Aims and objectives of the research: The thesis aims to find out the functions of practical situations, design practical situations, apply designed practical situations to organize cognitive activities for students in the process of teaching mathematics at high school, contribute to improving the quality of mathematics teaching at high schools towards student’s competence development.
Trang 1MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY
PHAM NGUYEN HONG NGU
ORGANIZING COGNITIVE ACTIVITIES FOR STUDENTS THROUGH EXPLORING FUNCTIONS OF THE PRACTICAL SITUATION IN TEACHING MATHEMATICS AT HIGH
Trang 2INTRODUCTION
1 The reason to choose this topic
In recent years, our education has been changing drastically to keep pace with international education development The teaching method which focuses on transferring knowledge and skills is not popular anymore Now, we need to tend to build intellectual development abilities for students
Resolution No 29-NQ / TW in November 4, 2013, approved by the 8th Central Conference Session XI on fundamental and comprehensive renovation of education and
training, stated that: “Need to transform education from knowledge provision to
comprehensive development of learners' capabilities and qualities Learning is linked with practice, theory associated with reality Schools need to be combined with family and social education” The goals of the High-School Education Program in the resolution are defined as
"focusing on intellectual and physical development, forming personal qualities, citizen
capacity, detecting and fostering talents and career orientation for students; Improving the quality of comprehensive education, focusing on education about ideals, traditions, ethics, lifestyles, foreign languages, informatics, competence practical skills and abilities to apply knowledge to real life; Developing creativity, self-study and encouraging lifelong learning; Completing the high-school education program after 2015; Ensuring that students at
requirements after secondary school High school students must be accessed to career orientation or well-prepared for post-high-school education; Improving the quality of
Accordingly, the 2018 General Education Program stated that the goals of high- school education in Vietnam are to "Help students develop themselves physically and
lifelong learning; have good qualities to become responsible citizens, cultivated,
requirements in building and protecting the nation in the era of globalization and new industrial revolution”
Currently, Math Education in our country has been changing in the direction of associating mathematical knowledge with practice, paying attention to abilities applying mathematical knowledge learned by students This is seen through the goals of the new Math Education and Development Program issued by the Ministry of Education and Training on December 26th, 2018 Specifically, Math helps form and develop students’ basic
qualities, capabilities and Math competence namely mathematical thinking and reasoning, modelling, abilities to solve math problems, abilities to communicate math, abilities to use mathematical tools and means; developing key knowledge and key skills and creating opportunities for students to experience and apply mathematics to real life Math education creates a connection between mathematical ideas, between mathematics and other sciences and between mathematics and real life
This goal is completely consistent with the current trend of mathematical assessment
in the world by educational organizations and programs such as Organization for Economic Cooperation and Development (OECD) with 15-year-old Student Assessment Program (PISA), Science Technology Engineering Maths (STEM), Trends in International Mathematics and Science Study (TIMSS), etc These programs,focus on assessing mathematical competence and use mathematical knowledge to explain practical activities; focus on abilities to modelize practical phenomena using mathematical language and
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notation, and to apply mathematics to practice
Those goals should be implemented in teaching mathematics with the idea that mathematics education should be associated with practice, mathematics education should be directed to educating dialectical materialist worldview of students Teachers in the teaching process, need to organize many cognitive activities for students to experience the interactive learning, apply their mathematical knowledge to the process of creating new knowledge; avoid teaching mathematics at high schools abstractly, lack of practical orientation
The above issues are the premise to guide us to implement the topic: Organizing
cognitive activities for students through exploring functions of the practical situation in teaching mathematics at high schools
2 Aims and objectives of the research
The thesis aims to find out the functions of practical situations, design practical situations, apply designed practical situations to organize cognitive activities for students in the process of teaching mathematics at high school, contribute to improving the quality of mathematics teaching at high schools towards student’s competence development
3 Scientific hypothesis
It is possible that finding out practical situations and using mathematical representations as an intermediary will drive students to activities of exploring, discovering, and achieving knowledge, contributing to the effective improvement of cognitive activities
in learning mathematics at high school
4 Research questions
To fulfil the objectives of the research, the thesis focuses on answering the following questions:
4.1 What is a practical situation in teaching mathematics?
4.2 Which functions of practical situations in teaching mathematics are of developing learners' competences?
4.3 How can we design and build practical situations in mathematics teaching? 4.4 How can students take part in cognitive activities and perform the functions of practical situations in mathematics learning?
