TEACHING HCF AND
LCM USING VIRTUAL
MANIPULATIVES
By:
Norazlinawati Hj A. Magon (11M8128)
Safiah binti Hj Yakup (11M8129)
Didinawati binti Hj Zunaidi (11M8131)
Introduction
• The new National education system
21st century in Brunei Darussalam
promotes the use of technology in
teaching and learning of mathematics
in schools.
• The mathematics curriculum supports
the developments in children by placing
clear emphasis upon the sensible use
of ICT in classroom (CDD, 2008 p.9)
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Emphasis use of ICT
Technology
assisted
instruction
Interactive
learning
Use of ICT in
concept
development
Problem
solving
Modelling
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Background
• This research investigated whether the
use of virtual manipulative could improve
pupils’ achievement and attitudes in
mathematics.
• The researchers explored the use of
technology in the teaching and learning of
factors and multiples in Year 6.
• The researchers used free virtual
manipulative software obtained online
from http: //nlvm.usu.edu/.
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Background
• The software ‘Factor Tree’ was used to
help the pupils to learn the topics found to
be difficult for pupils.
• Reasons of using Virtual manipulatives
to teach the Factors and Multiples :
a) Provide learners with visual images.
b) It can be manipulated as physical
manipulatives and provide students
opportunities for constructing
mathematical knowledge.
c) To promote effective teaching.
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Background
• Benefits of using Virtual
manipulatives :
a) It can help students visualize
relationships and engage students
learning actively.
b) Useful to help the students to
understand better.
c) It will arouse students’ interest in
learning mathematics.
d) Pupils can work out the answer faster
by using the virtual manipulatives.
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Background
• Benefits of using Virtual
manipulatives :
e) It can help students visualize
relationships and engage students
learning actively.
f) Useful to help the students to
understand better.
g) It will arouse students’ interest in
learning mathematics.
h) Pupils can work out the answer faster
by using the virtual manipulatives.
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Research Questions
• Are there any improvement in
students’ achievement or scores in
answering questions on factors
(HCF and LCM) after using Virtual
Manipulatives?
• Do students’ attitude towards
mathematics affect their
mathematics achievement?
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Literature Review
• Abstract concepts are essential to
understanding and performing
mathematics. They are also a
source of difficulty for many
students who struggle with
mathematics, many of whom find
even basic mathematics concepts
difficult to understand.
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Literature Review
• A popular approach to help students
understand abstract concepts is the
use of manipulatives.
• Manipulatives enable students and
teachers to represent concretely the
abstract concepts that they are
learning in mathematics class and to
link these concepts to prior
knowledge.
• Manipulatives are useful to introduce
new concepts to all students
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Literature Review
• For the most part, classroom use of
manipulatives has involved concrete, or
physical, manipulatives. However, with
the advent of the World Wide Web,
there is a new category of
manipulatives.
• Virtual manipulatives are basically
digital “objects” that resemble physical
objects and can be manipulated,
usually with a mouse, in the same ways
as their authentic counterparts.
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Literature Review
• Virtual manipulatives are a relatively
new technology and are modeled
after existing manipulatives such as
base ten blocks, coins, blocks,
tangrams, spinners, rulers, fraction
bars, algebra tiles, geoboards,
geometric plane and solids figures.
• They are usually in the form
of Java or Flash applets.
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Literature Review
• Virtual manipulatives :
1) allow teachers to allow for efficient
use of multiple representations and
2) to provide concrete models of
abstract mathematical concepts for
learners of mathematics.
• Research suggests that students may
also develop more connected
understandings of mathematical
concepts when they use virtual
manipulatives (Moyer, Niezgoda, &
Stanley, 2005)
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Methodology - instruments
• The researchers used both
quantitative and qualitative
methods:
a) pre- and post-tests
b) questionnaires on attitudes
towards mathematics
c) feedback on using the virtual
manipulatives
d) informal observation of pupils
attitude in class
e) Teacher’s reflection form
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Methodology - instruments
• Face validity, content validity and
reliability
 The test items were shown to our lecturer
to be checked and finalized.
 The items were constructed based on
Primary School Examination (PSR) format
questions. The final questions were
checked by the Year 6 mathematics
teachers before given to the pupils.
 The alpha reliability of the 10 items is
0.803, which rounded to two decimal
places is 0.80, indicating that the scale
had good reliability.
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Methodology- samples
• The researchers used one intact
class for the study. There were
twelve boys and sixteen girls in
the sample.
