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Prospective mathematics teachers sense making
of polynomial multiplication and factorization modeledwith algebra tiles
Gu nhan Caglayan
Published online: 5 February 2013 Springer Science+Business Media Dordrecht 2013
Abstract This study is about prospective secondary mathematics teachers understanding
and sense making of representational quantities generated by algebra tiles, the quantitative
units (linear vs. areal) inherent in the nature of these quantities, and the quantitative
addition and multiplication operationsreferent preserving versus referent transforming
compositionsacting on these quantities. Although multiplicative structures can be
modeled by additive structures, they have their own characteristics inherent in their nature.
I situate my analysis within a framework of unit coordination with different levels of unitssupported by a theory of quantitative reasoning and theorems-in-action. Data consist of
videotaped qualitative interviews during which prospective mathematics teachers were
asked problems on multiplication and factorization of polynomial expressions inx and y. I
generated a thematic analysis by undertaking a retrospective analysis, using constant
comparison methodology. There was a pattern which showed itself in all my findings. Two
studentteachers constantly relied on an additive interpretation of the context, whereas
three others were able to distinguish between and when to rely on an additive or a mul-
tiplicative interpretation of the context. My results indicate that the identification and
coordination of the representational quantities and their units at different categories
(multiplicative, additive, pseudo-multiplicative) are critical aspects of quantitative rea-soning and need to be emphasized in the teachinglearning process. Moreover, represen-
tational Cartesian products-in-action at two different levels, indicators of multiplicative
thinking, were available to two research participants only.
Keywords Additive reasoning Algebra tiles Cartesian product Concept-in-actionMapping structure Models and modeling Multiplicative reasoning Polynomialrectangle Prospective teacher education Quantitative reasoning RelationRepresentation Bijections
G. Caglayan (&)Department of Mathematics and Statistics/Department of Teaching and Learning, Florida InternationalUniversity, 11200 8th Street, Miami, FL 33199, USAe-mail: gcaglaya@fiu.edu
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J Math Teacher Educ (2013) 16:349378DOI 10.1007/s10857-013-9237-4
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Background
Manipulatives
Physical objects, also often referred to as manipulatives or instructional devices, can serveas essential representational models in the course of experiential learning. The Principles
and Standards for School Mathematics document (National Council of Teachers of
Mathematics [NCTM] 2000) has consistently emphasized the use of physical objects as
representational tools. Research has shown that the use of physical objects can be an
obstacle to mathematical progress in some cases (Howden 1986; Puchner et al. 2008).
Research by Suydam and Higgins (1977) and Aburime (2007), on the other hand, showed
that students mathematics achievement increased through the use of mathematics ma-
nipulatives. Work by Sowell (1989) indicated that even though for K-16 students, ma-
nipulatives were effective ways of modeling and understanding mathematics, the teachers
were not appreciative of their usage.As for the teachers, on the other hand, inexperienced ones favored their usage more
often than experienced teachers (Gilbert and Bush 1988). Moyer and Jones (2004) found
that the teachers with prior experience with manipulatives were the ones utilizing them
more in the instruction. In her study on middle grades teachers use of manipulatives for
teaching mathematics, Moyer (2001) found that using manipulatives was simply a recre-
ational activity where teachers had difficulty in explaining and representing the mathe-
matical topics themselves. She goes on to state Manipulatives are externally generated as
manufacturers representations of mathematical ideas; therefore, meaning attached to the
manipulatives by manufacturers is not necessarily transparent to teachers and students(p. 192). Prospective and practicing teachers often believe that manipulatives have an edu-
cational significance inherent in their manufacture. Having already experienced and made
sense of these instructional devices for a long time, the connection between the abstract
representation and the concrete representation becomes too transparent to them (Cobb et al.
1992; Meira 1998). Roschelle (1990) postulated that the transparency level of an
instructional device typically draws on the level of epistemic fidelity of the device. Meira
(1998) suggests epistemic fidelity as an obvious characteristic of an instructional device.
Uttal, Scudder, and DeLoache (1997) state that part of the difficulty that children encounter
when using manipulatives stems from the need to interpret the manipulative as a representation
of something else (p. 38). A reference to any kind of physical object brings with itself thenecessity to think about the object under consideration as some sort of quantity possessing a
referent, a value, and a measurement unit (Schwartz1988; Thompson1993,1995). Attending
to the quantitative nature of manipulatives may be an asset for students success in relating the
manipulatives to their written symbolic referents. The physical object itself cannot be a rep-
resentation of a written symbol without meanings projected into these concrete objects (Ball
1992; Clements1999). A successful mapping of the concrete to the abstract depends on
the manipulative itself and a family of meanings attached to these objects.
Multiplicative structures
Conceptual field theory (Vergnaud 1983, 1988, 1994) aims to present the complexity
inherent in the nature of simple tasks on additive and multiplicative reasoning. Research
indicates that the multiplicative conceptual field is very complex and has many concepts of
mathematics in its structure, other than multiplication itself (Behr et al. 1992; Harel and
Behr1989; Harel et al.1992). Additive reasoning develops quite naturally and intuitively
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through encounters with many situations that are primarily additive in nature (Sowder
et al.1998a, p. 128). Building up multiplicative reasoning skills, on the other hand, is not
obvious; schooling and teacher guidance are essential to acquire a profound understanding
and familiarization with multiplicative situations (Hiebert and Behr 1988; Resnick and
Singer1993).The study of multiplicative structures has been conducted by mathematics education
researchers since the 1980s. Behr et al. (1994) developed two representational systems
extremely generalized and abstractin an attempt to transcribe students additive and
multiplicative structures in which the notion units of a quantity plays the main role.
Confrey (1994) provides splitting, an action of creating simultaneously multiple versions
of an original (p. 292), as an explanatory model for childrens construction of multipli-
cative structures. Vergnaud (1988) claims that a single concept does not refer to only one
type of situation, and a single situation cannot be analyzed with only one concept
(p. 141). He argues that teachers and researchers should study conceptual fields rather than
isolated concepts. Vergnauds (1994) conceptual field theory asserts:One needs mathematics to characterize with minimum ambiguity the knowledge con-
tained in ordinary mathematical competences. The fact that this knowledge is intuitive and
widely implicit must not hide the fact that we need mathematical concepts and theorems to
analyze it (p. 44).
According to Vergnaud (1988),theorems-in-actionare mathematical relationships that
are taken into account by students when they choose an operation or a sequence of
operations to solve a problem (p. 144). He goes on to state To study childrens math-
ematical behavior it is necessary to express the theorems-in-action in mathematical terms
(p. 144). Concepts-in-action serve to categorize and select information, whereas theorems-in-action serve to infer appropriate goals and rules from the available and relevant
information (Vergnaud1997).
Previous research studies indicated that the use of algebra tiles positively impacted
students attitudes (Sharp1995). There was no difference between the two groups (those
who used algebra tiles vs. those who did not) based on test scores; however, written
comments of the majority of students indicated that the algebra tiles helped them learn the
material easily and meaningfully by providing useful visual aid (Sharp 1995). In another
study, Algebra I students that are taught using the traditional techniques outperformed
those that used Algeblocks (McClung 1998). Vinogradova discussed the use of algebra
tiles in the teaching of quadratic functions and in particular, the process of completing thesquare (2007). Johnson (1993) reported that both teachers and students understood poly-
nomial multiplication better by using algebra tiles.
Representations of algebraic expressions as areas of rectangles as a sum and as
a product have been investigated by various mathematics educators and researchers
(Huntington1994; Sharp1995; Takahashi2002). Modeling expressions such as 2x ? y ? 3
by using color tiles may not be as obvious. In the example of 2x ? y ? 3, the term 2xis a
collection of two units of x (two purple bars with the model), the term y is 1 unit of y
(1 blue bar with the model), and the term 3 is a collection of three units of 1 (three little
black squares with the model). Therefore, the expression 2x ? y ?
