Training and developing algorithmic thinking for students in technical universities though the course of descriptive geometry

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION HOANG VAN TAI TRAINING AND DEVELOPING ALGORITHMIC THINKING FOR STUDENTS IN TECHNICAL UNIVERSITIES THROUGH THE

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION

HOANG VAN TAI

TRAINING AND DEVELOPING ALGORITHMIC THINKING FOR STUDENTS IN TECHNICAL UNIVERSITIES THROUGH THE

COURSE OF DESCRIPTIVE GEOMETRY

Major: THEORY & METHODOLOGY OF MATHEMATIC EDUCATION

Code: 62 14 01 11

IN SCIENCE EDUCATION

HA NOI – 2016

The work was completed at:

Department of Mathematics - Hanoi National University of Education

Scientific supervior: Prof Bui Van Nghi PhD

Reviewer 1: Assoc Prof Trinh Thanh Hai PhD

Thai Nguyen University of Sciences

Reviewer 2: Assoc Prof Nguyen Xuan Thao PhD

Hanoi University of Science and Technology

Reviewer 3: Assoc Prof Nguyen Anh Tuan PhD

Hanoi National University of Education

The dissertation will be defended before the Council of dissertation assessment

or at: Hanoi National University of Education

At: on ……/……/ 2016

The dissertation can be further referred at:

- National Library of Vietnam

- Library of Hanoi National University of Education

PREAMBLE

1 Reason of study

+ Improvement of learner’s capability: Conference of UNESCO in 2003 presented a report which analyzed clearly significant changes on the need and demand of knowledge society for students, especially capability of problem solution and innovation of thought

+ Role of Descriptive geometry in Technical universities: Helping learners to present and read drawings, and build up the cooperation and creativity in career

+ Practical teaching of descriptive geometry shows that: Although this course is very essential for the profession, its results of teaching and studying are not high One of the reasons is the method of teaching and studying, of which students do not grasp the algorithm in each solution If an appropriate method is applied, this weakness will be improved to foster the effectiveness of teaching and studying

+ Development of thinking for students: To understand and solve problems

on descriptive geometry, students are not only good at spatial imagination but also

be able to solve problems in a logical and accurate manner and well apply procedures, basic mathematical problems and other rules of basic procedures and problems In addition, students are encougared to propose the alternative ways to solve mathematical problems by different procedures All those things create a type

of thought, called Algorithmic Thinking It is not only necessary for the course of descriptive geometry but also for the life

+ Regarding to the relevant researches: There are several researches on the development of innovative thinking, logical thinking, algorithmic thinking for students, but they do not mention about training and developing algorithmic thinking for students of technical universities

For the above mentioned reasons, the chosen subject is “Training and

developing algorithmic thinking for students of technical universities though the course of descriptive geometry”

2 Scientific theory

According to the theoretical and practical base on development of algorithmic thinking for learners, if during the course of descriptive geometry, trainers equip students the basic algorithm and create opportunities for them to propose algorithm as well as improve gradually the level of algorithmic application, students shall have better learning outcomes and develop their algorithmic thinking

3 Goal and mission of study

+ Goal: Proposing methods of training and developing algorithmic thinking for students of technical universities through the course of descriptive geometry,in order

to help students with better learning results and development of algorithmic thinking

+ Mission of study: To gain the above goals, the missions include:

(1) Brief introduction on thinking; algorithmic thinking and its role, through published scientific documents

(2) Practical investigation on the studying of descriptive geometry and development of algorithmic thinking for students of technical universities

(3) Proposing methods of training and developing algorithmic thinking for students of technical universities through the course of descriptive geometry,in order to help students with better learning results and development of algorithmic thinking

(4) Implementation of pedagogical experiment to evaluate the posibility and effectiveness of the study

4 Research method

Main methods applied in this thesis are:

+ Theoretical studies (performing tasks (1), (3));

+ Survey and Observation (performing tasks (2), (4));

+ Pedagogical experiment (performing tasks (4);

5 Objects and scope of study

- Object of study is a process of teaching descriptive geometry, training and developing algorithmic thinking for students of technical universities

- Scope of study: Content, teaching program of the descriptive geometry course in the technical universities

