CONFRONTATION OF TRADITIONAL AND COMPUTER SUBSIDIZE LEARNING OF FUNCTION 1

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1 Acta Didactica Universitatis Comenianae Mathematics, Issue 6, 006 CONFRONTATION OF TRADITIONAL AND COMPUTER SUBSIDIZE LEARNING OF FUNCTION MARTINA SANDANUSOVÁ Abstract. I used program Equation Grapher at the teaching of theme Function. The results of these eperiment I described in following article. The basic hypothesis that I verified was: The using of computers during teaching of mathematics could make education more effective and enhance motivation of pupils on mathematics lessons. Résumé. J ai utilisé le logiciel Equation Grapher dans l enseignement du thème Fonctions. Les résultats de cette epérience j ai décrit dans l article présenté. L hypothèse principale, que j ai vérifié, était: Les ordinateurs dans l enseignement des mathématiques peuvent améliorer le processus de l enseignement et augmenter la motivation des élèves au cours des mathématiques. Zusammenfassung. Das Programm Equation Grapher habe ich zum Unterricht der Funktionen benutzt. Das Ergebnis von diesem Eperiment habe ich in folgendem Artikel beschrieben. Die Basis-Hypothese, die ich verifizieren wollte, lautet: Die Computer während der Matheunterricht koenen der Unterrrichtprocess effektiver zu machen und die Schuele-rmotivation zu erhoehen. Riassunto. Ho usato il programma Equation Grapher per l insegnamento dell argomento Funzioni. Il risultato di questo esperimento è descritto nel seguente articolo. Le ipotesi di base da me verificate sono state: l uso dei computers durante l insegnamento della matematica potrebbe permettere un educazione più efficace e accrescere le motivazioni degli studenti durante le lezioni di matematica. Abstrakt. Program Equation Grapher som použila pri vyučovaní tematického celku Funkcie. Výsledky tohto eperimentu som popísala v nasledujúcom článku. Základná hypotéza, ktorú som overovala znela: Počítače vo vyučovaní matematiky môžu zefektívniť vyučovací proces a pomôžu zvýšiť motiváciu žiakov na hodine matematiky. This article was partly supported by European Social Funds JPD 3 BA-005/-063 and SOPLZ- 005/-5

2 64 M. SANDANUSOVÁ Key words: Confrontation, traditional, computer subsidize learning, function, Equation Grapher, motivation of pupils, effective learning INTRODUCTION The goal of the eperiment was to show that using the computer could be solve the plenty of issues more effectively. From three classes of the first grade on secondary technical school were chosen two classes one with worse results (.B. eperimental group) and one class with better results (.D. control group) from math. Lessons themes were functions and their properties. Division of themes: Total number of lessons: 0 What is a function? Definition Decreasing and increasing function Bounded function Even and odd function Periodic and inverse function Revision and written eam Initial conditions in the classes: Number of students in the class Semi annual average in math Total Semi annual average Teaching aids.d control group.b eperimental group 6 33,88,54,03,4 color chalk, blackboard newspapers, tetbooks (Matematika pre študijné odbory SOŠ a SOU. časť; Matematika pre. ročník gymnázií a SOŠ zošit Rovnice a nerovnice; Zbierka úloh z matematiky pre SOŠ a SOU. časť shareware Equation Grapher at computers, felt-tips and tetbook as in control group television

3 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 65 The students were taught 4/ lessons in a week (3 lessons had a theory of functions and +had a training). There were students divided to three groups in the eperimental group, so that each of them had a computer. The students had been taught traditional lessons before. The reason was that they would not be allowed to use the computers at the further eaminations in the future. Division of the themes in the eperimental group: Theory: What is function? Definition. Decreasing and increasing function Bounded function Even and odd function Periodic and inverse function Eperiment: Groundwork with Regression Analyzer and Equation Grapher Calculate problems from theory with computer Goniometric function Written eam with PC The students sat in two lines, in front of the room was a blackboard, and teacher s work at table with PC and television, where the students can see what teacher is doing on PC. TRADITIONAL LEARNING IN THE CONTROL GROUP (.D) Structure of lessons and time interval:. LESSON Theme: Function the domain of a function D(f) and the range of the function H(f). Goal: Definition of the function, D(f) and H(f).. For motivation of the students was used an eample of trajectory from physics where they already could see the function and its graph. They wrote the relationship and draw a diagram. 0 min. Definition of new ideas of functions. What is a function and how it is entered?

