Day 14 (12/18/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

           Again, because Professor Dolbin was not here today, I focused on completing my tasks for Professor Farag.

(Part 2 - Professor Farag)

(12:30 - 2:30)

           We spent all of today finalizing the works that I have completed so far. First I prepared the solution set to the class's final exam. Then, we reviewed all of the documents that I had prepared, including the official solution key to his exams, and we did so with a VERY fine-toothed comb. After all major corrections had been made and the documents were edited, we spent the rest of the time discussing the course in general. Overall, I have learned a lot throughout this internship, and I am very sad that this part of my internship will be over after this week. I am very thankful for Professor Farag's time and effort in mentoring me! From now on, I will be working with Professor Dolbin during all of my internship time!

Day 13 (12/11/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

           Because Professor Dolbin was not here today, I focused on completing my tasks for Professor Farag.

(Part 2 - Professor Farag)

(8:00 - 3:00)

           Now that the classes are finished and students are taking their final exams, I am approaching my culmination date for the mini project that I have been working on with Professor Farag. Today, I spent a lot of time with the professor, carefully reviewing the 60 pages of notes that I had typed for the class, in addition to the 40+ pages of solutions to various assignments and assessments. Thankfully, I had barely made any typos, and so the reviewing process went very smoothly. I am very thankful for this internship, which has allowed me to review my past knowledge in both mathematics and LaTeX, and learn even beyond that. Next week will be the last week I will work with Professor Farag, and I will certainly miss working with him afterwards!

Day 12 (12/4/13)

Unfortunately, I woke up with a terrible flu, and thus was not able to attend my internship. It's a shame that I'v e missed two days in a row!

Day 11 (11/27/13)

For this day, I was not able to go to my internship because I was at a math competition in Beijing, China for the entire week. My team placed 4th internationally, and I was a silver medalist! Woohoo!

Day 10 (11/20/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:00 - 12:00)

           Today, Professor Dolbin taught me some extra information in Galois Theory that was not covered in the textbook. I also began my studies in Lie algebras, in which several unusual properties hold. For instance, x+(y+z) is not always equal to (x+y)+z. I spent a majority of my time solving exercises from the text, which helped me get a better grasp in the topic. I am really glad to be in this internship, where I have the freedom to choose whatever I want to learn, guided by a professional in this field.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Afterwards, in Professor Farag's class, we discussed more ring theory and explored special kinds of polynomials. We separated into groups and discussed problem set questions, and another test was scheduled. Luckily for me, the topics I studied with Professor Dolbin again resurfaced in Professor Farag's class today, allowing me to compose notes and solution sets for the week more easily. In general I learned more knowledge in various types of algebras and I am very thankful for this internship!

Day 9 (11/13/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:00 - 12:00)

           Today I further studied the intricate structure of Galois groups. I learned the Fundamental Theorem of Galois Theory, which says that there is a direct correspondence between extensions of fields and subgroups of Galois groups. I also became used to drawing Galois correspondence diagrams, which shows this relationship pictorially. It’s amazing how the two types of mathematical structures, which seem entirely different at first, can be related in such a simple way. Moreover, I learned that all of this can be further related to the one type of groups called the symmetric groups, which are just the set of ways to permute a set of objects. I see this in two ways—mathematicians have either shown that these seemingly complicated structures in mathematics boil down to the simple group of rearrangements, or that the seemingly simple symmetric group is, in fact, monstrous and obscenely complex. I guess it depends on whether one is a pessimist or an optimist. Nonetheless, after studying these structures, I moved on to the proof for why there is no generic solution to the cubic equation, which utilizes the material that I have learned so far. In addition, under the guidance of Professor Dolbin, I have learned several interesting side details along the way. Nonetheless, by the end of the day I was able to finish yet another chapter of the text, thus taking a gigantic step towards my algebraic research project.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Next, in Professor Farag’s class, we further studied rings. We covered a special case of rings called ideals, which have a unique "absorption" property. Afterwards, we discussed some more practice problems and went over the last problem set, for which I had already written up solutions. Today I gained further knowledge about abstract algebra, and this class has been helpful especially for the first part of my internship, as one supports and adds more information to the other.