3 is a collection of theindividual irreducible representational units. One not only has to individually identify each
representational unit (one purple bar for the x, one blue bar for the y, and one little black
square for the 1), but also one has to reconcile a collection of these irreducible repre-
sentational units in order to demonstrate that 2x ? y ? 3 cannot be simplified any further
because 2x, y, and, 3 are unlike terms (representational quantities). Representation of
irreducible quantities as well as bigger ones made of these quantities is reminiscent of
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the unitizing process (Behr et al.1994; Lamon1994; Steffe1988,1992,1994). Algebra
tiles denoting a 1, an x, a y, an x2, an xy, or a y2, and their various
combinations1 serve for an essential theoretical construct, which is defined as Represen-
tational Unit Coordination (RUC) (Caglayan2007a).Smith and Thompson (2008) state:
Conceiving of and reasoning about quantities in situations does not require knowing
their numerical value (e.g., how many there are, how long or wide they are, etc.).
Quantities are attributes of objects or phenomena that are measurable; it is our
capacity to measure themwhether we have carried out those measurements or
notthat makes them quantities (p. 101).
In mathematics, we define the Cartesian product of two sets A and B as the set of all
ordered pairs in which the first component is taken from the first set, and the secondcomponent is taken from the second set. Using this analogy, one can say that a product
quantity can be coordinated (composed) as an ordered pair of the form (a, b), where a and
b are understood to be coming from the first set and the second set, respectively. All
possible orderings of the form (multiplier, multiplicand) with coordinates multiplier and
multiplicand generate the binary relation under consideration. In the example of the
polynomial product for instance, the coordination (x, 2y) is not the same as (x,y) or (x, 3).
There are various types of product quantities modeled with polynomial rectangles. In the
example of (x ? 1) (2y ? 3), we have the following product quantities: (Fig.1)The
product quantity (x ? 1) (2y ? 3), which is mapped as the area of the whole rectangle
(largest areal2
singleton) enclosed by its sides x ? 1 and 2y ? 3 (Multiplicative RUC).
The product quantities x2y, x3, 12y, 13 each being mapped as the area of thecorresponding boxes of the same color (This is also a Multiplicative RUC, yet prone to
be treated as pseudo-products, which necessitates a different RUC type in between
Multiplicative and Additive: Pseudo-Multiplicative RUC).
The product quantities xy(there are two of them), x1 (there are three of them), 1y(there are two of them), 11 (there are three of them) each being mapped as the areaof the corresponding irreducible areal unit (Multiplicative RUC).
Fig. 1 The x ? 1 by 2y ?3polynomial rectangle
1 Examples:
A 4 by 2 rectanglemade of 8 irreducible units of 1conceptualized as the unitizing of the evennumber 8
A 2x ? y ? 3 by x ? 1 rectanglemade of 2 irreducible units of x2, 5 irreducible units of x, 3irreducible units of 1, 1 irreducible unit ofy, 1 irreducible unit ofxyconceptualized as the unitizing ofthe polynomial expression.
2 Areal is an adjective meaning of or pertaining to area.
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Teachers knowledge of algebra and multiplicative structures
The number of research studies investigating teachers knowledge of algebra has been
scarce (Kieran1992,2007). The Principles and Standards for School Mathematics (NCTM
2000), the RAND Mathematics Study Panel (2003), and the National MathematicsAdvisory Panel (2008) highlighted the importance of algebra as a strand of mathematics
and the significance of teachers knowledge of algebra. Research on teachers knowledge
of algebra is limited to the investigations of functions, algebraic expressions, equations,
graphs, slope, and covariation (Cooney and Wilson 1993; Doerr 2004; Kieran 1992;
Leinhardt et al. 1990; Norman1993; Stump2001; Zbiek1998).
According to Shulman (1986), content knowledge for teaching can be classified into
three categories: subject matter, pedagogical, and curricular. In an attempt to call attention
to the mathematics that teachers utilize to carry out practice-oriented tasks, Ball and
colleagues introduced the view of mathematical knowledge for teaching (Ball and Bass
2000; Ball et al. 2001; Ball et al.2008). In a study involving U.S. and Chinese teachersperformance in solving problems (subtraction with regrouping, multiplication, fraction
division), Ma (1999) introduced the notion of knowledge packages to account for the
stronger performance of the Chinese teachers. A knowledge package consists of various
interconnected situations that support the teaching of the main concept. This is in line with
Vergnauds view ofconceptual field, a set of problems and situations for the treatment of
which concepts, procedures, and representations of different but narrowly interconnected
types are necessary (1983, p. 128). In particular, Vergnaud views the multiplicative
structures, a conceptual field of multiplicative type, as a system of different but interrelated
concepts, operations, and problems such as multiplication, division, fractions, ratios, andsimilarity. Mas (1999) study provided a detailed description of teachers mathematical
knowledge of additive and multiplicative structures. She reported that the Chinese teachers
in her study were more successful than the U.S. teachers in their ability to help their
students connect the new content to the previous content.
Fischbein et al. (1985) research on teachers knowledge of rational numbers and mul-
tiplicative structures offered a frame for multiplication, which was based on repeated
addition. A set of subsequent studies involving elementary school teachers indicated that
the teachers struggled in solving word problems involving multiplication with decimals
(Graeber et al.1989; Harel and Behr 1995). Sowder et al. 1998breported the difficulties
that a middle grades teacher had in making suitable connections between multiplicationand division. Another set of studies illustrated teachers struggle in explaining the multi-
plication of rational numbers using rectangular area model (Armstrong and Bezuk1995;
Ball et al. 2001; Eisenhart et al.1993).
The present study contributes to the previous research on teachers multiplicative rea-
soning and the use of materials in several ways. First, although multiplicative structures
can to some extent be modeled by additive structures, they have their own characteristics
inherent in their nature, which cannot be explained solely by referring to additive aspects.
Research on how teachers reconcile additive and multiplicative structures based on
sum = product identities is missing in the literature. Second, the coordination construct,
though studied several times before, does not cover all possibilities. Levels of unit coor-
dination have been used in additive, multiplicative, and fractional situations before (Behr
et al. 1994; Lamon1994; Olive1999; Olive and Steffe 2002; Steffe1988, 1994, 2002).
However, there is no prior work on unit coordination arising from the geometry of the
numbers, in the form of identities, where the left hand side (LHS) of the identity stands for
the additive situation (area as a sum, in the geometry of the context) and the right hand side
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(RHS) of the identity stands for the multiplicative situation (area as a product, in the
geometry of the context). Both phrases, area as a product and area as a sum, stand for
the measure of the area of the rectangle enclosed by its sides. Area as a product is the
conception of seeing the area as an ordered pair of linear units (Multiplicative Type RUC),
whereas area as a sum is the conception of seeing the area as an ordered n-tuple of arealunits (Additive Type RUC).
Third, we know nothing about teachers understanding and sense making of
sum = product identities involving linear and areal quantities based on the algebra tiles
representational models. This present study suggests a framework on teachers reasoning in
the different categories of linear or areal quantities; teachers coordination of different
types of representational unit structures (multiplicative, pseudo-multiplicative, additive);
and teachers levels of understanding (additive, one-way multiplicative, bidirectional
multiplicative) arising from the polynomial multiplication and factorization content. To be
more specific, this study investigates prospective secondary mathematics teachers
understandings and sense makings of polynomial multiplication and factorization problemsmodeled with algebra tiles representational models.
Theoretical framework
As the concepts of units and quantities are the essential ideas guiding this research
study, I used unit coordination (Steffe 1988,1994) and quantitative reasoning (Thompson
1988, 1989, 1993, 1994, 1995) as the main theoretical frameworks. I also made use of
Schwartz adjectival quantities and referent preserving/transforming compositions (1988),which served as a meaningful perspective in looking at the interviews comparatively (e.g.,
studentteachers making use of a referent preserving composition vs. those making use of a
referent transforming composition).
Unit coordination has been previously studied by various researchers in the mathematics
education field (Lamon 1994; Olive 1999; Olive and Steffe 2002; Steffe 2002). In the
context of this study, it refers to the conception of unit structures in relation to smaller
embedded units within these unit structures, or bigger units formed via iteration of these
unit structures. Steffe, for instance, analyzed the coordination of different levels of units in
whole number multiplication problems, which is reminiscent of a key concept in multi-
plication, that is, the notion of composite units (1988). Steffe (1988,1992) postulated thatthe multiplication ofaby bcan be thought as the injection of units ofb (each being units of
1) into thea slots, each slot representing a 1. In this example, the conceptualization of each
singleton unit describing a unity, that is, 1, stands for a first level of unit coordination.
