Running head: MAKING SENSE OF INSTRUCTION ON FRACTIONS
Making Sense of Instruction on Fractions When a Student Lacks Necessary Fractional Schemes:
The Case of Tim*
John Olive and Eugenia Vomvoridi
The University of Georgia
Draft: Do not cite without permission of the authors.
*This research was conducted as part of the CoSTAR (Coordinating Students’ and Teachers’
Algebraic Reasoning) project, directed by Dr. Andrew Izsák and funded by the National Science
Foundation, grant No. REC 0231879. All opinions are those of the authors.
Making Sense of Instruction on Fractions 1
Abstract
This paper critically examines the discrepancies among the pre-requisite fractional concepts
assumed by a curricular unit on operations with fractions, the teacher’s assumptions about those
concepts and a particular student’s understanding of fractions. The paper focuses on the case of
one student (Tim) in the teacher’s 6th-grade class who was interviewed by one of the authors
once a week during the teaching of the unit. The teaching materials and the teacher’s instruction
were based on the assumption that students understood the concept of a unit fraction as being one
of several equal parts of a given whole. The teacher neither emphasized the need for equal parts
nor the part-to-whole relation. The teacher’s reasonable assumptions about her students’
understanding of fractions were severely challenged by the cognitive constructs that Tim
exhibited during his first 2 interviews. When she viewed tapes of the interviews she realized Tim
lacked essential constructs to make sense of her instruction. She subsequently made adjustments
in her instruction and built on tasks that we used with Tim in our interviews. These adjustments
helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This
study illustrates the importance of coming to understand a student’s mathematical activity in
terms of possible conceptual schemes and modifying instructional strategies to build on those
schemes. The coordinated design of the research study facilitated these instructional
modifications.
Making Sense of Instruction on Fractions 2
Theoretical Framework
As teachers, we all make assumptions about what our students know about the subject or
topic we are teaching. These assumptions are based on our understanding of the topic, our
experiences with teaching this topic to students in the past and our expectations of what students
bring to the topic from their prior learning experiences, along with our interpretation of what
students ought to know based on the materials we are using with students. Cohen and Ball
(1999, 2001) emphasized that instruction is a function of interactions among teachers, students,
and content as mediated by instructional materials. Teachers use their knowledge of the content
and experience with students to mediate students’ opportunities to learn through instruction with
specific materials. Students use their prior knowledge to assimilate, interpret, and respond to
instruction. Moreover, students’ prior knowledge and responses to the instruction and content
help determine what teachers can accomplish. In this study we attempt to examine these
interactions by coordinating classroom observations of instruction with a detailed study of one
student’s learning and interviews with the classroom teacher. The central focus of the paper,
however, is on the student’s learning.
Our framework for studying students’ learning is based on Piaget’s scheme theory as
interpreted by von Glasersfeld and used by Steffe and Olive (1990) in their study of children’s
construction of fractional knowledge. Piaget (1980) based the idea of scheme on repeatable and
generalized action. “All action that is repeatable or generalized through application to new
objects engenders . . . a ‘scheme’.” (p. 24) We identify schemes through observing children’s
recurrent use of a goal-directed activity on several different occasions in what to us are related
situations. Through such observations, it is possible to describe a scheme. According to von
Glasersfeld (1980), a scheme consists of three parts. First, there is an activating situation as
perceived or conceived by the child, with which an activity has been associated. Second, there is
the child's specific activity or procedure associated with the situation. Third, there is a result of
the activity produced by the child. Key schemes identified by Steffe and Olive in their research
on children’s construction of fractional knowledge that are relevant for this case study include
the following:

An equi-partitioning scheme (Olive, 1999; Steffe, 2002; Olive & Steffe, 2002) with
which children can mentally partition a continuous quantity into equal parts prior to
Making Sense of Instruction on Fractions 3
operating on that quantity.

A partitive unit fractional scheme (Steffe, 2002; Tzur, 1999) with which a child can
establish a unit fraction as one part out of a whole consisting of equal parts and also, any
one part can be used in iteration to re-establish the whole, or any sub-part of the whole.
Thus one eighth iterated three times produces three eighths of the whole.

An iterative fraction scheme (Olive, 1999, Olive & Steffe, 2002) with which a child can
use an iterable unit fraction to construct fractions greater than the whole.
The Context of the Study
This study is part of Project CoSTARi, a 3-year investigation of the interplay between
teachers’ and students’ understandings of shared classroom interactions and on ways that
teachers and students modify their teaching and learning of middle school mathematics as they
become more aware of those understandings over sequences of lessons. The project takes place
in Pierce Middle School (all names are pseudonyms). Pierce Middle School has 685 students
and is in a rural community outside of a large urban center in the southeast. Students in the
district are racially and economically diverse. The district has ranked in the middle on statewide
mathematics tests and is making a concerted effort to improve its mathematics education. As part
of this effort, teachers and district administrators began working together to transition from
traditional, skills-based materials to standards-based materials across grades K-12 starting in the
2001-2002 school year. For the middle school, the district adopted the Connected Mathematics
Projectii (CMP) materials.
The CoSTAR project studies the teaching and learning of CMP units that involve
multiplicative comparisons, such as ratio, rate, and slope. The primary purpose of the project is
to gain insight on the consequences (for teaching and learning) of the different understandings
teachers and students have of joint activity. The CMP materials, being “organized around
appealing and engaging problems” (Lappan, Fey, Fitzgerald, Friel and Phillips, 2002),
potentially affords means of accessing these understandings.
The specific context of this case study was the sixth grade CMP unit, Bits and Pieces II,
which focuses on fraction arithmetic. The sixth grade class consisted of 23 students: 7 African
American, 2 Hispanic, and 14 Caucasian/13 girls and 10 boys. The students had completed the
CMP unit, Bits and Pieces I, which focuses on establishing fractional relations between parts and
Making Sense of Instruction on Fractions 4
wholes. The teacher of the 6th-grade class, Ms. Moseley, is an experienced teacher who had
taught 7th grade mathematics for many years using traditional, skill-based textbooks. The same
year that Pierce Middle School adopted CMP, she was moved to teaching 6th grade mathematics.
This study took place during her second year of teaching 6th grade. Ms. Moseley was very
enthusiastic about teaching, learning and using CMP, although she did not always feel confident
with her content knowledge. She had not used problem solving/exploratory based textbooks prior
to CMP. She constantly seeks professional development and is genuinely concerned about her
student’s learning.
Methodology
Data Collection
Various members of the CoSTAR team, including the authors, were involved in the data
collection activities. The data for this case study were collected during the spring semester of
2003, with some confirmatory data collected during the fall semester of 2003. (Ms. Moseley
followed her students into seventh grade.) The last nine class lessons of Bits and Pieces I and all
class lessons during Ms. Moseley’s teaching of the Bits and Pieces II unit from CMP up to but
not including the final investigation on division of fractions were videotaped using two cameras.
One camera was focused on the students’ written work and the other on the teacher’s instruction
and interactions with students. Four pairs of students were identified by the classroom teacher to
provide us with a spectrum of mathematical understanding and skills that she saw in her
classroom. The pairs of students were also selected for their ability and willingness to express
themselves during interviews. The initial pairings by the teacher were thought to be between
students of similar abilities (however, in the case of Tim, this turned out to not be the case). Each
pair of students was interviewed by one of the authors once a week during the eight weeks of
data collection. These interviews were all videotaped, also using two cameras – one focused on
the written work or computer screen and the other on the students and interviewer. After the first
two interviews it was obvious that Tim and his partner, Angela, were working with different
fractional schemes so a more appropriate partner, Jennifer, was found for Tim. A CoSTAR
researcher (not listed as an author) interviewed Ms Moseley at least once a week during the eight
weeks. There were 11 teacher interviews in all. Ms Moseley’s interviews were also videotaped
using two cameras. Excerpts from classroom video (viewed on a laptop computer) were used in
Making Sense of Instruction on Fractions 5
both student and teacher interviews. In addition, video excerpts from student interviews were
used in the teacher interviews.
Additional data were collected during the fall semester of 2003 while Ms. Mosley was
teaching the seventh grade unit on Variables and Patterns from CMP. The data were collected in
the same manner as the ones from the spring 2003 semester. Tim was paired with another
partner, Kelly, as Jennifer had moved to a different classroom.
Data Analysis
There were three main stages to the data analysis process. The first stage took place
during the actual data collection and was carried out by various members of the CoSTAR team,
including the authors. On the same day following the classroom videotaping, the digital videos
from the two cameras were mixed, providing a single video source in which one picture was
inserted in a box within the other (picture-in-picture). The video sources could be switched on
the fly during mixing in order to show close-ups of relevant student work. “Lesson Graphs” were
created from this mixed video source that provided a written description of the classroom
instruction, highlighting key points in the instruction and interactions with case study students.