6 New contribution of the study

+ For theoretical Perspectives

- Generalize the domestic and abroad researches and systematize theoretical perspectives on algorithm, algorithmic thinking and development of algorithmic thinking in teaching mathematics

- Actual situations on training and developing algorithmic thinking for students in teaching and learning the course of descriptive geometry in technical universities

- Propose possible and effective solutions for training and developing algorithmic thinking for students in teaching and learning the course of descriptive geometry in technical universities

+ For pracical perspectives

- Study results contribute to the innovation and improvement of teaching and learning quality of descriptive geometry in technical universities

- It is a useful reference document for colleagues and students in technical universities

7 Defended issues

(1) There are domestic and aboard researches on algorithm, algorithmic thinking and development of algorithmic thinking in teaching Mathematics, Informatics, Computer Science, however the issues of training and developing algorithmic thinking for students in technical universities during the course of descriptive geometry has not been studied yet

(2) There are some shortcomings on teaching and learning descriptive

geometry in technical universities that affect the teaching effectiveness and quality

of this course

(3) Mesures to train and develop the algorithmic thinking for students in technical universities during the course of descriptive geometry proposed in this research are possible and effective

8 Study structure

Besides preamble and conculsion, this thesis consits of 03 chapters

Chapter 1: Theoretical and practical base

Chapter 2: Measures to train and develop algorithmic thinking for students in teaching Descriptive geometry

Chapter 3: Pedagogical experiment

Chapter 1 THEORETICAL AND PRACTICAL BASE 1.1 Brief of study

1.1.1 Abroad researches on algorithm and algorithmic thinking

1.1.1.1 For algorithm and teaching algorithm

* Research on the appearance of “algorithm”, Morten Misfeldt (2015) indicated that: The appearance of the algorithm is associated with the birth of Mathematics Evgeniy Semakin Khenner and Igor (2014) stated: The algorithm describes the sequence of actions (plan), which are performed strictly according to the instructions to solve the problems in a finite number of steps According to Robert J Sternberg (2000), in daily life, we have learned some algorithms and ocassionally created it to guide others to do something

* Research on teaching algorithm, Evgeniy Semakin Khenner and Igor (2014) stated: The algorithmic teaching has also appeared very early, in the form of puzzle or fun maths The book of Levitin Anany (2008) presented many algorithms and exercises with programming puzzles and algorithms The book of Thomas H Cormen (2009) introduced the algorithm 3E, which is used at many universities

worldwide Marasaeli, Jacob perrenet, Wim M.G zwaneveld jochems and Bert (2011) has proposed four abstract levels in the algorithmic thinking of students corresponding to those of algorithm as follows: (1) Implementation level; (2) Program level; (3) Object level; and (4) Problem level

Studies of algorithmic thinking in a foreign country are consistent with the concept of algorithm in Informatics According COMAP (Consortium for Mathematics and Its Applications) (1997): "Algorithmic thinking" is one kind of a mathematical thinking The expression of algorithmic thinking is: Application of algorithm; Development of algorithm; Analysis of algorithm; Noting the problem without algorithmic solution According to Gerald and Julia Moschitz Futschek (2011), algorithmic thinking is an important capability in Informatics that can be separated with the learning of computer programming

1.1.2 Domestic researches

1.1.2.1 For algorithm and teaching algorithm

In essence, each calculation, rules for calculation and solving the equations are algorithm In Geometry, there are some algorithms such as: drawing with a ruler and compass At university, algorithms are also found, for example: calculating the definite, higer equatations, matrix inversion and determinant… Nguyen Ba Kim and Vu Duong Thuy (1992) defined the algorithm as followings:

“The algorithm is considered as a descriptive rule of the clearly accurate instructions helps people (or machines) to perform a series of actions with the aim

of achieving its propsed goals or solving a certain problem It is not an exact definition but merely a statement which helps us to imagine the concept of algorithm intuitively” Bùi Văn Nghị (1996) used the definition on algorithm of the

two above authors and added the concept “algorithmic procedure” Vương Dương

Minh (1996) studied “Development of algorithmic thinking for students while teaching numeration system in high schools" The author has given a definition of

algorithm as follows: "Algorithm is an accurate and simple rule of limited numbers

of primary actions following a definite order specified on the object so that we will obtain desired results after perporming that procedure” Some authors also

identified the two concepts, "algorithm" and "algorithm" such as works of Chu Cẩm Thơ (2015), Nguyễn Chí Trung (2015)

