Improvement of thinking skills and abilities of recognition for students by using algorithm in teaching mathematics at schools

JOURNAL OF SCIENCE OF HNUE Educational Sci. 2011, Vol. 56, No. 4, pp. 38-42 IMPROVEMENT OF THINKING SKILLS AND ABILITIES OF RECOGNITION FOR STUDENTS BY USING ALGORITHM IN TEACHING MATHEMATICS AT SCHOOLS Nguyen Thi Tinh Hanoi National University of Education E-mail: tinhnt@hnue.edu.vn Abstract. Algorithmic thinking are very important for students studying Mathematics. It is believed that if teachers apply algorithms in teaching Mathematics and require students to work more on pencil-and-paper compu- tations, students will have more opportunities to build their thinking skills, the abilities of Mathematic recognitions and computational skills. This pa- per presents some examples in teaching integral parts where teachers can improve the abilities of mathematic recognitions and algorithmic thinking skills to students by providing procedures or algorithms of each problem to students and require them to do follow up activities. 1. Introduction An algorithm of a mathematic problem is a step-by-step procedure designed to obtain the results of the problem in a finite period of time. Sometimes, an algorithm has some steps that can be repeated in a number of times. It is believed that if teach- ers apply algorithms in teaching Mathematics and require students to work more on pencil-and-paper computations, students will have more opportunities to build their thinking skills, the abilities of Mathematic recognitions and computational skills. This paper presents some examples in teaching integral parts where teachers can improve the abilities of mathematic recognitions and algorithmic thinking skills to students by providing procedures or algorithms of each problem to students and require them to do follow up activities. 2. Content In the past, experienced teachers when teaching Mathematics at schools put more focus on improving the ability of recognition of students in finding out the way to solve the problem faced and applying algorithmic thinking to work out the steps leading the solution of the problem. However, these days the role of algorithms in teaching Mathematics at schools have been changing [2]. Perhaps, one of the main reasons for that is the availability of easy-to-use and powerful calculators 38 Improvement of thinking skills and abilities of recognition for students... and computers. As a result, many students are facing difficulties in carrying out simple algorithms on pencil-and-paper computation such as addition, subtraction, multiplication, division,. . . let alone to solve more complicated mathematic problems. There are often complaints that many students seem not to have the abilities of algorithmic recognitions in Mathematics [3]. The following are 3 examples, where teachers can use algorithms to show students the way to solve the problems. 2.1. Using substitution algorithm * Problem: Find the antiderivative I = ∫ 6x(3x2 + 3)3dx Students with good knowledge on derivatives and understand clearly the con- cept of anti-derivative can easily to recognize that the derivative of (3x2 + 3) is 6x and quickly find the result for the problem. However, in reality, there are a lot of students who do not immediately learn that. If so, the teacher can make some sug- gestions for them and to instruct them to work out the best way to find the result. The instructions or the algorithm for this problem can have the following steps: Step 1. Recognise that (3x2 + 3)’ = 6x. Let u = 3x2 + 3, Step 2. Find du dx = 6x, Step 3. Substitute u for 3x2 + 3 and 6x = du dx , we have I = ∫ 6x(3x2 + 3)3dx = ∫ u3 du dx dx Step 4. Simplify the integral: I = ∫ u3du Step 5. Antidifferentiate with respect to u: I = 1 4 u4 + C Step 6. Replace u with 3x2 + 3 we have: I = 1 4 (3x2 + 3)4 + C. This example is simple, however, teachers then need to give students more examples and emphasize that by using this algorithm, we can make some difficult integrals simpler and easier. Then the teacher can remark that: If we recognize that some part of the expression is the derivative of the other part, then we can use substitution algorithms. In the classroom, after teacher’s instructions by the above example, we can let students do the exercises by themselves, such as finding anti-derivative of the following functions: y = (6x5 + 1)/ √ (x6 + x); y = (x2 – 1) cos (3x- x3); y = sin2xcos3x,... At first, teachers can require students to work individually. Each student has to write all the steps of the algorithm leading to the solution. Then teachers can ask 39 Nguyen Thi Tinh students to work in groups of two or three to discuss and compare the results and the steps in the algorithms for each problem. Next, teachers can encourage students as volunteers to go to the board to present the algorithm. By these activities, all students can understand more on his or her algorithm and the ones provided by other students. At the same time, they can improve their algorithmic thinking, Mathematic recognition when solving those problems. Furthermore, they can learn from each other and also they can build up communication and presentation skills. Following are some more examples on the topic. 