Mathematical Knowledge for Teaching Fraction Multiplication 1
Running Head: KNOWLEDGE FOR TEACHING FRACTION MULTIPLICATION
Mathematical Knowledge for Teaching Fraction Multiplication
Andrew Izsák
The University of Georgia
Department of Mathematics and Science Education
105 Aderhold Hall
Athens, GA 30602-7124
(706) 542 - 4045
izsak@uga.edu
Word Count: 18, 800
Acknowledgements
I would like to thank Ms. Archer, Ms. Reese, and their students for allowing me to be
part of their classrooms; members of the CoSTAR project for a range of assistance
conducting the research; and John Olive, Les Steffe, and anonymous reviewers for
helpful comments on previous drafts. The research reported in this article was
supported by the National Science Foundation under Grant No. REC-0231879. The
opinions expressed are those of the author and do not necessarily reflect the views of
NSF. Correspondence should be addressed to Andrew Izsák, Department of
Mathematical Knowledge for Teaching Fraction Multiplication 2
Mathematics and Science Education, The University of Georgia, 105 Aderhold Hall,
Athens, GA, 30602-7124.
Mathematical Knowledge for Teaching Fraction Multiplication 3
Abstract
The present study contrasts mathematical knowledge that two 6th-grade teachers
apparently used when teaching fraction multiplication with the Connected Mathematics
Project materials. The analysis concentrated on those tasks from the materials that use
drawings to represent fractions as length or area quantities. Examining the two
teachers’ explanations and responses to their students’ reasoning over extended
sequences of lessons led to a theoretical frame that emphasizes relationships between
the teachers’ unit structures and pedagogical purposes for using drawings. The choice
of topic, fraction multiplication, and the methods developed for the study extend
research on knowledge specialized for teaching particular mathematics topics,
sometimes referred to as mathematical knowledge for teaching. The methods should be
of interest to those investigating teachers’ knowledge of mathematics and related
disciplines.
Mathematical Knowledge for Teaching Fraction Multiplication 4
Mathematical Knowledge for Teaching Fraction Multiplication
Research on teacher knowledge has expanded from studies of teachers’ subjectmatter knowledge of various content areas to the organization of teachers’ knowledge
for teaching particular content to students (e.g., Ball, 1991; Ball, Lubienski, & Mewborn,
2001; Borko & Putnam, 1996; Davis & Simmt, 2006; Ma, 1999; Sherin, 2002; Shulman,
1986). This expansion follows a generation of research that did not find clear
connections between teacher knowledge and student achievement and reflects
increasing awareness that teachers need not just content knowledge that many educated
adults have, but also knowledge specialized for teaching particular topics (see Ball et al.,
2001, for a review of relevant literature). In light of this insight, current discussions of
teacher knowledge are often framed in terms of subject matter, pedagogical, and
pedagogical content knowledge (e.g., Borko & Putnam, 1996). When introducing the
notion of pedagogical content knowledge, Shulman emphasized knowledge of
students’ thinking about particular topics, typical difficulties that students have, and
representations that make mathematical ideas accessible to students.
Within mathematics education, Ball and colleagues (Ball & Bass, 2000; Ball et al.
2001) have used a term related to pedagogical content knowledge, mathematical
knowledge for teaching, to emphasize the mathematics that teachers might use to
accomplish tasks central to their practice. Such tasks include using curricular materials
judiciously, choosing and using representations, skillfully interpreting and responding
to students' work, and designing assessments. In the most visible line of current
Mathematical Knowledge for Teaching Fraction Multiplication 5
research on mathematical knowledge for teaching, Hill, Schilling, and Ball (2004)
reported a multiple choice instrument that reliably measures elementary teachers’
knowledge of number concepts; operations; and patterns, functions, and algebra. Hill,
Rowan, and Ball (2005) demonstrated through a large-scale study that first- and thirdgrade teachers’ knowledge of these topics, as measured by the same instrument, had an
effect size comparable to SES on student gain scores on CTB/McGraw-Hill’s Terra
Nova measures of student achievement. The gains occurred over a one-year period in
urban and suburban schools serving higher poverty populations. These results established
a positive correlation between teacher knowledge and student achievement, a link that
proved elusive in prior research. At the same time, elaborating mathematical knowledge
for teaching particular topics remains a central challenge.
The present report uses contrasting case studies of two teachers to elaborate
aspects of mathematical knowledge for teaching fraction multiplication, a topic central
to the elementary curriculum but that research on teachers’ knowledge has yet to
investigate closely. In contrast to recent research that has administered tests to large
populations of teachers (Hill, Rowan et al., 2005; Hill, Shilling et al., 2004), the present
study examines mathematical knowledge that the two teachers’ used for one category
of tasks included in the definition of mathematical knowledge for teaching—
interpreting and responding to students’ work. By concentrating on particular cognitive
structures, the analysis will be more narrow in focus than recent studies that have
examined not only knowledge, but also beliefs and goals that teachers employ during
lessons (e.g., Schoenfeld, 1998, 2000, in press).
Mathematical Knowledge for Teaching Fraction Multiplication 6
Data for the present study came from sixth-grade classrooms in which the
teachers and students used a combination of fractions units from Connected Mathematics
(CMP; Lappan, Fey, Fitzgerald, Friel, & Phillips, 2002) and draft revised units that have
since been published as part of Connected Mathematics 2 (Lappan, Fey, Fitzgerald, Friel,
& Phillips, 2006). These are reform-oriented materials whose development was
supported by the National Science Foundation. The units ask teachers and students to
use manipulatives and drawings to reason about problem situations in which fractions
are embedded, often as length or area quantities. Teachers are to use students’ strategies
to help them develop general numeric methods.
The four drawings shown in Figure 1 for representing 2/3 of 3/4 are all based on
work produced by students in the case study classrooms. The range of approaches for
representing parts of parts motivates the overarching research question for the present
study: What mathematical knowledge did the teachers use when examining such
examples for opportunities to build general numeric methods? When looking for such
opportunities, teachers would need to decide which of the drawings suggest correct
answers for the current problem and which suggest methods that could extend readily
to further examples such as 3/5 of 4/7. Furthermore, building general numeric methods
from students’ strategies would require many teachers to adapt in response to student
strategies that were novel to them. I use the word adapt to mean adjustments that
teachers would make in their thinking about the content as a consequence of interacting
with students.
Mathematical Knowledge for Teaching Fraction Multiplication 7
Figure 1 About Here
The next section of the article positions the current study with respect to two
bodies of research, research on teachers’ understanding of fraction multiplication and
research on unit structures required to reason about fractions as quantities. The review
will point out different perspectives on knowledge found in these bodies of research.
The theoretical frame presented in the following section will articulate a perspective on
knowledge that builds more directly on some of these perspectives than others. In
particular, it will incorporate descriptions of unit structures as well as further aspects of
knowledge not well represented in past research. Then, after a brief description of the
classroom context, I will describe the data collection and analysis methods in detail
because an important part of the research was the construction of methods that afforded
coordinated analyses of knowledge that teachers and students evidenced over
sequences of lessons in which they participated together. Such methods were essential
for developing accounts of knowledge that teachers used when responding to their
students’ reasoning about problem situations. Finally, I will proceed with the two case
studies, examining one teacher at a time.
Background
Teachers’ Knowledge of Fraction Multiplication
Research on teachers’ knowledge has examined fraction division and decimal
multiplication more closely than fraction multiplication. Research on fraction division
Mathematical Knowledge for Teaching Fraction Multiplication 8
has reported that teachers can confuse situations that call for dividing by a fraction with
ones that call for dividing by a whole number or multiplying by a fraction (Armstrong
& Bezuk, 1995; Ball, 1990; Borko et al., 1992; Ma, 1999; Sowder, Philipp, Armstrong, &
Schappelle, 1998). Ball asked 10 preservice elementary and 9 preservice secondary
teachers to generate situations that would illustrate 1 3/4 ÷ 1/2. Seventeen could
compute the correct answer but only five were able to generate an appropriate word
problem or situation. Five others generated situations that would illustrate 1 3/4 ÷ 2,
one generated a situation that would illustrate 1 3/4 x 2, and eight were unable to
generate any situation, correct or incorrect. Ma reported similar results for U.S. teachers
after including the same task in her investigation of 23 U.S. and 72 Chinese inservice
teachers’ understandings of core mathematics topics taught in elementary grades. Borko
et al. reported a case in which a teacher candidate, Ms. Daniels, was asked by a student
to explain the invert and multiply rule for fraction division. The class had just
computed the answer to 3/4 ÷ 1/2. Ms. Daniels generated a situation and area
representation that illustrated 3/4 x 1/2, realized that her example showed fraction
multiplication instead of division, and was stumped. Sowder et al. reported that a
middle-grades teacher, Cynthia, had trouble making appropriate connections between
operations on fractional quantities and the arithmetic operations of multiplication and
division (pp. 49-50).
Ma’s (1999) study not only confirmed constraints reported by Ball (1990) on U.S.
teachers’ performance, but also provided a more elaborated description of teachers’
mathematical knowledge. In interviews, she asked U.S. and Chinese teachers to solve
Mathematical Knowledge for Teaching Fraction Multiplication 9
mathematics problems embedded in teaching scenarios. Topics included subtraction
with regrouping, multidigit multiplication, and fraction division. Ma developed the
idea of knowledge packages to explain the vividly stronger performance of the Chinese
teachers. A knowledge package consists of numerous subtopics that are connected to
one another and that support the teaching of the larger topic. To illustrate with the
example most relevant to the present study, Ma described Chinese teachers’ knowledge
packages for fraction division in terms of the meanings of addition, multiplication, and
division with whole numbers; the concepts of inverse operation, fraction, and unit; and
the meaning of multiplication with fractions (Ma, 1999, p. 77). Ma found that the
Chinese teachers in her sample had more elaborate knowledge packages than the U.S.
teachers, which allowed them to move flexibly among subtopics when explaining how
they would help students connect new topics to prior topics.
A second set of studies about teacher knowledge of rational numbers has built
upon a different perspective on knowledge that states people have intuitive models for
arithmetic operations (Fischbein, Deri, Nello, & Marino, 1985). The proposed model for
multiplication is repeated addition, which implies that the multiplier or operator needs
to be a whole number and supports the notion that multiplication always makes larger.
Fischbein et al.’s results were on fifth-, seventh-, and ninth-grade students, but further
studies have extended the results to elementary school teachers (Graeber & Tirosh,
1988; Graeber, Tirosh, & Glover, 1989; Harel & Behr, 1995; Post, Harel, Behr, & Lesh,
1991; Tirosh & Graeber, 1989). In these studies, teachers have had a harder time solving
word problems in which the multiplier was a decimal less than one. For instance,
Mathematical Knowledge for Teaching Fraction Multiplication 10
Graeber and colleagues (Graeber & Tirosh, 1988; Graeber et al., 1989) administered to
129 preservice elementary teachers a written test consisting of 26 word problems
adapted from those used by Fischbein et al. They found that a higher percentage of the
preservice teachers solved multiplication word problems correctly when the multiplier
was a whole number and often used division inappropriately when working on
problems like the following:
One kilogram of detergent is used in making 15 kilograms of soap. How
much soap can be made from .75 kilograms of detergent? (Graeber &
Tirosh, 1988, p. 264)
A few further examples in the literature suggest that teachers may also have a
hard time using drawings to explain the product of two rational numbers. Eisenhart et
al. (1993) reported that during an interview the same Ms. Daniels began, but could not
complete, an explanation for .7 x 2.35 using a rectangular region. Armstrong and Bezuk
(1995) gave inservice middle school teachers a word problem for which computing 1/3
of 3/4 would be appropriate. The teachers recognized that the situation called for
fraction multiplication but had a hard time explaining their thinking, drawing diagrams
to match algorithmic solutions, and understanding the appropriate unit or whole for the
problem. Sowder et al. (1998) reported difficulties that a middle-grades teacher, Linda,
had explaining fraction multiplication with an area model during interviews and
classroom instruction (p. 83-87). Finally, Ball et al. (2001) present a classroom vignette in
Mathematical Knowledge for Teaching Fraction Multiplication 11
which a teacher struggled to explain products of decimals using dimensions and areas
of rectangles.
The studies summarized in this section have emphasized errors and other
constraints on teachers’ performances, primarily on tests and in interview settings. Such
results suggest that, beyond knowledge of computation procedures, teachers’
understandings of fraction multiplication (and division) is limited. Those studies that
have described teachers’ mathematical knowledge in more detail have delineated
subtopics or have gathered evidence for the intuitive model of multiplication as
repeated addition, but they have not examined teachers’ cognitive structures for
reasoning about fractional quantities. Yet, reasoning with such quantities is essential for
interpreting and evaluating different approaches to representing parts of parts, such as
those shown in Figure 1. The next section summarizes results of further research, much
of it done with students, that has examined cognitive structures for reasoning with
fractional quantities. These results will inform the theoretical frame developed
subsequently.
