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The Construction of Kadyeisha’s Fractional Concepts
Virginia Polytechnic Institute and State University
Lorraine Feury
April 20, 2012
This research was conducted in order to construct a model of the mathematics of a
student who participated in a month long teaching experiment while in the sixth grade. With the
recordings of Norton’s teacher-researcher, interactions with two students, my research focuses
on the accommodations of Kadyeisha’s mathematical schemes when solving fraction problems.
By using retrospective analysis, I transcribed videotaped interactions and made inferences based
on the mathematical behavior and body language to comprehend the operations and schemes
Kadyeisha was currently using. Based on these current schemes and the modifications she
makes, I was able to create a mathematical construct of the schemes she can assimilate in future
problems. By reading literature from previous math education research, I formulated a
hypothesis to explain the schemes Kadyeisha used in order to resolve a fractional situation. My
research was guided by this question: Do Kadyeisha’s participatory stages in scheme
development guide her comprehension of fractional concepts? This hypothesis has allowed me to
study the current fractional schemes she uses and whether she is able to reorganize them to a
higher level of fractional concepts. The product of the research will allow me to better
understand Kadyeisha’s uses of fractional concepts and, in turn, construct a model of her
fractional behavior.
Mathematics is considered one of the hardest subjects to teach a child in K-12 education.
To make this subject more approachable to students, teachers must take time to understand the
operations and schemes their students are unknowingly using. Understanding students’
mathematics allows the student and teacher to have a healthier relationship towards mathematics.
Teaching experiments should encourage other teachers to become researchers of their students’
mathematics. By recognizing the operations and schemes students currently use in the classroom,
teachers would be able to address the problems students have. This research would aid more
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teacher-researchers to seek interest in using activities that properly engage students into
reorganizing the student’s current fractional theme. With this research, Kadyeisha's current and
potential fractions operations and schemes, based on participatory and anticipatory ways of
operating, can be used to determine how a student like her might grasp certain fractional
concepts.
Literature Review
Modeling a student's mathematics takes into consideration the ongoing and retrospective
analysis fractions schemes, operations, and language. Fractions operations and schemes help
explain the mental operations a student uses to assimilate fractional situations. By understanding
fractions operations, the teacher-researcher can recognize current and potential schemes a student
has. With schemes, fractional knowledge and mathematical communication between the student
and teacher-researcher is a significant indicator of level of construction a student has of a certain
operation and scheme. Fractional knowledge allows the teacher-researcher to form inferences of
the student's mathematics. The fractional knowledge of a student could not be fully understood
without understanding the components and building blocks.
Teaching experiments emerged in the United States in the 1970s with the purpose to
construct a model on the mathematical behavior of students (Steffe, 2000). Teaching experiments
attempt to provide the researcher with a firsthand experience of a student's mathematical learning
and reasoning. The teacher-researcher initiates activities and interactions between the student and
teacher. A witness video records the interactions during the teaching experiment. They ensure
that the teacher-researcher observes the interaction and provides assistance with the hypotheses
in ongoing analysis.
Steffe provides two concepts to help the teacher-researcher recognize the perspective of
their research. The "student's mathematics" and "mathematics of students" provide the teacherresearcher with two different perspectives of the student's zone of potential construction.
Mathematical realities of the student construct the range (zone) of activities the student activates
when approaching a situation. The "student's mathematics" is the mathematical realities of that
student. This concept is focused solely on the mathematical reasoning and procedures that are
realistic to the student. The "mathematics of students" is the interpretation the researcher makes
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based off of the mathematical behavior of the student that is observed. As a teacher-researcher, it
becomes difficult to fully comprehend the student's mathematics. With the aid of teaching
experiments, we can construct models of the student that describes schemes and concepts that are
quite parallel to the actual student's mathematics (2000).
Retrospective analysis allows the teacher-researcher to fully study and comprehend the
data collected during the teaching experiments. The benefits of a teaching experiment allow
teachers to better understand their students' thinking methods and therefore be able to properly
teach them the objective at hand. He stresses the points of how a researcher must be flexible
within his/her experiment. "A primary goal of the teacher in a teaching experiment is to establish
living models of students' mathematic (L. P. Steffe & Thompson, 2000, p. 284). Successful
teaching experiments have proposed an overall hypothesis that, if subject to alteration, may still
be able to be effectively tested during the experiment. The researcher must also be wary that they
are not imposing their mathematical knowledge and assumptions on the students, in turn
affecting how the students respond in the experiment environment. As a result of the teaching
experiment, observations are used to make inferences that complement the model of the student
that is being constructed. As a researcher, the model is best viable when it can be used as a base
for other teaching experiments.
Operations (Norton & McCloskey, 2008, p. 46) are mental actions that have been
abstracted from experience to become available for use in various situations. Operations are the
building blocks and indicators for certain schemes to be activated. As mental processes, they are
the first mathematical realities students use to recognize and process situations. Fractional
operations are specified to student using mental processes to assimilate fractional situations.
With all mathematical subjects, fractional operations are developed in hierarchical order.
Unitizing is the foundation of operations with the ability to treat an object or grouping of objects
as an entire whole. With recognizing the unitizing operation, the student develops the mental
process of separating a whole into equal parts activating the partitioning scheme. This operation
is a significant scheme that constructs multiple schemes. The disembedding operation is the
mental ability to imagine a fraction taken from the whole without altering the identity of the
whole itself. Iteration operation allows the student to mentally take a piece of the whole and copy
it as many times as desire. The most complex operation that has been known for a student to
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develop is splitting. With this operation, the student masters the operations of partitioning and
iterating so well that both operations can be simultaneously activated and mentally processed.
Fractional schemes embody the collection of fractional operations that are left to be
activated and accommodated by the student. This research has led to the prior research of the
first three schemes, in hierarchal order, out of the seven that Steffe has discovered. Simultaneous
partitioning is the least complex scheme with the uses of the unitizing and partitioning
operations. "Unitizing the whole, partitioning the continuous, whole using a composite unit as a
template" (Norton & McCloskey, 2008, p. 47) is the simplest perspective for a student to view a
fraction in a whole. As the fractional scheme becomes more exercised, it is reorganized and
develops into the part-whole scheme. While unitizing and partitioning the whole simultaneously,
the student disembeddes a fraction of the partitioned whole. This scheme provides the student
with the activity of recognizing the fraction and visualizing it apart from the whole. The last
scheme my research is focused on is the equi-partitioning scheme. The collection of unitizing,
partitioning, and iterating allows the activity of all three operations to be used while the student
identifies parts of the whole to each other parts within the whole.
To fully understand the type of research I was conducting, I started my readings with The
Teaching Experiment Methodology by Steffe and Thompson. Their article explains how teaching
experiments can be seen as a scientific process to form a model of a student’s mathematical
behavior. The purpose of a teaching experiment is to firsthand experience a student’s
mathematical learning and reasoning. Steffe introduces two terms to help the research recognize
the perspective of their research. The “student’s mathematics” is the mathematical realities of
that student. The “mathematics of students” is the interpretation the researcher makes based off
of the mathematical behavior of the student that is observed. The methodology of the teaching
experiment consists of a research, witness, students, and a recording of the interactions between
the students and researcher. The benefits of a teaching experiment allow teachers to better
understand their students’ thinking methods and therefore be able to properly teach them the
objective at hand. He stresses the points of how a researcher must be flexible within his/her
experiment. It is strongly recommended to start off with a hypothesis that, if subject to alteration,
may still be able to be properly tested during the experiment. The researcher must also be wary
that he/she is not imposing her mathematical knowledge and assumptions on the students, in turn
affecting how the students respond in the experiment environment. As a result of the teaching
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experiment, observations will be used to make inferences that complement the model of the
student that is being constructed. As a researcher, the model is best viable when it can be used as
a base for other teaching experiments (L. P. Steffe & Thompson, 2000).