5 Focus and scope of the research
5.1 Focus of the research
The research focus on the ways of designing and organizing cognitive activities for students through exploring the functions of practical situations in teaching mathematics
5.2 Scope the research
The scope of the research is mathematics education through exploring the functions
of practical situations at high school in Vietnam
6 Research methods
The thesis uses theoretical research methods, observational surveys, synetheses of pedagogical experiences and experiments
7 New contributions of the thesis
+ In theory, the thesis is:
- Proposing an approach to practical situation in teaching mathematics at high schools Clarifying the functions, the roles of practical situations such as the function of stimulating students to access to the knowledge, the funtion of discovering the laws of exploring mathematical rules, the funtion of consolidating the knowledge, the funtion of explaining the simulation of practical phenomena to exploit different applications of mathematics in practice, and the funtion of creating student’ mathematics culture;
Trang 4- Clarifying the viewpoint of organizing cognitive activities for students through practical situations Proposing some cognitive activities in teaching and learning mathematics through practical situations such as getting new knowledge, consolidating knowledge, and applying mathematical knowledge
- Clarifying outstanding difficulties teachers have in designing and organizing mathematics teaching in practice
+ In practice, the thesis is:
- Clarifying how to utilize the funtions of practical situations and provide new illustrations in practice
- Proposing processes for designing practical situations, organizing cognitive activities in typical situations composed of teaching mathematical concepts, theorems, and rules Developing some new illustrative examples for these processes
8 Issues in defence
The theoretical and practical approaches the thesis focus on:
- Explaining the difficulties teachers face in designing and using practical situation when teaching mathematics at high schools, and that students have in connecting mathematics with practice, clarifying the roles and functions of practical situations in teaching mathematics;
- Analysing the principles of designing practical situations, clarifying the process of designing practical situations in teaching mathematics;
- Clarifying the process of organizing cognitive activities for students via the designed practical situations;
- Assessing the feasibility of the thesis through pedagogical experiment
9 Structure of the thesis
In addition to the introduction, conclusion and recommendation, the thesis structure includes the following main contents:
Chapter 1 Theoretical and practical background
Chapter 2 Designing practical situations in teaching mathematics at high schools Chapter 3 Organizing cognitive activities via practical situations in teaching mathematics at high schools
Chapter 4 Pedagogical experiments
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Chapter 1 THEORETICAL AND PRACTICAL BACKGROUND
not been any detailed and profound research on practical situation in teaching mathematics
as a means to search for mathematical rules, as well as designing process of practical situation and organizing cognitive activities for students in teaching mathematics
1.1.2 Terminologies used in the thesis
+ Situations in teaching mathematics are defined as the situations containing
mathematical content to be discovered, that are designed or redesigned by teachers, and integgrated with learning tasks so that students can achieve mathematical knowledge by doing the tasks
+ Practical situations in teaching mathematics are understood as the situations
deriving from practice of our daily life in which mathematical contents or relationships can be found, designed or redesigned, in accordance with the lesson objectives in order for students to take part in these situations and acquire mathematical knowledge
1.1.3 Approaches of Practical Situation in mathematics teaching
Practical situations are approached from different angles, including cognitive methods in teaching mathematics, the connection between mathematical content and practice through modelization, assessment and evaluation, integrated teaching trend, math cultural education for students Because they all have in common that mathematics is derived from practice and today's teaching perspective is to consider the process, not just the result of learning activities
1.1.4 Functions of practical situation in teaching mathematics
In this section, we propose, analyze and clarify the functions of practical situations in teaching mathematics including: motivating students to approach practical situations and get knowledge, detecting laws, searching mathematical rules; inculcating knowledge; explaining the simulation of practical phenomena to exploring different applications of mathematics in practice; contributing to creating students’ mathematical culture
1.1.5 Mathematics cognitive activities
Theoretical and practical researches, show the relationship between the functions of practical situations and the cognitive activities of students in mathematics lessons Therefore, utilizing the functions of practical situations in teaching mathematics, should be implemented through mathematical cognitive activities In the thesis, mathematical
cognitive activities is understood as "The thinking process leading to the acquisition and
comprehension of mathematical knowledge, the identification of causal and other relationships between concept, relationships, mathematical laws, etc thereby mathematical knowledge may be applied to solve practical problems”
In the thesis, the cognitive activities associated with practical situations in teaching mathematics including: - Providing new knowledge; - Consolidating knowledge; - Applying
Trang 6mathematical knowledge such subactivities as observation, assimilation, moderation, modelization, discovery and object transformation
Organizing cognitive activities for students in teaching mathematics is considered the teachers’ performance of the most necessary mathematics teaching tasks to students think and achieve mathematical knowledge actively and creatively
