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.
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.
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 Calculus make 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.
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:
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 (firstname.lastname@example.org), 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.
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.
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.
Day 2: Friday, June 25, 2010. Trigonometric ratios is the topic of the day.
Day 3: Monday , June 28, 2010.
Day 4: Tuesday, June 29, 2010.
Fay 5: Wednesday, June 30, 2010.
Day 6: Thursday, July 1, 2010.
Day 7: Friday, July 2, 2010. First Presentation.
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.
Day 9: Thursday, July 29, 2010.
Day 10: Friday, July 30, 2010.
Day 11: Monday, August 2, 2010.
Day 12: Tuesday, August 3, 2010.
Day 13: Wednesday, August 4, 2010: Final Presentations.
Day 14: Thursday, August 5, 2010.
Day 15: Friday, August 6, 2010.
Here are some Hints for Success that I give to my students.
Views on the High School Curriculum.
Do you have some spectacular students? Let them know about MathPath. Can you do the Qualifying Quiz?