- Acta Universitatis Matthiae Belii series Mathematics
Contents
1. K EE CA Conversion FIGURE 1 Overall structure of a system The Visuals package contains the GeoCanvas class which is a special Canvas control that supports the geometrical coordinate systems besides the usual pixel based screen coordinate system The package also contains classes for graphical representation of geometrical notions as well as other graphical classes and user interface elements such as classes derived from WPF controls The Conversion package contains converters which perform data type conversions necessary for data binding 4 2 Interfaces and enums Positioning mode of visual elements is determined by the LocationMode enumeration The value Screen means that the location of an element is expressed in screen pixels while the value Geometry means that the location is expressed in geometrical coordinates and that a conver sion to screen coordinates is necessary before the element is drawn on screen public enum LocationMode Screen Geometry LISTING 1 LocationMode enumeration All visual elements must implement the IElement interface Table 4 2 which provides basic functionality for element positioning and visibility control The Valid property is used to control element s visibility based on the validity of the expression it is bound to For example let M Segment A B be a segment between the points A and B and P Perpendicular M M Midpoint a line perpendicular to M passing2. in terpreted public double Size Size of this VPoint in pixels protected override void Called by the WPF when the VPoint needs OnRender DrawingContext drawingContext to be drawn private DrawingVisual DrawIt Performs actual drawing of the VPoint in the specified DrawingContext public int VisualChildrenCount Required by the FrameworkElement spec public void AddVisual Visual v ification These methods must be im public int VisualChildrenCount plemented in all classes deriving from public int VisualChildrenCount the System Windows FrameworkElement class To implement a geometrical point we start by inheriting the FrameworkEle ment class The methods VisualChildrenCount AddVisual DeleteVisual and Get VisualChild must be implemented as specified in 8 We also implement the Element interface and the optional Location inter face Actual drawing of the point takes place in the DrawIt method which is called when needed from the overridden OnRender method The VPoint class has properties that determine its visual appearance We will consider only the Size property as implementation details are similar for all other such properties static VPoint RegisterProperties F private static void RegisterProperties FrameworkPropertyMetadata mdSize new FrameworkPropertyMetadata 8 0 FrameworkPropertyMetadata0ptions AffectsRender new PropertyChangedCallback OnSizeChanged SizeProperty DependencyProperty Re
3. the evaluator Figure 4 5 lt UserControl x Class HMS Geometrijica Visuals TrafficLight xmlns http schemas microsoft com winfx 2006 xaml presentation xmlns x http schemas microsoft com winfx 2006 xam1 Height 90 Width 30 gt lt Border BorderBrush FFDA1818 BorderThickness 3 3 3 3 gt lt Grid x Name LayoutRoot Background FFDDCECE gt lt Grid RowDefinitions gt lt RowDefinition gt lt RowDefinition Height gt lt RowDefinition gt lt Grid RowDefinitions gt lt Grid ColumnDefinitions gt lt ColumnDefinition gt lt Grid ColumnDefinitions gt lt Ellipse Fill 000000 Stroke FF000000 Grid Row 0 gt lt Ellipse Fill 000000 Stroke FFO00000 Grid Row 1 gt lt Ellipse Fill 000000 Stroke FF000000 Grid Row 2 gt lt Ellipse Fill FFFF3304 Stroke FF000000 Margin 3 3 3 3 Grid Row 0 x Name RedLight Opacity 100 gt lt Ellipse Fill FFDFE22A Stroke FFO00000 Margin 3 3 3 3 Grid Row 1 x Name YellowLight Opacity 100 gt lt Ellipse Fill FF1D8B35 Stroke FFO00000 Margin 3 3 3 3 Grid Row 2 x Name GreenLight Opacity 100 gt lt Grid gt lt Border gt 77 lt UserControl gt LISTING 6 Specification of the TrafficLight control in XAML One benefit from creating visual elements from existing controls is that these elements retain full functionality of the controls they are based on This way we can mix geometrical shapes with WPF co
4. through its midpoint Suppose that both M and P have their corresponding vi sual elements shown on screen Then if the points A and B are equal the length of the segment M is zero and the line P cannot exist In that case the value of 70 the variable P in the evaluator will be marked as invalid and the visual element corresponding to P should not be drawn This is accomplished by binding the Valid property of the visual element to the Valid property of the variable in the evaluator TABLE 3 Members of the Element interface GeoCanvas GeoCanvas The GeoCanvas object this element belongs to Elements use this reference to obtain information about geometrical coordi nates in the GeoCanvas bool Valid Determines whether the element is valid i e whether it should be drawn public Expression Result Result of evaluation of the variable bool Visible Controls visibility of the element void CalcScrLocation Calculates and updates the location of the visual element on the screen private GeoCanvas _geoCanvas public GeoCanvas GeoCanvas get return _geoCanvas set geoCanvas value public void CalcScrLocation if GeoCanvas null switch LocationMode case LocationMode Geometry ScrLocation GeoCanvas Geo2Scr Location break case LocationMode Screen ScrLocation Location break code omitted LISTING 2 A typical IElement implementation in a v
