using geogebra in secondary school mathematics teaching

Background GeoGebra: Free Dynamic Mathematics Software (DMS) Examples in Using GeoGebra Concluding Remarks

Using GeoGebra in SecondarySchool Mathematics Teaching

Towards Enhancing Higher Order Thinking Skills

Ateng’ Ogwel

Centre for Mathematics, Science and Technology Education in Africa

ICT in the Classroom ConferenceDurban Girls’ College: 24-26 September, 2009

OGWEL, JCA (CEMASTEA): "GeoGebra in Mathematics Teaching"

ICT in the Classroom Conference: Durban Girls’ College: 24-26 September 2009


Outline1 Background

Rationale and ObjectivesBrief on CEMASTEA

2 GeoGebra: Free Dynamic Mathematics Software (DMS)Elements of GeoGebraPotentials of GeoGebra

3 Examples in Using GeoGebraProperties of Plane FiguresPotentials for HOTS

4 Concluding RemarksRemarksFeedback




Rationale and Objectives

Rationale

Inadequate Modeling: There is persistent urge to align schooleducation with the fast changing needs of the society. Educationappears to be predominantly characterized by individualisticlearning of isolated facts and skills for examinations.

There is pressure to incorporate ICT in education. Emergenceof Courseware and tools are driven by technologists,programmers and mathematicians as educationists show laxityand resistance to techno-revolution in education

ICT is not a panacea for quality and efficient education, unlessproperly integrated and accepted. Therefore, there is need todevelop a community of educators who collaborate in the useand integration of ICT in education.





Objectives

By the end of the interactive session, a participant shouldbe able to

Download and install a Dynamic MathematicsSoftware, GeoGebra

Explore and use Menus and basic tools of GeoGebrato construct mathematical objects

Design activities and objects to demonstrateintegration of GeoGebra in school mathematics

Evaluate the potential of GeoGebra as a tool forintegrating ICT in mathematics education, and itspotential towards stimulating higher order thinking





Session Plan: 2 Hours

The session is organized as in the table below. You areencouraged to carry notebooks and geometrical sets.Ability to interact and share experiences would bevaluable for the success of the session.

Session Duration RemarksBackground 15 Minutes InformativeGeoGebra Introduction 45 Minutes InteractiveExamples in GeoGebra 45 Minutes InteractiveConclusion 15 Minutes Interactive




Brief on CEMASTEA

In-Service Education and Training (INSET)

CEMASTEA, a centre for in-service education, wasestablished after years of working on mathematics andscience teachers’ professional development. Factors thatled to the intervention – Strengthening Mathematics andScience in Secondary Education (SMASSE) include:

1 Attitude: Head-teachers, teachers, and subsequently pupils andparents.

2 Pedagogy: Lack of appropriate teaching methodology

3 Content: Inadequate mastery of subject content by teachers

4 Few or non-existent interactive fora for teachers

5 Missing link between primary and secondary school content




Brief on CEMASTEA




Brief on CEMASTEA

ASEI-PDSI Approach

SMASSE programmes have been based onActivity-based teaching which is Student-centred,incorporating Experiments and practical work, andwhere resources are Improvised (ASEI).The teaching process emphasizes Planninginstruction, Doing/ teaching the planned lesson,Seeing the learning process through formative andsummative evaluation and Improving subsequentinstruction (PDSI).Network (National and Regional): ASEI-PDSIapproach is shared by mathematics and scienceteachers in Kenya and other countries in Africa.




Brief on CEMASTEA




Brief on CEMASTEA

Some Basics of ASEI Lesson Plan

Rationale: A justification for the lesson in terms ofreal life application, further study in the subject andlinkage with other disciplines.Objectives: Precisely what is to be achieved by thelearners at the end of the lesson (SMART)Prerequisite Knowledge and Skills: Consciousness ofthe necessity of prior knowledge in the learningprocessTeaching/ Learning Resources and ReferencesBridge: Linking activities in a lesson to intendedconcepts in mathematics and sciences.




Brief on CEMASTEA

ICT Integration in Education

Currently, CEMASTEA is one of the partners in theMOE-VVOB programme on "Integration of ICT inEducation"The path of integrating ICT in education is complexand uncertain, and thus, require collaboration andsector-wide partnerships. CEMASTEA wishes to bea strategic partner with developers and providers ofeducational materials and services who areinterested in visibility.GeoGebra is the first among other ICT tools whichare currently being explored in order to achieveintegration of ICT in education




Elements of GeoGebra

GeoGebra

A Dynamic Mathematics Software, that integratesGeometry, Algebra (GeoGebra) and Calculus. It haselements of Dynamic Geometry Software (DGS) andComputer Algebra Systems (CAS). Runs on anyplatform that supports Java.It is freely available for online installation fromwww.geogebra.org/webstart or downloadable for localinstallation at www.geogebra.org.