The following year, Steffe brought up a theory that heavily influenced my paper. In A
new hypothesis concerning children’s fractional knowledge, the reorganization hypothesis was
tested in this paper to see that the numerical scheme was superseded. Scheme is not just an
operation but a compilation of various operations and activities activated by past experiences.
According to von Glaserfeld, a scheme consists of three parts that a student uses to execute a
problem. There's an experiential situation; an activating situation that is recognized by the
student. The student associates an activity (procedure) that they use to solve the problem. The
last part of the scheme is the result the student obtains from the activity that was used. The
numerical counting scheme is when the student uses numbers as placeholders to obtain a certain
quantity. Numbers become a result of a certain activity - they are not memorized- and are
abstract to the student. The equip-partitioning scheme was tested with the two students by seeing
how they broke a stick evenly into a requested amount. Steffe also introduced the idea that
connected numbers is used when approaching fractional problems. The student partitions the
whole while recognizing that the fraction is still connected to the composite unit. Necessary
errors were evident during the teaching experiments since the students had errors as the results of
the operation they used. His protocols showed that Laura had the simultaneous partitioning
scheme. Steffe also shows how the students lack the unit fractional scheme by explaining the
differences between splitting and equi-partitioning. By having the students explain what was
going through their heads, the production of fractional language is the consequences of their
current fractional schemes (Leslie P. Steffe, 2001).
Using Steffe’s Advanced Fraction Schemes, operations and schemes can be the main
components in understanding and modeling the mathematical operations of a student were the
main points. Before attempting to make a model, the researcher must have a based process for
observations and inferring information. The researcher must have a recognition template: a
foundation to be used to help identify the certain behaviors the student is displaying in the
situation. With a base to work with, the researcher must be very familiar with the several
operations and schemes used to describe the mathematics of the students. There are four fraction
operations: unitizing, partitioning, disembedding, and iterating. Each step that precedes a step is
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a higher and more complicated level of looking at fractions. These operations are used to
recognize the fractional abilities the student may have. There are several fraction operations:
simultaneous partitioning, part-whole, equi-partitioning, partitive unit, and partitive fraction.
Each scheme incorporates one or more of the operations to help the researcher try to recognize
the thinking methods of the student. With access to these operations and schemes, the researcher
can make strong inferences about the mathematics of the student (Norton & McCloskey, 2008).
Olive discussed in Children’s number sequences how different levels of interiorization of
a number sequences influences the way a student can assimilate a fractional situation. He
mentioned how number sequences are schemes that develop through levels of basic anticipatory
behaviors to reflective abstract. In order for a student to achieve a higher level of number
sequencing, they must re-interiorize the prior number scheme. In this article, he addresses the
various levels of number schemes in hierarchal order. In Pre-Numerical Counting Schemes
(PNS), the child does not see any correlation between the act of counting and the words they
assign to represent what they are counting (basically the lowest form of learning how to count).
Initial Number Sequences (INS) is when the child views numbers as a composite unit and is able
to count on from any number. Tacitly-Nested Number Sequence (TNS) allows the student to use
their fingers (beginning stages of anticipatory behaviors) to in order to keep track of what they
are "adding on" to a composite unit. Explicitly-Nested Number Sequence (ENS) re-interiorizes
TNS and allows the student to work on two unit levels. They see the numbers as groups
(composite units) rather than separate small problems that need to be solved in order to continue
on to the next one. These composite units allow numbers to become iterable or copies of "one".
Generalized Number Sequence (GNS) is the highest number sequence that re-interiorizes ENS
and allows the student to function on three unit levels. The student can take a group of numbers
(composite unit) and see iterate it as many times as they desire. They are able to symbolize that a
group of a certain number can be copied a certain amount of times, rather than seeing it as copies
of one ( O l i v e , 2 0 0 1 ) .
For Fine grain assessment of students’ mathematical understanding, Tzur studied a third
grade class in Israel on how certain stages help determine how well a student grasps a fractional
concept. He uses participatory and anticipatory stages to determine the level of comprehension
the student has about a concept. Tzur mentions the reflection on activity-effect relationship to
recognize how a student choses operations to assimilate a situation. He even notes conceptual
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"regress", where the student temporarily forgets to assimilate a situation with the new concept
they just learned. This taps into the different stages of how a student internalizes a new concept.
There are two stages Tzur focused on to help determine the levels of comprehension.
Participatory (first) stage is considered the "oops" stage. Students anticipate the situation by
doing an activity-effect relationship. When receiving an unexpected result, the student then
attempts to use the new concept learned. Anticipatory (Second) stage is when the student can
correctly assimilate the operation that properly addresses the fractional situation ( T z u r ,
2007).
In Mathematical caring relations in action, Hackenburg did an 8-month study on how
student-teacher relationships are formed and how they influence a student's mathematical reality.
This research was influenced by Nodding's care theory: an interaction that is an evolving
relationship that both the student and teacher participate in. Hackenburg believed that
mathematical caring relations (MCR) meant more openness to the teacher's intervention and
willingness to pursue questions and ideas of interests that may not have been pursued by the
student. Previous studies on teacher-student relationships have not focused on how they were
formed. The methodology used was the one from witness-researched used in Steffe's Teaching
Experiment. Hackenburg's results lead to the ideas that harmonizing with a student and paying
attention to subject vitality produces higher chances of the teacher making a stronger connection
with a student on learning mathematical concepts ( M a t h e m a t i c a l , 2 0 1 0 ) .
Hackenburg also did an 8-month teaching experiment that studied how four sixth-graders
approached reversible multiplicative relationships in Students’ reasoning with reversible
multiplicative relationships. Inhelder and Piaget (1958) figured that the ability to reverse one's
mental or physical actions caused equilibrium in their experience on all levels of cognitive
development. With this, Hackenburg also took notice on how anticipatory and participatory
schemes affected a student's ability to construct reversible multiplicative schemes. She
mentioned the many different fractional schemes, accommodations and perturbations that lead to
a student reorganize their current scheme. Reversibility is seen more of an internalized network
of operations (concepts) that can be activated all at once to assimilate a situation. Prior research
took note of how whole numbers can interfere with fractional knowledge and fractions are
perceived more as operators to a whole number divisional problem. Hackenburg wanted to see
how a student constructs the process of reversibility; seeing if the student had the ability to travel
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from the original state and back. In her results, only one student successfully used reversibility.
She became interested with what caused the other three students to not achieve such
reorganization. It led down to the level of anticipatory schemes each student had and the levels
of internalization and compensation being made throughout the situations ( S t u d e n t s ’ ,
2010).
