Draft Thesis

7/27/2019 Draft Thesis

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[SALIZA SAFTA ASSITI F103179] Thesis

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UNDERSTANDING THE MEANING OF TENS AND UNITS IN MULTI-DIGIT NUMBERS

I. Introduction

The place value has become the most difficult subject taught in the primary

school (Price, 2001). Kamii (1986) found that most of the first- and second-graders face

so many difficulties in understanding that 1 in 16 indicates that there is 1 ten, and also,

many of third- and forth-graders still do not understand about place-value. The pupils

know that there are sixteen items that is represented by 16, they also can write the

correct symbol 16 for sixteen items, but they do not constitute the understanding of

place-value (Kamii, 1986). The research of Cobb and Wheatley (1988) stated that there

is difference in the pupils thinking of ten as a collection of 10 single items, ten as a single

unit, and ten as a collection of 10 that can be counted as an item.

In Indonesia, it is common that the teacher emphasizes the teaching of

procedures, rather than considering the development of the pupils own strategies

(Marsigit, 2004; in Rumiati & Wright, 2010). The research about this topic involving

Indonesian pupils is also rare (Rumiati & Wright, 2010).

The learning environment is important to support the learning process of the

pupils. Learning is not following the teachers instruction and remembering the

procedure very well. The learning process occurs when the pupils face some problems

and they have to grapple with the problems (Murray, Olivier & Human, 1998). By

solving the problem by themselves, the pupils are challenged to compare their solution

to their friends and discuss it (Murray, Olivier & Human, 1998). They will reflect their

strategy within the discussion in the classroom (Murray, Olivier & Human, 1998). The

problems here are The Inventory Activity (Dolk & Fosnot, 2001), Sending The Cubes

(adapted from Yackel, Underwood & Elias, 2007) and Finding-How-Many. The last

activity is inspired by the traditional game from Indonesia the pupils are familiar with.

It is still hard for the first graders to represent the symbol of the numbers. They

even cannot order the number into the right order yet (Dolk & Fosnot, 2001). To solve

these difficulties, there will be three earlier activities before giving the contextual

problems to the pupils to be solved. These activities can make the learning process


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Thesis [SALIZA SAFTA ASSITI F103179]

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become more interesting for the pupils. The activities are (1) Counting-on; (2) Locating

the number; and (3) Jumping on the number line (adapted from Menne, 2004). These

three activities will promote the pupils to be able to understand the fundamental skills

to calculate numbers in the later stage of learning (Menne, 2004).

In the Counting-on activity, the pupils will be asked to count on up to 100. This

will be interesting for them since it is stated by Ginsburg (in Dickson, Brown & Gibson,

1984:201) that the pupils learn to count larger number by repeating the word patterns,

and this activity is giving them more pleasure. However, though the pupils are able to

count up to 100, they still confuse in putting the number into order (Dickson, Brown &

Gibson, 1984:201). To make them understand about the sequence of the number, there

will be the second activity where the pupils are asked to play with the number line. The

next activities will be finding the amount in which the pupils will work with real thing

like Dakocan, beads, cubes, books, and candies. These activities will promote the pupils

to structure a set of objects with a help of grouping, which will lead them to the idea of

unitizing.

The aim of the present study is to develop an instructional theory about

supporting pupils understanding of tens and units in multi-digit numbers by making a

group of ten in solving the contextual problems. In the present study, the researcher will

see how the pupils learn to understand the idea of unitizing by doing a grouping

strategy. Here is the research question for the present study: How can we support

pupils understanding of the meaning of tens and units in multi-digit numbers?

II. Theoretical framework

2.1. Place Value

Understanding base-ten numbers is one of the most important mathematics

topics taught in the primary school, and yet also one of the most difficult to teach and to

learn (Price, 2001). Resnick (1984) stated that in one of his studies, all of the third-grade

children he interviewed could count any single block denomination, but more than half

of the children became confused when two or more denominations were to be

quantified. Kamii (1986) cited results of research in the United States, Canada and

Switzerland which found that most pupils in the first and second years of schooling do


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not understand that the 1 in 16 indicates that there is 1 ten, and that many pupils in

Years 3 and 4 do not understand place value. Being able to put out the correct number

single units for a number such as 16, or writing the correct numeral 16 for sixteen

objects does not constitute place value understanding. Cobb and Wheatley wrote an

influential paper (1988) describing some in some detail the conceptions of ten held by

young children. Their research is particularly useful in pointing out the difference

between children thinking of ten as a collection of 10 single items, ten as a single unit,

and ten as a collection of 10 that can be counted as an item.

Place value means the value of the place a digit occupies, for example, in 57 the 5

occupies the tens place (Bloomfield, 2003). Ginsburg (in Dickson, et al., 1984) identifies

three stages in developing an understanding of the theory of place value, where the

written symbolization of number is concerned.

