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STUDY GUIDE 7 8 9

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FM-AA-CIA-15 Rev. 0 10-July-2020
Study Guide in ECED 106 Numeracy Development
Module No. 7
STUDY GUIDE FOR MODULE NO. 7
COUNTING: MORE THAN JUST 1, 2, 3
MODULE OVERVIEW
Whether at home or at school, naming numbers and counting has been at the core of preschool
mathematics. This study guide will teach you everything you need to know about teaching counting in
preschool.
MODULE LEARNING OBJECTIVES
At the end of this module, you should have:
1. explained the big ideas on counting;
2. demonstrated rational counting skills through authentic experiences;
3. highlighted number pattern and structure to advance rational counting skills;
4. used routines to practice counting;
5. created activities in counting more than just 1,2,3;
6. demonstrated engagement and enjoyment in facilitating mathematical activities to nurture and inspire
learner participation; and
7. designed an assessment activity for Counting.
LEARNING CONTENTS
QUANTIFICATION
Quantification is the ability to recognize that all numbers are associated with an exact quantity, and the
ability to recognize sets of objects, such as dots on a dice. This is sometimes referred to as subitizing.
SUBITIZING: A FOUNDATIONAL CONCEPT FOR MATHEMATICS SUCCESS
Subitizing is the ability to accurately and rapidly identify a small amount of items without having to
count. It is immediately knowing what number is rolled on a six sided dice. Most adults don’t need to count
the pips. The number comes automatically because the brain can subitize.
The following are three activities to teach subitizing:
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Quick Dot Images-Use colored dot stickers to make a set of cards with a set number of dots on each
card. For beginners, dots should follow the most easily recognized pattern, like the pattern of pips on
dice. Beginners should also start with small numbers, nothing higher than five or six. Select a card
and flash it before the students, showing them the image for no more than three seconds at most,
preferably shorter. The goal is to recognize the set as quickly as possible without having to actually
count.
Quick Dot Images Look Alike– Play the same games as above, but instead of students calling out
the number, have them use manipulatives to create the same set on their tray or desk.
Concentration-Make a double set of cards and have students play the matching game
concentration. If playing with young children, such as preschoolers through first grade, consider using
number sets up to three, but in multiple color sets. This way, the students gets lots of practice with
the smaller numbers without getting too frustrated with trying to differentiate the larger sets. Once
near mastery is evident, larger numbers can be added, as well as less familiar configurations.
COUNTING
Counting is the skill of matching set of objects with their corresponding number name. This is sometimes
referred to as one to one correspondence. It is that ability to count 1, 2, 3, while distinguishing that there are
indeed three objects that were counted, no more and no less.
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ONE TO ONE CORRESPONDENCE
Here are some inexpensive ideas for counters:
 buttons
 cleaned coins
 pom poms
 wooden disks
 floral marbles
 large beads
 dried pasta
 small erasers
 party favors
 dried lima beans
 small rocks
 large legos
NUMBER REPRESENTATION
Number representation is the ability to identify the number that corresponds with a quantity. This is also
called number identification or number recognition. For example, it is the ability to identify that the symbol
6 refers to the word “six” and the same quantity.
NUMBER MATCHING
Every household has a deck of number cards, right? Maybe not a traditional deck of poker cards, but
some card game with large numbers printed on colorful cards, like Phase Ten or Uno.
These decks of cards make great items to add to your teaching toolbox. They much more durable than
paper print outs and an incomplete set is still useful for teaching your little one about counting and numbers.
This is a simple activity, but a great one for number identification practice, sorting, and critical thinking. It is a
simple number matching game, with a few variations of mix things up a bit.
Supplies needed are a single deck of cards. William and I pulled out an old deck of Uno that had several
missing cards, which worked to our learning advantage. I pulled out the non-number cards and shuffled the
deck, then asked William to pull one off the top. 5.
