Numerics with Numerics with Geogebra in High Geogebra in High SchoolSchooldr Dragoslav Hercegdr Dragoslav Hercegdr Đorđe Hercegdr Đorđe HercegFaculty of Science and MathematicsFaculty of Science and MathematicsNovi Sad, SerbiaNovi Sad, Serbia

{hercegd | herceg}@im.ns.ac.yu{hercegd | herceg}@im.ns.ac.yu

22 of 18 of 18

Faculty of Science and Faculty of Science and MathematicsMathematics

Department of Mathematics and Department of Mathematics and Informatics offers undergraduate and Informatics offers undergraduate and graduate studies in mathematics, graduate studies in mathematics, mathematics of finance and computer mathematics of finance and computer sciencescience

Many of our faculty teach in the "Jovan Many of our faculty teach in the "Jovan Jovanović Zmaj" high school Jovanović Zmaj" high school

33 of 18 of 18

The Software that we useThe Software that we use

Mathematica 5Mathematica 5 Matlab 6Matlab 6 GeoGebra 2GeoGebra 2 C# (MSDN AA)C# (MSDN AA) JavaJava We develop course materials in We develop course materials in

Mathematica and GeoGebraMathematica and GeoGebra

44 of 18 of 18

Our textbooksOur textbooks

Many books on Numerical MathematicsMany books on Numerical Mathematics

55 of 18 of 18

Zmajtematika Zmajtematika

66 of 18 of 18

How we use GeoGebraHow we use GeoGebra

In a high school course in Numerical In a high school course in Numerical Mathematics, held for fourth grade Mathematics, held for fourth grade students during five years (2002-2007)students during five years (2002-2007)

As a teaching tool in the classAs a teaching tool in the class In the lab – for student assignmentsIn the lab – for student assignments How does using a computer in the class How does using a computer in the class

help the students learn more than help the students learn more than before?before?

... and what will they really learn?... and what will they really learn?

77 of 18 of 18

Course outlineCourse outline

A course in numerical mathematics, covering A course in numerical mathematics, covering standard topicsstandard topics

The textbook written by usThe textbook written by us A CD with interactive lessons in Mathematica A CD with interactive lessons in Mathematica

and programs to perform numerical methods and programs to perform numerical methods step by stepstep by step

GeoGebra drawings with geometrical GeoGebra drawings with geometrical interpretation of numerical methodsinterpretation of numerical methods

Let the students use the computer to solve Let the students use the computer to solve mathematical problems, not the programming mathematical problems, not the programming ones!ones!

88 of 18 of 18

Our goalsOur goals

To let students perform some tasks on To let students perform some tasks on their own, to experiment and vary the their own, to experiment and vary the solutionssolutions

To teach each student something (at least)To teach each student something (at least) Strict criteria – the solutions must match!Strict criteria – the solutions must match! Relaxed testing conditions – students may Relaxed testing conditions – students may

use whatever they have on hand, but not use whatever they have on hand, but not communicate among themselvescommunicate among themselves

99 of 18 of 18

TestingTesting

All students have/can use a computerAll students have/can use a computer But not too enthusiastic about learning But not too enthusiastic about learning

another programming languageanother programming language All exams based on textbook examples, All exams based on textbook examples,

slightly modifiedslightly modified Same grading criteria during 5 years, so Same grading criteria during 5 years, so

we were able to compare five generations we were able to compare five generations of studentsof students

1010 of 18 of 18

Grade averageGrade average

Same types of tests during 5 yearsSame types of tests during 5 years

The generation of 2004/05 consisted of The generation of 2004/05 consisted of seven students who were eager to learn, seven students who were eager to learn, while the other six were uninterested.while the other six were uninterested.

Year2002/

032003/

042004/

052005/

062006/

07

No. of students 24 22 13 30 28

Grade average 3.14 3.55 3.25 4.04 4.17

1111 of 18 of 18

2005/06 and 2006/072005/06 and 2006/07

A slightly different testing procedureA slightly different testing procedure Students split into two groupsStudents split into two groups

Two additional tests were given:Two additional tests were given: Graphical localization of rootsGraphical localization of roots Localization of solutions of algebraic equationsLocalization of solutions of algebraic equations

Year 2005/06 2006/07

Group I II I II

No. of students 17 13 14 14

Grade average 4.20 3.80 4.38 4.00

1212 of 18 of 18

Two different approaches to Two different approaches to teaching have been testedteaching have been tested

In the first test, the first group was encouraged In the first test, the first group was encouraged during lectures to experiment and try, using during lectures to experiment and try, using some of the programs from the CD, to find the some of the programs from the CD, to find the solutions of equations. No particular method solutions of equations. No particular method was favored and the choice was completely up was favored and the choice was completely up to the students.to the students.

The second group was taught in the more The second group was taught in the more "classical" way to first analyze the function "classical" way to first analyze the function theoretically, and then graph the function and theoretically, and then graph the function and show where the solutions are.show where the solutions are.

The roles were reversed during the The roles were reversed during the preparations for the second test.preparations for the second test.

1313 of 18 of 18

ResultsResults

Not surprisingly, the results were:Not surprisingly, the results were:

Obviously, letting a student chose her/his Obviously, letting a student chose her/his own way of solving a problem gives own way of solving a problem gives better results better results

The first test The second test

Group I II I II

No. of students 14 14 14 14

Grade average 4.18 2.62 2.50 3.57

1414 of 18 of 18

ConclusionConclusion

Computers free students of tedious manual Computers free students of tedious manual workwork

However, they shouldn't free them of having to However, they shouldn't free them of having to reason, draw conclusions and act on themreason, draw conclusions and act on them

The possibility to build the problem in reverse, The possibility to build the problem in reverse, starting from the desired solutionstarting from the desired solution

Personalized testsPersonalized tests Careful preparation of tests ensures that Careful preparation of tests ensures that

only the real knowledge is valuedonly the real knowledge is valued Trial and error – the good way to do math?Trial and error – the good way to do math?

1515 of 18 of 18

The role of GeoGebra?The role of GeoGebra?

GeoGebra provides essential support to GeoGebra provides essential support to the teacherthe teacher

A number of drawings were developed A number of drawings were developed and students were encouraged to try to and students were encouraged to try to duplicate themduplicate them

..\primeri\Njutnova metoda.ggb..\primeri\Njutnova metoda.ggb ..\primeri\Opsti iterativni postupak.ggb..\primeri\Opsti iterativni postupak.ggb

1616 of 18 of 18

What we would likeWhat we would like

What we would like to see in GeoGebra:What we would like to see in GeoGebra: ListsLists Nested functionsNested functions Calculated properties:Calculated properties:

AA: If[AA: If[boolexprboolexpr == 1, A, (1000, 0)] == 1, A, (1000, 0)]

More flexible external controlMore flexible external control

BIG FONTS AND DOTSBIG FONTS AND DOTS

1717 of 18 of 18

Work in progress...Work in progress...

Formula Editor, for entering and Formula Editor, for entering and displaying formulasdisplaying formulas

Accompanying software for the book Accompanying software for the book "Zmajtematika""Zmajtematika"

Numerics with Numerics with Geogebra in High Geogebra in High SchoolSchooldr Dragoslav Hercegdr Dragoslav Hercegdr Đorđe Hercegdr Đorđe HercegFaculty of Science and MathematicsFaculty of Science and MathematicsNovi Sad, SerbiaNovi Sad, Serbia

{hercegd | herceg}@im.ns.ac.yu{hercegd | herceg}@im.ns.ac.yu

Thank you!Thank you!