1. The first stage is where the pupil writes a number correctly with no idea as to why.2. The second stage is where the pupil realizes that other ways of writing a particular

number are wrong for example 31 is incorrect for thirteen.

3. Thirdly is the stage where the pupil is able to relate written notation of numbers tothe theory of place value.

Place value is extremely significant in mathematical learning, yet the pupils tend

to neither acquire an adequate understanding of place value nor apply their

understanding of place value when working with computational algorithms (Fuson,

1990; Jones and Thornton, 1989). In their extensive study of pupil understanding of

place value, Bednarz and Janvier (1982) concluded that:

1. Pupils associate the place-value meanings of hundreds, tens, and ones more interms of order in placement than in base-ten groupings.

2. Pupils interpret the meaning of borrowing as crossing out a digit, taking one away,and adjoining one to the next digit, not as a means of regrouping.

Fuson (in Baroody, 1990) stated that with well-designed instruction, several

weeks maybe enough to learn multi-digit arithmetics procedures in a meaningful

fashion. The research of Thompson (2000) suggests that pupils are still able to work

successfully with two-digit numbers, including the teens, without being explicitly aware

that the first digit stands for the number of tens.


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2.2. Realistic Mathematics Education (RME)

Freudenthal (2002, p. 55) stated that in guiding the pupils to grasp themathematical concept, the delicate balance between the force of teaching and the

freedom of learning is needed. Therefore, I designed some activities that can be given to

the pupils in order to promote them to structure a set of objects with help of grouping so

that they are able to understand the idea of tens and units in multi-digit numbers. Pupils

will use their common sense to develop their own strategy to solve the problems given.

Here, the pupils should be given as much opportunities as possible to find their own

levels and explore every path to go there (Freudenthal, 2002, p. 47). The first

advantages of the re-invention process are the knowledge and abilities will stick better

and more readily available than when imposed by others. The second one is the learning

process is enjoyable and it can be motivating the pupils. The last advantage is the re-

invention process fosters the attitude of experiencing mathematics as a human activity

(Freudenthal, 2002, p. 47).

As stated by Freudenthal (2002, p. 50), the first verbalized mathematics is

counting and the activity can be counting something. In the present study, the pupils will

be asked to count the amount of Dakocan, the beads, the cubes, and doing the inventory

activity. In these activities, the pupils will apply the sequence of numerals to the set of

objects. When they are applying the sequence of numerals, they are doing the horizontal

mathematizing.

In the present study, the designing process of the activities is guided by five

tenets of RME defined by Treffers in Bakker (2004). The description is as in the

following:

1. Phenomenological explorationThe pupils will be given a contextual situation in each lesson as a start for the

instructional activity. The situation is related to the pupils current reality and

appropriate for the horizontal mathematizing.

2. Using models and symbols for progressive mathematizationThe process of learning will progress from the informal to the formal level. When the

pupils have to find the amount, they will at first counting one-by-one all the time by

tagging the beads, for example.


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3. Using pupils own construction and productionsThe pupils can make their own productions. By doing the activities, they will know

how to work with multi-digit numbers, how to structure a set of objects, and the idea

of grouping and unitizing. They have their own freedom to choose the strategy they

can use in order to solve the contextual problems until they understand the idea of

tens and units in multi-digit numbers. The re-invention process is not only for the

solution, but also for the problems. The pupils can make their own problems and

discuss these with the others.

4. InteractivityThe discussion in the classroom is not only between the teacher and the pupils, but

also between the pupils. The interaction can be built by dividing the pupils into

groups and let them work in that group with as little guidance as possible from the

teacher. The pupils have to count a large amount of beads in which it will be hard for

them if they do it individually. They need other pupils to make a representation and

also to divide the beads into several groups and count them. A class discussion is

designed so that all pupils will share their ideas in finding the amount. They can

share their strategies and by doing the sharing, they can choose the best strategy

that can be used.

5. IntertwinementThe range of a mathematical idea shall be in connection with long-term learning

process (Freudenthal, 2002, p. 57). The pupils can apply the knowledge they get

from all the lessons in all aspect of their real life.

For the second tenet of RME, there are 4 level of emergent modeling. The

adaptation of the emergent modeling in the present study is described as follows:

1. Situational levelIn this level, pupils will work with a contextual situation in which they can apply

their informal knowledge and any strategy they like to use. The pupils will find the

amount ofDakocan, beads, cubes and doing the inventory activity as the start of the

instructional activities.

2. Referential levelThe model-ofsituation, the models and the strategies the pupils use to refer to the

situation, occurs in this level. They can use their own strategy, the informal one, to

find the amount. The pupils can count the stuffs one-by-one for the first time. The


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representation they make is also being considered. They can use the pictorial

representation to represent the amount before using mathematical symbols.

3. General levelIn this level, pupils develop a model or strategy that is applicable in different

situations. They can use this strategy in solving almost every problem related to the

place value. In the present study, structuring a set of objects with help of making a

group of ten can be generalized to different situation in multi-digit numbers.