We also propose organizing cognitive activities for students in teaching mathematics through utilizing the functions of practical situations is the teachers' design of practical situations providing activities for students so that these cognitive activities can stimulate their learning needs, understand the significance of mathematical knowledge, and applying mathematical knowledge into practice
1.2 Practical Basis
1.2.1 Practical element in the current textbook program in Vietnam
Based on the surveys of the current situation of Vietnamese textbooks, we find that in the current textbooks in Vietnam, the authors of books have some historical stories, tables, figures, drawings, examples, problems with practice to motivate students to study However,
according to Nguyen Chi Thanh's opinion: "there are few practical problems", not focusing on
the application of math to solve a practical problem after each lesson We also agree with Ha Xuan Thanh's opinion that exercises, examples in textbooks of high school mathematics, are mainly divided into two categories: pure mathematical problems and practical problems; in which the practical problems account for a very low rate (4.6%) in the total number of exercises, even for highly applicable such as inequality, Probability statistics,…
Therefore, we think that reviewing and supplementing practical element in the high school education programs in general and mathematics education in particular in our country at this time is urgent that requiring the cooperation and contribution of the educational management, teachers and the whole society It is praiseworthy that in the 2018 General Education Program in general and the 2018 Mathematics Education Program in particular, the program researchers emphasized the important meaning of practical connection in teaching mathematics, showing in requirement on skills and competencies for students from primary to high school in the new mathematics program through hands-on activities (Elementary to high school) and thematic activities (High school level)
1.2.2 Surveying the real situation of teachers and students' awareness of practical situation in teaching mathematics at high schools
In order to understand the real situation of teachers 'and students' awareness about practical situation in teaching and learning math, we designed the questionnaire (see the appendix) and conducted a survey with 134 teachers and 1057 students in high schools in many cities and provinces nationwide such as Quang Nam, Quang Ngai, Da Nang, Gia Lai, Kon Tum, Dac Lac, Quang Binh, Nghe An, Ha Tinh, Ninh Thuan,
- On the teacher's side, we consider the teacher's perception of practical situation, teacher difficulties when using practical situation, going over the regular lesson of the teacher as well as the awareness and experience of the teacher in building practical situation Based on the survey results as well as consulting experts, reputable educational researchers,
we recognize the difficulties that teachers face during the process of experimenting, exploring, building practical situation in teaching mathematics following these reasons:
Firstly, the limitation of teachers' awareness of theoretical basis They do not know where to start to be able to model a mathematical knowledge The nature of mathematics is abstract so a mathematical knowledge can be concretized in many different directions and teachers do not know what practical object to choose to represent that mathematical knowledge
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Secondly, the concept of teaching mathematics tied to the practice of teachers is different; they do not know whether the situation of teaching mathematics in practice is the situation associated with the phenomena that occur in reality or only internally in mathematics, or only in the relationship between mathematics and other subjects
Thirdly, most teachers teach math exactly like what are in textbooks, but in the current textbooks, the number of problems containing practical contents, or practical simulations, is both small in number
Fourthly, teachers have few researches on mathematical history, so they have not really seen the origin of mathematics, the need for developing mathematics, nor the ideology of methodology in teaching math which is teaching the relationships between chapters, sections, spaces and quantities
Another shortcoming of mathematics teacher nowadays is not paying adequate attention to their lessons; there are few activities in discussing and cooperating between teachers on a specific problem or teaching situation
- On the student's side, we collected students' perception of the necessity and need to understand the application of mathematical knowledge in practice
Based on the result of the student survey, we found that: Most of the students, when asked, want to know which mathematical knowledge learned in high school can be applied
in practice, how to apply it (more than 89%) There are only 41 students, accounting for 3.9%, do not want to know learning math for what, or to learn math because it is a compulsory subject and 6.6% of the students are indifferent, no opinion This shows that students are very interested in what to learn math for Currently, math is a main, compulsory subject in high school, but in reality, students do not understand why to learn math than facing exams When surveyed, 47.4% of students said they were regular, 15.8% of students very often used math in their daily activities However, they only use common summation, subtraction, multiplication, division, but have not seen the use of other mathematical content such as functions, derivative, integral in practice According to them, these are luxurious content and in fact never be used, although many students are studying grade 12th (they have learned quite a lot of mathematical applications such as the application of integral and numerical series) There are 39 students who think that they do not use any mathematical knowledge in their work, daily life, even simple arithmetic calculations (!?) Most students think it is very important to have skills to apply math into daily work; This is entirely consistent with question 2 about the desire to apply math knowledge in practice However, students think that they rarely learn math through practical situation The voluntarily solving
