ANALOGICAL REASONING IN TEACHING MATHEMEMTICS IN HIGH SCHOOLS THE CASE STUDY OF COORDINATE METHOD IN SPACE

MINISTRY OF EDUCATION AND TRAINING HO CHI MINH CITY UNIVERSITY OF EDUCATION –––––––––– BUI PHUONG UYEN ANALOGICAL REASONING IN TEACHING MATHEMEMTICS IN HIGH SCHOOLS: THE CASE STUDY OF COORDINATE METHOD IN SPACE Specialization: Theory and Methods of Teaching and Learning Mathematics Scientific Code: 62 14 01 11 SUMMARY OF DOCTORAL THESIS ON EDUCATIONAL SCIENCE HO CHI MINH CITY– 2016 THE THESIS COMPLETED IN: HO CHI MINH CITY UNIVERSITY OF EDUCATION Supervisor: Assoc Prof Dr Nguyen Phu Loc Dr Le Thai Bao Thien Trung Reviewer 1: Assoc Prof Dr Le Thi Hoai Chau Reviewer 2: Assoc Prof Dr Le Van Tien Reviewer 3: Dr Tran Luong Cong Khanh The Thesis Evaluation University Committee: HO CHI MINH CITY UNIVERSITY OF EDUCATION Thesis can be found at: - General Science Library of Ho Chi Minh City - Library of Ho Chi Minh City University of Education INTRODUCTION The reasons for selecting the topic 1.1 Using analogical reasoning in teaching mathematics was studied by many domestic and foreign researchers When solving a new problem, students often compare it with previous similar problems and then they find out how to solve the problem Using analogical reasoning in the teaching process can support students to use their previous knowledge to discover new knowledge; this process enhances their independent thinking, their critical thinking and their creative abilities Analogical reasoning plays an important role in science, in particular, in mathematical education Analogical reasoning is used to build the meaning of knowledge, formulate hypotheses in the teaching method of discovery, predict and prevent students’ errors, and solve mathematical problems Analogical reasoning was studied by many domestic and foreign authors such as G Polya, D Gentner, K Holyoak, H H Zeitoun, S M Glynn, Harrison, Coll, H Chung, N B Kim, D Tam, N P Loc, L T H Chau, L V Tien, D H Hai… 1.2 Analogical relationships between Coordinate method in space and in plane The coordinate method is an important content in the mathematical curriculum in high schools In the curriculum and textbooks, there are many concepts in the chapter “Coordinate method in space” similar to the ones in the chapter “Coordinate methods in plane” (mentioned in class 10); moreover, there are many similar problems and their similar solutions Therefore, we asked ourselves the following interesting questions: - Have the authors of current Geometry textbooks used analogical reasoning to present the contents in the chapter “Coordinate method in space”? - When using analogical reasoning in the textbooks, the teachers in high schools and the pre-service teachers use analogical reasoning as a strategy to enhance students' positiveness? - What kinds of errors have students committed when using analogical reasoning in the chapter “Coordinate method in space”? - What are effective methods used when teaching “Coordinate method in space” with analogical reasoning? From the above reasons, we chose the topic of the thesis as follows: “Analogical reasoning in teaching mathematics in high schools: the case study of Coordinate method in space” 2 Theoretical frameworks and research tasks Our study was carried out in the range of theories about analogy, analogical reasoning and teaching with analogy A number of didactic theoretical tools were used in our thesis: anthropological theory in mathematics didactic, teaching contracts, and the situation theory The purpose of research is to find out about analogy, analogical reasoning, the roles of analogical reasoning in teaching Coordinate method in space From the original questions, we restated into the following questions: Research question 1: What is the analogical relationship between Coordinate method in plane and in space like? What the type of task are mentioned in “Coordinate method in space” and “Coordinate method in plane”? What are conclusions of the situations using analogical reasoning are drawn in the current Geometry textbooks? Research question 2: How is analogical reasoning used on “Coordinate method in space” on textbooks? How is the practice of the teachers in high schools and preservice teachers influenced? Research question 3: What kinds of errors have students committed when using analogical reasoning to solve problems in Coordinate method in space? Research question 4: What are the solutions to promote the positive effects of analogical reasoning in teaching Coordinate method in space? What we need to verify the effectiveness of these solutions? Limitations of the study We chose to study analogical reasoning and applied it on some specific contents in the chapter “Coordinate method in space” In the thesis, we focused on using analogical reasoning both in plane and in space Research hypotheses H1: By using analogical reasoning, teachers can help their students to discover the mathematical knowledge in the chapter “Coordinate method in space” H2: By using analogical reasoning, teachers can help students to find out the solutions for mathematics problems in the chapter “Coordinate method in space” H3: When learning the contents in the coordinate method in space, students will commit errors when solving problems by using analogical reasoning The main contributions of the thesis 5.1 In terms of theory - Summing up many educational views about analogy, analogical reasoning, the roles of analogical reasoning in teaching, the classification of analogical reasoning and the teaching models with analogical reasoning such as GMAT model, TWA model, FAR model - Developing standards of evaluation of using analogical reasoning - Developing six solutions to promote positive effects of analogical reasoning - Developing six teaching processes of using analogical reasoning: teaching how to discover concepts, teaching how to discover theorems, teaching how to solve mathematics problems; predicting errors of students by analogy sources before teaching, analyzing and detecting errors and correcting errors 5.2 In terms of practice - Analyzing analogy and analogical reasoning used on the Geometry textbooks in the chapter Coordinate method in space - Clarifing the impact of analogical reasoning in textbooks for teaching with analogy of mathematics teachers and pre-service teachers in the chapter Coordinate method in space - Listing some errors of students when using analogical reasoning to solve mathematics problems in the chapter “Coordinate method in space” - Developing the solutions and teaching processes of using analogical reasoning in specific contents in the chapter “Coordinate