STUDENTS' OPPORTUNITIES TO ENGAGE IN ...

STUDENTS’ OPPORTUNITIES TO ENGAGE IN TRANSFORMATIONAL ALGEBRAIC ACTIVITY IN DIFFERENT BEGINNING ALGEBRA TOPICS AND CLASSES Michal Ayalon & Ruhama Even

International Journal of Science and Mathematics Education ISSN 1571-0068 Volume 13 Supplement 2 Int J of Sci and Math Educ (2015) 13:285-307 DOI 10.1007/s10763-013-9498-5

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Author's personal copy

MICHAL AYALON and RUHAMA EVEN

STUDENTS’ OPPORTUNITIES TO ENGAGE IN TRANSFORMATIONAL ALGEBRAIC ACTIVITY IN DIFFERENT BEGINNING ALGEBRA TOPICS AND CLASSES Received: 5 March 2013; Accepted: 30 November 2013

ABSTRACT. This study compares students’ opportunities to engage in transformational (rule-based) algebraic activity between 2 classes taught by the same teacher and across 2 topics in beginning algebra: forming and investigating algebraic expressions and equivalence of algebraic expressions. It comprises 2 case studies; each involves a teacher teaching in two 7th grade classes. All 4 classes used the same textbook. Analysis of classroom videotapes (15–19 lessons in each class) revealed that the opportunities to engage in transformational algebraic activity related to forming and investigating algebraic expressions were similar in each teacher’s 2 classes. By contrast, substantial differences were found between 1 teacher’s classes with regard to the opportunities to engage in transformational algebraic activity related to equivalence of algebraic expressions. The discussion highlights the contribution of the interplay among the mathematical topic, the teacher, and the class to shaping students’ learning opportunities. Specifically, the mathematical topic appeared to play a prominent role in certain situations, with the topic involving deductive reasoning generating high variation in classes of 1 teacher but not in the other’s. KEY WORDS: beginning algebra, classroom research, classes taught by the same teacher, deductive reasoning, equivalence of expressions, mathematical topic, opportunities to learn, transformational algebraic activity

Research suggests that students’ opportunities to engage in mathematics vary across different classes that use the same textbook, even when taught by the same teacher (Eisenmann & Even, 2009, 2011). However, how factors, such as the mathematical topic, the class, and the teacher contribute to shaping students’ opportunities to engage in mathematics, is not well understood. The study reported here examines this issue in the context of transformational (rule-based) algebraic activity—a central component of school algebra (Kieran, 2007). Scrutinizing two case studies, we compare students’ opportunities to engage in transformational algebraic activity between two classes taught by the same teacher and across two topics in beginning algebra. All four classes used the same textbook. International Journal of Science and Mathematics Education (2015) 13(Suppl 2): S285YS307 # National Science Council, Taiwan 2013

Author's personal copy S286

MICHAL AYALON AND RUHAMA EVEN

THEORETICAL BACKGROUND This section comprises two parts. The first deals with transformational algebraic activity in beginning school algebra. The second part reviews relevant research on students’ opportunities to engage in mathematics in different classes. Transformational Algebraic Activity in Beginning Algebra Transformational algebraic activity is a central component of many beginning algebra curricula. It is one of the three types of algebraic activity (i.e. generational, transformational, and global/meta-level) classified by Kieran (2007) as the core of school algebra. Transformational algebraic activities are “rule-based” activities, such as collecting like terms, simplifying expressions, factoring, and substituting numerical values into expressions. School algebra has traditionally centered on the transformational aspects of algebraic activity, emphasizing rule following and symbol manipulation, frequently without attention to conceptual understanding and meaning (Kieran, 2007; Usiskin, 1988). However, transformational algebraic activity need not be confined to rule following and rote symbol manipulation. Rather, it can play an important role in conceptual understanding and meaning in algebra (Kieran, 2007). For example, simplifying expressions might be presented as a unidirectional process, as an end in itself, promoting an “operational view” of the equal sign (i.e. “do something”) (Knuth, Stephens, McNeil & Alibali, 2006) and students’ tendency to conjoin or “finish” algebraic expressions (Tirosh, Even & Robinson, 1998). In contrast, manipulating expressions using properties of real numbers (i.e. simplifying and expanding expressions) could be presented as a principal means for generating and proving equivalence, reflecting a bidirectional process that represents a symmetric relationship between two expressions that denote the same object. Substitution of numbers into expressions also plays an important role in conceptual understanding and meaning in the case of equivalence of expressions. This transformational activity is closely connected to the essence of equivalence of expressions; namely that algebraic expressions are equivalent if and only if any substitution of numerical values into the expressions produces identical outcomes. Consequently, it serves as a useful way of proving non-equivalence (refutation by counter example). Transformational algebraic activity plays a smaller yet significant role in conceptual understanding and meaning also in the case of forming algebraic expressions, an activity that is essentially a generational algebraic activity (Kieran, 2007). For example, substituting numerical values into expressions

