asgmt. mt pre and y1

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1.0 PLACE VALUE

An analysis of representative literature concerning the widely recognized

ineffective learning of "place-value" by American children arguably also

demonstrates a widespread lack of understanding of the concept of place-value

among elementary school arithmetic teachers and among researchers

themselves. Just being able to use place-value to write numbers and perform

calculations, and to describe the process is not sufficient understanding to be

able to teach it to children in the most complete and efficient manner.

A conceptual analysis and explication of the concept of "place-value" points to a

more effective method of teaching it. However, effectively teaching "place-value"

(or any conceptual or logical subject) requires more than the mechanical

application of a different method, different content, or the introduction of a

different kind of "manipulative". First, it is necessary to distinguish among

mathematical 1) conventions, 2) algorithmic manipulations, and 3)

logical/conceptual relationships, and then it is necessary to understand each of

these requires different methods for effective teaching. And it is necessary to

understand those different methods. Place-value involves all three mathematical

elements.

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TEACHING OF MATHEMATICS:PRE-SCHOOL AND YEAR ONE

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1.1 DEFINITION OF PLACE VALUE

According to Webster's New Millennium Dictionary of English, place value

means the value of a digit as determined by its position in a number, the name of

the place or location of a digit in a number.

Place value also is the value given to the place or position of a digit in anumber1.

1.2 PLACE VALUE EXAMPLES

Example 1:

Numbers, such as 84, have two digits. Each digit is a different place value.

The left digit is the tens' place. It tells you that there are 8 tens.

The last or right digit is the ones' place which is 4 in this example. Therefore,

there are 8 sets of 10, plus 4 ones in the number 84.

8 4

| |__ones' place

|_________tens' place

Example 2:

The place value of 6 in 6,934 is thousands.

The place value of 5 in 523,089 is hundred thousands.

The place value of 4 in 1035.743 is hundredths

1http://www.aaamath.com/B/g12b_px1.htm

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Example 3:

Tens Ones

3

1

So, 3 + 1 0 = 1 3 or 1 tens 3 ones = 1 3

Teacher can distributes worksheet like example below to make pupils understand

better.

Fill in the blank spaces:

Tens Ones Answer

1 5

6 3

4 74

2 25

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2.0 DEFINITION AND EXAMPLES OF NUMERAL

There are a lot of definitions and examples of numeral. There are:

i) A symbol or name that stands for a number.2

Examples: 3, 49 and twelve are all numerals

ii) A symbol or mark used to represent a number.3

iii) Numeral is a symbol or word used to represent a number4

The numbers one through ten in different numeral systems

indian

Devanagari

Hebrew

Arabic 1 2 3 4 5 6 7 8 9 10

Malayalam

Chinese

Suzhou

Roman I II III IV V VI VII VIII IX X

Thai

2http://www.mathsisfun.com/definitions/numeral.html

3http://education.yahoo.com/reference/dictionary/entry/numeral

4http://www.answers.com/number%20

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iv) a figure, letter, or a group of any of these, expressing a number . The

numerals of the year of graduation of one's class in college, high

school, etc., awarded for participation in sports, activities etc.5

2.1 DEFINITION AND EXAMPLES OF NUMBER

There are a lot of definitions and examples of number. There are:

i) A number is a count or measurement.

They are really an idea in our minds. We write or talk about numbers using

numerals such as "5" or "five". We could also hold up 5 fingers, or tap the table 5

times. These are all different ways of referring to the same number.

There are also different types of numbers, such as whole numbers (1,2,3)

5 http://www.yourdictionary.com/numeral

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decimals (1.48, 50.5), fractions (1/2, 3/8), and more. 6

ii) a symbol or word, or a group of either of these, showing how many or which

one in a series:

Example:1, 2, 10, 101 (one, two, ten, one hundred and one) are called cardinal

numbers; 1st, 2d, 10th, 101st (first, second, tenth, one hundred and first) are

called ordinal numbers .7

2.3 DIFFERENCE BETWEEN A NUMERAL AND A NUMBER

6http://www.mathsisfun.com/definitions/number.html

7http://www.yourdictionary.com/number

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The number we call one is called satu, by some people, wahid by others,

cinque by others, and so on. Despite the different names, its still just the same

old number five.

We also use different symbols different numeral systems - for writing down

numbers. Even though they may look different, they are just the same old

numbers named in a different way. When people in computing refer to a binary

number, there is really no such thing, They mean a binary numeral, the number

written in a certain way.

