals and their history by reading this article. The activities that the authors shared were excellent. I would definitely consider using them in a classroom setting. The authors piqued my interest in fractals and convinced me that the study of fractals should be a part of the mathematics curriculum.
Keywords: Proof, Connections,
Ref: TinaM4
Author(s): Dodge, Walter; Goto, Kathleen; and Mallinson, Philip
Date: 1998
Title: "I Would Consider the Following to Be a Proof..." - www.nctm.org/mt/1998/11/soundoff.html
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 91, Num. 8
Reviewer: TinaM
Date of Review: February 20, 2000
This was a really short but interesting article about proof. The authors present several proofs presented in different mathematics classes, including pre-algebra, first year algebra, geometry, second year algebra, trigonometry, and calculus. Some of the proofs are technology based including the use of the TI-83 and TI-92 calculators, and the Geometer’s Sketchpad software package. Some of the proofs were solved using diagrams and others were solved using algebraic manipulations. Two points should be made from this article. First, although proof has generally been associated with geometry coursework, it is definitely useful in other math courses. The purpose of proof must be determined and whether it is the same in different classes. Second, there are many ways proofs can be presented. The instructor must decide what is acceptable as a proof as well as evaluate what the students find acceptable.
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, Algebra,
Ref: LeifN4
Author(s): Frorringer, Richard S.
Date: 2000
Title: (A+B+C)^3
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 1, pgs. 6-8
Reviewer: LeifN
Date of Review: 2/20/2000
I found this article to be quite interesting and informative. It is about the cubic equation (A+B+C)3 and the solution to that equation. We have all seen that we can use the area of a square to solve the quadratic equation (A+B)2 by breaking it into smaller squares and rectangles. Well in this article they use the volume of a cube to help the students find the solution to (A+B+C)3. First the students work with the quadratic equation and solving it with area, then they are introduced the cubic equation (A+B)3 before (A+B+C)3. They were asked if (A+B)3 could be represented in a manner similar to (A+B)2? Many students do not make the connection that the “cubing (A+B)” represents the volume of a cube whose dimension is A+B. Most students have not made the connection that the terms square and cube have two different meanings, one algebraic and one geometric. The idea is to give the students the blocks and let them see if the can come up with a to represent first (A+B)3 and then (A+B+C)3 three dimensionally as cubes. I thought this was a great exercise and one that I had not thought about before.
Keywords: Activities
Ref: AndreaA4
Author(s): Edwards, Thomas G.
Date: 1995
Title: Students as Researchers: An Inclined -Plane Activity
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7, pp. 532 - 535
Reviewer: AndreaA
Date of Review: 2/20/00
This article describes a class that investigates inclined planes. The students collect data in groups by experimenting with different inclined planes. They agreed on a list of variables: the time for the ball to reach the bottom of the plane, the mass/weight of the ball, length and height of the incline. The students are to use three different values that can be controlled such as three different masses, lengths and heights. After allowing the students to collect data in any manner they choose, the teacher may step in to suggest they organize the data in tables. At this point, the teacher has not assigned any questions to the students. As students collect the data, they think up questions on their own. The teacher has students come up with their own questions and answers from the data they have collected. This activity lasted a whole week but contained several mathematical and scientific tasks. Students actively collected data by measuring length, height, weight and time. They calculated averages and in using calculators needed to use number sense and rounding. Through working together in groups and problem solving, the students developed their abilities to communicate mathematically. Finally, the nature of the project engaged students to use their problem solving and other skills to solve a real life physics problem. The students combined all this practice and skill building in one problem which was much more meaningful than drill and practice.
Keywords: Curriculum
Ref: TracyA4
Author(s): Alper, Lynne; Fendel, Dan; Fraser, Sherry; Resek, Diane
Date: 1995
Title: Is This a Mathematics Class?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 88, Issue #8, pages 632 - 638
Reviewer: TracyA
Date of Review: February 20, 2000
What is your image of the Integrated Math Program (IMP)? Do you believe in the curriculum's ideas and direction? This article will try to convince you that IMP is the best thing since indoor plumbing. According to this article, students will take more math classes during high school and will graduate with more math knowledge because of the IMP curriculum. IMP is also the mathematical solution for students who think math is boring and difficult. The authors mentioned all the training required to be an effective IMP teacher. Unfortunatley, they never address the outcome of an under-trained teacher who tries to implement this curriculum. By the end, I was wondering if the authors were part of the team who created the curriculum.
I believe all students can learn math, and that math is for everyone. Not everyone will take advanced calculus, but everyone can learn! According to this article, "A major premise of IMP is that nearly all students are capable of thinking about mathematics and of understanding deep concepts" (page 635). So what happends to the students they excluded? Where do they go and how are they going to learn math?
This article did not encourage me, although it painted a really nice picture of the perfect math class. If you believe in the IMP program and would like to read about how wonderful the curriculum is, this is the article for you. If you have doubts about the system and don't think it is perfect, you would not enjoy this article. The choice is up to you.
Keywords: Geometry
Ref: ChrisW4
Author(s): Greive, Cedric;
Date: 1999
Title: The sum of i-squared and the Volume of a Cone
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 9, 825-827
Reviewer: ChrisW
Date of Review: 21 February 2000
In this article, Greive derives the formula for the volume of a right circular cone and then shows us a method for making this problem accessible to upper secondary students. Greive's derivation is fairly geometrical. It reminds us of using the disk method to calculate the volume of a solid of revolution in calculus, but, instead of calculus, this problem uses the series found in a high school analysis course to find the volume. By creating a tower of concentric cylinders one on top of each other (which some might say looks much like a baby's toy), Greive sets up a method which involves both geometry and algebra to derive the volume of a cone. The scaffold that Greive sets up for his students' is a table that includes a list of each cylinder, has a column for writing the radius of each cylinder, and a column for writing the volume of each cylinder. Greive suggests that a teacher provide the first three radii and first two volumes for their students. In the classroom, Greive suggests that the teacher start the class by asking their students how they would come up with the volume of a cone, and leading those students to think about modeling the cone as a tower of ever shrinking cylinders. After the model has been visualized, Greive recommends that the teacher give the students the table and demonstrate how the teacher calculated the radius and volume of the first and second cylinders in the stack. Greive's experience using this in the classroom has shown that when students have series as a context, they are able to recognize the pattern in the cylinder volumes and find the limit as n approaches infinity, which ! allows them to derive the volume. This method of deriving the volume of the cone gives students a "meaningful application of the topics of series and limits, and it makes an appropriate introduction to calculus." ( 826 Mathematics Teacher) I agree with this analysis, and think that this provides a good way of showing the importance that geometric visualization can have in other areas of mathematics.
Keywords: Algebra, Manipulatives, Geometry
Ref: AndreaB4
Author(s): Forringer, Richard S.
Date: 2000
Title: (A + B + C)^3
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, Issue 1, p. 6-8
Reviewer: AndreaB
Date of Review: 2/22/00
This article connects algebra to geometry. Many of us are familiar with representing (x + y)^2 with a square of length x + y, and the resulting squares. This article uses the same approach with the equation (A + B + C)^3. This approach allows students to see that a cube with side length of (A+B+C) is composed of smaller blocks, and that the faces look much like the square from (x+y)^2.
I found this article very enlightening. I am a visual learner, but had never seen this approach. This approach is much clearer than algebraic expansion of (A + B + C)^3.
Keywords: Geometry
Ref: LukeB3
Author(s): Bedford, Crayton W.
Date: 1998
Title: The Case for Chaos
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 4, 276-280
Reviewer: LukeB
Date of Review: 2/28/00
I thought this was a good article for teachers interested in chaos and in teaching a class on chaos. I am not familiar with chaos or fractals but after reading this article I am very interested. The author explains what chaos is and also explains what fractals are. He then outlines a course that he believes is important. He set the course up so that it covers the vocabulary of dynamical systems, fractals, orbit analysis, the logistic function, functions of complex numbers, Julia sets, mathematics of chaos, and the understanding of chaos. He also lists sources available for learning about chaos and appropriate textbooks. He believes that this course he designed will help his students see a new way of looking at the world and have a better understanding of mathematics.
Keywords: Geometry, Calculus, Connections
Ref: LukeB4
Author(s): Morriss, Patrick
Date: 1998
Title: Discovering a Geometric Volume Relationship in Calculus
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 4, 334-336
Reviewer: LukeB
Date of Review: 2/28/00
I thought this was a good article for teachers planning on teaching calculus. It connects geometry to calculus. He has developed a plan for deriving a theorem. The theorem is: the volume of a solid of revolution formed by revolving about the y-axis the region bounded by the graphs of y = ax^n, y = ar^n, and the y-axis is V = (n/(n + 2))* Vcyl, where a and r are positive constants and Vcyl is the volume of the cylinder circumscribed about the solid. He came up with a three day method using shell method ideas. On the first day, have the students find the volume of the solids of revolution formed by revolving about the y-axis the regions bounded by the graphs of (1) y = x^2, y = 0, and x = 2 and (2) y = x^2, y = 4, and x = 0. Then have them find the volume of the cylinder circumscribed about (2). For homework, have them generalize to (1) y = ax^2, y = 0, and x = r and (2) y = ax^2, y = ar^2, and x = 0. Then have them find the volume of the cylinder circumscribed abo! ut (2). On the second day, use the homework results to establish that the volume of a paraboloid is half the volume of its circumscribed cylinder, then use the same method to rediscover the cone volume formula. For homework have the students find the volume of the solids of revolution formed by revolving about the y-axis the regions bounded by the graphs of y = ax^n, y = ar^n, and x = 0 for n = 3, 4, 5, .... Then find the volume of each circumscribed cylinder. On the third day, use the homework results to establish the general result. For homework have them identify several problems where the theorem applies.
