Real Analysis and Trigonometry for Middle-Grade Teachers, PSMM 228

**DC FAME, **
**June 24** – **July 2 and July 28**
– August 6, 2010

PSMM 228 classes will be held 2:30 PM - 5:30 PM.

**Your instructor: **

My name is Fred Rickey. I teach at the U. S. Military Academy, West Point, NY. My primary interest is the history of mathematics and I have special interests in the history of the calculus and in using history to motivate students. To find out more about me you can look at my automathography, which was written in August of 1998 when I joined the USMA faculty.

**Textbooks:**

1. Gelfand, I. M. (Izrail Moiseevich) and Saul, Mark, *Trigonometry, *
Boston: Birkhäuser, 2001.

2. Toeplitz, Otto, *The Calculus: A Genetic Approach*, with a new
forward by David Bressoud. University of Chicago Press 2007. The first edition
was published in 1949 in German.

3. Stewart, James, *Calculus: Concepts and Contexts*, Thomson -
Brooks/Cole, 2006. Copies will be distributed on the first class day. Caution:
Not everyone will have the same edition of the book, so be careful with page
numbers. If the page numbers seem inappropriate, so some problems from the
sections we are considering.

- Something you can do that will help your students: As you browse through the Stewart
Calculusmake note of each time he uses some result from trigonometry; then look the result up in Gelfand/Saul and make a cross reference to each of the books. Similarly you should take note of algebraic facts that are used. Armed with this information, you can answer the question all students ask: "What are we going to use this for?" Also, make note of applications of the mathematics that are discussed; they provide a powerful answer to this question.

**Reference Books:**

1. Katz, Victor J., *A History of Mathematics: An Introduction*, Second
edition, Addison Wesley Longman, 1998.

2. Usiskin, Zalman; Peressini, Anthony; Marchisotto, Elena Anne; and Stanley,
Dick, *Mathematics for High School Teachers: An Advanced Perspective*,
Prentice-Hall, 2003.

3. I have brought a box of books along to consult as needed.

4. Any other books you may have.

5. Of course you should use the internet, but be cautious.

**Aims of this course: **

- To give life to your knowledge of mathematics.
- To provide an overview of trigonometry and calculus so you can see how your students will use the mathematics you teach them.
- To teach you how to use the library and internet (important tools for life).
- To show you that mathematics is part of our culture.
- To indicate how you might use the history of mathematics in your future teaching.
- To improve your reading and analytic skills, especially in a technical situation.
- To improve your oral and written communication skills in a technical setting.

**What is expected of you?**

**1. Automathography**: The first assignment is to send me via email an
automathography. You are to introduce your mathematical self to me. Tell me
about the courses you have taken, what your favorites were, what you find
hardest. Explain what your mathematical interests are. Tell me about your
teaching. Where and what do you teach. Reveal what you expect to get out of this
course. If your aims for taking this course are different than those stated
above, please let me know. If you have any anxieties about this course, or any
special problems or needs, let me know. You are encouraged to be creative in
your response; don't be pedantic and just answer the questions asked above;
include whatever you wish. Send your response by email by Monday, 28 June
2010. Since I will be unable to access my school email address (fred-rickey@usma.edu),
please send your automathography to my home address:
Fred.Rickey@gmail.com . 10% of grade,

**2. Keep a journal:** As you read the material
the night before it is to be discussed, make notes about the most important
points. More importantly, make note of what you don't understand and
write down questions you plan to ask in class. In
class take notes on the things that are not in your earlier notes. After class,
reflect on what you have learned. How does this material relate to previous
things you have learned. Do you have new questions? Are there things you would
like to know more about? What have you learned in discussing this material with
your classmates. Did you talk to your family about any of this material. Don't
just reflect about the previous day, go back to the beginning. Are things that
troubled you earlier now coming into focus? Reflecting on your learning, and
especially on your teaching, is the way to improve. Make notes about what you
could do better.