4. Formal levelHere, the pupils are already able to understand the meaning of tens and units in

multi-digit numbers. They can work with conventional procedure and the notation is

independent from the use ofmodel-formathematical activity.

2.3. Context problems in Realistic Mathematics Education (RME)

Freudenthal (1991, in Gravemeijer and Doorman, 1999) stated that

mathematics should start and stay connected within common sense. It is important to

start the learning activity by giving a context to the pupils to make them more involved

with mathematics. They will think that mathematics is a means to understand reality

(Boaler, 1993).

Solving the problems can enhance discovery and active learning, but personal

meaning is only attributed when pupils are able to determine the direction of activities

(Burton, in Boaler, 1993). Activities must be genuinely open and allow pupils to move in

the directions appropriate to their perception of the problem (Boaler, 1993). The

counting activity in order to find the amount, the pupils can choose their own strategy to

be used, for example, they can count one-by-one, count on from any number, and make a

group of ten. These activities will then be discussed at the end of the lesson, so that theycan compare their strategy with the other to find the best one that can be used

(Freudenthal, in Gravemeijer & Terwel, 2000). The guidance from the teacher is also

needed because different pupils respond to the same circumstances somewhat

differently (Planas & Civil, 2002, in Beswick, 2011).

Related to the framework of the present study, there are two sub-research

questions:

1. How can pupils learn to make a group of ten to understand the idea of unitizing?


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2. How can pupils learn to understand the meaning of tens and units in multi-digitnumbers using contextual problems?

III.Methods

3.1.Study approachThe aim of a design study is to develop theory about both the process of learning

and the means that are designed to support that learning (Gravemeijer & Cobb,

2006). I am going to design a study about place-value in multi-digit numbers focus

on the idea of unitizing. The aim of this study is to develop an instructional theory

about supporting pupils understanding of tens and units in multi-digit numbers by

making a group of ten in solving the contextual problems. In the present study, the

researcher will see how the pupils learn to understand the idea of unitizing by doing

a grouping strategy. The present study will be conducted to answer the research

questions.

3.2. Data collection

2.2.1Preparation phaseBefore conducting the present study, the researcher collects as many sources as

possible related to the place-value in multi-digit numbers for the first grade in

primary school. These data is used to support the idea of giving the contextual

problems related to the grouping strategy to determine the tens and units in

multi-digit numbers.

In this preparation phase, there will also be an observation about the classroom

climate and the pupils and an interview with the teacher. The participants that

will be observed are the pupils in the classroom in which the experiment will be

conducted. The researcher will come to the classroom before the preliminary

experiment to observe the condition of the classroom and any other details (see

Appendix 5.3). The situation in the classroom will be recorded using a small

camera held by the researcher and there will also be a field notes made during

the observation.

Another preparation for this study is conducting an interview with the teacher.

This will be conducted after the classroom observation. The teacher will be

asked several questions related to place value and the classroom organization


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and situation (see Appendix 5.2). The data will be recorded and the transcription

will be made after the interview.

Later, the researcher and the teacher will talk about the Hypothetical Learning

Trajectory (HLT) and the activities. The intention of this discussion is to see if

any changes should be made to make the conjecture become more precise, or

adjust the HLT to the level of understanding of the pupils, although what will

happen in the classroom is still unpredictable. The teacher can give some

suggestions to the researcher about the activities and the level of difficulty of the

problems. This will be a small discussion between the researcher and the

teacher before conducting the experiment.

3.2.2.Preliminary teaching experiment (first cycle)The preliminary teaching experiment will be conducted in the same school but

with a different class. There will be a small group consisting of 4 pupils from

another first grade classroom. The choice of a group of 4 pupils is because they

will be divided again into two smaller groups consist of 2 pupils. These 4 pupils

will be given the activities designed in the HLT.

The data will be collected in two ways, from collecting the written works of the

pupils and recording the activities during the lesson. These data will be used in

the retrospective analysis to test the assumptions described in the HLT in order

to make a better HLT for the teaching experiment (the second cycle). The

written work are the pre-test, post-test, and pupils work during the lesson.

Pupils have to hand in their written work to the teacher at the end of each

lesson.

There will be an interview between the teacher and the pupils about the

activities they do in the lessons. They will be asked about the problems given

and what their strategy are. The idea behind the choice of the strategy is

important, so that the pupils will also be asked about this.

3.2.3.Teaching experiment (second cycle)The teaching experiment will be conducted in a first grade of primary school.

This will be a whole class experiment and the participants are the pupils in that

class. In the first meeting, a pre-test will be conducted to see the pupils prior


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knowledge and to compare the development of their understanding from the

beginning of the lesson until the end.