5. values of type Point ValueConversion typeof EPoint typeof Point public class EPointConverter IValueConverter public object Convert object value Type targetType object parameter System Globalization CultureInfo culture EPoint ep value as EPoint if ep null amp amp targetType Equals typeof Point double x1 Number ep X Value double y1 Number ep Y Value return new Point x1 y1 F else throw new ArgumentException Invalid type EPoint expected value public object ConvertBack object value Type targetType object parameter System Globalization CultureInfo culture code omitted LISTING 3 A partial listing of the DoubleConverter class Listing 4 shows lines of code that bind the result of the evaluator variable A to the first point of the segment sg After this code has executed all changes in the result of the variable A will be immediately reflected on the drawing on screen sg new VSegment new Point 0 0 new Point 3 5 GeoPanel1 RegisterVisual sg Binding bA new Binding Result bA Source Evaluator Default Variables A bA Converter new HMS Geometrijica Visuals Conversion EPointConverter sg SetBinding VSegment AProperty bA LISTING 4 Binding of an evaluator variable to a visual element 73 4 5 Visual Elements Visual elements are objects that are actually drawn on GeoCanvas They can be simple dots and lines or
6. Acta Universitatis Matthiae Belii ser Mathematics 16 2009 65 79 Received 28 June 2009 Accepted 2 February 2010 THE USE OF WPF FOR DEVELOPMENT OF INTERACTIVE GEOMETRY SOFTWARE DAVORKA RADAKOVI AND DORDE HERCEG ABSTRACT The Windows Presentation Foundation WPF is a graphical subsystem in NET Framework 3 5 that uses a markup language called XAML for rich user interface development Interactive geometry software IGS are computer programs that allow one to create and then manipulate geometric constructions primarily in plane geometry Some of the free well known 2D IGS are Geogebra Cabri and Cinderella Besides the intended use as a means of teaching and studying geometry IGS are often used for other purposes such as development of mathematical games or as a part of other mathematical software e g mathematical drawing viewers Thanks to its JavaScript interface GeoGebra is often used in that role and controlled externally by some other software However there are some limitations to GeoGebra s usefulness in that respect since it wasn t developed primarily for that purpose Our aim is to offer a solution that can be easily used as a software com ponent for mathematical visualization and interaction The framework we developed called Geometrijica is simple straightforward and extensible It is based on the WPF which enables it to have a rich graphical appearance and interactivity In this paper we demonstra
7. anged a OnRender 39 RegisterProperties 3 OnScrLocationChanged Segment 1 overload 3 OnSelectionOpacityChanged 3 OnSizeChanged 59 RegisterProperties 5 Point 1 overload FIGURE 3 Visual elements VPoint VSegment and VButton The process of making a new visual element is different depending on what class is chosen as a starting point We can start from System Windows FrameworkElement and program everything by hand or we can start from either Control UserCon trol or one of the existing WPF controls and implement only a few necessary methods In either case the Element interface must be implemented 4 5 1 Creating a new visual element from FrameworkElement In this section the steps necessary to create a visual element from FrameworkElement will be explained The VPoint class represents a geometrical point and it is drawn as a small circle on screen The important members of the VPoint class are explained in Table 4 4 74 TABLE 5 Important members of the VPoint class public GeoCanvas GeoCanvas The GeoCanvas object this VPoint belongs to private static void RegisterProperties Registers dependency properties and their corresponding event handlers Called from the static constructor public Point Location Location of the geometrical point either in screen or geometry coordinates public LocationMode LocationMode Specifies how the location property is
8. complex drawings Besides that common WPF controls such as buttons check boxes and sliders can be turned into visual elements and placed on GeoCanvas while retaining all their functionality Even complex user controls with graphical effects and animations can be turned into visual elements and used Element IElement TElement ILocation TElement ILocation ILocation Point z Segment Button A TrafficLight 2 Class Class Class Class gt FrameworkElement gt FrameworkElement gt Button gt UserControl Fields Fields Fields Fields Properties E Properties Properties E Properties SF GeoCanvas a CP Geocanvas SP GeoCanvas Location fs Location CF Light gt LocationMode GeoCanvas LocationMode i Location ScrLacation FP scr PF valid LocationMode SelectionOpacity oP scrB SF Visible A valid oF size valid Methods SF visible amp valid Z Visible CaleScrLocation Methods ey Visible VisualchildrenCount 3 OnLocationChanged CalcScrLocation F VisualchildrenCount E Methods 3 OnLocationModeChanged 3 OnLightChanged E Methods Addvisual 34 RegisterProperties a RegisterProperties Addyisual CalcScrLocation 3 Button 1 overload a TrafficLight 1 overload CalcScrLocation DeleteVisual ee Deletevisual 34 Drawlt SY Drawlt Getvisualchild 9 GetVisualchild 3 OnaABChanged 34 OnLocationChanged 9 OnRender 39 OnLocationModeChanged 3 OnScrABCh
