Research purposes: Based on theoretical and practical research related to issues of language, mathematical language, thinking, the relationship between language and thinking, mathematical language and mathematical thinking, we propose some measures for developing mathematical language to contribute to improving the quality of teaching for pre-university students in the Central Highlands.
Trang 1MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY
KIEU MANH HUNG
CONTRIBUTING TO THE DEVELOPMENT
OF THE MATHEMATICAL LANGUAGE FOR PRE-UNIVERSITY STUDENTS
IN THE CENTRAL HIGHLANDS
Major: Reasoning and methodology of teaching mathematics
Code: 9140111
SUMMARY OF DOCTORAL THESIS IN SCIENCE EDUCATION
NGHE AN - 2020
Trang 2The thesis was completed at Vinh University
Supervisors:
1 Dr Nguyen Van Thuan
2 Assoc Prof Dr Nguyen Thanh Hung
Reviewer 1: Assoc Prof Dr Vu Duong Thuy Reviewer 2: Assoc Prof Dr Dao Thai Lai Reviewer 3: Dr Nguyen Huu Hau
The thesis will be defended at the Boards
of Examiners of Univeristy level, at Vinh University, No 182, Le Duan Street,
Vinh City, Nghe An Province,
Time: At , date month year 2020
This thesis can be found at:
1 National Library of Vietnam
2 The Information Center - Library of Nguyen Thuc Hao, Vinh University
Trang 3INTRODUCTION
1 Rationale
1.1 By studying theoretical and practical teaching, we find that pre-university
students master the knowledge and skills of mathematics Once they master the
mathematics language system, they will be able to use this language system in the thought process, reasoning in solving math problems and putting them into practice In addition, language difficulties are a significant barrier to the acquisition and application
of scientific and technical knowledge, especially for highly abstract scientific fields such as mathematics The results of the grassroots research topics that the author had done in 2009, 2013, 2015, 2016 and 2018 showed that pre-university students still have many limitations of expression when solving the problem
1.2 Flexibly understanding and applying concepts, theorems, consequences,
properties, to solve math issues successfully is not an easy task But how to present the theoretical content briefly, concisely and highlight content to facilitate the use of them in mathematical reasoning is much more difficult
1.3 The Math curriculum for pre-university students do not have content
specifically to introduce and teach knowledge related to mathematical language The
knowledge is introduced in an implicit manner in teaching process, in accordance with the students' level of knowledge in order to serve mathematical reasoning as well as apply it to other science subjects This shows that teachers have to pay attention to fostering self-studying skills for pre-university students so that these skills can be used
as a means to serve the thinking and reasoning process
1.4 The mountainous areas in our country in general and the Central Highlands
in particular are places where socio-economic conditions are still facing many difficulties There are still many children of ethnic minorities with low educational levels and uneven knowledge Through the teaching process at the pre-university Department of Tay Nguyen University, we found that the pre-university students in the Central Highlands are mainly ethnic minority people with many different languages, rituals and customs In general, they have many difficulties in learning subjects in general, and Maths in particular
Trang 4The lecturers teaching mathematics always try to let students know how to interpret the definitions, theorems and problems, from ordinary language to mathematics language and vice versa for the purpose of consolidation and applying knowledge However, in fact many pre-university students in the Central Highlands are still confused and encountered many mistakes when performing the above tasks This greatly influences the acquisition of knowledge, mathematical reasoning and the development of logical thinking
1.5 The task of the pre-university training system is to help students
consolidate, systematize and better understand the basic knowledge of the high school program, build learning methods and self-study methods In order to help pre-university training students confidently study Maths at the University and College levels later, it is necessary to practice and develop mathematical language during the studying period at pre-university From that awareness, the proposal of pedagogical measures in teaching to develop the language of mathematics in learning Maths is meaningful and practical work The study of this issue contributes to improving the learning results of Mathematics for pre-university students in the Central Highlands
in particular and pre-university students in general
Start from the above reasons we study the thesis “Contributing to the developing
mathematical language for pre-university students in the Central Highlands”
2 Research purposes
