Albert Mosley and Eulalio Baltazar
An introduction to the discipline of logic covering subjects from the structures of arguments, classical and modern logic, categorical and inductive inferences, to informal fallacies.
- Over 30 years of development provides a sound empirical based pedagogy throughout the text.
- Examples in ordinary language using familiar examples avoids the suggestion of an alien cultural imposition.
- A focus on the basic representational techniques of classical and modern logic.
- Students introduced to basic concepts of set theory, using Venn diagrams to represent statements and evaluate arguments.
- Students introduced to basic concepts of propositional logic and the use of truth-tables.
- Students introduced to basic concepts of predicate logic and the use of mixed quantifiers.
- Students introduced to the relationship between logic diagrams, circuit diagrams, and gate diagrams in computer science.
- Students introduced to the use of logic in ordinary and scientific contexts.
- Students provided a historical introduction to the development of modern probability theory and its relationship to logic.
- Students introduced to basic concepts of statistical inference, with non-technical treatments of hasty and biased statistical generalizations. And a unique treatment of stereotypical thinking in terms of statistical syllogisms.
- Students introduced to basic notions in analogical and causal inference.
- Exercises requiring both passive (recognition) and active (construction) skills.
- Exercises including locutions and examples from standard English and ethnic dialects of English (African-American, Hispanic-American, etc)
- Answers for sample exercises provided, making the text closer to a self-teaching module
James Callahan, David Cox, Kenneth Hoffman, Donal O'Shea, Harriet Pollatsek, and Lester Senechal
Designing the curriculum
We believe that calculus can be for students what it was for Euler and the Bernoullis: a language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in connections to other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines. We began our work with a "clean slate," not by asking what parts of the traditional course to include or discard. Our starting points are thus our summary of what calculus is really about. Our curricular goals are what we aim to convey about the subject in the course. Our functional goals describe the attitudes and behaviors we hope our students will adopt in using calculus to approach scientific and mathematical questions.
- Calculus is fundamentally a way of dealing with functional relationships that occur in scientific and mathematical contexts. The techniques of calculus must be subordinate to an overall view of the questions that give rise to these relationships.
- Technology radically enlarges the range of questions we can explore and the ways we can answer them. Computers and graphing calculators are much more than tools for teaching the traditional calculus.
- The concept of a dynamical system is central to science. Therefore, differential equations belong at the center of calculus, and technology makes this possible at the introductory level.
- The process of successive approximation is a key tool of calculus, even when the outcome of the process--the limit--cannot be explicitly given in closed form.
- Develop calculus in the context of scientific and mathematical questions.
- Treat systems of differential equations as fundamental objects of study.
- Construct and analyze mathematical models.
- Use the method of successive approximations to define and solve problems.
- Develop geometric visualization with hand-drawn and computer graphics.
- Give numerical methods a more central role.
- Encourage collaborative work.
- Enable students to use calculus as a language and a tool.
- Make students comfortable tackling large, messy, ill-defined problems.
- Foster an experimental attitude towards mathematics.
- Help students appreciate the value of approximate solutions.
- Teach students that understanding grows out of working on problems.
Impact of Technology
- Differential equations can now be solved numerically, so they can take their rightful place in the introductory calculus course.
- The ability to handle data and perform many computations makes exploring messy, real-world problems possible.
- Since we can now deal with credible models, the role of modelling becomes much more central to the subject.
The text illustrates how we have pursued the curricular goals. Each goal is addressed within the first chapter which begins with questions about describing and analyzing the spread of a contagious disease. A model is built: a model which is actually a system of coupled non-linear differential equations. We then begin a numerical exploration on those equations, and the door is opened to a solution by successive approximations. Our implementation of the functional goals is also evident. The text has many more words than the traditional calculus book--it is a book to be read. The exercises make unusual demands on students. Most are not just variants of examples that have been worked in the text. In fact, the text has rather few "template'' examples.
Shifts in Emphasis
It will also become apparent to you that the text reflects substantial shifts in emphasis in comparison to the traditional course. Here are some of the most striking:
How the emphasis shifts:
increase: concepts, geometry, graphs, brute force, numerical solutions
decrease: techniques, algebra, formulas, elegance, closed-form solutions
Since we all value elegance, let us explain what we mean by "brute force." Euler's method is a good example. It is a general method of wide applicability. Of course when we use it to solve a differential equation like y'(t) = t, we are using a sledgehammer to crack a peanut. But at least the sledgehammer works. Moreover, it works with coconuts (like y' = y(1 - y/10)), and it will even knock down a house (like y' = cos2(t)). Students also see the elegant special methods that can be invoked to solve y' = t and y' = y(1 - y/10) (separation of variables and partial fractions are discussed in chapter 11), but they understand that they are fortunate indeed when a real problem will succumb to such methods.
This publication was produced as a teaching tool for college chemistry.The book is a text for a computer-based unit on the chemistry of acid-base titrations, and is designed for use with FORTRAN or BASIC.computer systems, and with a programmable electronic calculator, in a variety of educational settings. The text attempts to present computer programs that are relatively free of reliance on specialized large computer systems programs. The case-study approach presented is highly research and laboratory oriented. Similar subject matter is conventionally taught in most introductory college chemistry courses, but this text material attempts greater depth of instruction through utilization of the computational resources of a computer.
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