Dr. Sarah's What is a Mathematician?     A Teaching Assignment (Presentation and Worksheet)

In this segment we will examine the way that mathematicians do research and the kind of problems that they work on. While we will mainly focus on the mathematics, you should try and identify with the mathematicians and their struggles and relate this to the way that you do mathematics, and you should also continue to think about what mathematics is, and the useful problem solving techniques that arise from its study. We will highlight the validity of diverse styles and diverse mathematical strengths and weaknesses. We will see that there are lots of different ways that people are successful in mathematics. We will also examine the changing roles of women and minority mathematicians over time.

I have worked hard to accumulate good references for you so that you won't have much research to do. I will give you both web and paper references. Your job is to learn the material and figure out how to teach it to the rest of the class. I am handing the assignment out early to give you plenty of time to examine the material and let it sink in and jell, and to allow you time to make it into office hours. This process is important for success on this topic, so do not leave this until the last minute! Instead, you should work on the mathematics a little bit at a time, reading and engaging it for about an hour or two, then putting it aside for a day or two, and then coming back to it and trying again and again and again.

The class will also need to repeat their exposure to each topic in order to reinforce learning. Before each presentation, the class will have been exposed to some readings on the topic. After your presentation, I will go over the mathematics. The length of time that I take to go over the mathematics is not necessarily an indicator that there was anything wrong with the presentation. A good presentation will answer the questions (below) and follow the presentation checklist points (see the web page link). Sometimes I will go over additional material or material at a deeper level than I would expect of a presentation. Everyone needs to learn the material that I go over in preparation for the final and so notes should be taken. The class will then complete the worksheet that you have created in order to engage them with the mathematics. For each worksheet the class will give constructive suggestions for improvement based on my worksheet checklist guidelines. Completion of the activities AND suggestions for improvement will both count as a homework grade. Because you will be receiving 2 worksheets every class, you will end up with numerous papers for this segment. This, in addition to notes you take when I go over material, will serve as your "text" for this segment.

Because this is a teaching assignment, we are using classroom worksheets that you create in order to satisfy part of the writing designator. In the effort to save paper (and trees!), you must return the paper references to me at the end of the semester. I will try to save paper when I can by using web readings.

Answer the Following Questions on the Mathematics that is Related to your Mathematician

Be sure that you include answers to the following questions about mathematics that is related to your mathematician in your presentation. Some answers may be brief so ask me if you are not sure of the expected depth. Be sure that you engage the class with the material in your worksheets, and that you explain the mathematics in your own words. I also recommend that you bring drafts into office hours and class to discuss them with me. You are responsible for the material from other people's presentations, worksheets and the material that Dr. Sarah highlights for WebCT quizzes and for the final exam. See below and the checklists for more pointers on the presentation and classroom worksheet that you will prepare.

Thomas Fuller (1710-1790) Speed of Mental Calculations, Calculator and Computer Time
  • How would we do Fuller's calculation of the number of seconds a man who is seventy years, seventeen days and twelve hours old has lived by hand?
  • What affects calculator and computer calculation time?
  • Compare Fuller's times to various calculator and computer times (the first calculator, the first computer, eniac, and modern calculators and computers).
  • Is there a limit to how fast a computer can calculate?

    Maria Agnesi (1718-1799) Witch of Agnesi and Calculus
  • What is the witch of Agnesi? How did it receive that (derogatory) name?
  • How do you construct it geometrically?
  • If the generating point is dragged far enough to the right, why won't the point generated on the curve be located at y=0?
  • How does it relate to Agnesi's mathematics?
  • How is it used in real life?
  • What are some of the applications of calculus to real-life?

    Sophie Germain (1776-1831) Sophie Germain Primes and RSA Coding
  • What is a Sophie Germain prime?
  • Why did she come up with Sophie Germain primes?
  • What is the largest Sophie Germain prime that has been found? How many digits does this have?
  • What is the definition of a mod b?
  • How do modular arithmetic and Sophie Germain primes relate to RSA coding?

    Carl Friedrich Gauss (1777-1855) Non-Euclidean Geometry
  • What is Euclid's 5th postulate?
  • What is the form of the 5th postulate that we learn in high school? What is the negation? What two geometry possibilities does this negation give rise to?
  • How does this relate to Gauss?
  • What are some other areas Gauss worked on?

    Georg Cantor (1845-1918) The Size of Infinity
  • What are the natural numbers?
  • What are the real numbers?
  • Using an argument that is similar to Dodgeball, show there more real numbers than natural numbers.

