Training students ability to detect, practice algorithmic and quasi-algorithmic rules in the process of dominating mathematical knowledge

JOURNAL OF SCIENCE OF HNUE 2011, Vol. 56, N◦. 1, pp. 77-85 TRAINING STUDENTS ABILITY TO DETECT, PRACTICE ALGORITHMIC AND QUASI-ALGORITHMIC RULES IN THE PROCESS OF DOMINATING MATHEMATICAL KNOWLEDGE Nguyen Huu Hau Dong Son 2 High school - Thanh Hoa Tran Trung Tinh(∗) Hanoi National University of Education (∗)E-mail: tinhtckh@gmail.com Abstract. The main purpose of this paper is to identify mainstream views to develop students thinking ability. For each view, we provide a process of researching, discovering, setting up and implementing algorithmic rules, quasi-algorithmic rules by giving out examples. Keywords: Algorithmic rules, quasi-algorithm rules. 1. Introduction In mathematics, there are many types of problems that could be solved by algorithmic rules and quasi-algorithmic rules. Researching, investigating new algo- rithmic rules, quasi-algorithmic rules to solve a specific problem, class of problems could contribute in developing intellectual manipulations for students. In fact, these activities have not been properly appraoched. The teachers are not very proficient in exploiting situations, teaching contents in order to develop students ability to think about algorithms. 2. Content 2.1. Activities of algorithmic thinking Methods of algorithmic thinking is embodied in the following activities: T1: Perform operations in a determined sequence that is appropriate with an algorithm; T2: Separate an activity into sub-activities that are performed in a determined sequence; T3: Generaliy a process occurs on some individual objects into a process wich occurs on a class of objects; T4: Precisely describe a process of conducting an activity; T5: Discover an optimal algorithm to solve a task [2;383]. 77 Nguyen Huu Hau and Tran Trung Tinh T1 activity demonstrates the capacity of performing an algorithm. From T2 to T5 activities demonstrate the capacity of building an algorithm. All 5 activities above are considered activities of algorithmic thinking. We found that in oder to develop algorithmic thinking for students in mathematic classes, teachers must orga- nize and control the activities of algorithmic thinking. Those activities should help students understand, strengthen the algorithmic rules as well as develop students ability of algorithmic thinking. 2.2. Some mainstream views 2.2.1. Views 1 In the process of impacting mathematical knowledge, the concern in construct- ing teaching processes is needed Forming a process is to form a methodical knowledge. Methodical knowledge is important in fostering students independence, self-discipline, self-test habits. For example, forming a process of solving a class of inequality √ f(x) > g(x). Example 1. Teaching process of solving a class of inequality √ f(x) > g(x) (f(x), g(x) are functions in x). To help students discover algorithms, teachers can use method problem-solving by conversation as follows: After introducing the class of inequality, teachers ask students to do the fol- lowing exercises: Solve inequalities: a) √ 2 + 3x > x, b) √ x2 − 1− x > 2. The teacher guides students to solve exercises (a) with some oriented questions: Please find domain of the given inequality? 2 + 3x ≥ 0. To remove square roots that contain varible, which cases must be considered? Case 1: x < 0; Case 2: x ≥ 0. In the case x < 0, right side of inequality is negative, left side of inequality is positive, find all solutions of given inequality? Solutions of inequality are also the solutions of 2 + 3x ≥ 0 and x < 0 (1). In the case x ≥ 0, both sides of inequality are positive, simplify the given inequality? 2x+ 3 ≥ x2 (2). Conclusion: The inequality √ 2 + 3x > x equivalent to (I) { 2x+ 3 ≥ 0 x < 0 or (II) { x ≥ 0 2x+ 3 > x2 Solutions of above inequality are union of solutions of two inequality system (I) and (II). 