When doing calculations, we really manipulate the symbols representing the

numbers not the numbers themselves. Try multiplying xvii by xxv its just

seventeen by twenty-five, but we havent learnt how to manipulate the symbols

for numbers written in this way. Choosing an appropriate numeral system is

important - it can make calculations much easier.

Key idea : We manipulate symbols for numbers, not the numbers themselves.

Choosing the right numeral system is important the wrong choice

can make calculations difficult.

It is important that the emphasis below is on understanding, rather

than techniques. You will have relatively little need for techniques in

this area later on, but will need a good understanding

The system we use today was invented in India, and spread into Europe through

the Arab world, so it is called the Hindu-Arabic system. In the first two systems,

the value of a symbol was essentially fixed for example V always meant five.

In the Hindu-Arabic, the value of a symbol depends on its position.

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For example 5 Means five ones, or five

50 Means five tens, or fifty

500 Means five tens of tens, or five

hundred

As we move the symbol to the left, its value goes up ten times for each step. We

use a special symbol, 0, to show that the 5 has been moved one to the left, or

two to the left, or whatever. This symbol was known as zipher in Arabic, hence

the word cipher, which originally meant to do arithmetic.

Realise that the same number can be represented in many different ways, for

example in unary, in Roman, or in the Hindu-Arabic system with various different

sizes of pairs of hands. Its still the same old number, written using different

numeral systems. The numeral system determines how the number is written

down, and also the tricks needed for addition, multiplication and so on. Its the

numeral systems and the tricks that are different, not the numbers.

3.0 TEACHING AND LEARNING STRATEGIES OF COUNTING TWO-DIGITNUMBERS

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The teaching challenge here is to extend students' factual knowledge (the

names of numbers), conceptual understanding (linking number names to base

ten and place value properties) and strategic skills (to plan methods of counting

efficiently).

Activity 1: Verbal sequence assists students to extend their verbal counting

sequence, especially bridging the decades.

Activity 2: 'What's missing?' helps students to move away from rote learned

sequences to begin counting from any number.

Activity 3: More efficient strategies for counting assists students to develop

efficient strategies for counting large numbers of objects.

Activity 4:Number rolls is a favourite with children learning to count.

Activity 1: Verbal sequence

Learning the number names to 100 is done is conjunction with developing place

value and base ten knowledge.

Students will first learn to count and read two digit numbers (in the teens,

twenties and possibly beyond) without explicit attention given to the grouping

into tens. For example, when young children see 24, they see it only as 'twenty

four' and probably as the number after twenty three, but not as 2 tens and 4

ones.

Later the base ten understanding begins as they learn to count by tens (ten,

twenty, thirty, etc.) and then to fill in the numbers between (e.g. first learn forty,

fifty, sixty etc and then forty one, forty two etc.).

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Reading and writing numerals is developed simultaneously with the higher

verbal sequence.

A hundreds chart is a very flexible visual aid to support verbal work. It is

particularly important because the numbers are arranged in groups of ten, so

the base ten patterns are evident.

The most difficult aspect of counting to 100 is bridging the decades (for

example, from 59 to 60), so this needs additional attention.

Teachers should be encouraged to take every opportunity for counting together

both out loud and silently, with and without moving or touching objects, with and

without writing numbers.

Activity 2: 'What's missing?'

'What's missing?' is an activity that can be used for many counting tasks. A

hundreds chart , on the wall or made with tiles on a frame, provides the number

sequence in order. Ask the student to look away while you turn one number face

down/hide it/turn tile over. Ask the child to say what number they think is missing

then let them check. This can be extended to hide adjacent sets of numbers,

rows of numbers, etc.

There are many patterns on the hundreds chart that support these activities.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

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21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

Activity 3: More efficient strategies for counting

Students need plenty of practice to develop efficient strategies to count large

numbers of objects. There is much more involved than counting small numbers.

A large number of Unifix blocks are placed on the table. Students estimate (best

guess) how many blocks there are altogether. They are then set the task of

counting them. Initially they count, or attempt to count the number of Unifix by

ones. However, soon students realize they could join ten Unifix together and

then count by tens rather than by ones. This is more efficient and more likely to

be correct, and easier to check.

Initially the estimate will be guesses, but with practice students will become

quite proficient at estimating the number of Unifix. They develop a better idea of

how big numbers are.