Moreover, a and b can be conceptualized (as composite units of 1) as a 9 1 and b 9 1,
respectively, as a second level of unit coordination. The product a 9 b, which denotes
a (composite) units ofb (composite) units of 1, can be conceptualized as a third level of
unit coordination.
Some other researchers also studied unit coordination in a fractional situation (e.g.,
Lamon 1994; Olive 1999; Olive and Steffe 2002; Steffe 2002). Additionally, work onintensive (e.g., miles per hour) and extensive quantities (e.g., number of hours) reflect unit
coordination as well (Kaput et al. 1985; Schwartz 1988). Olive and Caglayans (2008)
work on quantitative unit coordination and conservation also takes the unit coordination
issue into account. According to Steffe, for a situation to be established as multiplicative,
it is always necessary at least to coordinate two composite units in such a way that one of
the composite units is distributed over the elements of the other composite unit (1992,
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p. 264). When dealing with polynomial multiplication and factorization problems using
algebra tiles representations, unit coordination can be formed via linear units, areal units,
areal subunits, and areal subsubunits, which is in agreement with Steffes three levels of
unit coordination. However, the structure of these units is different in that an emphasis in
the different dimensions (linearity and arealness), the quantitative character, and thequantitative operations taking place is necessary, in an attempt to establish identities of the
form area as a sum = area as a product based on the growing rectangles created with
algebra tiles.
Context and methodology
I was interested in investigating prospective mathematics teachers sense making of dif-
ferent types of units and quantities arising from the use of algebra tiles. I was hoping to
reveal the foundations supporting these studentteachers mathematical thinking andreasoning associated with the polynomial multiplication and factorization activities per-
taining to these manipulatives. In that regard, I chose to use a qualitative design because I
would have more opportunities to probe on these ideas in an attempt to explain the
participating studentteachers understanding and sense-making processes (Denzin and
Lincoln2000).
I conducted this study with (2 middle and 3 high-school mathematics) prospective
teachers enrolled in the Mathematics Education Program in a university in the Southeastern
United States, whom I interviewed individually three times. Duration of each session was
about 90 min and each interview session was videotaped using one camera. The firstsession with each participant was based on the representations of prime numbers, com-
posite numbers, and summation of counting numbers, odd natural numbers, and even
natural numbers with magnetic color cubes on the white board. The second and third
sessions were based on polynomial multiplication and polynomial factorization problems
modeled with algebra tiles, respectively, which is the focus of this article. My overarching
goal was to collect data on studentsteachers sense making and understanding of these
growing rectangles and how those understandings shaped their interpretations of the dif-
ferent types of units (e.g., linear vs. areal, additive vs. multiplicative).
I selected my participants from two different undergraduate level mathematics educa-
tion classes. The studentteachers in these classes were racially, socially, and economicallydiverse, with an approximately equal distribution of gender. Ben, Sarah, and John vol-
unteered from a secondary mathematics education concepts class of 11 enrolled student
teachers. This was an advanced level content course offered by the mathematics education
department; designed for prospective high-school mathematics teachers; and it consisted of
the basic concepts in the secondary mathematics curriculum, including concepts from
algebra, functions, shape and space, and number systems. The prerequisite of this course
was Integral Calculus, offered by the mathematics department. Nicole and Ron volunteered
from a geometry methods class of 22 enrolled prospective middle-school mathematics
teachers. This course had a corequisite, the geometry content class that was offered by themathematics department. All names of participants are pseudonyms.
The participants of this present study were in their junior year as full-time student
teachers, only one semester behind their teaching practicum. I selected my research par-
ticipants from the aforementioned classes because I needed research participants that had
already completed an algebra content course for secondary mathematics teachers. All these
participants had successfully completed an algebra content course for prospective
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secondary mathematics teachers that was offered by the mathematics department. None of
these studentteachers were familiar with algebra tiles; it was a new challenge for them.
The focus of this present study is on problems on identities of the form prod-
uct = sum for products and factorizations of polynomials modeled with algebra tiles. In
this model, each little black square tile represents the number 1, purple bar represents the x,blue bar represents they, purple square represents the x2, blue square represents they2, and
green rectangle represents the xy. The 1, the x, and the y are called irreducible linear (or
areal, depending on the context) quantities; whereas the x2, the y2, and the xy are called
irreducible areal quantities. Prospective teachers constructed rectangles with specified
dimensions of the form (ax ? by ? c), wherea,b, andc were natural numbers. They were
also asked to write their answers for the area of the polynomial rectangle both as a product
and as a sum.
Polynomial multiplication tasks were based on three types:
Multiplication of polynomials of the form p (x) and q (x), where, p (x), qx 2 ZX.Example: p (x) = 2x ? 5, q (x) = x ? 1.
Multiplication of polynomials of the form p (x) and q (y), where px 2ZX andqy 2ZY. Example: p (x) = 3x ? 2, q (y) = 4y ? 7.
Multiplication of polynomials of the form p (x, y) and q (x, y), where, p (x, y),
qx;y 2ZX; Y. Example: p (x, y) = 4x ? 5y ? 10, q (x, y) = 2x ? 8y ? 3.A polynomial rectangle is defined as a rectangle representing a specific polynomial
made of different sized color tiles. Representationally speaking, various integer number
combinations of irreducible quantities 1, x, y, xy, x2, y2 that are represented by different
sized color tilesalso referred as algebra tiles or algebra models in the literatureare usedto generate polynomial rectangles (Fig.2). For instance, it is not possible to represent the
real coefficient polynomial 0:25 23x ffiffiffi2p y y2 by using these tiles. In this present
study, I focused on integer coefficient polynomials in one variable as well as integer
coefficient polynomials in two variables.
Polynomial factorization tasks were based on:
Factorization of a polynomial of the formpx 2ZX. Example:px 2x2 3x 1: Factorization of a polynomial of the form qx;y 2 ZX; Y. Example: q (x, y) =
2x2 ? 7xy ? 3y2 ? 5x ? 5y ? 2.
The rationale for collecting interview data with prospective teachers was mainly tounderstand how they establish sum = product identities involving linear and areal
Fig. 2 Irreducible quantities
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quantities based on the algebra tiles representational models. I also wanted to determine
whether they were able to reason at the different categories of linear or areal quantities
associated with growing rectangles generated by algebra tiles. Table 1below summarizes
the interview outline that I developed based on a semistructured interview model (Bernard
1994).I started each interview by introducing the irreducible tiles (Fig. 2) and the multipli-
cation mat to the studentteachers. Studentteachers worked the tasks using algebra tiles
along with pencil and paper for recording their answers for area as a product and area
as a sum. I probed on studentteachers thinking and interpretations on these problems,
but did not interfere during the problem-solving process, nor did I correct errors or propose
instructional help. I used one camera to record studentteachers hand gestures, con-
structions with the algebra tiles, written comments, and verbal descriptions. Each partic-
ipant solved six problems (three multiplication and three factorization problems)
individually.
The interviews were consecutive; no analysis was done between interviews. A retro-spective analysis (Cobb and Whitenack 1996), using constant comparison methodology
(Glaser1992; Glaser and Strauss1967), was then undertaken during which the interviews
were revisited many times in order to generate a thematic analysis (Boyatzis 1998). These
analyses were conducted in an integrated fashion as follows. After the end of the 3 weeks
of data collection, I first generated an outline for each interview, from which I obtained a
summary for each studentteacher. This written summary also contained comments about
any significant events and screen shots from the video when needed for clarification or
highlight. I then reviewed each interview data along with the written summaries for sig-
nificant events, that is, hand gestures, constructions with the algebra tiles, and verbaldescriptions that substantiated notions I interpreted as being related to types of different
units (e.g., linear vs. areal, additive vs. multiplicative) relevant to the research questions. I
Table 1 Interview outline
Polynomial multiplication
Directions:
Multiply two polynomials using as generic rectangle by placing one of the polynomials as the top, andthe other, on the side of the generic rectangle
Indentify the area and the dimensions of the rectangle for a polynomial productProbing questions:
What is the area of each polynomial rectangle as a sum? As a product?
What are the length and the width of each polynomial rectangle?
What are the (linear) units associated with the dimension of the polynomial rectangle?
What are the (areal) units associated with the area of the polynomial rectangle?