Keywords: Geometry, Manipulatives,
Ref: LukeB5
Author(s): Kennedy, Joe; McDowell, Eric
Date: 1998
Title: Geoboard Quadrilaterals
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 4, 288-290
Reviewer: LukeB
Date of Review: 2/28/00
I thought that this was a good article. It demonstrates a project the author did in class with geoboards. This project was designed to help students recognize geometric shapes and to learn their special properties. The students start out with a three by three geoboard. Then the students are to find and count the number of noncongruent squares, then nonsquare rectangles, nonrectangular parallelograms, and trapezoids. After going over the examples and nonexamples, have the students find more general quadrilaterals. First restrict them to convex quadrilaterals, and then move to concave. After going over these results, have them try different size geoboards. Have them discuss their answers and find systematic ways of counting the different quadrilaterals without counting the same figure more than once.
Keywords: Geometry, Activities, Connections
Ref: LoriLu5
Author(s): Johnson, Carl
Date: 2000
Title: Human Coordinates and Floor Tiles
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 93(1), pg. 13
Reviewer: LoriLu
Date of Review: 02/28/00
This is a short article from the "Sharing Teaching Ideas" column of The Mathematics Teacher in which Johnson certainly offers a new slant on a familiar subject. He sees the square-foot tiles of his classroom as forming the coordinate system on which human geometric models can be constructed. Johnson creates the axes by sticking masking tape to the floor. He then labels the coordinates with a marker.
Johnson recommends starting with some basic mathematical illustrations. For example, students stand at assigned coordinates representing the four vertices of a square. Students are asked to describe the shape and find the lengths of its sides and diagonals, using a tape measure to verify. Areas can also be discussed.
Johnson then moves on to transformations. For example, he instructs human points to translate three units to the right, rotate 90 degrees, or reflect about an axis. Other transformations include dilations such as "double just your x-coordinate" or "double both coordinates". Students working in groups are asked to make up and demonstrate their own transformations. Johnson also incorporates the arts by having volunteers perform a popular line dance to music, using the coordinate grid to space themselves. Onlookers are asked to describe what they saw in terms of transformational geometry. As a follow-up project, students can be given the option of choreographing their own transformational dance.
This is a great activity! Johnson has come up with a creative way to involve students and help them visualize the transformation process. Students are also able to see the mathematical connections to the arts. I bet this is one lesson students will not soon forget; maybe it will even "transform" them--pun intended. What an ingenious way to get around scarce resources--use your own classroom and the students themselves as manipulatives. So you say your classroom is carpeted??
Keywords: Geometry, Activities, Manipulatives
Ref: AndreaB5
Author(s): Malloy, Carol E.
Date: 1999
Title: Perimeter and Area Through the Van Hiele Model
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, Issue 2, p. 87-90
Reviewer: AndreaB
Date of Review: 02/27/00
This article summarizes the Van Hiele levels and applies them to the study of perimeter and area. An activity is provided, along with the responses from three students. Each of these students were at different levels. Guiding questions and extensions are included.
I have studied the Van Hiele levels and believe I have a decent understanding of them. However, this article was enlightening because it not only included activities and questions, but it also showed how your students might go about solving the same problem in different ways.
Keywords: Geometry, Activities,
Ref: LeifN5
Author(s): Smith, Lyle R.
Date: 1999
Title: Using Dragon Curves To Learn About Length and Area
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No.4, pg 222-223
Reviewer: LeifN
Date of Review: 2/29/2000
This article was a short one but I thought it described a great activity. It is an activity that uses dragon curves, something I have not heard of, to learn about area and length. It would be a good activity that I think the students would enjoy and be interested in doing. It also doesn't require a lot of prior knowledge, just area formulas for a circle and a square and the circumference of a circle. It would be a great activity to use when students are just starting to learn about pi, circumference, and area of circles. The students can use the formulas they have to find the areas of each three types of tiles (a blank tile, one with one arc and one with two arcs in opposite corners) and then the area of the patterns they build using the tiles. The students use the three types of tiles to build as creative patterns as they can. Once the have completed their patterns the students are asked to find the area and the length of them. This activity could be used to reinforce the formulas for area and circumference or to allow them to discovery them to an extent. It could also be used for area estimation, the students could estimate the area of their patterns then check their estimation against the actual area of the pattern.
Keywords: Problem Solving, Geometry, Algebra
Ref: MiriamN5
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: 92(2), Feb.'99, 114-116
Reviewer: MiriamN
Date of Review: 2/29/00
The authors of this article argue that current mathematics curricula focus too heavily on obtaining correct solutions (emphasis on “how”) and not enough on mathematical reasoning (emphasis on “why”). Often, the only course in which students are expected to provide explicit chains of reasoning is geometry. The authors argue that reasoning should be promoted in all mathematics courses, and that students should gain experience with this process prior to taking a geometry course. In this article they present four problems promoting exploration and reasoning, which involve aspects of algebra and geometry, but which could be given to students prior to a geometry course. The problems deal with finding the dimensions of a rectangle of fixed perimeter that maximize its area, and with comparing areas of squares and circles with equal perimeters. A variety of approaches are suggested to solve these problems, including exploration of tables and graphs (graphing calculators are recommended), use of calculators to compare areas expressed in decimal form, and formal algebra involving formulas for perimeter and area of rectangles and circles.
I agree with the authors that this type of exploration problem should become an integral part of all math curricula, not just geometry. Such problems promote reasoning and communication skills; they lend themselves to meaningful and appropriate uses of technology; they involve deep and interesting mathematics; they make connections among different mathematical topics; and they help students appreciate the necessity of building a repertoire of knowledge and techniques in diverse areas of mathematics. Perhaps most importantly, the regular use of such problems helps students understand that mathematics is not just about “how” to solve things, but it is also about investigating “why” things happen.
Keywords: Geometry, Connections, Curriculum
Ref: LoriLa4
Author(s): Kane, Jill A.
Date: 1999
Title: A Book of Creative Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 9, pgs. 800-801
Reviewer: LoriLa
Date of Review: Feb. 29, 2000
At Indian Hills High School, the curriculum strongly enforces writing skills. Math class is no exception. Students are given journaling or creative-writing assignments in which they have the opportunity to sort through what they have learned, to expand on the ideas that have been presented in class, or to discover that they may not completely understand a concept. Teacher Jill A. Kane gives her students the task of putting their creative skills to work. Using Edwin Abbott's book Flatland as an introduction, she has them create their class Book of Creative Geometry. Students include their interpretations of geometry through computer graphics, comic strips, poems, and nursery rhyme illustrations, to name a few. All of these are made by the students and are "published" at the end of the year. Each student is given a copy as well as several administrators. The author hopes that this assignment will take the students beyond the traditional proofs, theorems, and problems of geometry to develop a sense of wonder and amazement at the relationships within mathematics and across disciplines. I personally feel that giving students creative projects in math class is a great way to grab their attention, while also providing an opportunity for higher-level thinking and learning. By giving students a creative project the teacher is tapping into hidden talents while students are making connections and discovering relationships. Chances are that the student will definitely remember what they learned and they will better understand the concepts with which they are working. This project also creates a sense of ownership with the material, as well as generating some class pride in learning Geometry.
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: 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
Ref: TracyA5
Author(s): Barnes, Sue
Date: 1996
Title: Perimeters, Patterns, and Pi
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol, 89, Issue 4, pages 284 - 288
Reviewer: TracyA
Date of Review: March 1, 2000
Some articles are very interesting and keep your attention, and others you can't wait to finish, if you even finish at all. For the most part, this was an article I couldn't wait to finish. Don't get me wrong, there were a few good parts.
Most of the article was centered on the students discovering pi and presenting their findings to the rest of the class. That was the part I didn't care for. The way they had the students pull together the information was useful. I liked how it showed them how to organize their information, and how to logically proceed. Those are useful skills life skills everyone needs.
Unless you need organizational assistance or are interested in reading about how to generate pi, I wouldn't read this article.
Keywords: Activities, Geometry, Measurement
Ref: TracyA6
Author(s): Iovinelli, Robert
Date: 1999
Title: Discovering Optimum Networks in Triangles
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, Issue 6, pages 534 - 539
Reviewer: TracyA
Date of Review: March 1, 2000
This article provides a wonderful classroom activity that introduces the students to graph theory. This is an area not usually covered in high school math course. I would high recommend this article and don't want to give too much away, but here is a brief and selective summary of the article.