method in space”, and mathematics in general The points to defend - The views of analogy, analogical reasoning and its roles in teaching - Analogical reasoning used on current textbooks and teaching situations with analogy designed by teachers and pre-service teachers in the chapter “Coordinate method in space” - Some results about students’ errors when using analogical reasoning in chapter “Coordinate method in space” - The strategies for using analogical reasoning in teaching Coordinate method in space and empirically verifiable results The structure of the thesis Besides introduction and conclusion, the main contents of the thesis consisted of chapters: Chapter Theoretical frameworks; Chapter The methods and research designs; Chapter Research on analogical reasoning in the chapter “Coordinate method in space”; Chapter Research on the teaching practice of using analogical reasoning; Chapter Research on the real errors of students when using analogical reasoning; Chapter Solutions to promote the positive effects of analogical reasoning in teaching mathematics and pedagogical experiment CHAPTER THEORETICAL FRAMEWORKS This chapter analyzed, synthesized, and systemized the views of analogy, analogical reasoning, the roles of analogical reasoning in teaching mathematics and teaching with analogy models 1.1 The concept of analogy and analogical reasoning 1.1.1 What is analogy? The thesis mentioned analogy concepts according to G Polya, H Zeitoun, D Gentner, in which we paid special attention to and considered G Polya’s concept of analogy as a theoretical basis According to G Polya (1997), analogy is a certain similarity type These objects which fit together in relationships is defined as analogy objects The two systems are analogical if they clearly fit together in the relationships and identify between two respective parts For example, a triangle is analogical to a tetrahedron 1.1.2 What is analogical reasoning? The thesis presented the analogical reasoning views of the authors Hoang Chung, Hativah, Gentner, and Holyoak In logic, Hoang Chung (1994) defined analogical reasoning as a reasoning based on some similar properties of two objects, then draw conclusions about the other similar properties of two objects According to Hativah (2000), analogical reasoning is defined as “A comparison between different things, but they are strikingly alike one or more pertinent aspects” The thing which functions as the basis for analogy is called the source In the other hand, the thing explained or learnt through analogical reasoning is called the target The expected conclusions drawn by analogical reasoning are just hypotheses, so their correctness needs to be verified clearly In the thesis, we considered analogical reasoning as a kind of reasoning from the same characteristics of source and target, then draw the other same characteristics 1.1.3 Analogical reasoning under the perspective of philosophy and psychology 1.1.4 Thinking operations related to analogical reasoning Analogical reasoning has a close relationship with other thinking operations such as analysis, comparison, and generalization 1.1.5 Classifications of analogical reasoning a According to Nirah Hativah (2000), one might consider three types: analogical reasoning with sources and targets in the same domain, analogical reasoning with sources and targets in different domains, analogical reasoning with sources based on the experience of students b According to Helmar Gust and el al (2008), there are three types: analogical reasoning form of (A: B) :: (C: X), speculated analogical reasoning and analogical reasoning to solve problems c According to Nguyen Phu Loc (2010), analogical reasoning is divided into analogical reasoning based on properties and analogical reasoning based on relationships d According to Orgill and Yener, analogical reasoning presented in textbooks can be classified as below (see Table 1.1) The classification is used in analyzing textbooks in chapter Table 1.1 The classification of analogical reasoning used in textbooks The analogical relationship between source and target The presentational format The level of abstraction of the source and target concepts The position of the analog relative to the target The level enrichment of The limitations of the analogy Structure: source and target concepts share only similarities in external features or object attributes Function: source and target concepts share similar relational structures, or the behavior of the source and target is similar Structure-Function: source and target concepts in the analogy share both structural and functional attributes Verbal (in words): The analogy is presented in the text in a verbal format Pictorial-Verbal: The analogy is presented in a verbal format along with a picture of the source Concrete-Concrete: Both source and target concepts that students might see, hear, or touch with hands are clear Abstract-Abstract: Both source and target concepts are abstract Concrete-Abstract: The source concept is concrete, but the target concept is abstract Advance organizer: The source concept is presented before the target concept in the text Embedded activator: The analog concept is presented with the target concept in the text Post synthesizer: The analog concept is presented after the target concept in the text Simple: A simple analogy is a simple sentence that the source is similar to the target Enriched: a statement with explanations, set up a correspondence between the source and the target Extended: analogies with clear relationships or authors use it multiple times in the same book Misunderstandings are showed Misunderstandings are not showed 1.2 The roles of analogical reasoning in teaching mathematics Analogical reasoning is used to construct the meaning of knowledge, build hypotheses, solve maththematics problems and detect and correct errors of students 1.3 The teaching models of using analogical reasoning 1.3.1 GMAT model (The General Model of Analogy Teaching) GMAT model suggested by H Zeitoun (1984) consists of nine steps The author emphasized the importance of a plan before using analogical reasoning to help students to learn new knowledge and assess the effects of analogy 1.3.2 FAR model (Focus-Action-Reflection) Before and after teaching with analogy, teachers need to analyze the analogy FAR model in order to teach more efficiently Focus: Concept Students Analog Action: Similarities Differences Reflection: Conclusion Reflection Is the concept taught difficult, unfamiliar or abstract? What knowledge have students already known about the concept? What things are familiar with students? Discuss