1.1.2.2 For algorithmic thinking and development of algorithmic thinking

There are domestic researches on development of algorithmic thinking for students For instance, a research of Vũ Quốc Chung (1995) on fostering capacities

of thinking for students in the final grade of primary school; a work of Nguyễn Thái Hòe (1997) on training the thinking for students via mathematic exercises; works of Nguyễn Đình Hùng (1996), Nguyễn Văn Thuận (2004) on developing logical thinking for students; awork of Tôn Thân (1995), Trần Luận (1996) on fostering creative thinking for students

Among the domestic researches on algorithm and algorithmic thinking, it can be counted for Trần Thúc Trình (1975), Nguyễn Bá Kim (1992, 2011, 2015), Vương Dương Minh (1996) và Bùi Văn Nghị (1996)

Nguyen Ba Kim (2011) suggested that algorithmic thinking is shown in the

following activities: (i) Implementing the activities following the certain order in accordance with a provided algorithm; (ii) Anlalyzing an activity based on performance of its components in a certain order; (iii) Describing exactly the process of conducting an activity; (iv) Generalizing an activity on a group of objects from an activity; (v) Comparing different methods to perform the same work in order to find the optimal solution

Based on the research results on algorithm and algorithmic thinking, the conclusion is summarized as follows:

- The domestic and abroad authors agree with the concept of algorithm in Computer Science and Informatics However, the researchers in mathematics education in domestic schools only concern about the concept of algorithm in

intuitive manner Meanwhile, researchers in Computer Science and Informatics can not stop at this limit, especially when they need to prove the non-existence of an algorithm to solve a problem; an algorithm based on the Turing machine or recursive function are required

- It is nesscessary to distinguish algorithm in science from algorithm in daily life If a solution process does not consit of specific and clear actions to gain a good result, it only is considered an algorithmic-like process

- Many abroad authors assumed “algorithmic thinking” in the meaning of strict in Computer Science and Informatics; some domestic authors considered algorithmic thinking as an algorithmic-like process

1.2 Concepts on algorithm and algorithmic thinking in this thesis

1.2.1 Algorithmic concepts

In this thesis, we assume: The algorithm is considered as a descriptive rule of the clearly accurate instructions helps people (or machines) to perform a series of actions with the aim of achieving its propsed goals or solving a certain problem

1.2.2 Algorithmic thinking concepts

We assume that: Thinking is a cogitative way to perceive things, phenomena, and the natural and social relationships and human that is expressed through notion, judgments, and inference These concepts do not concentrate on the psychological nature of the cognitive process, but appearance (more intuitive) on the thinking Algorithmic thinking is applied to solve problems through not only algorithm but also “algorithmic process" or “algorithmic-like process"

1.3 Descriptive geometry course in technical universities

1.3.1 Brief history of descriptive geometry

Descriptive geometry was introduced by Gaspard Monge (1746-1818) and used in French education system since Century XVIII In Vietnam, since the year 60s of the previous century, when the first universities was established, descriptive geometry was taught officially in Univerity of Technology and Science

1.3.2 Brief introduction of descriptive geometry

Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures This course equips the leaners knowledge and skills to understand and draw the technical drawings Knowledge of descriptive geometry is basic, compulsory and minimum for a student in technical universities

In descriptive geometry, each point A in the space is represented by only a pair of projection (A1, A2) on two planes of perpendicular projection And vice versa, each pair of projection (A1, A2) on two planes of perpendicular projetions identifies point A in space Thus the representation of spatial projection on two planes of perpendicular projections shall totally define the size and shape of geometrical figures All problems of descriptive geometry are problems of the formatting image; every problem has only one answer Hence, application of algorithm to solve the problems of descriptive geometry can be considered

1.3.3 The expression , the level of algorithmic thinking of students and the opportunity to develop algorithmic thinking in teaching descriptive geometry at the Technical University of block

