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 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.

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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
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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
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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.
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