9. e actual drawing of the point at the screen coordinates 4 5 2 Creating visual elements from existing controls WPF controls already have full functionality in other words they know how to draw themselves and to react to user interaction such as keyboard actions and mouse clicks It is much easier to create visual elements from existing WPF controls than to code all drawing and behavior logic by hand as is the case with visual elements based on Frame workElement In order to make a visual element from the Button class we only need to implement the Element interface in the inheriting class If we want to be able to place the button on geometrical coordinates as well as on screen coordinates we should implement the Location interface too Figure 4 3 shows the VButton class which was created in the described way 76 Window O FIGURE 4 Sequence diagram for the RegisterVisual method 4 5 3 Creating visual elements from UserControl As WPF controls can be spec ified in XAML it is also possible to create visual element in that way Listing 6 shows the specification of a traffic light control with three controllable lights which can be switched on and off by setting the TrafficLight Light dependency property Since the TrafficLight control also implements the Element interface it can be placed on GeoCanvas in the same way as all other visual elements and its appearance can be controlled by a variable from
10. e reflected in the geometrical drawing We have also demonstrated that any property of a visual object can be bound to an arbitrary expression in the calculation engine which is a step further from what GeoGebra offers in this respect Also we have demonstrated how new visual objects can be made either by programming them from scratch or by inheriting existing controls By following a few simple rules new visual objects can easily be created and used in geometrical drawings Geometrijica can easily be used as a component in other programs REFERENCES GeoGebra http www geogebra org accessed on 8 7 2009 Cinderella http www cinderella de accessed on 8 7 2009 Cabri http www cabri com accessed on 8 7 2009 Mathematica http www wolfram com accessed on 8 7 2009 Adobe Flash http www adobe com products flash accessed on 8 7 2009 OpenLaszlo http www openlaszlo org accessed on 8 7 2009 WPF Data Binding MSDN Library Online emhttp msdn microsoft com en us library ms750612 aspx accessed on 8 7 2009 MacDonald M Pro WPF in C 2008 Apress 2008 9 Hohenwarter M GeoGebra educational materials and applications for mathematics teaching PhD thesis 334 pages University of Salzburg Austria 2006 10 Hohenwarter M and Preiner J Creating mathlets with open source tools Journal of Online Mathematics and its Applications ID 1574 vol 7 2007 11 Hohenwarter J Hohenwarter M and Lavicza Z In
11. ea displayed in the GeoCanvas can be panned and zoomed e GeoCanvas supports both screen coordinate system and geometrical Carte sian coordinate system Visual elements that are placed on a GeoCanvas decide which coordinate system they will use Objects using screen co ordinates do not move when the geometrical coordinate system moves This facilitates mixing of user interface elements with the elements of a geometrical drawing 4 IMPLEMENTATION 4 1 System overview The structure of a program built on the Geometrijica framework is shown in Fig 4 1 The scope of this discussion is limited to the Vi suals Conversion and Algebra packages which correspond to appropriate names paces in Geometrijica The Algebra package contains classes discussed in the section Expression evaluator and parser A partial list of the classes is shown in Table 4 1 TABLE 2 A partial list of classes in the Algebra namespace Number String Logical EColor Data types Neg Plus Times Arithmetic operations Sqrt Power Sin Cos Mathematical functions EPoint Segment Line Circle Geometrical shapes Evaluator Calculation engine Var A variable used in the calculation engine 69 WPF Windows Silverlight Web Application browser Application T T Vieira ee hat peak De Pee eS OO an ee ae l Visuals T j i fossa oo oe en i i Parser 7 7 Algebra
12. ers could change its color or border width when all the appropriate elements represented as points are placed inside its bounds e GeoGebra can be controlled from an external program by means of its JavaScript interface However this interface in its current state provides only the basic functionality We would like to be able to control every aspect of the geometrical drawing and to react to all the events such as mouse clicks keyboard pressed or object overlapping e Properties of objects in GeoGebra are accessed by means of special func tions unlike properties of objects in object oriented programming lan guages which we feel is a more natural way For example to obtain the x coordinate of a point one needs to type x A instead of A x This 66 method is awkward when there are a large number of properties since each property requires a special function to access it e For mathematical game development we often need to create customized graphical objects such as coins fruits traffic lights houses cars etc While this is possible in GeoGebra it can be awkward and time consum ing Furthermore it is not possible to create more than one instance of a customized graphical object needed in any other way but by drawing each instance separately For that reason we decided to develop a new framework which will solve these problems while retaining all good aspects of GeoGebra Since we already had developed a mathematical expre
