Journal Entries APRIL 1999

We spent the class discussing and exploring the euclidean GSP circle constructions in the hyperbolic plane. It was very interesting and motivating. The lesson was extremely informative, organized, and flowed nicely. You were patient with us slow learners and always ready and available to answer our questions. I appreciate your willingness to stay after to help me with my constructions.

I spent a great deal of time working on the initial constructions before class to have a general understanding of what to do in the homework assignment. This was beneficial and added to the class discussion. I understood what you were doing much more had I not done some of the work beforehand.

The hyperbolic geometry is really coming together now by doing these constructions. They make the connections. I am finding it rather difficult to keep going back and forth between the two different geometries. I have to stop and think constantly making sure I am in the right model. This experience is a great learning experience.

Spherical geometry is next on the agenda. I do not know if it can compare to hyperbolic geometry. We shall see. You have my interest.

We discussed spherical geometry today. The lesson was a basic introducation in the main concepts of spherical geometry. We discussed euclidean concepts of point and line and what they represent in speherical geometry. We also looked at the euclidean axioms and how these parallel or do not parallel with spehreical geometry.

This comparison of the two geometries and their axioms was most interesting. I had never worked with sphereical geometry and found this to be a challenge. I can say have understand the basic underpinnings of the spherical geometry such as point, line, plane, distance and angle measure.

Comparing and contrasting the two geometries is powerful. It helps me to understand euclidean geometry even more so by having to examine a new model. For this very reason, I argue that some form of non-euclidean geometry should occur in the middle or very least secondary school mathematics curriculum. I think from an adolescent's perspective that non-euclidean geometry would be interesting and highly motivating. I know how much I am enjoying from our exploration of other models other than euclidean.

Your use of the globe and masking tape props were helpful for my to visualize the model. I have no doubt that a high school student could understand the basic underpinnings of these two models. One problem might be their reluctance to do something new. This could work both ways, for the good or bad, depending on the type of student. I do think you would have to be ready to explain to the students WHY learning new models will be helpful and be clear in your definitions in all of the geometries from the beginning. Also, GSP would be a must, especially in the euclidean and hyperbolic geometry model. Most of us, even us more mature students, need visual models to help with the basic conceptual underpinnings.

Class discussion centered on more conceptula underpinnings of the spherical geometry.We discussed what a circle and ray is in sphereical geometry and also the concept of betweenness.

A portion of class revolved around Girard's theorem which states that the area of the spherical triangle is the angle excess in radian measure multiplies by the radius squared. This holds assuming that the sides of the regions (spherical triangles) are arcs of great crcles. This theorem was incredibly fascinatng to think about.

The concept of betweeness was also intriguing. I gleaned from our discussion in class that since a line is a great circle in spherical geometry, it really doesn makes sense to discuss betweeness. We did however come up with some notion of a definition for betweeness which was based on order of the points.

A consequence of Girard's theorem was the sum of the measures of a triangle is greater than 180. This was really interesting when we looked at the triangle that had angles summing to 270. This was hard to mentally visualize. But I could see it clearly on the globe.

Overall, I am finding non-euclidean geometry quite fascinating. I wish we had much more time to learn more.

Friday, April 9

Today's discussion was a review for the last test that will cover Spherical and Hyperbolic Geometry. We compared and contrasted the undefined terms and axioms in each model. This strategy was helpful in reviewing. It allowed us to view each model with an overall picture in mind. The table was a great way to see the connections between each model. Since these models are based on euclidean geometry, I find that I am understanding euclidean geometry better and seeing connections across all three geometries.

In my EMAT 8020 class, we have been discussing the preparation for secondary mathematics teachers. We read MAA's Call For Change. This document is there recommendations for the preparation of mathematics teachers. This document focuses more on content preparation. They discuss that conceptually understanding models is extremely important. MAA also mentions that mathematics teachers should have exposure to different models and work with different models. MAA also discussed what should be covered in terms of mathematical content which included euclidean and non-euclidean geometry. It also discussed how this should be taught and what content should be taught. I found as I was reading the document that your geometry class and teaching style has definitely meet their standards for the preparation of teachers of mathematics. I will bring you a copy of this document for your reference.