Here is the basic problem. "Picture three cities, each 200 miles from the other two. Each pair of cities can be connected to the others by using a total of four hundred miles of fiber-optic cable"(page 534). The students need to figure out the best way to connect the cities, using as little cable as possible. First the students are introduced to the problem on paper. Then they use a spreadsheet to solve the problem. Lastly, they use the Geometer's Sketchpad computer program to find a solution.
This is an informative article and interesting to read. I really liked how the students used so many methods to solve the "basic problem", and it had a great real life application. No on in the class would ask, "where is the math in this?" or "when would I ever need to use this in my life?" You really need to take 15 minutes to read this article, it would be worth your time.
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: 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: Activities
Ref: AndreaA6
Author(s): Kelley, Paul
Date: 1999
Title: Build a Sierpinski Pyramid
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 5, pp. 384 - 386
Reviewer: AndreaA
Date of Review: 3/5/00
Students from Anoka High School built a nineteen-foot tall Sierpinski pyramid at the Minneapolis Convention Center for NCTM's 75th annual meeting in April 1997. The students began their unit on fractal geometry by looking at fractals that can be created by hand. They detailed the basic characteristics of most fractals - self similarity and iteration. The three fractals they looked at are called Cantor Dust, Purina Dog Chow, and Koch Snowflake. They all involve cutting the figure's sides into thirds and taking out the middle third. The class formed a Sierpinski pyramid by cutting out many templates of a pyramid from card stock. They assembled all of these into individual pyramids. They would then assemble four of these to make the start of a pyramid. Then take four of the new pyramids and assemble those into a larger pyramid. They continued this way until they had made a stage six pyramid. This contains 4096 of the original templates and is 224 inches tall. (Each template is 3 1/2 inches tall.) Before the class made this pyramid for NCTM, they had been perfecting their process for five years. Their first pyramid was more than nine feet tall but toppled the next day. They improved the process each year until the pyramid stayed until they decided to take it down. For this final pyramid, they needed a place with a ceiling at least 19 feet high and they used corner bead to reinforce the attachments. The article suggests having several classes work together on this to avoid taking too much class time. It says that to make a stage 5 pyramid, it takes about 10 - 12 47 minute class periods so if you can have 3-4 classes work on it, you can complete it in 3-4 days. I think I'd like to try this but maybe start with a stage three or four pyramid to get the process down.
Keywords: Activities, Games, Geometry
Ref: TinaM5
Author(s): Foshay, John D.; Wells, Wendy L.
Date: 1997
Title: "Table Tennis Anyone?" Using Ping-Pong to Teach the Coordinate Plane
Journal or Publisher: Mathematics Teacher online www.nctm.org/mt/1997/12/90.09.tabletennis.htm
Volume, Issue, Pages: Vol. 90, Num. 9
Reviewer: TinaM
Date of Review: March 5, 2000
John Foshay and Wendy Wells used the game ping-pong to teach their students about the coordinate plane. Foshay teaches first- year algebra and Wells teaches students with moderate mental retardation and helps them with various skills such as hand-eye coordination. They recognized that the Ping-Pong table could make the coordinate plane, the net could represent the x-axis and the center line could represent the y-axis. The rules of the game, as well as basic definitions of the coordinate plane, including the concept of ordered pairs were introduced to the students. The students were split into two teams. Although only four students could actually play the game at any given time, the other students (academic players) were matched up with an active player. Each time a point was scored an academic player placed a red dot on the approximate spot where the ball last hit. The teacher would then ask a question regarding the ordered pair. At first, if the team that scored the point answered incorrectly, the teacher would correct the response. The students, however, encouraged the change in the game so that if an incorrect response was given, the other team would have the opportunity to answer the question correctly and ‘steal’ the points from the other team. In this respect, the students had the opportunity to participate in the game as well as have input regarding rules of the game. At the end of the unit, students reflected on their experience. Many said that the method was enjoyable and that they could “see” the concept of coordinate geometry in the ping-pong table. I believe this was a very interesting lesson/activity for the students as well as the teachers. It allowed them to learn and have fun at the same time. It also appealed to the needs of the students with different learning styles.
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: Inquiry, Connections, Technology
Ref: MiriamN6
Author(s): Iovinelli, Robert
Date: 1999
Title: Discovering Optimum Networks in Triangles
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, Sept., 534-39
Reviewer: MiriamN
Date of Review: 3/6/00
The author describes an exploration activity in which the problem is posed to students: What is the shortest possible length of cable needed to link 3 cities that are at specified distances apart from one another, and what is the shape of this optimum network? First simple cases are demonstrated to the whole class: an isosceles triangle is shown, and students are introduced to the concept of minimum spanning trees (paths spanning the lengths of the shortest 2 sides of a triangle). Next students are asked whether a different type of connection, for example, a T-type junction (segment connecting 2 vertices, with another segment from the 3rd vertex to the middle of the first segment), might result in a shorter length of cable. Finally, students launch into the investigation, in which they attempt to discover the type of junction that will result in the minimum length of cable. Eventually they are led to the conclusion that a Steiner tree (Y-type junction with central angles of 120 degrees) results in the optimum network (true for triangles with all angles less than 120 degrees, which is what these students are given to work with). Students move from exploration with paper and rulers to computer spreadsheets, in which cable lengths are calculated using the distance formula. The author then describes how this activity was used in a teachers’ in-service workshop. The activity was generalized to finding optimum networks in arbitrary triangles, and the investigation was conducted using Sketchpad. Participants discovered that in triangles with all angles less than 120 degrees, Steiner trees are the optimum networks, whereas in triangles with an angle larger than 120 degrees, the optimum network is the minimum spanning tree.
This activity is a great way to introduce students to graph theory, which they normally wouldn’t see in traditional curricula. It can easily be embedded in the context of geometry, since it employs many geometric concepts, yet the use of the distance formula also provides connections to algebra. It is a rich investigation which could be further extended to finding optimum networks of different types of shapes, and it employs meaningful and appropriate use of technology. My opinion, however, is that the experience might be enhanced for high school students if they could conduct at least part of their investigation on Sketchpad. It seems an obvious medium for this activity, and I see no reason why high school students couldn’t use it just as teachers in the workshop did.
Keywords: Activities
Ref: TomD7
Author(s): Naylor, Michael
Date: 1999
Title: Exploring Fractals in the Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (4), 360 - 364
Reviewer: TomD
Date of Review: 03/04/2000
In this article, Michael Naylor has six investigations that he does/suggests to do with students to further investigate and explore fractals. The first investigation that he talks about in his article is fractal trees. The student starts with a tree trunk, and extends two branches off the trunk, then two more branches off the existing branches, then two more branches off each of the two new branches, and so on. During this activity the students have sheet in which they will determine the number of new branches and then the total number off branches for each row of branches. From their observations, the students must try to come up with the number of new and total branches after n rows. If they counted correctly they should come up formulas for the number of new branches (2n) and for the total number of branches (2n-1 - 1). Another investigation that Naylor discussed doing with his students was what he called the Sierpinski Carpet. In this activity the students will! first start with a square, then divide inside of the square into 9 congruent squares (tic-tac-toe fashion). Then the middle square is considered "removed", and the process is repeated for each of the 8 remaining squares, then the other eight remaining squares, and etc. The key discussions that come out of these activities is how does the area of the congruent squares change as n (the number of squares) increases [area = (1/9)n], and what pieces will never be removed from the square (it will be the edge).
I have very little information and/or experience with fractals and the formulas that come out of investigating different types of fractals. I found this article to be very intriguing because the activities that Naylor was doing in his classroom were both activities I could follow and understand (considering my lacking knowledge of fractals), and activities that the students would follow and be intrigued by. I also enjoyed how Naylor had in each of his investigations what formulas he wants his students to derive, some key discussion questions that would make the students think about what they were doing and some homework tips for the students so they could venture on their own after class.
Keywords: Geometry, Activities, Manipulatives
Ref: ChrisW5
Author(s): Smith, Lyle
Date: 1999
Title: Using Dragon Curves to Learn about Length and Area
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 4,
Reviewer: ChrisW
Date of Review: 26 February 2000
Smith writes about using dragon curves to help introduce and reinforce middle school students to the concept of pi, circumference, and area. The dragon curves that Smith talks about are not derived from the shape and movement of mythical animals, but, rather are sets of square cards (typically 2x2) where arcs of circles connect the midpoints of two adjacent sides of the card. There are three variations of cards: single arc, double arc, and blank. Using these cards, students are able to create a variety of shapes. Once students have created their shapes, the challenge is to calculate the perimeter of the shape and the area which is enclosed by the shape. Smith suggests that teachers can help their students discover the length of the arc on one card and the area on either side of that arc by having their students initially construct a circle of radius 1 out of four single arc cards. With careful questioning, and the use of the circumference and area formulas for circles, teachers can help their students discover important information about the arc length and separate areas on their dragon curve cards. Once students have discovered these pieces of information about the dragon curve cards, they are able to determine the perimeter and enclosed area of their more interesting shapes. This activity provides students with an opportunity to use newly learned information about circles and pi in a hands-on and mildly creative way. It seems like this activity could be used to help students to better understand pi and the area and circumference formulas for circles.