13. gister Size typeof double typeof VPoint mdSize code omitted public static DependencyProperty SizeProperty public double Size get return double GetValue SizeProperty set SetValue SizeProperty value private static void OnSizeChanged Dependency0bject obj DependencyPropertyChangedEventArgs e VPoint t VPoint obj t DrawIt F private DrawingVisual DrawIt DrawingVisual vis DrawingVisual visuals 0 using DrawingContext dc vis RenderOpen selection circle Brush sbr new SolidColorBrush Colors Orange sbr Opacity Selection0pacity Pen sp new Pen sbr 5 5 dc DrawEllipse null sp ScrLocation Size 4 Size 4 shape of the point Brush br Brushes Blue Pen p new Pen br 2 0 dc DrawEllipse br p ScrLocation Size Size return vis F LISTING 5 Implementation of the Size property and the Drawlt method Figure 4 4 shows the sequence diagram for the RegisterVisual method When a new visual element VPoint in this case is added to the GeoCanvas via the Reg isterVisual method the GeoCanvas control invokes the CalcScrLocation method from the Element interface Since the VPoint in question is in the geometry positioning mode it calls the Geo2Scr method of the GeoCanvas in order to transform its geometrical coordinates into screen coordinates After that the OnRender method is invoked which in turn calls the DrawIt method This method performs th
14. ion assigned to the variable public Expression Result Result of evaluation of the variable public bool Valid Indicates whether the result is valid public event PropertyChangedEventHandler Event from the INotifyPropertyChanged PropertyChanged interface which must be implemented in order to use this class as a data source Evaluator is a computational engine that keeps a set of named expressions and maintains dependencies between them ensuring that when one expression changes all dependent expressions get reevaluated It also maintains the expres sion set in a consistent state by preventing creation of circular dependencies and by deleting all dependent expressions of a deleted expression 3 DESIGN GOALS The requirements placed before the Geometrijica framework are the following e Mathematical notions that can be drawn on paper such as points lines circles and graphs of functions can be shown on screen For example by defining a variable M Segment A B we are also creating a graphical representation of the segment M which is drawn on screen When the value of the variable changes the image on the screen also changes e Geometrical drawing is a 2D image consisting mostly of but not limited to geometrical shapes such as points lines and circles It is kept in computer memory as a list of visual elements with their respective coordinates and other properties such as color size and border width e Any property
15. isual element class The ILocation interface Table 4 3 should be implemented by visual elements which can choose the positioning mode between LocationMode Screen and Loca tionMode Geometry Most visual elements that represent geometrical shapes do not implement this interface On the other hand user interface elements such as buttons check boxes and sliders which can be placed either on fixed location on screen or bound to geometrical coordinates implement the ILocation interface 71 TABLE 4 Members of the Location interface Point Location Location of the visual element either in screen or geom etry coordinates LocationMode LocationMode Specifies how the element s location is interpreted 4 3 GeoCanvas The GeoCanvas class extends the System Windows Controls Canvas class It represents a view of a 2D plane in the Cartesian coordinate system GeoCanvas has four properties named X0 YO X1 and Y1 which determine the region of the 2D plane that is shown on the GeoCanvas Visual elements are added to the GeoCanvas by calling the RegisterVisual method The Geo2Scr method is used to convert geometrical coordinates into screen coordinates This method is called by child visual elements when they are re quested by the GeoCanvas to determine their locations Obviously only the elements in the geometry positioning mode use this method GeoCanvas Class gt Canvas Fields E Properties 2 x