Polynomial factorization
Directions:
Build a rectangle enclosing the tiles corresponding to the polynomial expression
Identify the dimensions (length and width) of the polynomial rectangleProbing questions:
What is the area of each polynomial rectangle as a sum? As a product?
What are the length and the width of each polynomial rectangle?
What are the (linear) units associated with the dimensions of the polynomial rectangle?
What are the (areal) units associated with the area of the polynomial rectangle?
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then transcribed these aforementioned significant events from audio files that were created
from the videotapes of the interviews. By doing so, my overarching goal was to generate
possible themes for a more detailed analysis.
I also benefited from generalized notation for mathematics of a quantity (Behr et al.
1994) and theorems and concepts-in-action (Vergnaud 1983, 1988, 1994) framework as
data analysis tools from which I developed a data analysis framework of my own: Rela-
tional notation and mapping structures duo (Caglayan 2007b). This analytical tool is
essentially an extension of Behr et al.s notation in such a way as to cover identities that
equate summation and product expressions of representational quantities. In this notation,
the product a 9 b, in general, is denoted as (a, b)namely as an ordered pair of linear
units a and b. For example, the product 2x 2y is denoted as (2x, 2y). The additivecounterpart uses square brackets [] instead of parentheses. For example, the sum
xy ? xy ? xy ? xyis denoted as [xy,xy,xy,xy]. Moreover, the quantities that are listed in
the square brackets are of areal nature. In this notation, the ordered pair (a, b) of linear
units and the ordered n-tuple a1;
a2;
. . .
;
an of areal units are reconciled via mappingstructures, which is the essence of what is meant by sum = product identities in this
present study. Area as a product coincides with area as a sum at the end, thanks to
these mapping structures (Fig.3).
Results
Polynomial multiplication
On the first polynomial multiplication task, my instruction was Use the algebra tiles to
multiply the polynomials x ? 1 and 2x ? 3 on the multiplication mat. Ben first placed
the dimension tiles on the side and at the top. He then followed a filling process during
which he tried to fit the areal tiles in the polynomial rectangle outlined by the dimension
tiles. Rather than a pairwise multiplication, he relied on a filling in the puzzle strategy, a
concept-in-action, indicative of his additive thinking; despite the fact that he was asked to
multiply these polynomials. Figure4a depicts Bens polynomial rectangle, which he
obtained by the filling in the puzzle concept-in-action. Figure4b depicts what he would
have produced if reasoned multiplicatively. Figures4c, d depict, another prospective
teacher, Ronsfilling in the puzzle strategy, while commenting Any chance of fitting this[green tile] there [right next to the purple square]?
Neither Ron nor Ben used the linear quantities on the perimeter of the figure to determine
the resulting areal quantities. However, Ron was able to interpret the resulting areal tileson
their own as well as with reference to dimension tiles, which was missing in Bens case.
Rons statement when you put this length and that length together can be modeled with
Fig. 3 Equivalence of mappingstructures
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the ordering (1, 1) that corresponds to the resulting areal 1 unit. His second statement it
makes a two dimensional shape, which is this and this length and width shows that he
not only was aware of the resulting areal tile as suggested by the words two dimensional
shape, but he also saw the resulting areal tile as an ordered pair, as suggested by his
language which is this and thislength and width. It is also possible to postulate that
both Ben and Ron seemed to think of the areal quantities as arrangements as opposed to
representation of multiplicative links between the two dimensional expressions.
In fact, in the notation (1, 1), the linear 1 and the linear 1 are sort of put together, in a
specific order, which calls for an ordered pair notation. The multiplicative nature of unit
coordination in this context is much different from the unit coordination described in the
literature. Rons phrase put this length and that length together is really about an
ordering; it is like an ordered pair. RUC in this present study is more of a relational typeas opposed to the unit coordination in the literature, which is of distributive type (Steffe
1992).
John, when working on the second task on the x ? 1 by 2y ? 3 polynomial rectangle,
produced a polynomial rectangle with blue squares, blue bars, and black squares only (i.e.,
the polynomial rectangle was independent ofx). John started with blue squares instead of
green rectangles, which indicated that what he was doing was definitely not term wise
multiplication (Fig.5a). Below the two blue squares, he placed 3 blue bars (Fig. 5b). Right
next to the blue square at the top, he placed 3 blue bars (Fig.5c).
Figures5ac stand as visual evidence that John was not using multiplication. In fact,
John said I am making the rectangle by parts. Therefore, Johns statement validates myprevious hypothesis that the filling in the puzzle strategy seems to be related to an area
as a sum strategy, namely calling for an additive nature. John was aware that there was
something wrong. He decided to revise his figure (Fig. 5c) by removing the three blue bars
in the second column and suggested replacing them with a blue square. The following
protocol illustrates this point.
Fig. 4 Ben and Rons filling in the puzzle strategy
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Protocol 1: Johns struggle with the puzzle
J: These two [blue squares] fit here but this one [he locates another blue square
among the tiles and tries to fit it right next to the blue square at the top] is too long for
here (Fig.6a). Likewise cant put another one of these [he then removes the same
blue square and tries to fit it right below the blue squares on the first column] here
(Fig.6b) its too long So Ill use as many of these [blue squares] as I can to
simplify
He did not like his last attempts and shifted back to his previous figure (Fig. 5c). He
then went on with the filling in the puzzle strategy again by placing three more blue bars
right below the three blue bars at the top (Fig. 7a). Finally, he placed 9 black squares right
below the previous three blue bars, hence completing his puzzle (Fig.7b).
Though he obtained a totally different polynomial rectangle for this second task, Johns
written answers and verbal descriptions were consistent in that he was always referring to
his y-dependent-only polynomial rectangle. Because the initial instruction was to make a
polynomial rectangle with lengthx ? 1 and width 2y ? 3, at some point he had to write an
identity in the last column of the activity sheet (Fig. 8).
Johns written answer warrants disconnect as well, in that John was unable to write an
area as a productexpression (LHS) based on the actual dimensionsof his rectangle. If he
Fig. 5 Johns filling in the puzzle strategy
Fig. 6 Johns attempts to fit the y squared areal tile
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was able to refer to theactual dimensionsof his rectangle, the correct identity would then be
(y ? 1) (2y ? 3) = 2y2 ? 3y ? 6y ? 9 instead of (x ? 1) (2y ? 3) = 2y2 ?
3y ? 6y ? 9. The following protocol takes this issue into account and reflects how John
reconciled the equivalence ofx- and y-dependent LHS with the y-dependent-only RHS:
Protocol 2: John establishes the LHSRHS equivalence
Interviewer: Are they equal? [about the LHS and the RHS of his identity]
John: I mean theyre equal they have to be equal
Interviewer: Do you want to verify?
John: Do you want me to multiply that [the LHS] out? [I then ask him to do it on theboard. Figure9a illustrates the first step of his verification.]
Interviewer: Is there something wrong?
John: No Its just that we dont know what x is so if you knew what x was
youd probably x probably equals [He looks at his figure] It looks like x equals
y plus 2 [He then substitutes x = y ? 2 and completes his verification (Fig. 9b).]
Interviewer: So it works with the condition that
John: With the condition that x equals y plus 2.
At the beginning of the conversation, John was so certain about his equality that he did not
feel the need to question it. Upon my request to verify his findings, he obtained
Fig. 7 Johns complete rectangle made of blue and black tiles
Fig. 8 Johns equation
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2yx ? 3x ? 2y ? 3 = 2y2 ? 9y ? 9 (Fig.9a). At this point, he realized that the RHS
is y-dependent-only, whereas the LHS has xs and ys, and deduced that he somehow
had to get rid of the x on the LHS. He then referred to his figure made of tiles; he
actually measuredthe xat the top of his figure using the y and the 1 tiles. In order toget rid of the x on the LHS, he substituted x = y ? 2 (Fig.9b), based on his mea-
surements. In other words, John made sense of the dimension tiles for the first time . For
him, the dimension tiles do not stand as irreducible linear quantities whose term wise
multiplication yields the corresponding irreducible areal quantity, though. They rather
stand as some sort of measurement tools helping John establish the LHSRHS equivalence
of his written identity. The table below illustrates studentteachers written answers for the
area of the boxes of the same color as a product for the x ? 1 by 2y ? 3 polynomial
rectangle and the nature of their answers.
