ASSESSING
THE STUDENT’S CONCEPT IN MULTIPLICATION AND DIVISION IN YEAR 4.
RESEARCH
PROPOSAL
NAME: NUR NADIA HALIANI BT HALID
CLASS: ED2295B
NO ID: 2010647738
DATE: 20 DEC 2012
LECTURE’S NAME: DR. TEOH SIAN HOON
CHAPTER
1
THE
STUDY
1.0 BACKGROUND
OF STUDY
Mathematics is
one of the core subjects in the Secondary School Integrated Curriculum (KBSM)
outlined by the Ministry of Education which needs to be mastered by all
students in primary and secondary school. Hence, the provision of quality in
mathematics education from an early age in the education process is important.
. It emphasizes on the use of
mathematics in real life situations and aims at enhancing problem solving
skills as well as promoting logical thinking and critical thinking practices
among students. Mathematics as outlined in the KBSM curriculum has also
emphasized on the balance of understanding the concepts and of mastery of
skills.
Competency in
Mathematics is very important if a student wants to pursue his/her studies in
institutes of higher learning, as the student will need to fulfill the
requirement of earning at least a ‘credit’ in mathematics. To be able to
graduate with flying colors in a mathematics-based program, students must
master the fundamentals of mathematics. Students who fail to master the basic
of mathematics will experience difficulties in solving mathematical problems as
they pursue their higher studies.
Mathematics is often
associated with numbers, form and relevance. Numbers, form and relevance are
the basics that students should master in learning mathematics. Multiplication and division are important concept to teach in primary
school because students need to understand about operation and how they relate
to one another. (Greer, 1992; Harel & Confrey, 1994; Hiebert & Behr,
1988; Sowder et al., 1998).
Multiplication and
division are introduced into the Mathematics
(KBSR) curriculum as early as in Primary 3. It was considered as a
difficult topic to understand by students that these
students have just learned numeracy as a concept. Based on Ontario Curriculum, “The study of mathematics
equips students with knowledge, skills, and habits of the mind that are essential
for successful and rewarding participation in such a society”. (Ontario
Curriculum pg.3). Besides that, the quote “All students do not necessarily
learn mathematics the same way, using the same resources, and within the same
time frame” from Ontario Curriculum explains the students have many ways of
learning so that students can really understand what they learn.
1.2 RESEARCH OBJECTIVES
The purpose of this study is to investigate the level of the Year
Four students when performing multiplication and division problem.
Specifically, the study aims to investigate and analyze the students’ skills in
solving multiplication and division problems on:
·
Conceptual
understanding of multiplication and division
·
Identify properties
of multiplication and division
·
Words problem
involves multiplication and division
1.3
RESEARCH QUESTIONS
The research questions areas follow:
·
What is the level
of understanding of Primary 4 students in Multiplication and division test?
·
Identify the
mistakes done by Primary 4 students on solving multiplication and division
problem.
1.4
SIGNIFICANCE
OF STUDY
Multiplication
and division are integrated in the Mathematics curriculum early in the primary
school syllabus because of its importance in an individual’s
daily life. Therefore, students start to learn multiplication and division from
Year 3 where students learn the basic concept such as addition and subtraction.
As students move to a higher standard in the primary school, students are
required to solve problem involves multiplication and division.
Moreover, multiplication and
division had strong connection. A
researcher, John A Van De Walle state, “How important it is to combine the
teaching of multiplication and division facts”. It because multiplication and
division had relationship how to do with understands the both concept.
Mostly, students had
misconception in multiplication and division when they solve the problems. The
students maybe confuse in multiplication properties such as commutative property,
associative property and zero property. Besides that students make error when
solve division problems.
1.5
OPERATIONAL
DEFINITIONS
1.5.1 Multiplication
The
basic idea of multiplying is repeated addition.
1.5.2 Division
The action of separating something into parts,
or the process of being separated.
1.5.3 Multiplication concepts
Multiplication is a mathematical operation that
is faster and more efficient than addition.
1.5.4 Division concepts
Division is the inverse of multiplication.
1.5.5 Conceptual knowledge
Well defined as understanding
relationship that was connected to other mathematical ideas and concepts (Aksu,
2001).