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: Connections, Geometry,
Ref: MiriamN7
Author(s): Little, Catherine
Date: 1999
Title: Geometry Projects Linking Mathematics, Literacy, Art, and Technology
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 4, February, 332-335
Reviewer: MiriamN
Date of Review: 3/8/00
For her eighth-grade geometry unit, the author developed an extended 6-week project that draws upon the students’ interests, creativity, and connections between geometry and other subjects. Students were offered a choice of one of three possible projects: 1) writing a manual with instructions and diagrams on how to do some basic constructions using Geometer’s Sketchpad; 2) creating a children’s picture book, in which one of the main characters learns a geometry concept as part of the plot; and 3) writing a short report on Escher’s contribution to mathematics and creating an Escher-style tesselation. A simple rubric based on a 0-10 scale was devised with three scoring criteria per project (shown in article; criteria deal mainly with aspects of the presentation rather than quality or accuracy of the mathematical content). The rubric was provided to students at the beginning of the unit along with the project descriptions. The project was assigned to four classes, including one for gifted students and one for learning-disabled students, and the results for all classes were very positive. Most students chose the picture book, but the results of all three projects were highly imaginative (several samples are presented). Some students expressed afterwards that the assignment had increased their interest in the subject of geometry. In my opinion this is the most important outcome of the project. By allowing students to discover creative connections between mathematics and other areas of interest and talent, the subject will come alive for them. As the author puts it, “a good project can help us to let [their] potential shine.” I would definitely consider assigning this type of project for a middle-school geometry class. Similar extended projects could also be used for high-schoolers, but the project options must appeal to that age group - perhaps they might perceive a children’s picturebook as too babyish, for example. For the high school level, I also believe the rubric should contain more explicit criteria for evaluating mathematical content than this one does.
Keywords: Technology, Geometry,
Ref: AndreaB6
Author(s): Purdy, David C.
Date: 2000
Title: Using The Geometer's Sketchpad to Visualize Maximum-Volume Problems
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, Issue 3, p. 224-228
Reviewer: AndreaB
Date of Review: 3/5/00
This article shows how to explore the Maximum-Volume problem for a box made from a square piece of paper using Geometer's Sketchpad. It can be used in conjunction with a graphing calculator and actual construction of boxes. It is not recommended as a stand-alone activity.
I really liked the way this introduced an interactive aspect to the problem of Maximum-Volume problems. This is also gives a new way to solve an old problem using Geometer's Sketchpad.
Keywords: Activities, Games, Geometry
Ref: TracyA7
Author(s): Woodward, Ernst and Hamel, Thomas
Date: 1992
Title: Polydron, Activities in Two- and Three- Dimensional Geometry
Journal or Publisher: J. Weston Walsh
Volume, Issue, Pages:
Reviewer: TracyA
Date of Review: March 11, 2000
Every math teacher needs to have this learning tool, even if they are not teaching geometry. Polydrom Activities in Two- and Three- Dimensional Geometry comes with an activity/Lesson booklet, a book of teacher notes, and a bag of interlocking triangles, squares, and pentagons in different sizes. This activity sets offers a wonderful way to teach Tesselations, Polyominoes, Nets, Pyramids and Prisms, and Polyhedras just to name a few.
Most students enjoy learning through hands on activities and discovery. Who wouldn't like to do a fun lesson, play with different interlocking pieces, and create cool shapes? By using activities like these in the classroom, math class will be fun and interesting for everyone present.
Keywords: Activities, Technology,
Ref: StaceyS4
Author(s): Reinstein,D.; Sally,P.;Camp,D.R.
Date: 2000
Title: Generating Fractals through Self-Replication
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90(1); 34-38,43-45
Reviewer: StaceyS
Date of Review: 3/10/00
This article discusses a hands-on activity of fractal geometry, "geometry of nature." The basis of this geometry is the limiting result of an infinite number of self-replications. Students are given 3 activity sheets and a link to a TI-82 program. Through hands-on experience, technology, and geometric visualization, the students can explore fractal geometry in a cooperative classroom. In order that this activity run smoothly, students must have prior knowledge of series and sequences. Otherwise, the teacher can give some examples before the investigation takes place.
I liked this article because it provides the worksheets and a clear insight to an investigation of fractals. This would be a great break from the traditional geometry book or even an integrated curriculum to explore a topic that most students do not explore until college. This activity is a wonderful way to incorporate discovery, creativity, technology, and the desire to find mathematical connections.
Keywords: Resoning, Connections,
Ref: StaceyS5
Author(s): Perham,A.E.; Perham, B.A.; Perham,F.L.
Date: 1997
Title: Creating a Learning Environment for Geometric Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90(7); 521-526
Reviewer: StaceyS
Date of Review: 3/10/00
This article describes how students in a 10th grade geometry class discovered relationships that led to the development of conjectures, theorems, and directions of proofs regarding the centroid of a triangle. This is accomplished through 3 different means: manipulative experiences, software, and the graphing calculator. The author explains the benefits of each method and what can be accomplished.
I really enjoyed this article because of the clear analysis of two related theorems and how students can investigate its properties through a paper-pencil method, Geometer's Sketchpad and Mathcad, and finally, the graphing calculator. Of course, a teacher would not have enough class time to use all of these learning methods for each topic, but I think that by varying the investigation and conjecturing of theorems and postulates should be done with each at one time or another.
Keywords: Technology, Problem Solving,
Ref: StaceyS6
Author(s): Purdy,D.C.
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(3); 224-228
Reviewer: StaceyS
Date of Review: 3/10/00
This article discusses integrating various areas of mathematics into traditional courses. Technology including graphing calculators and the Geometer's Sketchpad can further a student's investigation and allows them to start working with problems involving advanced algebra, pre-calculus, and calculus. The problem posed by the author is the "Maximum-Volume Box Problem." The objective is to find the largest possible volume of a box constructed from a 10 in-square piece of flat metal. The stipulation is that the equal squares must be cut out of the 4 corners of the original metal. This can be explored through paper models, organized charts, graphing calculator, and the Geometer's Sketchpad in order to discover a pattern for solving the problem. Once solved, students can generalize this problem to a rectangular sheet.
I think this article has some great ideas for exploring patterns and utilizing technology. However, I am trying to imagine this in one of the classrooms I have done a practicum at and I cannot see where one could find the time to complete an activity such as this one. Even though the author thought this would be appropriate for students in grades 9-12, I think it would work best in a pre-calculus or calculus class.
Keywords: Problem Solving
Ref: StaceyS7
Author(s): Gannon,G.E.; Martelli,M.U.
Date: 2000
Title: The Prisoner Problem-A Generalization
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(3); 192-193
Reviewer: StaceyS
Date of Review: 3/10/00
This article gives a great approach to the discovery of the "Prisoner Problem" and then how to generalize the problem. In the initial problem there are 3 prisoners and each wears one of 3 white hats or 2 black hats. The prisoners are in a line facing the wall and if they know (not guess) the color of their hat, they can go free. The prisoner furthest from the wall removes his blindfold and can see the two prisoners in front of him. The middles prisoner can only see the prisoner closest to the wall and the last prisoner cannot see either of the other two. Once the student solves this problem, they can formulate a problem involving 4 prisoners. This can lead to a generalization of n prisoners. The authors recommend this problem as a way to emphasize to students the final step in a problem solver's kit - considering possible generalizations when a particular problem has been solved.
I loved this article! This is a fun way to investigate patterns and could work at any level in high school getting more involved as the student gets older.
Keywords: Geometry, Activities,
Ref: RyanV4
Author(s): Lege, Steve
Date: 1999
Title: Why Not Three Dimensions?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 7, pp. 560-563
Reviewer: RyanV
Date of Review: 3/12/00
This is an article addressing the topic of studying three dimensions in high school geometry courses. He begins by discussing how poorly his high school curriculum (among others) addressess this concept. Then he begins talking about why he feels three dimensional geometry should be studied. He feels too many students have trouble understanding rotations that produce solids (among some other things). Because of this, he focuses the rest of the article on rotations and solids.
After his introduction he proceeds to give some good examples of lesson ideas for three dimensional geometry topics. He suggests using three wooden skewers (like those used for barbecues) to represent the axes. From there students are given instructions to create certain solids using only 3x5 inch and 4x6 inch cards, scissors, tape, and the axes (skewers). Each group presented their project to the class and hung them on the ceiling for easy reference. Many students said they enjoyed this activity and felt more confident in their visualizations of complex figures.
Keywords: Activities, Manipulatives,
Ref: RyanV5
Author(s): Naylor, Michael
Date: 1999
Title: The Amazing Octacube
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 2, pp. 102-104
Reviewer: RyanV
Date of Review: 3/12/00
This article deals with some great models that can be made for a math classroom by students. First, he starts with a small discussion about polyhedral duals and what they are. He also gives some good examples as well as visual models of what they look like. From here he moves into a discussion of quasi-regular polyhedra. Quazi-regular polyhedra are formed if corners of regular polyhedra are truncated through the midpoints of the edges. Then he talks about some interesting figures that occur as a result of these formations. Finally, he concludes the article with a section entitled "Building the amazing octacube."
In this final section he lays out the plans for constructing an octacube. He also tries helping the reader visualize this process in a very interesting and creative way. With the great diagrams, many pictures, and detailed directions, I found this article very intriguing. It is also a project I plan to try with my future classroom.