    Srinivasa Ramanujan (1887-1920) Chebyshev's Theorem and Estimating the Number of Primes Less than a Given Number.
  • What is a prime number? Why are they of interest?
  • What is the statement of Chebyshev's Theorem? How does this relate to Ramanujan?
  • What were Gauss' estimates of the number of primes less than or equal to a given number? How does this relate to Ramanujan?
  • Why are people still interested in Ramanujan's notebooks?

    Paul Erdos (1913-1996) The Party Problem
  • What is the statement of the party problem? How does this relate to Erdos?
  • Why does a 6 sided polygon colored red with a 6 sided embedded star with edges colored blue prove that 5 people at a dinner party is not enough to ensure that there are at least 3 people who are either complete strangers or acquaintances?
  • To show that if there are 6 people at a dinner party then there are at least 3 people who are either complete strangers or acquaintances, once we reduce to just looking at 4 of those people (3 who all either know or don't know the first person), how do we complete the argument from here?

    David Blackwell (1919-) What is Game Theory?
  • What is the prisoner's dilemma? How does this relate to what David Blackwell worked on?
  • Give an array of payoffs filled with numbers that represents the prisoner's dilemma, and explain how to read the array.
  • If a person is deciding what to do, why does it make sense (when looking at the possible cases) for him to confess?
  • What are some applications of game theory to real life?

    Mary Ellen Rudin (1924-) What is Topology?
  • What is topology? How is it different from geometry and projective geometry?
  • Why are a basketball and a football the same in topology?
  • Why are an iron and a mug with one handle the same in topology?
  • Why is a vest and a mug with two handles the same in topology?
  • Why are objects with different number of holes different from each other in topology?

    Frank Morgan (195?-) The Double Bubble Problem
  • Why is the sphere the least area way to enclose a given volume for a package?
  • If a sphere and a box have the same surface area, then which will have the largest volume?
  • What is the Double Bubble Problem?
  • What did Frank Morgan prove about it?
  • Did he have to check every possible double bubble?

    Ingrid Daubechies (1954-) Wavelets
  • How can images be stored on a computer?
  • What is image compression? Why do we need it?
  • How are wavelets related to image compression?
  • Why are wavelets better then JPEG?
  • What are other applications of wavelets to real life?
  • What do wavelets have to do with Ingrid Daubechies?

    Worksheet - Counts as the Major Writing Assignment for this Topic - copies for the class DUE at the beginning of class on the day of your presentation.

    Your job in this assignment is to create an effective classroom worksheet that engages the rest of the class with the mathematics for your mathematician and satisfies the checklist. The first draft of this worksheet is worth 50% of the grade for this teaching assignment and is due at the beginning of your presentation. Bring copies to hand out to the class. These are DUE at the beginning of class on the day of your presentation, but I am happy to make the photocopies for you if the completed worksheet is given to me a day in ADVANCE during office hours. Otherwise you must make the copies yourself. Be sure to follow the directions on the worksheet checklist. The final version of the worksheet is worth 50% of the worksheet grade and must be turned in a computer file attached onto WebCT (see the directions on the worksheet checklist) along with your graded original and graded checklist. The Carolyn Gordon worksheet is a good model for you to follow. You will receive another worksheet on Andrew Wiles as a second model.

    15-22 minute PowerPoint Presentation - Counts as a Major Topic Exam Grade - DUE before 1:30 pm on the day of your presentation, NOTES DUE after your presentation.

    Prepare a 15-22 minute PowerPoint presentation. The computer file (whatever.ppt) must be sent to Dr. Sarah as an attachment on WebCT before 1:30 pm on the day of your presentation. It is your responsibility to make sure that this is received by Dr. Sarah and runs correctly. Do not expect to load your presentation from a disk - the computer does not have a disk drive - but do also have the presentation backed up on disk so that you could send it from school if necessary. This presentation counts as a major exam grade, so be sure to follow the directions, the presentation checklist and to prepare great presentations! In the past, to supplement their presentations, groups have brought in manipulatives, and one group even created a claymation video about the mathematics! Poster are not a good idea since they are hard for the class (someone in the back row) to read. You should present the answers to the above questions very slowly - as you are planning your pace for the class, remember that it took a while for the material to sink in for you.

    Oral presentations my be summed up as follows: "Tell them what you're going to tell them. Tell them. Then tell them what you told them". In the introduction you tell them what you are going to tell them. In the main portion of the talk you tell them. In the conclusion you tell them what you told them. Don't be scared of this repetition. Sometimes repetition is the only way to clarify misconceptions. Naturally, this means that you should repeat things in different ways, and not quote yourself verbatim.

    Group Work

    Group work on major assignments will be self evaluated and these evaluations will be taken into account in the determination of the final grade. So, your job is to make sure that you do your part to make sure you are working in a group effectively. Inequalities in group work WILL be addressed.