Unit Structures
Conceptual units of various types have played a central role in research on
children’s understandings of whole and rational numbers. The most important point in
the following discussion is the distinction between two and three levels of units, which I
explain first in the context of whole numbers by summarizing Steffe’s (1988, 1994)
analyses of emerging multiplication schemes.i According to Steffe, a child who can form
Mathematical Knowledge for Teaching Fraction Multiplication 12
two levels of units can understand a whole number such as five simultaneously as five
separate units (one level of unit) and as one group of five understood to be a single
entity (the whole group is a second level of unit) (Figure 2a). Steffe states that such a
child has formed composite units. Through interiorization and coordination of
composite units, a child can produce three levels of units by nesting composite units
within composite units. As an example, a child with such an operation can assimilate a
display of 20 blocks as a single group (the whole group is one level) composed of five
units (a second level), where each of the five units is also a composite unit composed of
four separate units (a third level) (Figure 2b). Reasoning with three levels of units
requires attending to all three levels simultaneously.
Figure 2 About Here
In subsequent work, Steffe and Olive (Olive 1999; Olive & Steffe, 2001; Steffe
2001, 2003, 2004) have examined how elementary students modified their knowledge of
whole-number counting and multiplication to construct knowledge of fractions when
engaged in tasks involving lengths and rectangular areas. The students in Steffe’s and
Olive’s reports solved tasks by coordinating two and three levels of units and by using
three mental operations called disembedding, iterating, and partitioning. For the
present article, iterating and partitioning will be the most important, and I illustrate
them in the next paragraph. Differences in students’ ability to coordinate levels of units
and contexts in which students used their disembedding, iterating, and partitioning
Mathematical Knowledge for Teaching Fraction Multiplication 13
operations led to significant differences in learning. The most successful students
constructed two new operations, recursive partitioning and splitting, that were
fundamental to the construction of schemes for commensurate fractions (e.g., 1/3 is
equivalent to 5/15), improper fractions, fraction composition (e.g., finding 1/3 of 1/4),
and addition of unit fractions.
I describe recursive partitioning to illustrate the role three levels of units can play
in reasoning about fractions and because it will be important to one of the case studies
reported below. Steffe (2003, 2004) has defined recursive partitioning to be taking a
partition of a partition in the service of a non-partitioning goal. Consider a task that
asks for the result of taking 1/4 of 1/3 using lengths. Students might first construct the
requested quantity by partitioning a unit length into three pieces (reasoning with two
levels of units), narrowing their attention to just the first of those pieces, and
partitioning that piece into four further pieces (reasoning with two levels of units a
second time) (Figure 3a). Determining the size of the resulting piece is a nonpartitioning goal, and students could accomplish this in more than one way. Students
might simply iterate the resulting piece and count to see that 12 copies exhaust the
original unit (Figure 3b). This solution involves reasoning with two levels of units a
third time (one unit containing 12 twelfths). Alternatively, students might recursively
partition by subdividing each of the remaining thirds into four pieces (Figure 3c). In
contrast to the first solution, recursive partitioning involves reasoning with three levels
of units in which four of the smallest units are nested within each mid-level unit, and so
there must be 3 x 4 = 12 units in the whole unit. Note that recursive partitioning
Mathematical Knowledge for Teaching Fraction Multiplication 14
incorporates aspects of distribution because each of the thirds is repartitioned into the
same number of smaller units.
Figure 3 About Here
In a second line of research also focused on the formation and reformation of
nested levels of conceptual units, members of the Rational Number Project have
examined five rational number subconstructs—part-whole, quotient, ratio number,
operator, and measure (e.g., Behr, Harel, Post, & Lesh, 1992). Of particular relevance to
the present study is the operator construct, in which a fraction is thought of as a
function applied to a number, object, or set. Behr and his colleagues (Behr, Harel, Post,
& Lesh, 1993) decomposed the operator construct further into three subconstructs—
duplicator and partition-reducer, stretcher and shrinker, and multiplier and divisor.
The role of unit structures in the first two subconstructs is most important for the
present discussion. When reading the descriptions below, those familiar with research
on multiplication and division will notice that the duplicator and partition-reducer
subconstruct is based on partitive division, and the stretcher and shrinker construct on
quotitive division.
Behr et al. (1993) used the problem 3/4 of 8 to contrast the duplicator and
partition-reducer subconstruct with the stretcher and shrinker subconstruct. In the
duplicator and partition-reducer subconstruct, the numerator duplicates by specifying
how many copies to make of the original set. In the present example, three sets of eight
Mathematical Knowledge for Teaching Fraction Multiplication 15
objects are created (Figure 4a). The denominator partitions by determining among how
many sets the resulting individual objects are shared—in this example 24 objects are
shared among four sets—and reduces by taking one of the resulting sets of six objects as
the answer. Behr et al. also explained that one could partition first and then duplicate.
Notice that the operations of duplicating and reducing transform the number of
composite units but not the size of those units.
Figure 4 About Here
In contrast, the operations of stretching and shrinking do act on sizes of units.
Here, the numerator stretches by exchanging each of the eight original objects for
composite units of three objects (Figure 4b). The denominator determines the size of the
groups into which the resulting individual objects are rearranged—in this example 24
objects are rearranged into groups of four—and shrinks by exchanging the resulting six
groups of four for six groups of one. Behr et al. (1993) also explained that one could
shrink first and then stretch. Note that, like recursive partitioning, the stretcher and
shrinker construct contains aspects of distribution in the sense that the numerator and
denominator operate on each subunit.
In a subsequent study, Behr, Khoury, Harel, Post, and Lesh (1997) interviewed 30
preservice elementary teachers on the operator subconstruct. This is the only other
study of which I am aware that analyzed unit structures to which teachers attend when
reasoning about fraction multiplication. The teachers were given 8 bundles of 4 sticks
Mathematical Knowledge for Teaching Fraction Multiplication 16
and asked to complete three tasks. The first task was simply to show a pile that has 3/4
as many sticks. For the second and third tasks, the teachers were asked to imagine that
each bundle of sticks represented boards that one carpenter used on a job in one day.
The second task stipulated that only 3/4 of the carpenters came to work one day, and
the third stipulated that all eight carpenters came to work but only worked 3/4 of a
day. In both cases the teachers were asked to arrange the boards for the carpenters.
The researchers sought to (a) characterize the strategies the preservice teachers
used as duplicator and partition-reducer, stretcher and shrinker, or other and (b)
determine the conceptual units that the teachers formed and transformed during their
solutions. They reported that even though the intent of the third problem was to elicit
stretcher and shrinker strategies (for instance by taking 3/4 of each bundle), only 12 of
the documented strategies were of this type. Nineteen of the remaining strategies were
consistent with the duplicator and partition-reducer construct. In these cases, teachers
tended not to offer a stretcher and shrinker strategy even when the interviewers
attempted to elicit such strategies. Apparently these teachers found it hard to distribute
fractions as operators across their conceptual units, which in turn suggested constraints
on the flexibility with which they formed and transformed conceptual units when using
sticks to find a fraction of a whole number.
The present study contributes in several ways to the bodies of research
summarized above. First, as already mentioned, research on teachers’ knowledge of
rational numbers has examined fraction division and decimal multiplication more
Mathematical Knowledge for Teaching Fraction Multiplication 17
closely than fraction multiplication. The few studies that do exist suggest that teachers
have trouble explaining fraction multiplication when fractions are understood as
quantities such as lengths or areas, but these same studies have not examined the
underlying knowledge that could account for such performances. The present study
concentrates on fraction multiplication and elaborates the underlying mathematical
knowledge that two teachers evidenced when reasoning with fractions as quantities.
Second, research that has described cognitive structures for reasoning about
fractions as quantities has highlighted, among other things, conceptual units organized
into multi-level structures. The present study will demonstrate that teachers need to
reason explicitly and flexibly with nested levels of units if they are to respond to the
variety of quantitative structures, such as those shown in Figure 1, that their students
might assemble. The analysis will demonstrate that the capacity to distribute fraction
operations across conceptual units is central to such flexibility.
Third, the present study examines not only teachers’ unit structures, but also the
pedagogical purposes for which they used drawings that depict visually perceivable
features of problem situations. To the best of my knowledge, researchers have not
delineated different goals, or purposes, for which teachers might use drawings during
lessons on fraction arithmetic or other topics. I focus on the use of drawings in teaching
not only because discussions of pedagogical content knowledge and mathematical
knowledge for teaching often make explicit reference to representations, but also
because reform-oriented curricula in the United States are placing new demands on
teachers and students to interpret and reason with a variety of inscriptions. The
Mathematical Knowledge for Teaching Fraction Multiplication 18
theoretical frame developed in the next section includes a discussion of different
possible purposes.
Theoretical Frame
In many respects, the perspective on knowledge I take in the present study is
consistent with others found in research on cognition in mathematical and scientific
domains (e.g., Izsák, 2004, 2005; diSessa, 1988, 1993; Piaget, 1970; Schoenfeld, Smith, &
Arcavi, 1993; Steffe, 1988, 1994, 2001). I emphasize three characteristics. First,
knowledge elements are often more fine grained than statements found in textbooks.
Summaries presented above of past research include descriptions of knowledge at
different grain sizes. To illustrate, Ma (1999) included the meaning of multiplication, the
concept of fraction, and the meaning of multiplication with fractions when elaborating
the knowledge package for fraction division; but, research by Steffe (1988, 1994, 2001,
2003, 2004) and by members of the Rational Number Project (e.g., Behr, Harel et al.,
1993; Behr, Khoury et al., 1997) decomposed such meanings and concepts into
numerous smaller cognitive structures. By attending to teachers’ nested unit structures,
I examine knowledge at a grain size similar to that analyzed by Steffe and members of
the Rational Number Project. Second, knowledge elements are diverse, which means
that a person’s knowledge elements related to a particular domain are not only
numerous but also of different types. In the present study, nested quantitative units and
pedagogical purposes for using drawings (described in more detail below) are different
types of knowledge. Third, knowledge is context sensitive and often tacit. A person
Mathematical Knowledge for Teaching Fraction Multiplication 19
who uses a given piece of knowledge when reasoning about one situation may not use
that same piece again in further situations where, from an observer’s perspective, it
might be applied productively. Furthermore, a person may only evidence some
knowledge elements by responding to situations through combinations of talk, gesture,
and inscription.
The perspective on knowledge just outlined implies that developing accounts of
knowledge used by teachers while teaching requires inferring diverse, fine-grained
knowledge elements that teachers activate (sometimes in response to students’
comments and questions) and how those elements function together as teachers try to
accomplish particular goals. Inferring knowledge elements is delicate because a teacher
may have, but not use, a particular piece of knowledge for such reasons as (a) the
teacher does not recognize a given situation as one where the piece of knowledge could
be applied productively or (b) connections among other activated knowledge elements
lead the teacher to consider but then not use a piece of knowledge. From my
perspective, if a teacher has but, for whatever reason, does not use a piece of knowledge
when responding to a range of situations that arise in the course of teaching—situations
where that knowledge could be applied productively—then for that teacher the
knowledge is not an instance of knowledge for teaching. In the analysis below, I restrict
my claims to knowledge that teachers appear to use when responding to their students.
I stop short of making claims about what they may or may not know more generally.
The theoretical frame developed here emerged through analyses of two teachers’
enactments of the CMP materials and, in particular, why the two teachers were more
Mathematical Knowledge for Teaching Fraction Multiplication 20
adaptive in response to some of their students’ comments and strategies than others.
The analysis led to three categories that serve to highlight the diversity of knowledge
elements that they used—numerical aspects of multiplication, unit structures, and
pedagogical purposes for using drawings. First, teachers evidenced several ideas
related to numerical aspects of multiplication. Examples included declarative
statements about whole-number factor-product combinations, that multiplication is the
same as repeated addition, and that a fraction times a number is the same as a fraction
of the number. Although each of these examples could have been anticipated, there was
one interesting wrinkle. Despite one teacher’s assertion that the words “of” and “times”
meant the same thing, she and her students apparently attended to parts of wholes
when thinking about a fraction times a number but to parts of parts when thinking
about a fraction of a number: Each word appeared to direct this teacher’s and at least
some of her students’ attention in different ways. Thus, I will not use the words “of”
and “times” interchangeably and will be careful to use the same word that the teachers,
students, books, and interviewers used in a given instance.
Second, as explained in the next section, the CMP materials provide extensive
opportunities to reason about quantities when developing fraction arithmetic, and
hence opportunities for teachers and students to employ their unit structures. Both
teachers appeared comfortable with the parts-of-a-whole interpretation of fractions and
the associated two levels of units. The extension of fractions from parts of wholes to
parts of parts created opportunities to reason with three-level unit structures, but I
emphasize that tasks involving parts of parts only created opportunities to reason with
Mathematical Knowledge for Teaching Fraction Multiplication 21
three-level structures. As the first solution to 1/4 of 1/3 discussed above illustrates, in
some cases it is possible to determine the result of taking part of a part by reasoning
with just two levels of units. Analyses of classroom interactions presented later in this
article will examine instances where a teacher had trouble responding to students’
reasoning about parts of parts, apparently because she focused on just two levels of
units. Further analyses will demonstrate that the ability to reason with three levels of
units explicitly is necessary but not sufficient if teachers are to adapt in response to the
variety of ways that their students might assemble three-level structures in a given
problem situation. The underlying flexibility with unit structures needed to compare
alternative structures that students might produce, such as those suggested in Figure 1,
rests in turn on reasoning with drawn versions of distribution.