There will be two kinds of observations, the whole class observation and an

observation of a small group consists of 4 pupils. This group consists of pupils

with an average level of understanding, not the smartest pupils or the lowest

achievers. The data will be collected during the whole lessons by using two

different cameras, one for the whole class observation and a small one for

observing the small group. The development of understanding of the pupils in

the small group will be recorded from the beginning until the end of the lessons.

This group will be observed during the whole lesson series.

Not only use the camera to record the pupils activity in the classroom, but also

the researcher will make some field notes about the classroom situation and

some crucial events that happens during the lesson. The pupils work will be

collected also to see their thinking process. In every lesson, the pupils will work

with an activity and they will write down their ideas on paper. At the end of the

lesson, they have to hand in their work to the teacher.

After the lesson, the pupils in the small group will be interviewed to see their

thinking process. They will be asked about the contextual problems and their

idea about the activities.

3.2.4.Post-testThe post-test will be conducted at the end of all the lessons. This is a written test

followed by all pupils in the classroom. The items in the post-test will be more or

less the same with those in the pre-test. The intention of this similarity is to see

the development of pupils understanding.

3.2.5.Validity and reliability(1)Validity

The validity of the data is divided into two, the internal and the external

validity. The internal validity includes the way of collecting the data,

collecting several data (data triangulation) contributes to the internal

validity, and the method of analysis. The data collected in different ways, by

recording the activities during the whole lessons and collecting the pupils

written works.


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The collected data will be tested with the HLT to see what really happen

during the lessons. The pupils will be interviewed to know their thinking

process during the activities in the classroom. This will be the way to know

what really happen behind the differences between the conjectures and the

reality.

(2)ReliabilityIt is important to pay attention on how the data is registered. To avoid

subjectivity, it is better to use the camera to record what happen in the

classroom than only making a field note. The data have to be in detail, every

crucial fragment in the video and audio will be transcripted and pupils work

will be collected and analyzed carefully.

3.3. Data analysis

3.3.1.Pre-testThe data from the pre-test will be used to see the pupils prior knowledge that is

important as a starting point of the whole lessons. These data will also be used

to compare pupils understanding before and after the lessons. Later, the data

from the pre-test will be compared with the data from the post-test at the end of

the lessons. The data will be collected from analyzing the pupils answer on the

written work. The strategies they use in answering the pre-test will not be

analyzed.

3.3.2.Preliminary teaching experiment (first cycle)An HLT is designed before conducting the lessons for the first cycle. The data

from this experiment will be tested with the HLT to compare the differences

between the conjectures and the reality in the classroom. Each lesson has its

own HLT. These HLTs then will be tested with the data collected in the

classroom. The differences are analyzed and the reason behind them will be

discussed. The result will be used in the retrospective analysis as a help in

making new conjectures for the next HLT for the second cycle.

3.3.3.Teaching experiment (second cycle)Here, the HLT that is made based on the result of the data analysis from the first

cycle will also be tested with the observation collected during the whole lessons.


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After each lessons, the researcher will analyze the data collected and compare

them with the conjectures made in the HLT. Every lesson in the second cycle will

be a mini-cycle and every mini-cycle will be analyzed so that the next lesson will

be adjusted based on the result of the previous one. If the goal of the previous

lesson is not being reached yet, then the goal for the next lesson should be

changed. The data of these mini-cycles will be analyzed continuously until the

last lesson.

At the end of the lesson, all the data will be analyzed to see whether the overall

goal is reached or not. The result will be used in the discussion later on to see if

the contextual problems can support pupils understanding of tens and units in


3.3.4.Post-testThe post-test will be given at the end of the lessons series. The data will be

collected from the whole pupils in the classroom. The format will be in written

data from the pupils work. The data is analyzed to see if the overall goal is

reached. The result will be used in answering the study questions

aforementioned by comparing the data with the pre-test result. By doing this,

the researcher can see the development of the pupils understanding of tens and

units in multi-digit numbers.

3.3.5.Validity and reliability(1)Validity

The validity of the data will not be discussed here.

(2)ReliabilityTo make sure that the data are reliable, there will be a process in which the

researcher will ask to a colleague about the data. This process is called inter-

subjectivity. In analyzing the video, the researcher will ask a colleague to

also see the video and look for an agreement in testing the HLT whether the

conjectures are happen in the classroom.


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IV. Hypothetical Learning Trajectory (HLT)

4.1.IntroductionThis is the HLT for my research about place value. The present research will be

conducted in the first grade of primary school in Indonesia and I will choose one of

the schools which already familiar with Indonesian Realistic Mathematics Education.

There will be six lessons in three weeks aimed to reach the main goal of the

research. Each lesson has its own learning goal that should be reached at the end of

the lesson.

The intention of these six lessons is to make the pupils be able to determine the tens

and units in multi-digit numbers. To reach this goal, the pupils have to be able to do

grouping and understand the idea of unitizing. In order to support the pupils to be

able to reach the goal, they will be given some contextual problems related to the

idea of grouping and unitizing.