Mathematics teachers that are not proficient in or not sure about representing (e.g., with
algebra tiles) a variable expression appropriately are highly likely to become a hindrancerather than an asset to students learning. Though it may save the moment for the teacher,
an explanation for why the LHS equals RHS for the above example based on the substi-
tution x = y ? 2 may create more confusion for students. Johns written expressions in
Table2can be used to hypothesize that John seemed to think of the areal quantities (same-
color-boxes) as arrangements. However, there is a slight difference between Johns
arrangement approach and Ben and Rons arrangement approach analyzed above. In Ben
and Rons case, this arrangement view manifests itself in the big picture, namely in the
design of the polynomial rectangle as a whole, which can be thought of as a consistent
approach with the filling in the puzzle strategy. In Johns case, however, the arrangementview appears in the same-color-boxes, yet, John does not seem to stray away from a
multiplicative interpretation. In fact, this multiplicative view is apparent in Johns written
expressions for the areas of these same-color-boxes as products. John is successful in the
sense that he is able to induce a multiplicative meaning to these same-color-boxes, despite
the fact that he obtains an incorrect, y-dependent-only polynomial rectangle.
On the third polynomial multiplication task, my instruction was Use the algebra tiles to
multiply the polynomials 2x ? y and x ? 2y ? 1. Both Nicole and Sarah, when placing
the dimension tiles, followed the x tile followed by the y tile followed by the 1 tile
ordering. As was the case with all the polynomial multiplication problems, both Nicole and
Sarah actually did each term wise multiplication carefully by pointing to the correspondingirreducible linear quantities and placed the resulting irreducible areal quantities accord-
ingly (Fig.10). Both Sarah and Nicole thought aloud and pointed to the irreducible linear
tiles at the top and on the side for each multiplication. The multiplicative nature of the
irreducible areal quantities seems to be warranted by Sarahs statements in the following
protocol:
Fig. 9 Johns work reconciling LHS and RHS
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Protocol 3: Sarahs reference to a representational Cartesian product of Type I
Sarah: This is [pointing to and placing the areal x squared tile] x [pointing to the linear
xtile on the side] timesx [pointing to the linearx tile at the top]. This one is also x times
x [in a similar manner]. This one is x times y [pointing to and placing the green tile
representing the areal unit xy]. And x times y [in a similar manner].
Interviewer: Where is the x times y?
Sarah:y [pointing to the linear y tile at the top] andx [pointing to the linear x tile on the
side]. Andx timesy [in a similar manner]. Andx timesy [in a similar manner]. And this
is x times y [in a similar manner]. And then this is y [pointing to the linear y tile at the
top] times y [pointing to the linear y tile on the side]. And y times y [in a similar
manner]. This isx [pointing to the linear x tile on the side] times 1 [pointing to the linear1 at the top]. And x times 1. And y [pointing to the linear y tile on the side] times 1
[pointing to the linear 1 at the top].
Sarah did not say x squared, nor y squared. She rather said this is x times x and
then this is y times y, that is, multiplicative in nature. Her language y and x is also
indicative of an ordered pair (y, x) of linear quantities. In this vein, both Sarah and Nicole
can be said to construct a representational Cartesian productof Type I. With relational
notation, Sarah and Nicoles verbal descriptions accompanied by their hand gestures can be
modeled with the following representational Cartesian product-in-action of Type I: {x, x,
y} 9 {x,y,y, 1} = {(x,x), (x,y), (x,y), (x, 1), (x,x), (x,y), (x,y), (x, 1), (y,x), (y,y), (y,y),(y, 1)}. When we discussed the area of the boxes of the same color as a product for the
same polynomial multiplication problem 2x ? y times x ? 2y ? 1, Nicoles written
answers, once again, were areas defined as the product of two quantities, that is, multi-
plicative in nature. The following protocol illustrates Nicoles multiplicative thinking.
Table 2 Areas of the boxes of the same color as a product for thex ?1 by 2y ? 3 rectangle
Student-Teachers Ben Nicole Sarah Ron John
Student-TeachersAnswers
Nature of Answers Additive Multiplicative Multiplicative Additive Multiplicative
Fig. 10 Nicole and Sarahs 2x ? y by x ? 2y ? 1 polynomial rectangle
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one is x times y [pointing to the corresponding linear tiles]. This would be y times
2y [pointing to the corresponding linear tiles]. And this would be y times 1 [pointing to
the corresponding linear tiles].
Interviewer: So the product each time you were doing the same thing tell me more
about that I just want to make sure that I understand thatSarah: I was using the area as a length times width where this is a length or and this
would be the width and basing it of like that otherwise I could have added the
insides [pointing to the areal tiles] the way I did it was length times width.
Protocol 5 indicates that Sarah was aware that what she was doing was term wise multi-
plication of the combined linear quantities, and not addition. Her statement otherwise I
could have added the insides combined with her gestures indicates that there are only two
possibilities: The areas of the same-color-boxes could be modeled either via multipli-
cation or via addition,representationally. But since she was asked about the areas of these
boxes as products, the other option, namely additiveness, was irrelevant as she respondedthe way I did it was length times width. From a teachers content knowledge perspective
(Shulman1986), Sarahs content knowledge evolved and this evolution manifested itself as
her ability to induce acounter-example(otherwise I could have added the insides) to falsify
a claim (the claim that Sarahs areas of the same-color-boxes as a product are of additive
nature) that was never made explicit. Sarah was able to take into account the never-
explicitly-stated claim, which she internally formed, and responded to that claim with a
counter-example. Using set notation, Sarahs descriptions can be modeled via a repre-
sentational Cartesian productof Type II defined as follows: {2x,y} 9 {x, 2y, 1} = {(2x,x),
(2x, 2y), (2x, 1), (y,x), (y, 2y), (y, 1)}. The table below illustrates studentteachers written
answers for the area of the boxes of the same color as a product for the 2x ? y byx ? 2y ? 1 polynomial rectangle and the nature of their answers (Table 3).
Polynomial factorization
In the polynomial multiplication tasks analyzed in the previous section, the dimension tiles
were always placed on two sides of the polynomial rectangle, and in both cases, student
teachers relied on a diversity of approaches (filling in the puzzle strategy, arrangement
approach, term wise multiplication of the irreducible linear tiles). I added this task on the
factorization of polynomials to the interview outline because I was trying to understand
whether studentteachers would be able to realize the multiplicative nature of the irre-
ducible areal tiles as well as the boxes of the same color without the presence of the
dimension tiles initially. In that sense, this task required quantitative reasoning at a more
advanced level. Some studentteachers were simultaneously placing the irreducible linear
Table 3 Areas of the same-color-boxes as products for the 2x ? y by x ?2y ?1 rectangle
Student-Teachers Ben Nicole Sarah Ron John
Student-TeachersAnswers
Nature of Answers Additive Multiplicative Multiplicative Additive Multiplicative
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tiles corresponding to the irreducible areal tiles generating the polynomial rectangle, which
was an indication of inverse reasoning. Other studentteachers preferred first completing
their rectangles, then placing the dimension tiles around the edges.
In the first problem Make a rectangle for the expression x2 ? 5x ? 6, then factor the
expression using the algebra tiles, all studentteachers first completed their rectangle andthen placed the dimension tiles representing x ? 2 and x ? 3 around two adjacent edges.
On the second task on polynomial factorization, my instruction was Make a rectangle for
the expression 2x2 ? 7xy ? 3y2 ? 5x ? 5y ? 2 first , then factor the expression
2x2 ? 7xy ? 3y2 ? 5x ? 5y ? 2 using the algebra tiles. Sarah was the only student
teacher to simultaneously place the pair of irreducible linear tiles corresponding to each
irreducible areal tile generating the polynomial rectangle, which was an indication of
inverse reasoning. In contrast, Nicole, John, Ron, and Ben first completed the rectangle
and then placed the dimension tiles around it. Sarah first collected all the pieces she
thought she would need. At the first stage, she placed the purple square representing the
x squared on the upper left corner. She then placed the pair of irreducible dimension tilesaccordingly. She said We start with that [about the purple box] the x times x.