4 66 M. SANDANUSOVÁ E.. What is the domain of the function f: y = 3, g: y = (the teacher solved the problems)? E.. What is the domain of the function (students solved the problems on the blackboard)? 0 min 3 u: y = ; 3, a) D(u), H(u) b) u(), u( 5) c) Decide if the points [0; ], [0; ] and [; 6] belong to the function u. E.3. Calculate D(f) of the function f : y = 3 (students solved the problems in the eercise book). 5 min 3. Homework from the tetbook. 5 min. LESSON Theme: Graph of the function. Goal: Introduce a definition of graph of the function, specify domain of definition from the graphs.. Revision of homework (three students from class). 0 min. Revision from last lesson, new eamples. 5 min E. Specify D(f), D(g), f(3), g(3) and solve the problem whether 5 H(f), H(g): + 5 f : y= ; g: y = 3+ ( 3).( + ) 3. Define the graph of a function; what is the graph of a function and give an eample. 5 min E. Draw a graph of given functions: { ] ] ] h y ] ] } { ]} g: y= ;3 ; 4; ; 3;5 ; : = 0;0 ;,5;7 ; 4;0 ; z: y = +, 3; ; ; 0;; ; 3 ; u: y= { E. Calculate the co-ordinates of intersection point of function h with, y ais 5 min h: y 4. Homework from the tetbook. 5 min

5 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING LESSON Theme: Decreasing and increasing function. Goal: Define decreasing and increasing function; determine them from the graph or by the aid of calculation.. Homework control. 0 min. Revision and new problems from last lesson. 5 min E.. Draw a graph of function f and calculate its D(f) and H(f) f : y= E.. Draw a graph of function u: y = + b, b { ; 3; 0; 5} and describe the difference. E.3. Draw a graph of function u: y = a, a { ; 0; } and describe the difference. 3. Definition of decreasing and increasing functions, its graph. The teacher solved the problems on the blackboard. 5 min f : y= ; g: y= 3 The third graph is not function. 4. Homework, draw a graph and properties of the function. 5 min f : y= y = f( ) = ; y = f( ) = ; y = f( ) = 3 4. LESSON Theme: Properties of functions increasing, decreasing, one to one function. Goal: Acquisition idea, when functions are increasing or decreasing and launch new properties, when a function is one to one.. Inspection and analyse homework responsibilities, because they had the problems with drawn of the functions. We repeated problems together with the students on the lesson. 0 min

6 68 M. SANDANUSOVÁ. Theoretical application of the new properties of functions one to one, how this property relates with another properties of function increasing and decreasing. Eplanation on the eamples. 5 min The fourth, sith and seventh graph are not functions. 3. Application of the typical problems from prais (service tetbook Matematika pre. ročník gymnázií a SOŠ zošit Rovnice a nerovnice p. 58 6). 5 min 4. Acquisition of survey about the graphs bring the biggest quantity of various graphs especially from the newspapers or magazines. 5 min 5. LESSON Theme: Properties of functions bounded and unbounded function. Goal: Application of the new properties of functions following own know-how and brought material.. Selected students presented articles with graphs from the newspapers development crowns towards euro, demographic development, change in temperature etc. 5 min. Definition of the function bounded from the bottom and bounded from the above. Eample: 5 min E. Write everything what do you know about displayed function. f : y= 0,5+ 4; g: y =,5; h: y= 3. Homework: Draw the graph of given function and specify all its properties. f : y = 0 min +

