The Role of the Computer in Learning Mathematics Through Numerical Methods
Costică Lupu
Faculty of Sciences, Department of Mathematics-Informatics and Education Sciences, Vasile Alecsandri University of Bacău, Str. Mărășești 157, România
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To cite this article:
Costică Lupu. The Role of the Computer in Learning Mathematics Through Numerical Methods. Science Journal of Education. Vol. 4, No. 2, 2016, pp. 32-38. doi: 10.11648/j.sjedu.20160402.13
Received: March 3, 2016; Accepted: March 10, 2016; Published: March 24, 2016
Abstract: The paper exploits the results of an experimental, formative-ameliorative research conducted by 30 3^{rd}-year students from the Department of Mathematics, Vasile Alecsandri University of Bacău, during their teaching practice at the Ștefan cel Mare National Pedagogical College from Bacău, involving 150 students from 6 11^{th}-grade classes with a real specialization profile. The research was based on the following hypothesis: if we use numerical methods to solve linear equations systems and to the graphical representation of functions in the instructive-educational process, then we shall enhance the efficiency of these activities and increase the students’ performance by enhancing intrinsic motivation. In order to achieve the objectives, there were presented various techniques for solving linear equations systems and for the graphical representation of functions through numerical methods, followed by the application of sets of tests on the different methods for solving mathematical problems integrated in the various moments of the lesson, either in teaching new content or in consolidating and checking it. The paper highlights the role and values of computer use in learning Mathematics, in the informative as well as formative-educational sense, in agreement with the taught objectives and contents, based on the tendencies of updating and upgrading the school activity, and enhancing its role in preparing students for life.The research objectives were: - knowledge of the students’ (initial) training level as a basis for implementing the experiment; - presentation of the theory on numerical methods; - evaluating the contribution of the methods for solving linear equations systems and the graphical representation of functions through numerical methods to the enhancement of school performance; - recording progress following the application of the progress factor, respectively the various methods for solving linear equations systems and the graphical representation of functions through numerical methods.
Keywords: Computer, Learning Mathematics, Formative-Ameliorative Research, Numerical Methods
1. Introduction
The algorithm for solving linear equations systems can also be computer programmed. This method relies on serially reducing unknowns, the system evolving into other equivalent systems, whose number of equations diminishes step-by-step; this method is called Gaussian elimination or row reduction.
For the gradual processing of the system, there are applied the following elementary transformations that generate equivalent systems: - swapping two rows (equations); - reordering the unknowns; - multiplying a row by a non-zero number; - adding a multiple of one row to another row. (Frumuşanu G., 2008),
By applying these operations, there is generated one of the situations: - the final system is triangular, its solution being unique (compatible determined); - the final system is trapezoidal, with several solutions (compatible non-determined); - the final system contains a contradiction, with no solutions (incompatible).
Practically, applying the Gaussian method consists in covering the following steps: - writing the extended matrix of the system (namely, the system’s matrix, to which we annex the column of free terms); - applying elementary transformations to this matrix, until it takes a triangular or trapezoidal form; - analysing the linear system to which this extended matrix belongs; - if there occurs a contradiction within this system, then the system is incompatible; - if there occurs no contradiction, then the system is compatible or compatible non-determined, according to whether it is triangular or trapezoidal; - the system’s solution is easily found by covering the path backwards (back-substitution), from the last equation (with the fewest unknown values) towards the first (with the most unknown values) (Mariș S., Brăescu L., 2007).
2. Theoretical Substantiation
1) Given and the real numbers
The equation system is called a system with m linear equations and n unknowns.
2) The matrix is called the system’s matrix or the matrix of the system’s coefficients.
3) Numbers b1, b2, ... , bm are called free terms; the matrix is called the column matrix of the free terms, and matrix obtained from the system’s matrix through bordering to the right with the column of free terms, is also called the extended matrix of the system.
3. Examples for Applying the Gaussian Method to Linear Systems
The Gaussian method consists in the equivalent transformation of the system through elementary transformations, in systems where the unknown occurs only in the first equation, whereas in the other equations it is eliminated.
