Monday, March 1
Class discussion centered around hyperbolic geometry. Several interesting ideas were discussed. The following are some of those ideas:
The sum of the measures of the angles in hyperbolic triangles is less than or equal to 180 and varies by sum.
When the hyperbolic triangle is very small, the sum of the measures approach 180.
Hyperbolic geometry satisfies all Euclidean axioms other than the parallel postulate. This helps explain why the the sume of the angles will never exceed 180 degrees.
We also spent a great part of the class time discussing distance in hyperbolic geometry. We questioned how one measures the distance between two hyperbolic points. We were given the formula for the distance in terms of ln. It was helpful to discuss how one arrives at the formula for the distance as it relates to ln. We questioned WHY LN? The rest of the time was spent giving a preview of how and why ln is involved in the formula for distance. We also discussed the two cases of distance between two hyperbolic points as well as verified that the Triangle Inequality holds in Hyperbolic geometry.
This discussion centered around the formula for hyperbolic distance was extremely interesting and challenging. You took great care to go slow and explain in detail each step. I am appreciative of this fact. I look forward to jumping deeper into this notion of distance on Wednesday.
Wednesday, March 3
A comment about the homework 8: It seemed so easy in class when you were doing the scripts on GSP. When I acutally sat down to construct my scripts, it was much harder to do. Doing the homework stressed how important it is to have students practice the concepts that are discussed in class. I am not advocating skill and drill. But constructing the scripts really enforced the main ideas of hyperbolic geometry that you had introduced in class.
Your discussion of the origins hyperbolic distance was well done. Pedagogically speaking, the class lesson was organized, paced appropriately, flowed, and extremely interesting. I also thought your attempting to show how the formula was developed revealed courage to go beyond simply giving us the formula that you know most of us would readily accept. I admire that from a pedagogical standpoint. Your teaching practices are a nice model to follow and aspire to when it comes to teaching mathematics.
Inchul and I are looking forward to our project presentation. It should be fun and interesting for all.
Friday, March 5
I thoroughly enjoyed these past three days of class. The class discussion has focused on showing where the formula for the hyperbolic distance comes from. It was interesting and a great review of some important calculus topics. The calculus became clear again after your review. The most intriguing thing that I have learned from this lesson is how it shows the interconnectedness of all or our school mathematics. The proof of this one formula links together topics from Euclidean and non-Euclidean geometry, algebra, trigonometry, and calculus. I am always amazed at the beauty of mathematics. Your great presentation also made the lesson interesting and fun.
Monday, March 15
Thank you for giving us the review notes for showing the hyperbolic distance formula and its origins. They will mesh nicely with mine.
Class discussion centered around checking and proving that all of the axioms of absolute geometry hold except for the parallel postulate. I followed the lecture for the most part. I am fuzzy on quite a bit of the material involving the proofs. Once I review my notes and work cooperatively with Robin, I should get a better grip on the material. I thought the lesson went well. I thought it to be fuzzy in some areas. This could stem from my past tendancy to have mathematical material told and spelled out more directly. You gave us a broad and general outline to help us get started. This approach is good and allows for each of us to think more independently and problem solve; something I always need more help in. I do think that there is a great deal of material to cover by Wednesday and then have the homework due on Friday seems a challenge. Nevertheless, the work will progress and evolve (I hope by Friday).
Wednesday, March 17
Class focused on a discussion centered around experimenting with GSP to help prove the axioms of absolute geometry hold in hyperbolic geometry. For the most part, I followed the lecture. Seeing the proofs clearly in my mind is challenging. I usually see how and why a proof works by talking and working ith another student. Doing the GSP investigations first help with the proofs. This is a rather lengthy assignment so I am making this journal entry brief. One more thing, your help after class was great and helped some ideas gel.
Friday, March 19
This week in Geometry has proven to be a most challenging one. Let me explain.
First, the assignment was difficult and lengthy. Most of my other work suffered due to the vast amount of time necessary to barely complete the homework assignment. I believe I deserve an A+++ for effort and willingness to learn a new and complex geometry. However, I feel like I have not grasped completely everthing you wanted in this assignment. I am fuzzy and feel insecure about this assignment.
Second, the material is incredibly interesting. Each day as you introduce new ideas about h-geometry, pandora's box is opened for a whole new set of mathematical ideas revolving h-geometry. It confirms the philosophy that the more mathematics you know the more you realize how much you don't know and still have to learn. This has overwhelmed me particularly this week. And now we turn to looking at transformation in h-geometry and more is revealed. I did enjoy working on GSP today and investigating the transformation (reflection) in h-geometry. I found this activity intriguing.