16. nd intuitive to use However the situation is not so simple when it comes to developing new stand alone software that should use an existing IGS as a component First of all the licensing mode of the IGS in question may not permit such use and even if it does there may be some technical limitations or interoperability prob lems On the other hand there are commercial software packages which are more than suitable for such development 4 but the prices for development and run time versions of these packages may prohibit their widespread use One of the possible approaches to this problem is to use tools such as Adobe Flash 5 or OpenLaszlo 6 which are not primarily intended for geometrical applications for development of mathematical teaching materials games and examples In our previous work we used GeoGebra to develop course materials 13 14 primarily because it is a free software and therefore accessible to our target audience However we encountered some of GeoGebra s limitations e Geometrical shapes in GeoGebra have properties such as color line width and shape of points which can only be changed via the user interface It would be much more useful if properties of geometrical shapes could get their values from the results of mathematical expressions That way we could have visual indicators that change their appearance based on the state of the geometrical drawing For example an oval representing a set of even numb
17. ntrols on a GeoCanvas control Furthermore dependency properties of those controls can be bound to results of arbitrary expressions in the evaluator provided that appropriate converters exist 4 6 Example Figure 4 5 shows a simple window containing a GeoCanvas con trol which in turn contains three TrafficLight controls at screen coordinates and one point one segment and one button at geometrical coordinates When the GeoCanvas is resized panned and zoomed the traffic light controls retain their positions while the other controls positions move accordingly E Geometnjica WPF demo FIGURE 5 Point segment button and three traffic light controls on a GeoCanvas 5 CONCLUSION Existing interactive geometry software IGS are used in teaching of geometry and mathematics in general GeoGebra is one IGS that has gained wide acceptance thanks to its intuitive use and a great range of features However GeoGebra cannot without difficulty be used as a component in other software products Therefore we developed the Geometrijica framework for geometry software de velopment A significant part of our framework is the graphical subsystem which can display geometrical shapes WPF controls and user controls at the same time 78 By using dependency properties and data binding infrastructure in the WPF we have managed to link the calculation engine with the graphical subsystem so that all changes in calculation results ar
18. o A x 2 yo oF yi Methods Y Geo2Scr Y GeoCanvas g3 GeoCanvas_SizeChanged 3Y Rearrange Y RegisterVisual FIGURE 2 The GeoCanvas class 4 4 Dependency properties and data binding The main idea in our work is to create visual elements which react dynamically to changes in expression values in the evaluator WPF data binding 7 provides a simple and consistent way of binding elements to data sources such as databases XML documents CLR objects etc In our case data sources are evaluator variables objects of type Var Ta ble 2 1 These objects implement the INotifyPropertyChanged interface which takes care of notifying the WPF infrastructure when a property of a variable 72 changes Typical scenario is as follows When a visual element is created its properties are bound to the appropriate properties of the corresponding Var ob ject The visual element is then placed on a GeoCanvas and thus displayed on screen Each subsequent change of Var object s properties causes the data bind ing infrastructure to change corresponding properties of the visual element which is displayed immediately on screen For this purpose one way data binding is used Since the data types used in the Evaluator are different from those used in the visual elements converters must be implemented for each pair of data types for which data binding is meaningful Listing 3 shows the EPointConverter class which converts values of type EPoint into
19. of a visual element can be bound to any variable of the appropriate type For example the location of a point in the Cartesian coordinate system can depend on a variable of the type Point When the value of the variable changes the position of the visual element is updated on the screen 68 e Visual elements can represent geometrical shapes mathematical no tions but they can also be controls such as buttons check boxes or slid ers They are implemented by inheriting from WPF controls and user controls or by inheriting from System Windows FrameworkElement Ex isting functionality of inherited controls is retained e Dependency properties of WPF user controls can be bound to any vari able of the appropriate type Data types are converted by special con verter classes For each pair of types there exists a converter class that provides conversion between them This enables creation of rich visual representations which can be controlled by expressions from the evalua tor For example one can develop a user control that displays a traffic light with a property that specifies which light is on and then animate the lights by binding the property to a variable in the evaluator e GeoCanvas is a WPF control that displays geometrical drawings Geo Canvas inherits from System Windows Controls Canvas The part of the 2D plane that is shown inside the GeoCanvas is specified by the coordi nates of the lower left and upper right corners The ar