The third category of knowledge, pedagogical purposes for using drawings, has
not been discussed widely in past research. I take drawings to be a particular kind of
inscription distinct from formal symbolic notations. The distinction between drawings
and symbolic notations as different types of inscriptions is important because reformoriented materials, like CMP, often use drawings to help students develop meaning for
formal notations and procedures for transforming those notations. A person must
interpret or use an inscription to reason about a situation before it makes sense to talk
about the inscription as representing.
Comparing the two teachers’ enactments of CMP to my interpretation of the
materials led to four different pedagogical purposes for which a teacher might use
drawings of quantities to solve fraction arithmetic problems. The first use is simply to
Mathematical Knowledge for Teaching Fraction Multiplication 22
illustrate solutions also arrived at using an alternate method, such as a numeric
computation. In this case, a teacher could use the computed answer as a guide when
developing an interpretation of the drawing. The second use is to infer a computation
method first by using drawings to determine solutions for various problems and then
by looking for patterns in the resulting numerical answers (for fraction multiplication
the pattern might be products of numerators are numerators of products and products
of denominators are denominators of products).
The third use for drawings is to deduce a computation procedure from the
structure of drawn quantities. In the case of fraction multiplication, a teacher might
focus on his or her own understandings of the whole, parts of the whole, and parts of
parts of the whole to determine a general computation procedure. For instance, a
teacher might partition a unit square horizontally to represent one fraction and
repartition the square vertically to represent the second. No matter what fractions are
used, this method generates a smaller array representing the part-of-the-part nested
inside a larger array representing the whole. The number of pieces in the smaller array
is always determined by the product of numerators (a x b in Figure 5), and that in the
larger array is always determined by the product of denominators (c x d in Figure 5).
Therefore, a general computation procedure can be based on products of numerators
and products of denominators. The fourth use is to adapt how one represents structures
of quantities in response to students’ thinking by (a) inferring students’ understandings
of the whole, parts of the whole, and parts of parts of the whole and (b) attending to the
Mathematical Knowledge for Teaching Fraction Multiplication 23
variety of ways that students might begin assembling three-level unit structures as
evidenced by their explanations and drawings.
Figure 5 About Here
I have already mentioned two reasons for attending to teachers’ pedagogical
purposes for using drawings: (a) Definitions of pedagogical content knowledge and
mathematical knowledge for teaching make explicit reference to representations, and
(b) reform-oriented curricula in the United States often place new demands on teachers
and students to interpret and reason with a variety of inscriptions. A third reason is that
the case studies presented below suggest that the unit structures teachers employed
shaped the purposes for which they used drawings when teaching fraction
multiplication. The first case is of a teacher who reasoned primarily with just two levels
of units and used drawings to illustrate solutions. The second case is of a teacher who
evidenced more consistent attention to three-level structures and used drawings to infer
a computation method. (A teacher who focused primarily on just two levels of units
might also be able to use drawings to determine solutions without first calculating
answers using numeric methods but consistent, explicit attention to three-level
structures would make this easier.) The third use for drawings would require the ability
to explicitly produce three levels of units, and the fourth use would require the ability
to produce such structures flexibility so as to compare alternative approaches for
producing three-level structures. Explicit attention to drawn versions of distribution
Mathematical Knowledge for Teaching Fraction Multiplication 24
supports such flexibility. Although I claim that core mathematical understandings
shaping how the two teachers’ enacted the CMP materials and responded to their
students over sequences of lessons can be organized into the three categories of numeric
aspects of multiplication, unit structures, and pedagogical purposes for using drawings,
I do not claim that these categories capture all knowledge that the teachers might have
used during their fraction multiplication lessons.
Connected Mathematics and Fraction Multiplication
The CMP materials shaped, but did not determine, the perspectives on fraction
multiplication and the purposes for drawings that the teachers used during their
lessons. The first fractions unit, Bits and Pieces I, introduces various interpretations of
fractions, decimals, and percents, including parts of a whole, measures of quantities,
and indicated division. The materials include fraction strips, number lines, and
rectangular regions for representing fractional quantities. Fraction strips are made of
paper subdivided by folding. The length of the entire strip is intended to represent one
whole.
The second fractions unit, Bits and Pieces II, develops fraction arithmetic by
building on the interpretations and drawings introduced in Bits and Pieces I. Each case
study teacher taught from a slightly different draft of the revised Bits and Pieces II unit
that has since been published (Lappan et al., 2006). At different points, the overarching
mathematical goals stated in the teacher’s edition appear consistent with the second,
third, and fourth uses for drawings discussed above. The goals include developing
Mathematical Knowledge for Teaching Fraction Multiplication 25
ways to model sums, differences, products, and quotients using fraction strips, number
lines, and rectangular areas; looking for and generalizing patterns in numbers (the infer
purpose); and developing algorithms (possibly the infer, deduce, or adapt purpose). The
introduction to the teacher’s edition states that Bits and Pieces II does not teach a
preferred algorithm. Rather, students are to develop solution strategies for problems in
which fractions and operations on fractions are embedded. Teachers are to develop
their students’ strategies into general algorithms (the adapt purpose).
Pierce Middle School, Ms. Archer, and Ms. Reese
The present case study was conducted as part of the Coordinating Students’ and
Teachers’ Algebraic Reasoning (CoSTAR) project. The project conducts coordinated
research on teaching and learning in grades six through eight at Pierce Middle School.
At the beginning of the study, the district had replaced traditional instructional
materials focused on skill in computation with the standards-based CMP (Lappan et al.,
2002) materials. Pierce Middle School is in a rural community outside of a large
southern city. At the time of the study, the school had approximately 700 students who
were racially and economically diverse: 35% were African American, 63% were White,
and 42% qualified for free lunch.
The first teacher, Ms. Archer, ii was in her first year as a full-time teacher, but she
was not new to teaching. She reported several previous non-teaching careers as well as
teaching in the district as a long-term substitute, mostly in high school classrooms. For
one assignment, she had taught the CMP seventh-grade Stretching and Shrinking unit in
Mathematical Knowledge for Teaching Fraction Multiplication 26
Pierce Middle School. Ms. Archer had a bachelor’s degree in mathematics and, at the
time of the study, was enrolled in a 2-year alternative certification program for teachers
who have been hired by a district and have a bachelor’s degree in an appropriate field.
The program assumes that teachers have the requisite content knowledge, emphasizes
other aspects of teacher preparation such as teaching methods, and requires that
certification candidates have an on-site mentor. Ms. Archer reported that her own
experiences as a student had emphasized drill and repetition, and her collegiate
mathematics appeared rusty when she reported having to “brush up” on functions for
an exam of content required for licensure.
The second teacher, Ms. Reese, was confident in her teaching. Prior to the study,
she had taught algebra to seventh-, eighth-, and ninth-grade students for approximately
10 years. Ms. Reese reported that her high school classes had focused on “traditional
mathematics” and algorithms, had rarely used manipulatives, and had included drawn
pictures only occasionally to introduce a topic or when there was confusion. Ms. Reese
was in her second year of CMP implementation, but at the time of the study was
teaching Bits and Pieces II for the first time. Izsák, Tillema, & Tunç-Pekkan (in press)
described Ms. Reese in greater detail.
CMP is an ambitious program that embeds mathematical ideas in problem
situations, and teachers often need significant support when first using the materials in
their classrooms. Prior to the present case studies, the district had provided limited
professional development opportunities to support teachers at Pierce Middle School as
they transitioned to reform-oriented materials. Ms. Archer and Ms. Reese participated
Mathematical Knowledge for Teaching Fraction Multiplication 27
in monthly, after-school professional development sessions provided by the CoSTAR
project but had to rely primarily on their existing mathematical understandings when
implementing the CMP materials. Of particular relevance to the present study, both
teachers reported that solving fraction arithmetic problems with drawings was new for
them. Both teachers seemed earnest about using the CMP materials successfully with
their students and combined aspects of more traditional and more reform-oriented
instruction.
Data and Methods
The CoSTAR project gathered data that afforded coordinated analyses of
understandings that the teachers and their students had of lessons in which they
participated together. We started by videotaping continuous sequences of enacted
lessons over entire units. When taping lessons, we only observed and did not interact
with teachers or students about the content. To the extent possible, our intent was to
document instruction in natural settings. Because comments made by teachers and
students during lessons were often insufficient for inferring with much confidence the
understandings that they were using, we augmented the lesson data with interviews
based on tasks and events from the observed lessons. We then analyzed both the lesson
and the interview data to infer knowledge that teachers and students may have used to
make sense of the lessons.
We used two cameras to videotape one teacher’s instruction daily during the
same class period. One researcher set the first camera in the back of the classroom and
Mathematical Knowledge for Teaching Fraction Multiplication 28
recorded the entire class, adjusting the levels of ceiling and wireless microphones to
hear all students during whole-class discussion and to zero in on conversations between
the teacher and individual students during group work. A second researcher used the
second camera to record written work, staying at the back of the classroom to record the
whiteboard during whole-class discussion and shadowing the teacher to record papers
that the teacher discussed with students at their desks during group or individual work.
This method of recording fit well with the perspective on knowledge outlined above
because it captured talk, gesture, and inscription as evidence for more explicit and more
tacit knowledge that supported students’ reasoning and teachers’ responses to their
students’ reasoning.
Later the same day, we combined the video and audio from the two cameras
using an audiovisual mixer to create a restored view (Hall, 2000). A restored view is a
wider view of activity than can typically be captured with an individual camera but is
still a selective view that reflects the researcher’s perspective. From the restored view,
we created a lesson graph, a document that parsed the lesson into episodes. Examples
of what we took to be episodes include a whole-class discussion of one problem and a
conversation between the teacher and a student during group or individual work. The
lesson graph included the start and end times of each episode, a brief description of the
problem being discussed, key comments made by the teacher and students, and screen
shots of any written work that was being discussed.
We used the lesson graphs to identify episodes where teachers and students
discussed drawings of quantities, as opposed to numeric methods. For the interviews,
Mathematical Knowledge for Teaching Fraction Multiplication 29
we gave priority to those lesson episodes where teachers and students apparently
struggled to make sense of how the other used or interpreted drawings. Such episodes
provided good opportunities to investigate what teachers and students attended to in a
particular drawing and how they understood the underlying topic. I emphasize that we
could not always anticipate when such difficulties might arise and that being present in
the classroom on a daily basis maximized our opportunities to capture episodes that
served as effective prompts in interviews.
I interviewed pairs of students from the same classes in which we gathered the
lesson videos (four pairs in Ms. Archer’s classroom and three in Ms. Reese’s). Each
teacher helped identify pairs that represented as best as possible a cross section of
achievement, a balance between boys and girls, and the racial composition of the class. I
interviewed each pair once a week for approximately 50 minutes starting the second
week of lesson taping. My overarching goal was to gather data on students’
understanding of the content and how those understandings shaped their
interpretations of the lesson excerpts.
I used problem-solving tasks and lesson video excerpts as prompts in
semistructured interviews (Bernard, 1994, Chapter 10). Typically, I prepared tasks that
included both the original tasks in the lesson excerpts and extensions of those tasks.
Students worked the tasks using pencil and paper and watched the lesson excerpts on a
laptop computer. I elicited students’ reasoning on the tasks and interpretations of the
lesson excerpts, but did not correct mistakes or offer instruction. I used one camera to
record students, and one to record written work and viewed lesson excerpts. Project
Mathematical Knowledge for Teaching Fraction Multiplication 30
members created a restored view that captured much of what the students said, wrote,
and watched and created interview graphs similar to the lesson graphs described
above. I used the graphs to locate excerpts for use in teacher interviews that revealed
aspects of students’ mathematical thinking and difficulties not evidenced during
lessons.
Finally, other project members conducted weekly, hour-long interviews with the
teachers to gain access to their interpretations of the CMP activities, their
understandings of their students, and the pedagogical decisions they made. One project
member interviewed Ms. Reese, and a second interviewed Ms. Archer. The interviews
were also semistructured and based on a combination of lesson and student interview
video excerpts selected in consultation with me. The consultations insured that lesson
excerpts used in the student interviews were also used in the teacher interviews.
Teacher interviews occurred after a completed cycle of student interviews and almost
always within one week of the original lessons.
When preparing teachers to view a given lesson or student interview excerpt, the
interviewers stated the mathematics task central to the excerpt. If the task was an
extension I had used in student interviews, teachers were sometimes asked to work the
task before watching the excerpt. As initial prompts, teachers were often asked to
discuss what mathematics was most important in the task, reconstruct what they may
have been thinking at a particular point during a lesson, interpret students’ reasoning,
or consider how they might respond to a particular student comment or question made
during a lesson or interview. As the case studies progressed, teachers developed a sense
Mathematical Knowledge for Teaching Fraction Multiplication 31
of the questions we were asking and sometimes responded spontaneously to lesson and
student interview excerpts. Follow up questions often directed a teacher’s attention to
particular aspects of a drawing or a particular student comment that we thought was
central to understanding how she and her students understood a given task. The
interviewers used one camera to record the teacher and one to record viewed lesson
clips. Project members created a restored view that captured much of what the teacher
and interviewer said and watched and created interview graphs similar to the lesson
graphs described above.