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Students need to learn to group into tens in many contexts. For example if

counting a large number of objects students must learn to group in tens. This

will commonly occur with coins. An efficient strategy is to make piles of ten, then

count the number of piles, then include the remaining objects not in a pile. To

efficiently count money, make piles of 10 coins of the same denomination, and

then count the number of piles, and then add the ones left over. This enables

easy checking and calculating.

Activity 4: Number rolls

Buy long strips of paper (e.g. paper rolls for cash registers from a newsagent, or

wide streamers). Children write numbers starting at 1 and continuing for as long

as they like. Many children will love the patterns that they see emerging, and will

be fascinated as they go higher and higher.

A calculator is a good support to help children find out the next number, e.g. as

they go past 999 etc. Students can add 1 using +, or they can use the repeated

addition sequence (1+1 = = = = = which produces numbers in order).

At this stage, it is not important that students can read out each numeral, or

have a strong idea of place value etc. The idea is simply to see the patterns that

emerge in the number sequence. If they reach a large number though, they will

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have a good idea of how big it is, because they know how far it is along the

number roll.

Decorate the classroom with the number rolls and talk about the patterns that

children see. This is also a good homework activity.

Some children will like to make number rolls counting in 2s or 5s or 10s etc.

4.0 DESCRIPTION OF ADDITION MISTAKE

Our first thought would be that misconceptions, once rooted in the students

memory, are hard to erase. The situation is somewhat more complex.

Researchers interest in student conceptions has been provoked by numerous

studies indicating that

1. Before formal study, persons have firmly held, descriptive, and explanatory

systems for scientific and logic-mathematical phenomena, that is, systems of

belief about mathematics.

2. These systems of belief differ from what is incorporated into the standard

curriculum.

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In Question 4, the child invented a simple algorithm by repeating the same

number when start adding . Below are detail descriptions about the

mistake/error.

1) 3 + 4 = 6

The child started adding by continue number from 3 , 4 , 5 , 6

* * * * = 4

2) 5 + 3 = 7

Again the child counted wrongly started from 5 , 6 , 7

* * * = 3

3) 4 + 5 = 8

The child made the same error started from 4 , 5 , 6 , 7 , 8

* * * * * = 5

4.1STRATEGY TO OVERCOME ADDITION MISCONCEPTION

Activities to explore addition of numbers

identifying the two numbers which have to be combined

adding one to achieve the next counting number

emphasizing the value of starting with the biggest number

Adding using doubles.

SONG

One little frog lonely and blue

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'Till another came along

Then there were two.

Two little frogs hopping with glee,Fred dropped in

And that made three.

Three little frogs still looking for more,

Another came along

And now there are four.

Four little frogs doing a jive,If you add one more

Then you've got five.

Five little frogs jumping over sticks

Here comes Clive

So now there are six.

Six little frogs, two more would be great

And here they come,

That's seven, eight.

Eight little frogs, is that all then?

No, here comes two more

So there are nine, ten.

Materials

Rhyme and frog finger puppets

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(1)Frog Activity

Give each student a copy ofthe ten frog finger puppets to colour, cut out

and assemble.

Act out the rhyme.

.

b) Play a game

Spin To Win

I.Materials - groups of four ( place the materials in bags for each group.)

Spin to Win Game - 2 spinners(paper plate with brass fastener as spinner.

Divide the plate into four sections with the #'s 2/3/4/5 representing a section),

cards 1-6, interlocking cubes(optional), and writing paper for each group.

Directions for each group - Found below

II. Procedures

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1. Review multiple step problems and say, "Now that we have reviewed our

skills we can play a multiple step problem game with our groups. Listen

carefully as I explain this: Each group will get a game packet that contains all

the materials you will need to play the game. (Demonstrate the following as

you speak) The way the game is played is that the first person spins each

spinner (spin both spinners) writes an addition problem using the two

numbers and finds the sum of the two numbers (I spun a 2 and a 5, 2+5=7).

Then the player draws a number card (draw a card) and subtracts the number

from the sum and writes it down (I drew a 3, so I subtract my sum, which is 7

from 3 and get 4.) I would circle the four and it would then be the next

persons turn. When everyone has gone, whoever has the largest number

circled gets a point. After four rounds, whoever has the most points in your

group is the winner. Lets try one together."

2. Have individual students come up and guide them through the steps. It is a

good idea to pick a student from each group to do the demonstration.