2.2. Using linear substitution If antiderivative having the form∫ f(x)[g(x)]ndx, n 6= 0, where g(x) is a linear function, that is, one of the type g(x) = ax + b, and f(x) is not the derivative of the g(x), the substitution u = g(x) is often successful in finding the integral. * Problem: Find the antiderivative I = ∫ x(x− 3)3/4dx. In this example f(x) = x, g(x) = x – 3 with n = 3/4. We can instruct students in the following steps: Step 1. Let u = x – 3 Step 2. Find du dx = 1 Step 3. Substitute u for x – 3, u + 3 for x and du dx for 1, we have: I = ∫ x(x− 3)3/4dx = ∫ (u+ 3)u3/4 du dx dx Step 4. Expanding the integrand: I = ∫ (u7/4 + 3u3/4)du Step 5. Anti-differentiate with respect to u: I = 4 11 u11/4 + 3. 4 7 u7/4 + C Step 6. Replace u with x – 3: I = 4 11 (x− 3)11/4 + 12 7 (x− 3)7/4 + C For those students who are very good at Mathematics, perhaps, this algorithm comes immediately and naturally. However, for many students, who find difficulties 40 Improvement of thinking skills and abilities of recognition for students... to solve the problem, again, teachers need to make suggestions to them and instruct them carefully. Then it is necessary for them to do more exercises of this type until they master the algorithm, such as∫ 2 x− 5dx; ∫ √ 4x+ 1dx; ∫ x(x+ 3)2dx; ∫ 2x(1− 2x)dx; ∫ (x− 2)(2x− 1)3/2dx; 2.3. Antiderivative involving trigonometric identities This type of integral requires students to remember derivatives of trigono- metric functions in order to recognize and find the way to substitute or transform the integrand to the easier form. Different trigonometric identities can be used to antidifferentiate sinnx or cosnx with n has the natural number. * Problem 1: Find the antiderivative I = ∫ cos2 x 2 dx Step 1. Use identity to change cos2 x 2 dx: I = ∫ 1 2 (1 + cosx)dx = 1 2 ∫ (1 + cosx)dx Step 2. Antidifferentiate by the rule: I = 1 2 (x+ sinx) + C Also, this problem is simple with those students who know well the trigono- metric identities, but we can’t guarantee that there are no students in any classes are confused when solving this for the first time. * Problem 2: Find the antiderivative I = ∫ sin3 xdx Step 1. Factorise sin3x as sinx and sin2x: I = ∫ sin x sin2 xdx Step 2. Use identity sin2x = 1− cos2x: I = ∫ sin x(1− cos2 x)dx Step 3. Let u = cosx so du = −sinxdx and the antiderivative rule can be applied, I = ∫ (u2 − 1)du I = 1 3 u3 − u+ C Step 4. Substitute u for cosx: I = 1 3 cos3x− cosx+ C Each step in the algorithm helps students develop thinking skills, deploying and linking the knowledge learned before (factorizing, trigonometric identities and substitution) to and applying them to solve the problem. Skills are only obtained with enough practice [1]. Students should be given enough exercises to work in the classroom and at home. As a result, they can improve the abilities of recognition and quickly building up and designing the algorithm for each problem. 41 Nguyen Thi Tinh 3. Conclusion Algorithms and thinking skills play very important roles in problem solving abilities of each student. Those skills can be built up in many different ways at school levels. It is said that mathematic teachers can help students highly develop those skills through requiring students to work out the algorithms for each problem and present it in the form of step by step. Then student should be given enough exercises to work in the classroom individually. At the same time, students should be required to work together, to compare algorithms of the same problem with other students. By these activities, students can learn from each other and deeply understand the algorithms they learned and improve their Mathematic recognition abilities and thinking skills. REFERENCES [1] Caroll, W.M., 1997. Mental and written computation: Abilities of students in reformed-based curriculum. The Mathematics Educator, 2(1): pp. 18-32. [2] Lorna J. Morrow, Margaret J. Kenney, 1998. The teaching and learning of al- gorithms in School Mathematics. Reston, VA : National Council of Teachers of Mathematics, c1998. [3] Edmonds, Jeff, 2008. How to think about algorithms. Cambridge; New York: Cambridge University Press. TÓM TẮT Nâng cao kĩ năng tư duy và khả năng nhận thức cho học sinh qua sử dụng thuật toán trong giảng dạy môn Toán ở trường phổ thông Thuật toán và những kĩ năng suy luận đóng vai trò rất quan trọng trong khả năng giải quyết vấn đề của mỗi học sinh. Những kĩ năng này có thể được thiết lập và tích lũy bằng nhiều cách từ bậc phổ thông. Các giáo viên Toán có thể giúp học sinh phát triển những kĩ năng này rất tốt thông qua việc yêu cầu học sinh tìm ra thuật toán cho mỗi bài toán và trình bày nó dưới dạng các bước giải. Bài báo giới thiệu một số thuật toán giáo viên có thể sử dụng để dạy cho học sinh cách giải các bài toán nhằm nâng cao kĩ năng tư duy và khả năng nhận thức cho học sinh. 42