Screen shots from the video were inserted into the lesson graphs to illustrate important events.
These lesson graphs were time-coded for easy retrieval of the video data source. Each lesson
graph along with the mixed video were used by the researcher who was conducting the student
interviews to review the day’s lesson and select episodes to use as stimuli for the student
interview on the following day. The researcher designed activities for the interview that could
potentially help probe for students’ deeper understanding of the learning issues arising from the
classroom instruction.
Following the student interviews, a similar process was carried out in order to create an
initial record of the important events and identify critical aspects of the students’ thinking and
learning. These records (lesson graphs and mixed video of student interviews) were subsequently
used by the researchers who interviewed Ms. Moseley to select excerpts for use in the teacher
interviews, along with selected excerpts from the classroom video record.
During the second stage of data analysis, the authors reviewed the lesson graphs and
video data from all three contexts (class lessons, student interviews and teacher interviews in a
retrospective analysis using a constant comparative methodology. Detailed transcriptions were
made of interviews that were seen as critical for the case study under construction. Links among
Making Sense of Instruction on Fractions 6
the teacher’s instructional actions, students’ classroom activity, students’ responses and
subsequent activity during the interviews, and the teacher’s reactions and reflections on all of the
above were established.
During the third stage of the analysis, the authors analyzed these links in order to posit
possible associations and interactions. We revisited the transcripts, lesson graphs and mixed
video sources many times during this third phase in order to develop our hypotheses concerning
associations among our emerging model of the student’s mathematical understanding of
fractions, the teacher’s acknowledgment and understanding of the student’s fractional knowledge
and her instructional strategies. For the purposes of this study we also used the data from the fall
semester to support our hypotheses.
Results of the Coordinated Analyses
Evidence of Ms. Moseley’s Assumptions
Based on our observations of her teaching of the last few lessons of Bits & Pieces I, we
concluded that Ms. Moseley assumed that her students understood the necessity for parts to be
equal when dividing a quantity into fractional parts. When working on a problem (from class on
March 10, 2003) involving sharing 8 pizzas among 10 people and asking how much pizza each
person would get, Ms. Moseley drew the following representation (figure 1):
Figure 1: Ms. Moseley’s illustration for sharing 8 pizzas among 10 people
Neither the pizzas (wholes) nor the slices (parts) were the same sizes respectively. Her roughly
drawn pictures were supposed to illustrate how students could divide each of the pizzas into 10
parts and each student would take one part from each pizza, ending up with eight tenths of a
pizza. This strategy assumes that all of the pizzas are the same size (otherwise, the resulting
eight tenths would not have a referent unit pizza) and that all ten parts in one pizza are the same
size. Ms. Moseley did not specifically state these conditions in her instruction, thus we conclude
Making Sense of Instruction on Fractions 7
that she assumed that students understood that these were necessary conditions and that her
drawings were just approximations to represent the strategy. In the interview that followed this
lesson, Ms. Moseley was asked if she learned anything new about the students’ understanding of
fractions from the problem.
Protocol A (from interview with Ms. Moseley on March 13, 2003)
Interviewer: Do you think that the, the problem told you anything about their understanding of fractions that
you wouldn’t have otherwise had?
Ms. Moseley: Well, again, I think, and I, this last little section of this book I’ve found, I’ve felt like the book has
not drawn a very straight path to where we’re trying to go, to seeing fractions as a proportion, to
seeing fractions as a division problem, which is what this was, and then seeing fractions as, um,
something that we can then turn into a decimal and a percent.
Ms. Moseley’s response does not really address the question of students’ understanding
of fractions. It does, however, illuminate her goals for the unit as seeing fractions as a
proportion, a division problem and as “something that we can then turn into a decimal and a
percent.” She does not mention fractions as equal parts of a unit whole in her response or in any
other part of this interview.
In her next interview, Ms Moseley was asked to compare the goals and approaches for
teaching Bits & Pieces I and II. She responded as indicated in Protocol B.
Protocol B (from interview with Ms. Moseley on March 19, 2003)
Ms. Moseley: Well, Bits 1 was all about trying to determine equalities, trying to determine different
representations of parts of a whole, you know doing very informal methods of how you add up –
you know – using all the fraction strips, I think you’re trying to get them to see how they could
possibly be added together if you have the same denominator. But, we spent an inordinate amount
of time on trying to get them to understand that if they’re different sized pieces they can’t be
combined. Also, informally trying to teach them the concept of the whole – what is the whole?
And that the whole is going to play a huge role in them understanding what a fraction is. So, that,
to me is Bit & Pieces 1 – understanding the whole; understanding the relationship between
fraction, decimal, and percent; understanding informally how – well not informally -- but
understanding how we decide which are bigger, which are smaller. Which, hopefully, then would
lead them into Bits 2, which if they have a solid understanding of what a fractional unit really
means, then they have some understanding, hopefully, of why they can’t just add 2/4 and 3/9 and
get 5/13ths.
Interviewer: Right.
Making Sense of Instruction on Fractions 8
Ms. Moseley: So, hopefully, with that base – that they understand fairly well what a fraction is, then you move
into, well – now that we know what they are, let’s see if we can learn how to work with them.
Interviewer: Given how your students did in Bits 1 and the approach that Bits 2 takes, which I assume you’re at
least somewhat familiar with at this point
Ms. Moseley: um hmm
Interviewer: Do you have a sense of how ready your students are to succeed in Bits 2?
Ms. Moseley: Not ready.
Interviewer: Where do you think that they are still unprepared?
Ms. Moseley: I still think they do not understand the idea that fractions are different size portions of a whole.
And, therefore, you cannot combine different sized portions in adding and come up with a sum. I
don’t think they get that.
Although Ms. Moseley did emphasize fractions as parts of a whole and the necessity to
understand what the whole is in the above protocol, her focus was on different sized parts of a
whole, and the students needing to understand that they can’t combine different sized parts. She
recognized that many of her students do not see a need for parts to be equal when combining
fractional parts of a whole but did not necessarily relate this difficulty to students’ lack of the
concept of a unit fraction as one of so many equal parts of the whole. Only after Ms. Moseley
remembers the difficulty students had in class in estimating the size of one fifth did she
specifically mention that some students might not see the necessity for equal sized pieces.
Protocol B (continued)
Ms. Moseley: I don’t know that they fully get that fraction strip being 1/5, 2/5, 3/5, 4/5, 5/5 because some of
them yesterday in estimating said ‘Oh, one fifth – that’s almost whole.’ Then, they just totally
missed that it’s a very small; it’s only one of that unit. I don’t know that they always agree that in
a given situation, every fifth is the same size… I think they don’t always know that if I have 1/5 of
this pizza pie and then he has 1/5 of the same pizza pie – I’m not sure that they always know they
have the same size piece.
Interviewer: They just, in your mind, you think that they just say that there’s five pieces and whatever size
piece…
Ms. Moseley: And they don’t think about what that means – that it’s five equal sized pieces. I don’t think a lot
of them really get that – that it’s five equal size pieces.
In the lesson that followed this interview on the same day, Ms. Moseley attempted to
draw a circle to illustrate fifths. Her first two attempts resulted in obviously unequal parts (one
part was closer to a third than a fifth). She eventually ended up with the following picture
(figure 2):
Making Sense of Instruction on Fractions 9
Figure 2: Ms. Moseley’s attempt to partition a circle into five equal parts
Even though Ms. Moseley commented on how difficult it was to draw fifths, she did not
explicitly state that she was trying to draw five equal parts in her circle. This resulting figure,
although much closer to equal parts than her first two attempts, still has one part that is more
than a quarter of the circle and two parts that constitute approximately half of the circle. While
we agree that it really is difficult to draw five equal parts in a circle, Ms. Moseley’s use of
approximate representations throughout her instruction, coupled with the lack of any explicitly
spoken intention to draw equal parts, may have unintentionally supported Tim’s lack of an
equipartitioning scheme for unit fractions (Olive & Steffe, 2002; Steffe, 2002, 2003).
Evidence for Tim’s Conception of a Fraction
Four issues concerning Tim’s conception of a fraction arose from his first two interviews:

Lack of necessity for parts to be equal (e.g. one small piece pulled out of a bar consisting
of three twelfths and three fourths was called “one sixth”).

Both the fractional whole AND the unit fraction were determined by ALL the parts (e.g.
a circle with six parts was both six sixths and one sixth).