Based on theoretical and practical research related to issues of language, mathematical language, thinking, the relationship between language and thinking, mathematical language and mathematical thinking, we propose some measures for developing mathematical language to contribute to improving the quality of teaching for pre-university students in the Central Highlands
3 Subjects and scope of the research
3.1 Research subjects: Measures to develop mathematical language for
pre-university students in the Central Highlands
3.2 Scope of the research
Scope of time: The Doctoral thesis collects data on students in two courses
K2016 and K2017 of pre-university Department of Tay Nguyen University
Trang 5Experiments on pre-universityclasses of blocks A and B, two courses of K2017 and K2018 of pre-university Department of Tay Nguyen University
Scope of space: The Central Highlands
Scope of content: Mathematical language in the curriculum of Mathematics
which is used for pre-university students
4 Research topic
- Theoretical study of language, mathematical language, thinking, mathematical thinking, the relationship between language and thinking, the relationship between mathematical language and mathematical thinking, development mathematical language
- Research content and curriculum of Mathematics using for pre-university students
- Researching the development of thinking and language of pre-university students in the Central Highlands
- Studying on the situation of the use of mathematical language in teaching mathematics at the pre-university
- Proposing a number of pedagogical measures to develop mathematical language for pre-university students in the Central Highlands in teaching Maths
- Pedagogical experiment to test the effectiveness and feasibility of the proposed pedagogical measures
5 Research method
5.1 Methods of theoretical research
We use a combination of research methods: collecting information, documents, analyzing, synthesizing, etc to study the theories of: language, mathematical language, thinking, mathematical thinking of Pre-university students of blocks A, B
At the same time researching Math content, programs and subjects of Maths which is used for pre-university students
5.2 Practical research methods
Coordinating the practical research methods to clarify the situation and test the effectiveness and feasibility of the Doctoral thesis
5.3 Information processing method
Trang 6Using statistical methods to process data after investigating the situation, data of pedagogical experiment process
6 Scientific hypothesis
In teaching mathematics for pre-university students, if building and implementing a number of teaching methods such as: Fostering knowledge of syntax and semantics (namely consolidating vocabulary, semantics, syntax, internal conversion capacity training in one language, converting from one Math language to another); Practice using mathematical language in typical teaching situations (specifically in conceptual teaching - theorem, in teaching rules - methods and in teaching math solving); Practice mathematical communication skills (listening, speaking, reading and writing skills); Developing mathematical language through active teaching methods (problem-solving method, role-playing method, game method and group work method) will contribute to university pre-university students' development of mathematical language, through then improve the quality of teaching
- learning Math for pre-university students in the Central Highlands
7 The contributions of the Doctoral thesis
Systematizing some theoretical issues about language, mathematical language, thinking, mathematical thinking, development of mathematical language
Analyzing the problem of mathematical language in the content of the mathematical program for pre-university students
Find out the situation of using the mathematical language of the pre-university students in the Central Highlands
Proposing 4 groups of measures to contribute to the development of mathematical language for pre-university students in the Central Highlands
8 Supporting contents for Thesis
- Concepts of language, mathematical language, thinking, mathematical thinking, development of mathematical language of pre-university students in the Central Highlands
- Pedagogical measures to contribute to the development of mathematical language for pre-university students in the Central Highlands
- The results of the pedagogical experiment
Trang 79 The structure of the Thesis
In addition to the Introduction, Conclusions and References, the Doctoral thesis
is presented in three chapters:
Chapter 1 The theoretical and practical basis
Chapter 2 Developing mathematical language for pre-university students in the
Central Highlands
Chapter 3 Pedagogical experiments
Trang 8CHAPTER 1 THE THEORETICAL AND PRACTICAL BASIS
In 1986, Andrew Waywood studied the influence of mathematical language on junior high school students
In 1986, Martin Hughes in the book "Children and Numbers" proposed a
perspective on the early efforts of children to understand mathematics He describes the incredible knowledge of numbers that children know before they start class Understanding of pre-school numbers is an obstacle to learning mathematical knowledge in the classroom