Now the rest of the game goes as follows: take turns flipping over cards. I specify to take turns and this
game along side your child because modeling your thought process has great academic advantages for your
youngster, and it is a skill that children learn in school. (The technical term is metacognition–which means to
think about your thinking process). If the card matches the first number drawn (5) lay it next to the first. If it
does not match, set it aside in a discard pile. Discriminating between numbers is an important skill in
emergent math.
Once all the cards have been used or discarded, count the number of cards that matched the first once
drawn.
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If there are multiple cards of the same color, those can be sorted into piles of their own, and then
counted. This simple activity can even be graphed so your child can recognize more and less.
WHAT IS PRESCHOOL COUNTING?
By definition, counting means to take an account of a group of items to come up with a total.
Preschoolers love to count. They relish in participating in counting games, songs, and rhymes, but to truly be
capable counters, children must be able to do much more than simply recite 1, 2, 3, and so on.
THE FIVE STAGES OF TEACHING COUNTING
In fact, according to the book Teaching Mathematics in Early Childhood by Sally Moomaw, there are
five principles of counting that children must master.
STABLE ORDER
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Stable order is rote counting in the correct order using number names. It is basic 1, 2, 3, 4, and so on.
When a child is deloping stable order counting skills he may skip numbers, 1, 2, 3, 5, 6… or may use the
same number more than once, 1, 2, 3, 2…
ONE-TO-ONE CORRESPONDENCE
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One-to-one correspondence is the understanding that each item is counted only once and one at a time.
For example, when counting a set of counting bears, the child does not count any one bear twice and
does not count any bear with another, assigning the same number.
CARDINALLY
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Cardinally is the understanding that counting is quantitative and the last number named is the total of the
group. For example, when a child is counting a group of five butterflies, she counts 1, 2…5 with the
understanding that the number of butterflies counted is five, not any of the previous numbers counted.
ORDER IRRELEVANT
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The order of counting does not affect the total. For example, when counting a line of buttons, the buttons
can be counted forward, backward, or in any order and the total amount will remain the same.
ABSTRACTION
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Eventually, children come to understand that both objects and ideas can be counted. One example of this
is actually in phonological awareness lessons where children are asked to listen for and count words in
sentences.
COUNTING IS REALLY IMPORTANT, AND HERE’S WHY
Research continues to stress the importance of building strong mathematical foundations in preschool
because the skills a preschooler brings with them into kindergarten has a strong influence on
their trajectory through elementary school. Children who tend to have the highest math scores at the end of
first grade are the same children who come into kindergarten knowing how to recite and count to 20.
Unfortunately, some studies state that as few as 10% of children entering kindergarten have proficient
counting skills.
The same study established that the reason for this might be because parents often think it the teacher’s
responsibility to teach counting, while the teachers pass the responsibility to the parents. Despite who is
actually responsible, there is no doubt that without adequate counting skills children have little chance of
being successful in mathematics.
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LEARNING ACTIVITY 1
Create a lesson plan for Numeracy Development following the sample format below.
I. OBJECTIVES:
At the end of the lesson, the students should be able to:
a. ;
b. ; and
c. .
II. SUBJECT MATTER:
Topic:
References:
Materials:
Values Integration:
III. PROCEDURE:
TEACHER’S ACTIVITY
PUPILS’ ACTIVITY
A. Preliminary Activities
a. Greetings
b. Prayer
c. Checking of Attendance
d. Setting of the Classroom Rules
e. Drill
f. Review
g. Unlocking of Difficulties
B. Development of the Lesson
a. Motivation
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b. Presentation
c. Lesson Proper
d. Generalization
e. Application
f. Values Integration
IV. EVALUATION:
V. ASSIGNMENT:
SUMMARY
The number recognition & related skills have vital roles to play in almost all walks of one’s life. The
number recognition skills build upon the initially developing number sense in a child, i.e., a child takes notice
of the number of objects in a group. Secondly, the child is supported in learning how to count, whereby
cardinality is acquired, practiced & exercised. Additionally, these skills are strong support in supplementing
the learning of arithmetic concepts.