For the system thus formed the first equation is kept unchanged and for the other m-1 equations there is applied the procedure for the unknown , keeping it in the second equation and eliminating it in the other equation. The procedure is repeated until in one equation of the system there remains only one unknown. Its value is transferred to the other equations and the other unknowns are determined. By applying the Gaussian method, the unknowns are successively eliminated.
1. Solve, in the set of real numbers, the following linear system, using the Gaussian method: .
Solution: We write the extended matrix associated with the system and by applying elementary transformations we give it a triangular form. By solving the system, we shall finally reach the solution: . The dotted line from the extended matrix has the role of rendering the system’s free terms visible, and the matrix of the final system (except the column of the free terms) has a triangular form, hence, from that moment on, it is known that the system is compatible determined. (Lupu C., 2014)
2. Solve, in the set of real numbers, the following linear system by using the Gaussian method: .
Solution: We write the extended matrix associated with the system and by applying elementary transformations we give it a triangular form. From the finally obtained system, there are obtained 2 contradictory equations: 35t = 71 and 96t = 186. Therefore, the system is incompatible.
4. Solving Linear Equations Systems Using Computers
The Gaussian method for solving linear systems with the matrix of coefficients as a column (set). Such a matrix attached to the system has all the elements null, except those from the main diagonal and some parallels to the main diagonal. The program uses a special method for memorizing the column matrix, further briefly presented.
Given B=(b_{i,j}) a matrix with n rows and n columns, having 2d+1 non-zero parallels to the main diagonal. The row i of this matrix is: (0,0,...,0,b_{i,i-d},b_{i,i-d+1},b_{i,i},b_{i,i+1},b_{i,i+d},0,0,...,0)
The meaningful information from this row may be memorized in the row i of a matrix A=(a_{i,j}), with n rows and 2d+1 columns: (a_{i,1},a_{i,2},...,a_{i,p},a_{i,p+1},...,a_{i,2d+1})
There may be observed that p=p(i,j)=i+d+1-j. There is applied to the matrix A=(a_{i,p(i,j))} the classic Gaussian elimination algorithm. (Nechita, E.; Muraru, C.-V.; Talmaciu, M., 2012),
The source program:
PROGRAM Test_Gauss_Elimin_Column;
USES Crt,Printer;
CONST
...nmax=200;{maximum number of equations}
...dmax=31; {number of non-zero diagonals}
TYPE
...UserType = Extended;{Double; Real}
...Vec = ARRAY[1..nmax] OF UserType;
...Mat = ARRAY[1..nmax,1..dmax] OF UserType;
VAR
...n,d : Integer;
...a : Mat;
...b : Vec;
...flg : Boolean;
FUNCTION p(i,j:Integer):Integer;
BEGIN
...p:=j-i+d+1
END;
FUNCTION Min(i,j:Integer):Integer;
BEGIN
...IF (i<j) THEN Min:=i
...ELSE Min:=j
END;
FUNCTION Max(i,j:Integer):Integer;
BEGIN
...IF (i>j) THEN Max:=i
...ELSE Max:=j
END;
PROCEDURE WriteMat(VAR a:Mat);
{Show system coefficients}
VAR
...i,j,n1,n2 : Integer;
BEGIN
...n1:=8; n2:=5; WriteLn;
...WriteLn(System coefficients:');