Keywords: Technology, Problem Solving, Activities
Ref: RyanV6
Author(s): Purdy, David C.
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 3, pp. 224-228
Reviewer: RyanV
Date of Review: 3/12/00
This is an excellent article dealing with a famous problem and the Geometer's Skethpad. The problem: a manufacturer wants to construct a box with the largest possible volume from a piece of flat metal (of some given dimension). The author begins this article discussing this problem and how he incorporated it into his teaching. Then after the lesson was taught, with some students still having visualization problems, he decided to explore this problem on the Geometer's Sketchpad. From there he continued with a very detailed description on how to use the Sketchpad for this activity.
His description requires some knowlede of Geometer's Sketchpad, but it follows very easily. Also, a section for the general analytical solution to this problem was given as well as an informal proof (which requires knowledge of calculus). He concludes this article with a discussion about the outlying implications of this project for students. The connections to the real world, the visualization aspect, and the mathematical content are all right on line with the Standards. This is an article perfect for any geometry teacher as well as a great application of technology.
Keywords: Geometry, Teaching Strategies,
Ref: LeifN6
Author(s): Chappell, Michaele; Thompson, Denisse
Date: Sept. 1999
Title: Perimeter or Area? Which is it?
Journal or Publisher: Teaching Mathematics in the Middle School
Volume, Issue, Pages: Vol. 5, No. 1, pg 20-23
Reviewer: LeifN
Date of Review: 3/13/00
As the title suggest this article is about Perimeter and Area. It talks about how area and perimeter are often taught to middle school kids and the problems that it causes. I found the article to be a very informative and well written one.
One of the main points of the article is that students don’t learn the concepts of area and perimeter because, “All too often, a fundamental understanding of these ideas is sacrificed while students learn the general formulas (pg. 20).” The article brings up many of the common misunderstandings that middle school students have about these topics. To demonstrate this that they conducted an action project. The majority of the article summarize the results of the project and talking about the why a student may have that misunderstanding.
I found the article to be a good help for me. It reminded me of something that I had thought to much about lately, to often we just hand students the formulas instead of actually teaching a concept. This is something that we really need to watch out for as teachers.
Keywords: Geometry, Algebra, Problem Solving
Ref: LeifN7
Author(s): Edward, Thomas
Date: March 2000
Title: Pythagorean Triples Served for Dessert
Journal or Publisher: Teaching Mathematics in the Middle School
Volume, Issue, Pages: Vol. 5, No. 7, pg 420-423
Reviewer: LeifN
Date of Review: 3/13/00
This as a very thorough article that talks about some interesting characteristics of Pythagorean triples. It is a response to an article, “Pythagorean Triples Served for Supper”, the was published in a 1997 Teaching Mathematics in the Middle School. This article talks about how answering three questions posed at the end of the original article.
The article explains how the table function of the TI-83 to investigate and answer the questions. The students can use algebra to set up the equations needed, plug those equations into the graphing calculator, and then examine the table to discover the answers to the problems posed.
This is a good activity to use because it integrates a geometry and algebra, use the technology available, is investigative and discovery driven, and it develops problem solving skills.
The last thing I really liked about this article was the it told the reader how the activity fit into the standards and how it can be used to satisfy some of the standards.
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.
Keywords: Technology
Ref: AndreaA5
Author(s): Foletta, Gina M.; Leep, David B.
Date: 2000
Title: Isoperimetric Quadrilaterals: Mathematical Reasoning with Technology
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 2, pp. 144-147
Reviewer: AndreaA
Date of Review: 3/14/00
A group of secondary math teachers got together to work with a problem they thought could be used with secondary school students as an investigation. They investigated the problem How does change in shape affect the area of the interior region of parallelograms with the same perimeter. Using Geometer’s Sketchpad, the teachers varied the length of one side and conjectured that among rectangles with equal perimeter, the square had the greatest area. With that restriction, the problem was easy to solve. They then expanded the problem to investigating nonrectangular parallelograms and then quadrilaterals in general. I think this might be an interesting demonstration or an activity to do as a class. However, I think it would be too confusing in the general case for the typical student.
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."
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
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, 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 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, 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: Geometry
Ref: LizA4
Author(s): Iovinelli, Robert
Date: 1999
Title: Discovering Optimum Networks in Triangles
Journal or Publisher: Mathematics Teacher, September
Volume, Issue, Pages: Vol. 92, No. 6, pages 534-539
Reviewer: LizA
Date of Review: 3-19-2000
This article gives an example of an activity in an area of mathematics I am less familiar, graph theory. The activity asks students to find the most efficient way to connect three points. The article introduces vocabulary. For example if you consider the triangle that would connect the three points, two legs of this triangle would connect all three points and is called a spanning tree. If you choose the two shorter legs, this is called the minimum spanning tree. But, students quickly discover that in many triangles this is not the most efficient way to connect the three points. They then explore connecting two of the points and then connecting the midpoint of this segment to the third point. Finally they explore different points on the median line they created in the step before. After this, I am not totally clear why, they begin similar experimentation with different equilateral triangles and discover what is called a Steiner point. The Steiner point is the! intersection of the perpendicular bisectors from each side of the triangle. Connecting this point to the three points appears to give the most efficient way to connect the points for an equilateral triangle.
The activity is conducted as a guided exploration lesson. One weakness I observed was that although the author introduces the problem as a model of a real-life situation, I thought that the real-life part could be more developed. A strength of the article is that the author explains how technology could be incorporated into the lesson and offers many extensions. I, personally, would have to study a little more about graph theory before feeling comfortable teaching this lesson.
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.
Keywords: Geometry, Problem Solving,
Ref: LizA5
Author(s): Jones, Graham A., Thronton, Carol A., McGehe, Carol A, Colba, David
Date: November/December 1995
Title: Rich Problems - Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 1, No. 7
Reviewer: LizA
Date of Review: 3-19-2000
In this article, the author offers an example of a rich problem and how it developed into a learning experience in his classroom. The problem has to do with an architect who is designing a hotel. Each room in the hotel opens onto a walkway overlooking a central atrium, which is rectangular in shape. The design includes a brass railing around the edges of the overlook. But, brass is expensive and the architect can only afford 650 feet of railing around each floor. The question is, what should the dimensions be to maximize the area of view to the atrium below? The middle school students begin this problem by just trying different rectangles with a perimeter of 650 and looking for the largest area. Different students notice patterns and one student even makes a connection to a problem they did the day before. What the students discover is that a square with a set perimeter gives the maximum area. The article is also helpful because it includes extensions and how to integrate technology in the form of graphing calculators or excel. One strength of the problem is that it is based on a real problem that had arisen when an architect designed an actual hotel. The author points out that the problem is also rich because it leads to many extensions, there are many different solutions and it may cause students to wonder what generalizations can be made about using a square in architecture.
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.
Keywords: Discrete, Technology,
Ref: LizA6
Author(s): Iovinelli, Robert C.
Date: February 2000
Title: Chaotic Behavior int eh Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 2, pages 148-152
Reviewer: LizA
Date of Review: 3-20-2000
This article presents an activity that helps students to begin to see the nature of chaos. It begins with students thinking of chaos as frenzy or disorder. This activity helped students recognize that the study of chaotic behavior begins with a well-defined system that is sensitive to initial conditions and that iteration in this system may not lead to uniform results.
The author first begins by explaining why the function y=ax(1-x) is a function that could be used to represent population growth. He then asks the students to assign a value of 2.5 to a and begin with an x value of .02. The students then use the outcome as the new x value and continue to follow this pattern to create an iterated function. Students discover that this function eventually approaches .60 and becomes constant. The teacher then discusses how these numbers relate to the environment's carrying capacity. The next step is that students try different starting values on an activity sheet prepared by the teacher. With the next starting value, they find this sequence approaches and becomes constant at .65. Students will think they are beginning to see a pattern. But, when students try the function with a=3.3 the function approaches two different points and oscillates between them. Then when students try a=3.54 the function oscillates between four points. Finally, w! hen students try a= 3.9 they find that the points appear to have no pattern at all. The whole activity has lead up to this graph, which is an example of a chaotic model of growth.
Strengths of this activity are that the author seems to have a deep understanding of chaos and population growth and gives clear explanations of how the mathematics relates to real life. He also gives clear instructions of how to enter iterated functions into the calculator. I believe this would be an excellent activity that could be used while students are studying population growth.
Keywords: Geometry, Manipulatives, Activities
Ref: LizA7
Author(s): Hopley, Ronald B.
Date: May 1994
Title: Nested Platonic Solids: A Class Project in Solid Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 87, Number 5, pages 312-318
Reviewer: LizA
Date of Review: 3-21-2000
This project helps students learn the five Platonic solids, create 2D nets for the 3D solids, and use trigonometry to find how the solids fit together. The activity can take up to a week to complete. Students begin by making a tetrahedron. They then create a cube that will fit inside the tetrahedron, an octahedron that will fit around the tetrahedron, a dodecahedron that fits around the octahedron, and finally an icosahedron that fits around the dodecahedron. The author has started a chart of how these shapes could fit together different ways and challenges the reader to complete the table.