3. Allow the game to begin. You will need to visit each group to make sure they

have the concept. Once they get it, they seem to really enjoy the game!

Spin to Win Directions:

1.) Give each person in your group a piece of paper.

Place the number cards in the center of the table face down.

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2.) Give the person whose name has the most letters in it the Spinners. This

person will go first

3.) Start the first round

4.) The first player spins the two spinners and adds the two numbers together on

their paper.

5.) Then the first person takes a number card and subtracts the number from the

sum of the answer to the last problem. The player must write the problem and

circle the answer. Put the number card on the bottom of the pile.

6.) Now it is the next persons turn. They do the same as the first person.

7.) Do steps 1-5 until all the players have had a turn

8.) After everyone has had one turn, the person who has the highest circled

number gets to put a point on their paper.

9.) Play until the teacher says stop.

10) The person who has the most points when the teacher says stop is the

winner

5.0 DESCRIPTION OF SUBTRACTION MISTAKE

Subtraction is one of the four basic arithmetic operations; it is the inverse of

addition, meaning that if we start with any number and add any number and then

subtract the same number we added, we return to the number we started with.

Subtraction is denoted by a minus sign in infix notation.

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In Question 5, the child invented a simple algorithm by repeating the same

number when start subtracting . Below are detail descriptions about the

mistake/error.

1) 7 - 4 = 4

The child started subtracting by continue number from 4 , 5 , 6 , 7

* * * * = 4

2) 9 - 5 = 5

She wrongly counted from 5 , 6 , 7, 8 , 9

* * * * * = 5

3) 8 - 3 = 6

Again , she counted wrongly started from 3 , 4 , 5 , 6 , 7 , 8

* * * * * * = 6

4.1 STRATEGY TO OVERCOME SUBTRACTION MISCONCEPTION

(1) Fly Away Birds

Materials

Sticky Tape

Coloured Pencils

Activities

Ask students to draw ten bird shapes to colour and cut out. They can be folded

and taped to fit over the tips of their fingers.

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In pairs students can play games with the birds flying away and the other student

work out how many are left and then counting to find out if he / she is right.

"There are ten birds flying in the sky." Student hides fingers - e.g. "How many can

you see?" "How many have flown away?"

(2) Bowling

Materials

Empty, washed drink bottles.

Ball

Paper and pencil to keep scores.

Activity

Assemble the bottles in a group and the students can take it in turns to try to

bowl them over. They can keep scores after having turns.

Extend this activity by getting the students to record a number from 1 to 10 on

each bottle. They can then keep a record of which bottles are bowled over first

and this information can be recorded on a chart

5.0 CONCLUSION

Finally, many (math) algorithms are fairly complex, with many different "rules", so

they are difficult to learn just as formal systems, even with practice. The addition

and subtraction algorithms (how to line up columns, when and how to borrow or

carry, how to note that have done so, how to treat zeroes, etc., etc.) are fairly

complex and difficult to learn just by rote alone. Children do not learn these

algorithms very well when they are taught as formal systems and when children

have insufficient background to understand their point. And it is easy to see that

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in cases involving "simple addition and subtraction", the algorithm is far more

complicated than just "figuring out" the answer in any logical way one might; and

that it is easier for children to figure out a way to get the answer than it is for

them to learn the algorithm. There is simply no reason to introduce algorithms

before students can understand their purpose and before students get to the

kinds of (usually higher) number problems for which algorithms are helpful or

necessary to solve. This can be at a young age, if children are given useful kinds

of number and quantity experiences. Age alone is not the factor.

5.0 BIBLIOGRAPHY

Ansary Ahmed (2007). HBMT 1203 Teaching of Mathematics: Pre- school and

Year One Kuala Lumpur: Open University Malaysia(OUM)

Baroody, A.J. (1990). How and when should place-value concepts and skills be

taught? Journal for Research in Mathematics Education, 21(4), 281-286

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Fuson, K.C. (1990). Conceptual structures for multiunit numbers: implications for

learning and teaching multidigit addition, subtraction, and place value. Cognition

and Instruction, 7(4), 343-403.

Jones, G.A., & Thornton, C.A. (1993). Children's understanding of place value: a

framework for curriculum development and assessment. Young Children, 48(5),

12-18.

Kamii, C. (1989). Young children continue to reinvent arithmetic: 2nd grade. New

York: Teachers College Press.

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APPENDIX

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asgmt. mt pre and y1