Day 8 (11/06/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:04 - 12:03)

            Today I continued to study from the textbook that Professor Dolbin had given me. I finished a brief chapter on an algebraic notion called “fields,” which are groups of elements under two operations. An example is the set of real numbers under addition and multiplication. After studying this branch of algebra, I moved on to studying Galois Theory, which studies the relationships between fields, field extensions, and groups. This was used to eventually show that there are NO ways to solve general equations of degree 5 or higher; for instance, the famous Quadratic Formula can solve any quadratic equation, but Galois showed that there are no such formulas for polynomials of degree 5 or higher. I find it fascinating how this is one of the greatest examples of how mathematics had proven the fact that something does not exist, which really sounds impossible at first thought. This will likely be included in my research later on. Professor Dolbin frequently checked up on me and guided me when needed, and I have already learned more than I possibly could have by myself in such a short amount of time. The past couple of weeks had been invaluable to me, as they showed me that I really can survive while doing just math all day, which is what most mathematicians in the field do. I am grateful that Professor Dolbin is willing to spend so much time to mentor me in this manner.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Next, in Professor Farag’s class, he taught us more about ring theory, which is another structure in algebra like “groups” or “fields,” as well as how rings relate to those other structures. Throughout his lecture today he emphasized the need to stop ourselves from invoking our previous knowledge on “numbers,” and the need to stick with the axioms given only. This mindset is crucial for the study of abstract algebra, as I learned through the various classes that our intuition may not always hold. For instance, x plus y might not always equal y plus x in a lot of algebraic structures. Boy, does this class give me a headache!

Day 7 (10/30/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:10 - 12:05)

            Today was the first day of my new internship with Professor Dolbin. I began studying from a textbook for undergraduates recommended by the professor, called “Abstract Algebra: an introduction” by Thomas W. Hungerford. I began studying the concept of field extensions, which are essentially fields that act as vector spaces over other fields. (It sounded just as complicated to me when I first saw it.) Throughout the day, the professor talked with me about the concepts, provided additional exercise problems for me to solve or prove, and clarified certain notions. At first, I had a difficult time grasping the higher-level concepts, but with some guidance I managed to understand an entire chapter of the textbook. I have already learned so much more about abstract algebra under his guidance, and I hope to continue this learning process so that I will be well-equipped to handle even more difficult problems in abstract algebra later on. I’m really thankful for Professor Dolbin, as he devotes a significant amount of time to help me learn. 

(Part 2 - Professor Farag)

(1:00 - 3:00)

            In class, we tackled another interesting and difficult problem in the field of group theory. We further discussed the "First Homomorphism Theorem," which relates groups that are deceptively different. Also, in the middle of class the students and the professor suddenly entered a gigantic, almost-philosophical debate about whether things are "well-defined"; the problem arose when a student suggested a function that might not always map a value of "x" to the same value of "y." However, after this heated discussion, we settled down and had a good, long laugh about it, and the rest of the day passed by normally. I finished the notes for the past week in time, although I have yet to write the solution set for the newest homework assignment. Surprisingly, I learned that these problem sets might not be as difficult as I had originally assumed!

Day 6 (10/23/13)

          Unfortunately, I was again too sick today to attend my internships. However, I gathered enough strength to go to the informal interview for my second, new internship at FDU, at which I would undertake a small mathematical research project and a directed study project with Professor Dolbin at the university; in short, I was accepted! More details to follow next week, but I am very thankful for this opportunity. Thank you, Professor Dolbin, and also thank you, Professor Farag, for helping me attain this internship! I do feel sad that I will not be able to work with Mr. Sincak at River Dell Regional HS, and I hope he understands.

Day 5 (10/16/13)

River Dell Regional High School

(7:30-12:00)

            Today I began creating my own lesson plan for class. Mr. Sincak provided me with reading materials and sample lesson plans from various teaching colleges, and I learned how teachers can prepare for each class. I specifically focused on creating a lesson on solving non-routine problems, such as those involving mathematical modeling or problems that might appear on math competitions. I designed the overall purpose of the lecture, outlined prerequisite knowledge and expanded on how the lesson would actually be carried out. I found example problems and practice questions for the class from the reading materials provided, and I specifically chose to incorporate problems that involved higher-level thinking in geometry and probability theory. Specifically, for example, I began the first day’s lesson plan with a coin-flipping game for students, in order to experimentally calculate the probability that more heads than tails show up when five coins are flipped. Then, I continued by asking students to think of ways to prove that their resulting probability was correct, with several different suggestions on how to approach the problem. Finally, I ended the lesson plan by demonstrating how one can theoretically calculate the required probability, emphasizing the methods utilized in attaining that answer.