REFERENCES
https://stayathomeeducator.com/one-to-one-correspondence-counting-activity/
STUDY GUIDE FOR MODULE NO. 8
NUMBER OPERATIONS: EVERY OPERATION TELLS A STORY
MODULE OVERVIEW
Many parents feel rather confident in teaching their child early mathematics skills such as the basics of
number recognition, one to one correspondence in counting, sorting activities, and knowledge of shapes. But
what about addition and subtraction?
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Here is a brief account of how an understanding of the operations develops in the preschool years. Young
children’s understanding of the operations develops out of their understanding of counting.
MODULE LEARNING OBJECTIVES
At the end of this module, you should have:
8. explained the big ideas on number operations through varied activities;
9. applied children’s strategies for problem-solving;
10. demonstrated engagement and enjoyment in facilitating mathematical activities to nurture and inspire
learner participation; and
11. designed an assessment activity for Number Operations.
LEARNING CONTENTS
Young Children Demonstrate Math Concepts in Everyday Activities
Young children, even infants, develop basic, non-verbal concepts of quantity: more/less, order, same, and
adding/subtracting. Children develop a rudimentary understanding of these concepts on their own, without
much adult help, often using them in everyday life, as in determining who has more or fewer cookies. As
young children develop understanding of counting they also learn about addition and subtraction.
Idea of adding as producing more and subtracting less. Children learn that that:
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When you add something to an existing set, the result is that you have more than you had at first.
If you start with two groups of the same number, and by magic (while the child is not looking), one set
is now more numerous than the other, you must have added to one or subtracted from the other.
Children don’t have to count to arrive at these judgments concerning more and concerning addition
and subtraction: they can solve the problem by reason alone.
Later instruction needs to build on all of these ideas when written numbers are introduced.
Everyday Numerical Addition and Subtraction
The story now is how concepts of more/less, order, same, adding and subtracting without exact number
(for example, adding means making a set larger without knowledge of the exact number), and enumeration
get elaborated to create numerical addition and subtraction. Children learn some of this on their own, but
adults can and should help.
Concepts to be learned to understand addition (subtraction is similar):
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Addition can be thought of in several ways, including:
Combining two sets, for example, pushing together 5 crayons and 3 crayons to get a new group of 8
crayons
Increasing the size of one set, for example, beginning with 5 crayons and then adding 1 crayon at a
time until 3 crayons have been added and the new group has 8 crayons
Moving forward on a number line, for example, moving 5 spaces on a number line and then 3 more to
get to space 8
Simple counting is also adding—1 at a time
The order of addition makes no difference (the commutative property)
Adding zero changes nothing
Different combinations of numbers can yield the same sum
Addition is the inverse of subtraction, for example, if taking away 5 from 8 yields 3, then adding 3 to 5
yields 8. This idea is crucial later on when children learn “fact families.”
Strategies used to add (or subtract):
Children often begin by using concrete objects and fingers to add but gradually learn mental calculation
and remember some of the sums
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Module No. 7
Using concrete objects, children may do the following to solve a simple problem like 3 + 2:
Count all: I have 3 here and 2 there and now I push them together and count all to get 5.
Count on from the smaller: I can start with 2 and then say, 3, 4, 5.
Count on from the larger: I can start with 3 and then say, 4, 5.
Approaching the problem mentally, children may solve the problem in these ways:
Building on what is known (“derived facts”): I know that 2 and 2 is 4, so I just add 1 to get 5.
Memory: I just know it!
More features of numerical addition and subtraction:
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It’s always useful to have backup strategies in case one doesn’t work: If unsure about memory, the
child can always count to get the answer
It’s important for the child to be able to check the answer
The child needs to learn different strategies for different set sizes: counting one by one is good for
adding small sets but tedious and inefficient for larger sets
The child should also be able to describe how he got the answer: self-awareness is one aspect of
“metacognition.” Of course, remembering what you just did is essential for describing it in words.
Language is vital for describing one’s work and thinking and to convince others; children need to learn
mathematical vocabulary
The child should be able to apply the math in real situations or to stories about real situations (like
word problems)
Groupings
Fair Shares. Young children are very much concerned with fairness. If you give Debbie one cookie, you
had better give exactly one to her twin Becky as well. If you give two to Becky, you must also give two to
Debbie. In this case, you are creating fair shares or groups—1 and 1, 2 and 2.
