Educational experiences about using different computer programs

in calculus courses at the University of Kaposvár

Anna Takács Klingné University of Kaposvar, Faculty of Economic Sciences,

Mathematics and Physics Department

Second Central- and Eastern European Conference on Computer Algebra- and Dynamic GeometrySystems in Mathematics Education11-13 July, 2009 RISC, Linz, Austria

University of Kaposvár

Faculty of Pedagogy

Faculty of Animal Science

Faculty of Economic Science

Faculty of Arts

My students study:

•Agribusiness and agricultural rural development programme•Agricultural engineering programme•Finance and accountancy programme

Subject and number of subject per week taught by Mathematic Department

1. semester 2. semester 3. semester

Finance BA Calculus I.

2+2

(Analysis)

Calculus II.

2+2

(Probability and Linear algebra)

Optimization

2+2

Agricultural rural development BSc

Calculus I.

2+2

Agricultural engineering BSc

Calculus I.

2+2

We find that our students have little success in mathematics. But why it is?

One of his reasons,

that the higher education became multitudinous

On the other hand:

the problem is that in the teaching-learning process the foundations of is left for higher education

Before starting studying the students are assisted from mathematics. We ask the number and function abstraction and about the model creation in the test. We reveal their deficiencies based on their solutions.

The pretest

We found out that our students have deficiencies in the following:

•The order of doing operations on numbers ( this is very important)

•The rules of the index laws

•Methods of fractions

•Etc…

•The teaching-learning process in damaged on the different levels of the education.

•How can we make up for there differences in higher education?

•I think this topic important because analysis of mathematics is a basic subject for our students and they have to know functional operations in order to be able to describe economic processes with the help of functions.

I have been dealing with the Bruner’s representational theory and I am trying to adapt it to my research. Bruner examined what the man is like with the help of codes stores the information arriving from the external world. All thought processes may happen on of three kinds of plane according to it:

•Material level (actual objective acts, activities)•Iconic level ( visual education, situation)•Symbolic level

•The 3 representation methods take part in each phase of the teaching process.

•To my mind the visual education is very important, that’s why I tried to provide everyday, lifelike illustrations to help the acquisition of the material.

I think the use of the computer is an opportunity to help the interaction between the cognitive levels listed above.

We recommend an optional subject to the students.It was called Teaching of mathematics using computer. This course was going in parallel with the Mathematics I-II (calculus) subject.

The subject had a threefold aim:

•The development and conditioning of the basis

•To link it closely with higher mathematics

•To link it with the use of computers.

In the last semesters we collected positive feedback during teaching this subject.

We used Excel and GeoGebra, too.

It was important to use a programme which is available for every student and they can use it during preparation. Within the frames of this subject there is a possibility for development. This is my opinion.

Teaching of mathematics using computer I.in parallel with

the Mathematics I. (calculus) subject.

Themes

1. Revision of some parts of secundary education curriculumAlgebraic identity, absolute value, solve of equations and inequalityRaising to a power, extraction of a root, definition of logarithm, identity of logarithmDefinition of function, function attributes, draw and elementary functions graphs Function transformations, operationsDefinition of sequnces, arithmetic, geometric sequnces2. Solving tasks which are closely connected to the syllabus of basic mathematics.Compared to the practice of Calculus it helps the student achieve the necessary level by solving task rows built on top of each other in a smaller and easier steps.3. Calculation and draw of elements of the sequnces, draw functions graphs in Excl and GeoGbra, geometrical illustration of Newton quotient, derivative in Excel and GeoGebra

Representacion of the series in Excel

4

3lim

2234 2

nn

n

a

nnna

C1=B1-2*A1

5

81lim

1195

3632

12

nn

n

nn

n

c

c

The graph representation of the function in Excel

We put lot of effort to draw graph of function with Excel because we experience that the students can solve function analysis problems well, except drawing the graph of function. They determine the 1st and 2nd derivative, their root, sign, but the drawing the graph of function is still causes trouble.We have to select the interval, on wich we draw the grapf of function, so we give here to subset of the domain of function. After this we choose step value, it is important how large an the step value, because it may happen on the case of a big step value, that everywhere differentiable functions have breakpoints on the graph. (We can correct this, when we select “smooth lines” diagram)As an example we selected a function, which has break point, extremal value, inflection point also. How do we choose the right interval?Which ones are the important, exciting points, which have to contain the selected subset of the domain of function?These are the singularity points, roots, extremal values, inflection points of function.

)(

1)(

2)( 2

xfxg

xxxf

)()(

25,0)(2

2

xfxg

xxxf

GeoGebra in the education of analysis

xxxxf ln42)( 2

22 ln)( xxxh

Observe the shape of the first derivative algebra!

Teaching of mathematics using computer II.in parallel with

the Mathematics II. (calculus) subject.

Themes

Linear approximation and approximation by Taylor series of the functions with GeoGebraRiemann sum of function (integral) with GeoGebra, with Excel not, because that is too difficultCalculation of faktorial and binomial coefficient. Permutations with and without repetitions, combinations with and without repetitionsModeling the random effects, frequency, relativ frequency. Probability distribution (discrete and continuous) Mátrix operations: multiplication, inverse. Determinant, solution to a system of linear equtions with computation inverse matrix and with Cramer's rule wih Excel and LINV pogrammI think the LINV program is used by our university.

Taylor-polynomial with GeoGebra

The more members are pictured, the Taylor polynomial approximates the function better.

Approximate amount of the lowerwith GeoGebra

Distributions exercise

Mátrix operations: multiplication, inverse. Determinant, solution to a system of linear equtions with LINV programm

LINV

Open LINV.ZIP

Here you can choose the number of rows and columns.

The program controls that we can make the product.

The produkt

Teaching of mathematics using computer III.in parallel with

the Research of operation subject

Themes

System of linear equtions, linear and nonlinear programming (LP, NLP) problems, transport optimization, sensibility analysis in Excel Solver and LINV program („home (self) made program”)

Solving the LP exercise

max42

352

0,, 20

302

321

321

32131

321

xxxz

xxx

xxxxx

xxx

with Excel

with LINV

We can solve with simplex method by LINVThe revised standard and the general LP task can be solved with the program.

Three variables LP exercise with Euler3D

Thank you for your attention!