7 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING LESSON Theme: Properties of function bounded and unbounded function its maimum and minimum. Goal: Eamination of student s knowledge and application of the concept of maimum and minimum of functions.. Inspection and analyse of homework, identification of problems with function displaying. Finding of the solution. 0 min. Set an eam to students, which implies theoretical and practical tasks. Students created definitions, described increasing and decreasing functions and drew graph of the function, which was one to one and which was not one to one. This eam has served only for feedback of teacher and it was not marked. 5 min 3. Application of new definition for maimum and minimum of functions. Demonstration on graphs and materials used at previous lessons. 5 min 4. Set homework from the book (Zbierka úloh z matematiky pre SOŠ a SOU,. časť). 5 min 7. LESSON Theme: Properties of functions even and odd functions. Goal: Application of the new properties reading from the graph and verification using the calculations.. Revision of homework, students had no problems with it. 5 min. Summarise of achieved knowledge, eplanation of limitations found in eam. 0 min 3. You draw these functions (use the graphs of functions that are drawn on the board) 5 min 3 3;3 f : y= ; g: y= ; i: y = ; j: y= 4. Calculate if the function is even or odd. 0 min 5. Students independently solved tasks. Unsolved tasks remain on homework. 3 4 u: y = ; v: + ; z: y = ; w: y = 3 ( + ); t: y = 0 min

8 70 M. SANDANUSOVÁ 8. LESSON Theme: Properties of function periodic and inverse function. Goal: Definition and dispersing of periodic function. Determination of inverse function by graphical and calculative ways.. Revision of homework on the lesson. 5 min. Summarise and solve some eercises before comple eam. 5 min 3. On the graph of goniometric functions f: y = sin, g: y = tg in the near future was illustrated the concept of periodic functions and the inverse function comple solution was demonstrated on function g: 0 min + 3 g: y = 5 4 p : y= + 0 min 4. Set of other eamples. ( ) 9. LESSON Theme: Repetition. Goal: Review of achieved knowledge on eamples. E.. E.. E.3. Define the graph and properties of given functions and define its point of intersection with ais ( ) t: y =, h: y =, v: y =, k: y = + 3 Determine if given prescriptions are functions, if they are find D(f) and decide if they are even or odd functions: y = + 0 y = 4 + y = y = 3 4 y = ( 4) ( 5) ( ) ( + 3) Find inverse functions to the functions: 7 3 y = y = y = A -6;0, B3;3. [ ] Construct graphs of the functions: E.4. Define all linear functions, which pass points [ ] E.5. y = + ( ) y = y =

9 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 7 0. LESSON Theme: Repetition written eam. Goal: Attestation of achieved results on written eam. The eam tasks were the same for all students. Maimum number of achieved points was 5 and took time 35 min. E.. Define D(f) of given functions: 5 points g: y= points 5+ 6 k: y h: y 3 = = points E Have the function g: y= 3 points Calculate g(5), g(). Calculate if g()=0 Calculate if is true that H( g) E.3. Determine D(f) and properties of given functions. 6 points ( + ) j: y = and draw the function j 4 points p: y = 4 3 points E.4. Calculate the co-ordinates of intersection points of function h with, y ais and inverse function to given function. o: y= Specification of evaluation of solution: E.. Function g determine correct conditions: > 0 determine correct D(g): Dg ( ) = (,) ( 3, ) Function k determine correct D(k): D( k) = R