We collected data on Ms. Reese first, and only during her enactment of Bits and
Pieces II. This began mid-March and continued to mid-May of 2003. The entire case
study consisted of 30 taped lessons, 20 interviews distributed over three pairs of
students, and nine teacher interviews. Roughly, the first month concentrated on fraction
addition and subtraction and the second month on fraction multiplication. The present
report concentrates on the 11 lessons, five student interviews, and two teacher
interviews that focused on fraction multiplication. (Izsák et al., in press, analyzed the
lessons and interviews that focused on fraction addition.) Although the present article
concentrates on the second month of instruction, claims made here that Ms. Reese could
reason with three-level unit structures are informed by the full dataset.
In analyzing data from Ms. Reese’s classroom, questions emerged about how she
and her students had discussed partitioning during the preceding Bits and Pieces I unit.
Thus, the next year we collected two rounds of data in Ms. Archer’s classroom. The first
round occurred in late January through February of 2004 and consisted of 13 lessons
Mathematical Knowledge for Teaching Fraction Multiplication 32
from Bits and Pieces I, eight interviews distributed over four pairs of students,iii and
four teacher interviews. These data contained numerous examples in which Ms. Archer
and her students determined products of fractions and whole numbers. The second
round occurred in late April through May of 2004 and consisted of 17 lessons from Bits
and Pieces II, 17 student interviews distributed over the same four pairs of students, and
five teacher interviews. In contrast to Ms. Reese, Ms. Archer touched on fraction
multiplication during her fraction addition lessons as well as her two lessons focused
exclusively on products of fractions using rectangular regions. We followed up on
instances of fraction multiplication instruction in 11 of the student interviews and three
of the teacher interviews. Thus, the analyses below are based on significant corpora of
data from each classroom.
At the end of data collection several episodes stood out in which Ms. Archer had
struggled when using drawings to respond to students. A particularly vivid example,
discussed in detail later in the article, occurred when she introduced her students to
products of proper fractions using number lines. This example raised questions about
Ms. Archer’s reasoning with fractional quantities and the pedagogical purposes for
which she used drawings, questions that provided initial direction for the analyses
presented below.
I reviewed all of the lesson videos from Ms. Archer’s classroom chronologically
for statements, gestures, or inscriptions that evidenced ideas I understood to be related
to multiplication of rational numbers. I found comments about using whole-number
multiplication to guide partitioning the whole; statements about the meanings of
Mathematical Knowledge for Teaching Fraction Multiplication 33
multiplication, fractions, and fraction multiplication; numeric computations with
fractions; and discussions of drawings that, to me, showed parts of parts. The
chronology was important to preserve in case the knowledge Ms. Archer used shifted
over the course of the unit, a distinct possibility given that she was teaching fraction
arithmetic with drawings for the first time. To be confident that my list of statements,
gestures, and inscriptions was complete, I went through the 30 lesson videos twice and
a second member of the CoSTAR project also went through the lesson videos to check
the list.
Next, I conducted a chronological analysis of knowledge that Ms. Archer
evidenced through statements, gestures, and inscriptions during her interviews. Here I
used verbatim transcriptions, as well as videos. I coordinated my analysis of knowledge
she evidenced during lessons and interviews using methods similar to those described
by Cobb and Whitenack (1996) and Schoenfeld et al. (1993) for analyzing longitudinal
sets of video recordings. These methods are based on analyzing small episodes,
generating hypotheses about those episodes, analyzing further episodes for evidence
that might support or undermine any of the initial hypotheses, and modifying
hypotheses as necessary to knit local accounts into consistent global accounts. I used the
same approach when analyzing the data from Ms. Reese’s classroom and interviews.
Given the delay of up to a week between the initial lesson taping and following
teacher interview, I used the interview data with great care when supplementing the
lesson data for evidence of the mathematical knowledge that each teacher used when
interacting with her students. I was most confident using the interview data when the
Mathematical Knowledge for Teaching Fraction Multiplication 34
teachers evidenced similar understandings in both contexts—for instance, when they
gave similar explanations for how to solve a problem first during a lesson and then
during an interview. In cases where the teachers evidenced possibly different
understandings in the lesson and interview contexts, I considered the possibilities that
they accessed different knowledge in the two contexts, that my initial attributions
needed to be refined or discarded, and that the teachers had learned something during
the time between the lesson and the interview. To illustrate, a teacher might evidence
different knowledge in the two contexts if she discussed aspects of the mathematics
during her interview that she did not discuss during her lesson. If she reported that it
did not occur to her to discuss those aspects during her lesson, then the data suggested
that she accessed different knowledge in the two contexts. If her comments during the
interview suggested a different interpretation of the original lesson data, then I changed
my account of the knowledge she used. If she had explicitly rejected a certain problemsolving approach during an earlier lesson, but then acknowledged the approach could
work during a subsequent interview, the data suggested she had learned something.
Finally, I examined the teacher’s editions of the CMP units to determine which
mathematical ideas they emphasized and how they presented the use of drawings. I
compared my understanding of the materials to my understanding of the observed
lessons to gauge where each teacher’s explanations were consistent with, or diverged
from, those in the materials. This was important because the materials could influence
explanations the teachers gave to students, and hence the knowledge they appeared to
access during classroom interactions. Instructional goals stated in the materials might
Mathematical Knowledge for Teaching Fraction Multiplication 35
lead a teacher to de-emphasize certain mathematical aspects that she understood about
a given task. In such cases, without an understanding of the materials, I might under
represent her mathematical knowledge for teaching. In other cases, where a teacher’s
explanations were particularly close to those in the materials, I might over represent her
mathematical knowledge for teaching if she simply repeated something she had read.
Results on Ms. Archer from Bits and Pieces I
The analysis of Ms. Archer proceeds chronologically to gather evidence for the
unit structures she used when responding to her students. The initial Bits and Pieces I
lessons concentrated on methods for partitioning fraction strips and for computing
fractions of whole numbers. Further lessons concentrated on numeric methods and
shed less light on the unit structures that Ms. Archer and her students apparently used
to reason about fractions as quantities.
I will make three main points. First, several tasks afforded opportunities to
reason with three levels of units, but whether Ms. Archer did so remained unclear: Her
comments made explicit reference to just two levels of units. Second, Ms. Archer and
her students discussed two different strategies for determining fractions of whole
numbers, a primary one that she and her students discussed most often, as well as a
student’s alternative. Ms. Archer’s quick recognition of the alternative demonstrated
her capacity to adapt to a student’s reasoning in one context. Third, if Ms. Archer
attended to three levels of units, she did not do so consistently in cases where
discussing a third level explicitly could have helped answer students’ questions. Data
Mathematical Knowledge for Teaching Fraction Multiplication 36
from Bits and Pieces I left open the possibility that Ms. Archer could reason explicitly
with three-level unit structures but, for whatever reason, did not. Subsequent data from
Bits and Pieces II would provide examples where she clearly struggled to relate three
levels of units successfully.
Whole-Number Multiplication and Partitioning
The Bits and Pieces I unit opens with a set of problems about fund-raisers and
introduces thermometers as representations for recording progress toward fund-raising
goals. Students are asked first to make fraction strips by folding 8 1/2 inch strips of
paper into halves, thirds, fourths, and so on up to twelfths. For prime numbers, there
are few alternatives to estimating directly the size of one piece and folding to check, a
strategy that emphasizes two levels of units. Composite numbers afford opportunities
to use knowledge of whole-number multiplication, to partition in stages, and to reason
with three levels of units. Ms. Archer clearly attended to whole-number multiplication.
The first day she made the following announcement to the whole class after students
had been creating fractions strips for several minutes: “Some of you need to think about
multiples, products, same stuff we had in Prime Time.iv Even though we are dealing
with parts of a whole, we can still apply some of the things that we learned in Prime
Time.” A few minutes later, she asked a student who had made thirds but was
struggling to make ninths, “So if you fold it in thirds, and I know that three times three
is nine, so what would I have to divide the thirds into?” (Ms. Archer also made a similar
comment about dividing fifths into fifteenths at one point during a subsequent
Mathematical Knowledge for Teaching Fraction Multiplication 37
interview.) The second day Ms. Archer listened as Damien demonstrated to the class
how he used factors to make a twelfths strip. He explained that half of 12 is 6, and that
half of 6 is 3, as he folded his strip in half, in half again to make fourths, and then in
thirds to make twelfths.
Whether Ms. Archer attended to three levels of units at these moments, however,
remained unclear. When summarizing the activity of creating fraction strips, she
referred to whole-number factor-product combinations two times, each time saying that
even numbers were easier because you could start by folding the strip in half. These
comments may have been influenced by a similar statement in the teacher’s edition and
by Damien’s explanation. Ms. Archer did not discuss connections between wholenumber factor-product combinations and taking partitions of partitions more generally,
but no task or comment in the CMP materials suggested that developing such
connections was a central instructional goal. Her concluding remarks to the class
emphasized that the strips were going to be used as measuring devices and that
students should remember to find equal-sized parts of a whole, comments that
emphasized two levels of units. Ms. Archer emphasized the same two points when
discussing these initial lessons during her interviews. These data left open the
possibility that Ms. Archer had capacities to reason with three levels of units that
remained largely latent.
Mathematical Knowledge for Teaching Fraction Multiplication 38
Two Strategies for Determining Fractions of Whole Numbers
The next tasks in the CMP materials ask students to solve problems about fundraising using thermometers that show progress toward specified goals. The
thermometers are also 8 1/2 inches so that students can measure them with their folded
strips. Ms. Archer’s students began determining fractions of whole numbers when
finding how many dollars had been raised at different points in several fund-raisers.
Over the course of two and a half days, students worked on 11 such problems in groups
and presented their solutions to the whole class. The vast majority of solutions relied on
determining the dollar value of one equal part of the whole number (interpreting
fractions as parts of a whole) and then using multiplication or repeated addition to
determine the final answer. As one example, Angie determined 3/5 of $300 by finding
that 1/5 was $60 and then calculating 3 x $60 = $180. Ms. Archer would use an
analogous strategy, applying repeated addition or multiplication to a unit fraction,
when introducing products of proper fractions during her subsequent enactment of Bits
and Pieces II.
One student, Damien, used an alternative strategy when determining 11/12 of
$240. He found that one of 12 equal parts would be $20 but, instead of using repeated
addition to build up to 11/12, he subtracted one part from the whole (i.e., $240 – $20 =
$220). Ms. Archer first observed Damien’s approach while he worked at his desk and
commented, “Oh. That is a good idea. Hold that thought so you all might share that
with the class.” During subsequent whole-class discussion, Ms. Archer called on
Damien to present his solution. Unfortunately, a debate emerged because some
Mathematical Knowledge for Teaching Fraction Multiplication 39
students had got $216 as the final answer using tenths strips. Ms. Archer ran out of time
and did not resume the discussion the next day. Nevertheless, that she complimented
Damien and remembered to call on him at the end of class provided evidence that she
understood his approach.
As with class discussions about folding strips, whether Ms. Archer attended to
three-level unit structures during class discussions about fractions of whole numbers
remained unclear. She referred to just two levels when emphasizing a whole divided
into equal parts but may have attended to a third level if, for example, she considered
300 divided into five groups, each of which consists of 60 ones. The data were
ambiguous, however, because the smallest level of unit was the one, and she may not
have attended to it explicitly. If she did attend to three-level structures here, data from
subsequent lessons discussed below made clear that she did not always assemble threelevel structures readily. Thus, it is possible that Ms. Archer reasoned with three-level
structures like that shown in Figure 2 but not in Figure 3. I will handle this uncertainty
by describing Ms. Archer as reasoning primarily with two levels of units.
Struggling to Respond to Students
The final episode from Bits and Pieces I that I discuss occurred two weeks after
those presented above. In the interim, we recorded two days during which students
used thermometers to answer questions about fund-raising goals. The discussions
focused on calculating fractions of whole numbers and emphasized numeric methods.
Then Ms. Archer took several days off from work for personal reasons. When we
Mathematical Knowledge for Teaching Fraction Multiplication 40
returned to her classroom, she had completed one lesson that used fraction strips to
understand equivalent fractions. This lesson afforded opportunities to discuss nested
levels of units but, if we missed such conversations, they would have been very
different from those we observed our first day back. On that day, Ms. Archer and her
students transferred the locations of fractions from all of the strips (halves, thirds,
fourths, and so on up to twelfths) onto one number line. Students lined up the left end
of their strips with the zero on the number line and made tick marks corresponding to
each fold, but they did not make explicit relationships among different levels of units.
Our second day back provided the first example in which we observed Ms.
