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Subject-Specific Research
Mathematics
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Welcome to the "Mathematics Research" page!  This is specifically devoted to making you a better mathematician and researcher.  Please complete the following steps:

  • Read the description. 
  • Examine the wonderful possibilities and implications of math research by watching the Math in Action video or reading the Andrew Wiles interview.  
  • To get some ideas about what you may wish to research, visit some of the websites in the resources tab.
  • Review the methodology section to get ideas on how to go about planning and conducting your research or proof.
  • Visit the resources section for miscellaneous information on gathering and reviewing information and some interesting or wacky math research facts.
description

Mathematics is the study of patterns, numbers, quantities, shapes, and space using logical processes, rules, and symbols.  Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.  Mathematicians investigate patterns, formulate new conjectures, and determine truth by drawing conclusions from axioms and definitions.   A mathematician can be an artist, scientist, engineer, inventor or straightforwardly, an independent thinker.   He/she is commonly more than one of these at once.

"Ethno-mathematician" Ron Eglash is the author of African Fractals, a book that examines the fractal patterns underpinning architecture, art and design in many parts of Africa. By looking at aerial-view photos -- and then following up with detailed research on the ground -- Eglash discovered that many African villages are purposely laid out to form perfect fractals, with self-similar shapes repeated in the rooms of the house, and the house itself, and the clusters of houses in the village, in mathematically predictable patterns. ......................................................................................
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Fractal
The above picture shows a fractal pattern.  A fractal is a shape made from repeating one pattern on different scales.  This repetition often makes the shape appear irregular. 

As he puts it: "When Europeans first came to Africa, they considered the architecture very disorganized and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."

His other areas of study are equally fascinating, including research into African and Native American cybernetics, teaching kids math through culturally specific design tools (such as the Virtual Breakdancer applet, which explores rotation and sine functions), and race and ethnicity issues in science and technology. Eglash teaches in the Department of Science and Technology Studies at Rensselaer Polytechnic Institute in New York, and he recently co-edited the book Appropriating Technology, about how we reinvent consumer tech for our own uses

 

Ron Eglash: Mathematician
Ron Eglash is an ethno-mathematician: he studies the way math and cultures intersect. He has shown that many aspects of African design -- in architecture, art, even hair braiding -- are based on perfect fractal patterns.

About this talk

'I am a mathematician, and I would like to stand on your roof.' That is how Ron Eglash greeted many African families he met while researching the fractal patterns he’d noticed in villages across the continent.
Typically, the study of math is divided into two major categories: pure mathematics or applied mathematics. Pure mathematical research involves significant mathematical exploration and the creation of original mathematics. Pure mathematics seeks to develop mathematical knowledge for its own sake rather than for any immediate practical use. Applied mathematics seeks to expand mathematical techniques for use in science and other fields or to use techniques in other fields to make contributions to the field of mathematics.  Boundaries between pure mathematics and applied mathematics do not always exist.    

math

What do the mathematicians have to say about math? 
Click the image above to find out!
Branches of Mathematics
algebra analysis application foundations geometry

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examples

Biographies of Mathematics Researchers

  • Google timeline of mathematicians and their discoveries from 300 BC to the present
  • Pythagoras (approximately 569B.C.)
  • Euclid (approximately 330 B.C.) Wrote comprehensive compilation of geographical knowledge used for over 2000 years.
  • Da Vinci (1452) Fifteenth century inventor/innovator who utilized math to detail and design inventions
  • Galileo (1564) Regarded as mathematic and scientific genius who was an innovator of the telescope that provided proof of Copernicus’ theory that earth revolves about the sun.
  • John Wheeler (1764) astrophysicist best known for coining the term, “black holes”
  • John August Roebling (1806) Pioneering architect of suspension bridges.  Designer of the Brooklyn Bridge, NY, NY.
  • Dr. Robert Goddard (1882) pioneering work with rockets is responsible for the NASA space program
  • Le Corbusier (1887) architect who developed the concept of modular human beings to design buildings based on anatomical geometric proportions
  • David Smith (1906) sculptor of geometric solids constructed from counterbalanced stainless steel shapes
  • Sylvia Earle (1935) invented “JIM”, a deep sea diving suit, requiring advance knowledge of engineering. 
  • Diana Eng (1983) fashion designer who integrates her knowledge of mathematics, science, technology and fashion to create a collection of “magical” clothing. 