FOR i:=1 TO n DO
...BEGIN
......FOR j:=1 TO n DO
.........IF ((p(i,j)>=1) AND (p(i,j)<=3*d+1))THEN
............Write(a[i,p(i,j)]:n1:n2)
.........ELSE Write(0.0:n1:n2);
......WriteLn
...END;
...WriteLn
END;
PROCEDURE Wait;
VAR
...c:Char;
BEGIN
...WriteLn;WriteLn('Press a Key !');
...REPEAT c:=ReadKey UNTIL (c<>'')
END;
PROCEDURE GaussElim(n,d:Integer;VAR a:Mat;VAR b:Vec;VAR flg:Boolean);
VAR
...i,j,k,ipvt : Integer;
...temp : UserType;
...apvt : UserType;
BEGIN
...flg:=False;
...FOR i:=1 TO n-1 DO
...BEGIN
......apvt:=0;ipvt:=i;
......FOR j:=i TO Min(i+d,n) DO
.........IF (apvt<Abs(a[j,p(j,i)])) THEN
.........BEGIN
............apvt:=Abs(a[j,p(j,i)]);ipvt:=j
.........END;
......IF (apvt=0) THEN
......BEGIN flg:=True; Exit END;
......IF (ipvt<>i) THEN
......BEGIN
.........FOR k:=i TO Min(i+2*d,n) DO
.........BEGIN
............temp:=a[i,p(i,k)];
............a[i,p(i,k)]:=a[ipvt,p(ipvt,k)];
............a[ipvt,p(ipvt,k)]:=temp
.........END;
.........temp:=b[i];b[i]:=b[ipvt];
.........b[ipvt]:=temp
......END;
......FOR j:=i+1 TO Min(i+d,n) DO
......BEGIN
.........IF (a[j,p(j,i)]<>0) THEN
.........BEGIN
............temp:=-a[j,p(j,i)]/a[i,p(i,i)];
............FOR k:=i+1 TO Min(i+2*d,n) DO
...............a[j,p(j,k)]:=a[j,p(j,k)]+temp*a[i,p(i,k)];
............b[j]:=b[j]+temp*b[i];
............a[j,p(j,i)]:=0
.........END
......END
...END;
...IF (a[n,p(n,n)]=0) THEN BEGIN flg:=True; Exit END;
...b[n]:=b[n]/a[n,p(n,n)];
...FOR i:=n-1 DOWNTO 1 DO
...BEGIN
......temp:=b[i];
......FOR k:=i+1 TO Min(i+2*d,n) DO
.........temp:=temp-a[i,p(i,k)]*b[k];
......b[i]:=temp/a[i,p(i,i)]
...END
END;
BEGIN {Main Prog}
...ClrScr;
...flg:=False;
...Write('n=');ReadLn(n); {Number of equations}
...Write('d=');ReadLn(d); {Number of diagonals}
...Date2(a,b);
...IF (n<50) THEN WriteMat(a);
...WriteLn; WriteLn('Free terms');
...ScrieSol(b);
...Wait;
...GaussElim(n,d,a,b,flg);
...Write(#7);
...WriteLn; WriteLn('System solution:');
...WriteSol(b);
...Wait
END {Test_Gauss_Elimin Column.
4.1. Application Examples of Applying Numerical Methods to Solve the Equation System
Solve the equation system:
0.39419·x1+0.25212·x2=0.64631
0.95007·x1+0.71412·x2+0.72213·x3=2.38632
0.68713·x2+0.57916·x3+0.76899·x4=2.03528
0.98697·x3+0.80641·x4+0.02246·x5=1.81585
0.83403·x4+0.22243·x5=1.05646
The result of running the program:
n=5
d=1
System coefficients:
0.39419 0.25212 0.00000 0.00000 0.00000
0.95007 0.71412 0.72213 0.00000 0.00000
0.00000 0.68713 0.57916 0.76899 0.00000
0.00000 0.00000 0.98697 0.80641 0.02246
0.00000 0.00000 0.00000 0.83403 0.22243
Free terms:
0.64631 2.38632 2.03528 1.81585 1.05646
System solution:
1.00000 1.00000 1.00000 1.00000 1.00000
4.2. Examples of Applying Numerical Methods to Graphical Representations
Likewise, these programs particular to numerical methods may be applied to graphical representations. We shall further present a program that enables the graphical representation of every function, the scaling of the graph being performed automatically. The axes of coordinates are traced only when they enter the display window. We have chosen to illustrate the graphical representation of functions . (Postolică, V.; Nechita, E.; Lupu, C., 2014)