Archer attempt but then abort the use of drawings when responding to students’
questions. Students were comparing 2/3 to 3/4 to determine whether 2/3 was closer to
1/2 or to 1. Some said that 2/3 and 3/4 were the same distance from 1 because both
were one piece away from making a whole. In an apparent attempt to direct students’
attention toward the size of each fraction, Ms. Archer drew two squares and
emphasized that both were the same size. She subdivided the squares and shaded
“three of four equal parts” and “two of the three,” again making explicit reference to
two levels of units. Some students continued to say that the two fractions were the
same. Rather than introduce a third level of unit that subdivided both thirds and
fourths in her drawing, Ms. Archer switched to numeric computations, asked for the
“least common multiple of 4 and 3,” and determined equivalent fractions. Figure 6
shows the white board as it appeared at the end of her explanation. Note that in writing
3 x 3/4, Ms. Archer asked how many times 4 went into 12 and then multiplied 3 times 3
Mathematical Knowledge for Teaching Fraction Multiplication 41
to get 9. Her discussion of 4 x 2/3 was similar. She then asked if 8/12 or 9/12 was
bigger. Many students called out, “three fourths.”
Figure 6 About Here
These lesson data provided initial evidence that if Ms. Archer could produce
three-level structures, she did not always do so readily. Further evidence that she did
not consider three-level structures during the episode came from a subsequent
interview. Ms. Archer watched an interview excerpt in which Emily and Angiev
reviewed the lesson excerpt summarized above. The students were asked if, starting
with 3/4, Ms. Archer could have made thirds. Angie explained that Ms. Archer “could
not make thirds exactly” but could make twelfths. Ms. Archer commented, “She’s right.
I could have, but I wasn’t thinking about that at the time.” Data from the initial Bits and
Pieces I lessons, during which students created fractions strips, made clear that Ms.
Archer connected whole-number multiplication to partitioning, but left unclear whether
she attended to three levels of units. Data from this last lesson suggested that if she
could produce three-level structures, they were not a central part of her mathematical
knowledge for teaching.
During the next three lessons that we recorded, Ms. Archer and her students
continued using fraction strips to place fractions on number lines and answered such
questions as how do you decide if a fraction is between 0 and 1/2, between 1/2 and 1,
or greater than 1? The discussions focused on comparing numerators and
Mathematical Knowledge for Teaching Fraction Multiplication 42
denominators. During the final two Bits and Pieces I lessons that we recorded, Ms.
Archer and her students discussed computation methods for multiplying fractions and
whole numbers that were based on repeated addition. The remaining Bits and Pieces I
lessons that we did not record developed connections among fractions, decimals, and
percents.
Ms. Archer Uses Number Lines to Explain Products of Proper Fractions
The Bits and Pieces II unit begins with lessons on estimating sums of fractions
using comparisons to 0, 1/2, and 1 as a guide. The unit then introduces fraction
addition through problems about buying and selling land in a rectangular town that is 1
mile by 2 miles. The task includes a map that shows the land divided among several
people. Students are told that there are 640 acres in one square mile and are asked to
find the fraction of one square mile and the number of acres that each person owns. The
strategy Ms. Archer emphasized was similar to the primary one students used to
answer questions about fund-raisers: Find how many acres correspond to an
appropriate unit fraction and then use multiplication or repeated addition to determine
the number acres a given person owns. Ms. Archer’s comments continued to emphasize
equal parts of a whole (two levels of units) and that a fraction of a number means a
fraction times a number. Immediately following these lessons, Ms. Archer used the
number line to introduce products of proper fractions.
Mathematical Knowledge for Teaching Fraction Multiplication 43
Using Two Levels of Units to Reason About Parts of Parts
Bits and Pieces II intends for teachers and students to work first with rectangular
regions when studying products of proper fractions, but Ms. Archer began instead with
an example that used number lines. The problem was to find one fifth times two thirds
and, as Ms. Archer explained in a subsequent interview, the drawing came from the
teacher’s edition (see Figure 7). The problem was not situated in any particular context.
Ms. Archer’s explanations of the solution, combined with her difficulties responding to
a student’s question, provided stronger evidence than did the Bits and Pieces I data that
she reasoned primarily with two levels of units
Figure 7 About Here
Ms. Archer began her explanation with the drawing already completed. She
reminded her students that a fraction of a number means a fraction times the number
and pointed out that the interval from 0 to 1 was divided into three parts. Next, she
covered all but the first third, and students commented that it was divided into five
parts. Ms. Archer then reminded students that the problem asked about two thirds:
So we have one fifth here (pointing to the shaded fifth of the first third)
and we have one fifth here (pointing to the shaded fifth of the second
third.) But this one fifth (pointing to the shaded fifth of the first third)
really is the same as what if we look across the whole number line?
Mathematical Knowledge for Teaching Fraction Multiplication 44
Emily explained that the answer was one fifteenth because “in the whole number line
there’s five equal parts in each section and you add them all together.” Ms. Archer
agreed, pointed again to the first shaded piece, and said this “really would be one of
fifteen equal pieces.” Although Emily’s comment suggested attention to three levels of
units, Ms. Archer’s comments referred explicitly to two levels at a time, interpreting
each shaded portion first as one fifth and then as one fifteenth.
A moment later, Ms. Archer had trouble responding to one student. Jeff may not
have attended to the distinction between one fifth of a whole and one fifth of a third
because he stated, “If you wanted one fifth, it would change to, it would be like three
fifteenths.” Jeff’s tone implied confusion, perhaps about discussing one shaded piece
both as one fifth and as one fifteenth. Ms. Archer did not address Jeff’s confusion
directly. Instead, she read a sentence on her overhead that also appears in the teacher’s
edition: “Each one fifth of a third is one fifteenth so that two parts marked would be
two fifteenths.” (I did not use these data to attribute three-level unit structures to Ms.
Archer because the explanation came verbatim from the teacher’s edition.) Jeff still
seemed confused when he stated once more that one fifth would be three fifteenths. Ms.
Archer may have thought Jeff attended to all three thirds because she stressed that this
problem required looking at just the first two. She then confirmed the answer by
multiplying numerators together and denominators together. Ms. Archer concluded her
demonstration by stating that “the number line shows why” and moved on to lessons
that introduced addition and subtraction of fractions and mixed numbers. The methods
Mathematical Knowledge for Teaching Fraction Multiplication 45
Ms. Archer and her students discussed during these subsequent lessons were numeric
and so provided little access to their unit structures.
Persistent Difficulty Responding to Jeff
The lesson data left unclear why Ms. Archer inserted the number line example,
what she understood about using the number line to solve fraction multiplication
problems, and what she meant by “showing why.” Furthermore, from my point of
view, the classroom discussion appeared to leave Jeff’s question unanswered. I
interviewed Jeff and Emilyvi about this lesson excerpt and, during her next interview,
Ms. Archer watched portions of the lesson and the student interview. To prepare Ms.
Archer for watching the lesson excerpt, the interviewer asked, “What was your goal
with showing the multiplication at this point?” She answered “to give [students] a taste
of what we’re going to be doing and to show them how there are other ways that things
can be done.” She explained that her students already knew where 1/3 was on the
number line and how to “divide things into parts.” Finally, she said that she had not
known how to use number lines to multiply fractions and found the example
interesting. These comments summarized her reasons for inserting the number line
example and suggested why she did not have students work further problems as part of
the lesson.
When the interviewer showed the lesson excerpt, including Jeff’s comment that
1/5 was the same as 3/15, Ms. Archer had difficulty relating three levels of units
appropriately. Talking over the video, she interjected:
Mathematical Knowledge for Teaching Fraction Multiplication 46
Maybe I should’ve said sixth fifteenths to let him know we talking about
three fifteenths plus three fifteenths. Yeah. And that would’ve helped him
to understand that now I’m listening to it.…He’s right. That’s three
fifteenths, but we’re concerned about two thirds. So therefore we have to
do three fifteenths plus three fifteenths, which is six fifteenths. And when
you reduce it, it still gonna give you two fifteenths.
When the interviewer followed up by asking, “Where are the three fifteenths you think
he’s thinking of?,” Ms. Archer first suggested that Jeff might be thinking of a fraction
equivalent to 1/5, then suggested that he might be thinking of a common denominator
for 1/3 and 1/5, and finally said she did not know.
One possible explanation for Ms. Archer’s comments is that she understood the
one fifths to be of the whole and, in a momentary lapse, assumed that 6/15 could be
reduced to 2/15. A second possible explanation is that she understood each one fifth to
be of one third and meant three fifteenths of one third plus three fifteenths of one third
makes six fifteenths of one third. Either way, the possibility that Ms. Archer could
establish an explicit three-level structure in which five fifteenths make up each of three
thirds of a whole seemed less likely a moment later: When the interviewer suggested
that Jeff might be confused by the distinction between a fifth of a whole and a fifth of a
third, Ms. Archer asserted once more that 3/15 plus 3/15 made 6/15 which “would
give you two fifteenths.”
The next video clip that the interviewer played came from an interview I
conducted with Jeff and Emily. In preparation for watching the student interview, Ms.
Mathematical Knowledge for Teaching Fraction Multiplication 47
Archer solved the extension problem 3/5 of 2/3, correctly. She first calculated 3/5 x 2/3
= 6/15 and 3/15 + 3/15 = 6/15, and then shaded three fifths of the first third and three
fifths of the second third. Next, Ms. Archer watched portions of the student interview
during which Jeff and Emily first produced incorrect solutions to 3/5 of 2/3, also on the
number line, and then reviewed the original lesson segment. (Their mistakes are
tangential to the present analysis.) At one point, I asked the students why Ms. Archer
shaded the way she did. Jeff proposed rearranging the two shaded pieces side by side
in the first third and explained, “You’re really thinking of two thirds, I mean, two fifths
of a third.” When Ms. Archer was asked, “Do you think that would work as well?,” she
replied “No” and explained:
This is one third by itself (pointing to the first third) and this is another
third by itself (pointing to the second third). And the concept is, was then
one fifth times two of three parts. So, you gotta have the two of three
parts. You can’t just put both of them down in there, then you won’t get
what you’re supposed to have.
Ms. Archer insisted that each third be treated separately once more near the end of the
interview. When the interviewer asked how she would build on what the students
appeared to understand after one lesson on fraction multiplication with number lines,
Ms. Archer said give other examples and that “if it was gonna be three fifths (i.e., 3/5 of
2/3), then you gotta do three sections of two separate things.” Thus, in both examples,
Mathematical Knowledge for Teaching Fraction Multiplication 48
Ms. Archer appeared to use the parts-of-a-whole interpretation twice in succession—
once to establish thirds and once to establish parts of one third—and then applied
repeated addition to the result. Note that this interpretation of Ms. Archer’s reasoning is
consistent with that of her reasoning for fractions of whole numbers during her Bits and
Pieces I lessons.
Ms. Archer’s rejection of Jeff’s alternative approach to 1/5 of 2/3, was in contrast
to compliments she paid Damien several weeks earlier when he devised a different
method for solving 11/12 of $240. A quick thought experiment based on these data
illustrates potential connections between teachers’ unit structures and using drawings
to adapt. Flexibility with just two levels of units might suffice to adapt in response to
Damien’s method but not to Jeff’s. Understanding that Jeff’s proposed diagram for 2/5
of 1/3 could also be interpreted as 1/5 of 2/3 would require sufficient flexibility with
three-level unit structures to see a length of 2/3 divided into ten equal pieces, the first
two of which were shaded. Recognizing that Jeff’s proposed diagram and Ms. Archer’s
could both represent 1/5 of 2/3 would require additional flexibility supported by a
visual understanding of the distributive property formally symbolized here as:
1/5 x 2/3 = 1/5 x 1/3 + 1/5 x 1/3.
Ms. Archer, however, reported earlier during the same interview that she had not
thought about the distributive property during the original lesson.
Finally, as a brief aside during the middle of the interview, Ms. Archer was asked
about “the value of showing [students] both the algorithm and the diagram.” She
explained, “I think the diagram showed to me, it showed them why one fifth times two
Mathematical Knowledge for Teaching Fraction Multiplication 49
third is two fifteenths, you know, and I just wanted to show them why.” In the next
exchange she continued:
[The number line] is just another way of doing things, you know, which I’ve
often told them there’s more than one way to do math problems. You know, if it,
that might help some of the students to understand things better. They may
wanna do it that way.
What Ms. Archer meant by “why” remained unclear at this point, but her tendency to
compute answers before using lengths to represent parts of parts, combined with her
difficulties producing three levels of units appropriately, suggested that she would
have trouble using drawings for purposes other than illustrating solutions first arrived
at through alternative numeric methods, the first use for drawings discussed above.
Data presented in the next section provided further evidence that her difficulties with
three-level unit structures precluded further uses for drawings.
Ms. Archer Uses Rectangular Regions to Explain Products of Proper Fractions
Ten days later Ms. Archer arrived at the CMP lessons intended to introduce
products of proper fractions. She continued telling students that the drawings “show
why” but initially blurred the distinction between part of a part and part of a whole
when interpreting problems as finding one fraction times another. These data suggested
that Ms. Archer continued to focus primarily on just two levels of units to reason about
fraction multiplication, and they continued to raise questions about what she meant
when saying that drawings “show why.”
Mathematical Knowledge for Teaching Fraction Multiplication 50
Blurring Part of a Part and Part of a Whole
The CMP materials use rectangular regions to introduce products of proper
fractions and begin with the example 1/2 of 2/3. Ms. Archer began her introduction
with just a few minutes left in the class period. She presented the problem as 1/2 times
2/3 and had students tell her to multiply the numerators together and the
denominators together. Having computed 2/6, she then showed her students two
transparencies, one showing a unit square partitioned into thirds with two parts shaded
and one showing a second unit square partitioned into halves with one part shaded.