Suppose that Debbie and Becky have a birthday party and 8 friends attend. To ensure fairness (and keep
the peace) the parent meticulously goes through the fair share routine of giving one grape at a time to each
child until all the grapes are gone. The result is that each of the children (ten in all, including the twins)
receives 10 grapes. In this case, the parent has (conveniently for our explanation of the math) created 10
groups of 10 each, so that there are 100 grapes altogether. The result is eminently fair and mathematically
useful, for it provides the basis for several important ideas.
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The groups of 10 form the foundation of our counting system. There are ten tens and we can count
them cumulatively as 10, 20, 30…100.
The grouping creates the basis for division. In this case, the parent has divided 100 items into 10
groups of 10 each. Of course, if there were 30 grapes, the division would involve 3 grapes for each of
the 10 children.
The situation allows for multiplication as well: If we know that there are 5 groups, all the same
number, with 4 in each, we can multiply to get 20 in all.
Of course we should not attempt to instruct young children in multiplication and division facts and
algorithms. But we can engage them in counting by tens, employing one-one correspondence to create fair
shares, determining the fairness of existing collections, and correcting unfairness when they see it.
Number Sense
Children need to develop number sense, a concept that is notoriously difficult to define in a simple and
exclusive way. I like to think of it as mathematical street smarts, which can be used in just about any area of
number, including those discussed above. Number sense, which helps the child to make sense of the world,
has several components, each of which undergoes a process of development.
Thinking instead of calculating. Number sense involves using basic ideas to avoid computational
drudgery, as when the child knows that if you add 2 and 3 and get 5, you don’t have to calculate to get the
answer to 3 and 2.
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Use what is convenient. Number sense involves breaking numbers into convenient parts that make
calculation easier, as when we mentally add 5 + 5 + 1 instead of 5 + 6.
Knowing what’s plausible or impossible. Number sense may involve a “feel” for numbers in the sense
of knowing whether certain numbers are plausible answers to certain problems (if you are adding 2 and 3 you
know that the answer must be higher than 3; anything lower is not only implausible but impossible).
Understanding relationships. Number sense involves intuitions about relationships among numbers—
this is “way bigger” than that.
Fluency. Number sense involves fluency with numbers, as when the child knows immediately that 8 is
bigger than 4 or sees that there are 3 animals without having to count.
Estimation. This involves figuring out the approximate number of a group of objects and is related to the
notion of plausible answers.
The Transition to Written, Symbolic Math
Formal, symbolic mathematics can provide students with more powerful tools and ideas than those
provided through their informal everyday math. Formal math (and its use of symbols) developed in several
cultures and is now virtually universal. Children need to learn it.
Everyday origins and formal math. Children encounter math symbols in everyday life: elevator
numbers, bus numbers, television channels and street signs are among the many. Often parents, television,
and software activities introduce some simple symbolic math—like reading the written numbers on the
television or on play cards.
Schools certainly have to teach formal math. But doing so is not easy. Even though competent in
everyday math, students may have trouble making sense of and connecting their informal knowledge to what
is taught in school. Teachers often do not teach symbolism effectively. If children get off on the wrong
symbolic foot, the result may be a nasty fall down the educational stairs. So the goal for teachers is to help
children—even beginning in preschool—to understand why symbols are used, and to use them in a
meaningful way—connecting already known informal mathematics to formal symbolic mathematics. The
teacher needs to “mathematize” children’s everyday, personal math—that is, help them connect their informal
system with the formal mathematics taught in school. It’s not ill-advised or developmentally inappropriate to
introduce symbols to young children, if and only if the activity is motivating and meaningful. On the contrary, it
is crucial for the teaching of symbols to begin early on, but again, if and only if it is done in a meaningful way.