Examples at www.geogebra.org/book/intro-en.zip andat the wiki www.geogebra.org/en/wiki. Thoughts, help andqueries are shared at the forum www.geogebra.org/forum



www.geogebra.org/webstart

www.geogebra.org

www.geogebra.org/book/intro-en.zip

www.geogebra.org/en/wiki

www.geogebra.org/forum



Menus: File; Edit; View; Options; Window; HelpTools: Move; Points; Lines; Loci/Constructions;Polygons; Circles and Circular arcs; Conics;Measurements; transformation; Slider; Visibility;Toolbar HelpViews:

Graphics/ Geometry View: Default view and drawingpad on which geometrical objects are constructed.Algebra View: Gives algebraic representation ofobjects.Spreadsheet View: Every cell has a specific nameand names of objects match the spreadsheet cells

Input Bar: Gives algebraic command as alternative tothe geometrical tools on the toolbar.





Activity 1: Menus, Tools and Views

1 Download, install and open GeoGebra application.2 Using Menus

Select the point style to a cross ("x") and labeling to"All New Objects"Activate only the "Algebra View".

3 Using ToolsPlace any three points "A", "B" and "C" on theDrawing pad and observe the "Algebra View". Usethe "Move" tool to manipulate the points and observechanges in the "Algebra View"Use the "Drop Down Arrow" to change the toolsappearing on the toolbar





Spreadsheet View

1 Save current file as "Menu1.ggb" in a folder "JCA25"Open a "New Window" and select "SpreadsheetView"

2 Type/ input the texts enclosed by quotation marks inthe given cells (ENTER to confirm input):

"1" (A1); "(2,5)" (A2); "b=5" (B1); "p=(3,1)" (B2);"(-1,0)" (C1);"Segment[A2,C1]" (C2)

3 Manipulate objects and describe your observations

4 In the "Algebra View", click on, and manipulate A1.

5 Save the current file as "Spreadsheet1.ggb"





Algebraic Input Bar

1 Open a New Window and activate the "Input Bar".Deactivate the "Spreadsheet View"

2 Type/ input the following verbatim in the "Input Bar":a=12A=(1,2)b=Segment[A,a]B=(4,7)p=Polygon[A,B,5]"Area of polygon p =" +p

3 Manipulate the objects and describe yourobservations. Save the current file as "Input1.ggb"





Object Properties

1 Open the file "Input1.ggb"2 Type c=Polygon[A,B,a] in the Input Bar3 Right Click

a in the Geometry window and select "Animation On"(first, highlight a in the Algebra Window)p, select "Object Properties" and change the colorand style of the objectChange the color and font size of the text "Area ofpolygon ..."

4 Select "View" and activate "Construction Protocol"Observations?

5 Save the current file as "Activity1.ggb"




Potentials of GeoGebra

Semiotic Tools

Collaboration: The previous activity introduced you tobasics of GeoGebra. Competence and confidence isgained through practice, explorations, review ofexamples and readiness to seek help and share withthe GeoGebra community.GeoGebra, like any other tool, mediates the learningprocess. In themselves, they don’t directlycommunicate the mathematical concepts.Novelty of GeoGebra, like other mathematical tools,may be exciting and fascinating at first. However,didactical use requires a reflection on the curriculumcontent, objectives and trends in education .




Potentials of GeoGebra

GeoGebra as a Pedagogical ToolGeoGebra has the potentials as a

Presentation Tool (Multiple representation)Graphical/Geometric RepresentationAlgebraic/ Functional RepresentationDynamic VisualizationsMaking Connections

Modeling ToolDynamic ConstructionsExperiments and Discovery learning

Authoring ToolCreation, use and sharing of materials with an onlinecommunityResearch on teaching and learning using GeoGebra




Properties of Plane Figures

Activity 2: From Paper and Pencil to DMS

The following activity establishes prerequisite skills in theuse of mathematical tools. It also elicits externalization ofsubconscious thinking often taken for granted by experts

1 On a plane paper, construct aTriangleSquare

2 Discuss with a colleague and enumerate, withexplanations, the procedure you would prescribe to anovice interested in constructing the twomathematical objects





Reflection on the "trivial"Why do we often erase protruding parts or extend shortersides when drawing a triangle





Modeling: Describe the tools and procedures thatartisans use to construct such structures in Africa,highlighting the mathematical issues involved





1 Open the "html" file Square and construct a squareusing the available tools

2 Use the tools in GeoGebra to convince yourcolleague that you have indeed constructed a square





Angles and Plane FiguresLearning Objectives

1 Use a protractor to accurately measure and constructangles

2 Solve problems involvingangles on a straight lineangles at a pointparallel lines

3 TrianglesState angle properties of trianglesRelate interior and exterior angles of a triangleApply angle properties of a triangle in problem solving