the similar characteristics of source and concept Discuss the different characteristics of source and concept Is the source clear, helpful or confusing? Consider the focus on the basis of conclusions 1.3.3 TWA model (Teaching-With-Analogies) The teaching process of using analogical reasoning in Glynn’s TWA model (1989) includes: Introduce target concept; Review source concept; Identify relevant features of source and target; Map similarities between source and target; Clarify the incorrect conclusions; Draw conclusions about the new knowledge 1.4 Some theories of mathematical Didactic We summarily presented some of theoretical tools in mathematical Didactic such as anthropological theory, situation theory and teaching contract 1.5 Conclusion of chapter Chapter covered the theoretical framework of analogical reasoning considered as a basis in the following chapters CHAPTER THE METHODS AND RESEARCH DESIGNS This chapter mentioned the research methods to answer the questions stated in the preface 2.1 Research on analogical reasoning in the chapter “Coordinate method in space” (reply to the research question 1) We analyzed the contents in the textbooks in order to: - Clarify the analogy concept, properties in coordinate method in plane and in space Analyze analogical reasoning used on current geometry textbooks according to the classification in Table 1.1 and analogical reasoning used in the chapter “Coordinate method in space” - Find out 30 mathematical organizations on the chapters “Coordinate method in plane” and “Coordinate method in space” according to the views of mathematical didactic: T is the task, τ is the technique, θ is the technology Specifically analyze several mathematics organizations as a basis for using analogical reasoning to solve mathematical problems and find out the errors of students 2.2 Research on the teaching practice due to analogical reasoning (reply to the research question 2) 2.2.1 The survey of teachers The purpose is to answer questions: Do teachers in high schools use analogical reasoning to help students to discover new knowledge? In the cases of using analogy in teaching, how they use analogical reasoning in their teaching? We surveyed 20 lesson periods in the chapter “Coordinate method in space” taught by 18 mathematical teachers in high schools in the Mekong Delta region In order to evaluate the level of using analogical reasoning, we used the rubric for evaluating in Table 2.1 Table 2.1 The rubric for evaluating using analogical reasoning in teaching Level Levels of using analogical reasoning in teaching Do not use analogical reasoning Only talk about the source Recall characteristics of the source, but not set up any correspondence between the source and the target Know how to set up correspondences between the source and the target Draw a conclusion about the analogy, mention the differences and similarities, have valuable conclusions due to analogical reasoning 2.2.2 The survey of pre-service teachers The purpose of the study is to answer the following questions: Do pre-service teachers at Can Tho University choose analogical reasoning in designing lesson plans related to the topics in the chapter Coordinate method in space? What difficulties can students meet when they use TWA model to design lesson plans in the chapter Coordinate method in space? What solutions help students to overcome these difficulties? * Survey 1: To answer the 1st question, we surveyed 52 pre-service teachers having a year before graduation Step 1: Pre-service teachers design lesson plans “Coordinate system in space” in a week Step 2: Pre-service teachers work in groups (from to students) to discuss about how to teach “Coordinate axis system in space” in 60 minutes *Survey 2: To answer the 2nd and 3rd questions, we surveyed 31 pre-service teachers having two years before graduation Step 1: We introduce analogical reasoning to pre-service teachers with TWA model and an illustrated example Then, they work in groups (from to students) in 60 minutes and use TWA model to prepare lesson plans for teaching concepts, properties and mathematical problems in Coordinate method in space Step 2: They discuss in groups to answer the following questions: In your opinion, what are strong and weak points of the TWA model for teaching mathematics? Please indicate the difficulties you have encountered in each step of applying the model to teach mathematics? According to you, what is the most difficult step? What factors enabled you to apply TWA model in an effective way? The rubric for evaluating analogical reasoning in teaching in Table 2.1 2.3 Research on some errors of students when using analogical reasoning (reply to the research question 3) From the right and wrong properties of source and target, there are types of errors involved with the target when students use analogical reasoning to solve problems: Error type 1: Students commit errors when solving source problems, so they also commit similar errors when solving target problems Error type 2: Students use some successful strategies to solve source problems, but when these strategies are applied in target problems, they make errors 11 In step 1, the average levels of using analogical reasoning in contents when they work individually were less than a In step 2, we compared the average level of using analogical reasoning when they worked individually and in groups The results showed that both working individually or in groups, students had to prioritize analogical reasoning in teaching the topics in the coordinate method in space b) The survey Step 1: Many pre-service teachers applied TWA model to teach new knowledge well However, some of them did not master this model, so they did not design suitable teaching activities Step 2: They pointed out both the advantages and disadvantages, then developed solutions to use TWA model effectively 4.3 Conclusion of chapter The surveys of teachers and pre-service teachers showed that using analogical reasoning in teaching Coordinate method in space had not been interested in This was due to the impact of the presentation of analogical reasoning in the current textbooks CHAPTER RESEARCH ON REAL ERRORS OF STUDENTS WHEN USING ANALOGICAL REASONING This chapter showed the results to answer the research question and verify the hypothesis H3 5.1 Research on the errors of students when they perform the task “Find the equation of a plane passing distinct points” 5.1.1 Priori analyses 5.1.1.1 Didactic variables (the selected values are marked with*) V1-1: The collinear of points: aligned*, misaligned V1-2: The equation types of a plane: parametric