Throughout the process, I learned and realized how I can also organize my thoughts and lessons for when I tutor students throughout the year. I have really enjoyed all of the processes and techniques associated with teaching that I experienced so far, and I hope to use these skills that I learned when I grow up.

Fairleigh Dickinson University

(1:00-3:00)

            Today’s class was devoted to reviewing for the test. We discussed problems from an old Midterm from the course, and several students presented solutions on the board. Professor Farag also outlined possible topics that would be on the test, and encouraged students to remember the main concepts and definitions.

            After the class, Professor Farag introduced me to Professor Dolbin, another mathematics professor at the university. While I could not speak with him for long, I learned that his interests in research were in Lie algebras, representation theory, and associated combinatorics, which I also find very fascinating. Professor Farag suggested that I ask him to mentor me in a research or independent reading project regarding that field of mathematics, and I am very thankful for that suggestion. I will soon email Professor Dolbin for an opportunity to work with him.

Day 4 (10/09/13)

Due to a sudden illness, I was unable to attend my internships in this week. Coincidentally, my mentor for River Dell Regional High School was sick as well.

Day 3 (10/02/13)

River Dell Regional High School

(7:30-12:00)

            Today, Mr. Sincak and I prepared a domino circuit activity to demonstrate logical operators and their effects on statements. Specifically, we designed a circuit for the logical statement “If P or Q is true, and if R or S is true, then T is true.” Although I felt that the business-casual outfit made me feel a bit out of place, the class and the domino exercise were enjoyable as a whole. After the class, I spent the day constructing methods of teaching indirect proofs and their applications to geometry. Mr. Sincak and I researched optimal strategies from various teaching textbooks, and he lent me a copy of one of the textbooks for me to study further if I chose to do so. I am very appreciative of how much Mr. Sincak cares about my career goal of becoming a teacher or professor, by constantly giving me advice and explanations on the best methods of teaching and working with other students. I feel like I am already attending a teaching school!

Fairleigh Dickinson University

(1:00-3:00)

            Today in class, Professor Farag taught about a special kind of maps from groups to groups called “homomorphisms.” We also learned about several branches of groups that revolve around these functions, as well as different kinds of homomorphisms. We began to discuss the importance of “isomorphisms,” or maps that relate each item in the first group to a unique element in the second group. He showed us that this is an especially important concept, as groups that are “isomorphic” to each other are essentially the same group. 

            After the class, my temporary office was moved to a new building, but the rest of the day was routine. I finished typing up the class notes and solutions for the past week accordingly, with ample time left for revisions.

Day 2 (9/25/13)

River Dell Regional High School

(7:30-12:00)

            Today, Mr. Sincak and I began to plan out the curriculum and lesson plans for his chapter on Boolean logic. We first devised several mathematical examples of statements involving Boolean functions such as “and,” “or,” or “not,” which we also related to previously learned topics such as basic algebra and geometry. We also devised a project for the students that involve constructing circuits with dominoes based on the various forms of logical statements—both simple and compound. And throughout the process, Mr. Sincak taught me various techniques within the realm of teaching that would be useful for engaging students who are struggling and “entertain” (as Mr. Sincak describes) those who are excelling at the topic. Such methodologies included demonstrations of reversals of two-column proofs and the inclusion of engaging technological tools such as Geogebra. Furthermore, I began to realize more and more how intricate and enjoyably complex each lesson plan for highs school students could be, as I learned numerous strategies and potential methods that could be applied for teaching (not only for high school students, but for students in general).

            After constructing the curriculum, lesson plans, and projects for students for the unit, I continued to work on the Geogebra program that would reveal which students are excelling and which students are not as proficient. For today’s specific program, I constructed a Java-based program through Geogebra that would create random lines and ask students to input the segment’s distance. Each correct answer would make the background of the screen greener, thus helping in revealing which students retained a solid understanding of the topic. Mr. Sincak could see all of the students’ screens through a screen-sharing program, and he was able to assign helpers for struggling students efficiently. My mini-project for the day had been completed successfully, as I became more fluent in the languages of Java and Geogebra scripts.

