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

- What kinds of errors have students committed when using analogical reasoning inthe chapter “Coordinate method in space”?. Research question 2: How is analogical reasoning used on “Coor

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 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 similarproblems 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 todiscover new knowledge; this process enhances their independent thinking, theircritical thinking and their creative abilities

Analogical reasoning plays an important role in science, in particular, inmathematical education Analogical reasoning is used to build the meaning ofknowledge, formulate hypotheses in the teaching method of discovery, predict andprevent students’ errors, and solve mathematical problems Analogical reasoning wasstudied 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 inhigh 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 andtheir similar solutions Therefore, we asked ourselves the following interestingquestions:

- Have the authors of current Geometry textbooks used analogical reasoning topresent the contents in the chapter “Coordinate method in space”?

- When using analogical reasoning in the textbooks, do the teachers in high schoolsand the pre-service teachers use analogical reasoning as a strategy to enhancestudents' positiveness?

- What kinds of errors have students committed when using analogical reasoning inthe chapter “Coordinate method in space”?

- What are effective methods used when teaching “Coordinate method in space”with analogical reasoning?

From the above reasons, we chose the topic of the thesis as follows:

“Analogical reasoning in teaching mathematics in high schools: the case study

of Coordinate method in space”.

2 Theoretical frameworks and research tasks

Our study was carried out in the range of theories about analogy, analogicalreasoning and teaching with analogy A number of didactic theoretical tools wereused 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 areconclusions of the situations using analogical reasoning are drawn in the currentGeometry 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 service teachers influenced?

pre-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 toverify 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 usinganalogical 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, theroles of analogical reasoning in teaching, the classification of analogical reasoning

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and the teaching models with analogical reasoning such as GMAT model, TWAmodel, 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 todiscover concepts, teaching how to discover theorems, teaching how to solvemathematics problems; predicting errors of students by analogy sources beforeteaching, analyzing and detecting errors and correcting errors

- Listing some errors of students when using analogical reasoning to solvemathematics problems in the chapter “Coordinate method in space”

- Developing the solutions and teaching processes of using analogical reasoning inspecific contents in the chapter “Coordinate method in space”, and mathematics ingeneral

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 withanalogy designed by teachers and pre-service teachers in the chapter “Coordinatemethod 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 inspace 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

CHAPTER 1 THEORETICAL FRAMEWORKS

This chapter analyzed, synthesized, and systemized the views of analogy,analogical reasoning, the roles of analogical reasoning in teaching mathematics andteaching 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 ofanalogy 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 areanalogical if they clearly fit together in the relationships and identify between tworespective 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 othersimilar 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 iscalled the source In the other hand, the thing explained or learnt through analogicalreasoning is called the target The expected conclusions drawn by analogicalreasoning are just hypotheses, so their correctness needs to be verified clearly

In the thesis, we considered analogical reasoning as a kind of reasoning from thesame 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

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sources and targets in different domains, analogical reasoning with sources based onthe 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 analogicalreasoning 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 analyzingtextbooks in chapter 3

Table 1.1 The classification of analogical reasoning used in textbooks

The analogical

relationship

between source

and target

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

Pictorial-Verbal: The analogy is presented in a verbal format

along with a picture of the source

The level of

abstraction of

the source and

target concepts

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

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

The level of

enrichment

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

The limitations

of the analogy

Misunderstandings are showed

Misunderstandings are not showed

1.2 The roles of analogical reasoning in teaching mathematics

Analogical reasoning is used to construct the meaning of knowledge, buildhypotheses, 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 authoremphasized the importance of a plan before using analogical reasoning to helpstudents 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 analogyFAR model in order to teach more efficiently

Focus:

Concept Is the concept taught difficult, unfamiliar or abstract?

Students What knowledge have students already known about the concept? Analog What things are familiar with students?

Action:

Similarities Discuss the similar characteristics of source and concept

Differences Discuss the different characteristics of source and concept

Reflection:

Conclusion Is the source clear, helpful or confusing?

Reflection 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

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CHAPTER 2 THE METHODS AND RESEARCH DESIGNS

This chapter mentioned the research methods to answer the questions stated inthe 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 inspace Analyze analogical reasoning used on current geometry textbooks according tothe 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 inplane” and “Coordinate method in space” according to the views of mathematicaldidactic: T is the task,  is the technique, is the technology Specifically analyzeseveral mathematics organizations as a basis for using analogical reasoning to solvemathematical 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 analogicalreasoning 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 Inorder to evaluate the level of using analogical reasoning, we used the rubric forevaluating in Table 2.1

Table 2.1 The rubric for evaluating using analogical reasoning in teaching

Level Levels of using analogical reasoning in teaching

0 Do not use analogical reasoning

1 Only talk about the source

2 Recall characteristics of the source, but do not set up any correspondence between the source and the target.

3 Know how to set up correspondences between the source and the target

4 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:

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 inspace?

2 What difficulties can students meet when they use TWA model to designlesson 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 ayear 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 teachershaving two years before graduation

Step 1: We introduce analogical reasoning to pre-service teachers with TWA modeland an illustrated example Then, they work in groups (from 3 to 4 students) in 60minutes and use TWA model to prepare lesson plans for teaching concepts, propertiesand 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 errorsinvolved 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

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The method of research: A priori analysis: specifically analyze some errors

when implementing tasks in Coordinate method in space A posterior analysis: asurvey forms for 309 students and interview 6 students to clarify how they useanalogical 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 Gradestudents (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 researchquestions 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 Coordinatemethod 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 analogicalreasoning on Advanced Geometry 10, 11, 12 for the advanced curriculum Weclassified these cases in table 1.1

3.1.3 Analogical reasoning in the chapter “Coordinate method in space”

The authors of textbooks used analogical reasoning in 4 cases: prove b n   0, theparametric equation of a straight line in space (Geometry 12), system of coordinateaxis 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 newformula without its proof; apply an earlier similar proofing way Moreover, the

authors of textbooks only used analogical reasoning to guide and present newknowledge, 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 inspace 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 thesolutions of new problems in Coordinate method in space

3.3 Conclusion of chapter 3

The above analysis showed that there were analogical relationships betweenconcepts, properties and mathematical organizations in Coordinate method in planeand in space The authors of textbooks used analogical reasoning to help students toreview old knowledge, guide and present new knowledge, but they did not mentionactivities of discovering new knowledge

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 analogicalreasoning: one lesson was about the general equation of a plane, another was aboutthe parametric equation of a straight line and the other three lessons were about theformula of calculating the distance from a point to a plane The contents whichteachers used analogical reasoning had many similarities with contents mentioned byauthors of textbooks The teachers used analogical reasoning to help students tohypothesize about new knowledge, explore and solve problems, create motivation tobegin the new lesson Some questions and activities were developed from thepresentations of textbooks It was clear that there was the impact of analogicalreasoning in the textbooks for the teaching process of teachers

4.2 The survey of pre-service teachers

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 resultsshowed that both working individually or in groups, students had to prioritizeanalogical 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 designsuitable 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 Thiswas due to the impact of the presentation of analogical reasoning in the currenttextbooks

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 astraight 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 wouldcommit 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