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 operations—referent preserving versus referent transforming compositions—acting 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 units supported 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 in x 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 student–teachers 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 multiplicative 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 reasoning and need to be emphasized in the teaching–learning process. Moreover, representational Cartesian products-in-action at two different levels, indicators of multiplicative thinking, were available to two research participants only.
Physical objects, also often referred to as manipulatives or instructional devices, can serve as 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 manipulatives. Work by Sowell (1989) indicated that even though for K-16 students, manipulatives 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 recreational activity where teachers had difficulty in explaining and representing the mathematical 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 educational 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 the necessity to think about the object under consideration as some sort of quantity possessing a referent, a value, and a measurement unit (Schwartz 1988; Thompson 1993, 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 representation of a written symbol without meanings projected into these concrete objects (Ball 1992; Clements 1999). 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 Behr 1989; Harel et al. 1992). “Additive reasoning develops quite naturally and intuitively 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 Singer 1993).
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 abstract—in 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 children’s construction of multiplicative 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. Vergnaud’s (1994) conceptual field theory asserts:
One needs mathematics to characterize with minimum ambiguity the knowledge contained 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-action are “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 children’s mathematical 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 (Vergnaud 1997).
Previous research studies indicated that the use of algebra tiles positively impacted students’ attitudes (Sharp 1995). 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 the square (2007). Johnson (1993) reported that both teachers and students understood polynomial 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 (Huntington 1994; Sharp 1995; Takahashi 2002). 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 2x is 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 the individual 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 representational 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 the “unitizing” process (Behr et al. 1994; Lamon 1994; Steffe 1988, 1992, 1994). Algebra tiles denoting a “1”, an “x,” a “y,” an “x2”, an “xy,” or a “y2,” and their various combinationsFootnote 1 serve for an essential theoretical construct, which is defined as Representational Unit Coordination (RUC) (Caglayan 2007a).
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 them—whether we have carried out those measurements or not—that 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 second component 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 arealFootnote 2 singleton) enclosed by its sides x + 1 and 2y + 3 (Multiplicative RUC).
The product quantities x · 2y, x · 3, 1 · 2y, 1 · 3 each being mapped as the area of the corresponding 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 x · y (there are two of them), x · 1 (there are three of them), 1 · y (there are two of them), 1 · 1 (there are three of them) each being mapped as the area of the corresponding irreducible areal unit (Multiplicative RUC).
Teachers’ knowledge of algebra and multiplicative structures
The number of research studies investigating teachers’ knowledge of algebra has been scarce (Kieran 1992, 2007). The Principles and Standards for School Mathematics (NCTM 2000), the RAND Mathematics Study Panel (2003), and the National Mathematics Advisory 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; Norman 1993; Stump 2001; Zbiek 1998).
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 teachers’ performance 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 Vergnaud’s view of conceptual 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, and similarity. Ma’s (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 multiplicative 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. 1998b reported the difficulties that a middle grades teacher had in making suitable connections between multiplication and division. Another set of studies illustrated teachers’ struggle in explaining the multiplication of rational numbers using rectangular area model (Armstrong and Bezuk 1995; Ball et al. 2001; Eisenhart et al. 1993).
The present study contributes to the previous research on teachers’ multiplicative reasoning 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 coordination have been used in additive, multiplicative, and fractional situations before (Behr et al. 1994; Lamon 1994; Olive 1999; Olive and Steffe 2002; Steffe 1988, 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 (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 areal units (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 problems modeled 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., student–teachers 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 multiplication, that is, the notion of composite units (1988). Steffe (1988, 1992) postulated that the multiplication of a by b can be thought as the injection of units of b (each being units of 1) into the a 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 × 1 and b × 1, respectively, as a second level of unit coordination. The product a × b, which denotes a (composite) units of b (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 on intensive (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 Caglayan’s (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, 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 sub–subunits, which is in agreement with Steffe’s 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 the quantitative 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 different types of units and quantities arising from the use of algebra tiles. I was hoping to reveal the foundations supporting these student–teachers’ mathematical thinking and reasoning associated with the polynomial multiplication and factorization activities pertaining 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 student–teachers’ understanding and sense-making processes (Denzin and Lincoln 2000).
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 first session with each participant was based on the representations of prime numbers, composite 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 students–teachers’ sense making and understanding of these growing rectangles and how those understandings shaped their interpretations of the different types of units (e.g., linear vs. areal, additive vs. multiplicative).
