This module provides a collection of Mathematica definitions and so-called "Palettes" that can be used

by lecturers for compiling electronic (interactive) mathematical lecture notes based on Mathematica 3.x and
by students for working with interactive mathematical lecture notes.

Interactive lecture notes bring up some new aspects that need no (or at least little) consideration when preparing conventional lecture notes:

Unique language: The mathematical language must be very precise, no usage of notations in different contexts with different meanings is allowed. However, the language should look as similar as possible to the language used in normal mathematics literature.
Algorithmic content: Some definitions and theorems must be formulated differently in order to become algorithmic, i.e. their content can be used as a computation rule.
Side conditions: The conditions under which a computation rule would be applied must be specified explicitely in order to guarantee that rules are applied appropriately.
Tracing computations: A computation done automatically must be traceable by the student, i.e. the student must be able to figure out why and how a certain result was produced. Moreover, the automated computation should be as similar as possible to the "computation with pencil and paper" in order to have a positive impact on the student's own performance.

As an example, it is shown how interactive lecture notes for probability calculus and statistics can be prepared using these tools.

Prerequisites

Both lecturer and studend should have a basic understanding of Mathematica because Mathematica is the user interface and all computations are done with Mathematica.
The distribution contains a Mathematica notebook "README.nb" that describes how Mathematica must be configured in order to make things work.

The Mathematical Language

Buttons and Palettes

Buttons are active elements in Mathematica notebooks, which - when pressed, i.e. clicked onto with the mouse - perform some action. Normally, the action is just to paste what is written on the button into a Mathematica notebook, thereby making input of complicated expressions easier. Another advantage of buttons is that the user need not know how to input particular symbols occuring in an expression, simply pressing the button suffices! For most of the supported notations we provide buttons to produce the respective expression. Palettes are Mathematica notebooks containing a collections of buttons (you find the palettes in the File->Palettes menu of Mathematica (provided that Mathematica is setup as described in the README).

Tools for Lecturers

Support for lecturers is given mainly through the possibility to assign names to environments (definitions, propositions, theorems, etc). If an environment has some algorithmic content then usually this is reflected in a computation rule that can be defined inside the environment. In Mathematica, ":=" is used to define computation rules, putting a (short)
label in parentheses on top of the ":=" assigns this label to the environment. Whenever this rule is used during a computation later, a message saying

Alma6::verbose: Unter Verwendung von label.

is printed, where label is a hyperlink, that will jump to the place where the environment has been defined. Basically, this is the way how tracing computations works, i.e. rules must carry labels (responsibility of the author), the produced trace contains links with which the student can look up the definition of the respective computation rule.

Most of the computation rules need conditions that guide the system to apply appropriate rules in appropriate situations. For this we invent a special syntax:

hs:=rhs <= cond

would mean that the computation rule "lhs goes into rhs" will only be applied when the condition "cond" is satisfied (see the examples for details). Typically, a definition or a theorem looks as follows:

(Note the label of the definition written as an overscript to the ":=" sign.)

After having evaluated the definition, a computation typically looks as follows:

(The blue lines are show the trace of the computation with labels of definitions or theorems as hyperlinks.)

Experience shows, that the author must design the properties very carefully in order to get "good results" in the sense that the right rules are automatically applied on the right occasions and that no infinite loops (cycles) are produced by applying a rule and it's "inverse" in a row. Probably, this is the crucial point in the entire business ...

Tools for Students

Of course, the main advantage for a student is that lengthy calculations "by hand" are not necessary anymore, which opens the field for "more interesting examples" since the restriction of limited computation power is not relevant anymore. Furthermore it is much easier to do many examples, which can also have a good training effect. An interesting side-effect is learning by checking whether the automated computation is correct or at least plausible.

When working with the lecture notes it is very often necessary to "announce facts or properties" s.t. computation rules with a side condition are applied appropriately even in situations where Mathematica is not able to infer that the side condition holds.

Probability Calculus and Statistics

Lecture notes for probability calculus and statistics have been prepared using the described tools. Additionally, the package mechanism of Mathematica is used to implement different sections of the topic, i.e. the lecture notes consist of a collection of Mathematica packages, each of them representing one section. This needs some organizational stuff to be wrapped around each section in order to make things work, which is, by our convention, put into a subsection "Package Administration I" at the beginning and a subsection "Package Administration II" at the end, respectively. The necessity of this lies in Mathematica's organization of packages. For the user of the lecture notes these parts are not relevant.
This is how a typical working session looks on the screen: what one can see is

in the center: a notebook containing a particular section of the lecture notes with definitions, theorems, text, and examples,
left margin in the background: a "scratch notebook" with own calculations and new examples,
right margin, partially overlapped: palettes for inputting special expressions.

We will show examples from different sections (in german):