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: 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
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: 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: 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: 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.