Here are key issues surrounding the introduction of formal math to young children:
Young children have a hard time connecting numerals and the symbols of arithmetic (+ and -) to
their own everyday math. They may add well but be confounded by the expression 3 + 2. It is as if the child
is living in alternate realities: the everyday world and the “academic” (in the pejorative sense) world. The
everyday world makes sense and the world of school does not. You think for yourself in the former and do
what you are told in the latter.
The = sign is a daunting challenge. The teacher intends to teach = as “equivalent,” and thinks she has,
but the child learns it as “makes” (e.g., 3 + 2 makes 5). This is a tale of how child egocentrism meets teacher
egocentrism but neither talks with the other.
The solution. We should not avoid teaching symbols but need to introduce them in a meaningful
way. This means taking account of what children already know and relating the introduction of symbols to that
prior knowledge. It also means motivating the use of symbols. Thus if you want to tell a friend how many
dolls you have at home, you need to have counted them with number words (symbols) and then use spoken
words (“I have five dolls”), written words (“I have five dolls” written on a piece of paper or a computer screen),
or written symbols (5) to communicate the result.
Manipulatives can help. Use of manipulatives can be effective in teaching symbolism and formal
math. But they are often used badly. The goal is not to have the child play with concrete objects but to use
these objects to help the child learn abstract ideas. The goal of manipulatives is to get to the point where
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children do not need them by putting them in the child’s head to use as needed in thought. For example,
suppose the child learns to represent tens and ones with base ten blocks. Given the mental addition problem
13 plus 25, the child may understand that each number is composed of tens (the 10 by 10 squares) and some
units (the individual blocks), and that solving the problem involves adding one ten and two more, which is
easy, and then figuring out the number of units. The mental images of the tens and ones provide the basis for
her calculation, part of which may be done by memory (one plus two is three) and part of which may be done
by counting on her fingers (five fingers and three more give eight).
LEARNING ACTIVITY 1
Make at least one (1) Instructional Material for Number Operations: Every Operation Tells a Story
SUMMARY
Children should have daily experiences with addition and subtraction strategies and activities. There are
many opportunities to teach addition and subtraction strategies and solve problems throughout the day. This
does not only happen at school in the classroom. These opportunities can be initiated as they arise in our
daily situations, such as a story in a book, setting the table, putting clothes away, and so forth.
There are many addition and subtraction strategies that should be taught when children are young.
Learning different strategies helps empower them to choose what works for them at that moment. When
children are young, they tend to rely heavily on using hands-on manipulatives to solve equations. As they
develop new skills and learn new strategies, they build their mental math skills and require math tools less
often.
REFERENCES
http://prek-math-te.stanford.edu/operations/what-children-know-and-need-learn-about-operations
STUDY GUIDE FOR MODULE NO. 9
PATTERN: RECOGNIZING REPETITION AND REGULARITY
MODULE OVERVIEW
Young children's ability to spot mathematical patterns can predict later mathematical achievement, more
than other abilities such as counting (Rittle-Johnson et al, 2016). Pattern awareness can vary a great deal
between individuals: we have all noticed children who arrange objects in radial patterns or make constructions
with reflective symmetry, while others pay no attention to pattern at all.
MODULE LEARNING OBJECTIVES
At the end of this module, you should have:
1. described and discuss patterns to build understanding of patterns;
2. recognized repetition and regularity through patterns;
3. explained the big ideas on patterns (regularity and repetition);
4. created activities for exploring patterns;
5. demonstrated engagement and enjoyment in facilitating mathematical activities to nurture and inspire
learner participation; and
6. designed an assessment activity for Pattern.
LEARNING CONTENTS
What is mathematical pattern awareness?
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Patterns are basically relationships with some kind of regularity between the elements. In the early years,
Papic et al (2011) suggest there are three main kinds:
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shapes with regular features, such as a square or triangles with equal sides and angles, and
shapes made with some equally spaced dots;
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a repeated sequence: the most common examples are AB sequences, like a red, blue, red blue
pattern with cubes. More challenging are ABC or ABB patterns with repeating units like red, green,
blue or red, blue, blue;
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a growing pattern, such as a staircase with equal steps.