She laid one unit square over the other as shown in Figure 8a, reproducing a drawing in
the teacher’s edition. She asked students how many parts were in the square and told
students to “look at the area that has the most lines in it.” This direction was similar to
one in the teacher’s edition to look at where “the shaded sections overlap.”
Figure 8 About Here
Ms. Archer returned to the same example the next day, reproduced the drawing
of two thirds from the previous lesson, and asked the class how to show two thirds
times one half. Several of the students’ struggled, and their attempts suggested that
they viewed both two thirds and one half as being of one whole. Ms. Archer stated once
more that the drawing in Figure 8a showed “why,” and told students to attend to “the
condensed area where the lines cross.” Jeff, the same student who asked about the
number line, asked what to do with the remaining three shaded parts. In her response,
Mathematical Knowledge for Teaching Fraction Multiplication 51
Ms. Archer began referring to the double shaded region as “half of two thirds,” but Jeff
remained confused.
A few moments later, Ms. Archer had students work on a series of problems in
the book that used squares to represent brownie pans and that asked for parts of parts.
As Ms. Archer worked with students at their desks, a few wanted to show 1/2 of 2/3 by
double shading one of their thirds (similar to Figure 8c). At first Ms. Archer redirected
these students towards drawings similar to the one shown in Figure 8a, but later she
demonstrated before the entire class two ways to show 1/2 of 2/3. The first solution
was based on cross partitioning but showed more clearly that the 1/2 was just of the
2/3, not the whole (see Figure 8b); the second solution double shaded one third (see
Figure 8c). Ms. Archer used hand gestures to show that the two “parts” from the first
method could be rearranged to coincide with “half” in the second method. Thus, for
this problem, she apparently adapted her perspective on parts of parts in response to
her students’ thinking and may have reasoned with three levels of units. For the balance
of the lesson, students used rectangular regions to determine 3/4 of 1/2. Classroom
data collection ended here because the school year was about to end, and Ms. Archer
moved on to fraction division.
Adaptive Representation of Parts of Parts Begins to Emerge
In conjunction with the member of the CoSTAR project who was interviewing
Ms. Archer, I planned a final interview to examine whether Ms. Archer might accept as
correct further ways that her students represented parts of parts and what she meant
Mathematical Knowledge for Teaching Fraction Multiplication 52
when she told students that drawings show “why.” The prompts included
reproductions of students’ class work for 3/4 of 1/2 shown in Figure 9 and excerpts
from the final round of student interviews. Ms. Archer’s responses suggested strongly
that when examining students’ work she focused primarily on whether the picture
showed part of a part and the correct numerical answer. I emphasize that Ms. Archer’s
use of the part-of-a-part language during her final interview may have been shaped by
her experiences helping students during the most recent lessons. Thus, I did not use
these data to infer understandings she may have used in prior lessons.
Figure 9 About Here
The interviewer asked Ms. Archer to review the set of examples in Figure 9 and
to indicate which ones students could use “to understand why the answer to three
fourths of one half is three eighths.” Ms. Archer indicated the first or second. She stated
that Angie’s method (Figure 9a) was what she had taught, but that Damien’s (Figure 9b)
was also possible. These data suggested that Ms. Archer now accepted as correct more
than one representation of parts of parts, but initially she rejected Theo’s solution
(Figure 9c):
That’s not right. Wait a minute. Four, eight, twelve, sixteen (counting
parts in the whole). That’s not right. Well, hold up. Three, sixteen. Two,
four, six, eight, sixteen. Four, eight, wait, four, eight, twelve, sixteen. Two,
Mathematical Knowledge for Teaching Fraction Multiplication 53
four, six (counting double shaded parts). Six, sixteen; three, eight. Okay.
All right. I see what he did.
After a few exchanges about the first two examples, Ms. Archer considered Theo’s work
once more, “One, two, three, four, five, six of sixteen and that’s three eighths. Yeah.”
Rather than analyzing relations among nested levels of units to see if Theo had
constructed 3/4 of 1/2, Ms. Archer apparently relied on reducing his numerical answer
to the one she knew.
For the balance of the interview, the interviewer replayed excerpts and showed
written work from the final round of student interviews. A primary goal for the student
interviews was to determine how they might extend Ms. Archer’s demonstrated
solution for 1/2 of 2/3 to other examples. Ms. Archer’s response to data from the
student interviews provided some evidence that she was beginning to attend more
flexibly to parts of parts and evidence for what she might mean by “why.”
During one interview, Damien and Theo solved 2/5 of 3/4 using brownie pans.
Figure 10a shows how Damien first divided the square into four parts. When he said
that he was not sure how to “divide into five,” Theo suggested drawing vertical lines.
Damien built on Theo’s suggestion, subdivided each fourth into five parts (Figure 10b),
and got stuck once more. Finally, Theo suggested taking two parts from each of the
three shaded fourths (Figure 10c) and explained that the answer would be 6/20 because
there are “four pieces divided up into five.” For me, the students’ drawings and
explanations evidenced their production of three levels of units and distributive
reasoning.
Mathematical Knowledge for Teaching Fraction Multiplication 54
Figure 10 About Here
The interviewer set up the clip by simply asking Ms. Archer to watch the
students work together on 2/5 of 3/4. At the point that Damien subdivided the fourths
into five parts (Figure 10b), Ms. Archer talked over the video and interjected, “Since
he’s done it that way, it’ll work, you know. He’s just gotta shade in two of each of those
fifths, which would give him, like, six twentieths, I think.” Although Ms. Archer also
evidenced distributive reasoning when anticipating Theo's suggestion, she still had
trouble describing the drawing appropriately. When the interviewer followed up by
asking, “Does that show why?,” Ms. Archer answered, “That is six twentieths of three
fourths. Yeah. That’s right.”
For the next example, the interviewer simply showed Ms. Archer a reproduction
of Emily’s work on the same problem. Emily partitioned the brownie pan vertically into
four parts and shaded three. She then repartitioned the three shaded fourths vertically
into five parts and double shaded two of the resulting parts (Figure 10d). Although
Emily’s construction showed part of a part correctly, her resulting drawing made it
difficult if not impossible to relate the double shaded region to the whole. When asked
if Emily’s approach was useful for determining “why it works,” Ms. Archer thought for
15 seconds before commenting:
Yeah. ‘Cause all she’s done is shown, what is that? One, two, I think, it’s
gonna be, what, three fifths? No, no, no, no. What is it? [Ms. Archer
Mathematical Knowledge for Teaching Fraction Multiplication 55
reduced 6/20 to 3/10 in the margin] …. Let’s see. How could she get three
tenths out of this? One, two. Well, yeah, I guess it could work because if
you count, it should be ten across when you get through counting and
then she would’ve done, this probably would be three if she put another
thing here. You know what I’m saying? And, that would be the same
concept.
That Ms. Archer sought to get a known numerical answer “out of” the drawing gave a
clearer picture than did previous data of what she meant by “why.” Her comments
focused on whether Emily might have arrived at the correct answer, not on whether
Emily’s drawing would allow her to relate multiple levels of nested units. These data
are consistent with using drawings to illustrate solutions also arrived at through an
alternate method, the first pedagogical use for drawings discussed above.
To summarize, Ms. Archer may have produced three-level structures in a limited
set of situations, but the data suggested strongly that, if she did, such structures were
not central to her mathematical knowledge for teaching fraction arithmetic. This, in
turn, precluded further uses of drawn representations discussed above. Ms. Archer’s
response to Damien’s method for determining 11/12 of $240 demonstrated that she
could adapt in response to a student’s reasoning in a case where reasoning primarily
with two levels of units was sufficient. Moreover, her focus on two levels of units, her
connection between a fraction of a number and times the number, and her connection
between multiplication and repeated addition were sufficient to use drawn
Mathematical Knowledge for Teaching Fraction Multiplication 56
representations for illustrating solutions to several fraction multiplication problems
from the Bits and Pieces I and Bits and Pieces II units. Although Ms. Archer became
somewhat more adaptive in response to her students’ reasoning with parts of parts at
the end of her enactment of Bits and Pieces II, data from both units provided strong
evidence that she did not produce appropriate three-level structures when responding
to her students’ reasoning across a range of situations.
Results on Ms. Reese from Bits and Pieces II
In analyzing data on Ms. Archer, a central challenge was determining whether
she reasoned with a third level of unit. The data on Ms. Reese’s unit structures were
clearer. Recall that she used a slightly different draft revision of the Bits and Pieces II unit
than did Ms. Archer. Only the version Ms. Reese used included lessons on adding and
subtracting fractions using the number line. Ms. Reese’s enactment of these lessons
provided several examples in which she used recursive partitioning, and hence
reasoned with three levels of units. vii Her multiplication lessons provided additional
examples in which she used recursive partitioning. Furthermore, her capacity to reason
with units was sufficient to infer a computation procedure based on answers to
numerous problems, the second pedagogical use for drawings discussed above. Yet, she
still had difficulty responding to some of the ways that her students produced nested
unit structures. Here the central challenge for analysis was determining whether Ms.
Reese attended to drawn versions of distribution. I proceed chronologically through her
six lessons on products of fractions.
Mathematical Knowledge for Teaching Fraction Multiplication 57
Representing Parts of Parts More Than One Way Using Rectangular Regions
In contrast to Ms. Archer, Ms. Reese introduced products of proper fractions as
the CMP materials intended—with brownie pans. She began with the example 1/2 of
2/3 and a drawing similar to that used by Ms. Archer. She introduced squares as
drawings of brownie pans, asked students how to draw two thirds of the brownie pan,
and followed a student’s suggestion to partition the square horizontally (Figure 11a).
She then asked how to represent 1/2 of 2/3, and one student said that half would be
one third because “to get two thirds you have to have one third plus one third.” Ms.
Reese accepted the solution immediately and said half of two thirds would be one of the
two shaded pieces (note the similarity to Figure 8c). She told students that the goal was
to figure out how to solve this problem using an algorithm and proceeded to divide the
square vertically (Figure 11b). Ms. Reese then asked students to “shade half” and after a
moment shaded half of the entire rectangle (Figure 11c). Like Ms Archer, Ms. Reese
reproduced the drawing in the teacher’s edition. She said that the double shaded part
represented half of two thirds and emphasized that there were no brownies in bottom
left hand corner. She then asked what the two pieces stood for as a fraction, and a
student said two sixths. Ms. Reese wrote “1/2 of 2/3 is 2/6,” asked if the two solutions
were the same, and called on students until one said that 2/6 could be reduced to 1/3.
Ms. Reese finished the lesson with a second brownie pan problem that demonstrated
3/4 of 1/2 is 3/8.
Figure 11 About Here
Mathematical Knowledge for Teaching Fraction Multiplication 58
Struggling to Respond to Students’ Approaches
Although Ms. Reese accepted different student approaches for representing 1/2
of 2/3, she had trouble responding to students’ reasoning the very next day. The central
issue was creating the first partition of the brownie pan. Ms. Reese reviewed the
previous lesson and had students work at their desks on the following problems: 1/3 of
5/6, 2/3 of 3/4, 1/4 of 1/3, and 1/4 of 2/5. At one point, she worked with a student
whose representation for 1/3 of 5/6 was similar to that shown in Figure 12c. The
student explained that the answer was 2/6 because the whole was broken into three
pairs of smaller rectangles. Even though she arrived at an incorrect answer, the student
may have reasoned with three levels of units if she took the whole pan as a unit that
was divided into thirds, each of which was further divided in half, making sixths.
Figure 12 About Here
Ms. Reese responded, “Something is going on here that is not right. I have got to
think about this a minute.” She questioned whether the student had shaded 5/6 and
counted the shaded pieces to check. That Ms. Reese counted the five shaded pieces
suggested that she was genuinely stuck. The student continued to explain when Ms.
Reese interrupted, “Oh. I gotcha. OK. So you just didn’t divide it up again. You just put
like this is one third (pointed to two pieces), that’s one third (pointed to another two
Mathematical Knowledge for Teaching Fraction Multiplication 59
pieces), and that’s one third (pointed to the last two pieces). OK. That’s fine.”
Apparently, Ms. Reese attended to three levels of units, even though she did not
address the problem with the student’s work.
A few moments later, Ms. Reese observed that another student had a similar
incorrect solution and stopped the class so that she could compare two strategies for
determining 1/3 of 5/6. The following discussion suggested that she may have figured
out, at least in part, what was wrong. Ms. Reese began by telling the class that it was
alright to have different answers so long as they were equivalent fractions. She
demonstrated her solution first, pointing out that she drew lines in just one direction to
partition the brownie pan into sixths (Figure 12a). She shaded five pieces, said she
wanted to take a third of just those pieces, and apparently attempted to produce three
levels of units when she continued:
I am thinking out loud. I am acting like a kid. I was thinking at first when
I saw this, well, maybe two of these could make up one of the thirds
where that would be like one third (pointed to the top two pieces), that
would be two thirds (pointed to the next two pieces) but Oh Oh (pointed
to the fifth shaded piece) there’s only one left. So that’s not going to work.
