vector calculus and solid geometry at teaching of mathematics

Acta Didactica Universitatis ComenianaeMathematics, Issue 6, 2006VECTOR CALCULUS AND SOLID GEOMETRYAT TEACHING OF MATHEMATICSAT SECONDARY SCHOOL 1LUCIA RUMANOVÁAbstract. Paper is presentation analysis of problem in the theory of didactic situation andthe evaluation of experiment with using statistical program CHIC. Problem analysis istask of stereometry for students from secondary schools. Aim of experiment was mentionon problems with application of knowledge of different part of mathematics at solution ofstereometry problem.Résumé. Cet article présente une analyse du problème dans la théorie des situationsdidactiques et la conclusion d´une expérience avec l´utilisation du logiciel CHIC. Le problèmeanalysé est une tâche de la stéréométrie donnée pour les étudiants de l´école secondaire. Le butde l´expérience est indiquer des problèmes de l´application des connaissances des thèmesdifférents mathématiques dans la solution d´un problème stéréométrique.Zusammenfassung. Der Artikel ist eine Vorführung der Analyse eines Problems mittelsder Theorie der didaktischen Situationen und des Auswertens mit Nützung des statistischenProgramms CHIC. Das analysierte Problem ist ein Beispiel der darstellenden Geometriefür Studenten der Mittelschulen bestimmt. Ziel des Experiments ist auf Problemeder Applikation der Kenntnisse von verschiedenen Thematischen Einheiten der Mathematikbeim Lösen eines Beispiels der darstellenden Geometrie zu zeigen.Riassunto. Il presente pone l’analisi di un problema teorico, di una situazione didatticae della valutazione di un esperimento con l’uso del programma statistico CHIC. L’analisidel problema è l’obiettivo di stereometria per studenti della scuola secondaria. Lo scopodell’esperimento è stato quello di soffermarsi su problemi con applicazioni di conoscenzedi diverse parti della matematica e soluzione di un problema stereometrico.Abstrakt. Článok je ukážkou analýzy problému v teórii didaktických situácií a vyhodnotenieexperimentu s využitím štatistického programu CHIC. Analyzovaným problémom jeúloha zo stereometrie určená pre študentov stredných škôl. Cieľom experimentu je poukázať1 This article was partly supported by European Social Funds JPD 3 BA-2005/1-063 and SOPLZ-2005/1-225

48L. RUMANOVÁna problémy s aplikovaním vedomostí z rôznych tematických celkov matematiky pri riešenístereometrickej úlohy.Key words: theory of didactic situations, analysis a priori, analysis a posteriori, statistical programC.H.I.C., solid geometry, synthetic geometry, vector calculus, analytic geometry1 INTRODUCTIONOne of the long – lasting problems in teaching mathematics at secondaryschool is the problem of relations among subjects, respectively the continuity ofmathematics content and the content of other subjects. However, a specific problemis the continuity or the co-ordination of individual parts of mathematics inthe process of education. In this work I want to pay attention to this part of teachingmathematics, specifically geometry − focus on the relation between solidgeometry from the vector calculus point of view and the synthetic approach toteaching of solid geometry.We know that the pupils are taught the basics of vector algebra and analyticgeometry already in the 8 th or 9 th grade at the primary school. Later on they getother information with the notion of vector and its operations (sum, odds...) in thefirst grade at secondary school, and this part is mostly used as a tool for knowledgeanalytic geometry, for example equations of the straight lines, planes, ... andlater, for the analytic geometry of conics, balls, and so on.Another problem is little attention paid to application of the vector calculusto solve the tasks of solid geometry. These tasks are taught in other parts onlyby the synthetic geometry, as well as little time spent for application tasks inother subjects (physics, geology, geography, etc.). Students usually don’t findrelative with other subjects at solution of solid problems whereupon their knowledgethey don’t know or don’t tried to apply.2 AIMS OF RESEARCHThe goal of research is therefore emphasize problems of students at secondaryschool with solving solid problems, with application their knowledge (from differentpart of mathematic) in solid geometry, as well as mention problems of studentsto find connection between parts of mathematics and other learning subjects.Then the aim of research is also experimental verify validity of the hypothesesH1–H4 and eventually suggest solutions of effectiveness education of analytical(vector) geometry with relating parts of mathematic and thereafter enhancementeducation between learning subjects at secondary school.

