Some methods to train competence of the reasoning reduction for students in

teaching Mathematics at high school

- Training students to change quickly and briefly, and omit intermediate arguments in

the process of familiar problem solving

- Organizing for students to visualize the general reasoning diagram or visualize the

results of the problem by reducing the analysis of arguments.

- Developing situations to help students quickly transform the form of mathematical

knowledge into different forms to realize the content of mathematical problems.

- Emphasizing the meaning of mathematical knowledge, focusing on semantic

interpretation of formal expressions.

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tion and imagination problem is the operational capacity of the students to link and
imagine mathematical objects and relationships with each other in specific cognitive
situations from which to mobilize relevant knowledge so that the students can solve the
mathematical problems. Training this competence may contribute to developing, expanding
knowledge and fostering exploration methods for students on the basis of existing knowledge.
Thus, the new is formed through the transform of associations and the speed of activation of
associations. Practicing activities to transform the associations between the objects for
students have the effect of developing thinking competence to help students appear intuitive
ideas and discover new knowledge.
1.2.6.2. Competence of the immediate generalization: Mathematical intuition is the
direct cognition of mathematical elements and objects, look like sudden flashes, which may be
the result of the movement of generalized modes and reduced structures. According to V. A.
Krutexki “In many cases, the sudden flash of a student's ability can be explained by the
unconscious influence of past experience, which basis is the ability to generalize mathematical
objects, relationships and calculations and thinking competence with reduced structures”.
Mathematical intuition is the immediate cognition of mathematical objects, relations and
problems due to the reduction of the analytical steps in learners’ awareness process. Therefore,
to want to develop the students’ MIC, the teacher need to train them to be able to generalize for
realizing the nature or the solution of a problem immediately.
1.2.6.3. Competence of the judgement and decision making: Competence of judgement
and decision making is the operational capacity of the students to make assumptions about
mathematical problems or propose an idea for a problem-solving strategy then choice the
appropriate solution for solve the problem. Psychologists have affirmed the role of intuition in
many areas, including diagnosis, creativity, decision making, reasoning, and problem solving.
The authors also emphasize the importance of using mathematical intuition in the problem
solving in unfamiliar situations, therefore learners will learn to think like mathematicians and
gain knowledge in a meaningful way.
1.2.6.4. Competence of the reduction of reasoning: According to Krutexki, “In a
number of the simple form structure only with the students that gifted at mathematics can
build to the relationship between the mathematical problem and its results”. External this
phenomenon seems a “missing reasoning” or “missing thinking”, on contrast, it is the height
of thinking. In this reductive reasoning required some necessary steps to be not missing, it is
only shrank more. This competence can help students shape the path of problem solving for
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new situations or see the results with the shrinking deduction process.
1.3. Organizing cognitive activities for students in teaching Mathematics
1.3.1. Mathematical cognitive activities: According to Dao Tam and Tran Trung,
“Mathematical cognitive activity is a process of thinking that leads to the comprehension of
mathematical knowledge, understanding the meaning of such knowledge: Identifying causal
relationships and other connections of mathematical objects studied (concepts; relations;
mathematical rules; ...); thereby using mathematical knowledge to solve practical problems”.
1.3.2. Characteristic of mathematical cognitive activity: Thinking to control students'
mathematical cognitive activities; Types of logic that adjust cognitive activities; Use forms of
reasoning in mathematical cognitive activities; Pay attention to the characteristics of
mathematical cognitive activities in teaching Mathematics; Enhance the self-awareness,
positive and independent role of learners under the organization and orientation of teachers.
1.3.3. Organization of cognitive activities in teaching Mathematics: Organization of
cognitive activities in teaching Mathematics is the process of teachers searching and selecting
pedagogical methods and methods to guide students to think, leading to comprehending