**3. Two presentations, each with a write up.** The
first presentation will deal with trigonometry and I have
some suggestions. However you may choose
another topic, but clear it with me first. You are to work in teams of two and
each is to do part of the presentation. You also need to write up your
presentation. The first presentation is on 31 July and it is worth 15% of your
grade. The second presentation is on August 8 is for 20% of your grade. Turn the
write up in at the same time; only one per team is necessary. The cadets at West
Point say "If you wait till the last minute, it only takes a minute," but you
should not follow their advice. Each of you should write a first draft, then exchange them and
edited them severely. Then rewrite, rewrite, rewrite.

**4. Homework:** Work together! Work face-to-face if
possible. This is the best way to learn. But work on the problems some before
you work together. There will be 5 homework sets, each worth 10% of your total
grade. Each homework assignment will consist of 5 problems. Doing three
correctly earns full points. Each extra one you do earns an extra point, i.e.,
1% of your total grade. You must write the problems up individually. Be
complete, be precise, be concise. Be sure to give a full explanation.

**5. What happened on this date?** Details below. 5% of
grade.

**Grading:** As in Olympic figure skating, your score is based both on
technical merit and on artistic expression.

**Daily Events: **

1. **What happened of mathematical significance on this date? **This
is a really great way to get students excited about mathematics. Spending five
minutes at the beginning of a class or somewhere during the day to discuss some
historical event is most rewarding. I will do this for the first day or so, and
then turn it over to you. For each day of this course, a group of two or three
you will spend a few minutes discussing historical events of the day. I want you
to present the material as if you were presenting it to your own class; at the
end everyone will have a chance to ask questions. To get you started I will give
you one date from my Calendar of Mathematical Dates. You can use one or more of
those events, or go out on the web to find something of your own choosing.
Remember the idea is to excite your students. The information I will give you
comes from my
Calendar of Mathematical Dates. One of my colleagues has created a web page
about
Today in Math History which you will be able to use in the fall.

2. **Stump the Prof.** This will be your opportunity to ask about
anything you wish. Perhaps you would like more detail about one of the
mathematical dates we have discussed. Perhaps you want to know about the
Archimedes Palimpsest. Perhaps you want to know how old I am. Everything is fair
game. As teachers you need
to be inquisitive about all of mathematics. My cadets at West Point have adopted
this practice on their own (is it to slow me down?). This should be
fun.

3. **Each day we will do some trigonometry and some calculus. **Details
on what to read are given below.

4. **Board Problems.** This is a tradition at West Point. Each day
each cadet gets a single sheet of problems to work on. We are blessed in that
there are blackboards all around the room so all of the students can go to the
board at once (the classrooms only have 18 seats). Since we don't have boards at AAAS,
you can work on these problems in groups at your seats and present solutions
using the overhead. These board problems will be a mixture of questions to make
sure we understand the basic concepts and then some to test your mettle.

5. **Quote of the day. **Another great way to get students
interested in mathematics. There are several collections of quotations to draw
from and there are many web sites that collect quotations. I encourage you to
*bring in your own favorites* to share with your colleagues.

**Detailed Syllabus: **

** **
Philosophy: Go as fast as possible, but as slow as necessary.

Consequently, this syllabus is subject to change. Learning is more important than a schedule.

**Day 1: Thursday, June 24, 2010. Get acquainted day.** This will be our day to get
to know one another. But we also want to do some mathematics. I will talk about
the two basic questions of the calculus and will also discuss the history of
trigonometry and begin the discussion of triangle trigonometry. If possible,
read the following before our class and come prepared to ask questions and to
discuss the material. On subsequent days, I expect that you will have done the
readings in advance.

- Read Gelfand/Saul: Preface, pp. xi-x, and pp. 1-12.
Read your book!
- What value is there in reading the Preface? Who are the people that are mentioned?
- What are the two most important triangles in trigonometry?

