Factorization With Tiles

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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: [email protected]

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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

350 G. Caglayan

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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

Polynomial multiplication and factorization 351

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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.

References

Aburime, F. (2007). How manipulatives after the mathematics achievement of students in Nigerian schools.

Educational Research Quarterly, 31, 316.Armstrong, B., & Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J.T. Sowder & B. P. Schappelle (Eds.),Providing a foundation for teaching mathematics in the middlegrades(pp. 85119). Albany: State University of New York Press.

Ball, D. (1992). Magical hopes. Manipulatives and the reform of math education. American Educator,16(1418), 4647.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach:Knowing and using mathematics. In J. Boaler (Ed.),Multiple perspectives on mathematics teachingand learning (pp. 81104). Westport, CT: Ablex.

Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolvedproblem of teachers mathematical knowledge. In V. Richardson (Ed.), Handbook of research onteaching (4th ed., pp. 433456). Washington, DC: American Educational Research Association.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?Journal of Teacher Education, 59, 389407.

Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. Grouws (Ed.),Handbook of research on mathematics teaching and learning(pp. 296333). New York: Macmillan.

Behr, M. J., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additiveand multiplicative structures. In G. Harel & J. Confrey (Eds.), The development of multiplicativereasoning in the learning of mathematics (pp. 121176). Albany, NY: SUNY Press.

Bernard, H. (1994). Research methods in anthropology (2nd ed.). Thousand Oaks, CA: Sage.Boyatzis, R. (1998). Transforming qualitative information: Thematic analysis and code development.

Thousand Oaks, CA: Sage Publications.Caglayan, G. (2007a). Representational unit coordination: Preservice teachers representation of special

numbers using sums and products. In T. Lamberg & L. Wiest (Eds.). Proceedings of the Twenty Ninth

Annual Meeting of the North American Chapter of the International Group for the Psychology ofMathematics Education(pp. 10571060). The University of Nevada: Reno, Nevada.

Caglayan, G. (2007b). Relational notation & mapping structures: A data analysis framework. In D. Pugalee,A. Rogerson, & A. Schinck (Eds.).Proceedings of the Ninth International Conference of the Math-ematics Education into the 21st Century Project. (pp. 112117). The University of North CarolinaCharlotte: Charlotte, North Carolina.

Clements, D. H. (1999). Concrete manipulatives, concrete ideas.Contemporary Issues in Early Childhood,1, 4560.

Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom vide-orecordings and transcripts. Educational Studies in Mathematics, 30, 2

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Factorization With Tiles