(Fig.11a). In a similar manner, she placed the second x squared areal tile and then one
linearx tile at the top, right next to the previous linearxtile (Fig.11b). She then placed two
green rectangles below the purple squares, and at the same time, she placed one blue bar
right below the x tile on the side (Fig. 11c). She continued this pattern, making sure that
each time she placed a box in the area, she also placed the relevant irreducible linear
tile(s) on the side and/or at the top. In that sense, Sarah worked with both the irreducible
areal quantities and irreducible linear quantities at the same time. Sarah was the only
studentteacher to associate each irreducible areal quantity with its dimensions, namely thecorresponding pair of irreducible linear quantities, in a polynomial factorization problem,
in the process of generating the polynomial rectangle under consideration. In this way,
Sarah established the multiplicative nature of the irreducible areal quantities. She was able
both to generate the correct polynomial rectangle (Fig. 11d) and to induce a representa-
tional Cartesian productvia inverse reasoning.
Sarahs behavior concerning her inverse reasoning and her induction of a representa-
tional Cartesian product calls for the notion of Invertible mapping structures. Starting
from the beginning, Sarahs first action (Fig.11a) can be notated with the relational
notation as [x2] ? (x, x). Her second action (Fig. 11b) can be modeled with the same
relational notation. Her third and fourth actions (Figs. 11c, d) are notated as [xy] ? (x,y).Her remaining actions can be notated in a similar manner. Sarahs areal-to-linear
decomposition can be summarized using an arrow diagram as in Fig. 12.
For a polynomial multiplication problem, on the other hand, everything stays the same
except that the arrows become inverted in Fig.12. In the polynomial multiplication
Fig. 11 Sarahs polynomial factorization steps via inverse reasoning
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problems, Sarah and Nicole, who constructed their rectangle via Term Wise Multiplicationof Irreducible Linear Quantities strategy, can be thought of making use of this model.
Sarah was the only studentteacher to refer to both types of mapping structures. In fact,
after constructing her polynomial rectangle via inverse reasoning as I described above,
Sarah then made use of mapping structures in her description of the same-color-box areal
quantities. The following discussion illustrates this point:
Protocol 6: Sarahs reference to mapping structures
Interviewer: How many different boxes of the same color do you see this time?Sarah: [counting and at the same time pointing to the same-color-boxes] One,two,three,
four, five, six, seven, eight, nine.
Interviewer: Now lets write the products [the areas of the same-color-boxes as a
product] again.
Sarah: Well this is gonna be 2x times x [pointing to the corresponding dimension tiles].
This ones gonna be x times y [pointing to the corresponding dimension tiles]. This is 1
times x [pointing to the dimensions of the box]. This is 2x times 3y [pointing to the
corresponding dimension tiles]. This one is y times 3y[pointing to the dimensions of the
box]. This one is 3y times 1 [pointing to the corresponding dimension tiles]. This one is
2x times 2[pointing to the corresponding dimension tiles]. This one is y times 2[pointingto the corresponding dimension tiles]. And this is 2 times 1 [pointing to the dimensions
of the box]?
My findings concerning Sarah show that a secondary mathematics prospective teachers
strength in successfully referring to a previously established fact (linear quantities mean-
ingfully generating areal quantities in a polynomial multiplication problem) while working
on a new problem (areal quantities can be meaningfully decomposed into pairs of linear
quantities in a polynomial factorization problem) indicates her capability to recognize and
use connections from a pedagogical content knowledge viewpoint (Grossman 1990;
Shulman 1986). In a classroom where a teacher stresses such connections and the inter-relatedness of the mathematical situations, students not only understand and meaningfully
connect the topics, but they develop a sense of the utility of mathematics (NCTM2000,
p. 63). Teachers that effectively facilitate students learning of the new material and build
on the previously learned mathematics through meaningful connections are the ones
equipped with consistent knowledge packages (Ma1999). It is of paramount importance
Fig. 12 Arrow diagram
summarizing Sarahs areal-to-linear decomposition
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for teachers to present the new material not as an isolated topic, but as an extension of and
a new knowledge building on previous knowledge.
Prospective teachers levels of understanding
Additive
In the polynomial multiplication tasks, Ben and Ron preferred the filling in the puzzle
strategy in the process of constructing polynomial rectangles, which was the indication that
what they were doing was addition, and not multiplication. Since a Term-Wise Multipli-
cation of Irreducible Areal Quantities strategy was nonexistent for them, representational
Cartesian product was not available, either. In fact, their additive thinking caused them to
(mis)interpret the structure inherent in the same-color-boxes when they were asked to
express the area of these areal quantities as products. Their answers were of the form
(a coefficient) times (an irreducible areal quantity) instead of the form (a combined linearquantity) times (a combined linear quantity), the former indicating a pseudo-product, a
concatenation of multiplicative meaning. In this sense, pseudo-multiplicative thinking is
equivalent to a repeated additive thinking in a polynomial multiplication problem when the
research participant is asked to express the area of a quantity as a product.
One-way multiplicative
Unlike Ron and Ben who constantly stuck to thefilling in the puzzlestrategy, Nicole relied
on the Term-Wise Multiplication of the Irreducible Areal Quantitiesstrategy by which sheestablished the Multiplicative RUC. Her proficiency in Multiplicative RUC resulted in a
representational Cartesian product. In particular, her statements in Protocol 4 above were
pure mathematical, establishing the existence of a representational Cartesian product. She
did not make any mistake in her expressions of the Area as a Product of the Boxes of the
Same Color. Her expressions were productsand not pseudo-productsof the form
(a combined linear quantity) times (another combined linear quantity). For Nicole, each
same-color-box was an areal singleton, unlike Ron and Ben for whom these same-color-
boxes were of repeated additive, rather than multiplicative nature.
In the process of constructing polynomial rectangles via algebra tiles, John was rea-
soning additively in the first two tasks. In his work with the x ? 1 by 2y ? 3 polynomialrectangle, spontaneous learning occurred and he shifted from filling in the puzzle strategy
to Term-Wise Multiplication of Irreducible Linear Quantitiesstrategy. John was unique in
that he was the only studentteacher to use both strategies. At times, he was also able to
make sense of the dimension tiles as some sort of measurement tools (e.g., he provided the
x = y ? 2 relation for his false identity (x ? 1) (2y ? 3) = 2y2 ? 3y ? 6y ? 9 in
an attempt to reconcile the LHS and the RHS).
After this task, it seemed that something happened as John started to act on and think
about the algebra tiles and the meanings projected onto them. To know an object is to act
on it. (Piaget 1972, p. 8). In contrast with his work on the first two tasks, while doingterm wise multiplication, John pointed to both the dimension tiles and the resulting areal
tile. I infer that this increase in Johns content knowledge resulted from his actions,
combined with a desire for reasoning quantitatively. Bert van Oers (1996) defined action as
an attempt to change some object from its initial form into another form (p. 97). I infer
that in Johns interpretation, the dimension tiles transformed into something more mean-
ingful from some sort of organizers. They were no longer purposelessly standing
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arrangements anymore. I infer that the action was Johns willingness to project some
meanings onto the previously useless dimension tiles. Like Nicole, John interpreted the
same-color-box areal quantities as areal in nature by providing true products of the form
(a combined linear quantity) times (another combined linear quantity). Both Nicole and
John came up with contradictory verbal proofs invalidating pseudo-multiplicative approachwhen dealing with the same-color-boxes.
Bidirectional multiplicative
In regards to algebra tile models, Sarah was the only studentteacher to exhibit a complete
multiplicative understanding in the process of constructing a polynomial rectangle for the
polynomial factorization tasks. The difference between Sarah and John, for instance, is that
John induced the representational Cartesian product after completing his rectangle (without
the dimension tiles placed around), whereas Sarah induced her representational Cartesian
product in the process ofgenerating the polynomial rectangle (by placing the dimensiontiles around), indicating a reference to inverse mapping structures. In that sense, Sarah
relied on a decomposition strategy, which can be thought as the inverse of the previously
discussed Term-Wise Multiplication of Irreducible Linear Quantities strategy. Both strat-
egies corroborate Sarahs multiplicative understanding at a sophisticated level. Table
below summarizes the meanings (multiplicative vs. additive) projected on the irreducible
areal quantities and same-color-box areal quantities by the interview studentteachers for
the cases in the process of and after the completion of the polynomial rectangles in
the polynomial multiplication and factorization tasks (Table4).