VECTOR CALCULUS AND SOLID GEOMETRY AT SECONDARY SCHOOL 49Abovementioned problems we summarized to following hypotheses:H1 The solid geometry is taught at secondary school separately, i. e. students areseparately taught axiomatic, separately deal with synthetic geometry, separatelyanalytic geometry, and so the sequence of various approaches is minimal orany.H2 In mathematics textbook there aren’t examples with unification character,which would promote elimination limitations (to fault) listed in H1.H3 Students of secondary school with the established educational schedule haven’tenough ability to apply their knowledge of the vector calculus in otherareas of mathematics, apart from analytic geometry, and that in this case onlyformally.H4 Students of secondary school are not able in ample measure to be aware ofthe continuity of synthetic and analytic geometry (vector calculus) in solutionof particular problem situations. Similar situation applies to University studentswho will be teachers of mathematics.3 PREPARATION OF THE EXPERIMENTIn accordance with the tenets theory of didactic situations frame: within theframe of the didactic situation S 3 (noosferic didactic situation) we made an analysisof math textbooks for secondary schools, an analysis of various mathematicalmaterials, where the goal was to choose a useful problem for students and whichwould help us to find out reply to already formulated hypothesis H1, H2, H3 andH4 in the introduction. Our goal was to seek such a problem, which the studentswere not able to solve with the learnt simple algorithms and in math textbookthere usually aren’t these types of problems.At last (after different reflections and after analysis of math textbooks forsecondary school) we chose problem from French textbook for secondary schoolMathématiques – Geometrie – Première S-E as application examples (Mathématiques– Geometrie – Première S-E).The task for the students in this experiment was to try giving an examplefrom the solid geometry with exploitation knowledge out of analytic geometryand vector calculus. (What we awaited results). Results of experiment have mentionproblems with solving solid problems, with application their knowledge(from different part of mathematic) in solid geometry, as well as mention problemsof students to find connection between parts of mathematics and other learningsubjects. Results of this experiment had to show how the students could use attainmentsfrom these units.

50L. RUMANOVÁThe task was:Given is a cube ABCDEFGH and K-point, L-point, M-point, N-point, sothat K-point is centre of upper surface EFGH, L-point is centre of theAB, M- point belong to AE, where AM = 1 AE and N-point belong3to BG BN = 1 BG . Are the points K, L, M, and N coplanary?3Based on the above-mentioned criteria the final sentence of the given taskcan be interpreted in several different ways, their experimental attesting will bethe part of our following research of this field.Students could solve problem in different theoretical frame with referenceto their level of knowledge and sciential level at secondary school. There arepossible strategy of students solution (analyse a priori of problem designation toexperiment) which relate with some parts of mathematic:Q 1 : Vector calculus − exploitation collinearity of vectors or coplanarityof vectorsQ 2 : Analytic solution − writing general equation of plane assigned threefrom four given points and then we prove incidenceof fourth point into planeQ 2´: Analytic solutionQ 2´´: Analytic solution− writing parametric equation of plane assignedthree from four given points and then we proveincidence of fourth point into plane− writing parametric equations of two lines fromgiven fourth points and then we fount out theirrelative position (if they construct of plane)Q 3 : Synthetic approach − construction section of plane of cube and then weprove incidence fourth point into planeQ 4 : Vector calculus − exploitation attributes of “barycentre“

VECTOR CALCULUS AND SOLID GEOMETRY AT SECONDARY SCHOOL 514 POSSIBLE STRATEGIES OF STUDENTS SOLUTION – ANALYSEA PRIORI OF PROBLEM DESIGNATION TO EXPERIMENTQ 3 : Synthetic approachFigure 1 1. ML ; M ∈ ABF ; L∈ABF ; ML∈ABF 2. P, Q; ML∈ABF ; BF∈ABF ; EF∈ABF P= ML∩BF∧ Q= ML∩EF3. PN ; P ∈ BCG ; N ∈ BCG ; PN ∈ BCG 4. X, Y; PN ∈ BCG ; BC∈BCG;FG∈BCG X = PN ∩BC∧ Y = PN ∩FG5. LX ; L∈ ABC ; X ∈ ABC ; LX ∈ ABC 6. QY ; Q ∈ EFG ; Y ∈ EFG ; QY ∈ EFG 7. Z; EH ∈ EFG ; QY ∈ EFG Z = EH ∩QY 8. MZ ; M ∈ ADH ; Z ∈ ADH ; MZ∈ADH9. MLXYZa) we construct the section plane LMN of cube ABCDEFGH and thanb) we attest: ? K∈ LMN ? (That they are single calculations – for examples:solution with exploitation follows Figure 2 or exploitation resemblance betweentriangle).