CHAPTER
2
LITERATURE
REVIEW
2.0 Introduction
This chapter is reviewing on the several findings from
previous research and academic writing that related to the students’ concept of
multiplication and division. This review of the literature will provide more
information and perspectives on teaching and learning the topic of
multiplication and division. Moreover, this chapter included five subtitles
which are background of multiplication, background of division, concept of
division and multiplication, misconception in multiplication and division, and
mathematical knowledge.
2.1 Multiplication
Multiplication is one of the
most important topics in Mathematics. This topic had been taught to children
since at the primary level. Nevertheless, this topic still seems very hard to
be comprehended by students in school nowadays. The topic of multiplication is
one of the basic compared to all other topics in Mathematics. In order to
master other topics in the Mathematics syllabus, the students need to first
master the topic on basic operation likes addition, subtraction and others.
This topic is actually easy but there are still a lot of misunderstandings
among students.
Wikipedia, multiplication has
several properties that we need to make sure the students understand such as
number line, equal sets, arrays and balanced equations. Besides that, in multiplication
also have several properties like commutative property, identity property and
zero property. Furthermore, students will explore problems involves
multiplication.
2.2 Division
Division is key
essential components in mathematics because related to the concept of addition
and subtraction. Moreover, this topic learns when in second grade. According Ball (1990), states
prospective teachers are often unable to properly explain the meaning insides
division problems and cannot represent with others problems. Besides that,
students and teacher have problems to understand the concept of division.
The concept of division is one that builds on previous
knowledge of addition, subtraction, and multiplication. Furthermore, in
division students will learn the basic division facts and the concept of
division. Students also need to understand about division so that can learn
algorithms of division.
2.3 Concept of Division and Multiplication
In multiplication there are several concepts
that the students need to understand so that the students can solve
multiplication problems. Concept about multiplication has several properties
based on wikipedia:
1) Commutative property of
multiplication
-
When
two numbers are multiplied together, the product is the same regardless of the
order of the multiplicands.
For example:
|
7 x 9 = 9 x 7
|
|
a x b = b x a
|
2) Associative property of
multiplication
-
When
three or more numbers are multiplied, the product is the same regardless of the
grouping of the factors.
For example:
|
a x (b x c) = (a
x b) x c
|
|
2 x (3 x 5) = (2
x 3) x 5
|
3) Identity property of
multiplication
-
The
product of any number and one is that number
For example:
|
a x 1 = a
|
|
7 x 1 = 7
|
4)
Zero property of
multiplication
-
Any number multiplied with
zero, the result will be zero
For
example:
|
a x 0 = 0
|
|
5 x 0 = 0
|
Division is the inverse of
multiplication. Therefore, skill in division depends on skill in multiplication.
So, in division there are also have several properties to understand the
concept of division. The properties are:
1)
Divisive identity property of
division
-
When any number divided by 1
will stay the same.
|
15 ÷1 = 15
|
|
a ÷ 1 = a
|
2) Zero
property of division
-
There has two rule
i)
If zero divided by any
number, the answer will be zero.
|
0 ÷ a = 0
|
|
0 ÷ 2 = 0
|
ii)
If
any number divided by zero, the problems cannot be solve
|
5 ÷ 0 = math
error
|
|
a ÷ 0 = math
error
|
2.4 Misconception
in Multiplication and Division
There
are studies that showed misconception in Mathematics such as Bell,
Fischbein, and Greer [1984]; Fischbein, Deri, Nello, and Marino [1985] that
misconception about their properties. Students especially who low academic will
misunderstand about mathematics conception because the acknowledged of students
itself. Most important, the teachers should really prepare before teach the
students so that the students will really understand. Besides that, that makes the students
involves in misconception about multiplication and division because they think
that when we multiply we ensure we will get larger value while when we divided
we will get smaller value. Moreover, when involves word problems, students will
try to find the clues whether the answers will get greater or smaller. The
students only use the concept of multiplication and division to apply them back
in their test or examination not to practice them in their daily life. Because
of the limited in early experiences, students knowledge only to whole-number
divisors and quotients.
Example
for multiplication from meridianschool.com and division misconception from
techfind.com:
1)
The
student may know the commutative property of multiplication but they cannot
apply it.
-
Student states that 9 × 4 = 36 with relative
ease, but struggles to find the product of 4 × 9.
2)
The
student sees multiplication and division as discrete and separate operations.
But, division is inverse operations of multiplication.
-
State
that 6 × 7 = 42 but fails to realize that this fact also tells him that 42 ÷ 7
= 6.