Methodology
This research was based on the recordings of the interactions with Norton, Kadyeisha and
Isaac with fractional problems. For one month, both students were presented with fractional
problems that provoked and strengthen current and new schemes. The videos ranged from
twenty-seven minutes to forty-two minutes of the teacher-researching asking questions, with the
students responding and vice versa. The majority of the videos involved the use of various
colored fraction rods to help the student visualize certain fractions and composite units. Sheets of
paper, scissors, and markers were also made available to the students to help them approach the
fractional problems requested. A variety of fractional problems were asked in order to evoke and
perturb the fractional operations and schemes that were current to them.
As an observer, ongoing analysis was the first step to construct a model of Kadyeisha. I
transcribed the interactions and fractional language used between Norton, Kadyeisha, and Isaac
(whenever he seem to influence Kadyeisha's thinking). Each video was tested with a hypothesis
that was used to make stronger inferences to propose a new hypothesis for the following videos.
These hypotheses questioned the current operations and schemes Kadyeisha was using during the
videos. Inferences were noted throughout to acknowledge potential growth in fractional schemes.
In turn, each video's hypothesis was used to collectively provide data and explanation for the
overall hypothesis of this research.
Retrospective analysis was used to make concrete inferences about Kadyeisha and test
original hypothesis. Body language, visual tools, and fractional language were taken in account
to make concrete inferences of Kadyeisha's current and potential operations and schemes. By
back tracking through the videos, the sub hypotheses were studied to recognize the development
of Kadyeisha's fractional operations and schemes. The body language was study to help verify
Kadyeisha's comprehension of the fractional problems presented to her. The visual aid provided
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during the teaching experiments where physical indicators use to predict current and future
mathematical realities.
Analysis
For the transcripts, A stands for Dr. Anderson Norton, K stands for Kadyeisha, and I
stands for Isaac.
Video One
Excerpt 1.1
A: This is a candy bar and this candy bar is four times as big as your candy bar. Can you make
me your piece?
K: She takes a regular computer printing paper sheet (8 x 11). She grabs the ruler and measures
the length of the paper. Then she uses the number she received from measuring and works a
divisional problem with it on the paper. She smiles and laughs a little. She puts her right hand on
her forehead and rubs it a little bit. Her eyebrows arch up. I don’t know.
A: (to Isaac) Do you think you know what she’s doing?
I: No…
A: Do you know why she wrote the 4 divided into 11?
I: Shakes head no.
A: (to Kadyeisha) Can you explain to Isaac?
K: This is how big the candy bar is; the length of the candy bar.
I: Oh. Oh yeah
A: What is she doing?
K: Continues to work the divisional problem. She counts her fingers underneath the table and
nods her head simultaneously with the counting.
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Feury I: She’s like um, the candy bar is like…umm she measured it and she got 11 bars and made it
four times bigger.
A: This, this candy bar is four times bigger than her piece and I wanted her piece.
I: Okay. So umm, she measured 11 over 4. She… (Turns into inaudible muttering.)
K: Makes a facial expression. She picks up the ruler and measures the length of the paper. She is
counting on her fingers.
A: Measured 11, and she’s dividing by four. Why would she divide the four into 11?
I: Because she wants to make 11 smaller.
A: Oh okay.
I: To get even parts (?)
A: So you divide by four to get a smaller piece?
I: yeah
A: About how big do you think her piece will come out to be?
K: She points at a measurement mark on the ruler. I think it’s seven. It will be seven.
A: Seven inches. Is there a way you can check that?
K: I did eleven minus four.
A: You did eleven minus four instead of eleven divided by four.
K: Nods head yes in agreement.
A: So why did you switch to eleven minus four?
K: I remember when we was in class and y’all gave us a problem like this and it was smaller so
the people in my class divided. And then…but…there’s no way for it to be even…four can’t go
into twenty-five.
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Feury A: Oh so when you divided four into eleven you got twenty-five. She divided four into eleven
and I think she wrote the two because the four goes into eleven twice. And so she said four times
two is eight. And then she went to subtract the eight from the eleven. And I’m thinking when you
subtract eight from eleven you should get (counts out loud on three fingers) nine, ten, eleven,
you should get three. Don’t worry about it right now but that might have been why your answer
didn’t work out the way you expected. Continues to explain and show the right way of how to
solve the specific division problem. So if you just fixed what you did here you would have got
two point seven five. Do y’all know how big that is?
K & I: Shake their heads no.
A: Is it bigger than two? Or smaller than two? Is it bigger than one? Or smaller than one?
K: Facial expression shows thought. Then she smiles. It’s bigger!
A: Bigger than. If someone says give me two point seven five gallons of milk. Two point seven
five gallons of milk. Would I get more than a gallon or less than a gallon? Would I get more than
two gallons or less than two gallons?
I: More.
A: (To Kadyeisha) What do you think? Two point seven five gallons, is it more than two gallons
or less than two gallons?
K: She looks down at the table. More.
A: Yeah, it’s going to be more.
I: (Points at division on paper) Yeah ‘cause you got two whole ones here.
A: Yeah, you got two whole ones and then there’s a little more. So would you get more than
three gallons of milk, or less than three gallons?
K: Looks off into the distance. I don’t know.
A: See, you got two full gallons, and I got point seven five. So you think you’ll get more than
three?
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K: Nods head yes and gives a small smile.
A: So if someone said to me two point seven five gallons of milk they might actually get more
than three gallons of milk?
K: Giggles.
A: Well, alright, if they say give me two dollars and seventy-five cents, would two dollars and
seventy-five cents would be more than three dollars?
K: Gives a confused look. No.
A: Which would you rather have, two dollars and seventy-five cents or three dollars?
K: Three dollars.
The question has given the student the whole. The student is expected to find their piece
of the candy bar. This stimulates the recognition that the piece of the candy bar is actually a
fraction of the whole candy bar, activating the unitizing, partitioning, and disembedding
operations. With the collection of these operations, the student activates the part-whole scheme.
The first problem is very hands on. Even though she did not get far in solving the problem, the
first thing she did was pick up a ruler (visual aid) and attempted to figure out how far apart each
partition in the candy bar should be. Since the length of the paper was not divisible by four, she
could not partition it. This led me to the assumption that Kadyeisha cannot visualize even
partitions on the paper first without making marks on it. She understood that the entire sheet of
paper was her whole candy bar. When she was told that her piece was four times as big as hers,
she realized that the entire candy bar is broken into four parts. She knew that each candy bar
partition, including her piece, had to be equally the same size as all the rest. In order to
effectively make her piece, Kadyeisha used the ruler to measure the length of the paper (11
inches) in attempt to split the paper evenly into four pieces. Since eleven is not evenly divisible
by four, she made the division problem of 11 divided by four. She has unitized the sheet of paper
as a whole candy bar. This gives her a visual of how big the actual whole is. Using the ruler to
divide the paper activated the activity of partitioning.
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Feury Excerpt 1.2
A: What does this say about her candy bar if she was ready to divide?
K: That her candy bar will be bigger…no smaller. I don’t know. Yeah, smaller.
A: About how big will it be? You can join in now Isaac, if you want. Y’all can work on this
together now. Don’t need to put on all the pressure on Kadyeisha.
I: Wait, what was the question?
A: This is my candy bar and it’s four times as big as your candy bar. Show me what yours looks
like. How is this (pointing to division problem on paper) going to help Kadyeisha? (To Isaac)
You want to show her?
I: He vertically places the ruler in the middle of the paper and draws a straight line with the
ruler. He continues to (his) left side of the paper. He vertically places the ruler approximately
equal distance from the edge of the paper and the first drawn line. He uses the ruler to make a
second straight line. I don’t know.
