Project by: Josh Sapira (12th Grade)
Project Advisor: Sergei
Student(s)’s Advisor(s): Jess
Description of the Project: In this project, I will be learning about proofs, including how to write them, interpret them, the origins of foundational ideas in math, and whatever other ideas come up naturally.
Final Product:
Below, I attached an example of a proof that I wrote. The task was to show that an invalid 10-digit ISBN can be detected by a computer. The 10-digit ISBN works by taking the weighted sum of the digits (multiplying the first digit by 10, the second by 9, and so on) and dividing it by 11. The remainder then must be equal to the last digit of the ISBN.
Final Reflection on Learning:
This trimester has been a blast. Though I had a lot going on between college applications, the Democracy Project, and other school events, it was nice to have some regularity meeting with Sergei every week to talk proofs. Transitioning from last trimester where we focused a lot on the principles of proofs and its foundation in logic, this trimester was centered on actual problem solving. In addition to continuing research and reading my book on proofs, I regularly completed practice sets and problems. As someone who plans to be a math major in college, this is incredibly helpful from a math perspective. Even more importantly, however, is how understanding proofs and progression of logic will help me in everyday life from syllogisms in the media to logical fallacies.
Update on Progress from Weeks 1-3 (include any photos or video if relevant):
At this point, I really don’t have much of an update. Sergei and I are continuing to meet every week. I am going over problems and readings between sessions. I am definitely enjoying the project and am looking forward to continuing my work on proofs for the rest of the trimester.
Update on Progress from Weeks 4-6 (include any photos or video if relevant):
Again, just going through a similar process as previous weeks. We are meeting and discussing chapters of the book, but are now transitioning to where I am beginning to spend more time solving problems than reading up on topics like the theory of games and infinity. There are only a few weeks in the trimester left so I am really looking forward to one last push to cram every opportunity with learning.
Update on Progress from Weeks 7-9 (include any photos or video if relevant):
Trimester 1 Update:
Final Product (e.g., documents, images, video, audio, poster, display, etc.): There is no final product to share at this time.
Final Reflection on Learning: This project was a lot of fun! I originally wanted to do this project just because this was a foundational topic in math and it was not covered in LREI’s math curriculum. However, throughout the trimester, I have really grown to appreciate it for its foundation in problem-solving and critical thinking. Though it has been very difficult at times, I definitely see myself improving and look forward to working on this project more in the second trimester.
Update on Progress from Weeks 1-3 (include any photos or video if relevant):
Given everything that is going on right now between college admissions and the 2020 federal elections, Sergei and I have not been meeting as regularly as we would have hoped. That being said, we have plans to ramp things up for the remainder of the trimester and I am really looking forward to it! We have discussed resources to use, types of proofs we want to explore, and have made a general plan for how we see this trimester going. Seeing the amount of polling and statistics being used in regard to the election has made me really interested in the potential applications of proofs in the real world, and I cannot wait to really ramp this project up this week!
Update on Progress from Weeks 4-6 (include any photos or video if relevant):
These past few weeks have really put into perspective how difficult this project is/will continue to be. Sergei and I have begun reading Proof and the Art of Mathematics by Joel David Hamkins. In it, I have read multiple proofs. Understanding these proofs are not very difficult because good proofs are clearly articulated and demonstrate a logical progression from one step to the next. What has been most challenging for me is the exercises at the end of each chapter. There is no algorithm or clear step-by-step process for proving mathematic theorems. Rather, it requires both creative and logical thinking. I know that I can get better at working with this mindset, but it will definitely take time for me to be able to create and write proofs on my own.
Update on Progress from Weeks 7-9 (include any photos or video if relevant):
In these past few weeks, I have continued working with Sergei. We have gone through a couple of chapters, including ones on graph theory and irrational numbers. Not too much to update on right now. I am just going to continue to work on this project through trimester 2 and am really looking forward to it! I’m finally starting to get into the groove of things with proofs.