The main goal of the whole activity is the pupils can differentiate the meaning of ten

as a collection of 10 single items, ten as a single unit, and ten as a collection of 10

that can be counted as an item (Cobb and Wheatley, 1988). In this HLT, I will explain

the assumptions about how the activities will support pupils understanding to

reach the overall goal.

4.2.First lessonThe first activity is a mix between three activities which are related each other. Here,

the pupils are asked to count on, to locate the number in the number line, and to

jump on the number line. At first, the pupils are asked to count from one until the

teacher asked them to stop counting at any number. Then the other pupils will

continue the counting activity. After that, the pupils will play on the number line in

which they have to locate the number on the number line. The last activity is

jumping on the number line. The pupils will be able to make a distinction between

tens and units but in a very early level of understanding.

Goals:

- The pupils learn how to count by saying the number in sequence- The pupils find the relation between numbers


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- The pupils understand how the numbers are built up1. Knowledge

- The pupils are able to count up to 100 in the correct order- The pupils are able to find the shortest solution to a given calculation- The pupils are able to know how the numbers are built up

2. Skills- Sequencing the number- Walking on the number line to find the location of the number- Saying the correct number in the number line after knowing the relation

between the numbers

- Making the distinction between jump and hop3. Attitude

- As a first lesson, this activity will make the pupils more engage with thelesson and they will become more eager to know the mathematical idea

between these activities.

Starting position:

1. Knowledge- Numbers up to 20- Early understanding of addition

2. Skills- Counting on- Making a jump on the number line- Making a hop on the number line

3. Attitude- The pupils value the mathematics as an important lesson related to their

daily life.

Conjectures of pupils thinking

No Activities Conjectures about pupils thinking and

reactions

1 Counting on

One pupil is asked to count from

one until he wants to stop

- The pupils will count correctly whenthey are in their turn.

- There are the other possibilities such as:


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counting, suddenly. Then

another pupil touched by the

first pupil has to continue

counting until the second pupil

wants to stop. This activity will

be over after the counting comes

to 100.

a. The pupils dont know what numbercomes after 29, 39, 49, and so on.

b. Some pupils skip several numbers,for example 44, 55, 66, and so on.

c. The pupils re-say the number theyhave said before. For example: 23,

24, 25, 25, 26, or else.

2 Locating the number on the

number line

The pupils guess the in which

number they, or someone else,

are standing on with help from

their friends about the position

of the number in the number

line.

- The pupils will guess several times untilthey know what the number is.

- After several trials, they will guess thenumber quicker than before

3 Jumping on the number line

One pupil standing in front of the

class on the number line drawn

on the floor. He makes several

jumps or hops, or combines

between jumps and hops, and

the other pupils guess what

number he arrives at.

One jump is equal to ten steps,

and one hop is equal to one step.

- The pupils will understand how thenumbers are built up by knowing that 37

is built from 3 jumps and 7 hops, for

instance.

- The pupils will make the variationbetween jumps and hops starting from

any number.

4.3.Second lessonThis activity is inspired by the traditional game in Palembang, Indonesia the pupils

are familiar with. The game is called Dakocan in which each Dakocan has different

value based on its size. It has different size and also different shape, and each size

has different value. The bigger the size, the greater its value. In this activity, the

pupils will be given a small bag of Dakocan and they have to count the amount of it


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in the bag. Each group will receive different amount so that every group has to count

their own stuffs.

Goal: The pupils can structure a set of objects with a help of grouping

1. Knowledge- The pupils can make a representation of the solution- They are able to do one-to-one correspondence- They are able to do addition- They can understand the idea of cardinality- They know the meaning of numbers- They build the sense of grouping.- They construct their initial understanding of the idea of unitizing

2. Skills- Tagging one-by-one- Counting on by two- Counting on by four- Making a group of friendly number

3. Attitude- The pupils will be more engaged with some contextual problems in the next

activity

Starting positions:

1. Knowledge- The meaning of numbers- Tagging one-by-one- The meaning of amount, that the amount wont be different no matter how

the pupils arrange the stuff

2. Skills- Counting from one- Counting on from any number- The sense of making a group

3. Attitude- The pupils believe that they can solve the problem- The pupils are interactively engaged with the classroom environment


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Conjectures about pupils thinking:

No Activities

Finding-How-Many

Conjectures about pupils thinking and

reactions

1 The pupils will be given a bag of

Dakocan and they have to count

the amount of it. Every group

receives different amount of

Dakocan in every bag.

- Some pupils will come up with the ideaof counting on one-by-one.

- They do tagging one-by-one butsometimes they tag one thing twice.

- The pupils find out that there is a betterstrategy instead of counting one-by-one.

The strategy is making a small group of

friendly number such as a group of two.

- The pupils in the higher level will be ableto count in a larger group, such as a

group of four, or even ten.