equation and general equation* V1-3: Problem requests: finding equation*, proof, multiple-choice questions, V1-4: Technical tools: pocket calculator*, a computer with mathematical software V1-5: The working style of students: work individually* or in a group 5.1.1.2 The analogical task (source) We considered the analogical task in plane: “Find the general equation of a straight line passing distinct points A and B” Based on the strategies of the source, we predicted an error (type 2) by using analogical reasoning that students would commit when finding the general equation of a plane passing points (in the case of collinear points): Error 1: Students replace the coordinates of normal vector 12 uuur uuur r r n =  AB; AC  = into the plane equation, and there exists a rule of teaching contract R1: Students have no responsibility to check the collinear of three points before finding the general equation of a plane 5.1.1.3 Carrying out experiments Students had to solve the problem: In space Oxyz, we have A(4;1;2) B(5;-2;1), C(3;4;3), D(1;-2;5) Find the general equation of a plane: a The plane (ABD) b The plane (ABC) In question a, A, B, D are non-collinear; but in question b, A, B, C are collinear Then, we interviewed students who made errors to find out how they used analogical reasoning 5.1.2 Posterior analyses In question b, by using analogical reasoning with the strategies of the source and the solutions of question a, nearly 70% of students had error (replacing the r r coordinates of n = into the general equation of a plane) Moreover, many students did not check the alignment of A, B, C (existed rule R1) because when they solved problems in classroom and textbooks, they did not need to check anything 5.2 Research on the errors of students when they perform the task “Find the equation of a plane passing point and two parallel straight lines d and d’ ” 5.2.1 Priori analyses 5.2.1.1 Didactic variables (the selected values are marked with*) V2-1: The relative positions of d, d ': parallel*, skew The variables V2-2 (The equation type of a plane), V2-3 (Problem requests), V2-4 (Technical tools) and V2-5 (The working style of students) are similar to section 5.1.1.1 5.2.1.2 The analogical task (source) We considered the analogical task in plane: “Find the general equation of a straight line passing a point and being parallel to a line d” Based on the strategies of the source, we predicted an error (type 2) by using analogical reasoning that students would exhibit when finding the general equation of a plane passing a point and being parallel to lines d and d’ (in case d//d’): Error 2: Students replace the coordinates of normal r r r r vector n = [ ud ; ud ' ] = into the plane equation, and there exists a rule of teaching contract R2: Students have no responsibility to check the relative position of straight lines before finding the general equation of a plane 5.2.1.3 Carrying out the experiment Students had to solve the problem: 13 Find the general equation of plane ( α ) passing A(3;2;-4),  x = 8+t x − y − z −1  = = a and being parallel to d :  y = + 2t , d ' : −7  z = 8−t  r r 1  b and being parallel to vector u = ( −3; −4; ) , v =  ; ; −1 ÷ 2  r r In question a, d and d’ are skew, but in question b, u , v have the same direction Then, we interviewed students who made errors to find out how they used analogical reasoning 5.2.2 Posterior analyses The results showed that students used analogical reasoning from the strategies of the source and the solutions of question a (d, d’ crosswise) to infer the solutions of question b (in case d //d’): 63.75% of students had errors (replacing the coordinates r r r r of n = [ u , v ] = into the equation plane) Moreover, many students did not check the relative position of d, d’, so the existence of rule R2 was confirmed 5.3 Research on the errors of students when they perform the task “Find the equation of a straight line ∆ in space passing a point and being perpendicular to straight line d” 5.3.1 Priori analyses 5.3.1.1 Didactic variables (the selected values are marked with*) V3-1: How the line d is given: known parametric equation*, general equation, passing points*, V3-2: The types of equation ∆ : general, parametric*, Cartesian equation The variables V3-3 (Problem requests), V3-4 (Technical tools) and V3-5 (The working style of students) are similar to section 5.1.1.1 5.3.1.2 The analogical task (source) We considered the analogical task in plane: “Find the parametric equation of a straight line ∆ passing a point and being perpendicular to a line d” Based on the strategies of the source, we predicted errors (type 2) by using analogical reasoning that students would commit when finding the parametric equation of ∆ in space: r Error 3: Students infer vector ud = (a; b; c) being perpendicular to ∆ , so r r u∆ = (b; −a; c) or u∆ = (−b; a; c) is a direction vector of ∆ ; Error 4: Students deduce that the direction vector of d is also a direction vector of ∆ ; Error 5: Students find 14 the parametric equation of ∆ passing point A, intersecting and being perpendicular to line d 5.3.1.3 Carrying out the experiment Students had to solve the problem: Find the parametric equation of a straight line ∆ passing point M(1;3;-2) and a being perpendicular to line d : x −1 y + z − = = −1 b being perpendicular to a line which passes points A(3;1;-2), B(-1;-2;1) We interviewed students who made errors to find out how they used analogical reasoning 5.3.3 Posterior analyses By using analogical reasoning with the source solutions in plane, many students had errors: 56.65% of students inferred that a direction vector of ∆ equal to the direction vector of d or AB About 30% of students found a direction vector of ∆ by a r similar way in plane: “reverse abscissa and ordinate of vector ud = (a; b; c) , adding a minus” Approximately 13% of students had additional condition line ∆ intersects line d Thereby, it was clear that there existed the errors 3, 4, 5.4 Research on the errors of students when performing the task “Calculate angle between a straight line and a plane” 5.4.1 Priori analyses 5.4.1.1 Didactic variables (the selected values are marked with*) V4-1: How the line d and the plane ( α ) is given V4-2: Problem requests: calculate angles*; multiple-choice questions;… The variables V4-3 (Technical tools) and V4-4 (The working style of students) are similar to section 5.1.1.1 5.4.1.2 The analogical task (source) We considered the analogical task in plane: Calculate angle between straight lines d and d’ Based on the strategies of the source, we predicted errors (type 2) by using analogical reasoning that students would