A: For some reason you drew two lines. How is that going to help you?
I: Picks up the ruler and places it vertically on the (his) right side of the paper. The ruler is
equidistance from the edge of the paper and the first line drawn. He uses the ruler to draw a
third straight line. He points at the (his) left most portion of the paper. Cuts selected piece with
scissors and place it in front of Kadyeisha. That’s the candy bar.
A: (to Kadyeisha) Do you think he’s right?
K: Smiles and looks at piece.
A: Take a new sheet of paper. Places the paper Isaac drew on and the piece Isaac cut out above
the new sheet of paper. Here’s a new sheet. This was supposed to be four times as big as your
candy bar. And Isaac drew some lines and he pulled out this one and said this is how big yours
would be. Think he’s right?
K: Looks back and forth between the three difference pieces of paper. Nods head yes.
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A: Why do you think he’s right?
K: 'Cause he divided it into four then he toke away…he divided it and he toke away one so that
he could get the other part?
A: One part? One part out of how many parts?
K: Four.
Norton providing a visual whole next to the fraction strengthened Kadyeisha’s
simultaneous partitioning and introduced the part-whole scheme. Even though it is not exactly
“hands-on”, Kadyeisha still used a tangible object to verify whether Isaac’s answer was correct.
She was able to visually see the fraction Isaac made to the whole candy bar that was given to her.
With the fraction pulled out, she visually saw Isaac partitioned the sheet of paper into four parts
and disembed the requested candy bar piece. She was able to identify that her candy bar is a
fraction out of the whole. With the visual aid, she could physically see her candy bar become
disembedded. Seeing the fraction one out of four parts, she realized that all four pieces are a
collection of candy bars that made a whole candy bar.
Excerpt 1.3
A: You, Kadyeisha, get to make up a problem for Isaac but has to be similar to the one that I
asked.
K: Points at a whole sheet of paper. There’s your candy bar. Five times as big as your candy bar.
A: So whose candy bar is this? The whole thing is yours or his?
K: His.
A: This is your candy bar. And she wants you to make one…
K: As five times as big yours.
A: That’s five times as big as it.
I: So bigger than this?
15
Feury A: Say it again.
K: As five times as big as it.
I: So five times as big as this? Points at the whole sheet of paper. He places a sheet of paper to
the (his) right of the piece. Another above the piece. Another adjacent to the piece and a fourth
sheet above the adjacent piece.
A: That’s your finally answer?
I: Yeah.
A: Do you think he’s right, Kadyeisha? Is that what you wanted him to do?
K: I don’t know….just thought of a question.
A: Well let’s think about it now whether he’s right. This was your candy bar and his was
supposed to be five times as big as hers.
K: Looking at the sheets of paper on the table. Takes one of the sheets Isaac placed on the table.
She takes the sheet of paper Isaac used in a previous problem and places it in front of her.
Proceeds to make four lines on her paper, counting the spaces in between periodically.
A: (to Isaac) Do you know what she’s thinking?
I: Shakes head no.
A: I think I know why she did that. See if you could figure out why she did that. (to Kadyeisha)
And then you, when you’re done with what you’re doing, you have to figure out why he did that.
Y’all did two totally different things and I want y’all to understand each other.
K: Uses her index finger and points at each part in the paper. Looks off into the distance.
A: (to Kadyeisha) What are you trying to do?
K: I don’t know. I’m just counting up the five pieces of the candy bar. And…looks around.
A: (to Isaac) Can you describe what she did? Can you figure out why she did it?
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I: She drew the four lines to make five boxes.
A: Were those boxes the same size?
I: Yeah.
A: Pretty close. So she made five equal sized pieces, five equal parts. So her question to you was,
this was her candy bar (points at a new sheet of paper) and you were supposed to make one five
times as big. Now you explain why you did what you did.
I: She said five times as big as this (points at one sheet of paper). So I put five out.
A: So can you explain why this is five times as big as hers?
I: Well they’re all the same.
A: Do you think yours is five times as big as yours?
K: Head nod yes.
A: But she had something else in mind. You were thinking something else. Remember how I was
asking the question? What she said was this is my candy bar and yours is five times as big as
mine. And you did make something we all agree is five times as big as this. But I think she
wanted to ask the question in a different way. Do you know how you would have to ask the
question to make this kind of answer? What question should she have asked if she wanted to do
this kind of thing?
I: Your candy bar is right here (points at one part Kadyeisha drew on her paper)…sort of like
what you asked but with five.
Kadyeisha continues to use a tangible object to guide her assimilation of the situation.
This has been the second time she has practiced participatory behavior. Kadyeisha did not use a
ruler this time to guider her measurements. To me, this suggests that she is taking a baby step
towards anticipatory behavior. I believe Isaac’s previous methods of partitioning the paper
prompted Kadyeisha to follow the same method. It seems that she is now starting to trust her
own judgment without having to physically carry it out. The fractional language she used
suggests that she sees parts of a candy bar as copies of a smaller piece to make the composite
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whole. She uses the method Isaac assimilated in the previous situation to approach the question
she asked by partitioning and iterating. When dividing the paper, she explained that she was
counting up the pieces to make sure they were five parts on the paper. She did not communicate
that she is seeing her piece as a fraction; more of a piece that can fit on the paper five times.
Video Two
Excerpt 2.1
The question is trying to see if she could mentally unitize, iterate, and create a partitioned
whole. This situation hints at the equi-partitioning scheme.
A: This cut out is a fourth of a candy bar (Places a piece of paper on the table in front on
Kadyeisha). This is one-fourth of a candy bar. And if this is one-fourth of a candy bar, can you
show me what the whole candy bar might look like?
K: Pauses for a couple of seconds while looking at the piece of paper. Picks up a marker and
proceeds to measure out parts of the piece of paper by using her thumb and index finger as a
ruler. She then starts to draw three lines and shades in a part of the piece of paper. I’m thinking
that’s my candy bar and I had to break it into four pieces and I just took out one.
A: So what would you call this piece? (Points at the shaded part of the piece of paper.)
K: One-fourth.
Kadyeisha misunderstood the question. She identifies fraction as an entire whole and
approaches the situation in the opposite of what the question was asking. She interprets the
question as produce what one-fourth of the whole would look like. It implies that Kadyeisha has
difficulty imagining the whole; she selected a scheme that only could solve what a fraction is out
of a whole. She was able to disembed the fraction from what she identifies as a whole but could
not recognize that the piece given to her was physically disembedded from the whole that she is
requested to find. She activates the part-whole scheme by using the unitizing, partitioning and
disembedding operations. With this scheme, Kadyeisha easily identifies the piece of whole candy
bar.
Excerpt 2.2
18
Feury A: Places a brown fraction rod on the table. If this is a whole candy bar what would you call
these pieces? (Places a light green, red, purple, and white fraction rods on the table.) Why don’t
y’all work together on it?
K&I: Start placing the purple and green fraction rods along the side of the brown fraction rod.
I: (Places two purple fraction rods along the side of the brown fraction rod.) It’s a half of a
candy bar. (Places three light green fraction rods along the side of the brown fraction rod.) This
is one-third. (Places four of the red fraction rods along the side of the brown fraction rod). And
this is one-fourth.
A: So one of those is a fourth? Do you agree with that, Kadyeisha?