Test 3 covering Hyperbolic and Spherical Geometry.

The test was extremely fair. Each question we had covered at some point in class or in our homework. You spent a fair amount of time making comparisons between each of the geometries. These comparisons and contrasts between the geometries were covered on the test.

The proof was fine. We had proved this axiom, PI-1, as part of our homework. I, however, only glanced over the proofs in our homework, and spent most of my time studying and reviewing the construction homework especially the circle by center and point and circle by center and radius constructions in h-geometry. Thus, I am not quite sure how I did on the proof. I looked over my notes and some parts seem to coincide others do not.

The other parts of the test I did well, save for any careless mistakes or definitions that are not clear or missing certain important words. I did take special time to carefully write the definitions and be clear and concise.

Overall, I think the test was easy given the time we had spent in class and in our homework and based on my studying.

Today's class discussion revolved around the five regular polyhedra:

We defined what a regular polyhedra is and then discussed the details of the homework assignment. We discussed the relationship between the circumradius of a circumsphere of a cube and the edge of the cube. It was discovered that c = sqrt (3)/2 * e. This was hard to visualize initially. After a diagram and some thought, I understand the relationship. The others should be rather interesting. I have never made polyhedra before. Thus, the polyhedra homework assignment should be fun as well as a learning experience.

You also discussed the history behind regular polyhedra. I am also interested in the history of mathematics and found the discussion interesting. Much of the information I learned in a MAA history of mathematics institute in DC last summer. We took three courses: The history of Geometry, Algebra, and Calculus. But we covered so much material in the two weeks that I would have to refer to my notes for most of the material and information. I did recall our discussion around Euclid's Elements and his book about polyhedra. Most people do not realize that he discussed rather extensively the polyhedra. I was very surprised that there were people in our class that were unaware of the Elements. This is difficult to understand. How can you possibly teach or have taught mathematics, Geometry in particular, without knowing its origins? This was strange to me. This lack of historical awareness shows how important it is to discuss the history behind mathematical ideas when you introduce them. You cannot assume everyone knows the historical origins behind the particular mathematics you are discussing.

Class began with a brief illustration of the Virtual Reality website. This website was very interesting and amazing. Then we discussed questions revolving around the homework assignment. The majority of the period was spent showing why there are only five regular polyhedra.

The theorem is

The approach used to show the theorem could be used in a middle or high school geometry class. This approach was illuminating for me. I understood exactly why the theorem holds and feel confident I could lead a discussion with a group of middle or secondary mathematics students.

The last portion of the class you previewed the next two class discussions which wouuld be contered on exploring the symmetries in the five regular polyhedra.

I have constructed the five regular polyhedra. This was a fun learning experience. Your brief discussion about finding the relationship between the edge and the circumradius was helpful.

Class discussion revolved around Symmetries of Polyhedra. First we reviewed symmetries of regular polyhedra which were reflection and rotation by 360/p. Then we evolved to three dimension and investigated the three five regular polyhedra. After exploring the symmetry for each polyhedra we noted that there were connections between a cube and octahedron and between a dodecahedron and icosahedron. We learned that these polyhedra could be considered dual polyhedra based on their symmetries. The tetrahedron was unique however. It was also noted that two tetrahedra can be inscribed in a cube and also comprise a cube. Class ended hinting about group theory and multiplication table of the transformations.

Class flowed well and seemed to make sense in the way and order that you presented the material. I am sorry that I will miss the discussion about the reflections in polyhedra. I have a classmate picking up all of the handouts and taking notes for me while I am at the NCTM conference.

Attending the NCTM conference in San Francisco

Class discussion centered on reflections in regular polyhedra.

Attending the NCTM conference in San Francisco

Presentation by Robyn Bryant

Presentation by Theresa Banker

Presentation by Dawn and Inchul.

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