Computer Algebra Systems Supporting Teaching/ Learning Linear Algebra

Ana Donevska Todorova

International GeoGebra Conference for Southeast Europe January 2011, Novi Sad, Serbia

Overview

Introduction Comparison

Computer Algebra Systems Dynamic Software for Mathematics

Teaching/ Learning Experiences University Education

CAS Maxima and the online system moodle at the MIT University

Secondary Education Some examples

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Introduction Faculties of engineering and informatics at the

universities implement CAS: Mathematica Matlab during the contemporary lab classes in

mathematics.

First year students at universities are usually not familiar with any of the CAS or DGS and show lack of computer supported mathematics.

Some possibilities to help the upper secondary school students in overcoming this problem and prepare them for university mathematics into lab.

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Comparison

System Creator Development started First public releaseLatest stable

versionCost (USD)

Opensource

Maple Maplesoft 1980 1984 14 (April 2010)

$1,895 (Commercial), $1,795 (Government),

$995 (Academic), $239 (Personal Edition),

$99 (Student), $79 (Student, 12-Month term)

No

Mathemaica

WolframResearch

1986 19888.0 /November

2010

$2,495 (Professional), $1095 (Education),

$140 (Student), $69.95 (Student annual license)

$295 (Personal)

No

Maxima

MIT Project MAC

and Bill Schelter

et al.

1967 1998 5.22 (2010) Free Yes

Comparison of Computer Algebra Systems (CAS)General Information

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http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems

Comparison

SystemFormula

editor

Calculus Solvers

Graph theory

Number theory

Boolean algebraIntegra

tion

Integral Transfor

msEquations

Inequalities

Differential equations

Recurrence

relations

Maple Yes Yes Yes Yes Yes Yes Yes Yes Yes No

Mathematica

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Maxima No Yes Yes Yes Yes Yes No Yes Yes No

Comparison of Computer Algebra Systems (CAS)Functionality

http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems

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Comparison

System Windows Linux Solaris

Maple Yes Yes Yes

Mathematica Yes Yes No

Maxima Yes Yes Yes

MuPAD Yes Yes No

http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems

Comparison of Computer Algebra Systems (CAS)Operating System Support

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Comparison Comparison of Dynamic Software for

Mathematics (DSM) Operating System Support

SoftwareCost (USD)

Platforms

Cinderella 1.4 Free Windows, GNU/Linux, Mac OS X (Java)

Cinderella 2.0 69 US$ Windows, GNU/Linux, Mac OS X (Java)

DrGeo Free GNU/Linux, Mac OS X

GeoGebra Free Windows, GNU/Linux, Mac OS X

GeoNext Free Windows, GNU/Linux, Mac OS X

Kig Free GNU/Linux

Kgeo Free GNU/Linux

KmPlot Free GNU/Linux, Mac OS X

http://en.wikipedia.org/wiki/List_of_interactive_geometry_software#Comparison

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Comparison

Comparison of Dynamic Software for Mathematics (DSM) Functionality

GeoGebra Extras: Algebraic manipulations

SoftwareCalculations

Macros

LociAnimations

LaTeX export Web export Multilingual

Cabri II Plus Yes Yes Yes Yes No Yes Yes

Cinderella Yes Yes Yes Yes Yes (PDF) Yes Yes

GeoGebra Yes Yes Yes YesYes (PSTricks &

PGF/TikZ)Yes

Yes (51 languages)

GeoNext Yes No No Yes No ? Yes

Kig Yes Yes Yes No Yes (PSTricks) No Yes

Cabri 3D Yes No No Yes No Yes (limited) Yes

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Online system moodle at MIT University Skopje

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Teaching/ Learning Experiences

Results in Mathematics

34,71

74

49,12558

60,4

44,3345,29

31,04

0

10

20

30

40

50

60

70

80

2007/08 2008/09 2009/10 2010/11

Generation

Ave

rag

e sc

ore

d p

oin

ts

First Midterm

Second Midterm

*Resource http://moodle.mit.edu.mk/course/Matematika

University Education (MIT University Skopje)Scores in Mathematics of the engineering students at the Faculty of Computer Sciences and Technologies

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Implementation of the mathematics upgraded knowledge in other engineering subjects Quantitive linear models for

optimization Example 1: A company produces

three types of products in three different facilities (machines). For each product in each in each facility the required processing time is given in the following table:

How many peaces of each of the products can be produced if the first facility has a capacity of 3200 working hours per month, the second facility 1700 and the third one 1300 working hours per month?

Facilities Product 1 Product 2 Product 3

1 2 3 4

2 1 2 1

3 1 1 2

Solution (using wxMaxima)

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Implementation of the mathematics upgraded knowledge in other engineering subjects Laplace Transformation

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Teaching/ Learning Experiences Secondary Education

Properties of Determinants

1. Calculate the values of the following determinants:

2. Using CAS Maxima calculate the values of the determinants given in the previous assignment.

3. Compare the obtained results and the given determinants; and explain what you noticed.

4. Write the conclusion in your own words. 5. Write the property using mathematical symbols.

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43

21det A

21

43det B

42

31detC

Teaching/ Learning Experiences Secondary Education

Properties of Determinants

6. Using CAS Maxima calculate the values of the following determinants:

7. Compare the obtained results and the given determinants; and explain what you noticed.

8. Write the conclusion in your own words. 9. Write the property using mathematical symbols.10. Generalize the property for n-dimension

determinant.

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987

654

321

det A 654

987

321

det B 963

852

741

detC

Teaching/ Learning Experiences Linear programming in GeoGebra

Example: Two different types of products A and B can be produced on

the machines M1 and M2. The capacity of M1 is 12000 working hours and the capacity of

M2 is 6000 w. h. Required time for producing one product of type A on the

machine is M1 is 3w. h. and on the machine M2 is 2 w. h. Required time for producing one product of type B on the

machine is M1 is 3w. h. and on the machine M2 is 1 w. h. The needs of the market are 2500 products of type A and

3000 products of type B. The profit of the company is 4000 euros per one product A and

2000 euros per one product B. The management of the company has to create the optimal

plan for producing the products A and B in order to achieve the best profit.

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Graphical Solution Systems of inequalities

0,

3000

2500

60002

120033

21

2

1

21

21

xx

x

x

xx

xx

21 20004000 xxf

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References Literature

D. Todorova A.: The transition from secondary to teriary level mathematics emphasized in the course of linear algebra, International conference dedicated to prof. d-r. Gorgi Cupona, Ohrid, 2010.

Donevska-Todorova, A. (2010): Difficulties in Mathematics for the Students in the First Year at Higher Education; Zbornik na MIT Universitet, Skopje, Macedonia, p. 177-184.

Trencevski K.; Krsteska B.; Trencevski G.; Zdraveska S.; Linear algebra and analytic geometry for third year reformed gymnasium educatiom, Prosvetno delo, Skopje 2004.

Roegner K. (2008) Linear Algebra as a Bridge Course for First-year Engineering students, Department of Mathematics, Technische Universität Berlin, Berlin Germany.

Internet Recourses http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page http://www.geogebra.org/cms/ http://moodle.mit.edu.mk/course/Matematika http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems http://en.wikipedia.org/wiki/List_of_interactive_geometry_software#Comparison

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