I selected my participants from two different undergraduate level mathematics education classes. The student–teachers in these classes were racially, socially, and economically diverse, with an approximately equal distribution of gender. Ben, Sarah, and John volunteered 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 the mathematics 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 participants 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 secondary mathematics teachers that was offered by the mathematics department. None of these student–teachers 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 “product = 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 the y, purple square represents the x2, blue square represents the y2, 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), where a, b, and c 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), \( q(x) \in Z[X] \). Example: p (x) = 2x + 5, q (x) = x + 1.
Multiplication of polynomials of the form p (x) and q (y), where \( p(x) \in Z[X] \) and \( q(y) \in Z[Y] \). 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), \( q(x,y) \in Z[X,Y] \). 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 tiles—also referred as algebra tiles or algebra models in the literature—are used to generate polynomial rectangles (Fig. 2). For instance, it is not possible to represent the real coefficient polynomial \( 0.25 + \frac{2}{3}x + \sqrt 2 y + y^{2} \) 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.
Factorization of a polynomial of the form \( p(x) \in Z[X] \). Example: \( p(x) = 2x^{2} + 3x + 1. \)
Factorization of a polynomial of the form \( q(x,y) \in Z[X,Y] \). Example: q (x, y) = 2x2 + 7xy + 3y2 + 5x + 5y + 2.
The rationale for collecting interview data with prospective teachers was mainly to understand how they establish “sum = product” identities involving linear and areal 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 1 below 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 multiplication mat to the student–teachers. Student–teachers 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 student–teachers’ 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 student–teachers’ hand gestures, constructions with the algebra tiles, written comments, and verbal descriptions. Each participant solved six problems (three multiplication and three factorization problems) individually.
The interviews were consecutive; no analysis was done between interviews. A retrospective analysis (Cobb and Whitenack 1996), using constant comparison methodology (Glaser 1992; Glaser and Strauss 1967), 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 student–teacher. 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 significant events, that is, hand gestures, constructions with the algebra tiles, and verbal descriptions 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 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: Relational 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 × b, in general, is denoted as (a, b)—namely as an ordered pair of linear units a and b. For example, the product \( 2x \cdot 2y \) is denoted as (2x, 2y). The additive counterpart uses square brackets [] instead of parentheses. For example, the sum xy + xy + xy + xy is 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 \( \left[ {a_{1} ,\;a_{2} ,\; \ldots ,\;a_{n} } \right] \) of areal units are reconciled via mapping structures, 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).
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. Figure 4a depicts Ben’s polynomial rectangle, which he obtained by the filling in the puzzle concept-in-action. Figure 4b depicts what he would have produced if reasoned multiplicatively. Figures 4c, d depict, another prospective teacher, Ron’s filling 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 tiles on their own as well as with reference to dimension tiles, which was missing in Ben’s case. Ron’s statement “when you put this length and that length together” can be modeled with 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 this… length 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. Ron’s 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” type as 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 of x). 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).
Figures 5a–c stand as visual evidence that John was not using multiplication. In fact, John said “I am making the rectangle by parts.” Therefore, John’s statement validates my previous 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.
Protocol 1: John’s 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 can’t 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) it’s too long… So… I’ll 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).
Fig. 7
John’s complete rectangle made of blue and black tiles
Though he obtained a totally different polynomial rectangle for this second task, John’s 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 length x + 1 and width 2y + 3, at some point he had to write an identity in the last column of the activity sheet (Fig. 8).
John’s written answer warrants disconnect as well, in that John was unable to write an area as a product expression (LHS) based on the actual dimensions of his rectangle. If he was able to refer to the actual dimensions of 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 of x- and y-dependent LHS with the y-dependent-only RHS:
Protocol 2: John establishes the LHS–RHS equivalence
Interviewer: Are they equal? [about the LHS and the RHS of his identity]
John: I mean… they’re 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 the board. Figure 9a illustrates the first step of his verification.]
John: No… It’s just that… we don’t know what x is… so… if you knew what x was you’d 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 “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 “x”s and “y”s, 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 measured the x at the top of his figure using the “y” and the “1” tiles. In order to get rid of the “x” on the LHS, he substituted x = y + 2 (Fig. 9b), based on his measurements. 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 LHS–RHS equivalence of his written identity. The table below illustrates student–teachers’ 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 hindrance rather 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 substitution x = y + 2 may create more confusion for students. John’s written expressions in Table 2 can 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 John’s arrangement approach and Ben and Ron’s arrangement approach analyzed above. In Ben and Ron’s 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 John’s case, however, the arrangement view 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 John’s 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.
