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Trieu Thi Thu Hien, Ba Ria - Vung Tau Pedagogic College E-mail: [email protected]
IMPROVING KNOWLEDEGE MOBILIZATION ABILITY TO ENHANCING MATHEMATICAL PROBLEM-SOLVING COMPETENCE FOR PRIMARY STUDENTS
Abstract: As regards mathematics instruction in schools, math problem solving plays an integral part in shaping and enhancing mathematical skills for learners. In that process, knowledge mobilization has an indispensable role to play in students' attempts to give correct answers to their math exercises. It is; therefore, of paramount importance to find alternative ways of improving students' competence in mobilizing knowledge, all for the sake ofboosting their mathematical problem-solving skills.
Keywords: knowledge mobilization; ability in solving mathematical problems; math problem solving.
In the process of training problem-solving skills, fostering the ability of knowledge mobilization for students is considered as a major content. Therefore, studying the implementation measures to improve the quality of teaching mathematics in general and improving the ability ofknowledge mobilization in math problem solving in particular is essential and meaningful.
1. Steps of math problem solving according to the viewpoint of G. Polya
In "How to solve a math problem?" G. Polya proposed a method to solve a general math problem with four steps:
Step 1: Clearly comprehending about the math problem;
Step 2: Building a mathematical problem-solving program;
Step 3: Implementing the solving plan;
Step 4: Evaluating the answer.
With his theoretical conclusions, Polya created a general mathematical teaching method which not only aims to solve a single math problem, but also bring arguments and reasons in mathematical problem-solving process. In other words, educator G. Polya hopes to be able to help pedagogues who want to develop math skills for their students and help students to develop their math skills.
In 4 steps that G. Polya proposed, step number 2 "Building a mathematical problem-solving program" is actually a unified step between the dialectic logic and the formal logic. The formal logic addresses the "structure" of the plan (the form of the plan) while the dialectic logic indicates the specificity, feasibility and method of implementing the mathematical problem-solving plan (the content of the plan). Thus, building a mathematical problem-solving plan is mainly to propose the strategy to solve the math problem. For this process to be effective, students must think in a number of following orientations:
- Have I seen this math problem yet? Or this math problem is in a different form? Is there any problem related to this?
- Is this math problem related to the problem that I have solved? Can its solution and result be used?
- If this math problem has not solved yet, is there any related problem that is easier? Or this may be a particular case? Or a similar problem?
- Can I solve a part of the problem? Or should I retain a part of the condition and ignore the other part? Can I extract a useful factor from data?
- Did I use all the conditions? Did I pay attention to every major concept in the math problem?
In other words: To achieve mathematical problem-solving strategies, students need to choose the relevant knowledge and experience to solve the problem, build up a plan to reach the destination of the problem.
According to Polya, in order to mobilize and organize knowledge, students need to know how to:
- Zoning the knowledge corresponding to the conditions of the math problem.
- Identifying the concepts, properties, formulas and rules that relate to the requirements of the problem. What type of familiar problem does it involve or relate to?
- Reliving the concepts, properties, formulas, or similar types of problems and methods of solving them.
- Complementing a few elements to better understand the way of solving the problem.
Difficult and complex math problems may have details that were considered as the key, we can provisionally isolate this element to focus on it, and then combine it with the whole problem.
2. Improving knowledge mobilization ability for student in math problem solving
2.1. Reinforcing knowledge as the basis of thought in the process of solving math problems
The characteristics of mathematics are system and continuity. The mathematical knowledge is arranged in a strict system. The later knowledge is formed on the basis of existing knowledge. An earlier concept and its properties must be clearly understood in order to comprehend the later concepts. Therefore, students need to understand and grasp each knowledge as the basis for receiving other new knowledge: the process of continuous learning is a stage in the thinking process which is the basis of the math problem solving.
In mastering the knowledge, understanding and comprehending the concepts play a leading role. From the definition of each concept we can detect its property and its relation to other concepts in the same array of knowledge.
In many cases just because students do not understand the concepts, they have solved the problems wrong or could not find the solution. For example, students who do not know the formula of triangular area can not solve problems related to the triangular area or the length of triangular edge. Unable to grasp the method of numerical structure analysis, students have difficulty in solving numerical problems. In contrast, understanding the concepts and comprehending formulas will help students to solve math problems related to triangular area, calculate the length of edges by triangular area and compare the area. Students can even solve advanced problems such as calculating area of the shapes that can be inferred as triangular area; grafting the shapes base on their area; calculating the ratio area of the shapes.
2.2. Exploiting the applications of concepts, rules and formulas to increase the ability of using knowledge
Teachers can exploit the applications of concepts, rules and formulas by integrating plentiful application exercises. When exploiting the applications of concepts, rules and formulas, teachers need to pay attention to the level of each student to adjust the "dose of knowledge" and appropriate method of comprehending knowledge.
For example: Giving a sequence of numbers: 4, 8, 12, 16, 20, ...
a) Are numbers 182 and 64 included in the given numerical series?
b) If they are, what are their order number in the sequence?
c) What is the 121th number of the sequence?
Students may make the following answers:
a) The above sequence of numbers defines the rule: "Each term of the series is divisible by 4."
Number 182 divides into 4 makes 45 and surplus is 2, that is, 182 is not divisible by 4.
So, the number 182 is not included in the given numerical series.
b) Again we have: "Each term of the series equal to its order number multiplies by 4".