Knowing how is as critically important as knowing why mathematical propositions existto be true (Ma 1999). Mewborn (2003) suggested that By and large, teachers have a
strong command of the procedural knowledge of mathematics, but they lack a conceptual
understanding of the ideas that underpin the procedures (p. 47), which is in agreement
with the findings presented above. Strengthening prospective secondary teachers content
knowledge in a manner that emphasizes the ideas underpinning this knowledge is crucial.
As postulated by Shulman (1986):
The person who presumes to teach subject matter to children must demonstrate
knowledge of that subject matter as a prerequisite to teaching. Although knowledge
of the theories and methods of teaching is important, it plays a decidedly secondaryrole in the qualifications of a teacher (p. 5).
Table 4 Meanings projected on irreducible areal quantities (IAQ) and samecolor-boxes (SCB)
Task During versus after AQtype
Ben Ron John Nicole Sarah
Polynomialmultiplication
In the process of constructing thepolynomial rectangle
IAQ SCB NA NA NA NA NA
After the completion of the polynomialrectangle IAQ SCB Polynomial
factorizationIn the process of constructing the
polynomial rectangleIAQ NA NA SCB NA NA NA NA NA
After the completion of the polynomialrectangle
IAQ SCB
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It is essential that teachers be able to integrate a variety of relevant knowledge packages
when dealing with particular mathematical situations. The notion of profound under-
standing of fundamental mathematics (Ma 1999) plays an essential role in teachers
pedagogical content knowledge development.
Discussion
Unit coordination levels
According to Steffe (1988), children who are on a unit coordination pathway start by
constructing singletons representing unities from which they achieve more sophisticated
unit coordination schemes (e.g., composite units, iterable units). As an adult, I can say
that multiplication of whole numbers is an operation that is based on repeated addition
(Steffe 1988, p. 128). It is the shift from operating with singleton units to coordinatingcomposite units that signals the onset of multiplication (Singh 2000, p. 273). In all
activities concerning algebra tiles, the prospective teachers of this present study were able
to refer to singleton units, irreducible areal quantities, in their expressions of the area of the
polynomial rectangle. In what follows, I discuss research participants RUC pertaining to
the 2nd polynomial multiplication task Multiply x ? 1 by 2y ? 3 using algebra tiles for
the sake of the constant comparison analysis methodology. My results in the previous
sections indicate that there is more to add to Steffes definition of multiplication (1994,
p. 19). For instance, in the polynomial multiplication tasks, Sarah and Nicole, who relied
on the Term Wise Multiplication of Irreducible Linear Quantities Strategy, referred tomapping structures in generating their polynomial rectangle. The dimensions of the
polynomial rectangle, namely the Combined Linear Quantities, still possessed some sort of
composite units (namely the irreducible linear quantities) inherent in their structure;
however, in the process of multiplication, a relational aspect was evident, along with the
distributive aspect.
On the first level of unit coordination (Steffe 1994), students make sense of unity as
singleton units, each singleton unit corresponding to the number 1. In this present study,
there were six different singleton unit types: A 1-singleton, anx-singleton, ay-singleton, an
x2-singleton, a y2-singleton, and an xy-singleton. On the first level (Fig.13a), Ben, Ron,
and John interpreted the irreducible areal quantities as meaningless areal singletons.3 Sarahand Nicole interpreted these irreducible areal quantities as areal singletons4 resulting from
the multiplication of the corresponding pair of irreducible linear quantities, which cor-
roborates their Term-Wise Multiplication of Irreducible Linear Quantities concept-in-
action.
On the second level of unit coordination (Steffe 1994), students make sense of a as
a composite unit of singleton units, each singleton unit once again corresponding to the
number 1. On the second level (Fig.13b) Sarah, Nicole, and John5 were able to think
about these quantities both additively and multiplicatively. Pseudo-Multiplicative RUC,
demonstrated by Ron and Ben, is equivalent to Steffes view of unit coordination at the 2ndlevel.
3 Respectively as [1], [x], [y], [x2], [y2], [xy].4 Respectively as (1, 1), (1,x) or (x, 1) (1,y) or (y, 1), (x, x), (y, y), (x, y), or (y, x).5 It took John quite some time to realize that it was possible to express the area of a same-color-box as aproduct of two (combined) linear quantities.
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On the third level of unit coordination (Steffe1994), students make sense ofa 9 b as
thea composite unit ofb composite unit of singleton units, each singleton unit once again
corresponding to the number 1. The composite unit of composite unit of singleton
units notion corresponds to the biggest areal unit (the polynomial rectangle itself) in this
present study. In Steffes 3rd level of unit coordination, the composite units (addends) are
all bs, namely equal addends, whereas in this study, the research participants view of
RUC differed from Steffes 3rd level unit coordination. For Ben and Ron, the area of the
2x ? y by x ? 2y ? 1 polynomial rectangle was five composite unit of irreducible sin-
gletons, where each same-color-box was interpreted additively.6 For Nicole, Sarah, and
John, it was six areal singleton units, where each same-color-box was interpreted multi-plicatively.7 Table5illustrates the difference in these studentteachers thinking.
Quantitative operations
Though Steffes Unit Coordination was the essential theoretical framework, I also felt the
need to use sub-frameworks in order to respond to my research questions. Only the ref-
erents, or only the measurement units, or only the values of quantities involved in a
mathematical situation do not suffice to adequately reflect the nature of those quantities.
For instance, in a mathematical situation involving a pile of oranges, the coordination(oranges, weight of oranges in lb, 12) is not the same as (oranges, cost of oranges in $, 24)
Table 5 Constant comparison of teachers RUC for 2x ? y by x ?2y ? 1 polynomial rectangle
Colorof the SCB
Dimensionsof the SCB
Ben, Ron Nocole, Sarah, John
Purple 2x 9 x 2 composite unit of arealx2-singletons 1 unit of 2x 9 x areal singleton
Green y 9 x 5 composite unit of arealxy-singletons 1 unit ofy 9 x areal singleton
Green 2x 92y 1 unit of 2x 9 2y areal singleton
Blue y 92y 2 composite unit of arealy2-singletons 1 unit ofy 9 2y areal singleton
Purple 2x 91 2 composite unit of arealx-singletons 1 unit of 2x 9 1 areal singleton
Blue y 91 1 composite unit of arealy-singletons 1 unit ofy 9 1 areal singleton
Fig. 13 IAQ and SCB
6 With relational notation, this can be expressed as 2x y;x 2y 1 x2;x2; xy;xy;xy;xy;xy;y2;y2; x;x; y.7 With relational notation, this can be expressed as2x y;x 2y 1 2x;x; y;x; 2x; 2y; y; 2y;2x; 1; y; 1.
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or (oranges, number of oranges, 36). Schwartz (1988) called such quantities adjectival
quantities (p. 41). He stated that all quantities have referents and that the composing of
two mathematical quantities to yield a third derived quantity can take either of two forms,
referent preserving composition or referent transforming composition. (p. 41). Referent
preserving compositions (e.g., addition and subtraction) yield quantities of the same kind,whereas referent transforming compositions (e.g., multiplication and division) yield
quantities of a new kind.
The fact that some studentteachers (Ben, Ron, John) relied on the filling in the puzzle
Strategy and some others (Nicole, Sarah, John) relied on the Term-Wise Multiplication of
Irreducible Linear QuantitiesStrategy in the process of constructing polynomial rectangles
suggests that all these studentteachers were aware that they were dealing with areal
quantities; however, the latter studentteachers were able to operate with both referent
preserving and transforming compositions, whereas the former ones took the referent
preserving composition into account only. Ben and Ron were generating their polynomial
rectangles by adding the irreducible areal quantities, which were already areas; there wasno such thing as the creation of a quantity of a new kind. Nicole, Sarah, and John, on the
other hand, first multiplied the corresponding pair of irreducible linear quantities, where-
from obtained the corresponding irreducible areal of-a-new-kind quantities. They then
added these new quantities. For these studentteachers, each quantitative multiplication
operation (referent transforming composition) was immediately followed by a quantitative
addition operation (referent preserving composition).
The discussion in the paragraph above can be slightly modified for my research par-
ticipants sense making of the samecolor-boxes. When I asked them to express the area of
these same-color-boxes as products, Ben and Ron provided pseudo-products, which indi-cates that these two studentteachers were referring to a referent preserving composition,
the quantitative addition operation, operating on the irreducible areal singleton constituents
of the same-color-box. As for John, Sarah, and Nicole, on the other hand, I can conclude
that, because their (both written and verbal) expressions were products of the corre-
sponding pairs of combined linear quantities, they were making use of a referent trans-
forming composition: the quantitative multiplication operation. Each pair of combined
linear quantities, possessing a linear character, is being transformed into a quantity (same-
color-box) of a totally new (areal) kind via a referent transforming composition.