Table 2 Areas of the boxes of the same color as a product for the x + 1 by 2y + 3 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 corresponding irreducible linear quantities and placed the resulting irreducible areal quantities accordingly (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 Sarah’s statements in the following protocol:
Fig. 10
Nicole and Sarah’s 2x + y by x + 2y + 1 polynomial rectangle
Protocol 3: Sarah’s 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 x tile on the side] times x [pointing to the linear x 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] and x [pointing to the linear x tile on the side]. And x times y [in a similar manner]. And x times y [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 is x [pointing to the linear x tile on the side] times 1 [pointing to the linear 1 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 product of Type I. With relational notation, Sarah and Nicole’s verbal descriptions accompanied by their hand gestures can be modeled with the following representational Cartesian product-in-action of Type I: {x, x, y} × {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, Nicole’s written answers, once again, were areas defined as the product of two quantities, that is, multiplicative in nature. The following protocol illustrates Nicole’s multiplicative thinking.
Protocol 4: Nicole’s reference to a representational Cartesian product of Type II
Interviewer: You are saying 2x times x [About Nicole’s expression (2x) · (x), which she wrote on the activity sheet]. And why not 2 times x2?
Nicole: Because I just saw these as a pair together… [pointing to the linear 2x] same things you can group them together. So it’s just A [pointing to the linear 2x at the top] times B [pointing to the linear x on the side]. Because when you look at this whole thing, this whole purple area [pointing to the 2x by x “same-color-box”] as one area… so you look the length as one number, 2x, instead of 2 times x.
Interviewer: So what is the difference between this and the other way [I am asking her to compare the expressions (2x) · (x) and (2) · (x2)] representationally?
Nicole: I did it [About her expression (2x) · (x), which she wrote on the activity sheet] in terms of the length times the width. Now it [About the expression (2) · (x2) via which I am trying to challenge her] would be… talking about… how many of these [pointing to the purple squares] you have. In the other case, length times width gives the area of the whole thing [pointing to the 2x by x “same-color-box”].
Interviewer: Please do the same for this one… [pointing to her expression (2x) · (2y), which she wrote on the activity sheet] why not 4 times xy?
Nicole: Again I did this [pointing to the 2x by 2y “same-color-box”] as one area. I did it as length times width. Now this [about the expression 4 times xy I am trying to challenge her with] means I have 4 of them [meaning 4 green rectangles] and each one is an xy.
Nicole showed a mathematically fruitful performance in creating a representational Cartesian product of Type II, in her comparison of “2x times x” versus “2 times x2.” The “pair” in Nicole’s statement “I just saw these as a pair together” refers to the pair of linear “x”s in “2x” and not to the ordered pair (x, 2x) of linear quantities. In other words, at the initial stage of defining a representational Cartesian product of Type II, she first identified the elements of the two sets of linear quantities {2x, y} and {x, 2y, 1}. Her later usage “So it’s just A [pointing to the linear 2x at the top] times B [pointing to the linear x on the side]” signaled the onset of a representational Cartesian product of Type II, another concept-in-action, an indicator of Nicole’s multiplicative reasoning. In this way, she established the existence of her concept-in-action: Nicole first picked an element, “A,” from the set {2x, y} and then picked another element, “B,” from the other set {x, 2y, 1}. She then formed pairs (A, B) of combined linear quantities by which she generated her representational Cartesian product. Nicole did not describe the expressions (2)·(x2) and (4)·(xy) as representationally multiplicative, as opposed to Ben and Ron, who embraced the filling in the puzzle strategy. Although (2)·(x2) and (4)·(xy) are representationally additive, as Nicole explains in Protocol 4 above, for Ben and Ron, these expressions are misinterpreted as of multiplicative nature. The algebraic symbols (2)·(x2) and (4)·(xy) can be deduced only within an additive context, according to Nicole, representationally.