64 = Order number (64) x 4 So the order number of 64 is: 64: 4 = 16 c) Due to "Each term of the series equal to its order number multiplies by 4", the 121th number of the sequence is: 121: 4 = 484
2.3. Practicing the typical math problems solving to increase the ability of knowledge mobilization Typical mathematics is the basic part of the math problem solving in elementary school which aims to introduce some basic math problems and the solving methods, so practicing this type of math helps to increase the knowledge to solve math problems and practice mathematical problem-solving skills for students. This is one of the factors that help students increase their ability to mobilize knowledge. In this respect, typical math problems can be considered as an additional problem or an intermediate step in the process of solving other problems.
When students study a particular form of typical math problem, the teacher should pay attention to explore all the specific forms of the problem, arrange them logically and reasonably in the form of a series of related problems to help students increase their ability of knowledge mobilization, improve their mathematical problem-solving competence.
For example, when students learn the math form of Sum - Difference in grade 4, we design a series of exercises:
Exercise 1: Finding the two numbers when their sum is 70 and their difference is 10
This is a basic problem. In this type of exercise, the problem is to know the sum and the difference of two numbers, it requires to find that two numbers (hidden). When solving this problem, a basic requirement is that students must grasp the basic characteristics of the original problem and the original problem model, then master and apply the original problem solving method to solve the same problems.
Exercise 2: Finding the two two-figure numbers whose sum is 54. Knowing that if in turn grafting the big numbers onto the left and the right of the small
number, we have two numbers with 4 digits and their difference is 2376.
This is the Sum - Difference problem. If we keep the data that their sum is 54 and solve the problem in the direction of finding the difference of two numbers, this type of problem is to known the sum and the difference is hidden. If we keep the data that the two four-figure numbers with their difference is 2376 and based on the remaining data to find their sum, this type of problem is to known the difference and the sum is hidden.
With this problem, the teacher can suggest that students zone and recognize the requirements of the problem which is related to the "Sum - Difference" problem, recall how to solve the "Sum - Difference" problem, add the information that we have the difference of the two four-figure numbers to analyze the numerical structure. From this point we isolate to determine their difference (ab-cd) which is the crux of the problem. Then we link the difference to know whether it is related to the sum of two given numbers and two finding numbers or not. Obviously yes. That is the ability to solve the problem.
In the case that students add the sum of the two numbers (ab + cd) is 54 as a basis to calculate the sum of the two four-figure numbers (abcd + cdab), the isolation is to determine the sum (abcd + cdab). And we link this sum with the given elements and elements that must be found to bring out different solution to the problem.
Exercise 3: Finding the two numbers whose sum is the largest three-figure number and if we add the figure 9 to the left of the small number we get the big number.
This is a Sum - Difference problem with both the sum and the difference are hidden. Based on the basic form of Sum - Difference problem and hiding one of the two data, students easily solve the problem.
When students learn a particular form of typical math problem, the teacher need to explore all the specific forms of the problem, arrange them logically and reasonably in the form of a series of related
problems, organize students to practice math problem solving according to pedagogical scientific guidance to help students increase their ability ofknowl-edge mobilization and improve their mathematical problem-solving competence.
2.4. Improving the ability of knowledge mobilization through organizing students to study relevant math problems
Practicing math problem solving not only helps students to deepen their knowledge and familiarize students with many different types of math problems, but also help them to accumulate and practice skills of mobilizing knowledge.
Organizing students to solve relevant math problems will helps them to form associations and mobilize their knowledge thoroughly and thereby gradually increase their ability of knowledge mobilization.
The ability of mobilizing knowledge will be enhanced if students continue to practice hard and complex math problems. Since along with the process of mobilizing knowledge, psychological and intellectual functions are also mobilized at a high level, facilitate students to study and find solutions to the problem.
For example: With the problem: "Giving 6 points in which no 3 points are connected to a straight line, how many triangles are there that their tops are the given points?" (*). Students can solve this problem if they practice and solve the following two problems: Exercise 1: Giving a quadrilateral. a) How many line segments connect 2 of the 4 tops of the quadrilateral?
b) How many triangles are there that their tops are the given points?
Exercise 2: "Giving 6 points in which no 3 points are connected to a straight line, calculates the number of line segments in which each line segment joins two of the 6 given points.
Generalizing from Problem 1 and based on Problem 2, students can solve problem (*) as following steps:
The number of straight lines connecting 2 of the 6 given points is: 5 + 4 + 3 + 2 + 1 = 15 (line segments) Getting a line segment as the bottom, connecting with the 4 remains peaks, we have 4 triangles.
From 15 line segments, the number of triangles are: 15 x 5 = 75 (triangles)
Since each triangle is computed in triplicate, the number of triangles are: 75: 3 = 25 (triangles) 3. Conclusion
Knowledge mobilization plays a crucial role in the thinking process to find solutions. In order to mobilize knowledge, students must practice math problem solving on a regular basis according to scientific methods developed by the teachers' pedagogic intention design. If students have a learning process that regularly learn from experience, the process of mobilizing knowledge would be quicker and the knowledge that they have mobilized is indeed the necessary knowledge. Researching to find solutions to improve the ability of knowledge mobilization to increase the mathematical problem-solving competence for students is always a big question that need to find the answer.
References:
1. Tran Dien Hien. 10 specialized subjects to foster advanced students in grade 4, 5. Education Publishing House.-2001.
2. Krutecski V. A. Psychology of mathematical competence of students. Education Publishing House.-1973.
3. Polya G. How to solve a math problem. Education Publishing House.- 1975.
4. Dao Tam. Practicing method of teaching mathematics in elementary school. Education Publishing House.- 2005.