• The researchers worked together
with the class teacher in delivering
the lessons.
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Methodology- samples
• The researchers found that at the
beginning, out of 28 Year six
pupils, majority were not able to
correctly find the LCM and HCF of
given set of numbers.
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Findings and Results
• Initially the pupils were quiet,
listening to a demonstration by the
class teacher and then later the
researchers noticed that the
students were enthusiastic and
deeply engaged with the activities
in the lessons.
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Findings and Results
• The researchers found at the end
of the study, Cohen’s effect size, d
= 1.83, indicating a large effect at
the 96.4th percentile.
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Findings and Results
Table 2 shows the mean score for pre-test and post-test of 28
pupils. The mean score for pre-test = 0.07 and post-test = 3.68.
There is a significant difference in the mean scores between the
pretest and posttest at p < 0.025 level (2-tailed). The pupils
performed significantly better in their posttest scores. This
implies that the use of virtual manipulative (Factor Tree) may
be effective in enhancing pupils’ academic performance.
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Findings and Results
Table 2 shows that female pupils (M=4.00, SD =
2.98) performed better than male pupils (M = 3.18,
SD = 2.48). But Cohen’s d=0.3 indicated only small
effect.
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Findings and Results
There is no significant relationship between attitude towards
mathematics and mathematics scores (r= 0.232, p = 0.24).
Pupils’ attitude towards mathematics did not affect the total
mathematics scores.
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Findings and Results
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Conclusion
On attitude toward mathematics:
• 75% like learning mathematics.
• 69% (19) said mathematics is fun to
learn.
On Feedback on Virtual Manipulatives
• 86% of the students like virtual
manipulatives; 61% find it easy to play;
50% know how to find the prime
factors, 64% LCM and 68% HCF using
the manipulatives; 79% like the topic
and 82% enjoy the lesson.
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Conclusion
Teacher reflection form:
• The benefits of using the virtual
manipulatives as:
a) useful to help pupils to understand
better
b) arose pupils’ interest in learning
mathematics
c) pupils can work out the answers faster
• The objectives of the lesson were
achieved as the pupils could use the
virtual manipulatives software
effectively in answering the questions
given
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Conclusion
Teacher reflection form:
• The software also worked in
developing the lesson plan as pupils
use the virtual software in expressing a
given number as a product of prime
factors.
• She also stated that the new method
(using the software) in finding the GCF
(HCF) and LCM is simple and easy to
understand.
• The software is also only need simple
working to get the answer.
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Conclusion
From the mean scores of 0.7% correct
responses in the pre-test and 36.8% in
the post-test, it can be seen there is an
improvement in student’s achievement
in answering questions on factors (HCF
and LCM) after using virtual
manipulatives.
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Recommendations
• The concepts and skills to be
learned from the virtual
manipulatives activities (in
teaching the Factors and Multiples)
are:
• Can make connections between the concepts,
extension and generalizations in order to
solve the problems.
• Identify prime and composite numbers.
• Find factors and multiples of whole number
less than 100 and identify which are prime or
composite.
• Identify common factors of a set of whole
numbers.
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Recommendations
• The concepts and skills to be
learned from the virtual
manipulatives activities (in
teaching the Factors and Multiples)
are:
• To manipulate and change the
representations, thus increasing exploration
possibilities to develop concepts and test
hypotheses
• Know how to use technology.
• Learn by using factor tree method through
games.
• To develop their relational thinking and to
generalize mathematical ideas
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Recommendations
• Related Technological Activities to
Factors and Multiples
• http://www.mathgoodies.com/factors/fa
ctor_tree.asp
• http://www.math-play.com/Factors-andMultiples-Jeopardy/Factors-andMultiples-Jeopardy.html
• http://www.toonuniversity.com/flash.asp
?err=499&engine=14
• http://www.mathplayground.com/factort
rees.html
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Bibliography
• Curriculum Development, Ministry of
Education, Brunei Darussalam (2009).
Framework and Guidelines for Curriculum and
Assessment Mathematics Year 6.
• National Library of Virtual Manipulatives
(2006). Retrieved March 3, 2012 from
http://nvlm.usu.edu/en/nav/vlibrary.html. Utah
State University.
• Yuan, Y. (Dec, 2009).Taiwanese elementary
school teachers apply web-based virtual
manipulatives to teach mathematics. Journal
of Mathematics Education, 2 (2), 108 - 121.
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teaching hcf and lcm using virtual manipulatives