Children who are highly pattern aware can spot this kind of regularity: they can reproduce patterns and
predict how they will continue.
Why is pattern awareness important?
Spotting underlying patterns is important for identifying many different kinds of mathematical
relationships. It underpins memorization of the counting sequence and understanding number operations, for
instance recognizing that if you add numbers in a different order their total stays the same. Pattern awareness
has been described as early algebraic thinking, which involves:
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noticing mathematical features
identifying the relationship between elements
observing regularities
The activity Pattern Making focuses on repeating patterns and suggests some engaging ways of
developing pattern awareness, with prompts for considering children's responses. Children can make 'trains'
with assorted toys, make patterns with twigs and leaves outside or create printing and sticking patterns in
design activities.
It is important to introduce children to a variety of repeating patterns, progressing from ABC and ABB to
ABBC. In the past we have tended to emphasize alternating AB patterns with young children: however this
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can result in some children thinking that 'blue, red, red' can't make a pattern. They say things like, 'That's not
a pattern”, because you can't have two of the same color next to each other' (Papic et al, 2011).
Copying patterns is important, according to Papic et al (2011), who suggest that the subsequent
conversation between teacher and child, comparing their construction with the original pattern, is crucial to
developing pattern awareness. It helps them to focus on 'What is the same and what is different?' and to see
the underlying pattern structure. Some young children will find this difficult, while others can replicate the
pattern using different colors or objects.
Some children will copy a pattern successfully, but one cube at a time, and are unable to continue it.
Those who isolate the 'unit of repeat' in an AB pattern may continue the pattern by picking up a red and a blue
cube together. The significance of identifying the 'unit of repeat' is in seeing a group of objects as one item,
which is a sophisticated development for young children. This leads to multiplicative thinking, which involves
counting groups instead of single items, as in counting the number of pairs of hands. Recognizing the 'unit of
repeat' also involves part-whole awareness, in simultaneously seeing the unit as a whole and the items within
it. This is important for understanding number composition and seeing numbers as made up of other numbers.
Children who are good at 'pattern spotting' can see the overall structure and identify the pattern rule.
When children make their own patterns, a helpful question is, 'What is the rule for your pattern?' or with
younger children, 'What is your pattern called?' Children who have understood a pattern structure can
translate it to different materials, generalizing the pattern. One five year old, when told his pattern was an
ABBC pattern, said, 'So it could be dog, cat, cat, sheep?'
Children can also spot a mistake in pattern and some may instantly see how to repair it, while others may
have to start from the beginning to make the correction or make a 'local' adjustment which just moves the
error along.
Younger children enjoy making up action patterns, like 'head, shoulders, knees and toes' and delight in
spotting errors in the sequence. Five year olds can devise their own ways of recording action patterns, which
involves 'translating' the pattern into a different mode, using pictures and symbols.
Older children also appreciate growing patterns, like the 'staircase' pattern which underlies traditional
stories and rhymes, such as The enormous turnip, The gingerbread man, There was an old woman who
swallowed a fly, The twelve days of Christmas and books such as the Hungry Caterpillar and many others.
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Patterns are also an important element of spatial thinking and geometry, for instance with reflective and
rotational symmetry. Once children begin to spot patterns they see them everywhere, not only in the
environment, but also in daily routines and all kinds of regular behavior. Most important of all, children find
pattern activities engaging and so they can help to develop positive attitudes and access to mathematics for
all children.
LEARNING ACTIVITY 1
Make at least one (1) Instructional Material for Pattern: Recognizing Repetition and Regularity
SUMMARY
To talk about patterns is to talk about regularities. Regularities are everywhere and it’s a special skill to be
able to recognize them. Regularities exist in natural and social phenomena; the search for regularities is the
base of science. Just by observing your surroundings, you can find groups of elements that have been set in
a certain way that follow a rule.
A pattern is a succession of elements (they could be auditory, gestural, or graphic) that is formed by
following a rule; that rule can be either repetition or reoccurrence.
REFERENCES
https://nrich.maths.org/13362
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