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Calculus
The OLI Calculus course is currently
under development and we are making the course available while it is under development. The course material you may access here will undergo significant changes over the next several months. As part of the standard OLI use-driven design process, we are currently testing our approach and presentation with learners and modifying and extending the instruction in accordance with those evaluations. We invite you to explore different parts of the course as it develops and give us feedback.
Following a review of the calculus education literature, an evaluation of the strengths and weaknesses of the teaching approaches of existing online calculus courses, and a review of common mistakes and misunderstandings that learners demonstrate following traditional instruction, we chose to take a unique approach to teaching the concept of the limit. We also chose to cover a number of analytical methods pertaining to graphing and to functions. Typically, such methods are not taught as standalone topics and may not be explicitly presented at all.
The material is organized in eleven units, marking major thematic subdivision. Each unit contains several modules of material. These units are:
- Unit 1: Introduction
- Unit 2: Graphs
- Unit 3: The Function: The Most Fundamental Calculus Tool of All
- Unit 4: Important Functions
- Unit 5: Local Behavior and Asymptotic Behavior of Functions:
- Unit 6: Pointwise and Global Properties of Functions
- Unit 7: The Meaning of the Derivative
- Unit 8: Computing the Derivative
- Unit 9: Graphing Using Derivatives
- Unit 10: Anti-derivatives and Proper Integrals: “Un-doing” Differentiation
- Unit 11: Improper Integrals
The first four units are essentially pre-calculus material. Inclusion of the first four units will significantly expand the potential audience for the course, because many institutions include pre-calculus material in their calculus curriculum. The four units could also serve as the basis of a separate pre-calculus course.
The concept known as the limit is the cornerstone upon which every branch of calculus is built. Mastery of this concept is absolutely central to the successful study of calculus. Yet many texts devote but a few modules to this core concept. Units 5 and 6 cover the limit with considerable thoroughness. We have placed the concept within a broader theoretical framework, that of analyzing function behavior.
The practice of teaching topical material is universal; teaching analytical skills is more uncommon. Our approach places the limit within a unique presentation designed to inculcate in learners a high level of confidence in their own analytical proficiency. We guide them away from reliance on recipe-type mathematical procedures so that they may carry out analysis with a higher degree of cognitive sophistication. The course’s first six units provide a conceptual framework within which we can later place the more technical aspects of differential and integral calculus.
By reviewing our organization of topics, and our presentation of the material, a mathematics educator would notice two significant departures from traditional calculus coverage:
- Between the pre-calculus material (Graphs and Functions) and the differential calculus material, we have two units devoted to analytical methods which, as mentioned, treats the limit with considerable thoroughness, and includes a preview of a number of function properties such as boundedness, monotonicity, continuity, differentiability, and integrability. In most calculus curricula, only continuity is discussed at this juncture.
- The limit is covered in a unit called Local and Asymptotic Behavior of Functions. We have chosen language which cues the learner to approach the limit in a very conceptual and analytical matter, understanding that mechanical/computational ability supports analysis but cannot replace it.
A traditional presentation of the limit intertwines the following two learning objectives:
- The learner should appreciate the variation in type of function behavior which may occur locally or asymptotically, and be able to describe it.
- The learner should master proper use of notation involving the lim symbol.
Our unique approach separates these two objectives. We gear the approach toward #1 first, because, whereas limits may not always exist, function behavior always does and can always be described somehow. We focus, then, on ways learners may describe function behavior verbally and understand it conceptually Objective #2 is addressed after learners have encountered a wide enough variety of behavioral phenomena; learner are then shown how to encode descriptions in proper notation. For this reason, we do not even use the language “the limit” until very late in the presentation of this material.
| Traditional Approach |
OLI Calculus Approach |
| Begin by introducing the limit, a mathematical object which may, or may not, exist in any given instance |
Begin by introducing function behavior, a notion which we cannot define but which we can demonstrate, a phenomenon which always exists |
| Introduce the term "the limit" from the outset, possibly including a precise definition and the traditional notation, i.e., the lim symbol |
Delay introduction of the term “the limit”, its definition, and its traditional notation, i.e., the lim symbol, until the end of the presentation of this material |
| Speak of a limit existing or not existing at x = a. Give varied examples of limit existence or nonexistence to show the different ways either may occur |
Use the term local behavior to describe function behavior near x = a; have learner describe this behavior informally so that they may concentrate on the actual concept rather than on notational conventions |
| Speak of limits "existing" or "not existing" at infinity |
Speak of asymptotic behavior and distinguish between local behavior and asymptotic behavior |
| Include functions continuous at a point in the existence examples. Introduce one-sided limits late in the presentation of the material. |
Concentrate on the notion of function behavior initially. Give varied introductory examples of how a function may behave on one side of x = a |
Just prior to beginning the material on local and asymptotic behavior of functions, we cover Advanced Graph-Sketching Techniques in more detail than is usually taught in many pre-calculus courses. This is so that learners are able to handle a larger array of functions (including piecewise-defined functions) and can thereby be exposed to a wider variety of behavioral phenomena demonstrating the principles of local behavior, and of asymptotic behavior. We can explore much more interesting examples if learners are well versed in the more advanced aspects of graphing.
Units seven through eleven will cover standard material on differentiation (techniques and applications) and integration (techniques and applications). These units will focus more on computation and methodology and refer back to the conceptual framework presented in the earlier units. As the first units take such care to provide the learner a conceptual framework within which to place the later material, by the time learners reach the later units, the most (conceptually) challenging aspects of learning calculus will be behind them. Additionally, the learner will have developed a fine sense of context for the later material, so that the theoretical basis for later techniques will be well understood.
Our course focuses on supporting the learner to develop proficiency in five areas:
- Computation: Some drill is required for the student to be able to apply differentiation rules, and integration rules, and to learn which of the many such rules to choose for the problem at hand. Diagnostic features are built into the course to alert the student to weaknesses and to direct them to areas for further practice. This is akin to the basic physical fitness one must have before complex athleticism can follow.
- Graphical aspects of calculus: Calculus is the mathematical study of change, which is why motion problems so often appear. Graphs offer a visual depiction of change and illustrate how two related quantities change together. With experience, a student of the subject learns how to translate algebraic formulas into graphs.
- Conceptual understanding: The curriculum we have suggested here focuses on three major concepts of calculus: the limit, the derivative, and the integral. Some appreciation of these concepts on an abstract level is required, so that the student develops an internal, intuitive mechanism whereby the correctness of computational and/or graphical results can be gauged.
- Modeling: To implement the mathematical tools from calculus, one must understand the distinction between the abstract world of calculus and the “real world” of phenomena, and understand how to build a bridge between the two. Modeling involves translating empirical observations into mathematics, and then using the mathematics to obtain theoretical results. These results must then be compared to the real, observed phenomena to see if the theory is vindicated. If not, one must decide whether the theory should be discarded or refined. If refinement seems appropriate, then another cycle of mathematical work ensues, hopefully bringing improved theoretical results. Learners who fully appreciate the modeling process are on their way to becoming skilled scientific practitioners.
- Synthesis: In order to master the material in calculus, a student must develop the ability to connect the various topics and see the interrelations among them. Especially if the student will be required to use calculus in an applied discipline, the student should acquire a sense of expertise, the confidence and the ability to recognize when, and to what extent, the computational, the graphical, or the conceptual aspects of the subject, should be applied. As wide an array as possible, of classical and contemporary applications of the calculus material is presented in the course. We hope that learners will be able to envision applications beyond those presented.
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