K: Nods her head yes.
A: Okay. So maybe…what is this? (Picks up a purple fraction rod).
I: Half.
A: Picks up a red fraction rod.
I: Fourth.
A: Picks up a white fraction rod.
I: Eighth?
K&I: Take turns lining up the white fraction rods along the side of the brown fraction rod.
I: (Points his finger at each white fraction rod while counting). One, two, three, four, five, six,
seven, eight.
K: Counts along with Isaac but silently to herself.
A: Y’all guessed that! How did y’all guess that?
I: Well, we…I used them a lot.
A: So how did you guess that?
Feury 19
K: I did it a lot at my other school.
She is quite familiar with the fraction rods so some of the fraction rods identities were
memorized facts rather than results of an activity. Even though Isaac was very confident about
his activity and hands-on, Kadyeisha still happened to check Isaac’s math. When she selects a
fraction rod to identify, she displays very strong partitioning and iterating skills. She could easily
tell Andy what fraction the fraction rod was to the whole. She has a strongly developed partwhole scheme. Throughout the video, Kadyeisha and Isaac continue to take turns identifying
fraction rods to different wholes. These situations continued to strengthen Kadyeisha’s partwhole scheme.
Excerpt 2.3
The question presents the students with a fraction of the whole and they must figure out
what the whole looks like. The students must unitize, partition, and iterate in order to obtain the
expected results Andy was looking for. This question requests for the equi-partitioning scheme to
be activated.
A: (Holds up a brown fraction rod) This is only half of a candy bar. If this is half of the candy
bar, can you show me what the whole one looks like?
K: Takes one of the brown fraction rods and starts to place the red fraction rods along its side.
I: it would be like that. Takes two of the brown fraction rods and places them next to each other.
K: Stopped working on the problem and looked at what Isaac did.
A: So if this is (picks up a brown fraction rod) half of a candy bar, Isaac thinks that this one is it.
(Points at the work Isaac did.) Do you think he’s right?
K: Yeah.
A: Can you explain why he’s right?
K: ‘Cause, if you say that’s a half, then this would be the other half. And it’s the same as this
one. So a half equals a half.
Feury 20
Since Kadyeisha’s part-whole scheme has not been reorganize, she automatically goes to
solving the situation how she previously done in the first video. Since the first question in the
second video is similar, her recognition template decides to use the same operations as she done
earlier. She treats the fraction as a whole, and using what she recently identified as the whole,
attempts to find the fraction presented to her. While looking at what Isaac did, she stopped
working and remembered that she and Isaac were not on the same page when Norton asked them
a similar question earlier. She recognizes she has not activated the proper scheme and operations
to approach her situation. She seems perturb due to her body language and attitude shows
somberness. With this epiphany, she knew that a degree of accommodation has to be made to her
current scheme to obtain the results she expects.
Excerpt 2.4
A: Okay. And I know that two of these fit in there (places two white fraction rods next to the red
fraction rod). If this is an eighth, and you know that two of these fit in here, does that tell you
how big one of these is?
K: Yeah. Half of an eighth is four.
A: So you think this is going to be one-fourth?
K: Yeah.
A: So you said (pointing to Kadyeisha) you should take half of eight to get four. And he’s saying
you should take eight and multiply it by two and get sixteen. Can y’all figure out a way to see
who’s right?
A: You said half of eight is four. She took half of eight. (to Isaac) But you didn’t take half of
eight. So what did you do?
I: I pretty much multiplied by eight.
A: And did that tell you how big this piece will be?
I: Yeah. One-sixteenth.
A: Explain why you did that.
Feury 21
I: Because, while this is an eighth and you can add two more… So two times eight.
A: (to Kadyeisha) Does that make sense?
K: Yeah, I understand.
A: All right Kadyeisha, so you say that this is a sixteenth now? Can you explain why it’s a
sixteenth?
K: ‘Cause if you line it up this many times into the bar you would get sixteen. (Points at the
whole – two brown fraction rods.)
Kadyeisha was presented with another equi-partitioning scheme question. She treats the
red fraction rod as a whole that is made up of two pieces (the two white fraction rods). She
knows that the white fraction rod is half of the red fraction rod. The white fraction rod is smaller
than the red but did not realize that the identity she gave to the white fraction rod was bigger than
the identity of the red fraction rod. I infer that Kadyeisha is able to disembed confidently with a
visual aid since she uses physical gestures (like pointing or estimating fraction’s lengths with her
fingers) to count parts. Kadyeisha continues to struggle with the concepts of inferring a fraction’s
identity compared to another fraction when asked a similar question.
Video Three
Excerpt 3.1:
The students begin the teaching episode by asking Norton a fractional situation. This
activates splitting.
I: This is your candy bar (pulls out a white fraction rod). Show me how many times can your
candy bar fit in here? (Places three orange fraction rods next to each other.)
A: This is my candy bar (lifts up the white fraction rod) and you want to know how many times
does my candy bar fit into there (points at the three orange fraction rods)? Okay. I think I’m
going to do a shortcut. (Starts placing the white fraction rods along the side of one of the orange
fraction rods)
I: You could do the reds.
Feury 22
A: That’s a good idea (starts taking out red fraction rods and places them along the side of one
of the orange fraction rods). Since that was Isaac’s idea, I’m going to ask Kadyeisha for help in
doing it. So Kadyeisha, I’ll line them up so this is my candy bar (picks up the white fraction rod)
and he said this whole thing he wants to know how many times it fits in there. And he said the
reds were going to help me somehow.
K: Two, four, six, eight, ten. (Counts out loud while pointing at each red fraction rod. She uses
her index and thumb to measure out the lengths of the remaining two orange fraction rods.)
Thirty times.
A: Thirty times. (To Kadyeisha) Is that our final answer?
K: Yes.
A: (to Isaac) Yeah we say thirty. It makes sense to me.
The situation initially created by Isaac called for the activation of the simultaneouspartitioning scheme. With Andy asking for Kadyeisha assistance in solving the problem, the
situation was elevated to a complex stage of part-whole scheme. She initially recognized that
some rod, whether red or white, can be fit a certain number of times to complete the requested
whole. As seen in the previous videos, Kadyeisha has strong iteration skills once the whole is
physically present. Andy’s use of the red fraction rod instead of the white fraction rod
encouraged Kadyeisha to recognize that the white fraction rod equivalency to the red fraction
rod. This stimulates the partitioning and iterating operations, causing accommodations to
previous schemes to activate the equi-partitioning scheme. As advanced as this may seem for
Kadyeisha, she paid more attention to the amount of red rods in the whole as opposed to the fact
that two white fraction rods equate each red fraction rod. She used the figures to measure the
lengths of the red fraction rods. In turn, her “marked spaced” helped guide her counting in order
to solve the situation. I believe that Kadyeisha recognized the fractional values of the white to
the red fraction rods and the red to the orange fraction rods, but I am not definite that she
developed the correlation between the white and orange fraction rods in general.
Excerpt 3.2:
Feury 23
A: I’m going to ask you a few questions about some different ways, ‘cause y’all were trying to
remember the tricky ways I was asking questions. All right I’ll start with one similar to what you
ask me. This is one-sixth (holds up a white fraction rod) of Kadyeisha’s whole candy bar. This
is just a piece of the whole candy bar, one-sixth. Show me what the whole piece looks like.