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methodology

Overlap also exists in research in the field of mathematics. However, for our purposes, we will subdivide the fields of study by their primary purposes. In the chart below, the following purposes are outlined by Jeff Suzuki in But How Do I Do Mathematical Research?

Suzuki, Jeff. "But How Do I Do Mathematical Research?”." Mathematical Association of America, 2010. Web. 26 July 2010. <http://www.maa.org/features/112404howdoido.html>.

Proof:  This research seeks to justify a conjecture using logical reasoning.  This category also includes finding alternative justifications for previously-proven theorems. 
Extension: This research seeks to expand current mathematical concepts.
Application: This research takes an existing idea or theorem and applies it to a new area. 
Characterization: This research seeks to characterize or classify a mathematical object or concept.
Historical: This research seeks to systematically collect and objectively evaluate data related to past occurrences of a theorem or area of mathematics. 
Existence: This research seeks to prove existence of an object, which is a form of characterization. 
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Step
 Steps in the Math Proof Process - Adapted from Math Research FAQ
1
Look for patterns or interesting phenomena.
2
Craft/Recraft the proposed conjecture.
3
Turn all givens/statements to be proved into useful mathematical statements(abstraction).  To understand abstract concepts, try using examples.
4
Decode/Analyze the definitions.
5
Determine the areas of math as well as axioms and other theorems that may be helpful in the proof.
6
Locate similar proofs and compare/contrast proof methods.
7
Choose or experiment with a variety of proof methods.
8
Ask peers/professor to suggest ideas and/or check the logic in the proof.
9
Review logical progression of steps.  Clean up the proof to make it concise and beautiful.
.It may also be appropriate, especially for applied math research, to follow the steps typically used in the scientific method
Methods of Proof in Pure Math
 P---> Q Direct Proof Use deductive reasoning to work from a given statement (hypothesis) to the proposed conclusion.
P ^ ~Q ---> Error Proof by Contradiction Assume the hypothesis and the negation of the conclusion are true. Then, use deduction to arrive at a statement that is false. If the deductive reasoning is not flawed, then the contradiction resulted from the false assumption that conclusion was false Therefore, the conclusion is proved true.

n=1 True

Case n----> Case n+1

 Proof by Induction Show that the intended theorem is true for the first case. Assume the nth case is true and use deduction to show the (n+1)th case is true. If this can be done, it means the theorem is proved true for all cases.

P1----> Q

P2----->Q     -----> Q

P3-----> Q

 Proof by Cases: Divide the situation into non-overlapping cases which exhaust all possibilities for the problem.  Prove the statement using each individual case as a given.
~Q ------> ~P Contrapositive Proof Assume the negation of the conclusion and deduce the negation of the hypothesis.

X and Y work --> X=Y

 Uniqueness Proof Assume that two items satisfy the conclusion and show that the two items are identical.
Note: In the statements above, we use the term "hypothesis" to indicate the first part of an if-then statement and the term "conclusion" to indicate the second part of the if-then statement.  The term "hypothesis" can also be used to refer to the theorem that is being proved overall.  

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resources

General Resources

Statistics as a Research Tool

How to Find Ideas for Research Projects

Conducting Pure and Applied Math Research

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writing Writing  a Math Paper

Proof Writing

Software Tutorials for Conducting, Writing and Displaying Math Research

MS Excel Geometer's Sketchpad MATLab 
LaTex Maple Mathematica
Minitab SAS SPSS

 

 

 

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glossary

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teacher resources

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