mathematical knowledge, grasping the meaning. of such knowledge, thereby applying
mathematical knowledge to solve practical problems.
1.4. Opportunities to develop students’ mathematical intuitive competence in
teaching Mathematics at high school
1.4.1. Some theories that direct teaching Mathematics toward to the developing
students’ mathematical intuitive competence at high school such as H. Bergson's Intuition
Theory, Brouwer's Theory Intuition in Mathematics, Multi-Intelligence Theory by H. Gardner
and the perspective of L. Vygotsky's near developing area.
1.4.2. Some teaching ideas toward to developing students’ mathematical intuitive
competence at high school
In order to organize activities to promote the characteristics of general education for
students through teaching mathematics, we emphasize the role of teachers in creating excitement,
stimulating their learning motivation and exploring their learning situations. In addition to
selecting, exploiting and designing teaching contents suitable for problematic and unfamiliar
teaching situations to organize cognitive activities for students, teachers need to focus on some
teaching ideas towards developing general capacity for students at high schools are as follows:
(1) Creating appropriate learning situations for students to implement their task:
Through motivations, teachers need to create learning situations that contain difficulties and
obstacles with learners’ previous knowledge and experience for them to solve the problem in new
situation.
(2) Creating opportunities for students to visualize, think quickly and propose
judgements for the problem before realizing: In teaching Mathematics, through teachers’
questions and requirements, students are discovered problems and judgments about the way
to solve and described about perceiving problems immediately before taking reasoning steps.
Teachers’ encouragement helps students to make different judgments for the problem with
many solutions, different aspects of the problem, finding breakthrough and creative ideas
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from learners. Sometimes teachers have to accept their naive, wrong ideas.
(3) Focusing on developing students with pre-logical thinking skills in mathematics:
through activities, students are able to use imagination, association, generalization and inductive
reasoning.
(4) Shaping a habit for students to grasp the nature of the problem and the solutions, to ignore
detailed reasoning and intermediate steps: Teachers should instruct to students how to present
thinking outline, how to describe the problems’ structure through their reasoning reduction, and
Deepen the meaning, nature of mathematical knowledge for students.
(5): Emphasising the intuitive activities implemented before the deductive activities: In
teaching Mathematics, teachers focus on mathematical intuition as the goal that needs to be realized
first to orient the problem-solving strategy, while logical and deductive reasoning as the means to
examine the results of intuition.
1.4.3. Opportunities to develop mathematical intuitive competence for students in
teaching Mathematics at high school
In studying mathematics, the content and basic learning activities of students are linked
together, aiming at forming their competence. Students' MIC in teaching Mathematics is
demonstrated in the ability to link the content of knowledge of mathematics through learning
activities to visualize the problem, grasp the problem, judge the problem solving situation,
transport situation applied mathematical knowledge. From the above requirement, we found
that the following characteristics need to be considered in order to find opportunities to develop
MIC for students through teaching Mathematics at high school:
- Opportunities to develop competence of immediate generalization and competence of
judgment and decision making are expressed through organizing for students to consider
individual and specific events in the receiving situations. be aware of the problem, know how to
generalize the problem from individual cases, know how to generalize from the general known
to the general unknown; able to judge, detect the general rules of events in math content learning,
be able to offer problem solving orientations, make hypotheses and predictions for math
problems, then choose choose appropriate decisions on problem solving strategies.
- Opportunities for developing competence of association and visualization knowledge
is shown through students' ability to visualize and relate math problems with existing
knowledge and experience , know how to connect and visualize visual images or detect
compatible models of abstract math problems, situations that require learners to discover
knowledge and solutions related to problems new, the ability to imagine their images and
properties, visualize the problem solving path in the face of new situations.
- Opportunities for developing the competence of the reduction of reasoning when