I: Places the white fraction rod next to two red fraction rods then adds another fraction rod at
the end of fraction rods.
A: (to Kadyeisha) He’s right?
K: Yes.
A: That’s what you’ll do?
K: Yes.
A: Would you have done it the same way?
K: Yes.
Kadyeisha’s recognition template to recognize the equi-partitioning scheme has been
supported with the continuous use of the fraction rods. With Isaac solving the situation, it gave
Kadyeisha an opportunity to follow and understand why this scheme was activated. In a previous
situation, Kadyeisha, along with Isaac, were encouraged to recognize the identity the white
fraction rods compared to red fraction rods. This substitution created the accommodation for
Kadyeisha to recognize that fraction rods that are bigger than another can be substituted to
complete a composite unit. This situation encouraged the activation of iteration, disembedding,
and partitioning operations. This has led me to infer the reorganization of Kadyeisha’s partwhole scheme is starting to become less dependent on participatory behavior. Kadyeisha seems
to be becoming more comfortable about having situations solved without her having to
physically carry it out. She realizes that two fractions with different identities still have relativity
to the same composite unit. These are the early stages of her part-whole scheme becoming more
anticipatory in order to create room for reorganization into the equi-partitioning scheme.
Excerpt 3.3
Feury 24
A: So what I said was that this piece (picks up a white fraction rod) is one-seventh of a whole
candy bar. If this is one-seventh of a whole candy bar, show me what the whole candy bar looks
like. So you build it this time, Kadyeisha. If that’s one-seventh of a whole candy bar, show me
what the whole candy bar looks like.
K: Pulls the red fraction rod whole closer to her.
A: You don’t have to use that. I just wanted it to be there.
I: What was the question?
A: You can use this (points at the red fraction rod whole) if you want, but you don’t have to use
it. I‘m saying this is one-seventh. This is a piece of your candy bar (picks up the white fraction
rod), one-seventh piece. If this is one-seventh of your whole candy bar, what does the whole
candy bar look like?
K: Takes an orange fraction rod and lines it up against the red fraction rod whole [that Isaac
made earlier] and then places a purple fraction rod next to the orange rod. I’m not sure.
A: Do you want me to ask you a different question and we come back to this? Or do you want to
keep working? (to Isaac) You want to help her out?
I: This is one-seventh. (Takes six other white fraction rods and place them next to each other in a
line.)
A: (to Kadyeisha) Did he do it?
K: She smiles.
Kadyeisha seems a bit perturb at how to assimilate the situation. She tries to approach the
situation using the scheme she used in excerpt 3.2, but realizes that this approach to the situation
does not work. It seems she is forcing herself to activate a scheme that her mind has not learned
to process fully without participatory behavior. Due to this complication, Kadyeisha seems to
skip over how the one-seventh piece’s identity relates to the red fraction rods. She becomes more
concerned with trying to find fraction rods that use less rods to make the requested composite
whole. Kadyeisha is trying to activate her equi-partitioning skills through participatory behavior
25
Feury but does not properly address the situation at hand. Even though she has recognized the
activation of this new scheme, it implies that her part-whole scheme has not been fully developed
to carry on antipaticipatory schemes. I make this inference due to the fact that Kadyeisha did not
seem to recognize that her part-whole scheme had to be carried out first mentally in order to
recognize the correlation the white fraction rod had to the red fraction rod. This has led me to
believe that most of her schemes are participatory dominant. Once provided some form of visual
aid, Kadyeisha will feel more comfortable approaching different fractional schemes to solve the
situation. However, Kadyeisha seemed lost when it was time to find the correlation between the
white fraction rods and the red and orange fractions rods she set up earlier. With this
perturbation, she stopped working and said “I’m not sure”. With the approach Isaac made,
Kadyeisha uses this as her recognition template to assimilate the following questions that
activated iterations of the one-seventh fraction. She is more successful with assimilating the
situations and even identifies that seven-sevenths is a whole. She has strongly developed the
concept of identifying fractions, partitioning, and iteration through participatory behaviors.
Video Four
Excerpt 4.1
A: So my first question is, if this is one-eighth (picks up a white fraction rod) of a candy bar, can
you show me what the three-eighths of the candy bar would look like?
K: Take two white fraction rods and places them next to the first white fraction rod.
A: You’re pretty fast. Last time y’all didn’t do it that fast. Can you tell me how you did that?
K: You said one-eighth so you would need two more.
A: If this is one-eighth (holds up a white fraction rod), can you show me what four-eighths of a
candy bar looks like?
K: Takes another white fraction rod and adds it to the three white fraction rods on the table.
A: Is there another name for this?
K: Remains quiet for a couple of seconds.
Feury 26
A: What were you thinking?
K: Shrugs her shoulder.
A: You’re just trying to think of one? Did you get any ideas?
K: Shakes her head no.
A: If this is one-eighth, can you make seven-eighths of a candy bar?
K: Takes four more white fraction rods and places it next to the three white fraction rods on the
table.
A: Do you know why that’s seven-eighths?
K: Counts silently to herself while point at each white fraction rod on the table. ‘Cause I added
six more.
A: How do you know that they’re eighths?
K: ‘Cause you said that this one is one-eighth and all the rest of them are too.
A: Can you show me what the rest of the candy bar would look like?
K: Pick up a yellow fraction rod then set it back down in the box. Picked up a blue fraction rod
then put it back down in the box. Puts a white fraction rod on the table apart from the work she
did earlier. This?
A: Is that the whole?
K: I don’t know. Yes?
A: What did we call this one before?
K: One-eighth.
Kadyeisha has difficulties relating the various values the same fraction can have without
a visual aid. She is able to iterate via fraction rods and recognizes identity. Due to the limits of
27
Feury her participatory behaviors, she cannot recognize that four-eighths carry the same value as onehalf. It is difficult for me to judge her iterating operation due to the fact that she figures the
difference between one-eighth and seven-eighths is that she added six more fraction rods. This
may be another sign of how a Kadyeisha participatory behavior limits her activation of certain
fractional schemes and operations. When asked to figure out what the whole candy bar look like,
Kadyeisha strived to find a single fraction rod. Her desire to find a whole is strong due to two
attempts of finding a fraction rod that was not long enough. Now in the previous video,
Kadyeisha communicated that she realized a whole consisted of a numerator and denominator
having the same number. But the rods she was looking for were to find the lengths of the seveneighths. This may have been a mistake. What I found interesting was that Kadyeisha needed to
see a whole as a whole. I am not sure if she realized that a whole can also be compromised of a
collection of partitions, in this case eight white fraction rods.
Excerpt 4.2
A: This is one-fifth of the candy bar (points at the red fraction rod), can you show me the whole
candy bar?
K: Take three red fraction rods and lines them up next to the first red fraction rod. She started to
reach for another red fraction rod in the box but did not pick it up. This is it.
A: Okay. Explain how you know that.
K: ‘Cause you said one-fifth and I just… (Looks at her current red fraction whole. Takes a red
fraction rod from the box and adds it to her whole)
A: Now you got it?
K: Yes.
A: You sure this time?
K: Yes.
A: Okay, can you explain?
Feury 28
K: You said that’s one-fifth and I just added four more so it’s five.
A: If this is one-fifth, what is two-fifths look like?
K: Pulls two red fraction rods away from the whole.
A: Three-fifths?
K: Adds another red fraction rod to the two red fraction rods on the table.
