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Mystery Group Project
Mystery Group Project
The best way to understand what mathematical research is like is to do it yourself. I want to give you the opportunity to: get lost in a problem; experience the frustrating step function that is mathematical progress, and the extreme thrill of mathematical discovery and insight; attain the mastery that comes from formulating conjectures and applying concepts learned in class to answer questions of your own asking. As Morris Kline so eloquently stated,
“The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, ‘refuge from the goading urgency of contingent happenings,’ and the sort of beauty changeless mountains present to sense tried by the present-day kaleidoscope of events.”
So to that end, you will be involved in your own research investigation this semester. During the first week I will give you your very own group to study/investigate/explore. We will be studying group theory for the first 2/3 of the class, examining examples, structures, properties, etc. As we introduce each definition, theorem, or property, your job is to apply this new concept to your group. Does your group have that property? How does this theorem apply? Using this new definition, can you answer questions about your group, or formulate new questions? Of course, if you claim something about your group, you should back it up with a proof (or at least make a very thorough attempt). Throughout the term I will be doing the same with my group and using it as an example in class. Some properties and investigations will be easy and clear for you to apply, others will take time, intense thought, and may lend themselves to numerical investigation. Some questions may be impossible to answer.
Research Journal Grading Criteria
Research Journal Grading Criteria
You will record your investigations in a research journal. This journal should have at least 50 pages, and may be as fancy or bland as you wish (I have found graph paper works best for me, but you should choose a journal format that appeals to you). You are welcome to use Google documents or a well documented python worksheet if that format works best for you. This journal is the record of your investigations into your group. In it you should neatly record calculations, propositions, questions, conjectures, and proofs pertaining to your group. Some things to remember:
Each entry should start with the date
Your entries should be legible and well organized
Before each calculation/investigation there should be a heading or explanation of what you are attempting
After each calculation/investigation there should be a conclusion
Before each proof there should be a clear statement of the proposition
At the end of each research session you should summarize your results, findings, new questions, conjectures, or theorems.
Think of your research journal as a source that you would want to use for future reference several years later if you were doing research on your group again. A nice way to conclude each research session is to answer the following questions:
What new conclusions can I draw about my group (or in general)?
What new questions do I have about my group (or in general)?
Your journal will be graded largely on effort, creativity, and your accurate application of class topics to the study of your group. To that end, be sure to write down evidence of your explorations, even if you don’t think they have led to new insights or discoveries. Often discoveries in mathematics are realized in hindsight, or what seemed like trivial observations at the time become crucial details later on.
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