students learn to reason and estimate quickly, have the ability to briefly discuss how to change
the problem, reduce intermediate steps in problem solving process,visualize deductive
diagrams to see the problem solving approach.
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CHAPTER 2
PRACTICAL FOUNDATION
2.1. The purpose and object of the survey
2.1.1. The purpose of the survey
- Research the awareness of maths teachers at high school about the concepts and some
activities of mathematical intuition, the elementary components of MIC, the levels of aware
and application intuitive activities through teaching Mathematics at high school.
- Find the ability of using mathematical intuition of students in the learning
mathematics at high school.
- Find out some else activities or competences expressed such as the elementary
components of MIC through teaching Mathematics at high school.
- Research and analyze the advantages and disadvantages of teachers’ activities in the
process of teaching students access to knowledge in the development of MIC.
2.1.2. The object of the survey: 98 mathematics teachers and 142 students grade 10
at some high school in Dong Thap province.
2.2. The content and organization of the survey
2.2.1. The content of the survey
- Through observation, interview: observe some maths lessons at high school.
- Through teacher survey: design survey forms for teachers.
- Through students’ questionnaire: design mathematical exercises for students grade 10.
2.2.2. The organization of the survey
- Design survey forms for maths teachers.
- Design math homework exercises for grade 10 students.
- Time of survey fom October 10th 2017 to November 12th 2017.
- Collect questionnaires, summarize and analyze data.
- Conduct qualitative and quantitative assessments. Draw initial conclusions about the
development of teacher and teacher's MIC after the survey.
2.3. Result of the survey
2.3.1. Result of the survey: Through observation of some lessons, survey forms for
maths teachers and results students’ questionnaire.
2.3.2. Analyzing result of the survey
Advantages: The majority of teachers have more than 10 years of teaching experience
and some of them have postgraduate qualifications with approaching new research directions.
Teachers have not only suitable contact in teaching but also have effective problem solving
and class organization skills.
Limitations: Teachers are still applying in traditional ways. They are familiar with
introducing and presenting mathematical knowledge without paying attention to explaining the
meaning to help students understand the nature of mathematical knowledge. They are not
required to motivate their students to find solutions, instead of, they usually provide immediate
solution, indicating the step-by-step procedure for new mathematical problem. Many exercises
only require the use of formulas, skills of calculation and manipulation. There are lacks of
questions and exercises visual observation training and prediction detection, quick reasoning
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Step 1
Step 2
skills, not yet fully exploited the situation can develop the ability out of school.
Reason: - For teachers: Teachers are influenced by traditional teaching methods, so
they often teach with their psychology of attempt in short-time to convey more knowledge for
students. Especially in teaching assignments, they only focus on solve as many exercises with
applying the available procedure as possible. The selection of content to develop pre-logic
thinking, especially intuition and imagination is still difficult and not interested properly.
- For students: Students have generally quite low knowledge level, so they are also
afraid of differences and making mistakes. When facing to solve the new problem, their study
habits is expected on solutions or direct instructions from their teacher. Most students still
think in the stereotypes way, learning by memorized knowledge.
- In terms of teaching materials: The literature on mathematical intuition and its
application to the teaching process in Vietnam is not much. There are no research to show
how to implement teaching for the development students’ MIC at high school.
CHAPTER 3
ORGANIZING COGNITIVE ACTIVITIES FOR DEVELOPING STUDENTS’
MATHEMATICAL INTUITIVE COMPETENCE IN TEACHING MATHEMATICS
AT HIGH SCHOOL
3.1. Orientations for the organizing cognitive activities for developing students'
mathematical intuitive competence in teaching Mathematics at high school: Orientation
in contents, teaching methods, orientation with teacher and learner.
3.2. The procedure of organizing cognitive activities for developing students’
mathematical intuitive competence in teaching Mathematics at high school
3.2.1. Science foundation of proposing the organizing cognitive activities for
student: Philosophical foundation, Piaget’s Theory and Vygotxki’s Theory, orientations of
innovation in Program education Mathematics 2018.
