University of Education (PH) Weingarten, Germany
For the Design of a Computer integrating Geometry curriculum
The present state of educational software development does have some standardized products, mostly for plane geometry: Dynamic Geometry Systems equipped with international user surfaces. These systems represent a common core curriculum for (middle) secondary geometry independent of specific cultural influences and different styles of interaction (corresponding systems for spatial software are yet less developed). These systems could be considered as a base for a worldwide standard for basic geometric contents and methods, which is available by internet and could be communicatively developed there.
0 DEFICIENCIES OF THE TRADITIONAL geometric TOOLS
The traditional tools: compasses, straightedge, set square, rule protractor, paper and pencil and pocket computer as tools of exploration and reconstruction determine largely the usual way of acquiring elementary geometry at school.
The traditional construction, calculation and visualization tools show marked deficiencies as means of exploring and reconstructing elementary geometry by the student. Such deficiencies of general kind are:
a lack of support of
- epistemic behavior
- individualized learning
- economic working
- the formation of flexible and functional thinking
- the development and application of intellectual techniques and heuristic strategies
- modeling real world etc.
By using adequate dynamic graphics systems, these deficiencies can be compensated. The use of such graphics systems however can only complete and not replace the traditional tools, because
- ultimately the importance of handling the traditional analogue tools for the tactile acquisition of fundamental knowledge in geometry cannot be assessed (an experiment where the effects of learning geometry only by means of the computer are explored, is not acceptable for humanitarian reasons)
- working with the traditional tools of construction and of measuring is a cultural technique that is not only very important in the mathematical-historical context;
- the definition of constructive modules with the computer is based essentially on constructive relations as they appear on constructing with the straightedge and the compasses
- there is a risk that only those geometrical subjects are chosen that are especially adequate for being presented and developed with the computer (which is, of cours, also true for the traditional tools in geometry teaching and learning)
- the continuity of specific curricula has to be guaranteed
- the amount of media required in the form of hardware and software causes considerable problems (for example the students require the necessary hardware and software tools to be able to do their homework)
- a worldwide non-verbal communication standard is inherent in these simple tools
- the analogue tools are indispensable for tactile application (do-it-yourself activities etc.).
1 DYNAMIC GEOMETRY SYSTEMS FOR PLANE GEOMETRY
1.1 FACILITIES OF GEOMETRIC COMPUTER SYSTEMS
An interactive graphics system that is considered to be suitable for geometrical construction and calculation in global teaching and learning has to meet some geometrical, organizational and general educational facilities.
1.1.1 GEOMETRIC FACILITIES
Generating geometric configurations
The possibilities of producing geometrical configurations are:
(1) The indirect or passive generation by designing procedures with a graphics programming system or a platform independant script language; the input variables are the determining pieces of construction. We get a formal model of the geometrical construction process like the traditional description of the construction written in a normalized language. The configuration is represented by a program. The procedure called up with actualized parameters produces a corresponding example of the configuration as screen drawing.
(2) The direct or active generation with a graphics system alternately by input of specified commands or options by the user and execution by the system (interactive construction). Two types of graphics systems for interactive construction can be distinguished:
(2a) the command-driven systems
(2b) the menu-controlled systems.
The result of interactive construction is in both cases the target configuration of
geometrical construction as a screen drawing.
The systems like (1) can completely describe a configuration using all the control structures for automatic repetition and selection, while by systems of kind (2) only sequential construction algorithms can be executed. The application of the system types (1) and (2a) requires that all the graphical objects to be designed should be named to be able to take reference to them; operation is made with the keyboard and requires the usual command of the semantic and syntax rules of a formal language as in any programming language. Such systems support mainly a graphic handling of constructive elementary geometry aiming at an objective and standardized description of construction. These kinds of representation do not favour selfcorrection by the student but they are very conducive for learning to plan and to imaginate the drawing by anticipation.
The menu-controlled systems (2b) also meet the demands of a "naive user" who uses such tools only occasionally. Menu-driven graphics systems - where direct access to the graphic objects by the user is possible by means of direct manipulation is guaranteed - support a mainly spontaneous approach to elementary geometry which aims at an individual acquisition and exploration. An interactive construction protocol can be issued. There is an option for repetition of construction processes.
The gap between configuration and drawing is bridged by installation of drag mode, which allows easily to represent a configuration by a large variety of isomorphic screen drawings.
Figure generation by direct interaction:
Construction processes traditionally carried out with compasses and ruler
In order to carry out construction processes traditionally realized with compasses and ruler the graphics system must be able to refer to and to generate the following elementary graphical objects:
- straight lines
- circles around a center and through a peripheral point
- point(s) of intersection of straight lines, straight line and circle. and of circles
- points on straight lines and circles.
(If this facility is met, the "method of geometric loci" for construction problems to be solved with compass and ruler can be realized.)