52L. RUMANOVÁFigure 2Q 2 : Analytic solutionFigure 3This is parametric equation of plain MLN 1 : x = t31y = t+s21 1z = − s; t, s ∈ R3 3( M , ML , MN )

VECTOR CALCULUS AND SOLID GEOMETRY AT SECONDARY SCHOOL 53⎡1 1 ⎤ And then the question is: K⎢, ,1 ∈ MLN2 2 ⎥ ?⎣ ⎦1 1We solve the system of equations:2 = 3 t1 1= t+s2 21 11 = − s . 3 3For parametersK∈ MLN .3t = and s =− 2 , the equations have a solution, there out resulting:2Q 1 : Vector calculus (exploitation collinearity of vectors or coplanarity)Figure 4 The problem is: Are the vectors LS and LK collinear? (point S is a centreof line segment MN ). If the vectors are collinear, so the K-point, L-point, M-point, N-point are coplanar.For the vector LS resulting these terms: LS = LB + BN + NS ; 2.LS = LB + LA + BN + AM + NS + MS .By means of substitution relations and simple reforms resulting term: 1 LS = ( BG + AE).6

54L. RUMANOVÁIn like the manner for the vector LK ⇒ 1 LK = ( BG + AE).2 1 And from the relationship of vectors LS , LK following: LS = LK and so,3 the vectors LS and LK are collinear.Q 4 : Vector calculus – exploitation “barycentre”Figure 5K-point is a barycentre of E(1), G(1), L-point is a barycentre of A(2), B(2),M-point is a barycentre of A(2), E(1) and N-point is a barycentre of B(2), G(1).And then, the question is: What is barycentre G of this points set {A(2),B(2), G(1), E(1)} ?From the facilities of barycentre ⇒ G is barycentre of {M(3), N(3)} and{K(2), L(4)}. 3 2 So, the points K, L, M, N are coplanary, because MG = MN and LG = LK .66Analysis a-priori problem had been formulated and teacher’s activityin a-didactic situationThe base of didactic research is the test students´ reactions in the given didacticsituation. So as we could make the test to realize at the first we had to dothe full analysis of didactic situation itself.My working come out from the theory of Guy Brousseau, Yves Chevallard,A. Sierpinska, the foundation of it is the theory of didactic situation. The French

VECTOR CALCULUS AND SOLID GEOMETRY AT SECONDARY SCHOOL 55school of didactic under leading of G. Brousseau processed the theory of didacticsituation, which is useful to make analysis of students´ works.This theory suffers us to analysis the listed problem from the several aspects.And just the analysis of problem in the individual levels of didactic situationforms the basis didactic researches.Part of the experiment is the integration individual phases of the problemsolution to system of levels in the analyse didactic situation.This is the tablet composition of milieu and with it associated types of didacticsituations (Margolinas 1994).M 3Constructional milieuM 2Project milieuM 1Didactic milieuE 1Reflective studentP 3Teacher −didacticP 2 Teacher −constructorP 1 Teacher −designerS 3Noosferic situationS 2Constructional situationS 1Project situationM 0Milieu of learningM -1Modeling milieuE 0StudentE -1 Cognizantintellect studentP 0TeacherP -1 Teacher −scrutatorS 0Didactic situationS -1Situation of learningM -2Objective milieuE -2Active studentS -2Modeling situationM -3Material milieuE -3Objective studentS -3Objective situationWe formulated analysis a-priori problem before students´ solutions (our experiment)and analysis includes descending analysis (analysis of teacher’s work),ascending analysis (analysis of student´s work) and we also formulated analysisa-priori problem for statistical program CHIC.5 ANALYSE OF TEACHER’S WORK (DESCENDING ANALYSIS)S 3 – noosferic situation – on this stage we analysis the math textbook for secondaryschool (2nd or 3rd class), analysis various mathematical materials (educationalschedule…), specifically study of solid geometry, vector algebra and analyticgeometry. We want to choose problem, which the students were not able tosolve with the learnt simple algorithms and students can use knowledge fromdifferent parts of mathematics and other learning subjects. Finish of noosfericsituation will be the milieu for the next situation.