Ms. Reese used this impasse as motivation for partitioning her brownie pan vertically
into thirds (Figure 12b). She shaded the first third of the whole, and reminded students
Mathematical Knowledge for Teaching Fraction Multiplication 60
that the bottom piece was not included. She counted five double shaded pieces and,
with some prompting about the size of the pieces, students said the answer was 5/18.
Ms. Reese then recounted the alternative solution by partitioning a second
brownie pan as shown in Figure 12c and told the class that she remembered reading in
the teacher’s guide that “it wont always work if you divide like this,” meaning cross
partitioning for the first fraction.viii Although comparing the two approaches provided
an opportunity to talk about taking a third of each sixth, and hence a drawn version of
distribution, Ms. Reese did not do so. Instead, she had students examine the class
solutions thus far: 1/2 of 2/3 is 2/6, 3/4 of 1/2 is 3/8, and 1/3 of 5/6 is 5/18. After
students offered a tentative pattern based on multiplying numerators together and
denominators together, Ms. Reese asked if the second proposed solution to 1/3 of 5/6
(Figure 12c) fit the pattern. Some students commented that 2/6 was equivalent to 6/18.
Ms. Reese pointed out that the two methods gave truly different answers but did not
state which one was correct. (Ms. Reese reported working problems in her lessons
ahead of time and probably knew the correct answer.) This discussion provided initial
evidence of Ms. Reese’s use of drawings to infer a computation method, the second use
described above.
As Ms. Reese had students continue to work with their partners on the
remaining problems, one discussion about drawing 2/3 of 3/4 suggested constraints on
her attention to parts of parts. Jack and Kate had similar work (Figure 13a and 13b).
Kate explained that she had divided her brownie pan into fourths. To me it appeared
that Kate had continued by shading two of the three fourths, correctly, but Ms. Reese
Mathematical Knowledge for Teaching Fraction Multiplication 61
commented that Kate was going to run into the same problem illustrated in Figure 12c
and had her start over. Jack partitioned a new brownie pan into fourths all in one
direction. As he did so, Ms. Reese commented, “OK. We need to take two thirds of that.
You might be able to answer it right there like that. I’m not sure.” When Jack shaded
two of the three fourths (Figure 13c), Ms. Reese said, “That will work.”
Figure 13 About Here
Ms. Reese may have overlooked Kate’s correct solution to 2/3 of 3/4 because
she was genuinely stuck when two students found that 1/3 of 5/6 was 2/6 and because
the teacher’s edition warned against cross partitioning for the first fraction. The data
evidenced Ms. Reese’s capacity to reason with three level of units in some situations but
left unclear whether she attended to drawn versions of distribution. The third day, Ms.
Reese and the students used area models to solve several more problems by
partitioning horizontally for one fraction, partitioning vertically for the second, and
determining how many parts were in the whole and in the part of a part. The day after
that, she introduced linear representations for fraction multiplication. I discuss those
data next because the final interview with Ms. Reese examined the area examples just
discussed and a linear example discussed below.
Mathematical Knowledge for Teaching Fraction Multiplication 62
Products of Fractions Using Thermometers
Recall that Ms. Archer skipped the Bits and Pieces II lessons that used
thermometers to develop fraction multiplication because she was running out of days in
the school year. In contrast, Ms. Reese used thermometers to find several examples of
parts of parts and to infer a computation method. In so doing, she used recursive
partitioning, and thus produced three levels of units. Furthermore, she clearly attended
to drawn instantiations of the distributive property, even though she did not discuss the
property explicitly with her students. These lesson data raised questions about Ms.
Reese’s attention to drawn versions of distribution.
Recursive Partitioning and Using Drawn Representations to Infer a Computation Method
As she had done with rectangular regions, Ms. Reese had students use lengths to
determine parts of parts. When students came upon apparent counterexamples to the
emerging numerical pattern based on products of numerators and products of
denominators, she had them consider equivalent fractions. The first problem Ms. Reese
posed was couched in a context where students had raised two thirds of their $720
fundraising goal in four days. Ms. Reese sketched a fundraising thermometer and
shaded two thirds. She asked students for the fraction that had been raised each day.
One student suggested dividing the shaded two thirds into four parts, and Ms. Reese
did so (Figure 14a). Another student said the answer was one sixth because the
unshaded part should be divided into two pieces. Ms. Reese partitioned the unshaded
part and wrote “1/4 of 2/3 = 1/6” underneath. Although this answer appeared
Mathematical Knowledge for Teaching Fraction Multiplication 63
inconsistent with the emerging pattern, she solved the problem a second time using
rectangular areas to show that 1/4 of 2/3 was also 2/12 (Figure 14b). Students then
explained that 1/6 and 2/12 were equivalent fractions.
Figure 14 About Here
For the second and third problems, Ms. Reese led the class through a similar
sequence of steps, and the questions she posed suggested attention to recursive
partitioning. For the second problem, 1/3 of 1/2, she partitioned one half into three
parts and had a student explain that the second half should also be divided into three
parts to see that 1/3 of 1/2 is 1/6. For the third problem, 2/5 of 1/2, she partitioned one
half into five parts, shaded two, and had a student explain that the second half should
also be divided into five parts to see that 2/5 of 1/2 is 2/10.
Attention to Drawn Instantiations of the Distributive Property
During her fourth lesson, Ms. Reese’s comments evidenced attention to the
distributive property in the context of finding a fraction of a mixed number. The
problem was to determine 1/3 of 2 1/2 pounds of cheese. Ms. Reese drew three
thermometers, shaded two and half, and divided each piece into thirds (Figure 15a). A
student suggested dividing everything into sixths, which Ms. Reese did. Ms. Reese then
commented that she needed “to get a third of each thing.” She shaded a third of the half
and, as a rhetorical move to invite students’ contributions, said that she was not sure
Mathematical Knowledge for Teaching Fraction Multiplication 64
how to get a third of the wholes. A student suggested shading two pieces, and Ms.
Reese shaded two pieces of each whole (Figure 15b). Ms. Reese counted five double
shaded pieces, and students said the pieces were sixths.
Figure 15 About Here
Ms. Reese’s instruction was interrupted by state testing. Afterwards, several
lessons relied largely on numeric methods to develop products of mixed numbers. The
one place Ms. Reese used drawings was to help students convert mixed numbers to
improper fractions. Two weeks after the cheese example, when reviewing for test, Ms.
Reese used lengths and rectangular areas for determining products of fractions once
more. During the review, she may still not have attended to distribution in the context
of rectangular areas: She stopped a student who began working on 1/3 of 5/6 by cross
partitioning a brownie pan into sixths and said, “Remember we had a long discussion
in class about why you cannot cut them this way and this way because your answer
might be close to the right answer but it is not going to be the right answer.” Classroom
data collection ended here because the school year was about to end, and Ms. Reese
moved on to fraction division.
Ms. Reese Explains Her Lessons and the Distributive Property During Interviews
When analyzing lesson and interview data, I pursued two hypotheses for why
Ms. Reese directed students away from cross partitioning brownie pans for just one
Mathematical Knowledge for Teaching Fraction Multiplication 65
fraction even in cases where students appeared, from my perspective, to progress
towards correct answers. The first was that the incorrect 1/3 of 5/6 is 2/6 solution,
combined with comments in the teacher’s edition, were sufficient for Ms. Reese to
understand that partitioning horizontally for one fraction and vertically for the second
was more reliable, even if she was still figuring out why. The second was that Ms. Reese
understood that the distributive property could be used to make either approach to
partitioning work but did not want to have that discussion with her students. That Ms.
Reese demonstrated a drawn instantiation of the distributive property in the
subsequent cheese problem context, combined with comments that she wanted to direct
students toward multiplying numerators and multiplying denominators with a
minimum of confusion raised the possibility that she had, but chose not to use, relevant
knowledge.
Data from the final interview did not allow clear discrimination between these
hypotheses but did provide evidence for three claims. First, Ms. Reese was less
confident about drawing versions of the distributive property in some contexts than
others. The interviewer first asked about two examples from lessons in which students
used numeric methods incorrectly when multiplying mixed numbers. In both cases, the
students only multiplied the whole numbers together and the fractions together (e.g., 3
1/3 x 4 2/3 = 12 2/9). Ms. Reese recognized the mistake as one that several students in
each of her classes had made and at one point asked, “So how could we represent that
with pictures to come up with a …?” These data demonstrated constraints on Ms.
Reese’s understandings of drawn versions of distribution, at least in cases of the (a + b)(c
Mathematical Knowledge for Teaching Fraction Multiplication 66
+ d) pattern. The interviewer reminded Ms. Reese that they had discussed how to use
rectangular regions for this purpose earlier in the year during a professional
development session. (Recall that the CoSTAR project conducted professional
development for teachers at Pierce Middle School).
After a discussion about what sixth-grade students might be able to understand
about the distributive property, the interview brought up the cheese problem discussed
above and asked, “What is one thing you might hope that [students] do?” Ms. Reese’s
answer was consistent with explanations she gave during the lesson summarized above
and appeared to reflect a much clearer understanding about drawn versions of the
distributive property in this context:
I would hope that they would see that, first of all, if you’ve got two
wholes and a half more, that they, a third of that amount total, so you
could make it easy and do a third of the half and figure that out, and then
you could do a third of the two wholes and figure that out and combine
those two.
Second, by the time of the interview, Ms. Reese could describe the role of
distribution in the 1/3 of 5/6 example but expressed reservations about discussing this
with students. The interviewer set up the lesson excerpt described above by simply
saying, “You put four problems: 1/3 of 5/6, 2/3 of 3/4, 1/4 of 1/3 and 1/4 of 2/5 and
then you start moving around the room.” He then showed the excerpt in which a
student used brownie pans to determine that 1/3 of 5/6 is 2/6. Ms. Reese first recalled,
“We talked about why it gets, you get close to the answer but its not exactly the right
Mathematical Knowledge for Teaching Fraction Multiplication 67
answer because you’re not dividing the pieces in the right number of pieces.” An
exchange later she elaborated: “So she’s not actually taking a third of each piece of the
five sixths.” Ms. Reese restated this explanation at a few other points during the same
interview. Although she understood why the student’s approach did not work by the
time of her last interview, Ms. Reese also commented, “I wasn’t really sure what she
was thinking. I was thinking well this isn’t 5/6, well yes it is 5/6. Well why, you know,
well she’s taking two pieces of the whole thing but not of the 5/6.” This comment lent
some weight to the hypothesis that during the initial lesson Ms. Reese did not use her
understanding of distribution to diagnose the student’s difficulty.
Third, Ms. Reese appeared still to be developing flexibility with three levels of
units when the interview replayed the lesson excerpt in which Jack and Kate worked on
2/3 of 3/4. The interviewer asked, “Would it have been okay if they had started with
3/4 doing horizontally and vertically?” Ms. Reese looked at Jack’s initial work
reproduced in Figure 13a and stated, “As long as they had made each piece into thirds
and shaded two of each piece. But see, he just shaded all of that.” Ms. Reese described
taking two pieces from each fourth several other times. When the interviewer asked if
one needs to divide each fourth into three pieces, Ms. Reese first said “Yes,” then asked
for clarification of the question, thought for a moment, and finally pointed out that Jack
could have taken just two of the three shaded pieces. These data suggested that Ms.
Reese’s ability to produce three levels of units, though sufficient to infer numeric
methods, was not sufficiently flexible to respond quickly and securely to the variety of
ways that her students might assemble such structures during lessons.
Mathematical Knowledge for Teaching Fraction Multiplication 68
To summarize, in contrast to Ms. Archer, Ms. Reese apparently employed threelevel unit structures effectively at several points during her lessons, for instance when
motivating the need to partition rectangular regions in separate directions for each
fraction (Figure 12) and when recursively partitioning linear representations. Thus,
reasoning with three-level unit structures appeared to be part of her mathematical
knowledge for teaching fraction multiplication with drawings. At the same time, the
knowledge remained tacit and was used only in some contexts. Ms. Reese’s use of
three-level structures was sufficient to use drawings for inferring a general numeric
method for multiplying proper fractions with proper fractions and proper fractions
with improper fractions. Although she attended to drawn versions of distribution when
reasoning with linear representations, attending to drawn versions of distribution also
when reasoning with area representations would have supported greater flexibility
with three-level unit structures. Such flexibility could have supported, in turn, fuller
adaptation in response to the range of ways her students began to assemble such
structures.
Discussion
Analyses of data on Ms. Archer and Ms. Reese led to a frame for describing
mathematical knowledge for teaching in one domain, fraction multiplication. The frame
was informed by a perspective on knowledge that emphasizes diverse, fine-grained
knowledge elements. The three categories—numerical aspects of multiplication, unit
Mathematical Knowledge for Teaching Fraction Multiplication 69
structures, and pedagogical purposes for using drawings—highlight the diversity.
Comparing the range of structures identified in the present study to a single construct
such as the “meaning of multiplication with fractions” used by Ma (1999, p. 77)
illustrates the finer grain size. I emphasize that describing such diverse, fine-grained
knowledge elements was motivated primarily by the desire to explain why the two
teachers adapted in response to their students’ thinking more readily in some situations
than others.