In 1988, in the works "Second international handbook of mathematics
education", two mathematicians Stigler and Baranes mentioned the use of
mathematical language of elementary students in Japan, Taiwan, South Korea and the United States
Pimm (1987), Laborde (1990), Ervynck (1982) confirmed that the use of mathematical language of students in maths is a barrier because the mathematical language is much different from the daily use language
In 1993, Diane L Miller concluded that developing mathematical language has
a profound influence on the development of mathematical concepts [78]
In 1995, Eula Ewing Monroe and Robert Panchyshyn studied the lexical problem of mathematical language, the need for vocabulary in the development of mathematical concepts
Trang 9In 2007, Chard Larson emphasized the role of mathematical vocabulary in the understanding and learning of junior high school students He believed that mathematics is a language and students who want to master it must be able to use and understand vocabulary By using vocabulary quizzes and vocabulary-related activities taken from math, students will better acquire an understanding of mathematical concepts [75]
In 2008, Charlene Leaderhouse studied the mathematical language in the subject
of Geometry He studied the mathematical language of 6th grade students in learning geometry and concluded that the ability to understand and correctly use mathematical terms will help them master the mathematical concept To study geometry well, children need to have many opportunities to discuss ideas and practice in teaching which uses mathematical language [80]
In 2008, Bill Barton [74] concluded that everyday mathematical ideas were expressed differently in different languages Diversity occurs in the way language expresses numbers, the language that describes the position of numbers and the grammar of mathematical content expressions
In 2009, Rheta N Rubenstein researched the issues of how to help teachers teach mathematics in high school to recognize the challenges that students often encounter with mathematical symbols to propose teaching strategies that can alleviate those difficulties The study proposed solutions to help teachers know how to use different symbols and identify common difficulties that students often encounter when they speak, read and write symbols; At the same time, he also provided teaching methods to avoid or overcome these difficulties [93]
1.1.1.2 Domestic studies
In 1981, Pham Van Hoan, Nguyen Gia Coc and Tran Thuc Trinh affirmed that the correct expression of the relationship between “mathematical ideology content” and “mathematical language form” wais the basis of important methodology of mathematical education [30, p 93]
In 1990, Ha Si Ho presented some concepts and characteristics of mathematical language Accordingly, the language of mathematics is primarily the language of
Trang 10using signs, not the language of "speech" as in the language of mathematics The major mathematical language is the "written" language, which is both tight and flexible [31, p 45]
In 1992, Hoang Chung studied mathematics language and teaching mathematical notation in high school
In 1998, authors Ha Si Ho, Do Dinh Hoan and Do Trung Hieu mentioned many aspects of mathematical language Accordingly, it is necessary to have a language suitable for expressing mathematical content, and at the same time overcome the disadvantages of mathematical language [32]
In 2004, in the Thesis "Contributing to developing the capacity of logical thinking and correct use of mathematical language for high school students in algebra teaching", the author Nguyen Van Thuan proposed the pedagogical measures: Set students to express some definitions and theorems in different ways; Train students to use correct transformations; Practice using terms and symbols of mathematical logic
to express mathematical propositions [57, p 82-135]
Recently, there are many direct and indirect studies on languages in teaching high school mathematics, such as Tran Ngoc Bich [4], Vu Thi Binh [5], Thai Huy Vinh [63],
pre-university students in the Central Highlands are mainly ethnic minority people in Ede In recent years, there have been many studies on Ede language from a linguistic perspective, such as: Malyo - Polynesian languages in Vietnam of Romal Del and Truong Van Sinh [21]; Doctoral thesis of Doan Van Phuc (2009) with the topic Phonetic of Ede language [46]; The doctoral thesis of Linguistics by Truong Thong Tuan on the subject of Comparative method in the customary language of Ede [61]; Doctoral thesis of Linguistics by Nguyen Minh Hoat (2012) with the subject of type nouns in the Ede language
The doctoral thesis of Linguistics by Doan Thi Tam (2012) with the topic of the system of human words in Ede language However, these works are only studied from the perspective of the Ede language - the mother language of most of the pre-university students in the Central Highlands
