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: 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: 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: 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: 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: 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.