commit when calculating angle between a straight line and a plane: Error 6: Students infer the formula of calculating angle (r r ) between a straight line and a plane is cos ( d , ( α ) ) = cos ud ; n( α ) ; Error 7: Students r r r find a normal vector nd = (b; −a; c) of d, then calculate cos(d,(α )) = cos( nd ; n( α ) ) 5.4.1.3 Carrying out the experiment 15 Students had to solve the problem: Calculate angle between a straight line d and a plane ( α ) when: a line d parallel to z-axis and the equation of ( α ) : x = x− y+1 z −1 = = and the equation ( α ) : x − y − z − = −2 −3 We interviewed students who made errors to find how they used analogical reasoning 5.4.2 Posterior analyses From the papers of students, more than 50% of students did not answer this problem correctly because they had not been introduced by textbooks or teachers They tried to use analogical reasoning from solutions of the problem of calculating angle b the equation d: (r r ) between straight lines: 25% of students used the formula cos ( d , ( α ) ) = cos ud ; n( α ) r and 9.06% of students found the normal vector nd = (b; −a; c) of d, then calculated r r cos(d,(α )) = cos( nd ; n( α ) ) The existence of error 6, was clear 5.5 The task “identify the equation of a circle and a sphere” 5.5.1 Piori analyses 5.5.1.1 Didactic variables (the selected values are marked with*) V5-1: Forms of the quadratic equations f ( x, y ) = g ( x, y ) for a circle and h(x,y,z)=l(x,y,z) for a sphere V5-2: Problem requests: proof, multiple-choice questions*, short answers*, The variables V5-3 (Technical tools) and V5-4 (The working style of students) are similar to section 5.1.1.1 5.5.1.2 Analogical forms in identifying the equation of a circle and a sphere Analyzed specific forms of tasks “identify the equation of a circle and a sphere” in analogical relationship and predicted errors type and type which students would exhibit when they used analogical reasoning to perform these tasks 5.5.1.3 Carrying out the experiment Students had to solve the problem: In plane Oxy, are the following equations the equation of a circle? If true, find the center and radius In space Oxyz, are the following equations the equation of a sphere? If true, find the center and radius 1a ( x − ) + ( y + 3) = 16 1b ( x − ) + ( y + 3) = 16 2a ( + x ) + ( − y ) = 25 2b ( + x ) + ( − y ) + (2 + z ) = 25 3a ( x − 1) + ( y + 3) = 36 3b ( x − 1) + ( y + 3) + ( 3z − ) = 36 4a ( x − 1) + ( y + ) = 36 4b ( x − 1) + ( y + 3) + ( z − ) = 36 2 2 2 2 2 2 2 2 2 16 5a x + y − x + y + = 2 5b x + y + z − x + y + z + = 2 6a x + y − x − y + = 2 6b x + y + z − x − y + z + = 2 7a x + y − x − y − = 2 7b x + y + z − x − y + z − = 8a ( x + y ) − x − = y + xy 2 8b ( x + y ) − x − − xz = y + xy − ( x − z ) 2 We interviewed students who made errors to find how they used analogical reasoning 5.5.2 Posterior analyses The results showed students committed errors (type 1, 2) due to using analogical reasoning to identify the sphere equation based on solutions of identifying the circle equation 5.6 Conclusion of chapter The results showed that the students committed a lot of errors (type and type 2) when using analogical reasoning, and there was the existence of the rules of the teaching contract This confirmed the hypothesis H3 CHAPTER SOLUTIONS TO PROMOTE THE POSITIVE EFFECTS OF ANALOGICAL REASONING ON TEACHING MATHEMATICS AND PEDAGOGICAL EXPERIMENTS This chapter developed pedagogical solutions to promote the positive effect of analogical reasoning on teaching some contents in the chapter “Coordinate method in space” and conducted experiments to test the hypotheses H1, H2, H3 6.1 Solutions to teach with analogical reasoning 6.1.1 Solution 1: Exploit and improve the textbooks activities of using analogical reasoning to promote students' positiveness 6.1.2 Solution 2: Develop some processes of teaching typical mathematical situations by analogical reasoning 6.1.2.1 The process of discovering the new concepts Table 6.1 The process of discovering concepts with analogy (improved from TWA model) Step 1: Motivate students at the beginning of lesson and towards the target; Step 2: Review the source knowledge; Step 3: Students indicate correspondent properties between the source and the target; Step 4: Teachers indicate incorrect conclusions, characteristic properties of new concept; Step 5: Students state the definition of the new concept; Step 6: Teachers conclude about the new concepts and give some examples and applied exercises 17 We offered illustrative examples: discovering the sphere equation, the general equation of a plane and the parametric equation of a straight line in space * Teaching the sphere equation: Step Motivate students at the beginning of lesson and towards the target: Teachers give the following questions for students to think and discuss in groups: Question 1a Recall the solutions of problem: In plane Oxy, find conditions to point M (x; y) belonging to the circle with the center I (1; 2), radius R = 3? Question 2a Similarly, solve the problem: In space Oxyz, given a sphere (S) with the center I (1;2;0) , radius R=3 Find conditions for point M(x; y; z) belonging to (S)? Question 1b Recall the definition of the circle and how to find the equation of the circle (C) with the center I ( x0 ; y0 ) and radius R? Question 2b Solve the problem: In space Oxyz, given a sphere (S) with the center I ( x0 ; y0 ; z0 ) , radius R Find conditions for point M(x; y; z) belonging to (S)? Step 2: Review the source knowledge: - Teacher (T): Let's analyze the analogical relationship of above questions? - Students (S): A circle and a sphere have many similar characteristics, so how to find the condition for M belonging to a sphere is similar to how to build the circle equation Step Students indicate correspondent properties between the source and the target: - T: Let’s recall the definition of a circle and a sphere? - S: State the definition - T: In plane Oxy, what is the circle equation with the center I ( x0 , y0 ) and radius R? - S: The circle equation: ( x − x0 ) + ( y − y0 ) = R 2 - T: In space Oxyz, is the condition for point M(x;y;z) belonging to a sphere with the center I ( x0 ; y0 ; z0 ) and radius R similar to that of the circle equation? - S: Compare and make prediction: ( x − x0 ) + ( y − y0 ) + ( z − z ) = R 2 2 Step 4: Indicate properties of the new concept: - Teacher request students to verify predictions * Indicate incorrect conclusions about the coefficients of x , y , z the equation ( x − a) + ( y − b ) = R in space is not a sphere equation Step 5: Students state the definition of the new concept: - S: State the definition of the sphere