Lack of necessity for referent wholes to be the same size (e.g. five of six equal parts of a
circle constituted one fifth).

Inability to conceptually disembed a part from the whole.
In the interview with Tim and Angela conducted on March 18, 2003, the students were
using a computer program, Fraction Barsiii (Orrill, 2003), to represent and name fractions of a
unit bar made by joining two unit fractional pieces of the bar (e.g. one third and one fourth). Tim
set up the next problem. He used the computer actions to partition an on-screen unit bar into four
equal parts. He then used the computer actions to partition the first part on the left into three
equal parts. The resulting bar consisted of six unequal parts that we would regard as 3 onetwelfths and 3 one-fourths. Tim then pulled out one of the small pieces (1/12) and placed it
Making Sense of Instruction on Fractions 10
below the unit bar (see Figure 3). [Pullout Split was a computer action that disembeded a part
from the whole while leaving it in the whole bar.] The following protocol begins at this point in
the interview.
Figure 3: Screen shot from video of Tim’s work with Fraction Bars on March 18, 2003
Protocol I (from interview with Tim and Angela on March 18, 2003)
Interviewer: Any idea what that is (pointing to the small piece below the original bar)?
Tim:
A half?
Angela:
See if you split that one piece in thirds then the other pieces, since those pieces are smaller, I guess
the other pieces will be in thirds also (pointing to the three 1/4-pieces in Tim’s bar).
Interviewer: O.k. So any idea what fraction this is of the unit bar (pointing to the small 1/12-piece that Tim
pulled out of the bar)?
Angela:
One twelfth? One twelfth I think.
Tim:
One seventh? Is it one seventh?
Interviewer: (to Tim) Do you think one seventh? Why do you think one seventh?
Tim:
Cause there’s seven in all and we pulled one out.
Interviewer: Show me the seven that you have.
Tim:
(Pointing to each of the parts in his partitioned bar) one, two, three, four, five, six. Oh six! One
sixth.
Based on Tim’s activity of counting each of the parts in the bar to arrive at his answer of
one sixth, we infer that when he initially said that it is one seventh he counted all the pieces -- the
six that make up the unit bar and the one he had pulled out that was the piece he was trying to
name. Alternatively, he could have miscounted just the pieces in the original bar. Either
explanation is an indication of Tim’s lack of an equi-partitioning scheme (the pieces he was
counting were not the same size – some were 12ths and others were 4ths).
The next protocol suggests that Tim’s concept of a unit fraction was based primarily on
the number of parts that were present (regardless of size) and did not take into consideration the
part-to-whole relation of one part to a referent whole. The students had been asked to work with
Making Sense of Instruction on Fractions 11
a set of plastic “fraction circles” similar to ones they had used in the previous class lesson. The
set consisted of one whole white disk, one disk made up of two half-disks in orange, one made
up of three congruent blue sectors, one made up of four congruent yellow sectors, one made up
of six congruent green sectors and one made up of eight congruent red sectors. When the
students attempted to make a circle using the green sectors, they found that one sector was
missing, thus they had an incomplete disk with five green sectors (see Figure 4). The protocol
begins at this point in the interview.
Protocol II (from interview with Tim and Angela on March 26, 2003)
Figure 4: The 6ths disk from the Fraction Circles with one piece missing
Interviewer: So how much of the circle is that (referring to the five green slices that all together represent five
sixths of the pizza)?
Tim:
One fifth, just right there, but we’re missing one.
Interviewer: Say that again Tim.
Tim:
Like the fraction right now is one fifth, but we’re missing one.
Interviewer: O.k. can you show me why you think it’s one fifth? You can use the pencil.
Tim:
The circle is like that, (draws a circle) and you cut them into sixths (draws six slices in the circle)
and like five are shaded but one is missing…
Interviewer: O.k. and you said that that was how much? Write it down how much that was. How much of the
circle is there? I want to know how much of the circle is there. (Tim writes
1
on his paper) You
5
think one fifth of the circle is there.
The above suggests that, even though Tim is aware that the whole disk would have six slices
(sectors) he still calls the amount made up of the five remaining sectors “one fifth” because there
are only five pieces. Such reasoning is evidence that for Tim there is a lack of necessity for
referent wholes to be the same size. “One fifth” refers to a quantity made up of five parts of a 6part circle and does not include a part-to-whole relation. The continuation of this protocol
reinforces our hypothesis regarding Tim’s concept of a unit fraction.
Making Sense of Instruction on Fractions 12
Protocol II (first continuation)
Interviewer: I think Angela’s bursting to say something else.
Angela:
I think it’s five sixths.
Interviewer: O.k. …. Can you use the blue pencil? Cause Tim is using the regular pencil and you demonstrate
why you think it’s five sixths.
Angela:
(draws a circle with six slices and then shades five of them) O.k. so this one is missing. These are
shaded in and these are six. Yea, six. And since that would be…um five of them are shaded in, but
there’s six pieces, so that would be five sixths.
Interviewer: Can you write it down as a fraction? Tim I want you to show me with the fraction pieces, and write
down…you use the pencil…what is one sixth of the circle…what would one sixth of the circle
look like?
Tim:
One sixth…they will all be shaded, like…they would all be shaded if it was one sixth…(Tim
draws a circle with six slices)
Interviewer: They would all be there if it was one sixth?
Tim:
Cause it’s six in all…
Interviewer: Six in all. O.k. what is the whole thing?
Tim:
Six pieces.
Interviewer: O.k. and how would you write the whole thing as a fraction?
(Tim writes 6
)
6
Interviewer: And what fraction is that?
Tim:
Six over six, cause if you shaded all of them in they will be six shaded over six pieces.
Interviewer: And that would be…say it again what the fraction is called…
Tim:
Six over six.
Interviewer: Six over six… What would one sixth be? Can you write one sixth for me?
(Tim has difficulty drawing the one sixth as Figure 5 indicates.)
Figure 5: Tim’s drawings of one fifth, one sixth, and six sixths from video of interview
Making Sense of Instruction on Fractions 13
The bottom circle with six very unequal pieces is Tim’s attempt to show “one sixth.” The shaded circle
above that is the first circle he drew to illustrate “one fifth”. The circle at the top of the picture is Tim’s
attempt to show “six sixths.” The circle on the right (in blue) is Angela’s attempt to represent five sixths.
Tim:
One sixth would be…probably the same thing, cause if you shade six and there is one whole of
them out of six pieces…
At this point in the interview, the interviewer asked Angela to show one sixth of the disk. She
picked up one of the green sectors and said that this would be one sixth.
The most intriguing aspect of the above protocol is Tim’s use of the same representation
for both six sixths and one sixth of a circle. In Tim’s words: “One sixth would be…probably the
same thing, cause if you shade six and there is one whole of them out of six pieces…” Tim’s
statement indicates to us the lack of a part-to-whole relation with regard to unit fractions. It also
suggests that unit fractions for Tim must consist of ALL the pieces (all five pieces for one fifth
and all six pieces for one sixth). This necessity to have all the pieces present indicates a lack of
any disembeding operation for establishing unit fractions as one part out of several equal parts
that constitute the given whole. There were many confirming instances in the continuation of
this interview for our model of Tim’s fractional concept. In the following protocol, Tim was
asked to show one-half plus one-fourth using the fraction circles.
Protocol II (second continuation)
Interviewer: Now how would you show 1/4 + 1/2?
Tim:
Like writing it down or with these (points at the plastic slices)?
Interviewer: With the circles.
Tim:
You can just put like that like that (places a whole circle made out of the four yellow pieces next to
the one orange half-circle and makes a “+” symbol with his finger between the orange and yellow
pieces).
Interviewer: And where is your one fourth?
Tim:
Right here (pointing at the whole circle made out of 4 yellow pieces). (He pauses for a couple of
seconds) Oh! (He lifts up one yellow piece.)
Interviewer: O.k. this is important, right. So put them together. Put the one-fourth and the one-half together
(Tim joins the 1/4-circle with the 1/2-circle).
Interviewer: O.k. Now what Mrs. K was trying to show you…if you write down Tim for me…if you write
down one-fourth plus one-half (Tim writes
should equal?
Tim:
One fifth? I think…
Interviewer: Why do you think it would be one fifth?
1
4

1
2
on his paper)…now… what do you think that
Making Sense of Instruction on Fractions 14
Tim:
Cause a fourth and a half will equal five.
Interviewer: Explain that one to me…
Tim:
You just like…I don’t know how I got it…it’s in my head, that one-fourth plus one-half is five,
one fifth.
