Instructor: James Madden, 213 Prescott Hall, madden@math.lsu.edu, (225)-978-3525 (cell).
This course will use as a central resource a series of lectures given by Herb Gross at MIT in 1970 in a course called "Calculus Revisited". Here is a link to the course resource page:
MIT OPENCOURSEWARE: Calculus Revisited
MNS candidates enrolled in this course will watch selected lectures on line through the MIT Open Course Website at home and will study the accompanying notes independently. Detailed assignments are given below. Each assignment will require an email to me, which you should send before midnight on the specificed date.
The class will meet on four Saturdays in Spring 2011. At these meetings, we will discuss the contents of the lectures and make connections between them and the classroom challenges you face as teachers in middle and high school. The dates of the Saturday meetings are: February 5, February 26, March 12, April 9 and May 2 (make up date).
The course has the following five goals (which are keyed to the NSF MSP Key Features):
These are listed with the current assignment first, and then in reverse chronological order.
Watch Lecture 2: Functions (40 minutes). Email me when you have finished watching it. Include a 25-word response to the question below.
This lecture presents the idea of a function in a formal way. You may think this is all very simple, but it was not until the 20th Century that funtions were thought of in the manner that Professor Gross presents. Mathematics is not just about numbers (arithmetic and algebra) and shapes (geometry). Functions themselves are objects in the mathematical universe. We do things with functions that we cannot do with numbers or shapes (e.g., compose them, differentiate them, integrate them) and use them for purposes that are new and different from those for which we use numbers and shapes. You can download my notes on this lecture here.
TO DO. The Common Core State Standards for 8th-grade math include the following statement: “Students [in 8th grade] grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.” While watching the lecture, note the parts of the lecture where Professor Gross builds on ideas that are in middle- and high-school math. Question: What does he add, that a high-school student might not be familiar with?
Watch Lecture 1: Analytic Geometry (37 minutes). Email me when you have finished watching it.
The main theme of this lecture is the way in which geometric representations help us understand and represent what Professor Gross calls "analytic" concepts. This is a central theme in middle- and high-school algebra.
While watching the lecture make a list of the illustrations of the main theme that Professor Gross provides. After watching the lecture, go to the Common Core State Standards and download the math standards. Look for references to the specific illustrations that you saw in the lecture . Observe that the Grade 8 standards contain some of the very topics that Professor Gross discusses. Where are these themes currently in the curriculum you are using? Write up your obserbvations on a single sheet of paper, and bring that with you this coming Saturday.
Watch the Course Introduction. This is an overview, and includes descriptions of the most important concepts of calculus. Some of you may not understand everything being talked about. If there are some parts that are unclear, take note of the time stamp on the video. As you watch, write down questions you might want to ask me or your classmates, with references to the time, if appropriate. Bring these questions with you to the first Saturday meeting.
After watching the lecture, write summary in 25 or fewer words, and email your summary to me. I will compile all the summaries, remove names and return the list to you. You will cast votes for the most precise, most informative summary. I will announce the winning entry during the week of January 31.
RESULTS of the voting
|
Summary |
votes |
1 |
Instantaneous speed and distance traveled represent the two main branches of calculus, differential and integral calculus. They are related by the Fundamental Theorem of Calculus. |
|
2 |
Calculus has two main parts. Differential calculus studies rate of change and integral calculus studies area under a curve. Limits are the common thread. |
4 |
3 |
Calculus is broken into two areas, Differential and Integral Calculus, which are related by the area under a curve. |
|
4 |
The focus of the lecture was to revisit the concept that the 2 main branches of calculus, differential calculus and integral calculus, are related by area under a curve. |
1 |
5 |
Calculus is the study of limits. Differential calculus (study of instantaneous speed) and integral calculus (study of area under a curve) are its essential branches. |
1 |
6 |
The two branches of calculus are defined by rates of change and areas, related through area under a curve, and hinge on the limit concept. |
1 |
7 |
CALCULUS, the study of change, uses the limit process to approximate instantaneous slopes and areas under curves by inverse operations of differentiation and integration. |
1 |
8 |
The two branches of calculus are defined by rates of change and areas, related through area under a curve, and hinge on the limit concept. |
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