equation Step 6: Teachers conclude about the new concepts and give some applied exercises: Find the sphere equation with the center I(1;2;-2) passing A(2;-1; 3) 18 Find the equation of a sphere (S) passing A(0;-1;4), B(1;,-5;1), C(0;7;0), D(-3;3;-5) 6.1.2.2 The process of discovering theorems Table 6.3 The process of discovering theorems with analogy (improve from TWA model) Step 1: Motivate students at the beginning of lesson and towards the target; Step 2: Review the source knowledge and the related knowledge; Step 3: Teachers offer suggestions and instructions for students to discuss Students discuss together to analyze the characteristics of the source and establish a correspondence between source and target knowledge, then set up a hypothesis; Step 4: Teachers guide students through testing the hypothesis; Step 5: Teachers state theorems accurately and give applied exercises Two examples: teaching theorems of the coordinates expression of vectors operations in space and the formula of calculating the distance from a point to a plane * Discovering theorem of the formula of calculating the distance from a point to a plane Step Motivate students at the beginning of lesson and towards the target: - T: If we know the coordinates of a point and the general equation of a plane, Can we calculate the distance from that point to the plane? Step Review the source knowledge and the related knowledge: - T: Let's recall the formula of calculating the distance from a point to a straight line Ax + By0 + C in plane Oxy? - S: d ( M , ∆) = A2 + B - T: Let’s state the proof of this formula? uuuuuur - S: Point M’ is the projection of point M on line ∆ , from M ' M , uuuuuur same direction and M ' ∈ ∆ to find M ' M r n having the Step Students discuss to analyze and set up hypotheses: - T: Let’s discuss in minutes in groups, each group consists of students to solve the problem: In space Oxyz, given (α ) : Ax + By + Cz + D = and a point M ( x0 ; y0 ; z0 ) Calculate d (M , (α )) - S : Predict: d ( M , (α )) = Ax + By0 + Cz0 + D A2 + B + C Step 4: Teachers guide students through testing the hypotheses: Students discuss in groups 19 Step 5: Teachers state theorems accurately and give applied exercises Calculate the distance from M (1;-2; 3) to ( α ) : x − y − z + = Calculate the distance between ( α ) : x + y + z + 11 = and ( β ) : x + y + z + = 6.1.2.3 The process of solving the problems Table 6.3 The process of solving the problems with analogy (improve from TWA) Step 1: Understand a problem (the target problem); Step 2: Find a known analogical problem (the source problem); Step 3: Analyze the similarities and differences of two problems; Step 4: Think about the solutions to the target problem; Step 5: Present the solutions; Step 6: Review and study the solutions We gave examples of applying this process of solving problems: find the general equation of a plane and the parametric equation of a straight line in space * Teaching solve mathematics problem “Find the general equation of a plane ( α ) passing points M(2;0;-1), N(1;-2;3), P(0;1;2) Step 1: Understand a problem (the target problem) - T: Give the problem Let’s find assumptions and requirements of the problem? - S: Assumption: ( α ) passes M, N, P Requirement: find the general equation of ( α ) Step 2: Find a known analogical problem (the source problem) - T: Let’s find an analogical problem with the stated problem? - S: Find the equation of a straight line ∆ passing A, B in Oxy Step 3: Analyze the similarities and differences of two problems: - T: Let’s compare these two problems - S: Assumptions: passing points (or points); requirements: find the equation - T: Let's recall the solutions of the problem in Oxy? uuur r r - S: ud = AB = ( a; b) ⇒ nd = (b; − a) , infer the equation of ∆ Step 4: Think about the solutions to the target problem: - T: Similarily, let’s infer the solutions to solve the problem in space uuuu r uuur - S: ( α ) passes M, N, P, so MN , MP are direction vectors of ( α ) Thus, select uuuu r uuur r n( α ) =  MN , MP  , infer the equation ( α ) Step 5: Present the solutions: Students present a complete solution Step 6: Review and study the solutions: Teacher comments on students’ solutions 20 6.1.3 Solution 3: Promote the positive effects of analogical reasoning in predicting, preventing and correcting the errors of students a Before teaching the knowledge, teachers need to predict analogical sources that students can use and lead to errors, then develop measures to prevent these errors when using analogical reasoning Table 6.4 Process of predicting students' errors by analogy sources before teaching in class Step 1: Consider the possible analogy sources for the target, Step 2: From each analogy source, find the correct and wrong conclusions, Step 3: Find measures to prevent students from making wrong conclusions b In the teaching process of knowledge, teachers should explain for students about the wrong conclusions to help students to avoid errors c After teaching the knowledge, teachers need to make notes, summarize the learned knowledge to help students to avoid errors in the future 6.1.4 Solution 4: Creating a chance for students to analyze, detect and repair errors caused by analogical reasoning a Provide opportunities for students to analyze and detect errors Table 6.5 The process of analying and detecting errors Step 1: Students are exposed to the problems with the wrong answer Step 2: Students determine errors in the answer Step 3: Students find the causes of errors: neglectfulness, wrong calculations, misunderstanding about concepts, the incorrect use of analogical reasoning, etc Step 4: Students find a way to correct the errors caused by using analogical reasoning: students analyze the source knowledge → infer the analogy characteristics of the target knowledge → infer the solutions to given problem Step 5: Students solve the problem again b Detecting and correcting errors when using analogical reasoning Table 6.6 The process of correcting errors when using analogical reasoning Step 1: Students are exposed to the target knowdege; Step 2: Teachers find out the source knowledge; Step 3: Students use analogical reasoning to establish a correspondence between source and target, then infer hypotheses; Step 4: Students test hypotheses and reject the incorrect hypotheses; Step 5: Students find the reasons of errors; Step 6: Students draw conclusions about the target knowledge 21 Two illustrative examples: find the orthocenter coordinates of a triangle in space and find the general equation of a plane passing point A and being parallel to (d, d’) * The problem of finding the