Sir Victor Raymundo-9b-The-Concept-of-Number_Operations-on-Whole-Number.pptx
1. How Children Learn Math:
The Concept of Number
(Operations)
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2. Is Arithmetic = Mathematics
?
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Arithmetic is the study of numbers.
Mathematics is a way of thinking in
logical or organized way.
*Math is not a subject or a set of facts that needs to be
memorized for a test. Math involves solving problems.
3. THE
FOUNDATION:
COUNTING
Understanding of arithmetic
evolves from children’s early
counting experiences.
Informal concept of addition
(adding more) and
subtraction (taking away
something) guide children’s
efforts to construct informal
arithmetic procedures.
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4. When knowing ‘facts’ is the objective, children are
taught techniques to get ‘facts’ and are drilled to
internalize them. (traditional addition)
When a child learns to add quantities by using his own
logic, repeats the same action every day in interaction
with other people (e.g. while at play), he will inevitably
remember the result, without adult pressure.
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ADDITION as an OBJECTIVE
5. The objective in subtraction, as in addition,
should be to encourage children to think and to
remember results of their own thinking.
Once children have constructed sums and have
committed them to memory, they are able to
express this knowledge in subtraction.
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SUBTRACTION as an
OBJECTIVE
6. THE LOGIC OF
ADDITION & SUBTRACTION
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The understanding of addition and
subtraction involves:
1. the logic of inclusion
2. reversibility of thought
7. The Concept of
Three
(Concept Level)
“Ilang kulisap ang
nasa kaliwang
kamay?
Ilang kulisap ang nasa
kanang kamay?
Ilan lahat ng kulisap?”
Hand Game
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8. The Concept of
Three
(Concept Level)
“Ilang kulisap ang
nasa ibabaw ng
mangkok?
Ilang kulisap ang nasa
ilalim ng mangkok?
Ilan lahat ng kulisap?”
Lift the Bowl
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9. The Concept of
Three
(Concept Level)
“Ilang kulisap ang
nasa labas ng
bahay?
Ilang kulisap ang nasa
loob ng bahay?
Ilan lahat ng kulisap?”
Peek Thru The Wall
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10. Stages of Concept
Development
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Level Teacher … Children …
Concept
Level
... tells number
stories or gives
verbal instructions
(in the mother
tongue)
... manipulate
concrete
materials
11. Stages of Concept
Development
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Level Teacher … Children …
Connecting
Level
... continues to give
verbal instructions or tell
number stories
... introduces writes
mathematical symbols
(e.g. numerals)
... introduces
mathematical language
(e.g. number words in
English)
... continue to
manipulate concrete
materials
... read
mathematical
symbols
12. The Concept of Three
(Connecting Level)
“Anong numero ang ilalagay
mo sa kaliwang kamay?
Anong numero ang ilalagay
mo sa kanang kamay?
Basahin natin ang nakasulat
sa counting mat.”
Hand Game
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13. The Concept of
Three
(Connecting Level)
“Anong numero ang ilalagay
sa ibabaw ng mangkok?
Anong numero ang ilalagay
sa ilalim ng mangkok?
Basahin natin ang nakasulat
sa counting mat.”
Lift the Bowl
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14. The Concept of
Three
(Connecting Level)
“Anong numero ang ilalagay
sa labas?
Anong numero ang ilalagay
loob?
Basahin natin ang nakasulat
sa counting mat.”
Peek Thru The Wall
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15. Stages of Concept
Development
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Level Teacher … Children …
Symbolic Level ... continues to give verbal
instructions or tell number
stories
… introduces and writes
more mathematical
symbols (e.g. + − =)
… introduces more
mathematical language
(e.g. plus, minus, equals)
... continue to
manipulate concrete
materials
... continue to read
symbols
... begin to record or
write symbols
18. Peek Thru the Wall
worksheet
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19. More Counting:
Counting
Groups
“May 2 lababo sa
kusina.
Sa bawa’t lababo ay
maroong tig-4 na
maruruming plato.
Ilang plato lahat ang
kailangang
hugasan?”