3.2.2. The procedure of organizing cognitive activities for developing students’
mathematical intuitive competence in teaching Mathematics at high school: including the
following steps: (1) Creating cognitive situations contained new knowledge, (2) Organizing
for imagine the ideas and conjecture the way to solve the problem, (3) Using deductive
inference to test the results of intuition, (4) Drawing conclusions about new knowledge, (5)
Selecting apply to new situations.
Creating cognitive
situations
contained new
knowledge
Targeting,
selecting,
motivational
learning
Identify students’
problem
Identify problem
space, associate
and mobilize
knowledge
The procedure of
organizing cognitive
activities for developing
students’ MIC
Teacher’s
activities
Student’s
activities
13
Step 3
Step 4
Step 5
3.3. Some ways of organizing cognitive activities for developing students’ the
components of MIC in teaching Mathematics at high school
3.3.1. Some ways of organizing cognitive activities for developing students’ competence
of association and imagination problem in teaching Mathematics at high school
3.3.1.1. Purpose of organization: The organization of training activities to build
association and imagination in order to build a solid foundation to create opportunities for the
appearance of the MIC when the subject realizes before a specific consideration problem. In
new knowledge discoveries, the practice for students to relate to ideas and mobilize knowledge
quickly and effectively helps students have good and accurate intuition from which to form and
develop MIC in teaching Mathematics.
3.3.1.2. Some methods to train competence of association and imagination for students
in teaching Mathematics at high school
- Exploiting similar mathematical properties and objects to help students quickly
identify associate related to the problem.
- Using learning situations to help students develop spatial imagination in order to
detect associations suitable to the problem quickly.
- Exploiting the cause-effect relationship of mathematical knowledge, the relationship
between mathematical knowledge helps visualize the way of solving mathematical problems.
- Using the learning situation for students to transform associations from one object to
another to help them appear ideas and discover new knowledge.
Organizing for
imagine the ideas
and conjecture the
way to solve the
problem
Selecting
approciate
teaching
methods
Organizing
cognitive
activities
Using deductive
inference to test
the results of
intuition
Detect problems,
appear oriented
solutions
Prove, explain
the problem
Fail
Guide to use the
manipulation of
inductive
Drawing
conclusions about
new knowledge
Implement the
solution
Assert new
knowledge
Find new
knowledge and
way to solve
problem
Selecting apply to
new situations
Assess students’
the use of new
knowledge
Apply to solve in
the new situation
Conduct
corresponding
intellectual
activities
14
- Developing situations contained a given visual image to intuitively discover the
nature of a mathematical problem or imagine a way to solve a mathematical problem.
- Developing situations to help students visualize the visual image of the problem,
thereby finding a way to solve mathematical problems.
3.3.1.3. Illustrative examples: Example 3.4. Tthrough teaching the problem solving:
“Give A(-1; 2; 3), B(3; 0; -1), C(1; 4; 7) và the plane (P) – 2 2 6 0x y z . Find the
coordinates of point M of (P) such that
2 2 2T MA MB MC has the smallest value”.
1) Step 1: Create cognitive situation contained new knowledge: The problem of
determining the coordinates of a point on a plane satisfies the given condition, which is quite
familiar to students, but the difficulty of expressing a given minimum value is quite complex.
2) Step 2: Imagine the idea and conjecture the way to solve the problem
+ Students can visualize the position of the point M on the (P) plane relatively close to
A, B, and C so that the small-valued of T, since the point M on (P) goes away from the points
A, B, C the longer the distance from M to those points, the greater the value of T.
+ With the three given points A, B and C, the middle point G of triangle ABC are fully
defined. Observation of the expression has given students the ability to associate to
3MA MB MC MG . Thus, the expression contained G must be transformed into a
simpler expression that defines the point M.
+ Students associate and mobilize the familiar problem solving method “Let point A
not belong to the plane (P). Find the point M on (P) for the smallest MA2”. Find the point M
on (P) such that the smallest MA2 M on (P) such that the MA has smallest value M is
the projection of A on the plane (P).
+ Students discover intuitively problem that the point M needs to find can be the
projection of point G of the triangle ABC on the plane (P). Thus, the solution of the problem.
3) Step 3: Use deductive inference to test the results of intuition
2 2 2
T MA MB MC
2 2 2 2
3 2MG GA GB GC MG GA GB GC