The source program:
PROGRAM Graphical representation;
USES Crt,Graph;
CONST
...nmax=640; { Number of points }
TYPE
...Vec = ARRAY[1..nmax] OF Real;
...St4 = STRING[4];
VAR
...xp,yp : Vec;
...n,brx,bry : Integer;
...xmin,xmax : Real;
...ch : Char;
...fld : Boolean;
PROCEDURE Date;
VAR
...x,dx : Real;
...i : Integer;
...c1 : Char;
FUNCTION f(x:Real):Real;
BEGIN
...f:=Sin(x)+Sin(3*x)
...{f:=1/(1+Sqr(x))}
END;
BEGIN
...c1:='x';
...REPEAT
......ClrScr;
......WriteLn('1...Number of points: ',n);
......WriteLn('2...xmin : ',xmin:6:3);
......WriteLn('3...xmax : ',xmax:6:3);
......WriteLn('4...brx : ',brx);
......WriteLn('5...bry : ',bry);
......WriteLn('0...Exit');
......REPEAT c1:=ReadKey UNTIL (c1 IN ['0'..'5']);
......CASE c1 OF
.........'1' : BEGIN
...............Write('Number of points: ');ReadLn(n)
...............END;
.........'2' : BEGIN
...............Write('xmin: ');ReadLn(xmin)
...............END;
.........'3' : BEGIN
...............Write('xmax: ');ReadLn(xmax)
...............END;
.........'4' : BEGIN
...............Write('brx: ');ReadLn(brx)
...............END;
.........'5' : BEGIN
...............Write('bry: ');ReadLn(bry)
...............END;
.........'0' : BEGIN
...............dx:=(xmax-xmin)/n;
...............FOR i:=1 TO n DO
...............BEGIN
..................x:=xmin+(i-1)*dx;
..................xp[i]:=x;yp[i]:=f(x)
...............END
......END
...END
...UNTIL c1='0'
END;
PROCEDURE Wait;
VAR
...cc:Char;
BEGIN
...REPEAT cc:=ReadKey UNTIL cc<>''
END;
PROCEDURE GrInit;
VAR
...Gd, Gm : Integer;
BEGIN
...Gd:=Detect;
...InitGraph(Gd,Gm,'c:\BP\BGI');IF GraphResult=grOK THEN Exit;
...Halt(1)
END;
PROCEDURE GetSize;
VAR
...i:Integer;
BEGIN
...maxa:=a[1]; mina:=maxa; maxb:=b[1]; minb:=maxb;
...FOR i:=2 TO n DO
...BEGIN
......IF maxa<a[i] THEN maxa:=a[i];
......IF maxb<b[i] THEN maxb:=b[i];
......IF mina>a[i] THEN mina:=a[i];
......IF minb>b[i] THEN minb:=b[i]
...END;
...sx:=(GetMaxX-2*brx)/(maxa-mina);
...sy:=(GetMaxY-2*bry)/(maxb-minb);
...stpx:=Int(Ln(maxa)/Ln(10)-1);
...stpx:=Exp(stpx*Ln(10));
...stpy:=Int(Ln(maxb)/Ln(10)-1);
...stpy:=Exp(stpy*Ln(10))
END;
BEGIN
...GrInit; mx:=GetMaxX;my:=GetMaxY; GetSize;
...IF (minb*maxb<=0) THEN
...BEGIN
......MoveTo(0,Yplot(0)); LineTo(mx,Yplot(0));
......LineTo(mx-8,Yplot(0)+8);
......MoveTo(mx,Yplot(0));
......LineTo(mx-8,Yplot(0)-8);
......OutTextXY(mx-8,Yplot(0)+10,txtx);
......xx:=stpx;
......WHILE Xplot(xx)<=mx-10 DO
......BEGIN
.........MoveTo(Xplot(xx),Yplot(0)-4);
.........LineTo(Xplot(xx),Yplot(0)+4);
.........xx:=xx+stpx
......END;
......xx:=-stpx;
......WHILE Xplot(xx)>=10 DO
......BEGIN
.........MoveTo(Xplot(xx),Yplot(0)-4);
.........LineTo(Xplot(xx),Yplot(0)+4);
.........xx:=xx-stpx
......END
...END;
...IF (mina*maxa<=0) THEN
...BEGIN
......MoveTo(Xplot(0),my); LineTo(Xplot(0),0);
......MoveTo(Xplot(0),0); LineTo(Xplot(0)+8,8);
......MoveTo(Xplot(0),0); LineTo(Xplot(0)-8,8);
......OutTextXY(Xplot(0)+10,0,txty);
......OutTextXY(Xplot(0)-10,Yplot(0)+8,'0');
......yy:=stpy;
......WHILE Yplot(yy)>=10 DO
......BEGIN
.........MoveTo(Xplot(0)-4,Yplot(yy));
.........LineTo(Xplot(0)+4,Yplot(yy));