Author's personal copy OPPORTUNITIES ENGAGING TRANSFORMATIONAL ALGEBRAIC ACTIVITY

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can help develop a feeling for the behavior of expressions. Substitution can also assist in showing that an expression does not represent a generalization of a given pattern, by testing specific cases. Moreover, choosing wisely how to manipulate expressions using properties of real numbers can assist in engineering algebraic expressions for a desired purpose; an important characteristic of symbol sense (Arcavi, 1994) and a significant component of algebraic problem solving (Boero, 2001). As exemplified above, in the case of equivalence of algebraic expressions, a key role of the two transformational algebraic activities—manipulating expressions using properties of real numbers and substitution of numerical values into expressions—is related to proving the equivalence or nonequivalence of expressions. Engaging in these proof-related activities requires extensive use of deductive reasoning (logically inferring conclusions from known information), which is known to be difficult for students (e.g. Harel & Sowder, 2007). Instead, students often employ inductive reasoning (generalizing from a pattern or observations made in specific cases), which is considered to be the simplest and most pervasive form of everyday problem-solving activities (Nisbett, Krantz, Jepson & Kunda, 1983), and is often students’ preferred way to form and test mathematical conjectures (Harel & Sowder, 2007). Thus, students often mistakenly embrace substitution of numerical values into expressions as a means of proving equivalence (Smith & Phillips, 2000)—a specific case of supportive examples for a universal statement as mathematically invalid. In contrast, in the case of forming and investigating algebraic expressions, substitution of numerical values into expressions supports the development of meaning for algebraic expressions as algebraic objects. In this case, substitution of numerical values into expressions plays an important role in developing a sense about the behavior of expressions (e.g. Even, 1998). Students’ Opportunities to Engage in Mathematics in Different Classes Research suggests that students’ opportunities to engage in mathematics vary across different classes even when they use the same textbook (e.g. Even & Kvatinsky, 2009, 2010; Gresalfi, Barnes & Cross, 2012; Manouchehri & Goodman, 2000; Tirosh, Even & Robinson, 1998). These studies highlight the prominent role that teachers play in shaping curriculum enactment and underscore their influential role in determining the nature of the learning experiences provided to students, as well as the mathematical ideas that students learn—a role that no curriculum program by itself can fulfill. Research has also suggested that students’ opportunities to engage in mathematics vary across different classes that use the same textbook, even



when taught by the same teacher (e.g. Eisenmann & Even, 2009, 2011). Eisenmann and Even found significant differences between two classes of the same teacher, which used the same algebra textbook, in the number of mathematical tasks related to transformational activity worked on in class. The studies presented above indicate that students’ opportunities to engage in mathematics are shaped by factors such as the textbook, the teacher, and the class. Building on this literature, the present study incorporates a focus on a new factor—the mathematical topic—and examines how students’ opportunities to engage in mathematics are shaped by the mathematical topic, the class, and the teacher. For this purpose, we used data collected as part of Eisenmann & Even’s (2009, 2011) two case studies, and compared students’ opportunities to engage in transformational algebraic activity. Unlike Eisenmann & Even’s study (2009, 2011), which has taken transformational activity as a whole without distinguishing among various types, the current study unpacked the transformational activity into different roles, purposes and characteristics, as revealed during work in the classes. In addition, the current study compared students’ opportunities to engage in transformational algebraic activity across two beginning algebra topics: equivalence of algebraic expressions and forming and investigating algebraic expressions. These topics were chosen because, as described earlier, transformational algebraic activity plays different roles in each topic and requires different kinds of reasoning. This paper compares students’ opportunities to engage in transformational algebraic activity between two classes taught by the same teacher and across two topics in beginning algebra. To this end, we compare the percentage of time and the distribution of each kind of transformational-related work in the teaching sequence between two classes taught by the same teacher and across two topics: forming and investigating algebraic expressions and equivalence of algebraic expression.