The frame emphasizes relations between teachers’ unit structures and purposes
for using drawings. That Ms. Archer attended primarily to two levels of units explained
why she could adapt in response to Damien’s strategy for determining 11/12 of $240
but not to Jeff’s questions about fraction multiplication on the number line.
Furthermore, when enacting the Bits and Pieces I and Bits and Pieces II units, her primary
attention to just two levels of units constrained the pedagogical purposes for which she
might use drawings to illustrating numeric answers. Ms. Reese provided clearer
evidence of reasoning with three levels of units, and she could apparently use lengths
and areas to determine answers without using numeric computations as a support.
Although her facility with three-level unit structures was sufficient to infer a numeric
method for computing products of fractions, she was not sufficiently flexible to adapt in
response to some examples of her students’ reasoning. Thus, her capacity to reason with
fractional quantities also placed constraints on the pedagogical purposes for which she
might use drawings, including building general numeric methods from students’
strategies as called for by the CMP materials.
Mathematical Knowledge for Teaching Fraction Multiplication 70
Taken together, the two case studies suggest that teachers’ capacity to produce
three levels of units is necessary but not sufficient for interpreting and evaluating the
variety of ways that students might begin to evidence multi-level structures when using
drawings to reason about fraction multiplication situations. Attention to drawn
versions of distribution is also essential. This result, combined with that reported by
Behr, Khoury, et al. (1997) in which preservice elementary teachers found it hard to
distribute fractions as operators across their conceptual units (see discussion above),
raises questions for future research about the extent to which teachers recognize the
distributive property when it is not represented symbolically as a(b + d) = ab + ad. The
ability to recognize distribution when represented in different ways seems an important
aspect of mathematical knowledge for teaching whole-number and fraction arithmetic
with drawings and manipulatives. Further research should continue examining the
extent to which teachers recognize distribution when represented in different ways.
Explicit, flexible attention to three levels of units may be central mathematical
knowledge for teaching further topics where fractions are treated as quantities. For
instance, the examples in which Ms. Archer struggled to compare 2/3 and 3/4 (Figure
6) and to recognize Theo’s strategy for representing 3/4 of 1/2 (Figure 9c) illustrate how
attending to only two levels of units can hamper using drawings to generate equivalent
fractions. The capacity to generate equivalent fractions, in turn, is central to comparing,
adding, and subtracting pairs of fractions with unlike denominators. Again, Izsák et al.
(in press) analyze the role of recursive partitioning, and hence three levels of units, in
Ms. Reese’s fraction addition and subtraction lessons.
Mathematical Knowledge for Teaching Fraction Multiplication 71
Explicit, flexible attention to three levels of units can also support using length or
area quantities to determine answers to fraction division problems. Here, flexible
reasoning with nested levels of units requires not only attention to drawn versions of
distribution but also the capacity to make different choices of unit for each level as a
solution unfolds. To see this, consider again the problem used by Ball (1990) and Ma
(1999) discussed above. To generate a situation that illustrates 1 3/4 ÷ 1/2, one might
ask how many halves are in 1 3/4 and generate an initial unit structure in which an
interval of length 1 3/4 is the largest unit, the interval from 0 to 1 is the mid-level unit,
and intervals of length 1/2 are the smallest unit. Because the intervals of length 1/2 do
not partition the length of 1 3/4 evenly, answering how many halves are in 1 3/4
requires a new unit structure in which a length of 1 3/4 is again the largest unit, but
intervals of length 1/2 are now the mid-level unit and intervals of length 1/4 the
smallest unit. At this stage, the interval from 0 to 1 moves to the background and is
important primarily for describing sizes of pieces, not multiplicative relations among
the levels used directly to answer the question. Answering how many halves are in 1
3/4 requires recognizing that the smallest interval is half of mid-level interval and so
there are 3 1/2. Sowder et al. (1998) reported a group of middle-grades teachers who
had trouble making such choices for units when using drawings to reason through the
problem 1 ÷ 3/5 (pp. 56-59). Future research on teachers’ mathematical knowledge for
teaching whole and rational number arithmetic, particularly in contexts where numbers
are treated as measures of quantities, should examine the extent to which other
Mathematical Knowledge for Teaching Fraction Multiplication 72
populations of teachers also have trouble reasoning explicitly and flexibly with nested
levels of units.
Results of the present study extend past research on teachers’ mathematical
knowledge not only by examining how they employ unit structures when responding
to students over sequences of lessons, but also by teasing apart different pedagogical
purposes for which teachers might use drawings. That little is known about teachers’
pedagogical purposes for using drawings is a significant omission in research on
teacher knowledge, especially given the increasing use of reform-oriented materials in
classrooms. Results from this study raise the possibility that curriculum developers,
teachers, and researchers may not be communicating clearly about the role drawn
representations can play in reform-oriented mathematics instruction. Recall that the
CMP materials suggested three of the four purposes for using drawings discussed as
part of the theoretical frame. Meanwhile, comments made by Ms. Archer, Ms. Reese,
and other teachers with whom the CoSTAR project has worked suggest that they
understand the purpose of multiple representations and strategies to be providing
alternatives from which each student can understand and use just one. This is different
from having students work with multiple representations to develop an integrated
knowledge base, for instance by using drawings to develop meaning for operations
with formal symbolic notations. The notions of illustrating, inferring, deducing, and
adapting may serve as useful tools for analyzing the design of reform-oriented
materials and for future research on teaching and learning with such materials.
Mathematical Knowledge for Teaching Fraction Multiplication 73
Finally, methods used in the present study may be of interest to those studying
teachers’ knowledge of other topics in mathematics. The methods do not replace those
used in other studies of teachers’ knowledge, but they do respond differently to some of
the challenges studying mathematical knowledge that teachers use in practice. Some of
these challenges are that (a) classroom observations alone can make it hard to
discriminate between competing hypotheses about what knowledge a teacher may be
using, (b) teachers’ self-reports of what they and their students think and do during
lessons may be unreliable, and (c) teachers may use knowledge when responding to
items on survey instruments designed to measure teachers’ knowledge, but not use that
knowledge when responding to their students’ thinking during lessons. The methods
used in the present study start with classroom events, particularly those where teachers
appear to have trouble responding to their students’ thinking, and then try to gather
further evidence for the knowledge that teachers may be using. Interviews provide
additional data on students’ reasoning both about the original task discussed in class
and about extensions of those tasks. Further interviews provide data on teachers’
reasoning about the original task discussed in class, about extensions of those tasks, and
about their students’ thinking. Examples of extension tasks used in the present study
include asking Ms. Archer to solve 3/5 of 2/3 with the number line and asking students
to determine 2/5 of 3/4 with rectangular regions. Combining lesson and interview data
in this way can help discriminate between competing hypotheses about teachers’
knowledge with greater confidence, determine to what extent teachers’ self-reports
appear consistent with classroom events (from researchers’ perspectives), and lead to
Mathematical Knowledge for Teaching Fraction Multiplication 74
insights about teacher cognition that can help us better understand explicit and tacit
mathematical knowledge that teachers’ use when responding to their students’ thinking
during instruction.
Mathematical Knowledge for Teaching Fraction Multiplication 75
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Mathematical Knowledge for Teaching Fraction Multiplication 81
Footnotes
1. Throughout his work, Steffe explains cognition and learning using theoretical
constructs based on the work of Piaget. He describes cognitive structures in terms of
schemes and operations and learning mechanisms that include interiorization,
assimilation, and accommodation.
2. All names are pseudonyms.
3. A graduate student conducted three interviews in my presence with one of the
student pairs during the first round of data collection and two more interviews in my
presence with the same pair of students during the second round.
4. Prime Time is a CMP unit that develops elementary number theory concepts
including factors, products, prime numbers, and composite numbers.
5. These students were the ones the graduate student interviewed in my
presence.
6. Jeff replaced Emily’s first interview partner, when Angie elected to stop being
interviewed.
7. Izsák et al. (in press) analyzed these lessons.
8. The Bits and Pieces II Teacher’s Guide does state that cross partitioning “does not
lead to the kind of partitioning that suggests multiplication of numerators and
denominators” (p. 8, Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006).
Mathematical Knowledge for Teaching Fraction Multiplication 82
Figure Captions
Figure 1. Four ways to draw 2/3 of 3/4.
Figure 2. (a) The number 5 understood with two levels of units. (b) The number 20
understood with three levels of units.
Figure 3. Determining 1/4 of 1/3. (a) Constructing part of a part. (b) Using iteration and
two levels of units. (c) Using recursive partitioning and three levels of units.
Figure 4. Determining 3/4 of 8. (a) Using the duplicator and partition-reducer
subconstruct (b) Using the stretcher and shrinker subconstruct.
Figure 5. Using a drawing to determine a general numeric method for computing
products of proper fractions.
Figure 6. Ms. Archer’s white board when comparing 3/4 and 2/3.
Figure 7. The number line showing 1/5 of 2/3 that Ms. Archer copied from the
prepublication version of Bits and Pieces II Teacher's Guide (Lappan, Fey, Fitzgerald,
Friel, & Phillips, 2006).
Figure 8. Ms. Archer uses brownie pans to show 1/2 of 2/3. (a) Her first drawing. (b)
Her second drawing. (c) Her third drawing.
Figure 9. Reproductions of students' drawings for 3/4 of 1/2. (a) Angie. (b) Damien. (c)
Theo.
Figure 10. Students’ drawn solutions to 2/5 of 3/4. (a) Damien divides the pan into four
and shades three parts (reproduction). (b) Damien divides each fourth into five parts
(reproduction). (c) Theo shades two parts in each shaded fourth (students’ original
work). (d) Emily creates all partitions vertically (reproduction).
Mathematical Knowledge for Teaching Fraction Multiplication 83
Figure 11. Ms. Reese determines 1/2 of 2/3. (a) She draws 2/3 of a brownie pan. (b) She
divides the square vertically. (c) She determines 1/2 of 2/3 is 2/6.
Figure 12. Ms. Reese demonstrates two approaches to 1/3 of 5/6. (a) She partitions
horizontally. (b) She partitions vertically. (c) She presents two students’ approach.
Figure 13. Three drawings of 2/3 of 3/4. (a) Jack’s initial work (reproduction). (b) Kate’s
initial work (reproduction). (c) Jack’s revised work (reproduction).
Figure 14. Ms. Reese determines 1/4 of 2/3 two ways. (a) She draws a thermometer and
divides the shaded 2/3 into four parts. (b) She solves the problem a second time using
rectangles.
Figure 15. Ms. Reese shows 1/3 of 2 1/2 is 5/6 (a) She draws 1/3 of 2 1/2 pounds of
cheese. (b) She divides each whole into sixths and shades five pieces.
Mathematical Knowledge for Teaching Fraction Multiplication 84
Figure 1
TOP
Mathematical Knowledge for Teaching Fraction Multiplication 85
Figure 2
TOP
(a)
(b)
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Figure 3
TOP
(a)
(b)
(c)
Mathematical Knowledge for Teaching Fraction Multiplication 87
Figure 4
TOP
Duplicate
Reduce
(a)
Stretch
Shrink
(b)
Mathematical Knowledge for Teaching Fraction Multiplication 88
Figure 5
TOP
d
b
a
b
c
a b ab
of 
c d cd

Mathematical Knowledge for Teaching Fraction Multiplication 89
Figure 6
TOP
Mathematical Knowledge for Teaching Fraction Multiplication 90
Figure 7
TOP
Mathematical Knowledge for Teaching Fraction Multiplication 91
Figure 8
TOP
(a)
(b)
(c)
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Figure 9
TOP
(a)
(b)
(c)
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Figure 10
TOP
(a)
(b)
(c)
(d)
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Mathematical Knowledge for Teaching Fraction Multiplication 95
Figure 11
TOP
(a)
(b)
(c)
Mathematical Knowledge for Teaching Fraction Multiplication 96
Figure 12
TOP
(a)
(b)
Mathematical Knowledge for Teaching Fraction Multiplication 97
(c)
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Figure 13
TOP
(a)
(b)
(c)
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Figure 14
TOP
(a)
(b)
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Figure 15
TOP
(a)
(b)
Mathematical Knowledge for Teaching Fraction Multiplication 101
i
Throughout his work, Steffe explains cognition and learning using theoretical
constructs based on the work of Piaget. He describes cognitive structures in terms of
schemes and operations and learning mechanisms that include interiorization,
assimilation, and accommodation.
ii
All names are pseudonyms.
iii
A graduate student conducted three interviews in my presence with one of the
student pairs during the first round of data collection and two more interviews in my
presence with the same pair of students during the second round.
Prime Time is a CMP unit that develops elementary number theory constructs
iv
including prime numbers, composite numbers, factors, and products.
v
These students were the ones the graduate student interviewed in my presence.
vi
Jeff replaced Emily’s first interview partner, when Angie elected to stop being
interviewed.
vii
Izsák et al. (in press) analyzed these lessons.
viii
The Bits and Pieces II Teacher’s Guide does state that cross partitioning “does not lead
to the kind of partitioning that suggests multiplication of numerators and
denominators” (p. 8, Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006).