of actual problems is very small, accounting for 32.8% of the surveyed students Nearly 70% of surveyed students said that they rarely and often do not find out the real problems themselves if they are related to mathematics Most students like to learn math through practical situation, only 3.4% do not like it and 5.5% have no answers yet
Trang 8Conclusion of chapter 1
In chapter 1, we highlighted:
- Overview of a number of research works of domestic and foreign authors about the relationship between mathematics and practice, the necessity of mathematics in practice, the situation of teaching mathematics associated with reality, the use of mathematical modelling
in teaching mathematics associated with practice, the contents of educational transformation
in association with practice, the assessment and tendency to apply mathematical knowledge into daily life of students, assessment of students' mathematical understanding
- The concept of practical situation in teaching mathematics, clarifying the functions
of situation in teaching mathematics at high schools according to the author's approaches (Cognitive methodology approach, approach to the content connecting mathematics with practice through modelling, approaching the trend of testing and assessing, approaching the trend of integrated teaching, approaching mathematical cultural education for students)
- Difficulties of teachers, students in teaching and learning mathematics associated with reality; assessing the awareness of teachers and students about the need, desire to teach and learn math associated to practice We also identify the causes that lead to difficulties for teachers in experiencing, exploring, and building practical situations such as: time constraints, difficulties in finding relationship between mathematical knowledge and practical facts, the difficulty of transforming the content of mathematics to be taught into real model,
- The concept of cognitive activities in teaching mathematics, analysing cognitive activity of students into component activities: observation activities, object transformation activities, modelling activities, facilitation activities, assimilation activities At the same time, the thesis also mentioned the positive teaching methods that teachers need to use to organize cognitive activities for students (this will be specified in the examples of organizing cognitive activities for students in the chapter 3 of the thesis)
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Chapter 2 DESIGN OF PRACTICAL SITUATIONS IN MATHEMATICS TEACHING
AT HIGH SCHOOL
In chapter 1, the thesis presented the necessity of connecting practice with mathematics in teaching, the functions of practical situation in teaching mathematics; as well as the difficulties that teachers are currently facing in order to experience, explore, and build practical situation Based on that basis, the thesis designed a few practical situations in teaching mathematics to support teachers in teaching mathematics associated with their practice
2.1 Principles of designing practical situation in teaching mathematics
Principle 1: The design of practical situation must ensure the contents of the current
textbook programs
Principle 2: The design of practical situation in teaching mathematics must be
suitable with the real life of students
Principle 3: Practical situation is designed to ensure creating an active environment
for all students to participate in the learning process to form mathematical knowledge
Principle 4: The design of practical situation must be suitable with the level,
capacity and psychological characteristics of students
Principle 5: The practical situation design should clearly show the pedagogical
intention of the teacher, evaluating the ability to understand and apply math to practice of students
2.2 Designing process of practical situation in teaching mathematics
Based on theoretical and practical research in chapter 1, referring to the researches on the use of practical factors in teaching algebra, calculus, geometry of previous researchers, inheriting the design process of learning case in previous documents, in this thesis, we
propose the process of designing practical situation in teaching mathematics including 6
steps as following:
Step 1: Researching the goal and lesson content
Teachers need to study deeply the curriculum in textbooks, teachers' books, reference books to identify lessons, identify standards of knowledge, skills, competencies and qualities that students will gain after studying Identifying the key knowledge of lesson, the knowledge that students have, the knowledge that students need to be supplemented to understand the lesson; identify the skills that students are having, the skills that students need to gain, what is the applications of lesson in real life
Teachers need to study textbooks to review the situations in textbooks if they have ensured the connection between teaching and practice, considering which mathematical knowledge can integrate, adjust and supply practical elements in teaching From there, teachers need to determine what knowledge is capable of designing into a situation, as well as the necessity and benefits of the situation with the lesson objectives At this step, the teacher also envisages what functions of the practical situation can be gained to guide the design of practical situation
Step 2: Observing practices
After identifying the lesson objectives, teachers must observe real life For example, they can go to local production facilities to search and select practical models related to the necessary content to design the situation in step 1, which is suitable to their teaching conditions At this step, empirical knowledge is very beneficial for teachers, based on their existing experience, teachers will quickly determine which practical form contains
Trang 10mathematical knowledge in step 1 to observe Sometimes, teachers can observe and refer to practical models that have been developed by educational researchers for reference
Step 3: Selecting a model that is suitable for the teaching goal, building situation
Teachers select a mathematical model from those they observed that are suitable for teaching purposes to build the situation They should select the knowledge and skills that students should gain to outline the situation to be built
