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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 8t
x 3 y 1 z 1
a. and being parallel to d : y 5 2t , d ' :
.
7
2
3
z 8t
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.

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