In the above protocol, Tim initially pointed to the whole circle of four yellow pieces in
response to the interviewer’s question “And where is your one-fourth?” He later named the sum
of one-half and one-fourth “one fifth” because “a fourth and a half will equal five.” We interpret
this statement as referring back to his initial representation for one-half plus one-fourth with the
circle pieces: he had one orange piece (the half-circle) and four yellow pieces, making five
pieces in all. An alternative (though complementary) explanation for his response can be based
on Tim’s interpretation of unit fractions as consisting of all the parts (a fourth is four parts) and
the special meaning he attributes to one-half (it is one of two pieces). Thus, “one-half plus onefourth is five.” He further indicates the equivalence (in his mind) for five parts and one-fifth
when he said: “one-fourth plus one-half is five, one fifth.”
We infer that Tim’s correction for showing one-fourth (holding up just one yellow piece)
was a reaction to Angela having picked up one green piece to indicate one-sixth of the circle.
Later in the continuation of this activity, Tim stated: “If that was together, [places all four yellow
parts together and creates a whole circle] this [lifts up one yellow piece] would be counted as one
fourth.” This action is confirming evidence for our hypothesis that Tim lacks a disembeding
operation. Tim later referred to the three yellow pieces that he had laid on top of the white unit
circle (to show the equivalence of one orange piece and one yellow piece using just yellow
pieces) as “one third.” We infer that he called the quantity “one third” because there were now
three pieces, further indication of his association of a unit fraction with the number of parts
rather than one part out of a whole.
In the interview with Ms. Moseley in which she viewed the above videotape episodes
with Tim, she agreed with our hypotheses concerning Tim’s lack of basic fractional concepts, as
the following protocol indicates. The protocol begins after the interviewer and Ms. Moseley have
viewed the first part of Protocol II, pausing the videotape after Tim has named the five green
parts of a circle as “one fifth.”
Protocol C (from interview with Ms. Moseley on March 28, 2003)
Interviewer: What does that tell you?
Making Sense of Instruction on Fractions 15
Ms. Moseley: That he doesn’t understand. That he does not understand what the concept is of the unit fraction.
What the 1/6 is. And that it – it doesn’t – well, it does have to do with the number of pieces, but
you can’t say one is missing, it became fifths now.
Interviewer: Right. That was one of my questions. Do you think that this might have been – well – I guess my
question is, do you think this is a conceptual problem for him , a problem that is related to the use
of a manipulative that he’s not particularly used to, or some combination of those two things kind
of playing together.
Ms. Moseley: Probably, a combination. Because I think, if I had to think about his thinking all year long –
there’s just gaps. Because I know they’ve done circle graphs before. I know the elementary school
uses that kind of manipulative. So, I don’t think it’s that he was unfamiliar with it. I just think he
does not get the idea of what a fraction means. That it’s 6 pieces of the same size and the size
didn’t change. Now, he’s saying the size did change when you took the one out. But, how he gets
it to 1/5 – see he thinks it’s one of something. He doesn’t see each of those little pieces as an
individual part of that whole.
Ms. Moseley has identified the key aspects of fractional knowledge that are problems for
Tim: That a unit fraction is one of so many equal-sized parts and that the referent whole has not
changed just because a piece is missing. From her point of view, however, Tim changed the size
of the parts when he named the five remaining parts as “a fifth.” For Ms. Moseley, the referent
whole was still the whole circle, whereas for Tim it was the five remaining parts of the 6-part
circle. From our point of view Tim did not change the size by naming the remaining pieces “a
fifth” because, for Tim fractions are determined by number of pieces, not a part-to-whole
relation. Ms. Moseley seemed to agree with this hypothesis when she said: “He doesn’t see each
of those little pieces as an individual part of that whole.”
In the continuation of this interview, Ms. Moseley indicated that Tim was not the only
one in her class having difficulties with basic fraction concepts and that the CMP unit Bits &
Pieces I made the assumption that students would come to the unit with these basic concepts
firmly in place.
Protocol C (first continuation)
Ms. Moseley: [Dr. Olive] said, ‘you’re really going to have to watch Tim. He doesn’t understand unit fraction.’
And, I thought to myself, ‘yes, Tim doesn’t, but neither do the other 17 kids that I didn’t pick for
you to interview because they don’t get it either.’ That’s the whole problem. It’s not just Tim, it’s
a huge number of kids do not get that. And, we’ve, I mean, that whole Bits 1 book just goes in
such depth with that and they came out of that 6 weeks later still blank.
… (Two minutes later)
Making Sense of Instruction on Fractions 16
Ms. Moseley: It is a very difficult concept – and I understand that. I only know so many ways to approach it and
I do think the book is – I think the book, overall, has made a lot of assumptions about a lot of
things – that these kids come with a real solid basic knowledge that they don’t have.
The next continuation of Protocol C begins after Ms. Moseley has watched the first
continuation of Protocol II in which Tim drew the same representation for 1/6 as for 6/6 (a circle
with six unequal parts) and said that 1/6 and 6/6 were the same.
Protocol C (second continuation)
Ms. Moseley: So, again, I guess he’s thinking about the one as being the whole thing.
Interviewer: Um-hmm
Ms. Moseley: That’s all I can figure he’s thinking about.
Interviewer: That’s what it looked like.
Ms. Moseley: Yeah.
Interviewer: So, 1/6 is one thing that’s…
Interviewer & Ms. Moseley together: …that’s cut into 6 pieces.
Interviewer: and if one piece is missing, then it’s 1/5 because it’s one thing that’s…
Interviewer & Ms. Moseley: got 5 pieces.
Interviewer: Even though it’s not whole.
Ms. Moseley: Even though it’s not whole. Yeah, yeah.
Interviewer: So, say you were still in Bits & Pieces I and you had all the time in the world. What would you do
for Tim at this point? Do you have any idea?
Ms. Moseley: Well, I think we’d – we would – just like we were talking about going back and talking more
about what a unit fraction is and how it relates to the whole. And, spend a whole lot – and I
probably would do more manipulatives – circles as well – maybe even cut up the fraction strips if
we needed to help them see that those are pieces of a whole. And, you know, we went over and
over one sixth, two sixths, three sixths, four sixths, five sixths, six sixths – that’s one whole. But,
Tim’s missed that. So, I think we just have to repetitively go through – let’s go through halves,
let’s go through thirds. How many pieces? Each of them is one. One half, one half, two halves:
whole. One third, one third, one third, three thirds: whole. And maybe make him go through the
whole set that we deal with. Maybe we could work it with the strips. Maybe we could come over
and work it with the circles and maybe it would be helpful to show him a one-half circle this size
and a one-half circle this size and understand that it’s the number of pieces that you’ve got that
make up the whole that determine your denominator.
Ms. Moseley listed many important strategies that could help Tim (and other students in
her class) to construct the basic fraction concepts that Tim appeared to lack. She appeared to
realize the importance of establishing a part-to-whole relation and suggested cutting up fraction
Making Sense of Instruction on Fractions 17
strips to make that relation more apparent for the students. Her emphasis on developing a
counting sequence for unit fractions that ends with one whole could help students construct an
iterative unit fraction concept that our prior research (Olive, 2002; Olive & Steffe, 2002) has
shown to be a critical construct in children’s fractional reasoning. Her final comment: “maybe it
would be helpful to show him a one-half circle this size and a one-half circle this size and
understand that it’s the number of pieces that you’ve got that make up the whole that determine
you’re denominator.” may, however, reinforce Tim’s way of naming unit fractions as it puts the
focus on the number of pieces rather than on a part-to-whole relation.
In the following week of classes (after spring break) Ms. Moseley did emphasize
combining unit-fractions to produce common and even improper fractions as multiples of a unit
fraction. For instance, she wrote
3( 14 ) 14  14  14
on the white board to emphasize that 3/4 was
three 1/4ths in the context of buying 1/4th-ounce bags of spice in the CMP investigation “Visiting
the spice shop.” Even though we had not discussed our prior research in any detail with Ms.
Moseley, her teaching strategies during the spice shop investigation were closely aligned with
what we had learned from our research on children’s construction of fractions (Olive, 1999,
2001, 2002, 2003; Olive & Steffe, 2002; Steffe, 2002, 2003) so we decided to build on these
strategies in the student interviews by using discrete quantities (bags of spice) in the context that
was introduced in class. Representing a unit fraction as a small bag of spice helped Tim to both
unitize a unit fractional quantity and to establish a multiplicative relation between a unit fraction
and the whole of which it was a part (e.g. 5 bags of nutmeg each weighing 1/5 of an ounce make
one ounce of nutmeg). Because the bags were drawn as discrete objects, Tim could establish the
one-fifth relation between a bag of nutmeg and one whole ounce of nutmeg without using an
equi-partitioning scheme (that he lacked). He was able to use his whole-number multiplicative
operations to establish the part-to-whole relation. Establishing this part-to-whole relation
however, was not easy for Tim, as the following excerpts from the interview held on April 9,
2003, indicate. The first excerpt begins after Tim’s partner Jennifer had drawn ten 1/10-oz bags
of cinnamon to indicate how many bags she would need to make a whole ounce of cinnamon.