- Things to read if you have time:
- Face to Face: Teachers and Mathematicians, by Mark Saul.
- "What is Area." In Usiskin, pp. 478-484.
- What was the Greek concept of Area? What does Quadrature mean? Read Katz, section 1.5.

- Homework #1 (due Monday, 28 June): From Gelfand/Saul: p. 12, #3; p. 23, #1; p. 26, #3; p. 30, #5; and p. 31, #1.
- Board Problems 1.
- The Quote of the Day is from Euler.
- We will have a pre-test today to see what you already know about trigonometry and calculus and to judge what you learn. I suspect the mode of the scores will be 0 of 4. This has no bearing on your grade, but do your best.

**Day 2: Friday, June 25, 2010. Trigonometric ratios*** *is the topic of
the day.

- Read Chapter 1 of Gelfand/Saul, omitting sections 9 and 10. This is a lot of pages, but what you need to know is the definition of the sine, cosine, and tangent. Do enough exercises so that you have these definitions down cold. Nice exercises are #2 p. 25, #3 p. 27, and #1 p. 31.
- Read "Etymology of the word 'sine'
".
- Why is it called the
*co*sine?

- Why is it called the
- Read Stewart: pp. 2-9.
- What do you tell your students about Area?
- How are the tangent problem and velocity related?
- What are inverse problems?

- Read section 3, The Exhaustion Method of the Greeks, pp. 11-14 in the Toeplitz book.
- Board Problems 2.
- You know the formula for the area of a circle, and the formula for the
circumference of a circle. Both involve . But how do you know that the in the two formulas are the same? The first proposition in
*Measurement of a Circle*by Archimedes will explain. Read it in advance; then we will read it together in class. Admittedly, this is a hard read. - The Quote of the Day is from Birkhoff.
- Can you stump the Prof?

**Day 3: Monday , June 28, 2010. **

- Read sections 1-4 and 10-11 of Chapter 2 of Gelfand/Saul (of course you
can read more if you are inspired).
- Why do we prove trigonometric identities?

- Calculus topic of the day:
**Derivatives**. Read Stewart, pp. 93-97 and 139-143. - Read section 18, Tangent Problems, pp. 77-80, in Toeplitz.
- Board Problems 3.
- Homework #1 is due today.
- Read "The Product Rule."
- Read "Why the Sine has a Simple Derivative."
- The Quote of the Day is form Kemeny.
- Have you ever heard of Arnold Ross?

**Day 4: Tuesday, June 29, 2010. **

- Read sections 1-5 and 9 of Chapter 3 of Gelfand/Saul.
- Exercise: Translate Euclid II.13 into a formula.

- Read "Bolzano's Intermediate Value Theorem."
- Board Problems 4.
- The Quote of the Day is from Copernicus.
- Homework #2 will be assigned today, due Thursday (I want to hear your reactions about the first HW).

**Fay 5: Wednesday, June 30, 2010. **

- Read Chapter 4 of Gelfand/Saul.
- Why do we speak of odd and even functions? Look at Stewart, p. 611.

- Calculus topic of the day:
**Integrals**. Read Stewart, pp. 343-352. - Read section 11, The quadrature of the parabola by Archimedes, pp. 43-51 in Toeplitz.
- Read section 15, The concept of the definite integral, pp. 62-68, in Toeplitz.
- Read "The Integral Sign."
- Read "Fermat's Integration of Powers."
- Board Problems 5.
- The Quote of the Day is from Euler.

**Day 6: Thursday, July 1, 2010. **

- Read Chapter 5 of Gelfand/Saul, sections 1-4.
- HW #2 due; HW #3 will be assigned, due the day after the break.
- Board Problems 6.
- The Quote of the Day is from MacLane.

Day 7: **Friday, July 2, 2010. First Presentation. **

- Here are some problems to sink your teeth into.
- The Quote of the Day is from Whitehead.
- Can you stump the Prof?

Summer Break: I am off to Greece and Italy. You can keep up with where I am if you wish.