I liked this article for many reasons. One reason is that manipulatives used to learn about polyhedra can be very expensive. The activity described is a lot less expensive and students will be able to keep their finished product. The author also includes clear instructions on how to make flaps so the shapes can open and close. Another reason I like the activity is because it involves good use of trigonometry and even includes the golden ratio. Also, there are many more extensions a teacher could do in this area like investigate what polygons can form platonic solids, consider why there are only five platonic solids, discover Euler's formula, or investigate Archimedean solids. Since this activity takes so much time it may be difficult to add into an established curriculum. It could be done in a math club or as an enrichment activity if the class has time.
Keywords: Proof, Geometry, Problem Solving
Ref: AndreaB7
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, Issue 7, p. 565-567
Reviewer: AndreaB
Date of Review: 03-12-00
This article describes one teacher’s method to teaching proofs. He started with a cooperative learning idea, and tailored it to fit what he wanted. Although his grouping method does not fit with the heterogeneous groups called for by cooperative learning, it seems to work well in his class. In this class students describe working on proofs as challenging, intense, exciting, and fun.
This method could do wonders for many geometry classes. Making math fun yet challenging is a great goal. In a class of my own, I would like to see if this approach works better with the heterogeneous groups recommended by cooperative learning leaders such as Professor Johnson.
Keywords: Connections, Assessment,
Ref: LoriLa5
Author(s): Williams, Nancy B. and Wynn, Bryan D.
Date: 2000
Title: Sharing Teaching Ideas: Journaling in the Mathematics Classroom: A Beginner's Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 2,pgs. 132 - 5
Reviewer: LoriLa
Date of Review: March 22, 2000
Many of today's mathematics educators are incorporating journal writing into their classrooms to break away from the traditional assessment approach of tests, quizzes, and worksheets. The authors of this article decided to test the waters for journaling in their classrooms and share their experiences with Mathematics Teacher readers. The authors each launched their experiments with one class. They specifically chose morning classes where the students generally held a positive attitude towards math. In the beginning, two journal assignments were given per week. One of the assignments was an affective entry, asking general questions about education, and the other was a mathematical entry, asking students to communicate their understandings of the topics in written, paragraph form. One teacher gave five minutes for journaling at the beginning of class, while the other gave ten minutes. Grading was based on a specific rubric where half of the grade was on effort and the other half on mathematical content. Initially, the authors received numerous complaints about writing for math class; however, the teachers were gaining insight into the comprehension of their students. Later, they decided to cut down the workload for the students and themselves by assigning one journal entry per week. They also decided they would give students ten minutes to work on their entries in class, since students dislike being interrupted in the middle of an assignment. By the end of their experiment, students didn't complain as much about the writing. In fact, students said that the journal entries helped their grades, they learned how to communicate mathematically more effectively, and the feedback helped them to understand the concepts. The teachers decided that the experiment went well, since the students benefited from the journal entries and they were better able to have insight into students understanding through the journal assignments. They decided they would continue this practice, but make a few suggestions: 1) Start small. 2) Only assign one journal entry per week. 3) Ask both affective and mathematical questions for variety. 4) Give students class time to work on the journals. In my personal opinion, I think that journal writing is an asset to the mathematics classroom. It enhances students ability to communicate mathematical concepts effectively and also allows them to practice their writing skills. I also think that journal writing gives a teacher immediate feedback to see whether or not the students fully understand the concepts before the teacher forges on to the next topic. I also like the authors suggestions that teachers should start small and only assign journal entries once a week. In that way, students are given some variety (without the assignment of journal writing becoming too monotonous.) Also, this gives teachers enough feedback without breaking their backs correcting journal assignments in addition to all the other work.
Keywords: Connections, Activities, Geometry
Ref: LoriLa6
Author(s): Kelley, Paul
Date: 1999
Title: Build a Sierpinski Pyramid
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol.92, No. 5, pgs. 384 - 6
Reviewer: LoriLa
Date of Review: March 23, 2000
A Sierpinski triangle is made by taking a shaded equilateral triangle and connecting the midpoints of each side to form four equilateral triangles. Then, remove the inner triangle. Repeat this process for several iterations on the new triangles. Students at Anoka High School built a Sierpinski Pyramid in conjunction with the NCTM's 75th Annual Meeting April 1997 as an extension of their work on fractal geometry. The pyramid was a culmination of the work done over a period of five years. After a few trials, they came up with a successful design, which they planned to use for next year's pyramid, which would be twice as tall. That pyramid materialized at the Minneapolis Convention Center in April, reaching a height of nineteen feet. The pyramid took eight and a half hours to construct by 30 students. The article gives instructions as to how to construct a Sierpinski Pyramid of the same nature as the nineteen-foot phenomenon constructed by Anoka High School students. (The height of the pyramid depends on the size of the template made for constructing the pyramids.) Basic materials needed are a template as shown in the article, sturdy paper, scissors, tape, and glue. (Wooden or metal reinforcements may be necessary, depending on the height of the pyramid.) I think this is an excellent display of ingenuity and problem-solving. I am impressed by the work of the Anoka High School students. Its marvelous to see students displaying their mathematical knowledge outside of the classroom. This is an activity I may want to take on with some of my math classes; however, I would probably construct it on a smaller scale!
Keywords: Connections, Geometry,
Ref: LoriLa7
Author(s): Wenninger, Magnus J.
Date: 1966
Title: Polyhedron Models for the Classroom
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages:
Reviewer: LoriLa
Date of Review: March 23, 2000
I reviewed the book entitled "Polyhedron Models for the Classroom." It is an excellent source for incorporating polyhedron models into your classroom. They are aesthetically pleasing and encourage a connection to the world of art. The also aide in acquiring skills in visualization. Students become motivated to work on math because they are constructing something that they find intriguing. Students tend to create polyhedrons with care and accuracy. Polyhedron models are also a helpful tool for discussing symmetry and congruency, not to mention adding some pizzazz to your institutional classroom. The book outlines how to create The Five Platonic Solids, The thirteen Archimedean Solids, Prisms, Antiprisms, and Other Polyhedra, The Four Kepler-Poinsot Solids, Other Stellations or Compounds, and Some Other Uniform Polyhedra. It also provides templates for some of the models, as well as outlining color schemes for your polyhedron models. Many students created polyhedron models in my high school, but I was never introduced to the concept of constructing one, or even what it was. They have always intrigued me, so therefore I believe the other when we says that polyhedron models are motivating and challenge students to take care in the accuracy of their work. I also think students will be motivated to design their own polyhedron models once they are given the background on how to construct such a fascinating geometrical shape.
Keywords: Connections, Teaching Strategies, Technology
Ref: TinaM6
Author(s): Purdy, David
Date: 2000
Title: Using the Geometer's Sketchpad to Visualize Maximum-Volume Problems
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93 No. 3
Reviewer: TinaM
Date of Review: March 22, 2000
Maximum-volume problems are common in advanced algebra, precalculus and calculus classes, however, recently they have found themselves in high school geometry texts such as Discovering Geometry: An Inductive Approach. This article highlights this problem as well as several approaches to solving it. The students start out by actually cutting out paper boxes to model the problem. They created charts to organize and display the data. They recognized patterns in their data and were able to come up with a volume function. They finally graphed the function and came up with the maximum size for their cutouts. This problem can also be looked at using The Geometer’s Sketchpad. This tool can either be used by the teacher to demonstrate the construction or the students can use the tool to perform the constructions themselves. The advantage of using such a tool is that the students can vary the dimensions of the construction and see the affects of these changes on the maximum volume. They can then test these results using their graphing calculators. “One hallmark of a good problem is that it defies instant and complete solution yet yilds parts of its solution over time.” (Purdy, March 2000, p. 228) Purdy points out that this problem can be extended over many years, each time tackling a different aspect of the problem. For instance, algebra geometry students can tackle the problem as described in the article; calculus students can use the derivative to solve this same problem. Such a problem, if approached by integration over time, will allow students to see strong connections among mathematics topics including algebra, geometry, and calculus. The use of technology only enhances their learning process.
Keywords: Geometry
Ref: LukeB6
Author(s): Quinn, Anne
Date: 1997
Title: Using Dynamic Geometry Software to Teach Graph Theory: Isomorphic, Bipartite, and Planar Graphs
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90, 4, 328-332
Reviewer: LukeB
Date of Review: 3/23/00
I thought this was a good article for teachers planning on teaching graph theory. The author talks about isomorphic graphs, bipartite graphs, planar graphs, and nonplanar graphs. She gives definitions of each and gives examples of each type of graph. She then describes ways to use technology, such as the Sketchpad, to help students draw the graphs. This is a lot easier than drawing them out by hand. It helps students visualize the problems that she poses. She provides several examples of problems that she gives to her students. The use of the Sketchpad gives students a better insight into proving if a graph is planar or not. They can use the functions of the Sketchpad to hide parts of the graph or to move different points around. This use of the Sketchpad is also consistent with what the NCTM Curriculum and Evaluation Standards are trying to implement.