Protocol III (from interview with Tim and Jennifer on April 9, 2003)
Interviewer: Do you know what one tenth means?
Jennifer:
You have ten bags that equal one ounce.
Making Sense of Instruction on Fractions 18
Interviewer: That’s right. And that’s why each bag is one tenth of an ounce – because you have ten bags that
equal one ounce. Very good, Jennifer, that’s a great way to think about it. Now Tim, given what
we’ve just done with the cinnamon, do you want to rethink what we need to make an ounce of
nutmeg?
Tim:
An ounce?
Interviewer: Yes, A whole ounce of nutmeg.
(Tim draws ten 1/5-oz bags for one ounce of nutmeg.)
Interviewer: How many bags of nutmeg did you draw?
Tim:
Ten. That’s a whole ounce.
Interviewer: O.K. That’s a whole ounce. Do you agree, Jennifer?
Jennifer:
Uh, umm (meaning “yes”).
Interviewer: O.K. How many fifths of an ounce do you have there?
Tim:
(Pause) ten fifths?
Interviewer: Ten fifths. Can you write that down for me?
(Tim writes
10
5
next to his ten bags.)
Interviewer: Is that a whole ounce?
Tim:
(Mumbles)
Jennifer:
(Mumbles at the same time with Tim)
Interviewer: How many fifths do you think would make one ounce? How many fifths of an ounce
would
make one ounce?
Tim:
Two.
Interviewer: Only two? Only two fifths make one ounce? Only two of these bags (pointing to Tim’s pictures of
bags of nutmeg) make one ounce?
Tim:
Oh no! You need ten of them to make one ounce.
Interviewer: You do? Can you explain why you need ten of them to make one whole ounce?
Tim:
Cause ten of these bags will equal… Oh! I messed up bad!
Interviewer: O.K. Tell me what, why you think you messed up.
Tim:
Cause I was just trying to phrase what she did with the one tenth, but I have a different fraction.
Interviewer: You do. So do you want to rethink how many of your bags… Are your bags bigger or smaller than
Jennifer’s? …Which weighs more: One bag of Jennifer’s or one bag of yours?
Tim:
Hers…no, wait…one of mine.
Interviewer: One of yours. O.k. how many of Jennifer’s will it take to equal- if we put them on a pan balanceyou see a pan balance (imitates a pan balance with his hands)- do you know what a pan balance is?
Jennifer:
A weight
Interviewer: You can put things on both sides
Tim:
And if they equal up they will be the same.
Making Sense of Instruction on Fractions 19
Interviewer: Right. So if we were to put…Let me draw the pan balance here (draws a pan balance). So this is
one pan on one side and one pan on the other side…Let’s draw an arrow here…My pan balance
looks like that…Is that o.k.?
Tim & Jennifer: Mhmm (agree)
Interviewer: O.k. If you put one of your bags over here…draw one of your bags in your pan (Tim draws one of
his bags on the right side of the pan balance). How many of your (asking Jennifer) bags would
you need to put in this pan so the pan balance would be level?
(Jennifer draws one of her bags on the left side of the balance)
Interviewer: Just one? Do you think that would be leveled?
Jennifer:
I guess so.
Tim:
Mine will be way down. It will weigh more.
Interviewer: Why?
Tim:
Cause a fraction of one fifth is bigger than one tenth.
In the above protocol, Tim apparently over-generalized Jennifer’s solution for the number
of bags of spice needed to make one ounce. He was probably led to his solution of ten bags by
the interviewer’s praise of Jennifer’s comment that “one tenth” means “You have ten bags that
equal one ounce.” It is interesting to note that Tim was able to write the improper fraction
10
5
to
indicate how many fifths he had drawn. What is even more intriguing is his use of this improper
fraction to determine that two fifths would make one ounce (we infer that he divided the 5 into
the 10). It seems that Tim has a disconnect between the numerical symbols he uses for fraction
notation and the quantities to which they refer. After the interviewer questioned his response of
“two” as meaning two fifths make one ounce, he immediately returned to the result represented
through his drawing “You need ten of them to make one ounce.” The request for him to explain
this result caused a moment of reflection in which he realized his over-generalization of
Jennifer’s result. He now realized that he was dealing with a different fractional quantity than
Jennifer. The interviewer decided to try and capitalize on Tim’s realization of the difference in
the two quantities of spice and introduced the idea of a pan-balance to help the students
determine how many of the 1/10-oz bags of cinnamon would weigh the same as one 1/5-oz bag

of nutmeg. Jennifer at first thought they would weigh the same but Tim stated that his bag would
be heavier “Cause a fraction of one fifth is bigger than one tenth.” Jennifer eventually
determined that two of her bags would weigh the same as one of Tim’s bags. Protocol IV begins
at the point in the interview where Jennifer had drawn two of her bags on the left –hand pan (of
Making Sense of Instruction on Fractions 20
the balance that the interviewer drew on their paper) to balance the one bag that Tim had drawn
on the right-hand pan (see Figure 6).
Protocol IV (from continuation of interview on April 9, 2003)
Figure 6: Drawing of pan balance from interview on 4/09/03
(Jennifer drew her spice bags with handles, so her picture on the left side of the pan
balance represents 2 spice bags, not 4.)
Tim:
Now it will be even (pointing at the pan balance).
Interviewer: So how many of your bags, Tim, make one ounce?
Tim:
(13-seconds pause) Like, (9-seconds pause) one?
Interviewer: Just one of your bags will make one ounce? How many of Jennifer’s bags make one ounce?
Tim:
Two
Jennifer:
Two (right after Tim)
Interviewer: No, make one ounce not one fifth.
Jennifer:
Oh!
Interviewer: How many of Jennifer’s bags made one whole ounce? She drew them all there (pointing to
Jennifer’s ten drawn bags on her paper).
Jennifer:
Ten.
Interviewer: Is that right?
Tim:
What did you say?
Jennifer:
Ten.
Interviewer: Ten of Jennifer’s bags.
Tim:
Ten. Ten of her whole… Ten of like one of them tenths; one tenth of like an ounce makes one
whole.
Interviewer: Right. So how many of yours?
Tim:
Five?
Interviewer: Can you tell me why?
Tim:
Cause half of ten is five. Like a whole of hers is ten, one tenth of an ounce, ten of those is a whole,
so you kind of reduce it down to one fifth and that makes it five.
Interviewer: O.k. so with the pencil draw… surround the five bags that are going to make one ounce from your
picture.
Tim draws an oval around five of his ten bags (see Figure 7).
Making Sense of Instruction on Fractions 21
Figure 7: Tim’s representation of ten 1/5-oz bags of nutmeg with five bags circled for one
ounce of nutmeg
Tim and Jennifer where both confused over what unit the interviewer was referring to in
his questions (one ounce or one bag of nutmeg). They both appeared to be regarding the one bag
of nutmeg on the right side of the pan balance as the referent unit. After the interviewer had
reinforced the notion of one ounce as the unit by referring back to Jennifer’s solution of ten 1/10oz bags making one ounce, Tim appeared to make the connection between 1/5-oz bag and one
whole ounce. His reference to half of ten being five may have indicated a connection between
the equivalence of two of Jennifer’s bags to one of his (so he only needed half the number of
bags). What became clear in the continuation of this interview (in which we switched to a
number line representation) was that Tim had formed a unit fraction concept that included the
multiplicative relation of the unit fraction to the whole: Ten of one tenth make a whole and five
of one fifth make a whole.
This notion of a unit fraction as a unit that could be iterated to produce the whole of
which it was a part was again reinforced by Ms. Moseley in her classroom instruction on the
same day following the above interview on April 9, 2003. Whole number multiplication of unit
fractions (e.g. 4 x 1/8) was introduced as repeated addition of a 1/8-unit segment. Tim was able
to use this notational procedure to find fraction names for 10 x 1/16 as well as 4 x 1/8 and 3 x
1/4. He was also able to find the sum of 1/2, 10/16 and 3/4 (having simplified 4/8 to get 1/2).
His solution was 30/16 (see Figure 8).
Making Sense of Instruction on Fractions 22
Figure 8: Clips of Tim’s work in class from video of April 9 2003
The left clip in figure 8 shows Tim’s work that he did by himself in class, with Ms.