Here is a quote for the break from John Adams.

**Day 8: Wednesday July 28, 2010. **

- This will be a catch up day. Come prepared to ask your unresolved questions.
- Chapter 8 of Euler's
*Introductio in analysin infinitorum,*is entitled "On transcendental quantities which arise from the circle." You should skim through this ahead of time, but we will read it in class as a group.- What are the new ideas here? Think about this as you read it. You will be surprised.

- Calculus topic: The Fundamental Theorem of Calculus. Read Stewart, pp. 377-383.
- Read section 23, The Fundamental Theorem, pp. 95-99 of Toeplitz.
- Read section 29, Trigonometric Functions, pp. 113-116 in Toeplitz.
- HW #3 is due; Here is HW #4, which is due Friday.
- The Quote of the Day is from Polya.
- Suggested topics for your second presentation.
- Board Problems 8.

**Day 9: Thursday, July 29, 2010.**

- Read Chapter 6 of Gelfand/Saul, sections 1-2, on Addition Formulas.
- Read the section of stewart on Maximum and Minimum Volumes (perhaps section 4.2, depending on you edition).
- Read "Torricelli's Trumpet."
- Board Problems 9.
- The Quote of the Day is from Pascal.

**Day 10: Fri****day, July 30, 2010. **

- Chapter 7 of Gelfand/Saul deals with trigonometric identities. Skim through the chapter and take note of these identities.
- Read "Saint Vincent and the Logarithm."
- Read section 12, Continuation after 1,800 years, pp. 52-57 in Toeplitz.
- HW#4 is due; Here is HW #5, due Thursday, August 5.
- Board Problems 10.
- The Quote of the Day.
- Can you stump the Prof?
- Suggested topics for next Thursday and Friday are most welcome. They need not be directly related to what we have done thus far, but can involve any topic that is of interest to several of you.

**Day 11: Monday, August 2, 2010. **

- Read sections 1-6, 10, and 14 of Chapter 8 of Gelfand/Saul, Graphs of Trigonometric Functions.
- Finally we get to the graphs of the trigonometric functions.
- We will use part of this day to work on your final presentations.
- Read "The Bridge and the Catenary."
- Board Problems 11.
- The Quote of the Day is by Conway.

**Day 12: Tuesday, August 3, 2010. **

- Read sections 1-3 of Chapter 9 of Gelfand/Saul, Inverse Trigonometric Functions.
- Read "History of the Brachistochrone."
- Here is an email about computing π. Is the guy a crank?
- Board Problems 12.
- The Quote of the Day is by von Neumann.

**Day 13: Wednesday, August 4, 2010: Final Presentations.**

- You may choose any topic you wish, with some reasonable restrictions.
- Your topic must be mathematical, involving either trigonometry or calculus.
- Work in groups of 2 or 3.
- Try to do something that you might be able to use in your classroom (or as a presentation at a mathematics meeting).
- Applications are great; History is encouraged.
- Here is your chance to
*talk*about mathematics. It is not necessary to do all the details.

- I will be happy to consult with you about your presentations. I will try to steer you away from projects that are overly ambitious or not appropriate.
- The Quote of the Day is by Shenitzer.

**Day 14: Thursday, August 5, 2010.**

- Finish up the presentations.
- Today and tomorrow will be devoted to whatever you wish. We can tie up loose ends, do some more problems, talk about history, whatever you wish. So think about this as our course goes on.
- There will be a post-test today to see what you have learned. Hopefully, everyone will get at least 3 of 4.
- The Quote of the Day is from Gilbert and Sullivan.
- HW #5 due.

**Day 15: Friday, August 6, 2010. **

- Today we resolve any and all outstanding issues.
- The Quote of the Day is by Caballero.
- Can you stump the prof?
- We will devote some time to reflection. What can you do with your new knowledge of mathematics?

Here are some Hints for Success that I give to my students.

Views on the High School Curriculum.

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