Teachers identify teaching objectives in accordance with the model (forming, applying, reinforcing concepts or theorems, rules, ) to build situation based on collected information Situation needs to be built systematically and logically, which is based on the principles of situation construction, ensuring accuracy, specificity, not too hard, not too easy for students and showing the functions of situation in the teaching process later
In the process of building situation, teachers need to analyze which is the cognitive level, experience, psycho-social characteristics of students to predict and confirm the difficulties of situation or the obstacles that students have to overcome Questions and necessary technical means to support students in solving situations
Step 4: Discussing and adjusting the situation
Teachers can share the situation they have just built with colleagues, research groups
to receive valuable suggestions about the content, rationality, coherence, appropriateness of the situation Through seminars or exchanges, interviews with experienced teachers to clarify the necessary requirements of situation design such as: Does situation clarify the meaning of knowledge to be taught? Does situation ensure excitement, stimulating for students? Is situation close to students' understanding? Can students easily convert situation into math model? Can students solve problem in mathematical model of situation? Can students explain the reality when there are results from the model? Referring to the predictions of teachers in the subject group and experienced teachers about difficulties and correct the situation
Step 5: Testing the situation
Teachers conduct a test case which has been edited in step 4 on a small group of students During the testing process, teachers need to observe to answer the following questions: - Are Students interested in the situation?, - How many questions can students solve in the situation?, - Can students modelize the situation?, - Have students seen the meaning of mathematical knowledge hidden in the situation ? From there to consider the feasibility of the situation, helping students if necessary, to solve the situation In this step, students must carry out the process of modelling from practical situation to the mathematical problems, solving problem which the situation gives
Step 6: Confirming the situation
Based on the results observed in the testing of the situations, the teacher will confirm whether the situation is feasible or not, if the student solves the situation (may need the support of the teacher), we accept the situation, otherwise in case students cannot solve situations (even though the teacher has helped), we can replace and redesign situations
2.3 The stages in teaching mathematics that are necessary to apply the process 2.4 Designing practical situation in teaching mathematics at high schools
Based on a 6-step process, we have conducted to design some practical situations in teaching mathematics, used in teaching concepts, theorems, and rules Due to the limited research conditions, these practical situations are carried out to discuss and collect comments through seminars at the division of method of teaching Mathematics, Department
of Mathematics at Quang Nam University; at the same time, collecting feedback (via email) from high school teachers at Nguyen Duc, Le Quy Don and Duy Tan schools in Quang Nam
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province; Nguyen Trai and Tran Cao Van schools in Khanh Hoa province; Le Thanh Phuong and Phan Boi Chau schools in Phu Yen province; Duy Tân school in KonTum province; Nguyen Cong Tru school in Quang Ngai province; Le Quy Don school in Binh Dinh province; Le Duan school in Dak Lak province; Nguyễn Chí Thanh school in HCM city Because of limited conditions, we have only tried to teach practical situation at a group
of students of class no 11/3 at Nguyen Duc High School (implemented by teacher: Cao Thi Lanh), and a group of students of class no 10C1 and class no 12C1 of Le Quy Don high school (by teacher: Huynh Thi Thu Phuong) in Quang Nam province
2.4.1 Practical situation used in conceptual teaching
Example 2.1 Designing conceptual teaching situation "Straight line perpendicular to plane" in lesson " Straight line perpendicular to plane" Geometry Textbook 11, page 98
Step 1: Research the goal and lesson content
We conduct research on textbooks, teachers' books, standard knowledge materials for high school math skills, documents to guide the implementation of knowledge standards of grade 11th to determine the content of knowledge and skills that students have as well as the
knowledge, skills students need to gain in this lesson are as follows: -About knowledge:
Knowing the definition and condition of straight line perpendicular to the surface,
perpendicular projection, Bisector plane, -About skill: Knowing how to prove that straight
line is perpendicular to the surface, straight line is perpendicular to other straight line, and determining normal vector of a surface, perpendicular projection, applying the theorem of 3 perpendicular lines, determining the middle angle of straight line and the surface
We conducted a study on the textbook of grade 11th to find that To give motivation into the lesson, the textbook mentions the image of the robe which is perpendicular to the floor that talks about the image of the straight line perpendicular to the floor, while not mentioning any practical elements in the theorem, properties
We think that we can design practical teaching situations in the following activities: - Concept forming activities, - Theoretical formation activities, - Theoretical manipulation activities, - Problem solving activities
Step 2: Observing reality
We observe practically that there are quite a lot of images about straight line perpendicular to plane in construction (plumb line), in design (revolving door), the image of flagpole in school yard, image of table legs, chair legs, picture of notebook on the table,
Step 3: Selecting the model that is suitable for the teaching and building goals
We choose a model of a notebook on the table with a pen Teaching objective is the concept of straight line perpendicular to a surface Building the situation as following:
Trang 12The perpendicular situation
1 Please observe the following image and can you comment
on the relative position of the "nape" of the notebook and pen?