In the same vein, I asked Sarah to outline the same-color-boxes on her 2x + y by x + 2y + 1 polynomial rectangle (Fig. 10d). The multiplicative nature of the areas of these “same-color-boxes” was prevalent, as reflected in the protocol below:
Protocol 5: Sarah’s reference to a representational Cartesian product of Type II
Sarah: This one is 2x [pointing to the linear 2x on the side] times x [pointing to the linear x on the top]. This one is 2y times 2x [pointing to the corresponding linear tiles in a similar manner]. This one is 2x times 1 [pointing to the corresponding linear tiles]. This 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 that…
Sarah: 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 multiplication 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 multiplication 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 responded “the way I did it was length times width.” From a teacher’s content knowledge perspective (Shulman 1986), Sarah’s content knowledge evolved and this evolution manifested itself as her ability to induce a counter-example (otherwise I could have added the insides) to falsify a claim (the claim that Sarah’s 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, Sarah’s descriptions can be modeled via a representational Cartesian product of Type II defined as follows: {2x, y} × {x, 2y, 1} = {(2x, x), (2x, 2y), (2x, 1), (y, x), (y, 2y), (y, 1)}. The table below illustrates student–teachers’ written answers for the “area of the boxes of the same color as a product” for the 2x + y by x + 2y + 1 polynomial rectangle and the nature of their answers (Table 3).
Table 3 Areas of the same-color-boxes as products for the 2x + y by x + 2y + 1 rectangle
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 student–teachers would be able to realize the multiplicative nature of the irreducible 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 student–teachers were simultaneously placing the irreducible linear tiles corresponding to the irreducible areal tiles generating the polynomial rectangle, which was an indication of inverse reasoning. Other student–teachers 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 student–teachers first completed their rectangle and then 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 tiles accordingly. 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 linear x tile at the top, right next to the previous linear x tile (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 student–teacher to associate each irreducible areal quantity with its dimensions, namely the corresponding 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 representational Cartesian product via inverse reasoning.
Fig. 11
Sarah’s polynomial factorization steps via inverse reasoning
Sarah’s behavior concerning her inverse reasoning and her induction of a representational Cartesian product calls for the notion of Invertible mapping structures. Starting from the beginning, Sarah’s 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. Sarah’s 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 problems, Sarah and Nicole, who constructed their rectangle via Term Wise Multiplication of Irreducible Linear Quantities strategy, can be thought of making use of this model. Sarah was the only student–teacher 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: Sarah’s 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 let’s 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 one’s 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 [pointing to 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 teacher’s strength in successfully referring to a previously established fact (linear quantities meaningfully 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 interrelatedness of the mathematical situations, students not only understand and meaningfully connect the topics, but they develop a sense of the utility of mathematics (NCTM 2000, 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 (Ma 1999). It is of paramount importance 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 Multiplication 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 linear quantity) 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 the filling in the puzzle strategy, Nicole relied on the Term-Wise Multiplication of the Irreducible Areal Quantities strategy by which she established 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 products—and not pseudo-products—of 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 reasoning additively in the first two tasks. In his work with the x + 1 by 2y + 3 polynomial rectangle, spontaneous learning occurred and he shifted from filling in the puzzle strategy to Term-Wise Multiplication of Irreducible Linear Quantities strategy. John was unique in that he was the only student–teacher 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 doing “term wise multiplication,” John pointed to both the dimension tiles and the resulting areal tile. I infer that this increase in John’s 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 John’s interpretation, the dimension tiles transformed into something more meaningful from some sort of organizers. They were no longer purposelessly standing arrangements anymore. I infer that the “action” was John’s 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 approach when dealing with the same-color-boxes.
Bidirectional multiplicative
In regards to algebra tile models, Sarah was the only student–teacher 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 of generating the polynomial rectangle (by placing the dimension tiles 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 strategies corroborate Sarah’s 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 student–teachers for the cases “in the process of” and “after the completion of” the polynomial rectangles in the polynomial multiplication and factorization tasks (Table 4).
Table 4 Meanings projected on irreducible areal quantities (IAQ) and same–color-boxes (SCB)
Knowing how is as critically important as knowing why mathematical propositions exist to 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 secondary role in the qualifications of a teacher (p. 5).
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 understanding 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 coordinating composite 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 Steffe’s 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 to mapping 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, an x-singleton, a y-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.Footnote 3 Sarah and Nicole interpreted these irreducible areal quantities as areal singletonsFootnote 4resulting from the multiplication of the corresponding pair of irreducible linear quantities, which corroborates 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 JohnFootnote 5 were able to think about these quantities both additively and multiplicatively. Pseudo-Multiplicative RUC, demonstrated by Ron and Ben, is equivalent to Steffe’s view of unit coordination at the 2nd level.