2 Class discussion - Every group will share their strategy infinding the amount.

- They will see the difference betweentheir strategy and their friends. This will

lead the pupils into the discussion of

finding the best strategy.

- The strategies can be:a. counting one by oneb. counting on from any numberc. making a group of friendly number

- Most of the pupils will see that groupingis the better strategy.

4.4.Third lessonThe teacher will bring a box of beads and the beads will be divided to each group of

pupils. They also have to count the beads in order to find the amount and after that

they are asked to make the representation of the amount. The teacher will provide

small bags to be used to help them count. These bags can be a tool to stimulate the

pupils to make a group in finding the amount.


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Goal: The pupils can structure a set of objects with a help of grouping. In this second

activity, their sense of grouping is deeper than in the previous activity.

1. Knowledge- The pupils can make a representation of the situation- They understand the meaning of amount- They can count on from any number- They are able to do addition- They are able to understand the meaning of cardinality- They can make a group of any number to structure a set of objects

2. Skills- Counting on from any number- Making a group of friendly number- Represent the amount with number

3. Attitude- The pupils are eager to know the amount of the stuffs given to them

Starting position:

1. Knowledge- The meaning of amount- Cardinality- Compensation- Addition

2. Skills- Counting on from any number- Grouping strategy

3. Attitude- Pupils will do the same activity with the one they usually do in the break

time

Conjectures about pupils thinking:

No Activities

Counting the beads


reactions

1 The pupils are given the beads in

larger amount

- They will directly count them one-by-one.


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- Some of the pupils already make a groupof friendly number and then add the

groups theyve made.

- The pupils who already make somegroups of friendly number are asked to

show their strategy to the other groups.

2 Class discussion - The pupils are asked to show theirstrategy. (Not every group is asked to do

this, but the teacher chooses one group

who already able to do grouping and

another one who cannot.)

- The pupils will compare their strategywith the other.

- Some of the pupils are still use theircounting one-by-one strategy in finding

the amount. Some other is already able

to structure the set of objects with a help

of grouping. These differences will be

discussed so that the pupils themselves

will find out the best strategy they can

use in solving this problem.

- The pupils will finally find out that thebest strategy to find the amount is

grouping.

4.5.Forth lessonHere, the pupils will work with packs of cubes and they have to pack or un-pack the

packs of cubes to send them to the buyer. There will be a contextual problem has to

be solved.

Goal: The pupils can construct their initial understanding about the tens and units in



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1. Knowledge- The pupils are able to do grouping and re-grouping strategy in solving the

contextual problem

- They have deeper understanding in tens and units- They understand the idea of unitizing- They are able to think in our base-ten number system- They are able to do the horizontal mathematization- They will understand about ten as a collection of ten single units that can be

counted as one

2. Skills- Packing and un-packing- Translate the contextual problem into the terms of mathematics

3. Attitude- The pupils will find the relation between real life problem and mathematics

Starting position:

1. Knowledge- Grouping strategy- Tens and units- The meaning of boxes, rolls, and ones- The meaning of packs of cubes

2. Skills- Thinking in group

3. Attitude- The pupils are eager to solve the problems

Conjectures of pupils thinking:

No Activities

Sending the cubes


reactions

1 Solving the contextual problem - The pupils will work in a group of four.- The pupils have to translate the problem

into mathematical terms.

- The pupils will understand that theyhave to pack or un-pack the cubes to be


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2. Skills- Making a group of ten- Adding the numbers- Determining the tens and units in multi-digit numbers

3. Attitude- The pupils will find the relation between their daily life experience with

mathematics

Starting position:

1. Knowledge- Counting on from any number- Making a group of friendly number

2. Skills- Forming a group and bundling it with rubber band- Tagging the group when counting

3. Attitude- The pupils know how to count their stuffs- The pupils will value the lesson because it is related to their daily life

Conjecture about pupils thinking:

No Activities

Counting the books in the

library


reactions

1 Working in the library - The pupils will do the same thing likethey did in some activities before, such

as counting one-by-one or counting on

from any number.

- Pupils in higher level of understandingwill make the books into group to make

it easier for the to find the amount.

- The pupils are already able to make agroup of ten.

2 Class discussion - This activity will take longer time thanthe two discussions mention earlier.


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- The pupils will be more focus on the ideaof tens and units in multi-digit numbers.

- The pupils will make a list of the amountof the books in which they will see the

tens and units in different columns.