Beginning Multiplication
(Concept Level)
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20. BEGINNING MULTIPLICATION
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• The process of multiplication requires
children to think about and count groups
of objects rather than single objects.
• Terms such as stacks of, rows of, groups
of, and NOT ‘times’, help children think
and visualize problems
21. More Counting:
Counting
Groups
“Mayroong 10 upuan
at 2 mesa sa hardin.
Ilang upuan ang
maaring ilagay sa
bawa’t mesa?”
Beginning Division
(Concept Level)
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22. BEGINNING DIVISION
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Two situations that call for division:
1. The grouping process is the dividing of a quantity of
objects into smaller groups of a particular size to
determines the number of groups that can be made.
1. The sharing process is the dividing of a quantity of
objects into a particular number of groups to
determine the number of objects in each group.
23. The Logic of
Multiplication and
Division
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The understanding of multiplication (and
division) involves:
1. an understanding of the one-to-one
correspondence or equivalence of two or
more sets (multiplicative equivalence)
2. reversibility of thought
25. More Number Concepts
(Division - Connecting
Level)
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Chairs Tables Chairs
in
each table
Remainder
8 2
7 2
4 2
6 2
9 2
27. QUESTIONS
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1. Why is it important to teach mathematics for
children in a logical way rather than teaching
a series of rules to memorize?
1. How do children learn the operation of
number using his own logic without adult
pressure?
28. ARITHMETIC ≠ RULES
Arithmetic should not be
taught as a series of rules
because:
• Rules are easy to forget
• Rules minimize thinking.
• Rules prevent visualizing
relationships.
• Rules do not work well for
problem-solving
2nd grade students selling potato
balls at the curriculum fair
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29. REFLECTIONS
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1. What developmentally appropriate activities
have you provided your learners in the past
months?
2. What gains might learners acquire if we provide
more hands-on/concrete activities?
3. What possible difficulties might be encountered
when providing more hands-on activities?
4. What classroom management skills would a
teacher need to have in order to manage a
class that is involved in more hands-on
activities?
30. 1. Develop a solid base (informal understanding)
before introducing written symbolism.
2. Structure informal calculational experiences to
promote discovery.
3. Help children see that formal symbolism is an
explicit expression of their informal knowledge.
4. Sequence formal mathematics to exploit children’s
informal knowledge.
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EDUCATIONAL IMPLICATIONS
31. THERE ARE NO
SHORT CUTS TO
LEARNING MATH
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32. References:
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• Association for Supervision and Curriculum Development. The Brain and Mathematics. Association
for Supervision and Curriculum Development Press, Alexandria, Virginia, 2001.
• Baratta-Lorton, Mary. Mathematics Their Way. Addison-Wesley Publishing Company, Menlo Park,
California, 1976.
• Baratta-Lorton, Robert. Mathematics a Way of Thinking. Addison-Wesley Publishing Company,
Menlo Park, California, 1977.
• Baroody, Arthur. Children’s Mathematical Thinking. A Developmental Framework for Preschool,
Primary and Special Education Teachers. Teachers College Press, New York, 1987.
Burns, Marilyn. About Teaching Mathematics, a K-8 Resource, 2nd edition. Math Solutions
Publications, Sausalito, California, 2000.
Copeland, Richard. How Children Learn Mathematics. Teaching Implications of Piaget’s
Research.
MacMillan Publishing Co., Inc. 1979
Hohmann, Mary and David P. Weikart. Educationg Young Children. Activve Learning Practices
for Preschool and Child Care Programs. High Scope Press, Ypsilanti, Michigan, 1995.
Kamii, Constance Kazuko. Young Children Reinvent Arithmetic. Implication of Piaget’s Theory.
Teachers College Press, New York, New York, 1984.
Moomaw, Sally and Brenda Hieronymus. More Than Counting, Whole Math Activities for
Preschool and Kindergarten. Redleaf Press, St.Paul, MN. 1995.
• Sousa, David. How the Brain Learns Mathematics. Corwin Press, California. 2008
• Stenmark, Jean Kerr, Virginia Thompson and Ruth Cassey. Family Math. Regents, University of
California, 1986.
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