10 7 M. SANDANUSOVÁ Function h determine correct conditions: > 0, > < 0 Dh ( ) =,0 determine correct D(h): ( ) E.. 3 Calculate g (5) = = 3, 5 4 0,5 point g() isn't definite at point 0,5 point 3 as g()=0 so = =,5 H( g) E.3. Drawing the graph of function j: determine correct D(j): ( ) ( ) D( j) = ;0 y = D( j) = 0; y= + properties increasing on full D(f) 0,5 point is odd 0,5 point isn't bounded 0,5 point isn't periodic 0,5 point calculate D( p) 4> 0 > ( ( D( p ) = ; ) ; ) properties increasing on the (, ) and decreasing on the (, ) 0,5 point even bounded below isn't periodic > 0 4 0,5 point 0,5 point 0,5 point E.4. Calculate intersection with ais 0,5; 0] and y 0; ] 0,5 point + calculation of inverse function o : y= 0,5 point

11 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 73 Valuation: Percentage of completion Mark Pointing valuation method 00 % 85, % 5,00,76 85 % 70, %,75 0,5 70 % 55, % 3 0,50 8,6 55 % 40, % 4 8,5 6,0 40 % 0 % 5 6,00 0,00 3 LEARNING PROCESS WITH THE SUPPORT OF EQUATION GRAPHER. LESSON Theme and Goal: Introduction to the Equation Grapher.. Acquaintance with software Regression Analyzer software for processing of statistical data. 5 min. Introduction to the Equation Grapher. I had translated the Main Menu of the application. We're passed together individual offers. Practically we had tried to put some functions f: y =, which we entered in the command line. Students worked easily with window Function Pad. Problems appeared with the length of ais. We set the right ais in windows Range. I left students work without my help while they did not familiarise with the icons of the main menu. Students had several problems: Malfunction of icon maim and minimum Malfunction of icon intersection Problem solution: Function hadn't got maimum either minimum, we could it verify in the window Log where was displayed the answer. We set function g: y = and using the icon for intersection, in the Log window we found the value of intersection point 30 min. LESSON Theme: Solution of specific eamples. Goal: Solution of eamples from theoretical lesson with using the computer.

12 74 M. SANDANUSOVÁ Students solved eamples with using of EG. E.. What is D(f) of the functions f: y = 3, g: y = /? u: y = 3 ; 3; ) a) D(u), H(u) b) u(), u( 0,5) c) Decide if the points [0; ], [0; ] and [; 6] belong to the function u. Students wrote out a command of the individual functions in the command line and from the graph they very simply verify the answer. Problems were with function u, because they didn t know how to set ^3. They were not familiar with Function Pad yet. E.3. Calculate D(f) of the function f : y= 3. Students set improper command of function e.g. y = sqrt ^ 3. Graphs looked like in the picture. Without using the brackets the computer calculate the second root only from ^ instead of whole epression ^ 3. E.4. Determine D(f), f(3) and solve problem 5 H(f) a) f : y= ; b) f : y= ( 3).( + ) Repeated error with brackets at setting the function y = /^ 3+. Correct notation is y = /(^ 3+). These two functions have different properties, which can we see on their graphs. Inscription of net function had to look like this y = ( 5)/ /(sqrt(( 3).( + )). It was important to eplain to students, that epression ( + 5) has to be in parenthesis, because we divide it with sqrt. There had to be parenthesis around the whole square

13 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 75 root, because we are making it from the product of both brackets. After solving problems with brackets, students simply solved the tasks using proper icons from the Log window, they read fast and accurate answer: y-cacl: Could not be evaluated E.5. Draw graphs: g: y= { ;3 ]; 4; ]; 3;5 ]}; h: y= { 0;0 ];,5;7 ]; 4;0]} z: y= + ; 3; ; ; 0;; ;3 E.6. { u: y= Calculate co-ordinates intersection points of h: y= with, y-ais. + Calculation in eercise book was replaced with automatic calculation in EG, using necessary icons. Eamples of incorrect notation of the function in the command line are e.g.: / + ( )/ + /( + ) Correct inscription had looked now y = ( )/( + ). The difference you can see on the graph. } 3. LESSON Theme: Solution of the specific eamples. Goal: Solution of eamples from theoretical lesson with using the computer. E.. Draw the graph of function u: y = + b, b { ;3;0;5}, and describe the difference between these functions.