METHODOLOGY This study comprises two case studies. In each, one teacher taught two 7th grade classes, each in a different school. All four classes (from four different schools) used the same textbook (Robinson & Taizi, 1997). The Curriculum Materials The 7th textbook used in the four classes was part of the Everybody Learns Mathematics curriculum program (Robinson & Taizi, 1995-2002), one of the


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mathematics curriculum programs (i.e. textbook series and associated materials such as teacher guides) for junior-high school developed in Israel in the 1990s in the spirit of the Standards-based curriculum programs in the USA. One main characteristic of this curriculum program is that students are expected to work cooperatively on small group investigation assignments for much of the class time. Following small group work, the curriculum materials (textbook and teacher guide) suggest assignments for whole-class work and discussions aimed at advancing students’ mathematical understanding and conceptual knowledge by involving them in the consolidation of important mathematical ideas. The curriculum materials comprise detailed lesson plans, including the period of time suggested for each assignment. According to this curriculum program, the first 15 textbook units (each designed for a 45-min lesson) for 7th grade center on two topics, which are the focus of this study: (1) forming and investigating algebraic expressions, and (2) equivalence of algebraic expressions. Units 1–3 deal with problem situations that require finding rules for visual patterns and forming algebraic expressions (e.g. task 1, Fig. 1). Units 4–5 involve investigating algebraic expressions, mainly by substitution of numerical values into expressions (e.g. task 2, Fig. 1). Units 6–7 deal with identifying, generating, and justifying equivalence and non-equivalence of expressions (e.g. task 3, Fig. 1). Units 8–13 are reinforcement units (e.g. task 4, Fig. 1). Units 14–15 are summary units. Participants and Setting Sarah (pseudonym) taught upper elementary school grades (up to grade 6) for 8 years. The year of the study was Sarah’s first year teaching in 7th grade and her first year teaching in both S1 and S2 schools. The S1 school was a selective Jewish religious girls’ junior-high school (grades 7–9). The S2 school was a secular junior-high school (grades 7–9) located in a rural area. Both S1 and S2 were categorized by the Ministry of Education to be in the upper 30th percentile in the socioeconomic index (SES). Active participation of most students characterized the S1 class (20 students). The S2 class (27 students) was characterized by a lack of student participation and frequent disciplinary problems (Eisenmann & Even, 2009). By and large, Sarah followed the teaching sequence suggested by the curriculum materials. Class work in both of her classes consisted entirely of work on assignments from the textbook. A typical lesson in Sarah’s classes included work on small group tasks for much of the class time, followed by whole-class work focusing on consolidation of the



1. Let's make trains from "matches". …

How many matches did you use for each "train"? How many matches are needed to build a train having r squares? Doron said: “For the first square, four matches are needed. Then three matches are added each time. We have r squares; therefore the expression is 4 + 3·r”. Is this an appropriate expression? Discuss and justify your claim. 2. Consider the algebraic expression 4 – k: Find a positive number and a negative number whose substitution yields a positive result. Is there a positive number whose substitution yields a negative result? Demonstrate it. Is there a negative number whose substitution yields a negative result? Explain why. 3. The following are pairs of expressions: 2· (a + 5) 2·a + 5 a· (a + 1)

4·a – 2

5·a + 5

3·a + 5 + 2·a

5 – 2·a

3·a

5· (a + 2)

5·a + 10

(a) Substitute the numbers 1 and 2 in the pairs of expressions. Cross out pairs of expressions that are not equivalent. (b) Can you tell for certain that the remaining pairs of expressions are equivalent? Explain. (c) Substitute the number 3 in the remaining pairs of expressions. Cross out the pairs of expressions that are not equivalent. (d) Substitute the number 4 in the remaining pairs of expressions. Are the remaining expressions equivalent? 4. Write equivalent algebraic expressions for the expression 6·k – 12. Which of these expressions easily show that substituting whole numbers into the expression results in a number, which when divided by 6, leaves no remainder?

Figure 1. Examples of textbook assignments (abbreviated from Robinson & Taizi, 1997)

mathematical ideas embedded in the small group tasks. In some lessons, in both classes, the whole-class work included revisiting tasks from the assigned small group work. Rebecca (pseudonym), the other teacher, had taught mathematics in the R1 school for 28 years, and in the R2 school for the previous 5 years as well. The R1 school was a secular elementary school (grades 1–8) whose


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students came mostly from five communal (kibbutzim) and cooperative (moshavim) settlements. This school was categorized by the Israeli Ministry of Education to be in the upper 30th percentile of the socioeconomic index (SES). The R2 school was a secular elementary school (grades 1–8) in a small town. The students’ families were from middle and low socioeconomic backgrounds, and the Israeli Ministry of Education categorized the school to be in the 50th percentile of the socioeconomic index (SES). The R1 class (20 students) was cooperative, with students appearing to be highly motivated during the observed lessons. In the R2 class (24 students), most students participated actively although at times they experienced difficulties in engaging with the mathematics (Eisenmann & Even, 2011). Like Sarah, Rebecca generally followed the teaching sequence suggested by the curriculum materials. Class work in both of Rebecca’s classes consisted almost entirely of work on assignments from the textbook. Rebecca’s lessons in both classes usually began with checking students’ homework. In R1, and in about half of the lessons in R2, homework checking was followed by small group work and then whole-class work focusing on consolidation of the mathematical ideas embedded in the small group work. In the other half of the lessons in R2, homework checking was immediately followed by whole-class work, mostly on textbook tasks suggested either for small group or for whole-class work. Interviews with the teachers revealed that they both liked the way the textbook presented mathematics to students (Eisenmann & Even, 2009, 2011). However, according to Rebecca, it lacked practice assignments; thus, she made changes regarding the way in which she enacted the curriculum materials in her classrooms, some of which were in accordance with students’ contributions to the lesson. In contrast, Sarah said that she hardly made any changes in the way she enacted the curriculum materials in her classrooms. Data Collection The data sources for this study were the data collected as part of Eisenmann and Even’s (2009, 2011) study. The main data source was videotapes of the lessons related to the teaching of the first 15 textbook units that center on the two topics—forming and investigating algebraic expressions, and equivalence of algebraic expressions—in each of the four classes. These videotapes showed that Sarah and Rebecca followed the core of the teaching sequence suggested by the curriculum program,