78 Training students ability to detect, practice algorithmic and quasi-algorithmic rules... With similar method, students could solve inequality (b) by themself. Based on archived result, teachers instruct students to form an equivalant transformation in order to solve inequality: √ f(x) > g(x) as follows: What must we do to solve an inequality? Form an equality without roots by removing square roots. How to remove square roots? Square both sides of inequality. To get an equivalent inequality by squaring both sides which conditions are required? Both sides of inequality must be positive. Square both sides of: { g(x) ≥ 0√ f(x) > g(x) ⇔   g(x) ≥ 0 f(x) ≥ 0 f(x) > g2(x) Is there any unnecessary condition? Please point out!   g(x) ≥ 0 f(x) ≥ 0 f(x) > g2(x) ⇔ { g(x) ≥ 0 f(x) > g2(x) Such conditions f(x) ≥ 0 is not necessary. Rewrite transformation! { g(x) ≥ 0√ f(x) > g(x) ⇔ { g(x) ≥ 0 f(x) > g2(x) To which conditions is this transformation corresponding? The left side to positive: g(x) ≥ 0. Could left side possibly be negative? g(x) could possibly be negative. As a consequence, inequality can not be squared. Could the solution still be concluded? { √ f(x) > g(x) g(x) < 0 ⇔ { f(x) ≥ 0 g(x) < 0 Combining the two possiblities, g(x) ≥ 0 and g(x) < 0, What do we have? √ f(x) > g(x)⇔ { g(x) ≥ 0 f(x) > g2(x) or { g(x) < 0 f(x) ≥ 0 Students could base on these above activities to build an algorithm to sovle inequality step by step. - Step 1: Convert inequality into the form of: √ f(x) > g(x); 79 Nguyen Huu Hau and Tran Trung Tinh - Step 2: Set the conditions f(x) ≥ 0. Find the values of x so that g(x) < 0, g(x) ≥ 0; - Step 3: Solve an inequality system { f(x) ≥ 0 g(x) < 0 the set of solutions is S1; - Step 4: Remove square root by squaring both sides; - Step 5: Solve an inequality system { g(x) ≥ 0 f(x) > g2(x) , the set of solutions is S2; - Step 6: Find S1 ∪ S2; - Step 7: Return the solutions. 2.2.2. View 2 Pay proper concern at impacting methodical knowledges of algorithmic think- ing while organizing, controlling training activities The authors in [3;57] proposed a methodical knowledge system of algorithmic thinking which includes: First, generally exploring the problem to find out partic- ular characteristic, distinct signs of the problem; Second: Analyze the problem to point out assumptions and conclusions of the problem. Put effort into finding vi- sual tools to represent the problem; Third: Analyze the problem to devide it into simpler problems; Fourth: research and predict solution by dividing problem into cases. Examine cases (with reasoning) by examining special cases, similarization, generalization, ...; Fifth: Transform unknown solving method problems into known solving method problems; Sixth: Check results. Find more reasonable solutions by overcoming irrational points of the old solution or change the problems point of view, apply good solutions for other problems, propose new problems. For exam- ple:generaly explore problem to find out particular characteristics, distinct signs of the problem. Example 2. Solve equation: (x+ 1)(x+ 3)(x+ 5)(x+ 7) = 9. Teachers guide students to solve the equation as follows: In this problem, the left side could be expanded to get a form of a full quartic equation: ax4 + bx3 + cx2 + dx + e = 0 (a 6= 0). Solving such equations would be difficult for students due to the fact that there is no general algorithm to solve full quartic equation. Comment about the coefficients in the left side: 1 + 7 = 3 + 5 = 8. Find an appropriate transformation that makes two expressions look similar. At the left side, multiply the first with the fourth expressions and the second with the third expressions. The result is (x2 + 8x+ 7)(x2 + 8x+ 15) = 9. What is the common denominater between two expressions? Share the same 80 Training students ability to detect, practice algorithmic and quasi-algorithmic rules... term x2 + 8x. Based on above property, propose a method to solve problem? Set t = (x2 + 8x+ 7), the equation becomes:(t + 7)(t+ 15) = 9. By abstracting specific numbers, teachers ask students to propose a general problem and build a method of solving it. Besides this methodical knowledge, teachers could organize for students to per- form some progresses of teaching methodical knowledge that has algorithm property explicit as follows: a, deductive process: To impart to students a method of solving a mathemat- ical problem that has algorithmic property we could follow these steps: - Step 1: Present the general problem that needs to be solved; - Step 2: Investigate and present a method to solve this problem; - Step 3: Examples, practice and consolidate methods. In order to develop students positive learning attitude, teachers should let students perform steps 2 and 3 by themselves. However, in Step 2 there are some cases that teachers should skip the process of investigating new methods (the process is inconvenient and time consuming) and introduce it directly. b, Inductive process. Require students to understand a method of solving a mathematical problem. Students perform the following steps under the organization of teachers. - Step 1: Ask students to solve some specific problems from the same class that the difficulty level is increased gradually; Step 2. Ask students to evaluate common features in solutions of these prob- lems. Students propose the genaral problem and its solving method (with teachers instruction) based on that evaluation; - Step 3: Consolidate and practice newly discovered methods by solving an- other specific problem from the same class. 