Third, I am worried about the upcoming test. I do not feel like I have a strong hold on the material centered on h-geometry.Yes, I understand the basics of h-geometry. But, the details I am fuzzy and unclear.
Lastly, with the test coing up and a new homework assignment, I feel even more overwhelmed. Tring to understand this new material and build on what I already know about h-geometry is concern for me, since I am unclear about the details of h-geometry.
I have no doubt that by next Monday, I will be better equipped to take a test on the material. Now, however, I feel like I am running in a fog.
Monday, March 22
As I stated on Friday, I had no doubt that today would be clear, not rainy and cloudy. The material today was clear, organized and made sense. It was as if I needed another dose of the conceptual underpinnings for the material to sink in. The class structure flowed well today. We discussed parts of the homework assignment, such as showing how the angle measure is preserved in hyperbolic geometry transformation, explored the table function, and reviewed the angle script.
The class lesson demonstrated how the teacher can always learn from the students. This pedagogical concept was apparent in the ways several students helped you understand some GSP functions, such as the table function and in making scripts. It is clear that teaching is a learning process as well.
I appreciate your willingness to try something new with the assessment on Friday. We will all learn something new in the process. I do agree that you should test what you value and what we do in terms of homework. This test will hopefully do that. I am looking forward to a new approach to test taking in this class with this test on Friday.
Thanks for your help after class today. I finally feel like I have a concept of the basic scripts and how to construct the basic concepts in hyperbolic geometry. Your help definitely improved my scripts.
Wednesday, March 24
Today the discussion centered around a review for the second exam. The discussion was helpful. I am still in need of some help on two of the proofs. The practice exam was beneficial. I would have not been able to do the proof without some effort unlike others in the class that saw a vision of the proof immediately.
I will say I am not as stressed or worried about this test on Friday since we are able to bring in all of our materials and tools. These act as psychological cushions. Though it may be easy for others to blow off the test and not study since we can bring in materials, I will take advantage of this opportunity and study more of the conceptual underpinnings not simply memorize definitions and axioms. I do believe this is your vision since you are opting for us to bring in notes, etc.
This has been the most interesting of all of the topics this entire semester. I believe this is so since the material was so new. I had never seen non Euclidean geometry before. This time was enlightening and engaging for me. I appreciate your willingness to delve into a new topic and challenge us mathematically. You are a wonderful mathematics teacher and model effective teaching practices.
Friday, March 26
Second Test was given today. It was challenging and a learning experience. The test did not seem to measure what I know about h-geometry. The proof seemed unfair and not representative of what we have spent the majority of time working on in terms of proofs.
Monday, March 29
I did as expected on the exam. I am grateful that you are not counting this exam and revising your course plan to include another test on h-geometry that is more representative of what we actually know. I believe we have learned, I know I have, a great deal of h-geometry. I have a good understanding of the basic concepts of this new geometry. I look forward to the next test to see how it will test our understanding of h-geometry. Before this unit, I had no understanding of h-geometry. Now I realize how much I have learned in a short time about a new and exciting branch of mathematics.
The discussion in class centered on an explanation of how to construct the perpendicular bisector of a h-segment given two points. (Problem 2a and b of the test). The discussion was helpful in understanding where I went wrong on the test. I have learned where my mistakes occurred and understand how to do the construction correctly. I spent time after class redoing the problem and feel much better knowing I understand. This is where learning occurs: seeing our mistakes, redoing the task correctly, and hopefully learning how to not make them over again.
You brought up a new idea of constructing h-circle. We will spend more time on this concept. I am looking forward to the new challenge.
Wednesday, March 31
We spent class time today discussing the GSP construction menu as it relates to hyperbolic geometry. We discussed each construction and how we can do each in hyperbolic geometry. We were able to cover all of the constructions except those related to constructing circles.
I thoroughly enjoyed the class discussion and exploration. It was informative. Tying hyperbolic geometry to the euclidean constructions is powerful. The geometry makes much more sense. This discussion reveals some of the beautiful connections between the two types of geometries. It also makes a clear statement about the beauty of mathematics particularly geometry.
It still amazes me that I am in my doctoral program and still learning so much about mathematics. This phenomena awakens me everyday to how little I know and how much I still have to learn. One can never learn all there is to know about mathematics or really any other subject or area for that matter. Your class and your philosophy of teaching mathematics conveys the message that there is still so much knowlegde to be discovered. This is apparent in your statements like "I don't know that." This is important for students to hear from their teachers. It allows students to understand that teachers do not know everything. They (you) are still in the process of lifelong learning. This is a powerful and important message to convey to students. I applaud your willingness to learn along with us.
I am eager to learn more about the construction of h-circles. Hopefully, the conceptual understanding will evolve and I will understand the more I see and grapple with the material.
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