Numerics with Geogebra in High School dr Dragoslav Herceg dr Đorđe Herceg Faculty of Science and...
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Transcript of Numerics with Geogebra in High School dr Dragoslav Herceg dr Đorđe Herceg Faculty of Science and...
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
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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
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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
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Our textbooksOur textbooks
Many books on Numerical MathematicsMany books on Numerical Mathematics
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Zmajtematika Zmajtematika
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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?
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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!
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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
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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
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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
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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
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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.
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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
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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?
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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
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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
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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!