APPENDIX
2.1
BACKGROUND OF MULTIPLICATION
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Page
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References
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Quotes
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Notes
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1
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Iliminations, All
About Multiplication, Resources 4 Teaching Math
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Explore several meanings and representations of
multiplication (number line, equal sets, arrays, and balanced equations).
They also learn about the order (commutative) property of multiplication, the
results of multiplying by 1 and by 0, and the inverse property of
multiplication. In addition, students write story problems in which the
operation of multiplication is required.
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Multiplication
have 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.
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2.2
BACKGROUND OF DIVISION
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Page
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References
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Quotes
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Notes
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1
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Lisa Baggio, Evelyn
Hwang, Yoojin Kim, Andrea Solorza (UCIrvine, August 2009), Pedagogical
Content Knowledge Project.
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division are key
components in mathematics and is used in all levels of math starting in the
second grade
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Division is key essential
components in mathematics because related to the concept of addition and
subtraction. Moreover, this topic learns when in second grade.
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9
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Leanna
Horton
May(2007), Understanding the Concept of Division
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The concept of
division is one that many students and teachers have problems understanding. Ball (1990a) found that prospective teachers
are often unable to properly explain the underlying meaning behind division problems and cannot generate representations
appropriate to some division problems
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According
Ball (1990a), he 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.
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1
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Houghton Mifflin Company(1999 ), Division
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As with addition, subtraction, and multiplication, students
progress by learning algorithms that allow them to perform operations beyond
basic facts. After students learn their basic division facts and the concept
of division, it is time to introduce algorithms that will allow them to
divide larger numbers. It is important to show the students that there is a
need to learn how to use algorithms to divide larger numbers.
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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
introduce to learn how to use algorithms to divide larger numbers.
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2.3 CONCEPT OF DIVISION AND MULTIPLICATION
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Page
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References
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Quotes
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Notes
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1
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What
are the properties of multiplication?
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Properties of
multiplication
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1
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Properties of Division
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Properties of
Division
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2.4 Misconception in multiplication and
division
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Page
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References
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Quotes
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Notes
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1
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Graeber,
Anna O (March 01, 1993 ), Misconceptions about
multiplication and division.
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Misconceptions or naive conceptions are commonly held ideas
or beliefs that are contrary to what is formally acknowledged to be correct.
Mathematics educators study misconceptions because if we can understand how
students are apt to see mathematical ideas, we may be better prepared to
offer instructional experiences that help them develop accepted conceptions.
The two misconceptions just described have been the subject of many research
investigations (see, e.g., Bell, Fischbein, and Greer [1984]; Fischbein,
Deri, Nello, and Marino [1985]; Greer [1987]).
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There are many
research that showed misconception in Mathematics such as Bell, Fischbein, and Greer [1984]; Fischbein, Deri, Nello,
and Marino [1985]; Greer [1987]). Students especially who low academic will misunderstand
about mathematics conception because the acknowledged of students itself. The
teachers should really prepared before teach the students so that the
students will really understand
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1
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Graeber,
Anna O (March 01, 1993 ), Misconceptions about
multiplication and division.
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The two misconceptions "multiplication makes
bigger" and "division makes smaller" are often noticed only
when students attempt to solve multiplication or division word problems
involving rational numbers less than one. Faced with a word problem, students
realize from the contextual clues in the problem that the answer should be
greater than or less than one of the numbers in the problem.
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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 answer s will
greater or smaller.
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8
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Misconceptions
and Errors Mathematics Navigator
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The student has over specialized his knowledge of
multiplication or division facts and restricted it to “fact tests” or one
particular problem format. Student completes multiplication or division facts
assessments satisfactorily but does not apply the knowledge to other arithmetic
and problem-solving situations.
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The student’s
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.
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1
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Graeber,
Anna O (March 01, 1993 ), Misconceptions about
multiplication and division
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Students' early experiences with division are also limited
to whole-number divisors and whole-number quotients. This limitation leads
students to place restrictions on division that are not necessarily true with
rational-number divisors and quotients.
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Because of the
limited in early experiences, students knowledge only to whole-number
divisors and quotients.
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1
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Misconceptions
and Errors Mathematics Navigator
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Example of
multiplication and division misconception
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1
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Errors and
misconceptions: Year 4 multiplication and division,
The national
strategies
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Example
involves knowledge and skills
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2.5 MATHEMATICAL KNOWLEDGE
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Page
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References
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Quotes
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Notes
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7
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Deborah
Loewenberg Ball, What Mathematical Knowledge is Needed for Teaching
Mathematics?
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teachers need
to know the same things that we would want any educated member of our
society to
know, but much more
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Before teacher
teach, they must really know about what will they teach for students
understanding.
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8
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Deborah
Loewenberg Ball, What Mathematical Knowledge is Needed for Teaching
Mathematics?
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knowledge for
teaching mathematics is different from the mathematical knowledge needed for
other mathematically-intensive occupations and professions
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Mathematics teacher’s
knowledge not same with others professional. So, what will teacher face in
problem will not same with others.
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8
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Deborah
Loewenberg Ball, What Mathematical Knowledge is Needed for Teaching
Mathematics?
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the mathematical
knowledge needed for teaching must be usable for those
mathematical
problems
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Any
mathematical knowledge that the students learn can solve their daily
problems.
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