            Hence overall, not only did I enhance my abilities to program basic applications for classroom settings, but I also learned various techniques and methodologies for teaching more efficiently and entertainingly. The domino demonstration especially struck out to me, as I never imagined that certain objects such as the little toys could ever be used for teaching mathematics. I have attained a more open, creative, and enlightened mindset for teaching, and I hope to utilize these skills as much as possible throughout my life and future careers.

Fairleigh Dickinson University

(1:00-3:00)

            During today’s lecture, Professor Farag shifted from teaching about groups to explaining the properties of their subgroups. We discussed and solved various selected problems from the textbook, and learned various theorems regarding cyclic subgroups, among others. Although I had missed the two previous classes (as the class runs on Mondays, Wednesdays, and Fridays), I was still able to follow the lecture comfortably, while typing notes.

            After the course, I moved to my regular, weekly responsibilities of typing and organizing class notes and solutions to homework problems. With my proficient skills in typing mathematical texts using the program LaTeX, I was able to type all of today’s class notes during the class itself. I then quickly typed up notes from the classes that I had missed throughout the past week, which were provided to me by a fellow student in the class, who attends Fairleigh Dickinson University and is also a staff member in BCA’s Math Team program. Afterwards, because the assigned homework problems were basic and elementary, I was able to type up the solutions as I solved them in my head simultaneously. While I perfected my knowledge in elementary Group Theory, I also discovered even more implications, extending to Number Theory and other branches of mathematics. In addition, I realized that I needed to study more about other forms of abstract mathematical structures to attain an even bigger grasp of abstract mathematics.

Day 1 (9/18/13)

River Dell Regional High School

(7:30-12:00)

            During this first day of my internship at the River Dell Regional High School, I met Mr. Sincak and the other teachers within both the mathematics and science departments. After introductions with various teachers and procedural activities, Mr. Sincak provided me with an office within the school for me to work in throughout the day. He and I discussed possibilities with utilizing the program Geogebra within his classroom, and he asked me to construct a program file that would interactively diagnose which students required assistance throughout his Honors Geometry class. Specifically, my task was to construct a program that would create random line segments on a coordinate grid, provide spaces for students to submit answers based on questions regarding those segments, and create an indicator on the screen to reveal which students are doing well and which students are struggling.

            Throughout the day, I worked on learning the Geogebra program and its association with JavaScript. I found it difficult to connect my experience with mathematics with my basic knowledge of computer science and programming, but I enjoyed it tremendously. I realized that the range of tools available for teaching purposes was larger than I ever expected, as I soon discovered how useful this application would be for the average classroom. These tasks, as well as my conversations with Mr. Sincak, have begun to inspire me into a career choice that connects to both technology and education, with the underlying factor still being mathematics; and although I still need to improve on my programming abilities, they nonetheless improved significantly on this first day. In the end, I was partly successful in performing my tasks regarding GeoGebra, but I fully associated myself with the other faculty members here. In addition, I also helped create challenging problems for Mr. Sincak’s Honors Geometry course. Overall, I plan to learn more about JavaScript in order to produce more fruitful results in the future.

Fairleigh Dickinson University

(12:56-3:00)

            On this first official day of the internship at Fairleigh Dickinson University, I discussed with my mentor, Professor Mark Farag, about my tasks throughout the semester: auditing the course, organizing class notes, preparing solutions to assignments, and other works related to mathematics. I also met with Professor Alfredo Tan, the director of the engineering department at the university, who was friendly and generous enough to speak with me during his busy schedule, albeit briefly. In addition, Professor Farag allowed me to explore the various sections and departments of the school, encouraging me to also consider careers related to applied mathematics (such as regarding finances), instead of solely focusing on theoretical mathematics.

            The class, for a course called “Abstract Algebra,” was very small in population, as only five students excluding me had shown up. Professor Farag explained that this phenomenon was due to the fact that the course was among the highest level classes provided at the university. Nonetheless, such a small class helped me fit in and become adjusted to the environment. I felt like an actual university student throughout the day.

            After the course, Professor Farag provided me with space to work on my assignments regarding class notes and solutions. Although I had a somewhat extensive knowledge of the mathematical typing program LaTeX before I began the internship, it allowed me to master my skills in the program. Because I had to practice typing notes throughout the class, in the format required by LaTeX, I became fluent, to the point where I can type in LaTeX as fast as I can type regularly. I also learned a significant amount of abstract algebra, even though it has only been one class so far. I discovered how college life and studies would feel, and I plan to use these learned skills throughout both high school and beyond.