.........yy:=yy+stpy
......END;
......yy:=-stpy;
......WHILE Yplot(yy)<=my-10 DO
......BEGIN
.........MoveTo(Xplot(0)-4,Yplot(yy));
.........LineTo(Xplot(0)+4,Yplot(yy));
.........yy:=yy-stpy
......END
...END;
...xp0:=Xplot(a[1]); yp0:=Yplot(b[1]);
...FOR i:=2 TO n DO
...BEGIN
......MoveTo(xp0,yp0); xp1:=Xplot(a[i]); yp1:=Yplot(b[i]);
......LineTo(xp1,yp1); xp0:=xp1; yp0:=yp1
...END;
...Wait;
...CloseGraph
END;
BEGIN {Main Prog}
...fld := False;
...brx := 20;
...bry := 20;
...n := 200;
...xmin :=-1;
...xmax := 2;
...REPEAT
......ClrScr;
......WriteLn('1...Initial data');
......IF fld THEN
......BEGIN
.........WriteLn('2...Function graphic')
......END;
......WriteLn('0..Exit');
......REPEAT ch:=ReadKey UNTIL (ch IN ['0'..'2']);
.........CASE ch OF
.........'1' : BEGIN
...............Date;fld:=True
...............END;
.........'2' : IF fld THEN Graf(xp,yp,n,brx,bry,'x','y');
.........'0' : Exit
.........END
...UNTIL False
END {Graphic Representation}.
The results of running the program are shown in the figures below (Lupu C., 2015).
5. Research Methodology
5.1. Research Methods and Techniques
The research was of an experimental type, using the test method. Other research methods and techniques used were: - Pedagogical observation; - The communication; - Analysis of school documents and student work products; - The interview; - Statistical techniques for data processing.
5.2. Research Description
The paper highlights the role and values of computer use in learning Mathematics, in the informative as well as formative-educational sense, in agreement with the taught objectives and contents, based on the tendencies of updating and upgrading the school activity, and enhancing its role in preparing the student for life.
5.3. Sample Description
The paper exploits the results of an experimental, formative-ameliorative research conducted by 30 3^{rd}-year students from the Department of Mathematics, Vasile Alecsandri University of Bacău, during their teaching practice at the Ștefan cel Mare National Pedagogical College from Bacău, involving 150 students from 11^{th}-grade classes with a real specialization profile.
5.4. Research Objectives
The research objectives were: - knowledge of the students’ (initial) training level as a basis for implementing the experiment; - presentation of the theory on numerical methods; - evaluating the contribution of the methods for solving linear equations systems and the graphical representation of functions through numerical methods to the enhancement of school performance; - recording progress following the application of the progress factor, respectively the various methods for solving linear equations systems and the graphical representation of functions through numerical methods.
5.5. Research Hypothesis
The research was based on the following hypothesis: if we use numerical methods to solve linear equations systems and to the graphical representation of functions in the instructive-educational process, then we shall enhance the efficiency of these activities and increase the students’ performance by enhancing intrinsic motivation.