But the mere simulation of compass, ruler and set square construction problems by means of the interactive tool computer in geometry teaching is not sufficient to justify this kind of computer application. Further facilities must offer new possibilities of learning geometry and offer to compensate the deficiencies of the traditional tools.
More basic figures to be generated
In order to be able to represent more general plane shapes the graphics system must be able to generate the following other elementary graphical objects:
- half-lines or rays
- angles (of different kind)
- arcs of circles
and the variety of geometric loci to be individually constructed as referencable objects.
Generation of complex figures
The generation of complex figures derived from basic ones is economically supported by the concept of already implemented or definable macro-construction:
- construction of a perpendicular bisector
- bisection of a straight line segment (i.e. construction of midpoint)
- bisection of an angle (i.e. construction of a line bisecting an angle)
- erection of a perpendicular on a line
- dropping of a perpendicular from a point to a line
- construction of a parallel line
- copying a line segment (replicating a length)
- copying an angle (replicating an angle)
Modification of configurations
Given or constructed figures can be modified regarding their position,orientation, size and shape preserving or changing their incidential structure.
Modification by dragging
Transformations performed by drag-mode are straight line- and circle- invariant. The following relations (if constructively defined within figures) are generally invariant during drag-mode transformations:
- part-proportionality (i.e. ratio of lengths)
- point symmetry (rotational)
- line symmetry (reflective)
- incidence (in general).
Geometry of drag-mode could be described as a hybrid from aquiformal and affine geometry.
Modification by mapping
Generating figures by application of mappings like congruencies and similiarities as compositions of elementary mappings preserving certain figure proporties.
Modification by redefining
Redefinition of objects serves the change of figure structure for economical construction and investgation, specifying and generalising figures.
Measurement and calculation of configurations
Besides construction of figures a main topic in elementary geometry is measuring and calculating of figures.
The following basic measurements, which have to be compatible to drag mode, are indispensible: measurement of
- distances (in relation to a relative length unit as a function of the screen)
- line segments
- perimeters of polygons and circles
- angles (of different kind)
- area of polygons and circles
Measurements can be gathered in tables. The input of given measurements is possible. The givens of construction problems can be measurements of angles and line segments. The student uses the measures of line segments or angles for marking off line segments and for laying off angles.
Calculations can be derived from measurements by generation of terms.
Calculated data can be gathered in tables.
The values of those tems depending on figure variation by drag-mode.
Analogous to the concept of macro construction, macro calculations with quantitative variables can be defined.
Interface for coordinate geometry
There is the possibility of embedding synthetic geometric configurations into a coordinate system and to issue point coordinates and equations of basic graphical objects like straight lines and conics.
Editing of figures
There are editing possibilities for reversable hiding auxiliary objects, for appearance (kind, colour, dimension, position) of objects and for denotion.
Denotion of objects (specifcation):
The graphics system should have a function for naming the graphical objects on the screen and must have the following features:
- freely selectable position of the denomination (automatic positioning of the denomi-
nation involves the risk of overwriting)
- denominations as usual in geometry
- optional: free or compulsory denominations.
1.1.2 Organizational facilities
Organizational facilities of an educational graphics system are essential for creating corresponding computerized learning environments and for self organized learning. These environments have to be adapted to the geometrical and instrumental competencies of the students. Modules for creating learning environments can be distributed via the Internet.
- Availability of documented figure files for demonstration and analysis by repetition
- Availability of documented macro-definitions of constructions and calculations for
support of problem solving
- Availability of menu configuration for adapting the system to specific learner groups
or to specific problem topics
- Availabilty of menus for non Euclidean geometries.
- Availability of topic orientated interactive worksheets (combination of configurations
to be treated according to an explaining text on the screen) with self control facilties
- Availability of information for using the system (self explanation by glossary and
- Availability of different language interfaces to be selected for communication
- Availability of platform independant tutorials with information systems equiped with
applets for distant learning.
DYNAMIC GEOMETRY SYSTEMS AS A BASE FOR TUTORIAL SYSTEMS:
The tutorial presentation and preparation of construction, calculation and proof tasks are a prerequisite for self controlled learning in computerized environments.
GEOLOG-WIN/GEOLOG 2000 (http://www.uni-giessen.de/~gcp3/geologde.htm)
is a first in Dynamic Geometry Systems, which integrates tutorial systems on knowledge based way for solving construction, calculation and proof tasks; a corresponding interface serves the tutorial preparation of tasks to be implemented and their sequencing. (Unfortunately, no platform independent version of this system which is programmed in Visual Prolog exists at the moment.)
CINDERELLA offers an interface for preparing and presenting tutorials for all of the compass and ruler based construction tasks at the moment; this tutorial treatment is based on a specific method of mechanical theorem proofing.
