# “What if Pascal’s triangle was a square instead of s triangle?”

Purpose:
You will write two Mathematics Discovery Papers this semester. The purpose of a MDP is to practice writing original mathematics results. You will identify a topic that interests you, pose an original question related to the topic, and explore answers to your question. Note that a MDP is NOT a report on mathematics that has already been done. You are NOT summarizing the existing findings of past/current mathematicians. Instead, this is an exercise in creating/discovering new mathematics.

Formulating a Question
New mathematics questions often arise from looking at another mathematician’s work and thinking, “What if instead…” or, “I wonder if…” For instance, we might work on the Fibonacci Sequence in class and you might find yourself thinking, “Would different patterns arise if the Fibonacci sequence started with 2, 2, instead of 1, 1?” Or, when we’re exploring number systems in different bases, you might think, “I wonder how addition would be different if our number system was base-5 instead of base-10.” The question you explore should arise from genuine interest. It is essential for a mathematician to be genuinely curious about the questions they solve. This will motivate you to keep working through the question even when you encounter difficulties. When determining a question to explore for your MDPs: if you find yourself wanting to keep playing around with something outside of class, this is a good sign that something related to the topic is a candidate for a MDP.

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Writing a Partner Paper:
MDPs are partner assignments and both partners will receive the same grade. Consider your “community contribution” when working with a partner. Would you want to be partnered with yourself on this assignment? Are you being a responsible and caring partner?

It is recommended that each partner SEPARATELY explore your question to start. Then, come together to compare methods and share findings. Together decide a specific route to explore further. DO NOT write up your findings as two separate explorations. Together, determine a way to share your findings cohesively. This may mean abandoning one partner’s work to focus instead on what the other partner has been doing. Basically, the reader should not be able to tell that there were two authors on the paper. It should read like one continuous piece.

Guidelines:
You are modeling your MDPs off of mathematics exploration papers published in Involve. Several sample Involve articles are posted on Canvas. These articles provide a model for what you will produce. If you find yourself stuck on how to write a certain section of your MDP, turn to the sample publications in Involve for guidance.

A template for the MDPs is provided on Canvas. Use this template to write your MDPs. The template contains detailed information about how to structure your paper and what to include. Read the template carefully before beginning your MDP.

A scoring guide for the MDPs is provided on Canvas. Read the scoring guide carefully before beginning your MDP.

Format:
The entire paper is a maximum of 10 pages long, using standard LaTeX formatting, not including the reference list. The template provides more detail on the contents of each section, but as an overview, your MDP will contain:

A brief abstract of 150 words of less that summarizes the contents of the entire paper
A 1-2 page introduction that reviews past work related to your topic, shares the importance of your work, and clearly defines your question for the reader
A results section with a maximum of eight pages which reports on what you found through your exploration of your question
A conclusion of approximately ½ a page to summarize what you did and what you found, and to state potential areas for continued research.
A reference list with all cited sources in APA format. Consult https://owl.purdue.edu/owl/research_and_citation/apa_style/apa_formatting_and_style_guide/general_format.html (Links to an external site.) for guidance.
As stated, more information on the contents of each section can be found in the template. You should also continue consulting the sample Involve papers.

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