Trang 111.1.2 Language
1.1.2.1 Concept of language
In this thesis, we agree with the concept of language of author Nguyen Thien
Giap in [22, p 28]: "Language is a special audio signal system, a means of
mechanical communication and the most important of the members in a community
of people; language is also a means of developing thinking, conveying historical traditions from generation to generation”
cultural-1.1.2.2 Function of language
Communicative function
Language is the most important means of human communication, helping people understand each other in the life and labor process; is a production tool, a class struggle tool
Function to show thinking
Language is the actual expression of thought, directly involved in the process of thought formation Human language exists in the form of speech (sound symbols in the brain) and writing Therefore, the reflection of the language does not only indicate when the language is spoken into the word, but when silent thinking or writing the paper
1.1.2.3 Nature of language
1.1.2.4 Characteristics of language
1.1.3 Mathematical language
1.1.3.1 A brief history of the development of mathematical languages
1.1.3.2 Concept of mathematical language
a Concepts
In this thesis, we agree with the viewpoint on mathematical language of author Nguyen Duc Dan in [20]: "Mathematical language includes symbols, terms (words,
Trang 12phrases), symbols and the rules that combine them as a means to express mathematical content in a logical, accurate, and clear manner Symbols include numbers, letters, alphabetic characters, mathematical signs, relational signs, and parentheses are used in maths Symbols include images, drawings, diagrams or models of specific objects ”
b Semantics and syntax
1.1.3.3 Function of mathematical language
a The language of mathematics is a direct realization of thinking
b The language of mathematics reflects thinking
c Mathematical language is a means of communication in mathematical activities
1.1.3.4 The role of mathematical language in the Math curriculum for college preparatory students
1.1.4 Thinking and Mathematical thinking
1.1.4.1 Thinking
1.1.4.2 Mathematical thinking
1.1.4.3 Some manipulations of mathematical thinking
a Analytical and general operations
These are two contradictory operations, analysis to find the solution of problems, synthesis is a process of discovering relationships that unify parts that seem to be separate This is a method of putting assumptions together to find solutions
b The similarity
According to G Polya: "The two systems are similar if they fit together in well
defined relationships between the corresponding parts" [26, p 23] According to
Hoang Chung: The similarity often means the same People often consider similar problems in mathematics in the following aspects: Two proofs are similar if the way and method of proof are the same; The two figures are similar if they have many similar properties, if their roles are the same in certain problems or if their respective elements are the same [16, p 8-9]
Trang 13C Generalization, specialty
Generalization and specialty are two different thinking actions that are contrary Specialization is the presentation of specific cases, particular cases of the problem Generalization is also generalizing the problem from specific cases and particular cases
d Comparison
Comparing mathematical objects helps students identify similarities and differences of two or more objects Regularly comparing students will have a more comprehensive view of the problem We often perform comparisons such as: compare two concepts, compare two definitions, compare two issues,
1.1.5 The relationship between language and thinking, mathematical language and mathematical thinking
1.1.5.1 The relationships between language and thinking
Language and thinking are a unified but not a consistent relationship The language exists in the form of material while thinking is the form of spirit Language
is perceived by humans by senses such as pitch, field, tone, etc and thinking is the inner awareness of the human brain in a certain logical order The language is nationalistic (the product of the nation) while thinking is humankind (all countries have the same products of thinking about the problem: sovereignty, peace, education, health, ) [22]
1.1.5.2 The relationship between mathematical language and mathematical thinking
Mathematical language is both a tool and a material shell of mathematical thinking Mathematical language and mathematical thinking are a unified but not consistent This is reflected in the fact that mathematical language exists in physical form, mathematical thinking exists in mental form Units of mathematical language are perceived by the senses and have physical properties such as pitch, intensity, etc And mathematical thinking is not perceived by such senses, there are no properties
of matter such as mass, weight, taste, etc The activity of mathematical thinking requires rational and logical while the mathematical language operates according to