mostly covering the same written textbook units. Four of the 15 units were not enacted in either of the classes (units 10–13), another one in S2 and R2 (unit 9), and an additional one in R2 (unit 8). The videotapes covered 19 lessons in S1, 15 lessons in S2, 16 lessons in R1, and 17 lessons in R2. Other data sources collected as part of Eisenmann & Even’s (2009, 2011) study were audiotapes of a semi-structured interview with each teacher that dealt with the ways the teachers perceived the curriculum program, and the differences in teaching it in the two classes, as well as lesson field notes. Data Analysis The analysis process focused on generating categories for the roles, purposes, and characteristics of the transformational algebraic activity in the four classes. To achieve that, we first watched the videotapes of the 67 lessons several times and, for each class, identified the teaching sequence of (1) the series of the lessons observed and (2) each lesson. For each lesson, we recorded the lesson topics, the problems and tasks on which the class worked, and the class organization. All the lessons in the four classes consisted almost entirely of work on problems and tasks from the textbook. Because of the large amount of data, detailed data analysis of the lessons included only the whole-class work. This included 225 min in S1, 185 min in S2, 312 min in R1, and 436 min in R2. For each class, we identified the whole-class work that was associated with transformational algebraic activity. This included substituting numerical values into expressions and manipulating expressions using properties of real numbers. For example, the work on assignments 2, 3, and 4 in Fig. 1, and the last part of assignment 1 in Fig. 1. We recorded (1) the length of each transformational-related whole-class work, (2) its place in the teaching sequence (i.e. the lesson number), and (3) the roles of the teacher and the students in the development of the mathematics. In both of Sarah’s classes and in both of Rebecca’s classes, whole-class work that included transformational algebraic activity comprised about one-half of the time that was devoted to whole-class work (see Table 1). We developed a coding scheme by examining the roles, purposes, and characteristics of transformational activity during whole-class work in the four classes. We used open coding (Strauss, 1987) to generate initial categories. The initial categories were constantly compared to new data and refined. In addition to the authors, two other researchers (one of whom was a leading developer of the curriculum program) participated in


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this process. About 30 % of the data was coded independently by each researcher, followed by a comparison of the codes. All disagreements were resolved by discussion and a consensus was reached. The process resulted in 14 categories: 2 categories were related to technical practice and 12 were associated with aspects, ideas, and issues connected to developing understanding and meaning making—some of which represent invalid mathematical ideas that were suggested by students. The 14 categories are discussed in detail below. Four of the categories were associated with meaning of algebraic expressions as algebraic objects. Three of these categories were related to different ways of using substitution of numerical values into expressions (one mathematically invalid), and one category was related to expanding and simplifying expressions. These categories are denoted by the prefix EX (for expression): EX1: Substituting numerical values into expressions as a means to show that an expression does not represent a generalization of a given pattern. EX2: Substituting numerical values into expressions as a means to prove that an expression represents a generalization of a given pattern (mathematically invalid)1. EX3: Substituting numerical values into expressions as a means to develop a sense about the behavior of expressions. EX4: Expanding and simplifying expressions as a means to engineer expressions for a desired purpose. Eight categories were associated with understanding of equivalence of algebraic expressions, mainly in the context of proving equivalence or non-equivalence. Six of these categories were related to different ways of

TABLE 1 Percentage of time (minutes) devoted to whole-class work that included transformational algebraic activity out of the total whole-class work (m), in each of Sarah’s classes and Rebecca’s classes Class

% Transformational work (min)

S1 (m=225) S2 (m=185) R1 (m=312) R2 (m=436)