- The pupils will understand bythemselves the idea of tens and units and

can determine the tens and units in



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Goal:

The pupils can

construct their own

understanding of the

meaning of tens and

units in multi-digit

Goal:

The pupils can

construct their initial

understanding of the

meaning of tens and

units in multi-digit

Goal:

The pupils can

structure a set objects

with a help of grouping

Goal:

-The pupils can say thenumbers in sequence-The pupils can find

the relation between

numbers

-The pupils canunderstand how the

numbers are built up

(Julie Menne, 2004)Counting up to 100

Locating the

number on the

Jumping on the

number line

Finding the

amount ofDakocan

Finding the

amount of the

Sending the cubes

(Yackel et al.,

Inventory activity

(Dolk & Fosnot,

2001)

Big Idea:

Numbers up to

100

Big Idea:

Grouping

Big Idea:

Grouping and

Unitizing

Tens and units

4.7.The visualization of the learning trajectory


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V. Appendices

5.1.Teacher guideThe teacher will be the guide in the learning process. Aforementioned, different

pupils may respond the same context differently (Planas & Civil, 2002, in Beswick, 2011)

so that the guide from the teacher will be important to help the pupils understand the

mathematical idea behind a context. The teacher should be able to orchestrate the

lesson to make the pupils get the idea in every problem. The guide for the teacher in the

present study is described in the following table.

No Activities and conjectures Teacher guide

1 Counting on

-The pupil who is in turn for countingcan be very excited so that he wont

stop.

-The pupil skips one number whencounting. For example, 21, 23, 24, 25,

(the pupil skip the 22).

-The pupil has no idea about whatnumber comes after 29, 39, 49, etc.

-Some of the pupils cant say thenumber in sequence.

- The teacher can ask the pupil to stopcounting and give the chance for his

friend.

- The teacher can help the pupil bywriting down the number in the

blackboard so that the pupil will see

what theyre saying.

- The teacher asks another pupil ifanyone knows the number.

- The teacher ask the small group whois still not able to do this to do this

activity again.

Locating the number on the number line-The pupils may become confused

because the number line is empty.

-The pupils need more number in thenumber line.

- The teacher shall write down thenumber if it is needed to make the

pupils see where their position are in

the number line.

Jumping on the number line

-The pupils may confuse indetermining the jumps and the hops.

- The teacher shall make a consensus inthe beginning of the lesson that one

jump consists of 10 hops.


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-They confuse when the startingnumber is not zero.

-The pupils find difficulties when theyhave to jump backward.

- The teacher can illustrate the situationin the blackboard by drawing the

number line, jumps, and hops.

- The teacher can draw a number line inthe blackboard, and write down the

starting number in it. Again, the

teacher draw the jumps and the hops

the pupils make.

- The teacher has to explain to thepupils that they actually do the same

thing, but in another way around.

They have to count backward instead

of counting forward.

2 Counting the Dakocan

-Most of the pupils are alreadyunderstand about the rule the game of

Dakocan. It will be quite easy for them

to count the amount ofDakocan.

-Some pupils maybe count it one byone.

- The teacher explains the rule in everyDakocan to make sure that all pupils

know about it.

Class discussion

-The pupils share their thinking in frontof the classroom.

- The teacher chooses two or threegroups to share their strategy in front

of the classroom.

- The first group will be the one whouses counting one-by-one strategy to

find the amount.

- The other group shall be the groupwho already able to structure the

objects with a help of grouping.

- The teacher has to be sure that thepupils know that in differentDakocan

it can be different value.


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3 Finding the amount of the beads

-The pupils tend to count all the beadsone-by-one.

-Some pupils already structure thebeads into groups.

-They put the beads everywhere, sothat it will distract them in doing

counting.

- The teacher can walk around theclassroom and see every strategy the

pupils use.

- The teacher can choose one group toshow their grouping strategy to

another group who still uses counting

one-by-one strategy.

- The teacher can provide small plasticbags to be used to promote the

grouping strategy.

- The teacher can ask the pupils to putthe tens in the left side, and the units

will be in the right side.

Class discussion

-The pupils again will share theirthinking process in the discussion.

- The teacher chooses a group of pupilwho works with grouping strategy,

and a group who still works with

counting one-by-one strategy.

- The teacher provides a table in theblackboard and makes a list of the

amount of the beads from every

group.

- The discussion will be orchestratedaround the strategy the pupils use. Is

it better to use grouping strategy?

4 Sending the cubes

-The pupils understand that there are10 cubes in every roll.

-When the pupils have unpacked thecubes, it is hard for them to re-pack

the cubes into rolls.

- The teacher gives some transparentplastic bags to make the packing

activity become easier.

- The teacher has to make sure that thepupils make a pack of 10 cubes.


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Class discussion

-There will be a discussion about theproblem whether all the pupils

understand about it or not. Then they

will translate the problem into

mathematics.

-The pupils can share all strategy theyhave in solving the problem.

- The teacher asks the pupil to sharetheir thinking process.

- The teacher provides a table in whichthe pupils can write down the number

of rolls and cubes in it.

- The teacher guides the pupils untilthey finally understand about the

meaning of tens and units in multi-

digit numbers.

5 Inventory activity

-Some of the pupils are already able tothink in groups to structure a set of

objects. Some other still counts one-

by-one.

- The teacher can ask the pupils whoalready works with grouping strategy

to show their initial strategy to the

others.