Explain?
2 If there is a ruler on the table, what is the relative position
between the "nape" of the notebook and the ruler? Explain?
3 What do you comment on the "nape of the notebook" with
the tabletop?
Step 4: Discussing and adjusting the situation
Most of the interviewed teachers said that the situation has clear, logical, and realistic goals However, we still receive the following comments: Need to clarify which "nape line"
in the situation is? Questions need to change the phrase "relative position" to "calculate the angular measurement" between two straight lines so that students use old knowledge and definition approaches
FEEDBACK ABOUT THE REQUEST OF PRACTICAL SITUATION IN TEACHING STRAIGHT LINE PERPENDICULAR TO PLANE
Request Agree Vote
(Ratio %)
Disagree Vote (Ratio %) Other comments The teaching objective of
situation is clear 121 (90%) 13 (10%) - Can use images of straight line perpendicular to plane
such as flagpoles, table legs
- Questions should focus on calculating the angle between two straight lines instead of determining the relative position, to help students, mobilize old knowledge and approach new concepts
- Clarify what is the "nape line"? Can explain in the picture
- During the teaching process,
it is recommended to use dynamic software to show the angle between the nape line and the line lying on the table
The figures in the situation are
logical and accurate 104 (77%) 30 (23%)
The question in situation is fit
Students can solve problems in
mathematical model of the
Students explain practical
problems after completing the
The perpendicular situation (Adjusted)
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1 Please observe the following image and indicate
the corner of the "nape line" (red arrow) of the notebook and
pen?
2 If there is a ruler on the table, what is the angle
between the "nape line" of the notebook and the ruler?
3 What about the angle between the "nape line" of
the notebook and the other straight lines on the table?
4 Predicting the relative position of the "nape line" of the notebook with the tabletop?
Step 5: Testing the situation
Observing students’ experience in the situation, we found that the students could not answer question 1 In question 2, students said that "if the ruler is close to the nape line of the notebook, the angle will be equal to 900, other straight lines cannot measure the angle” However, the students all predict the "nape line" perpendicular to the tabletop but could not explain why
We support students in solving problem and approaching definition, through the following questions:
Question 1: Ignoring the factors of thickness,
width of notebook, pen, ruler, table, called "the nape
line of the notebook" is the straight line d, the notebook
part, the tabletop is plane, the pen is straight line
Please determine the mathematical model of practical
situation
Expected answer: Given two rectangular
Determine the angle between line and plane
containing line
Question 2: How to determine the middle angle ?
Expected answer: Rotating plane around the axis so that
Question 3: What is the angle between line and any lines inside ?
Expected answer: Do the same as above, we determine the angle between line and any straight line in is
Question 4: What is the relative position of line with every line in ?
Expected answer: Line perpendicular to every line in
Question 5: Predict the relative position of d and plane ?
Expected answer: Line d perpendicular to plane
Step 6: Confirming the situation
Through the process that we observe while students test the situation, we realize that although there are students who have not solved the situation Thanks to the teacher’s support, students have solved the situation completely and they confirm the definition by themselves; so the situation is feasible and can be applied in teaching
2.4.2 Practical situation used in theoretical teaching
Example 2.5 Design teaching situation for theorem 3 about "sum of infinite