orthocenter coordinates of a triangle in space Step 1: Students are exposed to the target: - Teachers give the problem: In space Oxyz, give points A(1;0;1), B(2;1;2), C(1;-1;1) Find the coordinates of orthocenter H in triangle ABC Step 2: Teachers find out the source knowledge: - T: Let's recall how to solve the problem of finding the orthocenter coordinates of a triangle ABC in plane? uuuu r uuur uuur uuur - S: H(x;y) is a orthocenter We have AH BC = 0, BH AC = , infer x, y Step 3: Students establish a correspondence between the source and the target, then infer hypotheses: - T: Similarly, let's deduce how to solve the problem in space uuuu r uuur uuur uuur - S: H(x;y ;z) is orthocenter We have AH BC = 0, BH AC = , then infer x, y, z Step 4: Students test the hypothesis and reject the incorrect hypothesis: - T: Let‘s apply the mentioned solution to the above problem? - S: Apply the solution and realize errors Step 5: Students find the reasons of errors: - S: In space, there are countless lines AH ⊥ BC and BH ⊥ AC Therefore, it is impossible to find the coordinates of H We should complement an additional condition that H ∈ mp ( ABC ) Step 6: Students draw conclusions about the target knowledge: - S: Solve the problem again and restate the right solutions 6.1.5 Solution 5: Systematize knowledge by using analogy 6.1.6 Solution 6: Raising awareness of teachers in high schools and pre-service teachers about using analogical reasoning in teaching mathematics 6.2 Some notes when teaching by analogical reasoning Proposing some notes about selecting sources when choosing suitable sources, analyzing the characteristics of source, establishing a correspondence between source and target, testing the correctness of conclusions 6.3 Pedagogical experiments We conducted the following teaching experiments to test hypotheses H1, H2, H3 6.3.1 Experimental situation (discovering the sphere equation) 22 6.3.2 Experimental situation (discovering the formula of calculating the distance from a point to a plane) 6.3.3 Experimental situation (solving the problem of finding the general equation of a plane) 6.3.4 Experimental situation (detecting and correcting errors when finding the orthocenter coordinates of a triangle in space) The experimental situations showed that many students used analogical reasoning to discover the new concepts, new properties, solutions to problems and correct errors It was said that the hypotheses H1, H2 and H3 were precise 6.4 Conclusion of chapter In this chapter, we developed solutions and teaching processes of using analogical reasoning in teaching typical situations The experimental results confirmed their feasibility and effectiveness CONCLUSION The conclusion of thesis 1.1 The contributions of the thesis in terms of theory The main contributions of the thesis in terms of theory were: - Analyzing and systematizing the views of analogy and analogical reasoning concepts, classifications, applications and models of using analogical reasoning - Developing the rubric for evaluation of using analogical reasoning with levels It helped us to evaluate the teaching process of using analogical reasoning objectively and effectively - Developing pedagogical solutions to analogical reasoning in teaching mathamatics - Developing teaching processes of using analogical reasoning: discovering the concepts, discovering theorems, solving problems, predicting students’ errors by analogy sources before teaching, analyzing and detecting errors, correcting errors 1.2 The contributions of the thesis in terms of practice Besides the theoretical aspects, the thesis also gave some practical values: - The research results about using analogical reasoning of geometry textbooks showed that the authors had used analogical reasoning for many different objects, however, the authors had not proposed activities of using analogical reasoning that students would perform to discover new knowledge 23 - The research results about teaching practice of teachers in high schools and preservice teachers showed that using analogical reasoning in teaching topics of chapter “Coordinate method in space” had not been concerned about - The thesis pointed out many errors of students when applying analogical reasoning to solve problems in the chapter “coordinate method” Thus, it was necessary to design many learning situations to help students to recognize and correct errors - Pedagogical solutions and processes of using analogical reasoning applied to teach some specific contents in the chapter “coordinate method in space” which the thesis developed helped students to discover new knowledge, solve problems and correct errors; from which, the efficiency of teaching the chapter “coordinate method” was improved A number of research topics in the future The thesis also suggested new research topics: We will study internal errors in the chapter “Coordinate method in space” Also, we will teach by using both analogical reasoning and information technology to help students to discover new knowledge Additionally, we will use analogical reasoning in many different specific contents of mathematics in high schools Especially, we will find out both the advantages and disadvantages of teachers when using analogical reasoning THE PUBLISHED WORKS RELATED TO THE CONTENTS OF THE THESIS Articles on Journal of Science in Viet Nam Bui Phuong Uyen (2012), Sử dụng mô hình FAR vào dạy học tương tự toán học, Journal of Science, Can Tho University, No 22b (2012), p.63-70 Bui Phuong Uyen (2013), Các kiểu nhiệm vụ chủ đề phương trình mặt phẳng: nghiên cứu sở suy luận tương tự, Journal of Science, Can Tho University, No 27(2013), tp.108-115 Bui Phuong Uyen (2014), Dạy học khám phá công thức tính khoảng cách từ điểm đến mặt phẳng (Hình học 12) suy luận tương tự, Journal of Education, No 338 period (7/2014), tr 54-56 Bui Phuong Uyen (2015), Phân tích thực hành giảng dạy giáo viên qua tiết học công thức tính khoảng cách từ điểm đến mặt phẳng theo quan điểm didactic toán, Journal of Science, Can Tho University, No.36c(2015), p.1-7 24 Bui Phuong Uyen (2015), Sai lầm liên quan đến phương trình mặt phẳng từ cách tiếp cận suy luận tương tự hợp đồng dạy học, Journal of Science, Ho Chi Minh City University of Education, No 6(72) 2015, p 39 - 48 Bui Phuong Uyen (2015), Thực trạng sử dụng suy luận tương tự vào dạy học sinh viên sư phạm toán – Đại học Cần Thơ qua học phần tập giảng , Journal of Science, Can Tho University, No 39c (2015), p 1-6 Bui Phuong Uyen (2015), Nghiên cứu cách thức sử dụng suy luận tương tự vào dạy học phương pháp tọa độ không gian giáo viên toán trường trung học phổ thông, Journal of Science, Can