- The teacher can also provide thepupils with rubber ban or plastic bags

to promote them to use the tools as a

help when doing the grouping.

Class discussion

-The pupils share their strategy to theother.

- The teacher provides a table in theblackboard so the pupils can write

down the list of the books they count

in it.

- The teacher separates the column forthe tens and units so that the pupils

see at the end of the discussion the

position and the value of tens and

units. This is the crucial thing in the

present research. The pupils will

understand the meaning of tens and

units in multi-digit numbers.


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5.2.The teachers interview scheme

In the observation phase, there will be an interview with the teacher about theclassroom climate. The data from this interview will be used as a help in designing the

activities for the lessons series. The teacher will also help the researcher to choose one

group of pupils with average level of understanding to be a focus group in the present

research. The main thing wanted to be discuss with the teacher are about the pupils

understanding and behavior, and also the teachers experience in Indonesian RME. The

teachers interview scheme is described in the following:

(1)Pupils level of understanding and behavior in the classrooma. Is there any significant difference in pupils level of understanding?b. What are the difficulties the pupils usually face in the topic of place-value?c. Are the pupils accustomed with the kind of activities arranged by the

researcher?

(2)Teachers experience related to the domain of the present studya. What were the difficulties the teacher face about the topic of the present

study?

b. How could s/he solve the difficulties?c. How long will the lesson be until the overall goal of the present study is

reached?

d. Is the teacher accustomed with Indonesian Realistic Mathematics Education(IRME)?

e. How does the teacher arrange the discussion groups?f. What is the teacher idea about the pupils understanding of the domain of

the present study?

5.3.The classroom observation schemeBeside an interview with the teacher, in the preparation phase there will also be

a classroom observation that will be conducted before the interview. There will be an

observation about the interaction in the classroom, the organization of the classroom,


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the structure of the lesson, the classroom climate, and the method used in the classroom.

The following is the description about the things will be observed:

(1)Interactiona. Is it an interactive classroom or teacher-centered?b. Is the discussion always between the teacher and the pupils? Or is there also

among the pupils?

c. Is there any pupil that the teacher always points at?d. Who is the talkative pupil and who is not?

(2)Organization of the classrooma. Are the pupils already accustomed with working in groups?b. How is the arrangement of the classroom? Who is sitting with whom? Does

the smart pupil sit with the low achiever? Or else?

c. How are the groups constituted?

(3)The structure of the lessona. How does the teacher start the lesson?b. What comes after the opening of the lesson?c. Do they have a small break in the middle of the lesson?d. How does the teacher give the task to the pupils?e. Do the pupils hand in their written work to the teacher?f. Does the teacher give time to the pupils to think about the problem given?

(4)The classroom climatea.

Do the pupils always pay attention to the lesson?

b. Do the pupils ask everything directly or they raise their hands first?c. Can they go out of the classroom if they want to?d. Do they have to ask to the teacher whether they can go out or not?e. Do the pupils walk around the classroom in the middle of the lesson?f. Where is the position of the teacher in the classroom?g. Are there so many stuffs in the classroom? (Related to my topic, I will use

these stuffs if it is available)


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(5)The methoda. What kind of book the teacher use?b. Does the teacher hand out some paper every lesson?c. Does the teacher only write down the material in the blackboard?d. Is there any additional book the teacher use beside the textbook?

5.4.The pre-testThe pre-test will be given before the lessons. The items in the pre-test are as

follows:

(1)The first item is about counting the beads. The pupils have to find the amount ofthe beads in the picture. They can use every strategy they like. The motivation of

the item is to see the starting position the pupils have in the beginning of the

lesson by looking at the strategy they use in finding the amount. The pupils can

use grouping strategy, or just count one-by-one.


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How many beads are there?

(2)The second item is about structuring the beads. Here, the pupils are asked to putthe beads in order so that they can find the amount easier. They can make any

order they like. They can draw or write down their thinking process in the space

given below the picture.

Can you put the beads in order to make it easier to be count?

Explain your answer here:


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(3)The third item is to count the amount of candies in the picture. The amount ofthe candies has been designed in order to promote the pupils to make a group of

ten.

How many candies are there?



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How did you know the answer?

Budi

Suci


I counted the

candies one-by-one

I counted thecandies

based on the

Do you have your own strategy? Dont hesitate to write it downhere:


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(4)The forth item is structuring the candies. The pupils have to put the candies intoseveral bags consists of 10 candies and they have to think how many bags they

need. The intention of this item is to see pupils understanding about the idea of

grouping.

You have to put the candies into several bags. Every bag can be filled with 10

candies. How many bags do you need?

(5)The fifth item is about the formal mathematics. The item is designed to seewhether the pupils are already able to understand the meaning of tens and units

in multi-digit numbers.

Determine the meaning of number in every position below:

7 =

16 =

23 =

61 = 94 =


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Documents

Draft Thesis