Tài liệu 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: 1. Assoc. Prof. Dr. Nguyen Phu Loc 2. 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. 1 INTRODUCTION 1. 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, do 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 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 do we need to verify the effectiveness of these solutions? 3. 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. 4. 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. 5. 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 3 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. 6. 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. 7. The structure of the thesis Besides introduction and conclusion, the main contents of the thesis consisted of 6 chapters: Chapter 1. Theoretical frameworks; Chapter 2. The methods and research designs; Chapter 3. Research on analogical reasoning in the chapter “Coordinate method in space”; Chapter 4. Research on the teaching practice of using analogical reasoning; Chapter 5. Research on the real errors of students when using analogical reasoning; Chapter 6. Solutions to promote the positive effects of analogical reasoning in teaching mathematics and pedagogical experiment. 4 CHAPTER 1. 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 5 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 3. 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. 6 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: 1. Introduce target concept; 2. Review source concept; 3. Identify relevant features of source and target; 4. Map similarities between source and target; 5. Clarify the incorrect conclusions; 6. 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 1 Chapter 1 covered the theoretical framework of analogical reasoning considered as a basis in the following chapters. 7 CHAPTER 2. 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,  is the technique,  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 2 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 do 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 0 1 2 3 4 Levels of using analogical reasoning in teaching Do not use analogical reasoning. Only talk about the source. Recall characteristics of the source, but do 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: 8 1. 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? 2. What difficulties can students meet when they use TWA model to design lesson plans in the chapter Coordinate method in space? 3. 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 3 to 4 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 3 to 4 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: 1. In your opinion, what are strong and weak points of the TWA model for teaching mathematics? 2. 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? 3. 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 2 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. 9 The method of research: A priori analysis: specifically analyze some errors when implementing tasks in Coordinate method in space. A posterior analysis: a survey forms for 309 students and interview 6 students to clarify how they use analogical reasoning. 2.4. Research on solutions and pedagogical experiments to use analogical reasoning in teaching Coordinate method in space (reply to research question 4) The purpose of the study: Develop solutions to promote the positive effects of analogical reasoning in teaching mathematics and verify hypotheses H1, H2, H3. The method of the study: Experiment with four teaching situations for 12th Grade students (school year 2014-2015), at Pedagogical Practice High School, Can Tho city. 2.5. Conclusion of chapter 2 In the chapter, we proposed the research methods to answer four research questions stated. CHAPTER 3. RESEARCH 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 current Geometry textbooks 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: prove b .n  0, the parametric equation of a straight line in space (Geometry 12), system of coordinate axis in space, the formula of calculating the distance from a point to a plane (Advanced Geometry 12). The goals of cases are: introduce a new lesson; generate a problematic situation to help students to predict new knowledge; introduce a new formula without its proof; apply an earlier similar proofing way. Moreover, the 10 authors of textbooks only used analogical reasoning to guide and present new knowledge, none of the activities helps students to discover new knowledge. 3.2. Analogical mathematical organizations in Coordinate methods in plane and in space We presented 30 typical mathematical organizations in Coordinate method in space on analogical relationships with 30 mathematical organizations in Coordinate method in plane. They consist of 3 components: task T, technique  and technology  . This indicates that analogical reasoning can be used to help students to explore the solutions of new problems in Coordinate method in space. 3.3. Conclusion of chapter 3 The above analysis showed that there were analogical relationships between concepts, properties and mathematical organizations in Coordinate method in plane and in space. The authors of textbooks used analogical reasoning to help students to review old knowledge, guide and present new knowledge, but they did not mention activities of discovering new knowledge. CHAPTER 4. RESEARCH ON TEACHING PRACTICE USING ANALOGICAL REASONING This chapter showed the results in order to answer the research question 2. 