Computer algebra systems are programs that support manipulation of expressions and functions, handling of numeric data and plotting graphs in 2 and 3 dimensions. There may be some more sophisticated facilities like programming, animations, simulations and even direct interaction using some graphing parameters.
B. Dynamic Interactive Geometry/Algebra: Cabri Geometry (also 3d), Cinderella, Euklid DynaGeo, Geometer's Sketchpad, GeoGebra(3d promised to come, free and open source)
Dynamic geometry programs provide us with high level of
interactivity, but do not require programming skills when used for "pure"
geometric constructions. Even children - perhaps exactly them - are ready
to start from scratch and find amazingly quick applications of this graphical
facility. The basic objects are points put on the screen, and they can
be draggable or fixed, visible of hidden. Second level objects are other
points, segments, rays, lines, circles, arcs etc. defined using the basic
points. New objects depending on the previously defined ones can be created
using tranformations like translation, reflection, rotation and dilation,
and also defining the intersections of existing objects, e.g. the points
common to a cirle and a line.
More convenience is achieved if also more complicated
objects like triangles, quadrangles etc. are available.
If algebraic dependencies are allowed to be used in defining
new objects, one can also plot parameter and variable dependent graphs.
When the free points are moved, the whole construction
changes, keeping the defined relations invariant.
GeoGebra is a dynamic mathematics software for
education in secondary schools that joins geometry, algebra and calculus.
On the one hand, GeoGebra is a dynamic geometry system.
You can do constructions with points, vectors, segments, lines, conic sections
as well as functions and change them dynamically afterwards.
On the other hand, equations and coordinates can be entered
directly. Thus, GeoGebra has the ability to deal with variables for numbers,
vectors and points, finds derivatives and integrals of functions and offers
commands like Root or Extremum.
These two views are characteristic of GeoGebra: an expression
in the algebra window corresponds to an object in the geometry window and
vice versa.
C. Course Management Systems: Blackboard/WebCT, Stack, Wims, Moodle
There are numerous general purpose computer systems for
supporting course management. This means: delivering course materials in
electronic form, keeping track of students and their online-work like returning
essays and online-tests, allowing group work over the system, discussion
forums, chatting, etc. The systems runs on some server and it can be accessed
from anywhere.
The user interface is usually arranged on web browser
basis, which usually should allow dynamic figures to be embedded on any
tool on the system.
Moodle is a free course management system, open
source software package designed using sound pedagogical principles, to
help educators create effective online learning communities. The author
uses Moodle mainly for offering to his students online training materials
and small-scale online-tests concerning the definitions of basic mathematical
concepts like relation, function and binary operations in the general (Bourbakian)
sense that is needed in learning the axiomatics of abstract algebra and
linear algebra.
Own course pages (in Finnish): Introductory first year University course in Mathematics, Linear algebra and Discrete Mathematics.
Pedagogical research focus: fundamental abstract concepts in mathematics (relation, function, binary operations)
Binary Operation (Internal within a set, External with a scaling set)
Linear Functions (Basic in the Plane, Eigenvalues)
Complex Numbers (Complex 2nd order polynomial)
Function f, f(z) = z2 + bz + c, with zeros z1 and z2.
Double point if the center point 1/2 (z1
+ z2) of the segment joining the zeros of f
is inside the circle
Complex 2nd order
polynomial starting from the zeros
Had to be proved:
Proof
(pdf)
Eric Lehman and me in Caen in December 2007.