3)
The
student under generalizes the results of multiplication by powers of 10. To
find products like 3 × 50 = 150 or 30 × 50 = 1,500, she must “work the product
out” using a long method of computation.
- 300
000
0000
4)
Thinking
that division is commutative.
-
For
example 5 ÷ 3 = 3 ÷ 5
5)
The
student may know the associative property of multiplication but fails to apply
it to simplify the “work” of multiplication.
-
Student
labors to find the product of three or more numbers, such as 8 × 13 × 5,
because he fails to recognize that it is much easier to multiply the numbers in
a different order.
Besides that misconception,
there also have misconception in associated knowledge and skills.
Associated
knowledge and skills
|
Errors
and misconceptions
|
§
Students know multiplication facts
for 2, 3, 4, 5 and 10 times tables.
|
Students not confident in recalling multiplication facts.
|
§
Students know the division facts
which correspond with multiplication facts;
understand the inverse relationship connecting multiplication and division. |
Is confused about the connection between multiplication and
division facts, for example,
3 × 5 = 15 so 5 ÷ 15 = 3.
|
§
Understand the effect of multiplying
whole numbers less than a thousand by ten.
|
Describes the operation of multiplying by ten as ‘adding nougat’.
|
§
Can apply the distributive law to
multiplying, using partitioning and recombining, for example:
14 x 3 = (10 x 3) + (4 x 3) = 30 + 12 = 42 |
Does not apply partitioning and recombining when multiplying,
for example, 14 x 3 is calculated as (10 x 3) + 4 = 34 or 14 × 3 = 312, confusing the value of two-digit numbers. |
§
Recognize that the commutative law
holds for multiplication but not for division.
|
Assumes that commutative law holds
for division also. For example, assuming that
15 ÷ 3 = 5 so 3 ÷ 15 = 5. |
§
Understand the idea of remainder, and
when to round up or down after division.
|
Writes a remainder that is larger
than the divisor. For example, 36 ÷ 7 = 4 remainder 8.
-Discards the remainder, as does not understand its significance. |
§
Know how to record division as
repeated subtraction, with appropriate use of chunking.
|
Continues to subtract twos when
calculating 20 divided by 2 without using knowledge that 2 multiplied by 5
equals ten.
|
From
teachfind.com
2.5 MATHEMATICAL KNOWLEDGE
Nowadays,
teaching mathematics is a piece of mathematical work. So, mathematics’ teachers
must know mathematical knowledge. Firstly, before teacher teach, they must
really know about what will they teach for students understanding. Second,
mathematics teacher’s knowledge not same with others professional. So, what
will teacher face in problem will not same with others. Third, any mathematical
knowledge that the students learn can solve their daily problems.
CHAPTER
3
RESEARCH
DESIGN AND METHODOLOGY
3.0 Introduction
This chapter is divided into five parts that
describes how the study was conducted. The discussion covers design of study,
sample of study, instrumentation, data collection procedures and data analysis.
This study was conducted in Primary
school in Shah
Alam to investigate the understanding
and assessing concepts of multiplication and division among
Year Four
students. The
data were collected based on the test, and the questionnaire. The data were
then analyzed using the Microsoft Office Excel.
3.1 Design of Study
The methodology that was used
in this research is the quantitative methods which involve collecting data
through the paper and pencil test and questionnaire.
The paper and pencil test instrument consisted of 15
questions on Multiplication
and Division Test. (Refer Appendix A). Besides that, the
questionnaire is aim to measure the students understanding about the basic
knowledge on the topic.
3.2 Sample of Study
The population of this study
was students which Year
Four students in Shah
Alam. The sample was selected by using stratified random sampling. The sample
must have previous knowledge in Multiplication and Division.
3.3 Instrumentation
The instruments that were
used in the study were in English language and Malay language. The instruments consist
of tests and the questionnaires.
i.
Test
·
The
test papers consist of questions that only focus on Multiplication and Division
based primary school syllabus.
ii.
Questionnaires
·
The
questionnaires were given after the test. The questions were to identify the
problems and their opinions in the process of learning the topic.
3.4 Data Collection Procedures
The data were collected and
analyzed. The details of the test were summarized. The data collected
instruments were test and questionnaire
i.
Test
·
The test contains 15 questions which
subjective was be given to measure the students’ problems and understanding on
the concept of Multiplication and division. The test duration in 30 minutes.
ii.
Questionnaire
·
The
questionnaire would be given to ensure the students’ problems and understanding
on Multiplication and Division. The questionnaires were distributed after the
test.