5.6. Research Variables
The research hypothesis generates two research variables: - the independent variable, introduced through the numerical methods for solving systems and the graphical representation of functions; - the dependent variable related to enhancing the motivation for acquiring mathematical notions and school progress.
5.7. Research Stages
The research was conducted February 5^{th} – May 30^{th}, during the 2^{nd} term of the 2014-1015 school year. The paper exploits the results of an experimental, formative-ameliorative research conducted by 30 3rd-year students from the Department of Mathematics, Vasile Alecsandri University of Bacău, during their teaching practice at the Ștefan cel Mare National Pedagogical College from Bacău, involving 150 students from 6 11th-grade classes with a real specialization profile. The research was based on the following hypothesis: if we use numerical methods to solve linear equations systems and to the graphical representation of functions in the instructive-educational process. The paper highlights the role and values of computer use in learning Mathematics, in the informative as well as formative-educational sense, in agreement with the taught objectives and contents, based on the tendencies of updating and upgrading the school activity, and enhancing its role in preparing students for life.
6. The Results and Their Interpretation
Through the experiment carried out on an initial test with second year Math undergraduates of "Vasile Alecsandri" University of Bacău, it was proved that the teaching and the development of skills and abilities for assessment in high school are possible if we use various evaluation methods and procedures. This information was very useful in planning the following activities, taking into account the specificities of each student. Motivation for team learning consists (without the students being aware) of exciting activities, attractive, intuitive special materials, worksheets and modern teaching methods.
In terms of the second year students of the Faculty of Mathematics, it was found that through impact assessment, observed learning and assessment records, there was active participation on the part of the students, increasing the degree of intellectual effort, interest and curiosity with regard to mathematics.
This data was recorded in an observation grid. At the same time, the experiment results confirm the hypothesis that if we use various techniques for teaching-learning-assessment in all lesson stages, the teaching of mathematics in school will be more efficient, and the results of the pupils will improve.
In terms of the second year students of the Faculty of Mathematics, it was found that through impact assessment, observed learning and assessment records, there was active participation on the part of the students, increasing the degree of intellectual effort, interest and curiosity with regard to mathematics.
Analysing the results obtained by the students in the initial and final evaluation tests, there may be diagnosed a relevant progress in terms of problem-solving competences and skills, calculus abilities as well as correct use of mathematical concepts. Of the 15 students who got initial marks below 5, 10 succeeded in getting marks above 5 in the final test, applied at the end of the term. The raise in the number of students who obtained the marks 7, 8, 9 and 10 is relevant and may be followed in the table and frequency polygons below.
Nota | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Test initial | 30 | 20 | 34 | 20 | 24 | 12 | 10 |
Test final | 8 | 10 | 20 | 33 | 35 | 24 | 20 |
This comparative frequency polygon showing the results in the initial and final test highlights the fact that although the number of students who got 5 in the final test is much smaller than the number of students who got this mark in the initial test, the number of students who got marks above 8 increased significantly compared to the initial test to 20 marks of 10.
7. Conclusions
The progress of the school results is obvious, both at the individual and class level, an aspect reflected in the frequency of the marks obtained and the general means of the class, calculated at the two stages of the research: pre-experimental and post-experimental.
Regarding the initial evaluation, the mean of the class results were 6.42, an average that corresponds to below standard performances. In the final evaluation, the class average was 7.52, showing a relevant increase of 1,10 points. It is worth mentioning that a first progress was observed as of the stage of applying the experimental factor, the mean of the marks obtained by students in the formative evaluation being 6,85.
In the final evaluation test, 74% of the students obtained marks at least equal to 7, and 22 of the 30 students who got marks below 5 in the initial test succeeded in obtaining marks above 5 in the final test. The comparative analysis reveals the relevant growth of the students’ results at Mathematics, which validates the hypothesis of the experimental research. Besides the progress recorded at the level of the school results, it is also worth mentioning the progress on the motivational level, there being a greater number of active students, interested in the school activity, at the expense of the passive and disinterested ones.
References