52 58 50 48

% % % %

(118) (108) (155) (195)



using substitution of numerical values into expressions (one was mathematically invalid). Two categories were related to expanding and simplifying expressions. These categories were denoted by the prefix EQ (for equivalence): EQ1: Substituting numerical values into expressions as a means to prove non-equivalence. EQ2: Substituting numerical values into expressions as an inadequate means to prove equivalence. EQ3: Substituting numerical values into expressions as a means to examine the potential of equivalence. EQ4: Substituting numerical values into expressions as a means to prove equivalence (mathematically invalid)2. EQ5: Substituting all numerical values into expressions as a means to prove equivalence. EQ6: Substituting “representative” numerical values into expressions as a means to prove equivalence. EQ7: Expanding and simplifying expressions as a means to maintain/ prove equivalence. EQ8: Expanding and simplifying expressions as a means to prove nonequivalence. Finally, two categories were associated with whole-class work that comprised technical practice, in which substituting numerical values into expressions or simplifying algebraic expressions was explicitly called for, was presented as an end in itself, and was unrelated to underlying ideas, concepts and meanings, such as the behavior of a given expression or the generation of an equivalent expression. These categories were denoted by the prefix TP (for technical practice): TP1: Technical practice of substituting numerical values into expressions. TP2: Technical practice of simplifying expressions. We used the 14 categories to code all the whole-class work that included transformational algebraic activity (some work was associated with more than one category). We then compared the two classes taught by each teacher on (1) the percentage of time devoted to each kind (i.e. category) of transformational-related whole-class work and (2) the occurrences of each kind of transformational related whole-class work in the teaching sequence.


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Finally, we used the whole data corpus of videotapes, field notes, and final interviews to interpret the results.

FINDINGS Below, we compare the transformational algebraic activity in the two classes of each teacher regarding the two mathematical topics, first in Sarah’s classes and then in Rebecca’s classes. Each section begins with a general description of the whole-class transformational-related work associated with the specific topic, followed by a presentation of the percentage of time allotted to each kind of transformational-related whole-class work, and the distribution of each kind of transformational-related whole-class work in the teaching sequence of each of the teacher’s classes. Sarah’s Classes Analysis of classroom data revealed much similarity in Sarah’s two classes in the opportunities provided for students to engage in transformational algebraic activity, for each of the two mathematical topics. The first instance of transformational-related work associated with developing meaning for algebraic expressions occurred in both classes as part of an exploration that involved the forming of expressions that describe the general rule for a sequence of “match trains” (assignment 1 in Fig. 1). When examining the suitability of the expression 4+3·r to represent the number of matches that are needed to build a train having r squares, both classes substituted a number in the given expression, built and counted the number of matches in the corresponding train, and compared the two results, reaching the conclusion that the given expression is not suitable (EX1). A similar activity took place in both classes during the subsequent lesson. A few lessons later, in both classes, Sarah suggested the use of substitution of numerical values into expressions as a means to develop a sense about the behavior of expressions (EX3). Similar work took place in both classes later in the lesson as well as in a subsequent lesson. Much later in the teaching sequence, Sarah worked with her classes on choosing wisely how to expand or simplify expressions for a given purpose (EX4). For example, when working on assignment 4 in Fig. 1, Sarah asked students in both classes to suggest an expression that is equivalent to the expression 6·k−12, so that it would be easy to see that the result of substituting whole numbers is always divisible by 6. She then suggested using the distributive property to get the expression 6·(k−2) to fulfill the task’s



demand. Similar activities took place in both classes later in the lesson and in S1 in a subsequent lesson. The first brief instance of transformational-related work associated with developing understanding of equivalence of algebraic expressions occurred in both of Sarah’s classes rather early in the teaching sequence, as part of the work on assignment 1 in Fig. 1. After students suggested several expressions that represent the number of matches needed to build a train of r wagons (e.g. 3·r+1 and 4+3·(r−1)), Sarah demonstrated how to move from one expression to another, using properties of real numbers, such as the distributive property (EQ7). However, she did not present it explicitly as a tool for proving the equivalency of two expressions. The next occurrence of using transformational activity to develop understanding of equivalence of algebraic expressions was much more significant. It occurred in both of Sarah’s classes much later in the teaching sequence, when they worked on assignment 3 in Fig. 1, and its follow-up. Students in both of Sarah’s classes began the work on this problem by substituting the suggested numbers into the given pairs of expressions and crossing out the pairs of expressions that resulted in different values (EQ1). Then, in both classes, when left on the board with pairs not crossed-out, Sarah stated that substitution could not be used to prove that the two given expressions are equivalent (EQ2). She explained that there might be a number that had not yet been substituted, but its substitution in the two given expressions would result in different values. In both classes, Sarah presented this idea as a motivation to find a method to show equivalence, and introduced explicitly the use of properties of real numbers to manipulate expressions as a means for proving equivalence (EQ7). Later on, when asked to prove nonequivalence, students in both of Sarah’s classes adopted the method of expanding and simplifying expressions—which Sarah suggested for proving equivalence—also as a means to prove non-equivalence (EQ8). A similar activity occurred in a subsequent lesson. In later lessons, both of Sarah’s classes devoted considerable class time to using properties of real numbers to manipulate expressions as a means of proving equivalence (EQ7). Occasionally, students embraced this method to prove non-equivalence (EQ8). Table 2 and Fig. 2 present quantitative accounts of the above. Table 2 shows, for each kind of transformational related whole-class work, the percentage of time allotted to such work out of the total time devoted to transformational-related whole-class work, in each of Sarah’s classes. As can be seen in Table 2, the parts of the whole-class time devoted to each kind of transformational work were similar in the two classes, for each of the two mathematical topics.