20. rmed into SetVar A Plus MemberOf M X 3 Therefore all evaluation is actually performed by executing functions Expressions have data types Some common data types are Number String Logical Color Point Segment Line Circle and so on Arguments of functions are checked for data type compatibility at execution When an expression can not be evaluated for any reason it returns a value of the Error data type Any expression depending on that value also returns a value of the Error data type Variable is a named expression maintained by the evaluator A variable consists of an expression and its result 67 Variables can depend on other variables For example if M Segment A B is a segment between points A and B and the coordinates of the point A change then M must change accordingly Evaluation of variables is dynamic As soon as one variable changes its value all dependent variables are reevaluated Circular dependencies are not allowed Therefore the expressions A f B and B g A are not allowed at the same time Table 2 1 shows the most impor tant members of the Var class which is used to keep variables in the evaluator One important feature of the Var class is that it implements the INotifyProper tyChanged interface which enables it to notify data consumers of changed values TABLE 1 Members of the Var class public string Name The name of the variable public Expression Expr The express
21. ssion parser and evaluator in C we decided to base our framework on it However our solution can easily be adopted to use another computer algebra system Windows Presentation Foundation WPF was chosen as the graphical subsystem 2 EXPRESSION EVALUATOR AND PARSER We developed an expression evaluator and parser which are based on the same principles as the ones in GeoGebra Expression is any simple or complex expression which can be constructed by using constants variables arithmetic and logic operations properties of objects and function calls Supported functions include common mathematical functions such as power trigonometry and logical functions Basically an Expression is what we are used to seeing in most programming languages like C Complex expressions are built by combining simpler expressions using function compo sition Besides that expressions are used to describe geometrical notions their properties and relations For example if M Segment A B represents a segment between points A and B then Perpendicular M M Midpoint represents a line perpendicular to M passing through its midpoint Parser is tasked with accepting textual input and transforming it into expres sions The syntax resembles expression syntax in C with arithmetic operations function calls and member access Internally arithmetic and logic operations and member access are transformed into function calls For example the input A M X 3 is transfo
22. te how our framework when used together with a mathematical expression evaluator can be used as a starting point for developing interactive mathematical software 1 INTRODUCTION Today there are many interactive geometry software IGS products available 1 2 3 They are used mostly in teaching and studying geometry and some more advanced IGS can also graph functions and their derivatives perform algebraic and symbolic manipulations and so on The importance of IGS in today s teaching is widely studied and recognized 9 11 For that reason teachers team up with software developers in order to 2000 Mathematics Subject Classification 68N19 Key words and phrases WPF XAML IGS Geometry software Teaching software Com ponent development Mathematical games create interactive teaching and learning materials In order to reach wide audi ences such as elementary school pupils and teachers the resulting software must be affordable and able to run on various platforms GeoGebra is one such IGS which has gained wide acceptance due to several fac tors it is free runs on all modern operating systems it is constantly updated and improved its user interface and user manual have been translated into more than 40 languages there exist a number of examples and teaching materials freely available on the Internet and GeoGebra applets can be embedded and used in teractively in Web pages 10 Most importantly GeoGebra is easy a
23. troducing Dynamic Mathematics Software to Secondary School Teachers the Case of GeoGebra Journal of Computers in Mathematics and Science Teaching JCMST 28 2 2009 135 146 12 Hohenwarter GeoGebra From car design to computer fonts M Informatik Spektrum 32 1 2009 18 22 Springer 13 Herceg D Herceg D Numerical mathematics with GeoGebra in high school Teaching Mathematics and Computer Science 6 2 2008 363 378 14 Herceg D Herceg D Scalar product and numerical integration Pedago ka stvarnost LV 1 2 2009 88 102 NDOT WN HE 0 DEPARTMENT OF MATHEMATICS AND INFORMATICS FACULTY OF SCIENCES TRG DOSITEJA OBRADOVI A 4 21000 Novi SAD SERBIA E mail address davorkar dmi uns ac rs DEPARTMENT OF MATHEMATICS AND INFORMATICS FACULTY OF SCIENCES TRG DOSITEJA OBRADOVI A 4 21000 Novi Sap SERBIA E mail address herceg dmi uns ac rs 79
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