A: Four-fifths?
K: Adds another red fraction rod to the three red fraction rods on the table.
A: Five-fifths?
K: Adds another red fraction rod to the four red fraction rods on the table.
Kadyeisha is showing signs that she reorganized her part-whole scheme. The question
requests the activation of the disembedding, iterating, and unitizing fractional operations. She
has become more confident in acknowledge that even though a physical whole is not present, it
still exists. With this use of anticipatory behavior, Kadyeisha became more comfortable with
activating her part-whole scheme through anticipatory behaviors. She recognized that the
fraction was disembedde and that it needed to be iterated a certain amount of times to complete
the whole. Kadyeisha is showing confidence in the operations she activated due to the fact that
she was able to catch her own mistake when explaining her answer.
Excerpt 4.3
A: If this (holds up a light green fraction rod) is three-fifths of a candy bar, can you show me
what the whole candy bar looks like?
K: Grabs two light green fraction rods and places it next to the first light green fraction rod.
A: How do you know that?
K: Umm…You just add two more.
Feury 29
A: So this was three-fifths, and what would this be? (Pulls one light green fraction rod from the
rods Kadyeisha place earlier)
K: Four-fifths.
A: Adds the last light green fraction rods to the rest.
K: Five-fifths.
Kadyeisha realizes that the fraction has to be iterated to fails to connect the actual identity
of the fraction to the whole. Instead, she adds two more light green fraction rods. This infers that
she sees making composite units more as a number line rather than partitioned pieces of the
whole that are missing. Her strategy communicates that she realizes that there should be five
partitions in the whole and that she has already been given three parts of that whole. Due to her
participatory behaviors, she did not mentally recognize that the fraction given to her were three
individual partitions of the whole. I infer that her part-whole scheme is still being organized into
the equi-partitioning scheme. Her recognition template continues to automatically activate the
part-whole scheme rather than make accommodations. Kadyeisha’s participatory skills continue
to limit her operations. In the past videos, she has shown the ability to disembedde fractions, but
shows difficulty in mentally seeing the one-fifth in the three-fifths fraction rod.
Witness: This is the whole candy bar. Show us one-fifth of it. (Places one yellow and one orange
fraction rods on the table.)
K: Takes five light green fraction rods and places them underneath the requested whole. Then
she quickly takes them back and has a perturbed look on her face.
A: What did you want?
K: I wanted one five.
A: Well five of these (points at the light green fraction rods) fit in there. But you thought
something was wrong with that. What was wrong with that?
K: It was too big.
Feury 30
A: What did you want to do with it?
K: One-fifth of it. She grabs five white fraction rods and lines them underneath the whole.
There’s one-fifth of it.
Witness: Why do you think it’s one-fifth?
K: Umm….
A: Let’s come back to this one. Now I thought you had a good idea when you picked the green
one (points at the light green fraction rods). You saw that five of the green ones equaled that.
K: Looks at the line of five light green fraction rods. She takes away four and leaves out one.
That’s one-fifth.
Witness: Why is it one-fifth?
K: ‘Cause five of those equal the whole.
Witness: You have those white ones over there. Which one do you like better?
K: This one (points at the light green fraction rod).
Kadyeisha’s recognition template is strongly guided by her participatory behaviors. With
this situation, she conducted more of a trial and error method. She figured that the light green
fraction rod can be iterated into the whole five times evenly. She became perturbed when it was
time to physically disembedde the one-fifth from the whole. Since Kadyeisha could not
recognize the fraction, she moved on to using a different fraction rod. Instead of using the white
fraction rod to fill the whole, she just counted up to five and claimed that it was one-fifth of the
candy bar. The use of the white fraction rods lets me infer that Kadyeisha at times sees fractions
as a number line, therefore using basic math (like addition) to calculate how big the fraction
should be.
Video Five
Video: 051710 (K & I)
Feury 31
Excerpt 5.1
A: Which one is bigger, Kadyeisha? One-eight or one-ninth?
K: One-ninth….I mean one-eighth.
A: How do you know?
K: That’s…umm… ‘Cause as the number on the bottom gets bigger, it gets smaller.
A: How did you learn that? Was that something you just figured out on your own?
K: Yeah.
Kadyeisha finally understands that fractions cannot be viewed as if there are on a number
line. I consider this a big progress from participatory behavior to anticipatory behavior. She did
not need (nor was provided with) any visual aid to help her come to that conclusion. Through
enough experience, she has internalized this fact and it now serves as basic information to use
when activating a recognition template.
Excerpt 5.2
A: Suppose that this (places a small sheet of paper on the table) is my candy bar. And my candy
bar is four times as big as your candy bar. Imagine in your head what would yours look like?
K: I can imagine it. Like divide them into four, and the little piece is ours.
A: Okay, let’s do it. (Places scissors on the table).
I: Takes a ruler and draws three different lines on the paper.
A: I have a different way. I want y’all to see if my way works. (Grabs the piece of paper and
cuts it in half, then cuts the half in half). I think you’re candy bar would be this big.
K: Me too!
Kadyeisha has really shown some improvements on her anticipatory behavior, with
reference to the part-whole scheme. Presented the whole, she was able to properly visualize the
fraction, one-fourth, of the candy bar. She seemed very confident in her answer.
Feury 32
Excerpt 5.3
A: I’m going to ask y’all both a couple questions and I just want y’all to think about them in your
head. I don’t want you to draw anything at first. The first question is, suppose that this (places a
piece of paper on the table) is my candy bar. And my candy bar is as four times bigger than your
candy bar. Just imagine in your head what would yours look like.
K: Like we divide it into four. And this little piece (points at a small section of the paper) is ours.
I: Takes a blue Sharpie marker. He takes the ruler and uses it to measure four even partitions.
A: Now before you cut it, I want you to describe what you’ve done so far.
I: I measured it four times.
A: Okay I have a different way. I want y’all to see if my way works. So this is my candy bar and
it’s four times as big as yours (holds up the piece of paper, he cuts it in half, then cuts the halves
in half again). I think your candy bar will be this big (pulls out one of the four pieces).
K: Me too!
A: Really? Was that the one you were imagining?
K: Smiles and nods.
Kadyeisha is now able to visualize a fraction from a whole without having to physically
go through the steps. This is a big advancement towards anticipatory behavior for the
partitioning, iterating, and disembedding operations. She is becoming more comfortable
activating the part-whole scheme through anticipatory behaviors. This in turn is allowing
Kadyeisha to become less dependent on visual aids and become more confident in activating the
necessary fraction scheme to solve a situation.
Excerpt 5.4
A: Suppose this is a candy bar. And I said you can have three-fifths of this candy bar. Imagine in
your heads how much you would get. Okay, how big is it?
I: Uses his fingers to measure what the three-fifths would look like. That big
Feury 33
A: Kadyeisha, do you think that piece is going to be bigger or smaller?
K: Uses her fingers to measure what she thinks is three-fifths. Same size.
A: Explain why you think that is three-fifths?
K: Stares at the piece of paper. Can I change my answer? Measures with her fingers a different
length of what she considers three- fifths.
A: How about we draw it out?
K: Uses a green Sharpie to mark her idea of three-fifths.
I: Uses a red Sharpie to mark his idea of three-fifths.
A: All right. Now we are going to see who’s closer.
K: Takes a ruler and measures out the five partitions. She proceeds to shade in three-fifths of the
whole.