Moseley’s hand as she is about to draw the two boxes shown in the right clip. Ms. Moseley was
attempting to help Tim convert his improper fraction of 30/16 into a mixed number. What is
interesting to note in the left clip is Tim’s iteration of the unit fraction notation, without the “+”
signs, to indicate his composite fraction. In the right-hand clip Tim has used the “+” signs
between the three composite fractions that he is summing to arrive at 30/16. This subtle change
in notation could be indicative of Tim interpreting the “three one-fourths” literally as three
instantiations of the fraction notation “ 14 ” rather than as the sum
1
4
 14  14 .
Thus
10
16
may mean
1
16
ten times, which is what he has written on his paper. Such an interpretation could lead to an
iterative unit fraction concept for Tim if the quantity represented by the unit fraction was also
iterable.
Evidence of Tim’s modifications (reorganizations) of his fractional schemes
Construction of a unit fraction concept
In the interview on the following week, on April 16, 2003, Tim provided evidence that he
had solidified his concept of a unit fraction as one of so many equal parts of a whole. During the
interview, he corrected Jennifer when she was trying to mark sixths on a number line. Tim
insisted that the spaces between the tic marks had to be “even” (meaning the same length). The
interviewer asked Jennifer to show him one-sixth on her number line and she pointed to the first
tic-mark. The interviewer then asked the students what that was one sixth of. Neither student
could indicate that it was one sixth of the number line from zero to one, so the interviewer



Making Sense of Instruction on Fractions 23
decided to introduce a realistic context of a candy bar. Protocol V begins at this point in the
interview.
Protocol V (from interview with Tim and Jennifer on April 16, 2003)
Interviewer: Oh, so it’s going to be one sixth of something, right? O.k. So if I told you I’m going to give you
one sixth of my candy bar you would know what that meant, right?
Jennifer:
Um-hmm (yes).
Tim:
Right. If you cut it up into six pieces, you’d give somebody one and that would be one piece out of
six pieces.
(Meanwhile Jennifer is drawing a candy bar and divides it into six pieces)
Interviewer: Pieces…and that would be the one sixth. O.k. I like the way you drew that candy bar. Have you got
six pieces there? Almost; o.k. Now, would it matter if those pieces were all the same size or not?
Tim:
Yeah it matters.
Interviewer: Yeah.
Interviewer: Why does it matter?
Jennifer:
Because…
Tim:
They’ve got to be even pieces for it to be one-sixth cause some might be smaller, some might be
bigger, and the whole thing wouldn’t be equal to the same.
Interviewer: O.k.
Jennifer:
O.k. there’s one sixth (and writes one sixth underneath the candy bar) and we want…
Interviewer: Which is one sixth?
Jennifer:
This candy bar is one sixth (sliding her hand along the whole candy bar) nobody has gotten any
yet. Now you wanna give Tim some…and…
Interviewer: Do you agree with that Tim? Is the whole candy bar one sixth?
Tim:
Mm (meaning ‘no’)
Jennifer:
Well it’s six pieces.
Interviewer: What’s the whole candy bar?
Tim:
It’s six pieces but that (pointing at the first piece) represents one sixth. Each piece represents one
sixth.
Interviewer: Does that make sense to you?
Jennifer:
Aha!
Interviewer: So what’s the whole candy bar?
Tim:
Six-sixths, one whole.
Interviewer: One whole or…
Tim:
One whole candy bar.
Making Sense of Instruction on Fractions 24
After Jennifer had drawn a rectangle to represent a candy bar and partitioned it into six parts,
Tim again insisted that the parts “got to be even pieces for it to be one sixth.” Tim also clarified
what it meant to have “one sixth” when he explained, “that would be one piece out of six
pieces.” Jennifer appeared to have the same problem Tim had had in the second interview
(Protocol II) when she indicated that the whole candy bar was one sixth. When asked if he
agreed with Jennifer, Tim’s response was emphatic: “It’s six pieces but that (pointing at the first
piece) represents one sixth. Each piece represents one sixth.” He even went on to explain to
Jennifer that a unit fraction was one out of so many equal parts. Tim’s responses in Protocol V
clearly indicate equi-partitioning operations and a unit fraction concept for one sixth. Our
conjecture at this point is that the instructional emphasis on discrete unit fractional quantities and
the teacher’s strategy of iteration of unit fractions to produce the whole were key experiences
contributing to Tim’s construction of a unit fraction concept.
Determining if two fractions represent the same quantity
Later, during the same interview, even though Tim said he would rather have 15/20 of a
pizza than 3/4 of the same pizza (because 15 pieces is more than 3 pieces) he did seem to be
grappling with the idea that two fractions could represent the same quantity. The students were
working with a computer representation of two pizzas (see Figure 9). Protocol VI begins with
the interviewer trying to determine if the two circles representing the two pizzas on the computer
screen were the same size.
Protocol VI (from interview with Tim and Jennifer on April 16, 2003)
Making Sense of Instruction on Fractions 25
Figure 9: Two pizzas, one divided into 20ths (19 are yellow and one is orange) and the other is
divided into 4ths (3 yellow, 1 orange).
Interviewer: Now then lets make sure that these two pizzas are the same size (places one ‘pizza’ on top of the
other ‘pizza’).
Tim:
Nope
Interviewer: No they’re not. Lets make them the same size.
Jennifer:
No, a little bit bigger.
Tim:
They’re about the same…No, it’s sticking out.... a little bit smaller
(A couple more attempts and they get the circles to be the same size)
Interviewer: Are they the same size now?
Jennifer:
Yeah.
Tim:
Yeah (at the same time with Jennifer)
Interviewer: O.k. Now then, it’s just showing one slice of each.
Tim:
Ohhhhhh!
Interviewer: Could you show me? O.k. so if you had three fourths of this pizza which pieces would you have
(pointing to the circle on the right)?
Tim:
Three pieces
Interviewer: Can you point to the three pieces you would have?
Tim:
One, two, three (pointing at the orange 1/4-piece and two yellow1/4-pieces from the pizza on the
right).
Interviewer: O.k. What about all the same color? Are there three pieces there the same color you might have?
Tim:
Yeah, the yellow ones (the three yellow 1/4-pieces).
Interviewer: O.k. the three yellow ones. Do you agree that would be three fourths?
Tim:
I just found out something important!
Interviewer: Yeah?
Tim:
They’re both the same size! They have like the same amount of pieces. Cause like if I got one of
those (points at the orange 1/4-piece) it equal like…
Jennifer:
I know what you’re trying to say…
Tim:
Like four (pointing at the disk divided into 20ths)…five of those…
Tim had made the visual comparison of the two circles as they were checking to make
sure they were the same size. He suddenly realized that five of the smaller slices in the left-hand
pizza would be the same as one slice from the right-hand pizza. At this point, Jennifer asked if
she could move the fourths-circle on top of the twentieths-circle. The protocol continues after
she did so.
Making Sense of Instruction on Fractions 26
Protocol VI (first continuation from interview on April 16, 2003)
Jennifer:
All those little pieces equal that (moving her finger around the circles).
Tim:
So it wouldn’t really matter. You still get the same amount.
Jennifer:
All these pieces right here equal to that right there. (Jennifer points at the fourths-circle and then
at the twentieths-circle. She is not pointing at the orange quarter in particular)
Interviewer: So we can actually…I’m turning it so it will line up (rotates the upper circle so that the orange
slices overlap -- see Figure 10).
Figure 10: The 4-part circle superimposed over the 20-part circle with the orange parts lined up.
Jennifer:
All these pieces equal that amount right there I think. (Jennifer points at the pieces that make up
the orange 1/4-piece and then moves her finger around the circle)
Interviewer: So how many pieces is this (moving the mouse around the circle)?
Tim:
Twenty in all.
Interviewer: Yeah and how many would be the ones you ate if you ate three fourths?
Tim:
Five
Jennifer:
Yeah five.
Tim:
Pieces
Jennifer:
Five (pointing at the orange pieces). I ate five.
Interviewer: How much was those five pieces? That’s just the orange part.
Jennifer:
Was…It was…Me and Tyler got 5 pieces out of that 20…
Interviewer: Yeah and how much is that?
Jennifer:
Out of the whole pizza (finishing up the sentence she started)
Tim:
Uh half…It represents…that represents one fourth (pointing at the orange 1/4-piece).
Interviewer: One fourth. If you ate five pieces of this pizza (the 20-part disk), you would eat how much of the
pizza?
Tim:
One fourth?