Tho University, No 41c(2015), p.76-80 Articles on foreign Journal of Science Loc, N P & Uyen, B P (2014), Using Analogy in Teaching Mathematics: An Investigation of Mathematics Education Students in School of Education - Can Tho University, International Journal of Education and Research, ISSN: 2411-5681, Vol No July 2014, Contemporary Research Center, Australia Loc, N P & Uyen, B P (2015), A Study of Mathematics Education Students’ Difficulties in Applying Analogy to Teaching Mathematics: A Case of the “TWA” Model, American International Journal of Research in Humanities, Arts and Social Science (AIJRHASS), ISSN (Print): 2328-3734, ISSN (Online): 2328-3696, ISSN (CD-ROM): 2328-3688, 9(3), December 2014-February 2015, pp 276-280, USA 10 Loc, N P & Uyen, B P (2015), Using Analogical Reasoning in Teaching Mathematics: A Survey of Mathematics Teachers at Secondary Schools in The Mekong Delta–Vietnam, International Journal of Sciences: Basic and Applied Research (IJSBAR), ISSN 2307-4531, (2015) Volume 21, No 1, p.90-100, Jordan 11 Loc, N P & Uyen, B P (2015), Analogies in Geometry Textbooks for 12th Grade Students in Vietnam, American International Journal of Research in Science, Technology, Engineering & Mathematics (AIJRSTEM), ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629 , 10(1), March- May 2015, pp 73-78, USA 12 Loc, N P & Uyen, B P (2016), Students’ Errors in Solving Problem: A Case Study based on the Concept “Didactical Contract”, European Academic Research, ISSN 2286-4822, Vol IV, Issue 1/April, p.264-269, Romania 13 Loc, N P & Uyen, B P (2016), Students’ Errors in Solving Undefined Problem in Analytic Geometry In Space: A Case Study based on Analogical Reasoning, 25 Asian Journal of Management Sciences & Education, ISSN: 2186-845X ISSN: 2186-8441 Print, Vol 5(2) April 2016, p.14-18, Japan Articles in Conference, Workshop of Science 14 Bui Phuong Uyen (2014), Sai lầm học sinh giải toán tìm tọa độ trực tâm tam giác không gian từ cách tiếp cận suy luận tương tự, Conference of Science, School of Mathematics and Informatics (12/2014), Ho Chi Minh City University of Education 15 Bui Phuong Uyen (2015), Phương trình đường tròn phương trình mặt cầu: Sai lầm liên quan đến suy luận tương tự, Workshop of Science for Mater Students and Doctoral Students (10/ 2015), Ho Chi Minh City University of Education, VNUHCM Publishing House, HCM City Book Chapter 16 Nguyen Phu Loc & Bui Phuong Uyen (2016), Các xu hướng dạy học toán, Chương Dạy học với suy luận tương tự, Can Tho University Publishing House, Can Tho City [...]... ON ANALOGICAL REASONING IN THE CHAPTER COORDINATE METHOD IN SPACE This chapter reported the results of research to answer the research question 1 3.1 Analogical reasoning in the chapter Coordinate method in space 3.1.1 Analogy in Coordinate method in plane and in space We presented the contents in Coordinate method in space similar to Coordinate method in plane 3.1.2 Analogical reasoning on the. .. and models of using analogical reasoning - Developing the rubric for evaluation of using analogical reasoning with 5 levels It helped us to evaluate the teaching process of using analogical reasoning objectively and effectively - Developing 6 pedagogical solutions to analogical reasoning in teaching mathamatics - Developing 6 teaching processes of using analogical reasoning: discovering the concepts,... 6 teaching processes of using analogical reasoning in teaching typical situations The experimental results confirmed their feasibility and effectiveness CONCLUSION 1 The conclusion of thesis 1.1 The contributions of the thesis in terms of theory The main contributions of the thesis in terms of theory were: - Analyzing and systematizing the views of analogy and analogical reasoning concepts, classifications,... There were 8 cases in which the authors used analogical reasoning on Geometry 10, 11, 12 for the Basic curriculum; 15 cases in which the authors used analogical reasoning on Advanced Geometry 10, 11, 12 for the advanced curriculum We classified these cases in table 1.1 3.1.3 Analogical reasoning in the chapter Coordinate method in space rr The authors of textbooks used analogical reasoning in 4 cases:... using analogical reasoning, and there was the existence of the rules of the teaching contract This confirmed the hypothesis H3 CHAPTER 6 SOLUTIONS TO PROMOTE THE POSITIVE EFFECTS OF ANALOGICAL REASONING ON TEACHING MATHEMATICS AND PEDAGOGICAL EXPERIMENTS This chapter developed pedagogical solutions to promote the positive effect of analogical reasoning on teaching some contents in the chapter Coordinate. .. straight line passing 2 distinct points A and B” Based on the strategies of the source, we predicted an error (type 2) by using analogical reasoning that students would commit when finding the general equation of a plane passing 3 points (in the case of 3 collinear points): Error 1: Students replace the coordinates of normal vector 12 uuur uuur r r n =  AB; AC  = 0 into the plane equation, and there... teachers a) The survey 1 We considered the average level of using analogical reasoning according to the 0 +1+ 2 + 3 + 4 = 2 The obtained results were compared with a above rubric was a = 5 11 In step 1, the average levels of using analogical reasoning in contents when they work individually were less than a In step 2, we compared the average level of using analogical reasoning when they worked individually... target, then infer hypotheses; Step 4: Students test hypotheses and reject the incorrect hypotheses; Step 5: Students find the reasons of errors; Step 6: Students draw conclusions about the target knowledge 21 Two illustrative examples: find the orthocenter coordinates of a triangle in space and find the general equation of a plane passing point A and being parallel to (d, d’) * The problem of finding the. .. the advantages and disadvantages, then developed solutions to use TWA model effectively 4.3 Conclusion of chapter 4 The surveys of teachers and pre-service teachers showed that using analogical reasoning in teaching Coordinate method in space had not been interested in This was due to the impact of the presentation of analogical reasoning in the current textbooks CHAPTER 5 RESEARCH ON REAL ERRORS OF. .. knowledge, then set up a hypothesis; Step 4: Teachers guide students through testing the hypothesis; Step 5: Teachers state theorems accurately and give applied exercises Two examples: teaching theorems of the coordinates expression of vectors operations in space and the formula of calculating the distance from a point to a plane * Discovering theorem of the formula of calculating the distance from a point