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TABLE 2 Percentage of time (minutes) devoted to each kind of transformational whole-class work out of the total transformational whole-class work (m), in Sarah’s classes Class

EX1

EX3

EX4

EQ1

EQ2

EQ4

EQ7

EQ8

S1 (m=118)

3% (4) 4% (4)

27 % (32) 28 % (29)

22 % (26) 16 % (17)

9% (10) 10 % (12)

6% (8) 5% (5)

–

63 % (75) 57 % (62)

4% (5) 4% (5)

S2 (m=108)

1% (1)

Sometimes more than one kind of transformational-related whole-class work was addressed simultaneously

Figure 2 presents the distribution of each kind of transformational-related whole-class work, in the teaching sequence of each of Sarah’s classes. As can be seen, each kind of transformational-related work—either associated with developing meaning for algebraic expressions or with developing under-

i EXi

i=1–4

j EQj

j=1–8

Time (minutes)

(a)

lesson #

Time (minutes)

(b)

lesson #

Figure 2. Distribution of each kind of transformational-related whole-class work in the teaching sequence of each of Sarah’s classes: S1 (a) and S2 (b)



standing of equivalence of algebraic expressions—generally occurred at parallel places in the two teaching sequences. Rebecca’s Classes Analysis of classroom data revealed that the opportunities provided for students to engage in transformational algebraic activity associated with developing meaning for algebraic expressions were similar in Rebecca’s two classes. However, substantial differences were found between Rebecca’s classes with regard to students’ opportunities to engage in transformational algebraic activity associated with developing understanding of equivalence of algebraic expressions. As in Sarah’s classes, the first instance of transformational-related work associated with developing meaning for algebraic expressions occurred in both of Rebecca’s classes when working on assignment 1 in Fig. 1, with both classes using substitution to show that the expression 4+3·r was not suitable (EX1). In a subsequent lesson, students in R2 used substitution of numerical values into expressions several times as a means of proving that an expression represents a generalization of a given pattern (EX2). To demonstrate that this was mathematically invalid, Rebecca pointed out the possibility that equal results in such cases could be just a coincidence. Again, as happened in Sarah’s classes, when working on assignment 2 in Fig. 1 in a subsequent lesson, Rebecca’s two classes used substitution of numerical values as a means to develop a sense about the behavior of expressions (EX3). Similar work took place in both classes later in the lesson as well as in a subsequent lesson, and in another lesson towards the end of the teaching sequence in R1. The first instance of transformational-related work associated with developing understanding of equivalence of algebraic expressions occurred in R2 only at the beginning of the second half of the teaching sequence, but rather early in the first half of the teaching sequence of R1, when as part of the work on the “match trains” problem (Fig. 1, assignment 1), Rebecca introduced the use of substitution of numerical values into expressions as a means to examine the potential of equivalence (EQ3). Rebecca asked the R1 students to substitute numbers into (equivalent) expressions, to compare the results, and to sense their sameness. She emphasized that this activity was intended only to examine the potential of equivalence, and that in order to determine equivalence for certain, another method was needed. Later that lesson and in the next lesson, using the expressions students in R1 suggested as representing the number of matches needed to build a train of r wagons, Rebecca, like Sarah, demonstrated how to move from one expression to