Interviewer: One fourth. What would 15 of these pieces be (pointing at the 1/20-pieces)?
Making Sense of Instruction on Fractions 27
Tim:
Um, like…
Interviewer: If this is five pieces (pointing at the orange 1/4-piece)
Tim:
That would be three fourths.
Interviewer: It would be how much? Did you hear what he said?
Jennifer:
Three fourths
Interviewer: Three fourths. So is that the same amount? That’s what you said, isn’t it?
Tim:
Yeah, it’s the same amount.
Even though both students initially responded with “five pieces” for the number of small
slices (twentieths) that would make up three fourths of the pizza, Tim eventually corrected
himself and saw that 5 small slices was the same as one fourth of the pizza. The students’
response of “five” could have been the result of the visual overlap of five of the small slices with
the one orange slice in the 4-part circle (see Figure 10). Once Tim had established the relation
between five small slices and one fourth of a pizza he was able to correctly answer that 15 slices
would be three fourths of the pizza. What follows in the second continuation of Protocol VI
indicates that Tim attempts to coordinate these visual comparisons on the computer screen with
his numerical associations between the numerators and denominators of both fractions. The
interviewer returned to the video clip from the class lesson on April 11, 2004. The class was
working on the problem of 3/4 + 4/5. Figure 11 indicates what Ms. Moseley had written on the
Whiteboard.
Figure 11: Video clip from the class lesson on April 11, 2003, shown in interview on April 16,
2003.
Protocol VI (second continuation from interview on April 16, 2003)
Interviewer: So what can you tell me about this fraction and this fraction (pointing at the 3/4 and the 15/20 on
the screen)?
Tim:
They’re the same thing.
Interviewer: They’re the same thing. Do you agree?
Jennifer:
Uhu (‘yes’)
Making Sense of Instruction on Fractions 28
Interviewer: Now, what about this fraction? Four-fifths (pointing at 4/5 and 16/20 on the screen)?
Jennifer:
They’re the same thing.
Interviewer: And sixteen twentieths.
Tim:
Four goes into 16 and 5 goes into 20.
Interviewer: So are they the same thing?
Jennifer:
Yup (‘yes’)
Tim:
Yup (at the same time with Jennifer)
Tim:
So, I got a question. If both numerator and denominator (pointing at the 4/5) go into the other
numerator and denominator (pointing at the 16/20), they’re the same?
Interviewer: If they go into it by the same amount.
Tim:
Oh!
Interviewer: So 4 goes into 16 how many times?
Tim:
Four.
Interviewer: And 5 goes into 20 how many times?
Jennifer:
Four.
Tim:
Four.
Interviewer: Is that the same then?
Tim:
Um-hmm (‘yes’)…Oh, I get it!
Interviewer: If they go into the numbers the same amount of time…
Tim:
They’re the same.
Interviewer: They’re the same amount.
Tim:
Right.
Interviewer: Does that make sense Jennifer? Tim, explain to Jennifer what you just found out. It’s a very
important thing what you just found out. Very important.
Tim:
If the numerator and denominator go into, like another fraction you’re trying to compare together,
to their numerator and denominator, they’re the same.
Interviewer: If they go into those numbers the same amount.
In asking the question “If both numerator and denominator (of one fraction) go into the
other numerator and denominator, are they the same?” Tim may have been connecting the
numerical transformations that Ms. Moseley had performed in class with the visual comparisons
he had made with the computer representations of pizzas. The interviewer responded to his
question by adding that they would be the same “if they go into each other by the same amount.”
While we do not claim that Tim had constructed an equivalence relation between two fractions,
he was trying to make sense of fractions that represent the same quantity and was developing his
own algorithm for comparing fractions. Tim appeared to be focusing on the numerator and
denominator of one fraction being factors of the numerator and denominator of the other
Making Sense of Instruction on Fractions 29
fraction. What he may not have been aware of at this point in his development was the need for
what we would call a “common factor.” Focusing on the numerator and denominator as separate
numbers, however, indicates that Tim may still be thinking of fractions as pairs of whole
numbers rather than as part-to-whole quantitative relations.
Evidence from the classroom indicated that Tim could not use his strategy for finding
fractions that represented the same quantity in order to find a “common denominator” for two
fractions that he was operating on in additive situations. For example, during class, the same day
of the interview on April 16, 2003, Tim had to figure out what
3 45 3 34
was. He was initially
confused, as he didn’t know how to deal with mixed number addition. Ms. Moseley led him to
adding the fractions first. Tim suggested finding a common denominator. He said that the
common denominator would be 5 at first, but when Ms. Moseley asked him how he would turn
fourths into fifths, Tim said: “Oh! We did this in the interview today…did the multiples of 5, did
the multiples of 4 and the common denominator is 20.” The very next day in class, however,
Tim could not solve this same problem and again offered “five” as the common denominator for
three fourths and four fifths. Again, when Ms. Moseley reminded Tim of the strategy he had
used in the interview the previous day (multiples of fives) Tim said the common denominator
was 20. It appears that Tim could recall his activity when prompted but he had not yet
internalized that activity. It is also worth noting that the fractions involved (fourths and fifths)
were the same fractions used in the interview tasks. We doubt that Tim would have been able to
produce the common denominator using this strategy (of producing multiples of each
denominator) if the two fractions had been different from the ones used in the interview. Tim
may have guessed “five” initially for the common denominator because it was the larger

denominator and most of the examples in class up to this point had fractions for which the
common denominator was the larger denominator (e.g. fourths and eighths, fifths and tenths).
This explanation is consistent with our hypothesis that Tim initially regarded fractions as pairs of
whole numbers rather than as part-to-whole quantitative relations.
Making sense of a fraction of a fraction
The last investigation that we observed and interviewed students about was the
investigation on multiplication of fractions (fractions of fractions). To make sense of fractions of
fractions students need to reason with three different levels of units (units of units of units) and
Making Sense of Instruction on Fractions 30
be able to recursively partition units (Olive, 1999). Tim had progressed from only being able to
reason with one level of unit (the whole as the unit fraction) to being comfortable with reasoning
with two levels of units (the whole as a composite unit of unit fractions).
Our final two interviews with Tim and the classroom observations indicated that Tim
could follow the procedures for producing fractions of fractions. During the class on April 29,
2003 Tim successfully solved
3
4
of
1
2
. Ms. Moseley had drawn a representation of the pan of
brownies with half of the pan cut into four parts (see Figure 12).
Figure 12: Half a pan of brownies cut into four parts (from video of class on 4/29/03)
On the left of the pan representation Ms Moseley had written “3“and said that all students
had figured out that the answer would be three of something. She asked Tim what the something
was. Ms Moseley told him that he had already shaded three pieces and asked him what size each
of those pieces would be in the whole pan. Tim said that it would be eighth, explaining, “If you
cut the other one (referring to the right half that is un-shaded) into fourths and count them
together with the other pieces (with the fourths of the shaded half on the left) there will be eight
pieces.” Such reasoning with quantitative part-whole relations across two levels of units was
very different from the numerical associations he was struggling with in the previous
investigation dealing with additive combinations of fractions. It would appear that the area
model for obtaining a fraction of a fractional quantity was a meaningful context for Tim to
extend his reasoning to two levels of units.
During the interview on the next day (March 30, 2003) Tim was able to produce various
fractions of unit fractions using the rectangular area model that Ms. Moseley had introduced in
class. A problem that caused Tim difficulties was to represent 13 of
2
5
of a candy bar (a unit
fraction of a composite fraction). The interviewer provided Tim with a computer program that
allowed him to further partition one or more parts of a partitioned bar and to fill any parts with a
different color. Tim used the program to make five vertical parts in his on-screen bar and filled
Making Sense of Instruction on Fractions 31
in two of the five parts to represent the 2/5 of the bar. He then partitioned two of these five
vertical parts into three parts horizontally. He filled in one of the three parts in the first fifth and
said that this was one third of two fifths of his bar. Tim eventually realized that he had filled only
one third of one fifth after the interviewer asked him to identify the two fifths of his bar. Tim
filled in the top third of each of the two fifths but still had difficulty naming this resulting
fractional part as a fraction of the original bar. He asked if he could partition each of the
remaining fifths into three parts each. He did so and then counted all the parts in the bar,
obtaining 15. He named the two parts that he had pulled out of the bar as “two fifteenths” at this
point in the interview. As he wrote this result on paper, he made the following comment: “It’s
just like you multiply it. Three times 5 is 15 (pointing to the two denominators on his written
equation) and 2 times 1 is 2 (pointing to the two numerators).” Nevertheless, when he was asked
whether that will work everytime, he was doubtful. He did not apper to connect the
multiplication with the recursive partitioning of the bar that he had just completed. Tim could not
easily compare the result of the procedure to the original unit whole (that requires a comparison
among three levels of units). We did conclude, however, that such reasoning was within Tim’s
zone of potential construction (Olive, 1994).