On the third level of unit coordination (Steffe 1994), students make sense of a × b as the a composite unit of b 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 Steffe’s 3rd level of unit coordination, the composite units (addends) are all “b”s, namely equal addends, whereas in this study, the research participants’ view of RUC differed from Steffe’s 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 singletons, where each same-color-box was interpreted additively.Footnote 6 For Nicole, Sarah, and John, it was six areal singleton units, where each same-color-box was interpreted multiplicatively.Footnote 7 Table 5 illustrates the difference in these student–teachers’ thinking.
Table 5 Constant comparison of teachers’ RUC for 2x + y by x + 2y + 1 polynomial rectangle
Though Steffe’s 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 referents, 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) 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 student–teachers (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 Quantities Strategy in the process of constructing polynomial rectangles suggests that all these student–teachers were aware that they were dealing with areal quantities; however, the latter student–teachers 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 was no 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, wherefrom obtained the corresponding irreducible areal of-a-new-kind quantities. They then added these new quantities. For these student–teachers, 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 participants’ sense making of the same–color-boxes. When I asked them to express the area of these same-color-boxes as products, Ben and Ron provided pseudo-products, which indicates that these two student–teachers 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 corresponding pairs of combined linear quantities, they were making use of a referent transforming 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 to distinguish between a quantity and its measure hindered their ability to explicate relationships.” (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 cognitive obstacles to student–teachers’ quantitative reasoning in a representational situation (e.g., in comparing same-valued linear and areal quantities, in expressing the area of a same-color-box as a product).
Mapping structures
The analysis provided above shows that student–teachers’ additive approach in a multiplication task concatenates multiplicative meaning and it becomes something else—neither addition nor multiplication. Nicole, Sarah, and John’s successful interpretations could be attributable to the fact that they were able to reason quantitatively (Thompson 1988, 1989, 1993, 1994, 1995), paying attention to the referent-value-unit trinity (Schwartz 1988), 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 aspect duo as the main extension to multiplicative reasoning.
The analysis presented above recommends that mathematics teacher educators be aware of and emphasize the potential difficulties student–teachers 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 (Schwartz 1988), which will provide student–teachers with opportunities to develop a rich representational repertoire and assemble a solid knowledge of the content and the pedagogical content. Student–teachers should be provided the freedom to make, investigate, and revisit their own conjectures, while reflecting on the mathematical ideas that will support or refute their inferences. Through such conjecturing–justifying experiences and cycles, student–teachers 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 framework in the area of polynomial multiplication and factorization via algebra tiles, which is informed by the diversity of thinking and understanding levels—additive, one-way multiplicative, bidirectional multiplicative—exhibited 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 binomial identities 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, manipulatives, 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 polynomial multiplication and factorization provides students and teachers with opportunities 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 knowledge and 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 product model, partial products algorithm, foil algorithm). Excellence in partial products algorithm, for instance, could pave the way for proficiency in a teacher’s meaningful interpretation of the same-color-boxes arising from the polynomial multiplication and factorization situations.
Teacher education programs should provide opportunities for student–teachers 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 particular 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 thought of as totally irrelevant to set theoretical aspects, quantitative reasoning, or unit coordination. 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. multiplicative) is an important element of algebraic reasoning. In that regard, concepts-in-action and theorems-in-action formalisms are powerful instruments to illustrate and explain the continuing progress of student–teachers mathematical proficiency in a certain conceptual field (e.g., multiplicative, relational, mapping, quantitative, and algebraic structures). They also present a way to analyze, compare, and transform students’ knowledge intrinsic in their mathematical performance (e.g., hand gestures, actions, drawings, verbal descriptions) 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 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 their concept knowledge and increase their mathematical proficiency.
Notes
Examples:
A 4 by 2 rectangle—made of 8 irreducible units of 1—conceptualized as the unitizing of the even number 8
A 2x + y + 3 by x + 1 rectangle—made of 2 irreducible units of x2, 5 irreducible units of x, 3 irreducible units of 1, 1 irreducible unit of y, 1 irreducible unit of xy—conceptualized as the unitizing of the polynomial expression.
Areal is an adjective meaning of or pertaining to area.
Respectively as [1], [x], [y], [x2], [y2], [xy].
Respectively as (1, 1), (1, x) or (x, 1) (1, y) or (y, 1), (x, x), (y, y), (x, y), or (y, x).