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another using properties of real numbers (EQ7). However, she did not present it explicitly as a tool to prove that two given expressions are equivalent. By contrast, when working on the “match trains” problem in R2, Rebecca did not include any transformational-related whole-class work associated with developing understanding of equivalence of algebraic expressions. Instead, the whole-class work in R2 focused solely at this stage on generational-related work, generalizing visual patterns and generating corresponding algebraic expressions. The next occurrence of transformational algebraic activity associated with equivalence of algebraic expressions was much more significant and started rather similarly in the two classes. It occurred in both of Rebecca’s classes during the middle of the respective teaching sequence, when the classes were working on assignment 3 in Fig. 1, and its follow-up. As in Sarah’s classes, in both of Rebecca’s classes students began the work on this problem by substituting the suggested numbers into the given pairs of expressions and crossing out pairs of expressions that resulted in different values (EQ1). At this stage, students in both classes needed to determine whether the expressions in the remaining pairs that had not been crossed out were equivalent. However, in contrast to Sarah, who at this point suggested using properties of real numbers herself, in both R1 and R2 Rebecca encouraged her students to find a method by themselves that works. Consequently, different scenarios developed in each class, resulting in the occurrence of different kinds of transformational algebraic activity associated with equivalence of algebraic expressions. In R1, students continually referred to the idea that substitution per se is not an adequate means for proving equivalence (EQ2). Repeatedly, after substituting numbers in pairs of expressions and obtaining the same value, the class concluded that the pairs appeared to be equivalent but that it was impossible to know for certain. When facing the problem of determining whether the remaining pairs of expressions were equivalent, the R1 students tried to use substitution, but in mathematically sophisticated ways. On the basis of the students’ proposals, the discussion centered on two proposals both of which had a valid mathematical flavor but were either not doable or were incomplete: the possibility of proving equivalence by substituting all numbers (EQ5) or by substituting “representative” numbers (EQ6). Rebecca then changed the focus of the activity in R1 to looking for a connection between the two algebraic expressions in each pair, as a transitional move towards using properties of real numbers as a means for proving equivalence (EQ7). Occasionally, R1 students embraced this method to prove nonequivalence (EQ8).



In R2, in contrast to R1, during work on assignment 3 in Fig. 1 and its follow-up, over and over again, students embraced the mathematically invalid use of substitution into expressions as a means for proving equivalence (EQ4). They continued to claim that the expressions were equivalent because all the numbers they substituted resulted in identical numerical answers. Without explicitly providing the idea that substitution per se is not an appropriate means to prove equivalence, Rebecca asked the class to find new expressions that would be equivalent to the given ones. Eventually, R2 embraced the idea that equivalence can be determined by manipulating the form of expressions, using properties of real numbers (EQ7). However, substitution continued to be suggested as an adequate means for determining equivalence (EQ4) during the remaining work in this set of lessons. Occasionally, R2 students embraced the method of expanding and simplifying expressions as a means of proving non-equivalence (EQ8). Finally, as in Sarah’s classes, considerable class time was devoted at the end of the teaching sequence in R1 to using properties of real numbers to manipulate expressions as a means of proving equivalence (EQ7). By contrast, no transformational-related whole-class work associated with developing understanding of equivalence of algebraic expressions occurred in R2 at this stage of the teaching sequence. In addition to using transformational algebraic activities to help develop a sense for algebraic expressions, and understanding of the notion of equivalence of algebraic expressions, Rebecca assigned technical practice as part of whole-class work. The first occurrence of technical practice in both classes centered on practicing substitution of numerical values into expressions (TP1) (e.g. exercise in Fig. 3). This kind of technical practice occurred towards the end of the first half of the teaching sequence during a series of lessons that focused on the use of substitution of numerical values into expressions as a means to develop a sense about the behavior of expressions (EX3). A similar activity took place in R2 in a subsequent lesson and in both classes in another lesson towards the end of the teaching sequence. In addition to assigning technical practice that centered on practicing substitution, Rebecca assigned technical practice that dealt with simplifying expressions (TP2) (e.g. exercise in Fig. 4). This kind of technical practice occurred only in R2. Such practice work occurred repeatedly and extensively during the second half of the teaching sequence. Rebecca’s choice to devote a vast amount of time to this kind of technical practice in R2 seemed to be related to the difficulties encountered by R2 students in preceding work related to the equivalence of algebraic expressions, as described in the previous section.


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Substitute the numbers 0, 1/2, and -1 into the expression 4·y – 3 and find the answer. Figure 3. Example of technical practice focusing on substitution of numerical values into expressions

Table 3 and Fig. 3 present quantitative accounts of the above. Table 3 shows, for each kind of transformational-related whole-class work, the percentage of time allotted to such work out of the total time devoted to transformational-related whole-class work, in each of Rebecca’s classes. As can be seen in Table 3, the parts of the whole-class time devoted to each kind of transformational work associated with developing meaning for algebraic expressions were similar in the two classes. However, substantial differences were found between Rebecca’s classes in the parts of the whole-class time devoted to work associated with developing understanding of equivalence of algebraic expressions. In R1, as in Sarah’s classes, a considerable part of the time was devoted to the use of expanding and simplifying expressions as a means of maintaining/proving equivalence (EQ7), but much less so in R2. In addition, unlike in Sarah’s classes, a variety of kinds of work associated with equivalence of expressions occurred in Rebecca’s classes, with different kinds arising in each class: EQ2, EQ3, EQ5, and EQ6 in R1, and EQ4 in R2. They represent different ideas suggested by students in both classes, in response to Rebecca’s encouragement to propose their own ideas about how to prove equivalence, as exemplified earlier. As also shown in Table 3, substantial differences were also found between Rebecca’s classes in the parts of the whole-class time devoted to technical practice. Figure 5 presents the distribution of each kind of transformationalrelated whole-class work, in the teaching sequence of each of Rebecca’s classes. As can be seen, each kind of transformational-related work associated with developing meaning for algebraic expressions generally occurred at parallel places in the two teaching sequences. However, substantial differences were found in the occurrence of the work associated with developing understanding of equivalence of algebraic expressions. In R1, such work occurred throughout the whole teaching

