Mathematics: Home

Identify and Develop a Topic

Selecting a research topic is much like deciding on a travel destination. Once you have narrowed your ideas to an interesting subject, write down a brief statement about this topic. For example: "Rock groups of the 60s, their popularity and major influence on the music industry."  Once the topic is selected, write down specific questions that you'll want to answer. The research process will drive your destination. Your original topic may develop into something entirely different. You may choose to follow an alternate path and go down a different road.

Find Background Information

After determining the topic, you can map out your route. You must identify the types of sources that will provide the information needed, then determine where to find these sources. Types of sources that should be considered include books, periodicals, the Internet, and other libraries (through interlibrary loan).

The key to finding books is the online library catalog called Discovery.  Search in Discovery by selecting a keyword that best describes your topic. You can also search by title, author, subject, or keyword. In addition to books, the catalog allows you to search for periodicals, government documents, audiovisual material, and Special Collections.   

 If you get lost during your sight-seeing trip, stop and ask directions. The following reference sources will be most useful to acquire quick answers to any questions you may have.

•  Begin with Encyclopedias, then to get off the main drag, use Subject Encyclopedias.

•  What does it mean?  Use a Dictionary.       

•  How much, how many?  Find Statistical Information.

•  Who?  Find Biographical Information.

•  How can I get in touch?  Use a Directory.

•  Where do I go from here?  Bibliographies.

Find Journal Articles

Periodicals include newspapers, magazines, and journals. They are published regularly, daily, weekly, monthly, quarterly. Journals are periodicals containing articles written by experts in a particular field of study. If the researcher wrote the article, is it a primary source. If reporters write the article, such as in popular magazines, it is a secondary source. Typically, journal articles contain extensive bibliographies that lead to additional sources.

Journal List - If there is a specific journal that you are looking for, this will take you directly to McFarlin's holdings.

Discovery - If there is a specific article that you are looking for, you can search Summon with that article's title.

Database List - If you don't have a specific journal or article in mind, then McFarlin's databases will allow you to search multiple journals with a keyword.

Writing Proofs

Mathematics majors spend a significant portion of their studies learning how to construct and write rigorous mathematical proofs. Here are some key aspects of writing proofs in mathematics:

Understand the Statement: Carefully read and analyze the statement or theorem to be proven. Identify the given information (hypotheses) and the conclusion to be derived from it.

Define Relevant Terms: Explicitly state the definitions of all mathematical terms, variables, and notation used in the statement. This ensures clarity and avoids ambiguity.

Outline the Proof Strategy: Decide on the appropriate proof technique, such as direct proof, proof by contradiction, proof by induction, or others. Outline the general approach and the logical flow of the proof.

State the Initial Assumptions: Begin the proof by clearly stating the given information or hypotheses. These are the starting points from which the proof will proceed.

Write in a Logical Sequence: Construct the proof by presenting a sequence of statements, each logically following from the previous ones. Justify every step by citing relevant definitions, axioms, theorems, or previously proven results.

Use Proper Mathematical Notation: Express mathematical statements precisely using correct notation, symbols, and logical connectives (e.g., implies (⇒), if and only if (⇔), for all (∀), there exists (∃)).

Provide Explanations: Accompany each step with a clear explanation or justification, ensuring that the logical reasoning is transparent and easy to follow.

Consider Special Cases: Address any special cases or exceptions that may arise, ensuring that the proof holds for all possible scenarios covered by the statement.

Conclude the Proof: Explicitly state the conclusion, showing that it logically follows from the previous steps. Optionally, use a symbol like "Q.E.D." (quod erat demonstrandum, meaning "which was to be demonstrated") to indicate the end of the proof.

Writing Problem Sets

Writing problem sets is a great way to gain mastery of new skills learned in class. They also reflect the nature of the scientific process and help reinforce the explanatory power of the discipline. There are several key elements of a problem set:

Understanding the Concepts: Ensure you have a solid grasp of the mathematical concepts, definitions, theorems, and techniques covered in the problem set.

Reading Problems Carefully: Read each problem statement carefully, identifying the given information, unknown quantities, and the specific question being asked. Understand any assumptions or constraints provided.

Planning Your Approach: Before starting calculations, plan your approach to solving each problem. Consider which mathematical tools, strategies, or theorems might be applicable. Sketch diagrams or make notes if needed.

Showing Step-by-Step Work: As you work through each problem, show all your work and calculations step-by-step. Use proper mathematical notation, labeling variables and units clearly.

Justifying Steps: Provide explanations and justifications for each step in your solution. Cite relevant definitions, theorems, or properties you are using.

Including Examples and Diagrams: When appropriate, include specific examples or diagrams to illustrate your reasoning or clarify your solution.