2.2.3. View 3 Proficently combine practicing known rules, algorithms with building quasi- algorithmic rules in the teaching process. In oder to develop students creation, students shouldnt just apply known algorithms like a machine. In other words, students must investigate the algorithm and play a central role in problem-solving activities. Thus, even with problems that could be solved by an algorithm the could strengthen students’ self-learning skills and creativity ability under conditions that teachers instruct students to discover that algorithm. With such conditions mean 81 Nguyen Huu Hau and Tran Trung Tinh teachers need to get rid of the old teaching style: provide students general algo- rithms, give illustrative examples, ask students to apply provided algorthm to solve specific problems. Quasi-algorithmic rules and predicting methods are not explicitly teaching objects taught in high schools. These rules, methods are only hints to solve problems, not proper algorithms that could always solve the problems. So for students to be able to apply these rules, their flexibility, versatility needs to be trained; they should be able to control direction and change methods when necessary. It wouldnt be a problem if students fail to apply these rules, methods. The point is students realize that they have approaching in a wrong way and must change their methods. For example, build quasi-algorithmic rules: Use Cauchy inequality to find the minimum, maximum value. Cauchy inequality has many signs that we could base on to build an algorithm for a class of finding minimum, maximum problems. In this paper, we only outline some signs to build quasi-algorithmic rules. Signs: If the product of n non-negative numbers x1, x2, ..., xn is constant, the sum S = x1 + x2 + ...+ xn reach the minimum value when: x1 = x2 = ... = xn. For example: Find the minimum value of the function y = f(x). In particular, f(x) is the sum of non-negative numbers in which their product is constant or after some transformation steps f(x) is the sum of non-negative numbers in which their product is constant. * Below are some examples that help students to build quasi-algorithmic rules of finding maximum, minimum problems Example 3. Given function f(x) = x2 + 2 x . Find the biggest value of f(x) in the range K = (0;+∞). Teachers could guide students as follows: make comments that the expression of f(x) is given under the form of a summation of two non-negative numbers, but their product is not a constant. The expression of f(x) need to be transformed into a sum of non-negative numbers in which their product is a constant in order to apply the above sign. Transform: f(x) = x2 + 2 x = x2 + 1 x + 1 x Apply Cauchy inequality with three positive numbers: x2, 1 x , 1 x : x2 + 1 x + 1 x ≥ 3 3 √ x2 1 x 1 x = 3 Equal sign occurs if and only if: x2 = 1 x = 1 x ⇔ x = 1. 82 Training students ability to detect, practice algorithmic and quasi-algorithmic rules... Conclusion: min (0;+∞) f(x) = 3 achieved when: x = 1. Teacher guides students to describe, build steps to solve the problem. - Step 1: Identify domain of the function. - Step 2: Transform the fraction 2 x into such form that f(x) is a sum of non-negative numbers in which their product is a constant. - Step 3: Apply Cauchy inequality with the numbers found in step 2. - Step 4: Conclude the minimum value of f(x). Teachers could ask students to solve some similar examples but more difficult than Example 3 such as: Find the minimum value of function: f(x) = x+ 4 (x− 1)2 on the domain (1; +∞). * From the above examples we can build an algorithm for general problems as follows: Problem: Given a function: F (x) = a.fm(x) + b c.fn(x) . where a, b, c > 0; m,n ∈ N∗; f(x) > 0 on domain K and equatio fm+n(x) = bn acm have solution(s) on K. Find the minimum value of F (x). Algorithm: - Step 1: Identify the domain of the function. - Step 2: Transform F (x) into a sum of non-negative numbers in which their product is a constant. - Transform the term afm(x) into a sum of n non-negative numbers a n fm(x). - Transform the term b cfn(x) into a sum of m non-negative numbers b mcfn(x) . - Step 3: Apply Cauchy inequality with m + n positive numbers found in - Step 2. - Find equality case in Cauchy inequality: Solving: a n fm(x) = b mcfn(x) on K. Find set of solutions D. - Indicates the minimum value L of F (x) from Cauchy inequality. - Step 4: Conclus: min K F (x) = L. Achieved when x ∈ D. Some notes when using Cauchy inequality to find the minimum, maximum value of function: + Some skills that teachers need to train students while teaching this section: transform terms, factors in order to use the signs of Cauchy inequality. Some com- mon signs are: The problem contain factors in which their summation (or product) is 83 Nguyen Huu Hau and Tran Trung Tinh constant; Finding minimum, maximum problems that are given in the form of frac- tions in which the summation of their denominator has maximum value. Teachers base on that to help students build algorithms, quasi-algorithmic rules and flexibly apply them in solving problems. + One of the most important task of using Cauchy inequality is predicting the equality case. Students could find a reasonable analysis by the result of that prediction. Teachers instruct students to work in groups to discover algorithmic rules: - Teachers prepare a system of problems printed out to give students. - After self-studing these problems, students will discuss solutions in groups. Each group find step-by-step solving methods, ready comments, make a short pre- sentation. - Class discuss: A group report their solution in steps. Others give opinions, approval or disapproval, exchange ideas, support, question or present their own solution. - Teachers participate in discussion, evaluate and conclude the solutions of groups. Teacher could also hand out previously prepared solutions for students as a reference. 2.2.4. View 4 Concern about using ability grouping reasonably in process of training algo- rithmic thinking ability for students. In order to effectively use ability grouping to developing algorithmic think- ing ability, each students needs to be trained appropriate with their ability. Thus, leveling students ability of algorithmic thinking is required. Leveling task could be based on: level by average awareness, content of thinking about algorthmic activ- ities, complexity of thinking about algorthmic activities, quality of thinking about algorthm activities, complexity of objectives of thinking about algorthm activities. Example 4: When teaching about solving irrational equations, teachers could use following exercises: a, Solve √ (5− x)(x− 2) = x− 2 (2.1) b, Solve √ 3x− 3−√5− x = √2x− 4 (2.2) c, Based on (2)s solution, find solution of equation:√ 3x2 − 9x− 9−√5 + 3x− x2 = √2x2 − 6x− 4 (2.3) d, Build a solving method for following class of equations√ au(x) + b−√cu(x) + d =√mu(x) + n Depending on students awareness level, teachers ask students to work on ap- 84 Training students ability to detect, practice algorithmic and quasi-algorithmic rules... propriate problems. Exercise ”a” is a basic problem which, under teachers instruction will be sat- ifactory and adequate for students to solve. Exercise ”b” requires some basic skills. Could be solved by satifactory and good students. ”c” and ”d” exercises require more advanced skills, for good and excellent students. For exercise ”c”, (2.3) is equivalent to√ 3(x2 − 3x)− 9−√5− (−3x+ x2) =√2(x2 − 3x)− 4 With such transformation, students could easily find out that (2.3) is similar to (2.1) by replacing (x2−3x) with x. From solving methods of the above excercises, students could easily solve general problems ”d”: - Step 1: Identify domain of equation: au(x) + b ≥ 0; cu(x) + d ≥ 0;mu(x) + n ≥ 0. (*) - Step 2: With constraints (*), transform the equation as follows:√ au(x) + b = √ mu(x) + n+ √ cu(x) + d ⇔ au(x) + b = mu(x) + n+ cu(x) + d+ 2√[mu(x) + n)][cu(x) + d] Simplify expression by combining like terms, the equation has the form of√ f(x) = g(x). This form had algorithm. Constructing and applying classified exercises like above example in class not only allow students to study knowledge that is appropriate with their awareness but also inspire students self-confidence 3. Conclusion It is necessary to train students ability to detect, practice algorithmic rules and quasi-algorithmic rules in the process of dominating mathematical knowledge. Doing so is contribute to developing students mathematical thinking. However, all four mainstream views need to be implemented effectively in order to achieve this goal. REFERENCES [1] M. Alecxeep, V. Onhisuc, M. Crugliac, V. Zabotin, 1976. Developing minds of students. The Education Publishing House, Hanoi. [2] Nguyen Ba Kim, 2009. An approach of Teaching Maths. Pedagogic University Publishing House, Hanoi. 85