14 76 M. SANDANUSOVÁ It was very simple to draw these functions for the students. They eactly saw how the function is changed when the parameter b is changed. E.. Draw the graph of function u: y = a, a { ; 0; } and describe the differences. E.3. Draw graphs of these functions and describe the properties of these functions: f : y= ; y = f( ) = ; E.4. y = f( ) = ; y3 = f( ) = Problems are appeared at the notation of this function, and the correct notation should be: y = abs( ); y = abs( ) ; y3 = abs(( abs( ) ) and then students could describe the properties of function. Designate the properties of the following functions: 3 u: y = ; v: + ; z: y = ; 4 w: y = 3 ( + ); t: y = Concerning the problems with the notation of the brackets, they homework was to note down this function: f : y= LESSON Theme: Goniometric functions + repetition the written eam. Goal: Work with the goniometric functions, the correct choice of coordinate of the aes. E. Application of goniometric functions and properties of the periodic functions. For eample: y = sin and y = cos a) traditional adjustment of the number line Xmin 7; Xma 7 Xstep Ymin 5; Yma 5 Ystep

15 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 77 b) changing the interval on the multiplying π/ with Options Angles Radians, in the window Range Re Xmin 7,796 Xma 9,4; Xstep,57 Ymin 3,4; Yma 3,4 Ystep c) changing the ais to grades with Options Angles Degrees in the window Range with icon sin and Re Xmin 540; Xma 540 Xstep 90 Ymin 3,4; Yma 3,4 Ystep 5. LESSON Theme: Written eam. Goal: The verification of the student s knowledge same as in the.d. All groups of students wrote this eam at the same time. To avoid the facilitation, the teachers are not mathematicians. Students could use the computers, but the calculation should be also recalculated without using the computer. Students had 30 minutes for this eam. Solve with EGUATION GRAPHER E.. y y = ^3 y = / (sqrt (abs ()-)) g: y = /(sgrt (^ 5 + 6)) k: y = ^3 h: y = /(sqrt (abs() )) y = / (sqrt (^-5+6))

16 78 M. SANDANUSOVÁ E.. g: y = (+3)/( ) Y-Calc: y = 3.5 for = 5 X -Calc: y = 0 for =.5 X-Calc: y-value could not be found E.3. y 7 y = (+3)/ (-) y 5 y = / (sqrt (^-4)) y = ((+^))/ (abs ()) 4 3 j: y = ((+^))/(abs ) p: y = /(sqrt(^ 4)) E.4. o: y = X-Intersection = 0.5 Y-Intersection: y = o - : y = (+)/ Intersection: y = for = 4 ANALYSIS OF THE ACHIEVEMENT RESULTS For the relative comparison is processing their achieved results listed together..b. eperimental group (33 students).d. control group (6 students) In the Table are listed the results from the written eam. The results of the class.b. (using the computers) seem to be better like in the class.d. In the first line are listed the maimum points, which they could obtain for the each eample, also their total number (CP) of the mark (ZN), which student

17 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 79 got. The class.b. has the general average in the quantity-achieved points but in the eample is it opposite situation. Table 4..B D Č.Ž.e.e 3.e 4.e C.P. ZN Č.Ž.e.e 3.e 4.e C.P. ZN P.P P.P

18 80 M. SANDANUSOVÁ In the Table 4. you can see the mark from math at the end of the first halfyear (MAT/) and the mark from the written eam. Table 4..B. Č. Ž. MAT/ Eam Č. Ž. MAT / Eam.D P.P P.P

