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Instructional Design
The following document contains a brief outline of the sequence of topics I teach in my calculus course. I begin my course with an in-depth review of precalculus skills for two major reasons, the first, and most obvious, of which is that gaps in students’ understanding of these concepts, if not diagnosed, can cause undue difficulty in the assimilation of the concepts encountered in calculus. Moreover, these gaps, if not found early, can be masked as problems with calculus rather than what they actually are which makes them harder to fix later in the semester. The second reason is that this review allows me to treat subjects with which students are already familiar (my treatment of functions, for instance, is extremely general, requiring students not only to recognize relationships between non-mathematical sets as functions but also to properly describe such functions as being injective or surjective) at a greater degree of abstraction, thus allowing them to develop the abstract reasoning ability required for parts of calculus. Over the course of this review most students begin to realize that the ability to defend an answer and understand the underlying mathematics conceptually is valued more in my class than simply arriving at a correct answer through procedural knowledge. In addition, during this review I am able to stress the connections between functions and their graphs which I can then build upon in the calculus section of the course.
As regards my particular treatment of functions in this review, I first introduce functions as a specific type of a relation between sets using concepts from elementary set theory. After students are comfortable with this definition of a function (and the concepts of injective and surjective functions), I introduce the graph of a function as the locus of all points that are solutions to the given equation. This approach allows me to specifically emphasize the interplay between the graphical and numerical representations of the function and leads naturally into more advanced concepts such as the average rate of change of a function over an interval (which I approach both graphically and numerically) and elementary curve sketching techniques (where I can again emphasize the relationship between the numeric and graphical representations of a function through end behavior, an analytical concept). I find that this approach leaves students better able to deal with more advanced (often analytical) concepts relating to functions that arise later in the course such as the interpretation of the derivative and definite integral as functions (such as is found in the second fundamental theorem of calculus, for example). Throughout my review of functions (and later as related concepts arise) students are expected to be able to make verbal arguments of their solutions and approaches to problems and to communicate clearly (both verbally and in written assignments) both their positions on problems dealing with these concepts and, more importantly, the reasoning that led them to that position.
I try to continue this emphasis on conceptual understanding into the portion of the course in which we cover calculus. This takes several forms. Often, to introduce a new topic I first pose the topic as a subject for debate before the class. This helps both the speakers, who must come up with a convincing argument for their side, and those that don’t speak, who must find flaws in the speakers’ arguments, develop their reasoning skills. I also try to incorporate a more abstract understanding of the processes of calculus through extensions to topics. For instance, while my primary (conceptual) concern when we study limits is that students understand a limit as being able to get arbitrarily close to a value, I reinforce this understanding by covering the epsilon-delta definition of a limit. Though the use of that definition never shows up on one of my tests, I think it helps the students crystallize their more informal understanding of a limit.
While I believe that a true conceptual understanding of mathematics is necessary for students to be successful in my course and to retain their knowledge past my course, I am not a Luddite in that regard. Most of my students come to me with a basic working knowledge of how to use graphing calculators (which the school provides) and I incorporate those calculators whenever applicable. In problem sets on definite integrals, I usually throw in one or two functions not easily integrated so that students realize the value of the ability of the TI-83 to approximate such intervals. While I do have some assignments on which students cannot use calculators (to prevent students from becoming dependent upon them), many of my assignments not only teach students how to solve problems but also how to use the calculator intelligently (some students waste more time trying to do everything in a calculator than it would take to do the same task by hand) in doing so.
The book that acts as the primary resource for the class is the sixth edition calculus book by Larson, Hostetler, and Edwards. The school provides all students with TI-83 graphing calculators for use at school and at home.
Teaching Strategies
In my class I try not to simply teach but to facilitate learning. To this end, I frequently have students present problems to the class. After a homework assignment, I usually pick several particularly interesting problems from the assignment for students to present. If the assignment contains a particularly difficult problem, I sometimes have students work the problem as a class, with each student only allowed to contribute one step until everyone has contributed. I also assign research problems periodically throughout the course in which a student (or, in some instances, a group of students) is given a particular application of the topic we are studying which they must then research and then present to the class. These presentations (which include a written component as well as the actual presentation) are evaluated not by me, but by the other members of the class, who complete forms in which they not only rate the presentation but must then also comment on its particular strengths and weaknesses. As these presentations are usually on applications of calculus, the verifications that students provide of their solutions frequently make use of their graphing calculators; this is especially true of projects involving limits as students use everything from tables to graphs to programs they have written to illustrate (and provide verification of) their solution. I think that the use of calculators in this manner is particularly convincing to students and I encourage its use in these projects by having a ranking on the evaluation form for how convinced the observing student was of the accuracy of the solution.
Course Syllabus
I. Functions, Graphs, & Limits (8 weeks)
A. Functions
1) Functions as relations between sets (domain & range)
2) Definition of functions
3) Injective, surjective, bijective
4) Composition of functions
5) Inverses
B. Linear Functions
1) Slope as a rate of change
2) Parallel and perpendicular lines
3) Equations of lines
a) slope-intercept
b) point-slope
C. Non-linear Functions
1) Families of functions
(y=xn, y=|x|, y=1/x, y=x1/n)
2) Piece wise functions
3) Exponential functions
4) Logarithmic functions
a) Logarithmic functions
b) Properties of Logarithms
5) Trigonometric functions
6) Odd and Even functions
D. Limits
1) Intuitive definition of limits
2) Limit notation (including right and left hand limits)
3) Existence of a limit
4) Evaluating limits
5) Formal definition of limits
6) Limits involving infinity
7) Definition of Continuity
8) Discontinuity
a) Point discontinuity
b) Jump discontinuity
c) Infinite discontinuity
II. Derivatives (12 weeks)
A. Tangent lines
1) Local linearity
2) Secant lines
3) Tangent line at a given point
4) Instantaneous rate of change as limit of average rate of change
B. The Derivative
1) Definition of the derivative
2) Differentiability and continuity
3) The Power Rule
4) Properties of Derivatives
5) Derivatives of Trig. Functions
6) The Product of Quotient Rules
7) The Chain Rule
8) Derivative of exponential functions
9) Derivative of inverse functions
10) Derivative of logarithmic functions
11) Implicit differentiation
12) Related Rates
C. Applications of the derivative
1) Critical points and relative maxima
2) Graphical interpretation of the first derivative
3) Graphical interpretation of the second derivative
4) Applications of the derivative
5) Rolle’s Theorem
6) Mean value Theorem
7) Optimization
8) Linearization
III. Integrals (8 weeks)
A. Antiderivative
B. Approximating Area
1) Using a regular partition
a) Right and left endpoint
b) Midpoints
c) Using any point (proof of equivalence)
2) Riemann Sums
3) Definition of the definite interval
4) The Fundamental Theorem of Calc.
5) Definite Integrals and the Antiderivative
a. Average Value Theorem
b. Mean Value Theorem For integrals
6) The 2nd Fund. Theorem of Calculus
C. The Integral
1) Properties of integrals
2) Integration by parts
3) Approximating def. integrals
a. Trapezoidal rule
4) Area between Z curves
5) Volume
a. solids of rotation
b. solids with known cross-sections
6) Differential Equations
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