Keywords: Geometry, History,
Ref: LukeB7
Author(s): Gardner, Martin
Date: 1981
Title: Mathematical Games
Journal or Publisher: Scientific American
Volume, Issue, Pages: October, 23-30
Reviewer: LukeB
Date of Review: 3/23/00
I thought this was a good article about the history of Euclid's parallel postulate. This postulate says that through a point on a plane, not on a given straight line, only one line is parallel to the given line. Euclid himself could not prove it. He had to assume it was true. Many people thought that Euclid had to be right. Mathematicians have been trying to prove this theorem for hundreds of years. Many mathematicians have come up with proofs only to have them discarded because they assumed something that could only be proven by the parallel postulate. In trying to discover a proof, several mathematicians discovered new geometries. Instead of proving the theorem, a few mathematicians went a whole different way and assumed that there were an infinite number of parallel lines through the point. In this way they discovered hyperbolic geometry. The other geometry discovered was created in a similar manner. These mathematicians assumed that there are no parallel li! nes through the point. This was called elliptic geometry. There were arguments in favor of each geometry, but each geometry is equally "true" in the abstract.
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.
Keywords: Teaching Strategies, Research, Calculus
Ref: EoinO4
Author(s): Artigue, Michele
Date: 1999
Title: The Teaching and Learning of Mathematics at the University Level
Journal or Publisher: Notices of the American Mathematical Society
Volume, Issue, Pages: Vol. 46, No. 11, pp. 1377-1385
Reviewer: EoinO
Date of Review: 3/24/00
This is an article dealing with the issues that are the focus of mathematics education research at the college level. It is interesting in that the problems that are brought up in this article are the very same that are being looked at in k-12 mathematical education research.
They talk about the students moving through different levels of conceptualization. Though not directly tied to the Van Hiele levels (which should be expected due to the abstract nature of the concept) there is a strong connection. It is admitted that calculus being taught at the high school level cannot be taught from a completely formal approach because the students are not conceptually ready for that approach. Instead they rely on a dynamic concept of the limit (often rely on these dynamics too much, which is why you often see students plugging in numbers to find limits). For these reasons epsilon-delta proofs are often ignored or skimmed over. It is important to be aware of the level of conceptualization that the students are at, both to help them learn new material and to help them reach a deeper level of understanding.
Also discussed is the importance of integrating new knowledge with previous knowledge. In calculus this is tying new concepts to ideas learned in algebra and geometry. Research shows that most of the difficulties in learning calculus arise from failure to make these connections. From how the concept of the tangent relates to the derivative to how the Arclength integral is just related to the pythagorean theorem, the connections are what make calculus not only memorable but also possible.
The article goes on to discuss how it is important to have flexibility in both language and reasoning. This section is not as connected to the ideas that we covered in class, but it is discussing that we think about the same ideas in different ways in different situations. For example, the author cites research that found three different prominent modes of reasoning, "synthetic-geometric", "analytic-arithmetic", and "analytic-structural". It discussed that these modes of reasoning are used in defferent ways. Now while this research is not as relevent to the typical high school math teacher, it is important to recognise that there are a variety of approaches and that students may be more comfortable with some methods.
It was interesting to see how the issues that arose in an article about college math education were very similar, and in some cases identical to those dealt with in high school math education.
Keywords: Assessment, Teaching Strategies, Problem Solving
Ref: EoinO5
Author(s): Kroll, Diana; Masingila, Joanne; Mau, Sue
Date: 1992
Title: Grading Cooperative Problem Solving
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Col. 95, No. 8, pp. 619-627
Reviewer: EoinO
Date of Review: 4/24/00
Group work is becoming more and more prevelant these days as its powers in developing problem solving skills are being more widely recognised. One of the inherent problems with group work is how should it be assessed. Too often this problem arises because the teacher isn't assessing the work at all. But even when it is being assessed, there are the problems of whether this assessment is fair, efficient, and sufficient. This article provides some helpful suggestions about how to approach this issue. The first point that they are careful to make is "that assessment and grading are not synonymous." In pointing this out, the authors are trying to point out that there are many other reasons that a teacher may want to assess a cooperative learning situation other than assigning grades. They may want to decide on the direction the class will take in the future, both in terms of content and methods of intruction; they may want to use the assessment to make a decision or change in the classroom climate; or they may want to use the assessment to highlight an important concept. This said, the bulk of the article deals with assessment as a near synonym for grading. After providing a background by describing different types of cooperative learning, ther article returns to its main topic of assessment. The article suggests that an analytic grading approach be taken (numerical rubric) grade the groups work. To assess the individuals' work the authors suggest three individual questions that: assess understanding, parallel the group problem, extend from the group problem. The authors then go on to discuss the relative merits of giving the whole group the same grade or not. They did not have a definitive answer, but they felt it was still important to assess the individuals. They then go onto to discuss the importance of how you select problems to assess. They feel that it is important to make sure that it is complex enough to generate discussion, but that does not mean that the amount of mathematical difficulty needs to be increased. They also feel that it important that several approaches to solving the problem are avaiable. For the parallel problems for assessing the individual the suggest that the problem be modified by changing the context, "changing the numbers", "reversing given and wanted information", or a combination of the preceding. The article goes on to suggest some ways to organise rubrics, concentrating on understanding, planning a solution, and finding an answer. Their rubrics are very standard (though maybe, not quite so standard in 1992). The article then poses a number of questions posed by the authors that they feel that a teacher should consider in designing assessments. These deal with such ideas as, should you give different problems to different classes? "How should points be distributed?" and Will the layout of the class affect the assessment? Some of these questions seem obvious, but they are ones that need to be taken into consideration. Overall, I would say that the article provided little new information (again, possibly related to the age of the article). But many of the problems and questions that they discussed, though simple and somewhat obvious, are often overlooked.
Keywords: Assessment, Teaching Strategies, Problem Solving
Ref: EoinO5
Author(s): Kroll, Diana; Masingila, Joanne; Mau, Sue
Date: 1992
Title: Grading Cooperative Problem Solving
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Col. 95, No. 8, pp. 619-627
Reviewer: EoinO
Date of Review: 4/24/00
Group work is becoming more and more prevelant these days as its powers in developing problem solving skills are being more widely recognised. One of the inherent problems with group work is how should it be assessed. Too often this problem arises because the teacher isn't assessing the work at all. But even when it is being assessed, there are the problems of whether this assessment is fair, efficient, and sufficient. This article provides some helpful suggestions about how to approach this issue.
The first point that they are careful to make is "that assessment and grading are not synonymous." In pointing this out, the authors are trying to point out that there are many other reasons that a teacher may want to assess a cooperative learning situation other than assigning grades. They may want to decide on the direction the class will take in the future, both in terms of content and methods of intruction; they may want to use the assessment to make a decision or change in the classroom climate; or they may want to use the assessment to highlight an important concept. This said, the bulk of the article deals with assessment as a near synonym for grading.
After providing a background by describing different types of cooperative learning, ther article returns to its main topic of assessment.
The article suggests that an analytic grading approach be taken (numerical rubric) grade the groups work. To assess the individuals' work the authors suggest three individual questions that: assess understanding, parallel the group problem, extend from the group problem. The authors then go on to discuss the relative merits of giving the whole group the same grade or not. They did not have a definitive answer, but they felt it was still important to assess the individuals.
They then go onto to discuss the importance of how you select problems to assess. They feel that it is important to make sure that it is complex enough to generate discussion, but that does not mean that the amount of mathematical difficulty needs to be increased. They also feel that it important that several approaches to solving the problem are avaiable. For the parallel problems for assessing the individual the suggest that the problem be modified by changing the context, "changing the numbers", "reversing given and wanted information", or a combination of the preceding.
The article goes on to suggest some ways to organise rubrics, concentrating on understanding, planning a solution, and finding an answer. Their rubrics are very standard (though maybe, not quite so standard in 1992).
The article then poses a number of questions posed by the authors that they feel that a teacher should consider in designing assessments. These deal with such ideas as, should you give different problems to different classes? "How should points be distributed?" and Will the layout of the class affect the assessment? Some of these questions seem obvious, but they are ones that need to be taken into consideration.
Overall, I would say that the article provided little new information (again, possibly related to the age of the article). But many of the problems and questions that they discussed, though simple and somewhat obvious, are often overlooked.
Keywords: Geometry, Teaching Strategies, Activities
Ref: EoinO6
Author(s): Brodkey, Joseph
Date: 1996
Title: Starting a Euclid Club
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 5, 386-388
Reviewer: EoinO
Date of Review: 3/24/00
In this article, the author describes a Euclid Club. He tells how it got started, what it entails, and what the benefits of the club are.
The teacher decided to start this club when in a summer program at Saint John's in New Mexico he had to participate in a class studying Plato and Euclid and in studying Euclid they tried to prove the 48 of Euclid's 465 propositions.
He then returned to his high school where he started a club that had as its goal to run through all of the postulates. The format of the club is that people volunteer to present proofs of the postulates while the rest of the group questions to clarify, suggest alternatives, and to provide help. Between presentations, there are discussions about approaches, alternatives, and objectives. The structure of the club was that there was a leader (at the start this was the author) among peers, with everyone (students and teachers) on equal footing (and confusion).
The benefits to the students included furthering their understanding of geometry, enhancing their communication and discussion skills, and they were exposed to sophisticated logic in the proofs.