4.1. The survey of teachers There were 5 lessons (5 of 20 lessons) in which teachers used analogical reasoning: one lesson was about the general equation of a plane, another was about the parametric equation of a straight line and the other three lessons were about the formula of calculating the distance from a point to a plane. The contents which teachers used analogical reasoning had many similarities with contents mentioned by authors of textbooks. The teachers used analogical reasoning to help students to hypothesize about new knowledge, explore and solve problems, create motivation to begin the new lesson. Some questions and activities were developed from the presentations of textbooks. It was clear that there was the impact of analogical reasoning in the textbooks for the teaching process of teachers. 4.2. The survey of pre-service 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 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 2 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 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 STUDENTS WHEN USING ANALOGICAL REASONING This chapter showed the results to answer the research question 3 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 3 distinct points” 5.1.1. Priori analyses 5.1.1.1. Didactic variables (the selected values are marked with*) V1-1: The collinear of 3 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 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 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 6 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 1 (replacing the r r coordinates of n  0 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 1 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 2 lines d and d’ (in case d//d’): Error 2: Students replace the coordinates of normal r r r r vector n   ud ; ud '   0 into the plane equation, and there exists a rule of teaching contract R2: Students have no responsibility to check the relative position of 2 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    passing A(3;2;-4),  x  8t x  3 y 1 z 1    a. and being parallel to d :  y  5  2t , d ' : . 7 2 3  z  8t  r r 1 2  b. and being parallel to vector u   3; 4; 6  , v   ; ; 1  . 2 3  r r In question a, d and d’ are skew, but in question b, u , v have the same direction. Then, we interviewed 6 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 2 (replacing the coordinates r r r r of n   u , v   0 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 2 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 3 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  2 z  3   . 2 3 1 b. being perpendicular to a line which passes 2 points A(3;1;-2), B(-1;-2;1). We interviewed 6 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. 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    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 2 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 ,      cos ud ; n   ; Error 7: Students r r r find a normal vector nd  (b;  a; c) of d, then calculate cos(d,( ))  cos(nd ; n   ) . 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    when: a. line d parallel to z-axis and the equation of    : x  0 . x  2 y 1 z 1   and the equation    : 2 x  y  z  8  0 . 2 3 5 We interviewed 6 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 2 straight lines: 25% of students used the formula cos  d ,      cos ud ; n   r and 9.06% of students found the normal vector nd  (b;  a; c) of d, then calculated r r cos(d,( ))  cos( nd ; n   ) . The existence of error 6, 7 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 8 specific forms of tasks “identify the equation of a circle and a sphere” in analogical relationship and predicted errors type 1 and type 2 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  4    y  3  16 1b.  x  4    y  3  16 2a.  2  x    3  y   25 2b.  2  x    3  y   (2  z ) 2  25 3a.  3 x  1   3 y  3  36 3b.  3 x  1   3 y  3   3z  2   36 4a.  x  1   3 y  3   36 4b.  x  1   3 y  3   2 z  2   36 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 16 5a. x  y  2 x  4 y  6  0 2 2 2 5b. x  y  z  2 x  4 y  6 z  6  0 2 2 6a. 3 x  3 y  6 x  3 y  9  0 2 2 2 6b. 3x  3 y  3 z  6 x  3 y  3 z  9  0 2 2 7a. 2 x  y  6 x  4 y  8  0 2 2 2 7b. 2 x  y  z  6 x  4 y  2 z  8  0 2 8a. ( x  y )  6 x  8  4 y  2 xy 2 2 8b. ( x  y )  6 x  8  2 xz  4 y  2 xy  ( x  z ) 2 2 We interviewed 6 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 5 The results showed that the students committed a lot of errors (type 1 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 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 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 3 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 1. 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 3. 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 . 2 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  z0   R 2 2 2 2 Step 4: Indicate properties of the new concept: - Teacher request students to verify predictions. * Indicate incorrect conclusions about the coefficients of x 2 , y 2 , z 2 và the equation  x  a 2   y  b   R 2 in space is not a sphere equation. 2 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: 1. Find the sphere equation with the center I(1;2;-2) passing A(2;-1; 3). 18 2. 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 1. 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 2. 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 0  By0  C in plane Oxy? - S: d ( M , )  . A2  B 2 - 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 3. Students discuss to analyze and set up hypotheses: - T: Let’s discuss in 5 minutes in groups, each group consists of 3 students to solve the problem: In space Oxyz, given ( ) : Ax  By  Cz  D  0 and a point M ( x0 ; y0 ; z0 ) . Calculate d ( M , ( )) . - S : Predict: d ( M , ( ))  Ax 0  By0  Cz0  D A2  B 2  C 2 . Step 4: Teachers guide students through testing the hypotheses: Students discuss in groups.