I think we can build correct introductory analysis if we think about tangent as a basic concept.
how come? calculus I, is just limits and continuity, that is all about. the basic concept in calculus is the rate of change, it the instantaneous slope of the tangent.
Kati Munkacsy can you explain more about correct introductory analysis if we think about tangent as a basic concept?
take care.
In a project, we developed various tasks for introductory analysis; see:
Book Teaching calculus using dynamic geometric tools
In my deep conviction, in teaching Calculus, it is not the problem of limes or continuity, but the student's understanding of the set of real numbers and its properties.More specifically, the problem of teaching university students to topics in Calculus is the time foreseen for the set of real numbers and teaching technology in that teaching.
No... Calculus without Limits and Continuity will be then Calculus without Differentiation and so on.
No please. The concept of differentiation is defined through limits.
This is not so simply question. Continuity and limit are technical tools to define the tangent, as the linear approximation. If tangent is a basic concept, we do not need them. As we do not need them in nonstandard analysis.
Mohamed Milad Ahmed, thank you for you question.
We can teach geometry in secondary school. We do not ask: What does it meant area? we ask What is the area of a triangle? and so on. Students can use their knowledge.
This is the same in analysis, do not ask What is the tangent? ask What is the slope of the tangent? They can use pc. And students can solve minima-maxima problems and so on.
Numeric method and using your computer and the packages. looking for the pattern and to generalize. For regular form of geometry (usually the simple ones), constructing is an idea. It depends on levels of the students. But, my experience is for learning steps. So, we can do that.
Introductory Page of a Chapter in the Book: CALCULUS One and Several Variables (10th Edition) by Salas Hille Etgen.
My idea is not to make easier limits and continuity, but to make easier the whole calculus. We need a different way for students of mathematics and for non- mathematicians students.
I suppose you can "teach" them that way. But I worry about the loss of meaning.
I effectively teach a meaning of derivatives (limits) without calculus. Let me model an example of important meaning that comes with introducing the idea of limits outside of calculus.
Students walk away with the following operational definition of an instantaneous velocity:
1. If the motion appears smooth/uniform, pick an interval centered about the point you are interested in.
2. Find the average velocity (slope) in the first half of the interval, and the second.
3. If these average velocities are adequately equal, take the average velocity across the entire interval, and call that the instantaneous velocity at the center instant.
4. If they are not adequately equal, pick a smaller interval centered about the point you are interested in, and go back to step 2.
We also need to work hard to help them appropriately interpret the ratios which result.
I highlight a value of the limit, (in this case to compare motions of several objects) and the interpretation, when I ask students to evaluate (or correctly finish) the following conversation.
"But officer! How could I have been going 55 miles per hour in a school zone? I didn't drive for an hour!"
Arnold Arons identifies, "One of the most severe and widely prevalent gaps in cognitive development of students at secondary and early college levels is the failure to have mastered reasoning involving ratios.
. . .This disability, among the very large number of students who suffer from it, is one of the most serious impediments to their study of science." ( p 3, "A Guide to Introductory Physics Teaching," 1990, John Wiley & Sons)
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