1.2 METHODIC IMPACT FOR TEACHING AND LEARNING GEOMETRY
The use of dynamic graphics systems that are on the whole able to meet the already mentioned facilties leads to new methods of learning plane geometry, especially in
- solving geometric construction problems
- insolving geometric calculation problems
- the inductive acquisition of geometric theorems and concept formation
- application and investigation of transformations
- investigation of functional relations of geometric figures
- treatment of elementary functions
- modeling of applications
- simulation of motion in an applicational context
- the aesthetic design from geometric figures.
These new methods are based essentially on the following program features of the systems:
- automatic measurement
- generation of terms
- redefinition of objects
- generation of referencable loci
- interface for coordinate geometry
- editing of figures
- administration of results etc.
Using Dynamic Geometry Systems the following general methods are supported
- Trial and error
- Modular work
- Synthetic-Analytic transfer.
In the following sections the new methods of an explorative development of plane
geometry will be set against the deficiencies of working with the traditional tools.
1.2.1 Solving geomerical construction problems
Deficiencies on solving construction problems with the traditional tools:
- Little support in the heuristic phase of the construction process (e.g. a tentative ac-
tion in the sense of trial-error-correction is not supported)
- little chance of correcting construction results
- no possibility to change the position or size of the (partial) construction result
- little construction accuracy
- time-consuming processing in case of complex construction processes
- lack of clarity due to the inevitable auxiliar lines
- no possibility of repeating the construction process
- little support in construction due to the lack of construction modules (basic construction processes and user-definable macro-construction: one "great mental step" in a construction process is made up by many small manual steps that divert the students' attention from the aim of construction).
Solving construction tasks:
(1) construction of a corresponding plan figure
(2) adjust figure in drag mode so, that all of the givens match their predefined
(3) find a constructive solution by heuristic means
(4) simulation of compasses and ruler construction:
the construction steps are analoguely executed as constructing compasses and
(5) definition of the construction macro:
Initial objects are the given graphic and numeric objects; the figure to be
constructed from the initial objects consists of the target objects.
If the solution of construction problems is described by a three-phase process: heuristic phase - algorithmic phase - analytical phase all possibilities of the interactive construction tool come into play in these phases, but especially in the heuristic phase of solving construction problems (diagram: an arrow means: ".....supports ...")
1.2.2 Solving geomerical calculation problems
Deficiencies on solving calculation problems concerning drawings of figures with the traditional measurement tools and manual or pocket computer calculation :
- Little support in the heuristic phase of the calculation process (e.g. a tentative action in
the sense of trial-error-correction is not supported)
- little chance of correcting and reconstructing calculation results
- no possibility to change the position of the (partial) calculation result afterwards
- little calculation accuracy
- very time-consuming in case of complex calculation processes
- no possibility of repeating the calculation process
- little support in construction due to the lack calculation modules (basic calculation processes and user-definable macro-construction: one "great mental step" in a calculation process is made up by many small manual steps that divert the students' attention from the aim of calculation.
- Experimental calculation by using drag mode:
(1) Interactively construct an appropriate geometric figure
(2) Take trial measurements of both given data and target data (also calculated
(3) Vary the geometric figure until it matches the given data
(4) Read off the target data.
- Simulation of a calculation process step by step starting with measurements.
- Definition of calculation macros as formulae:
Form a term from measurements or already calculated values of terms concerning a figure drawing, then define a macro calculation which does automatically calculates the value of the formed term from the data given.
1.2.3 Inductive acquisition of theorems and concepts
Deficiencies of the inductive acquisition of theorems by traditional construction of configurations and measurements on configurations:
- time-consuming, often inaccurate construction of a sufficient set of suitable configu-
rations representing the theorem in question,
- only theorems that are based on less complex configurations can be developed,
- time-consuming and incorrect measurements or calculation,
- static configurations that could hitherto in most cases only be made flexible by
mental imagination (functional relationships and a dynamic relation of geometrical
quantities can be hardly represented).
Analoguely the same applies to the formation of shape concepts with elements from the appropriate geometrized concept ranges.
New method: interactive variation of configurations by changing the position of the constituent objects (so-called basic objects) in drag-mode. The initial objects of a construction can be moved freely in drag-mode, all the connected objects follow the movement according to the construction. The transition of the configuration from one state to another is continuous (i.e. in real-time processing) due to individual cursor movements.
The application of the drag-mode offers an opportunity of a real application of the following didactic principles for geometry-teaching:
Realization of the configurative mobility principle: in dragmode the inductive acquisition of theorems or concept formation can be developed by using the following possibilities of continuous variation of geometrical configurations:
Generate from a configuration (as realized theorem or concept) a wide range of many other isomorphic configurations (with continuous transformation. i.e. in real-time processing):