It took John quite some time to realize that it was possible to express the area of a same-color-box as a product of two (combined) linear quantities.
With relational notation, this can be expressed as \( (2x + y,x + 2y + 1) \equiv [[x^{2} ,x^{2} ],[xy,xy,xy,xy,xy],[y^{2} ,y^{2} ],[x,x],[y]] \).
With relational notation, this can be expressed as \( (2x + y,x + 2y + 1) \equiv [(2x,x),(y,x),(2x,2y),(y,2y),(2x,1),(y,1)] \).
References
Aburime, F. (2007). How manipulatives after the mathematics achievement of students in Nigerian schools. Educational Research Quarterly,31, 3–16.
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 middle grades (pp. 85–119). Albany: State University of New York Press.
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 teaching and learning (pp. 81–104). Westport, CT: Ablex.
Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433–456). Washington, DC: American Educational Research Association.
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. 296–333). New York: Macmillan.
Behr, M. J., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additive and multiplicative structures. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 121–176). Albany, NY: SUNY Press.
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 of Mathematics Education (pp. 1057–1060). 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 Mathematics Education into the 21st Century Project. (pp. 112–117). The University of North Carolina Charlotte: Charlotte, North Carolina.
Clements, D. H. (1999). Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood,1, 45–60.
Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics,30, 213–228.
Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind. Journal for Research in Mathematics Education,23, 2–33.
Confrey, J. (1994). Splitting, similarity, and the rate of change: New approaches to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291–330). Albany, NY: SUNY Press.
Cooney, T., & Wilson, M. (1993). Teachers’ thinking about functions: Historical and research perspectives. In T. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 131–158). Hillsdale, NJ: Erlbaum.
Doerr, H. M. (2004). Teachers’ knowledge and the teaching of algebra. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 267–290). Boston: Kluwer Academic.
Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding. Journal for Research in Mathematics Education,24, 8–40.
Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education,16, 3–17.
Gilbert, R., & Bush, W. (1988). Familiarity, availability, and use of manipulative devices in mathematics at the primary level. School Science and Mathematics,88, 459–469.
Graeber, A., Tirosh, D., & Glover, R. (1989). Preservice teachers’ misconceptions in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education,20, 95–102.
Harel, G., & Behr, M. (1989). Structure and hierarchy of missing value proportion problems and their representations. Journal of Mathematical Behavior,8, 77–119.
Harel, G., Behr, M., Post, T., & Lesh, R. (1992). The blocks task: Comparative analyses of the task with other proportion tasks and qualitative reasoning skills of seventh grade children in solving the task. Cognition and Instruction,9, 45–96.
Hiebert, J., & Behr, M. (1988). Introduction: Capturing the major themes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 1–18). Hillsdale, NJ: Lawrence Erlbaum Associates.
Huntington, D. (1994). Instruction in concrete, semi-concrete, and abstract representation as an aid to the solution of relational problems by adolescents with learning disabilities. Dissertation Abstracts International, 56(02), 512. (UMI No. 9520828).
Johnson, K. A. (1993). Manipulatives allow everyone to learn mathematics. Contemporary Education,65, 10–11.
Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. v. Glasersfeld (Ed.), Constructivism and mathematics education, (pp. 53–74). Dordrecht, Netherlands: Kluwer.
Kaput, J., Schwartz, J. L. & Poholsky, J. (1985). Extensive and intensive quantities in multiplication and division word problems: Preliminary report. In S. K. Damarin & M. Shelton (Eds.), Proceedings of the 7th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Columbus, OH.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York: Macmillan.
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age.
Kieren, T. E. (1995). Creating spaces for learning fractions. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 31–65). Albany, NY: SUNY Press.
Lamon, S. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89–120). Albany, NY: SUNY Press.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research,60, 1–64.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
McClung, L. (1998). A study on the use of manipulatives and their effect on student achievement in high school algebra I class. Masters thesis, Salem-Teikyo University. (ERIC Document Reproduction Services No. ED-425-077).
Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematical activity. Journal for Research in Mathematics Education,29, 121–142.
Mewborn, D. S. (2003). Teaching, teachers’ knowledge, and their professional development. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to the principles and standards for school mathematics (pp. 45–52). Reston, VA: National Council of Teachers of Mathematics.