3.5 Validity
and Reliability of the instrument
Validity was an evaluation of
sufficiency and appropriateness of the interpretations and it was used on the
instruments outcome.
3.6
Data Analysis.
The data obtained from the
paper and pencil test are examined will be analyzed by using Microsoft Excel whereas
mark is given based on the marking schema by Rubric Score Criteria. Data
obtained will be tabulated in the form of tables and charts. The data based on
questionnaire also will analyze using Microsoft Excel.
For test : For questionnaire :
Score
|
Criteria
|
0
|
Completely
incorrect / no answer
|
1
|
Understanding
|
2
|
Calculation/answer
not fully correct
|
3
|
Calculation
correct, wrong answer
|
4
|
Correct
|
Scale
|
Response
|
1
|
Strongly Disagree
|
2
|
Disagree
|
3
|
Uncertain
|
4
|
Agree
|
5
|
Strongly Agree
|
Table 3.5.2: Level
perception
Table 3.5.1: Scalar Rubric Score Criteria
|
|
|
ASSESSING THE CONCEPT OF MULTIPLICATION
AND DIVISION YEAR 4
Test for Students
|
PART A – DEMOGRAPHIC DATA
Please Tick
your choice or provide the relevant information in the
corresponding boxes provided.
1. Class : ________________________________
2. Age : ________________________________
3 Gender
Male
Female
4. Ethnic
Malay
Chinese
Indian
Others
5. Grade
obtained for Mathematics (Year 3) : _______________
PART B
IMPORTANT: Show your
work in EACH question by writing down in the space provided below each
question, using sentences, figures, mathematical expression or other forms of
illustration to find the answer.
1.
There are 28 monkeys in a zoo and 4 cages. How many
monkeys in each cage?
|
2.
Lucy shares 42
marbles between her and her 6 friends. How many does each child get?
|
3.
In each box
there are 6 eggs. How many in 6 boxes of eggs?
|
4.
Mrs. Jones
picks 3 flowers a day starting on Monday. How many does she have by Sunday?
|
5.
6 cats have 4 kittens each. How many kittens are
there in total?
|
6.
In a school
there are 320 children. There are 10 classrooms. How many children in each
class?
|
7.
I have 66
cookies and can fit 6 cookies into a box. How many boxes will I need?
|
8.
There are 9 smarties
in a mini box. How many smarties in 6 boxes?
|
9.
How many wheels
are there on 8 cars?
|
10.
There are 7
windows on a house. How many windows on 3 houses?
|
11.
15 children are
at a party. 3 go home. The others share 48 sweets. How many do they each get?
|
12.
9 groups of 9
children go to a sports day. 7 are sick. How many children take part?
|
13.
Each box has 8
pencils in it. Mr. Robert has 6 boxes of pencils but 4 pencils are broken.
How many does he have?
|
14.
There are 60
sheep in a field. 12 more arrive. The sheep are then shared equally among 6
farmers. How many sheep do they each get?
|
15.
24 children
take off their shoes next to the bouncy castle. The dog comes and runs away
with 3 shoes. How many shoes are left?
|
Questionnaire
Response of the students from the
Questionnaire
This
questionnaire elicited the level of students’ interest and ability in
understanding and applying the concept of ‘integer’ after the pre-test was
given.
Scale
|
Response
|
1
|
Disagree
|
2
|
Uncertain
|
3
|
Agree
|
4
|
Strongly
agree
|
No.
|
Statements
|
Scales
1 2 3 4
|
|||
1
|
‘Multiplication and Division’ is a
difficult topic to learn
|
||||
2
|
I understand the concept of ‘Multiplication
and Division’ since Year 2
|
||||
3
|
I am interested in learning ‘Multiplication
and Division’
|
||||
4
|
Written exercises in ‘Multiplication and
Division’ help me to understand the concept of these topic
|
||||
5
|
I am always confused with the properties
of these topic when doing questions.
|
||||
6
|
I can understand and solve problem questions
of ‘Multiplication and Division’
|
||||
7
|
I can understand and solve questions on
properties of ‘Multiplication’
|
||||
8
|
I can understand and solve questions on properties
of ‘Division’
|
||||
9
|
I can understand and solve questions on combined
question of ‘Multiplication and Division’
|
||||
10
|
Mathematics games will help me in
learning ‘Multiplication and Division’
|
||||