Angle Properties of PolygonsLearning Objectives

1 Rectangles:State angle properties of a rectangle and a squareApply angle properties of a rectangle and square inproblem solving

2 Parallelogram, Rhombus and Trapezium: State andapply angle properties of a parallelogram, rhombusand trapezium in solving problems

3 Kite: State and apply angle properties of a kite inproblem solving

4 General Formula: Derive and apply the formula forsum of interior angles of a polygon





Activity 3: GeoGebra and School Curriculum

1 Use GeoGebra to design activities that may be usedto address any of the previous curriculum objectives

2 Contextual ApplicationIdentify areas/topics in your local curriculum, whereGeoGebra may be used as a pedagogical toolWhat would be the advantages of using GeoGebrarather than the existing media and or teachingmethods in the identified content areas?

3 Outline steps, tools and objects that you will need inorder to address the identified local curriculumcontent, possibly indicating a time-frame for theactivity after the "ICT in the Classroom Conference".




Potentials for HOTS

Activity 4: Similar Figures




Potentials for HOTS

Similar Figures




Potentials for HOTS

Mathematical Properties"The Mid Points"

1 Construct any polygon P of side n (3�n�7)Use the "Midpoint" tool in GeoGebra to locate themidpoints of all the sides of the polygonJoin the midpoints to construct a polygon P’.Manipulate the objects and record observations

2 Construct any quadrilateral Q (not regular).Locate and join all the midpoints of Q to constructpolygon Q’. Manipulate objects and discussobservations

3 Open the html file Para inv and reverse theconstruction above from Q’ to Q

4 Justify the observable mathematical propertiesOGWEL, JCA (CEMASTEA): "GeoGebra in Mathematics Teaching"



Potentials for HOTS

Mathematical PropertiesReversing "The Mid Points"




Potentials for HOTS

Mathematical Properties"Justification"




Remarks

Remarks: Curriculum

Higher order thinking: The previous activities are alllinked and indicate mathematical connectedness.Solving some of them require analysis, synthesis andevaluation – elements of HOTSNew knowledge: GeoGebra affords generality ofconcepts and reveal mathematical structures withefficiency, unlike paper and pencil scenarios.Learning Need: GeoGebra has the potential to createlearning need, by stimulating situations in whichcurrent knowledge is inadequate in solving aproblem. ) discovery of mathematics, its processesor versatility of applications in real life




Remarks

Remarks: Educational Trends

Evidence: Use of information and communicationtechnologies in education requires evidence on theirefficacy to overcome the threshold of mistrust bypractitioners.

Collaboration: The widespread trend of collaborationin educational research, and in design of tools andcourseware need to be reflected in school educationby enhancing collaborative and cooperative learning.

Assessment reform: Assessment of learning with andusing ICT needs to be an immediate agenda ofeducational innovation in most countries




Remarks

Remarks: Modeling

Some Artisans (despite limited education)can estimate the cost of material from a clients’specification of dimensions of a structure.are able to determine the dimensions of a structurefrom the number of roofing sheets.

Societies face challenges whose solution requirecomplex thinking processes. These are oftenun-developed or underdeveloped by the currenteducation systems. There is need for tools andexperiences to develop modeling skills in the youth.ICT in education need to be meaningfully integratedwith modeling to enhance relevance of education




Feedback

ReferencesJones, K., Lavicza, Z., Hohenwarter, M., Lu, A., Dawes, M.,Parish, A. and Borcherds, M. (2009, March). BSRLM Geometryworking group: Establishing a professional developmentnetwork to support teachers using dynamic mathematicssoftware GeoGebra, Proceedings of the British Society forResearch into Learning Mathematics 29(1): 97-102Ogwel, A. (2008, November). Integrating ICT in mathematicseducation: Curricula challenges in the Kenyan system ofeducation. 1st Regional Conference on e-Learning: Increasedaccess to education, diversity in applications and managementstrategies. Kenyatta University: November 18-20, 2008Ogwel, A. (2007, July). Educating tomorrow’s mathematicsteachers: The role of classroom-based evidence. Modeling inmathematics learning: Approaches for classrooms of the future.Makerere University: July 23-25, 2007




Feedback

Free/ Libre Open Source Software

I am grateful to Markus and Judith Hohenwarter for theworkshop resources, some of which I have modified.Thanks to the community of GeoGebra developers andusers, and other developers of educational software.Thumbs up to the organizers and sponsors for this greatopportunity.

I am also a proud user of the following FLOSS:




Feedback

Feedback

Please give us some feedback bycompleting the given questionnaire. Kindly share yourexperiences and thoughts on use of GeoGebra as aDynamic Mathematics Software with others.

Thanks



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using geogebra in secondary school mathematics teaching

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