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: 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, 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: 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.