1.3.3.1 The expression, the level of algorithmic thinking of students expressed through descriptive geometry module

Thinking algorithm University students Technical block manifested in descriptive geometry module through the ascending levels of the following:

i) To comply with the basic algorithm known in the course of payment; ii) Imagine , performing the entire process of solving the problem, solve the problem according to the block diagram, process simulation or language, or algorithms written into the program;

iii) Know how to apply these algorithms known during problem solving; iv) May participate in the proposal, design algorithms in the process of accounting;

v) Can select the optimal algorithm in multiple algorithms and solve a problem

1.3.3.2 Opportunities to develop algorithmic thinking in teaching descriptive geometry at the Technical University of block

- Opportunities for the content knowledge in module

- Chance of cognitive abilities of students

- Opportunity to organize teaching methods

1.4 Practical Situation of teaching and learning descriptive geometry in technical universities

1.4.1 Advantages and disadvantages of students in learning descriptive geometry

* Advantage: Basic knowledge of descriptive geometry is based on basic knowledge of Euclidean geometry which was taught in high schools Some drawing softwares such as AutoCad, Cabri, GSP… can be used in teaching and learning descriptive geometry

* Disadvantage: When studying the descriptive geometry, the learners are required to have spatial imagination and logically reasoning ability

1.4.2 Investigating practical situation of teaching and learning descriptive geometry in technical universities

We have designed and used Questionnaire on teaching and learning the descriptive geometry for 250 2nd year students - term 57 and 58 at two educational institutions of the University of Mining and Geology (Hanoi and Vung Tau) in September (one month after learning the course of descriptive geometry) in 2013 and 2014

Results show that: When start learning the descriptive geometry, most of students (80%) reported that this is a difficult subject, the rest 20% stated that this subject is very difficult Many reasons were reported by students as follows For 10% of students, the reason is students must understand thoroughly the knowlgde

in high schools; 25% assumed that they has not found a proper learning methods, in which 15% for teacher’s teaching methods and 10% for timing isues; for 40% thought that it is difficult because of requirement of good spatial imagination Therefore, most of students are not interested in this subject; 20% feel normal and only 15% are excited with that

In conclusion, it is propably stated that the descriptive geometry is quite abstract and difficult for students in technical, civil engineering and architectural universities Also teachers has not concerned appropriately about formity and development of algorithmic process for students, leading to low effectiveness of teaching

1.5 Conclusion of Chapter 1

In the technical universities, descriptive geometry equips students the basic knowledge to understand and draw technical drawings, also contributes to develop spatial imagination, algorithmic thinking, creative thinking for students, engineers, architects, industrial art painter during their work Therefore, teaching the course of descriptive geometry in the direction of training and development of algorthmic

thinking for students of technical universities are justified

Chapter 2 MEASURES TO TRAIN AND DEVELOP ALGORITHMIC THINKING FOR

STUDENTS IN TEACHING DISCRIPTIVE GEOMETRY

2.1 Measures building orientation

(1) Orders of measures should be suitable to procedures of forming and developing algorithmic thinking for students

(2) Measures proposed should be suitable to students and perception process of leaners

(3) Measures should be feasible and effective

(4) Measures aim to innovate the present methods of teaching discriptive geometry

2.2 Basic definitions and knowledge in descriptive geometry

2.3 Methods to train and develop algorithmic thinking for students in teaching descriptive geometry

2.3.1 Method 1: select some basic algorithmics and train stydents to well apply them into basic maths in Descriptive geometry

2.3.1.1 Method base: base on the learners; base on the difficulty of

descriptive geometry course; base on the content of descriptive geometry

2.3.1.2 Method implementation approach

First and foremost, we need to select some basic algorithmics.Those are procedures that problems in Descriptive Geometry will be inferred to If students are trained to be skillfull in those basic Algorithmics, there are more chances for them to solve simple descriptive geometry problems

We selected the following basic algorithmics:

- Determine a point on a line;

- Determine the intersection point of a common line and the projected planes (trace of line);

- Determine the vertical projected plane (projected by) (P) contains a given line a (a 1 , a 2 );

- Determine the true magnitude of a line segment;

- Define a line perpendicular to the plane

Specifically,

Basic algorithmics 1: Determine a point on a line

In descriptive geometry, there are some common problems as follows: determine the intersection point of two intersected lines, determine a point of a given triangle or a given tetrahedron, and identify a point in a generatrix of a cylindrical or conical surface These problems are all defined as determining a point on a line Hence, we decided that the algorithmics to determine a point of a straight line is a basic algorithmics