With G(1;2;3) is the center of the triangle ABC.
We have
2 2 2T MA MB MC minimum 2MG minimum M is the
projection of the point G on (P). Finding the projection M of A on (P), we get 0;4;1M
(G,(P)) 3MG d
thus
2
min
3.3 48 75T .
4) Step 4: Draw conclusions about new knowledge and apply to new situations:
Students develop method knowledge by associating the familiar rules to solve the
problem. Teachers can train students with the ability to associate students with a view to
solving problems in a variety of perspectives on problem solving, the ability to develop new
15
problems from the original problem by associating the object with the other object.
3.3.2. Some ways of organizing cognitive activities for developing students’
competence of conjecture and decision making in teaching Mathematics at high school
3.3.2.1. Purpose of organization: The organization of cognitive activities to develop
the competence of judgment and making decisions to help students can immediately make
predictions about the nature of mathematical problems, to judge right away the solution or the
results of the problem, it leads to select their appropriate decisions for understanding and
grasping the problem, and the ways to solve problems, contributing to the shape and
development of the MIC for students.
3.3.2.2. Some methods to train competence of judgment and making decisions for
students through teaching Mathematics at high school
- Using situations for students to conduct intellectual activities such as comparison,
similar, generalization, specialization to judge hypotheses and make decisions.
- Organizing situations to help students use different kinds of reasoning to judge
hypotheses, ways to solve problems and make appropriate decisions.
- Using situations for students to visualize visual images of problems to make
judgments about how to solve problems.
- Creating opportunities for students to briefly use reasoning arguments to quickly
make effective judgments and decision choices.
3.3.2.3. Illustrative examples: Example 3.7. Through teaching of solving the equation:
22 4 6 11x x x x .
1) Step 1: Create cognitive situation contained new knowledge: Obviously, it is not
possible to use equivalent transformations to solve the above equation because it will raise
the exponent significantly, therefore the problem becomes more complex.
2) Step 2: Imagine the idea and conjecture the way to solve the problem
+ Judgment 1: Students have comments on the right side of the equation is always
greater or equal 2. From which the student makes the judgment by evaluating the two sides
of the equation.
+ Judgment 2: Observe the formulas in the radical in the left side of the equation. It is
obviously related to the expression of the right side. Therefore, students can make judgments
by using the sub-set method.
- Make the appropriate decision for the choice of how to solve the problem
+ For judgment 1: Find 2 4x x k , with constanst k by using Bunhiacopxki
inequality.
+ For judgment 2: The ability to hide subtracts the solving of the old equation solving
equation solvers can be solved.
+ For judgment 1: Find out 2 26 11 ( 3) 2 2VP x x x .
16
Students can discover 2 4x x in the equation, therefore they can associate to
use Bunhiacopxki inequality.
+ For judgment 2: Intuition with using the sub-set method
We have 2, 4 , , 0u x v x u v . Therefore 2 2 3VP u v . So, they can
solve the system of equations:
2 2
2 2
2
3
u v
u v u v
with 0, 0u v .
3) Step 3: Use deductive inference to test the results of intuition
+ For judgment 1: Condition of examined equation 2 4x .
Ta có 2( 3) 2 2VP x (1). Using Bunhiacopxki inequality, we have:
2
2 21. 2 1. 4 1 1 2 4 4x x x x 2 4 2VT x x (2).
Therefore
3
3
2 4
x
x
x x
.
+ For judgment 2: Put 2, 4 , , 0u x v x u v . We have system of equations
2 2
2 2
2
3
u v
u v u v
(I) with 0, 0u v .
Put ,S u v P uv we have
2
2
3 (1)
2 2 (2)
S P
S P
. Thus 3 2( 1)( 5 7) 0P P P P (*).
Solution of the system of equations is 1, 1u v so solution of the equation is 3.x
4) Step 4: Draw conclusions about new method knowledge and apply to new situations.
Students form method knowledge through solving problems of complex equations.
3.3.3. Some ways of organizing cognitive activities for developing students’
competence of immediate generalization in teaching Mathematics at high school
3.3.3.1. Organizational purpose: Intuitive activity immediately recognizes the
problem, the mathematical relationship or the discovery of the nature of the subject matter's
mathematical problem, obtained through the rigorous process, inductive deductive in a way
quick to catch the problem. In teaching Mathematics, teachers need to organize training for
students in awareness activities towards helping to observe, compare similar factors,
individual things and analysis to detect things in specific phenomena, using inductive
reasoning to detect the rule of the problem, cap

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