Thompson (1988) established several cognitive obstacles (p. 167) to students
quantitative reasoning. The most important cognitive obstacle was that students failure todistinguish between a quantity and its measure hindered their ability to explicate rela-
tionships. (p. 168). Another cognitive obstacle was that Multiplicative quantities of any
sort (products, ratios, rates) were commonly misidentified or given an inappropriate unit
(p. 168). Olive and Caglayan (2008) found that quantitative unit coordination and
quantitative unit conservation are essential constructs for overcoming these cognitive
obstacles when students reason quantitatively about word problem situations. The present
study established mapping structures as one such crucial construct to overcome cog-
nitive obstacles to studentteachers quantitative reasoning in a representational situation
(e.g., in comparing same-valued linear and areal quantities, in expressing the area of asame-color-box as a product).
Mapping structures
The analysis provided above shows that studentteachers additive approach in a multi-
plication task concatenates multiplicative meaning and it becomes something elseneither
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addition nor multiplication. Nicole, Sarah, and Johns successful interpretations could be
attributable to the fact that they were able to reason quantitatively (Thompson1988,1989,
1993,1994,1995), paying attention to the referent-value-unit trinity (Schwartz1988), and
attending to the mapping structures involved in these multiplication tasks. Research shows
that multiplicative reasoning is indispensable for proportional reasoning and in particular,in the context of fractional situations, decimal, ratio, rate, proportion, and percent problems
(Kieren 1995; Lamon 1994; Thompson 1994). According to Vergnaud, understanding
multiplicative structures does not rely upon rational numbers only, but upon linear and n-
linear functions, and vector spaces too (1983, p. 172). Although polynomial factorization
is intuitively thought to be an inverse operation for polynomial multiplication, my student
teachers did not refer to ideas of division; they rather worked with mapping structures. In
particular, Sarah was able both to generate the correct polynomial rectangle (Fig.10) and
to induce a representational Cartesian product via inverse reasoning. In that sense, she was
referring to bijections, namely invertible mappings represented as sets of ordered pairs of
some linear quantities. The research presented in this study suggests mapping structures
and relational aspectduo as the main extension to multiplicative reasoning.
The analysis presented above recommends that mathematics teacher educators be aware
of and emphasize the potential difficulties studentteachers may experience concerning the
RUC levels in solving polynomial multiplication and factorization problems with algebra
tiles. Instruction of these topics in methods classes could be organized through the lens of
transformations (Schwartz1988), which will provide studentteachers with opportunities
to develop a rich representational repertoire and assemble a solid knowledge of the content
and the pedagogical content. Studentteachers should be provided the freedom to make,
investigate, and revisit their own conjectures, while reflecting on the mathematical ideasthat will support or refute their inferences. Through such conjecturingjustifying experi-
ences and cycles, studentteachers will not only cultivate their algebraic reasoning skills,
but also will grow to appreciate diversity of approaches and understanding levels.
Mathematical knowledge for teaching
The analysis described above leads us to a mathematical knowledge for teaching frame-
work in the area of polynomial multiplication and factorization via algebra tiles, which is
informed by the diversity of thinking and understanding levelsadditive, one-way mul-
tiplicative, bidirectional multiplicativeexhibited by the participants of this present study.The ability to distinguish between these three levels of understanding appears to be an
essential characteristic of mathematical knowledge for teaching polynomial multiplication
and factorization with algebra tiles. There is a need for prospective secondary mathematics
teachers to examine multiple reasoning strategies in the multiplication and factorization
situations (their own and those of secondary students) in order to develop understanding of
the significance of various representations including manipulative materials for developing
students algebraic as well as multiplicative reasoning. Further research could investigate
the scope of teachers understandings and interpretations of binomial identities of the form
(x ? y
)
2= x2 ?
2xy ? y2
using area model based on algebra tiles as well as binomialidentities of the form (x ? y)3 = x3 ? 3x2y ? 3xy2 ? y3 using volume model based on a
different set of manipulatives.
The curriculum materials (e.g., textbooks, activity books, online modules, manipula-
tives, teacher guides) could emphasize the necessity of attending to the nature of the
quantities, their units, and the quantitative operations taking place on each side of identities
of the form sum = product. The use of algebra tiles as representational tools in teaching
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polynomial multiplication and factorization provides students and teachers with opportu-
nities to make better sense of and to explore and discover algebraic connections between
the sum = product identities and concrete operations. Explorations that incorporate
such manipulatives provide teachers with an easily accessible concept-building activity for
developing sum = product identities for polynomial multiplication and factorization.Sum = Product Identities do not solely apply to the mathematics context investigated
by the research participants of this study. Focusing on the big picture, one can find
Sum = Product Identities (or LHS = RHS identities in general) in various contexts
such as summation formulas, growing sequences and patterns, linear, quadratic, cubic
equations, equations involving derivatives, antiderivatives, and integrals. The findings of
this study imply that such content be written and guided by a framework based on RUC,
which pushes students and teachers to reason quantitatively, at the same time paying
attention to the relevant mappings and quantitative operations taking place.
Algebra tiles as a concrete way of teaching polynomial multiplication and factorization
problems could be a useful asset for prospective mathematics teachers content knowledgeand pedagogical content knowledge development. However, there are also limitations of
the use of algebra tiles as an instructional tool for teachers. Teachers should be proficient in
their understandings and sense makings of the different types of units arising from the use
of algebra tiles. For instance, being able to interpret the area both as a sum and as a
product, moving from uni- to bidirectional (inverse) multiplicative may require substantial
challenge. There is also the need to distinguish among teachers thinking at entry points as
a basis for selecting developmentally proper goals for their next learning. As an example,
teachers should be proficient and confident in a variety of multiplication models and
algorithms (e.g., repeated addition model, array model, area model, Cartesian productmodel, partial products algorithm, foil algorithm). Excellence in partial products algorithm,
for instance, could pave the way for proficiency in a teachers meaningful interpretation of
the same-color-boxes arising from the polynomial multiplication and factorization
situations.
Teacher education programs should provide opportunities for studentteachers to
explicitly engage in quantitative reasoning in a manner that leads to using all three levels of
unit coordination. This necessitates a focus on discrete mathematics content with a par-
ticular emphasis on sets, relations, Cartesian products, mapping structures, which by
definition encompass levels of unit coordination and quantitative reasoning in their
structure. In particular, at first, polynomial multiplication and factorization can be thoughtof as totally irrelevant to set theoretical aspects, quantitative reasoning, or unit coordina-
tion. However, as shown above, when prospective teachers engage in and want to make
sense of what they are doing, they end up performing mathematically, exhibiting set
theoretical aspects.
Distinguishing how quantities interact with one another (e.g., additive vs. multiplica-
tive) is an important element of algebraic reasoning. In that regard, concepts-in-actionand
theorems-in-action formalisms are powerful instruments to illustrate and explain the
continuing progress of studentteachers mathematical proficiency in a certain conceptual
field (e.g., multiplicative, relational, mapping, quantitative, and algebraic structures). Theyalso present a way to analyze, compare, and transform students knowledge intrinsic in
their mathematical performance (e.g., hand gestures, actions, drawings, verbal descrip-
tions) into the actual known and written algebraic identities and mathematical theorems. In
that sense, these tools help teachers and researchers get a better sense of how students
make sense of, reconcile, and shift among physical observables (Kaput 1991) at different
cognitive levels (e.g., algebraic expressions, their various representations, etc.). Using
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concepts- and theorems-in-action, teachers and researchers can come up with better
strategies to diagnose what students do or fail to understand, to reveal the source of their
misconceptions and conceptual flaws, and to help them see the internal and external
connections. In this way, students are provided with a set of more interesting, better-
prepared activities, and mathematically fruitful situations, which help them strengthen theirconcept knowledge and increase their mathematical proficiency.
Acknowledgments I would like to thank the five prospective teachers for being part of this research study,and the reviewers and the editors for their very helpful comments and suggestions.
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