56L. RUMANOVÁS 2 – constructional situation – teacher will try to find examples, that weredefined in noosferic situation S 3 and on the other side in situation S 1 , in whichthey will be able to realize. Goal was to choose a useful problem for studentswhich would help him to find reply to already formulate hypothesis H1, H2, H3and H4. They are examples which students can abet in examples solution Q 1 ,Q 2 , Q 2´, Q 2´´, Q 3 a Q 4 .S 1 – project situation – in situation S 1 , teacher writes a text of the exampleand he “projects” his solution. Student is one on teacher’s consciousness. Thisis a situation, which involves student’s activity, too. The student can solve problemin a way that he constructs the section plane of cube, and then he finds outif other point is point of plane; or the student solves a problem that he writesparametric equation of plain and he finds out if the fourth point is the point of theplane; or he applies exploitation collinearity of vectors or coplanarity or exploitation“barycentre” (but this solution assuming nothing).S 0 – didactic situation – in this situation we analysis and do institutionalisationof the new knowledge and we formulate the problem. Teacher takes care ofdesigned goals and he follows student’s solution, too. It is a situation where theanalysis of teacher’s work and analysis of student’s work meet, and the didacticsituation will be the result of the teaching process.6 ANALYSE OF STUDENT´S WORK (ASCENDING ANALYSIS)We introduce analysis of problem Q 3 (synthetic approach).S -3 – objective situation – the student gets acquainted with the problem andwith the material milieu. Material milieu are a cube ABCDEFGH and K, L, M,N – points, cognitive component of milieu are knowledge about incidence of points,lines, planes, geometrical construction new section of body, basis of vertical projection,the notions as skew lines, intersecting lines, etc. Social component of materialmilieu is mininal because other help isn´t allowing in our experiment (slef –activity of students).S -2 – modelling situation – the student solves the problem in milieu S -3 , i. e.he constructs the section plane LMN of cube ABCDEFGH (for example). Thestudnet makes use of the knowledge from solid geometry, he works withmaterial milieu, he applies visions and plot, he makes use of known relationsand practices. Work of student have just character of activity and there teacherdon´t interfere with students´ solution. It situation without teacher´s help andfeedback and so student must be able to inspect his work and solution.S -1 – situation of learning – in this situation the studnet takes the teacher´srole. He solves the formulated problem by means of question: „Are the given pointscoplanar?“, or he verify to validity K∈ LMN . The student abtains information

VECTOR CALCULUS AND SOLID GEOMETRY AT SECONDARY SCHOOL 57from reading text of example and he formulats his own results. Student loosecharacter of activity, he is more thinking and formulating results of his work. Theteacher is a scrutator and he tries to help, if the student has some absurdity,alternatively if he fails to solve it. In such a case he falls into position S 0 – didacticsituation.S 0 – didactic situation – in this situation the work of students is affected bythe teacher and takes his advice in form institutionalisation, which can help tostudent by solving given example, but teacher takes to into consideration student´ssolution, too. The teacher can help for example with individual elements ofsection cube or with the correct registration of solving.7 REALIZATION OF THE EXPERIMENT. QUANTITATIVEANALYSIS OF RESULTS RECEIVES FROM EXPERIMENTWITH EXPLOITATION OF STATISTICAL PROGRAM CHICExperiment consisted of one solid problem and was realized in two phases.First participated experiment students from different secondary grammar schoolsin Nitra (4 classes, in March 2002)). Second phase experiment was realized in oneof the secondary grammar schools in Bratislava and also problem-solved studentsat Faculty of math, physic and informatics in Bratislava (in February 2004). Allstudents had repeat parts of mathematics needs towards solution of our problem.In the aggregate experiment attended 108 students.On quantitative analysis of results receives from experiment we used statisticalprogram CHIC. In this statistical program were created graphs Similarity tree(Figure 6), Implicative tree (Figure 7) and Implicative graph (Figure 8).Figure 6

58L. RUMANOVÁFigure 7Figure 861 62 63 64CHIC indicated the fact that more frequently students´ solution was syntheticapproach and on the other side at least frequently students´ solution was vectorsolution, which didn’t appear on important level (for results of experiment) in thegraphs from CHIC.Analytic solution used 15 students and 10 of them solved problem with usingsynthetic approach, too. This fact we can read from Implicative graph althoughconversion between single variables is only 62 %. From graph the Similarity treeto see altogether equality between thereby that students chose analytic solution ofproblem and co-ordinate system {alternatively that cock doesn’t fight solve}.Vector solution appeared only in the graph Similarity tree, but not in importantlevel. But there validated similarity of variables namely student exploit vectorcalculus and student use notion vector in solution. Only 2 students solved problem