One reason why the 13 of
2
5
problem was more difficult for Tim than the
3
4
of
1
2
that he
had completed in class is that this latter problem is asking for a fraction of a unit fraction,
whereas the former problem is asking for a unit fraction of a composite fraction. Just as it is
much easier for a child to take 3/4 of one bar rather than 1/3 of 2 bars of candy (because the
thirding operation has to be applied to two unit objects) so it is with taking fractions of fractions.
It is easier to take a composite fraction of a unit fraction than it is to take a unit fraction of a
composite fraction. However, the difficulty in naming the resulting part was a problem for Tim
in both cases.
Evidence of Tim’s Development of an Iterative Unit Fraction
There were a few instances during the second round of data collection (fall 2003) in
which Tim demonstrated his capability to reason with two levels of units and his development of
an iterative fractional scheme. The next protocol is from an interview with Tim and Kelly (a new
interview student) conducted on December 8, 2003. The students attempted to find one-fourth of
Making Sense of Instruction on Fractions 32
55 in order to find the distance traveled in 5 14 hours at 55 mph. Tim made an estimate of 12 for
one-fourth of 55. Protocol V begins at this point in the interview.
Protocol V (from interview with Tim and Kelly on December 8, 2003)
Tim:
I was just trying to get close to 55 by 12, by multiplying it with something.
Interviewer: Yeah, by multiplying, right. What is it you’re trying to find?
Tim:
A quarter of 55.
Interviewer: O.k., how many quarters make a whole?
Tim:
Four?
Interviewer: So, if 12 is a quarter of 55, what do you have to do to it to see if it’s the whole, to make the whole?
Tim:
Times it by 4?
(Interviewer nods ‘yes’ and Tim multiplies 12 by 4 and gets 48)
Tim:
Forty-eight.
Interviewer: So it’s close, right?
Tim:
So I know it’s higher than 12.
Interviewer: Yeah…
(9-second pause)
Interviewer: So could you try another number?
Tim:
Fourteen?
Interviewer: O.k.
(Tim multiplies 14 by 4 and gets 56)
Tim:
It’s close.
Interviewer: Very close, (all three of them laugh) right?
Tim:
It’ll have to be like (3 second pause) 13 and 3 quarters?
Interviewer: Why don’t you write that down before you forget it?
Tim started by trying to estimate 1/4 of 55. He was aware of the fact that his estimate
would be the answer if he multiplied it by four and got 55. When multiplying his second
estimate, 14, by 4 he got 56, and realized that 14 was just a little bit too big. He offered 13 34 as
his final answer. Did he take 1/4 of the difference between 55 and 56 away from 14 to arrive at
this answer? Such reasoning would indicate that Tim regarded “one fourth” of a quantity as that
amount when iterated four times would give him the original quantity. Moreover, he could
possibly use this iterable fourth to adjust his estimate appropriately by subtracting one fourth of
the difference (between target and calculated result) from his estimate. This is strong
confirmation that one fourth was an iterable unit for Tim.
Making Sense of Instruction on Fractions 33
Discussion
In this paper we have attempted to examine the interactions among a teacher’s
assumptions about students’ basic fraction concepts, her instructional strategies and modification
of those strategies when faced with evidence that challenged her assumptions, and the
development of one student’s fractional schemes that resulted from those modifications. Over
several years of prior instruction, Tim had formed his concept of a fraction that was consistent
for him and with the representations used in many textbooks and classroom instruction – a region
indicating n parts with one or more parts shaded. Such representations do not discriminate
between a part and the whole. Thus, for Tim 1/n and n/n were the same thing.
Coming to understand “Tim’s world” was crucial in helping him modify his concept of a unit
fraction. The student interviews provided us and his teacher with a fine-grained view of Tim’s
reasoning. This fine-grained view enabled us to develop tasks and strategies (based on classroom
activities) to provoke perturbations for Tim. Through his efforts to neutralize his perturbations,
along with supportive activities in the classroom, Tim constructed a unit fractional scheme with
which he could reason about part-to-whole relations and develop strategies for solving problems
that involved operations on fractional quantities. Ms. Moseley was able to provide Tim with
support for his emerging unit fractional concept through the use of appropriate representations
and problem solving procedures.
Without this modification of his fraction concept, Tim would have joined the many
students for whom fractions remain a mystery and, consequently, have little chance of reasoning
algebraically. Tim’s case illustrates that it is not a lack of mathematical ability but ineffective or
sparse conceptual structures that limit students’ understanding. Once these conceptual structures
had been modified and enriched Tim was able to function within the context of his classroom
instruction. Teachers can help students like Tim by testing their assumptions about what students
bring to their instruction and providing reinforcement for basic conceptual structures in their use
of representations and problem solving procedures.
The coordinated research design, which is a central feature of the CoSTAR project,
enabled us to build our model of Tim’s conceptual understanding through observation of Tim’s
interactions in class and in-depth interviews that probed Tim’s interpretations of classroom
activities. The research methodology also enabled Tim’s teacher to better understand Tim’s
conceptual difficulties (and difficulties of other students in her class) and consequently adjust her
Making Sense of Instruction on Fractions 34
instruction to address those difficulties. In this way, the research had immediate impact on both
instruction and student learning.
References
Cohen, D. K., & Ball, D. L. (1999). Instruction, capacity, and improvement. (Consortium for
Policy Research in Education, Research Report Series No. RR-43). Philadelphia, PA: The
University of Pennsylvania, Graduate School of Education.
Cohen, D. K., & Ball, D. L. (2001). Making change: Instruction and its improvement. Phi Delta
Kappan, 83(1), 73-77.
Lappan, G., Fey, J. T., Fitzgerald, W., Friel, S. N., & Phillips, E. D. (2002).
ConnectedMathematics series. Glenview, IL: Prentice Hall.
Olive, J. (1994). Building a new model of mathematics learning. Journal of Research in
Childhood Education, 8 (2), 162-173.
Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis.
Mathematical Thinking and Learning, 1 (4): 279-314.
Olive, J. (2001). Connecting Partitioning and Iterating: A Path to Improper Fractions. In M. van
den Heuvel-Panhuizen (Ed.) Proceedings of the 25th Conference of the International
Group for the Psychology of Mathematics Education (PME-25), vol. 4, 1-8. Utrecht, The
Netherlands: Freudenthal Institute.
Olive, J. (2002). The construction of commensurate fractions. In A. D. Cockburn and E. Nardi
(Eds.) Proceedings of the 26th Conference of the International Group for the Psychology
of Mathematics Education (PME-26), vol. 4, 1-8. Norwich, U.K.: University of East
Anglia.
Olive, J. (2003). Nathan’s strategies for simplifying and adding fractions in third grade. In N. A.
Pateman, B. J. Dougherty & J. T. Zilliox (Eds.) Proceedings of the 27th Conference of the
International Group for the Psychology of Mathematics Education (PME-27) held jointly
with the 25th Conference of PME-NA, vol. 3, 421-428. Honolulu, HI: CRDG, College of
Education, University of Hawaii.
Olive, J. & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of
Joe. Journal of Mathematical Behavior, 20, 413-437.
Orrill, J. (2003). Fraction Bars [Computer software]. Athens, GA: Transparent Media.
Piaget, J. (1980). Opening the debate. In M. Piattelli-Palmarini (Ed.), Language and learning:
The debate between Jean Piaget and Noam Chomsky (pp. 23-34). Cambridge: Harvard
University Press.
Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge. Journal of
Mathematical Behavior, 102, 1-41.
Steffe, L. P. (2003). Fractional commensurate, composition, and adding schemes: Learning
trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior, 22, 237-295.
Making Sense of Instruction on Fractions 35
Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the
teacher’s role in promoting the learning. Journal for Research in Mathematics Education,
30(4), 390-416.
von Glasersfeld, E. (1980). The concept of equilibration in a constructivist theory of knowledge.
In F. Benseler, P. M. Hejl, & W. K. Kock (Eds.), Autopoisis, communication, and society
(pp. 75-85). Frankfurt, West Germany: Campus Verlag.
Footnotes
i
Coordinating Students’ and Teachers’ Algebraic Reasoning, funded by the National Science
Foundation, grant # REC 0231879
ii
iii
A reform curriculum developed with support from NSF and now published by Prentice Hall
None of the computer programs used in the student interviews were used by Ms. Moseley in
her classroom.