One question that was not discussed was whether and how this could be incorperated into a geometry class. Obviously, there are some severe problems, too abstract, no applications, limited (in some ways) subject matter, but could a watered down version of this be brought in? I don't know, but I would think that to some extent the answer would be yes. But exactly how? I don't know.
Keywords: Technology, Geometry,
Ref: EoinO7
Author(s): Watanabe, Tad; Hanson, Robert; Nowosielski
Date:
Title:
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: EoinO
Date of Review:
Keywords: Technology, Geometry,
Ref: EoinO7
Author(s): Watanabe, Tad; Hanson, Robert; Nowosielski, Frank
Date: 1996
Title: Morgan's Theorem
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 5, pp. 420-423
Reviewer: EoinO
Date of Review: 3/25/00
In the fall of 1993 a geometry class at Patapsco High School was working with Walter's theorem which deals with the hexagon formed by trisecting the sides of a triangle and connecting these points to the opposing vertex. One of the students was intrigued by this theorem, and wondered whether or not there were similar extensions for other n-sections of the side. This student, Ryan Morgan, explored this possibility with GeoExplorer, a computer geometry package. He noticed that the ratios formed by the hexagons and the triangle with other odd n-sections of the sides were not only constant (no matter what triangle) but they were also of an integer to one ratio. Also he found that he could generalize a formula for this ratio through regression. This only left the small problem of proving the conjecture.
Ryan shared his discovery with a larger math community at a colloquiem at a local college and later at a Maryland Council of Teachers of Mathematics. Many suggested approaches to proving the conjecture, but Ryan wanted to conquer it on his own.
The importance that this article has is neither the new theorem, nor the celebration of a promising young mathematician (both of which are nice), but instead a demonstrationn of the power of the new technologies. In previous years Ryan might have been intrigued by this possibility but been unable to explore it because the constructions would have been impossible to measure well enough, and he wouldn't have had the math to finish the job. But with the power of the geometry package he was able to make some progress and make a conjecture, which gave him a great incentive to find the proof (which at the time of the article he had a proof which was under review, though another proof had been previously accepted).
The article also has contact information for Ryan's teacher who has activity sheets that led to this exploration. Too often, people give up exploring because they are afraid of the proofs, but with the powers of technology you can go farther.
Keywords: Geometry, Activities,
Ref: EoinO8
Author(s): Samide, Andrew; Warfield, Amanda
Date: 1996
Title: A Mean Solution to an Old Circle Standard
Journal or Publisher: NCTM's Mathematics Teacher
Volume, Issue, Pages: Vol. 89, No. 5, 411-413
Reviewer: EoinO
Date of Review: 3/25/00
It is often true that when doing activities teachers have tunnel vision in not only what the solution is (in fact often there may only be one solution) but also in the manner that the problem is solved. Too often the teacher has very concrete expectations on how the problem is solved, and if a student says they found it a different way, they will blow it off as coincidence. Even though it is conceivable that the new method might be valid as well.
This article demonstrates this phenomenon and exemplifies how a teacher should consider proceeding in this situation. The problem the students were solving was: Given two circles of known radii (8 and 18) tangent to a line and each other (all three points of tangency are distinct), how far apart are the two points of tangencies on the line?
The teacher had a very straight forward method in mind relying on building a right triangle and using the pythagorean theorem. This is a very good method and very easy for students to come up with and find that the solution is 24. A student noticed that two times the square root of the product of the radii also gave 24 and this is where the teacher had to make a decision. Often a teacher will ignore a situation and say that it was a fluke. But was it?
The teacher in this case used this opportunity to intrigue the students. They had the students create more cases and check to see whether this new method worked for each case (which means that they had to check it against the solutions found in the more traditional manner). If it was a coincidence then it should become apparent in these cases. But it did not seem to be a fluke.
Further extension of the problem came from trying to prove this conjecture. It was easily proved by the class with an algabraic approach, but that was unsatisfying for a geometry. Eventually, they were able to prove it geometrically after thinking about how the conjecture was really two times the geometric mean of the radii, and what geometric mean actually means.
There were further explorations about extending this into other
situations, but that was not important to the real point of the
article. The real meat of this article is that you should never
dismiss out of hand the approach taken by a student, there may be
more there then meets the eyes.
Keywords: Proof, Problem Solving,
Ref: RyanV7
Author(s): Nissen, Phillip
Date: 2000
Title: A Geometry Solution From Multiple Perspectives
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 4, pp. 324-327
Reviewer: RyanV
Date of Review: 3-26-00
This is an article that deals with one of the problems we discussed in our problem set. The problem: WXYZ is a square, with M the midpoint of WZ; the lines XZ and YM partition the square into four portions marked P, Q, R, and S. Find the ratios of the areas P:Q:R:S.
The article begins talking about how students should have many opportunities to compare, contrast, and translate among synthetic, coordinate, and transformation geometry according to the NCTM standards. However, it's sometimes very difficult to find multiple representations of the same problem. Thus, this article examines four different proofs for this problem: a synthetic approach, a coordinate approach, a vector approach, and a transformation approach.
First off, each method has an intermediate goal of showing
that the altitude of the small triangle P (drawn perpendicular
from MZ to the vertex now called point U) is 1/3 the length of
the side of the square. Once this is shown, the problem
essentially becomes a calculation-of-areas problem. So each
method given in the article describes different ways to prove
that the altitude is of the given ratio. All in all, this was a
very interesting article that would have come in very handy for
our discussion of this problem. NOTE: this is a multiple
submission because the first one I sent did not have line breaks,
sorry about the confusion. - Ryan
Keywords: Connections, Geometry,
Ref: LoriLu6
Author(s): Smith, John P. III
Date: March 1999
Title: Preparing Students For Modern Work: Lessons From Automobile Manufacturinf
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(3), pp. 254-258
Reviewer: LoriLu
Date of Review: 03/28/00
This is an excellent article, in which Smith makes meaningful connections between math and the workplace. He looked for the mathematics of blue-collar work in workplaces involved in automobile manufacturing by visiting numerous job sites and observing workers at jobs open to high school graduates. He found that spatial and geometric reasoning are essential skills. Spatial visualization, translation between 2-D and 3-D perspectives, mastery of plane and coordinate geometry, and basic trig are necessary skills in many jobs. Workers use manual and digital tools to measure, compute, represent, or program. Computing dimensions, locations, and average values and estimating error are meaningful mathematical actions because product quality and performance depends on them. He goes on to describe in fascinating detail the mathematics of two broad categories of manufacturing jobs: Assembly and Machining.
Smith then discusses implications for curriculum and teaching. Much work in manufacturing calls for the mathematics of space, geometry, measurement, statistics, and numerical operations on measured quantities. It also involves the processes of visualizing, translating between representations, interpreting, checking results, and communicating. However, Smith feels that much of our curricula still emphasize numerical and algebraic computation without context. He recommends more attention be given to space and geometry, particularly 3-D geometry, given its importance in machining and such other professions as architecture, construction, and engineering. He lists sources of available curricula and additional resources for connecting school and workplace math.
I highly recommend this article. It was quite illuminating to
learn about the mathematics involved in these real life settings.
Bridging the gap between school and workplace may help motivate
students. As teachers, we should be able to address students'
skepticism when they ask "Why do we need to learn this?". As a
last aside, Smith suggests actually taking students to the workplace
to see the mathematics firsthand. What agreat idea! Unfortunately,
given time constraints, administrative rules, etc... this is
probably one of those great ideas that rarely get implemented.
However, taking math from the workplace and adapting it to the
classroom is definitely doable.
Keywords: Geometry, Problem Solving,
Ref: LoriLu7
Author(s): Nissen, Phillip
Date: April 2000
Title: A Geometry Solution from Multiple Perspectives
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(4), pp. 324-327
Reviewer: LoriLu
Date of Review: 03/28/00
This article is especially interesting since it looks at a problem (#11 from our Problem Set) that we are all recently familiar with as students. According to the NCTM Standards (1989), students should have many opportunities to compare, contrast, and translate among synthetic, coordinate, and transformation geometry. In addition, college-intending students are also required to apply vectors in solving geometric problems. The Draft 2000 Standards promote the importance of multiple representations, including vectors, for all students. Nissen takes a fresh look at an old problem and shows how it can be solved using all four approaches to geometry.
Recall the problem: WXYZ is a square, with M the midpoint of WZ; the lines XZ and YM partition the square into four portions marked P, Q, R, and S. Express the areas of P, Q, R, and S as fractions of the areas of the square. Hence, find the ratios of the areas P: Q: R: S.
Nissen presents the four separate solutions to this problem. He gives us his take on the relative merits of each approach, for this particular problem. But, he then points out that it is informative for students to see that no one approach is the best. Students should be encouraged to try a variety of approaches when attempting a solution to any problem, experimenting and discussing which method is helping them find an answer.
I enjoyed this article a great deal. It was very informative to
take a problem, especially one that I had previously worked
(synthetically), and see alternative ways of approaching it. This
article really reinforced for me how important it is to take the
time to discuss as a class the alternative approaches that different
students employ in their problem solving efforts. This process is
at the heart of mathematics. Multiple perspectives provide new
insights and allow students to experience the creativity of math.
It would be a great exercise to have students solve the same
problem using all four approaches.