Working by Calculating Rules |
Idea and Meaning |
|
Determination
of tangent inclinements and derivatives by syntactic rules |
Derivation as
idea of transition from the average to the local changing rate |
|
Integration for
the concrete calculation of areas and of integral functions by syntactic
rules |
Integral as
idea of reconstructing a function from local changing rates |
|
Performing the
Newton procedure by a pocket calculator or a computer |
Idea of
approximation of zeros by an iteration procedure and analysis of the
convergence behaviour |
|
“Curve
sketching” as application of calculating rules to function and their
derivatives |
“Curve
sketching” as capable analysis of properties of functions |
|
Formal solving
of systems of equations |
Representation
of geometric figures (straight lines, planes, circles, ellipses,…) by the aid
of analytical methods |
|
Mathematics as a Product |
Mathematics as a Process |
|
Putting over a
calculus procedure and it’s application |
Developing a
calculus procedure and understanding of it |
|
Communicating
knowledge, putting over connections |
Set-up of
knowledge, discovering of connections |
|
To aim at self
contained problems |
Deliberately
allow openness |
|
From the
structure to the application |
From the
problem to the structure |
|
Study a given
model |
Modelling
reality |
|
Isolated problems
with unique solution |
Fields of
problems in a network with versatile solutions |
|
To give
notions, to prove theorems formally |
To develop
notions, to find theorems, to demonstrate plausibly |
|
Convergent and
result oriented guidance during the class |
Open process
oriented guidance during the class |
|
Mistakes as a
sign of insufficient capability of the product |
Mistakes as a source of constructive improvements |
Further point,
promoted in Adolf J I Riede, Mathematical modelling as an integrated part of
the class on calculus:
|
Lack of understanding a sign for missing ability |
Lack of understanding as a source of didactical
improvements |