VECTOR CALCULUS AND SOLID GEOMETRY AT SECONDARY SCHOOL 59with using vector approach and one of them used synthetic and analytic approach.Hence vector solution didn’t exist in graph Implicative tree and Implicative graph.No one of 108 student didn’t try solve problem with using facilities of barycentreand so variables of barycentre didn’t appeared in no case of graphsImplicative tree, Implicative graph or Similarity tree. Students usually don’tfind relative with other subjects at solution of solid problems whereupon theirknowledge they don’t know or don’t tried to apply. Students don’t find relativewith physics and mathematics, although notion barycentre students know fromlessons of physics from 1 st class.8 CONCLUSIONProblem at teaching solid geometry is sequence single parts of mathematicswithin solid geometry. With this problem relate too solutions of solid geometryproblems. Concrete, students have problem solve problem, students could solveproblem in different theoretical frame with reference to their level of knowledgeand sciential level at secondary school, but which the students were not able tosolve problem only with the learnt simple algorithms.The goal of research is mention problems with application students´ knowledgeat solving problem from different part of mathematic in solid geometry andalso experimental verify validity or invalidity of the hypotheses with using pedagogicalexperiment.One of theoretical basis of this research was the educational schedule. Educationalschedule goes through some development on the basis of different influences.Our work gives survey of this development and also changes fromyear 1868 to recent educational schedule issued in 1997. We presented the mostimportant changes, concentrated on educational schedule of teaching geometry(solid geometry) at secondary schools.By analysis of pedagogical materials, included educational schedule ofmathematics for grammar schools we found out that various parts of geometry insecond and third year of study are thought isolated. (There are synthetic geometry,vector calculus and analytical geometry.) This means that sequence of variousways of teaching geometry is minimal. That verified our hypothesis H1.From analysis of educational schedule mostly from analysis of mathematics’textbooks for secondary schools came validity of stated hypothesis H2, thatmeans that in mathematics’ textbooks are very little tasks with unification character.Tasks of this type could help remove problems stated in hypothesis H1, inconcrete to find out relations between various themes of solid geometry or tohelp by applications of knowledge at solving of problem tasks in all thematicparts of solid geometry.

60L. RUMANOVÁFrom educational schedule came out that there is not enough time for solvingof such, which can help students to be aware of relations of thematic partsor between different teaching subjects. Task given in experiment could havebeen solved with usage of mass points’ centre of gravity, but no one of studentstried to solve tasks with usage of knowledge from physics, what we predicted.Continuance of this work should be from that reason also to use mentionedapplication’s tasks during teaching process, to acquaint students with these tasksand to solve them in mathematics‘ lessons. On approval we put some tasks fromsolid geometry with their solutions or with instructions how to solve them intotasks’ collection. This collection can be use at teaching of various parts of solidgeometry. Tasks can be solved with using of planimetry in space or with findingof some relations between objects in space. We would like also to show, if solvingof such tasks would improve teaching of solid geometry, and if students duringsolving of tasks would find relations between various parts of solid geometry.In experiment took part 108 students. We predicted, that students woulduse mostly synthetic and analytic solution of tasks, what showed to be true.Problem didn’t solve 10 students, synthetic solution used 92 students, analyticsolution used 15 students and vector solutions used 2 students. In experimentwere some students, which wised up to something, that problem having moresolutions. So, 10 students solved problem with 2 methods (analytic and synthetic),but only one of them solved problem with using 3 methods (analytic,vector and synthetic). No one of 108 students didn’t try to solve problem withusing facilities of barycentre.From these facts follow that students didn’t have ability where to appliedknowledge from vector calculus just at analytic geometry, for example equationsof the straight lines, planes, ... and later, for the analytic geometry of conics, balls,and so on. And so, in solutions some problem of solid geometry students didn’tapply their knowledge in sufficient measure. Therefore confirmed validity hypothesisH3 and H4.Experiment was realized also in university (future teachers of mathematics)and confirmed us that there is similar problem with application students´ knowledgefrom different parts of mathematics in solid geometry and also from othersubjects. Whereupon, teacher prefer concrete parts in teaching (in lessons ofmathematics) and so teacher often disuse examples with unification character.Conclusion of the work includes collections of the tasks and we want (ofcourse) expand any interesting application problems (tasks) from solid geometry,which we can improve on lessons of mathematics. Consequently, correctionof collections of these tasks for practical use as part of texts in books of mathematicsfor secondary schools. In continues of experiment we want to try makingeducational texts some parts of solid geometry. Educational texts will be especiallyorientated to coordination between individual parts of solid geometryand rather mention relations among subjects, i. e. mention possibility to applied

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