Moyer, P. S., & Jones, M. G. (2004). Controlling choice: Teachers, students, and manipulatives in mathematics classrooms. School Science and Mathematics,104, 16–31.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
Norman, F. A. (1993). Integrating research on teachers’ knowledge of functions and their graphs. In T. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 159–187). Hillsdale, NJ: Erlbaum.
Olive, J., & Caglayan, G. (2008). Learners’ difficulties with quantitative units in algebraic word problems and the teacher’s interpretation of those difficulties. International Journal of Science and Mathematics Education,6, 269–292.
Piaget, J. (1972). Development and learning. In C. S. Lavatelli & F. Stendler (Eds.), Readings in child behavior and development (3rd edition, pp. 38–46). New York: Harcourt, Brace, Jovanovich. (Reprinted from R. Ripple & V. Rockcastle (1964) (Eds.), Piaget rediscovered (pp. 7–19). Ithaca, NY: Cornell University Press).
Puchner, L., Taylor, A., O’Donnell, B., & Fick, K. (2008). Teacher learning and mathematics manipulatives: a collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons. School Science and Mathematics,108, 313–325.
RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education (MR-1643-OERI). Santa Monica, CA: RAND.
Resnick, L. B., & Singer, J. A. (1993). Protoquantitative origins of ratio reasoning. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 107–130). Hillsdale, NJ: Erlbaum.
Roschelle, J. (1990). Designing for conversations. Paper presented at the annual meeting of the American Educational Research Association, Boston.
Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Hillsdale, NJ: Lawrence Erlbaum Associates.
Sharp, J. M. (1995). Results of using algebra tiles as meaningful representations of algebra concepts. Paper presented at the annual meeting of the Mid-Western Education Research Association, Chicago, IL.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher,15, 4–14.
Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics,43, 271–292.
Smith, J., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95–132). New York: Lawrence Erlbaum Associates.
Sowder, J., Armstrong, B., Lamon, S., Simon, M. A., Sowder, L., & Thompson, A. (1998a). Educating teachers to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education,1, 127–155.
Sowder, J., Philipp, R., Armstrong, B., & Schappelle, B. (1998b). Middle-grade teachers’ mathematical knowledge and its relationship to instruction: A research monograph. Albany: State University of New York Press.
Steffe, L. P. (1988). Children’s construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 119–140). Hillsdale, NJ: Lawrence Erlbaum Associates.
Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–39). Albany, NY: SUNY Press.
Suydam, M., & Higgins, J. (1977). Activity-based learning in elementary school mathematics: Recommendations from research. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. (ERIC Document Reproduction Service No. ED 144 840).
Takahashi, A. (2002). Affordances of computer-based and physical geoboards in problem-solving activities in the middle grades. Dissertation Abstracts International, 63(11), 3888. (UMI No. 3070452).
Thompson, P. W. (1988). Quantitative concepts as a foundation for algebraic reasoning: Sufficiency, necessity, and cognitive obstacles. In M. Behr (Ed.), Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 163–170). Dekalb, IL.
Thompson, P. W. (1989, April). A cognitive model of quantity-based algebraic reasoning. Paper presented at the Annual Meeting of AERA, San Francisco.
Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics,25, 165–208.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). Albany, NY: SUNY Press.
Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. Sowder & B. Schapelle (Eds.), Providing a foundation for teaching middle school mathematics (pp. 199–221). Albany, NY: SUNY Press.
Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology,18, 37–54.
Van Oers, B. (1996). Learning mathematics as a meaningful activity. In L. P. Steffe, P. Nesher, P. Cobb, J. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 91–114). Hillsdale, NJ: Lawrence Erlbaum Associates.
Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). New York: Academic.
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 141–161). Hillsdale, NJ: Lawrence Erlbaum Associates.
Vergnaud, G. (1994). Multiplicative conceptual field: What and why? In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 41–59). Albany, NY: SUNY Press.
Vergnaud, G. (1997). Towards a cognitive theory of practice. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity, an ICMI study (Vol. 2, pp. 227–240). Dordrecht: Kluwer Academic Publishers.
Zbiek, R. (1998). Prospective teachers’ use of computing tools to develop and validate functions as mathematical models. Journal for Research in Mathematics Education,29, 184–201.
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.
Author information
Authors and Affiliations
Department of Mathematics and Statistics/Department of Teaching and Learning, Florida International University, 11200 8th Street, Miami, FL, 33199, USA
Caglayan, G. Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles.
J Math Teacher Educ16, 349–378 (2013). https://doi.org/10.1007/s10857-013-9237-4