19 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 8 Students from the class.b had worse average from math than class.d, but they were in opposite situation at the beginning of this eperiment. Eam.B Table 4.3 Moments N 33 Sum Weights 33 Mean Sum Observations 66 Std Deviation Variance 0.85 Skewness Kurtosis Uncorrected SS 58 Corrected SS 6 Coeff Variation Std Error Mean Table 4.4 Basic Statistical Measures Location Variability Mean Std Deviation Median Variance Mode Range Interquartile Range In the Table 4.4 you can see the average mark of eam, median and modus, and its value is two. This value of the mark you can find in the Table 4.5, where 95 % of the total number of students has mark between the interval,68038,396. Table 4.5 Basic Confidence Limits Assuming Normality Parameter Estimate 95 % Confidence Limits Mean Std Deviation Variance

20 8 M. SANDANUSOVÁ Figure 4.. Graphical distribution of student s marks in the class Eam.D Table 4.6 Moments N 6 Sum Weights 6 Mean Sum Observations 75 Std Deviation Variance Skewness Kurtosis Uncorrected SS 55 Corrected SS Coeff Variation Std Error Mean The average value of the eam mark is.88 and it is worse than in the class.b. Table 4.7 Basic Statistical Measures Location Variability Mean Std Deviation.4344 Median Variance.5465 Mode Range Interquartile Range.00000

21 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 83 Table 4.8 Basic Confidence Limits Assuming Normality Parameter Estimate 95 % Confidence Limits Mean Std Deviation Variance In the Table 4.8 you can see, that 95 % of students got the mark from the interval,3838 3, Figure 4.. Graphical distribution of student s marks in the class 5 STATISTICAL COMPARISON OF THE ACHIEVED RESULTS Table 5. Variable B N Mean Minimum Maimum MAT/ MAT/ MAT/ Diff ( ) Eam 33 4 Eam Eam Diff ( ) E

22 84 M. SANDANUSOVÁ Variable B N Mean Minimum Maimum E E Diff ( ) E E 6 3 E Diff ( ) E E E3 Diff ( ).409 E4 33 E E4 Diff ( ) 0.93 Points Points Points Diff ( ).7984 The Table 5. shows the comparison between both classes. Column B designated by number is the class.b, by number is designated the class.d and Diff ( ) means the difference between these two classes. Column N shows the number of students in each class. The fifth, the sith and the seventh column are average min and ma mark or number of points. In the Table 5. you can also see that class.b have about worse total average marks from mathematics than.d. But the class.b reached about better average marks from the written eam than class.d. Simple eamples: E. E. minimal number of achieved points in the eperimental sample is from the total of number of 5, errors, which occurred here, were caused by the wrong notation of function h in the command line, class.d was more successful by calculation of this eample, because students in class.b did not correct understand the computer report, E.3, E.4 better eperimental sample. Points better eperimental group about The main hypothesis that I verified was the effect of using of computers in the teaching process.

23 CONFRONTATION OF TRADITIONAL AND COMPUTER LEARNING 85 Advantages by using of computers: high motivation of students, individuality, high objectivity, time savings, better results from the written eam. Disadvantages, which I remarked in use of computers: lack of lessons using the computers different level of computer skills of each student, in the further education useless (without agreement of the subject commission, students can not use computers in the higher school years or graduation). REFERENCES Hecht, T.: Matematika pre. ročník gymnázií a SOŠ, Bratislava, Orbis Pictus Istropolitana Jirásek, F.: Zbierka úloh z matematiky pre SOŠ a študijné odbory SOU. časť, Bratislava, SPN 986 Kudláček, L.: Matematika pre. a. ročník štúdia na stredných priemyselných školách pre pracujúcich, Bratislava, SPN 976 Kutzler, B.: Úvod do Derive 6, Linc, Gutenberg Druck 003 Odvárko, O.: Matematika pre študijné odbory SOŠ a SOU. časť, Bratislava, SPN 993 Sandanusová M.: Počítače v procese vyučovania matematiky, rigorózna práca, FMFI UK, 003 MARTINA SANDANUSOVÁ, Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia tinasa@pobo.sk