Transcript Educational experiences about using different computer

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 Geometry
Systems in Mathematics Education
11-13 July, 2009 RISC, Linz, Austria
University of Kaposvár
Faculty of Pedagogy
Faculty of Economic
Science
Faculty of Animal
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.
Optimization
2+2
2+2
(Probability and
Linear algebra)
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 curriculum
Algebraic identity, absolute value, solve of equations and inequality
Raising to a power, extraction of a root, definition of logarithm, identity
of logarithm
Definition of function, function attributes, draw and elementary
functions graphs
Function transformations, operations
Definition of sequnces, arithmetic, geometric sequnces
2. 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
a n  4n 2  3n  2  2n
lim
a 
n n
3
4
C1=B1-2*A1
32 n  6  3n 1
cn 
5  9 n  2  11
81
lim cn 
n 
5
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.
f ( x)  x 2  x  2
1
g ( x) 
f ( x)
f ( x)  0,5 x 2  2 x
g ( x)  f 2 ( x)
GeoGebra in the education of analysis
f ( x)  x 2  2 x  4 ln x
h( x)  x ln x
2
2
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 GeoGebra
Riemann sum of function (integral) with GeoGebra, with Excel not,
because that is too difficult
Calculation of faktorial and binomial coefficient. Permutations with
and without repetitions, combinations with and without repetitions
Modeling 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 pogramm
I 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 lower
with 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 produkt
The program
controls that
we can make
the product.
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
x1  2 x2  x3  30
x1  x3  20
x1 , x2 , x3  0
2 x1  x2  x3  35
z  2 x1  x2  4 x3  max
with LINV
We can solve with simplex method by
LINV
The revised standard and the general LP
task can be solved with the program.
with Excel
Three variables LP exercise with Euler3D
Thank you for your attention!