A: So what was yours closer to?
K: Five-fifths….a whole!
A: So show me three-fifths of the candy bar.
K&I: Points at the shaded part of the paper.
A: Which one is bigger: three-fifths or one-half?
K: A half. I think it is a half. The bigger the number gets, the smaller the piece is.
Kadyeisha still continues to have trouble when trying to visualize fractions that are more
than one-nth of a whole. She resorts back to the number line, but instead realizes that when a
number in the denominator becomes bigger than the fraction is smaller. When comparing the
one-half to the three-fifths, she was more focused on the denominator opposed to the numerator.
She sees the two fractions as two separate identities.
Feury 34
Conclusions
In the beginning, there were some difficulties of trying not to superimpose my own
knowledge of fractions during the studying of Kadyeisha. In teaching experiments, it is very
important to strictly use your own knowledge outside of the study to decrease the chances of
selective study and avoid attributing your knowledge to the student. I use the term selective study
as overlooking certain behaviors of the student that could compromise the study or ignoring a big
turning point in the student’s fractional concepts. At times, it was very easy to pinpoint what
Kadyeisha was doing or attempting to do, and other times I was perplexed with her style of
thinking. With the repetition of watching the teaching videos that retrospective analysis calls for,
I was able to discover Kadyeisha strengths and weaknesses during the study with Dr. Norton.
In the first video, it was evident that Kadyeisha is a visual learner when it comes to
fractional problems. If a visual aid is not produced for her, she has difficulty assimilating the
situation and attempts to provide her own visualization. When making a mental visualization,
Kadyeisha uses classroom and real world experiences as a reference to solve the problem (as
seen with the 2.75 gallons of milk and $2.75). Since she made attempts of using the part-whole
scheme, I have inferred that she has frequently used the simultaneous-partitioning scheme prior
to this teaching experiment. Kadyeisha does well with problems that request her to divide a
candy bar evenly among a certain amount of people, provide that she sees the actual candy bar
(in this instance piece of paper). She attempts to separate parts of the fraction from the whole,
but only when she can tangibly do it.
Video two showed how Kadyeisha seemed comfortable around situations that activated
the part-whole scheme. The fraction rods made it easier for her to approach the situations since
they were tangible objects. However, there were some instances where Kadyeisha had issues
with using the proper fractional language to communicate her ideas. These necessary errors led
me to have a deeper understanding on how Kadyeisha viewed things. For example, when
Kadyeisha showed Isaac that this is one-third of his candy bar, she asked what is the one-third
instead of the whole. This showed that even though she realized that she wanted the whole to be
partitioned in three parts, she still saw her provided fraction as the actual whole instead of a piece
to the composite unit. Kadyeisha still continues to strengthen her part-whole scheme but is
recognized that she needs to reorganize it to approach situations that request higher schemes.
Feury 35
In video three, Kadyeisha is assimilating her newly developed equi-partitioning more
often. Her disembedding operations are slightly impaired if there is not a physical visual to guide
her recognition templates, limiting this operation to pure participatory behaviors. She is more
willing to make accommodations to schemes to assimilate a situation once a visual aid is
provided to guide reorganization. She activated the splitting operation for the first time
throughout this video, with the help of Isaac using the fraction rods. Kadyeisha seems more
confident with her mathematical concepts to the point that she allows herself to reorganize and
accommodate operations and schemes to obtain results.
Throughout video four, Kadyeisha has shown trouble on deciding which of the two
answers provides the actual fraction. This suggests that her participatory behavior tends to either
limit or confuse her on which recognition template to use. Her current fractional behaviors
support that her activation heavily relies on participatory behaviors. As her part-whole scheme
continues to get less participatory. Kadyeisha still has to make accommodations in situations that
call either for a strongly developed part-whole scheme or equi-partitioning.
Kadyeisha’s style of approaching fractional situations was very unique. She used a
combination of visual aid, classroom rules, and the number sequences. Olive’s hierarchal order
of number schemes let me understand the level of thinking Kadyeisha used to approach fraction
situations that asked her to compare one fraction to another. In the beginning of the teaching
videos, she thought that the bigger the number was in the denominator, the bigger that actual
fraction should be. With the part-whole scheme, the Initial Number Sequence and progressed to
the Explicitly-Nested Number Sequence (ENS). Kadyeisha would at times think that fractions
were numbers that continued in a regular number line. ENS become more first nature when she
could easily visualize, with the assistance of a visual aid, whenever comparing fractions to the
composite whole.
Initially, Kadyeisha’s fractional situations were strong only through participatory
behaviors. The first three videos, Kadyeisha would play off of whatever Isaac said or did at
times. For example, in excerpt 2.3, Kadyeisha was not confident in the operations she should use
until she used what Isaac did to activate a recognition template. Without having any visual aids,
it becomes very difficult for her to try to understand the situation and how to approach it.
Feury 36
Throughout the five teaching videos, Kadyeisha reorganized her part-whole scheme to
equi-partitioning. Between the third and fourth videos, she started to become less participatory
with her actions when using part-whole schemes. During the fifth video, I saw Kadyeisha
become more frequent in activating the equi-partitioning using participatory behaviors. In
excerpt 5.1, I correctly guessed that Kadyeisha would initially get the answer wrong and correct
herself. This video showed early signs of the part-whole scheme being activated with
anticipatory behaviors. It eventually gave space for Kadyeisha to activate the equi-partitioning
scheme, solely on participatory behaviors. She still relied on the visual aids to help guide her
thinking process and determine which operations should be used to execute the fraction situation.
In all, Kadyeisha proved my hypothesis that she relied quite heavily on her participatory
behaviors in order to assimilate the fractional situation. Any chance she had, she would rely on
visual aids, whether it was fraction rods, markers and papers, or even her fingers to count and
mark spaces. Starting from using the part-whole scheme, Kadyeisha was able to reorganize her
fraction scheme into the equi-partitioning scheme. As much as she used her participatory
behaviors to guider her choice of using certain operations, Kadyeisha eventually did feel
comfortable enough to use visual aids as a resource to check whether her method is successful as
oppose to using the visual aids to strictly guide her recognition templates. This research shows
that Kadyeisha was very reliant on approaches that used participatory schemes.
References
Hackenburg, A. J. (2010). Mathematical caring relations in
action. Journal for Research in Mathematics Education, 41(3), 236273.
Hackenburg, A. J. (2010). Students' reasoning with reversible
multiplicative relationships. Cognitive and Instruction, 28(4), 383432.
Norton, A. H., & McCloskey, A. V. (2008). Modeling Students' Mathematics Using Steffe's
Fraction Schemes. Teaching Children Mathematics, 15(1), 48-54.
Feury 37
Olive, J. (2001). Children's number sequences: an explanation of steffe's
constructs and an extrapolation to rational numbers of
arithmetic. The Mathematics Educator, 11(1), 4-9.
Steffe, L. P. (2001). A new hypothesis concerning children's fractional knowledge. The Journal
of Mathematical Behavior, 20(3), 267-307. doi: 10.1016/s0732-3123(02)00075-5
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying
principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in
mathematics and science education, 267-307. doi: citeulike-article-id:8002929
Tzur, R. (2007). Fine grain assessment of students' mathematical
understanding: participatory and anticipatory stages in learning a
new mathematical conception. Education Student Mathematics, 66,
273-291.