draft thesis
TRANSCRIPT
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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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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
multi-digit numbers.
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
Conjectures about pupils thinking and
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
multi-digit numbers.
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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
Conjectures about pupils thinking and
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
Conjectures about pupils thinking and
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
multi-digit numbers.
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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?
Explain your answer here:
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How did you know the answer?
Budi
Suci
Explain your answer here:
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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