Simplify the algebraic expression 5·a – a. Figure 4. Example of technical practice focusing on simplifying expressions

2 % (3)

2 % (4)

R1 (m=146)

R2 (m=195)

29 % (45) 26 % (49)

–

2 % (4)

EX3

EX2 7% (10) 6% (11)

EQ1 2% (3) –

EQ2 3% (5) –

EQ3

11 % (22)

–

EQ4 3% (4) –

EQ5

Sometimes more than one kind of transformational-related whole-class work was addressed simultaneously

EX1

Class 1% (2) –

EQ6

48 % (70) 13 % (26)

EQ7

3% (5) 2% (5)

EQ8

8% (12) 15 % (29)

TP1

28 % (55)

–

TP2

Percentage of time (minutes) devoted to each kind of transformational whole-class work out of the total transformational whole-class work (m), in Rebecca’s classes

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TABLE 3

Author's personal copy



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T ime (minutes)

(a)

1 1

lesson #

Time (minutes)

(b)

1 2

1

1

2

2

lesson #

Figure 5. Distribution of each kind of transformational-related whole-class work in the teaching sequence of each of Rebecca’s classes: R1 (a) and R2 (b)

sequence, whereas in R2 it occurred only during the middle of the teaching sequence. As also shown, considerable differences were found between Rebecca’s classes with regard to the occurrences of technical practice.

DISCUSSION This study compared students’ opportunities to engage in transformational algebraic activity between two classes taught by the same teacher and across two topics in beginning algebra. All four classes used the same textbook. In general, the opportunities to engage in transformational algebraic activity related to forming and investigating algebraic expressions were found to be similar in Sarah’s two classes as well as in Rebecca’s two classes. Nevertheless, substantial differences were found between Rebecca’s classes with regard to the opportunities to engage in transformational algebraic activity related to equivalence of algebraic expressions.



One of these significant differences between Rebecca’s classes emerged when students had to prove the equivalence of given pairs of expressions. R2 students, unlike R1, repeatedly suggested substituting numerical values into expressions (a specific case of supportive examples for universal statements as mathematically invalid). Another related difference was found with regard to the experiences related to simplifying and expanding expressions provided in each class. The difficulties encountered by R2 students seemed to lead Rebecca, who was attentive to her students and in particular to their mathematical behavior and performance (Eisenmann & Even, 2011), to provide each of her classes with different experiences. R2 had considerably fewer opportunities than R1 (in terms of time and distribution during the teaching sequence) to engage in simplifying and expanding expressions as a means of maintaining or proving equivalence, a core idea in transformational work in algebra, which is known to be difficult for students (Kieran, 2007). Instead, R2 was given extensive technical practice of simplifying expressions. Hence, R1 experienced the use of simplifying and expanding expressions mainly as a means for generating, maintaining, or proving equivalence, reflecting a bidirectional process that represents a symmetric relationship between two expressions that denote the same object (Knuth, Stephens, McNeil & Alibali, 2006). In contrast, R2 experienced the use of simplifying expressions mainly as a unidirectional process, as an end in itself, promoting an “operational view” of the equal sign (Knuth et al., 2006), and students’ tendency to conjoin or “finish” algebraic expressions (Tirosh, Even & Robinson, 1998). A possible explanation for the differences between Rebecca’s classes with regard to proving equivalence may have to do with the interrelationships between the nature of the specific mathematical topic, the specific teacher, and the specific classes. Work associated with proving equivalence involves extensive use of deductive reasoning, which is known to be difficult for students and counteracts students’ tendency to use inductive reasoning (e.g. Harel & Sowder, 2007). Rebecca’s students, who were encouraged to propose their own ideas about how to prove equivalence, indeed suggested ideas that resembled inductive reasoning, involving substitution of numerical values into expressions. Whereas in R1 these ideas had a valid mathematical flavor (e.g. substituting all numbers or “representative” numbers), in R2 the prevailing idea was substitution per se. Rebecca tried to encourage R2 to find a different method, but she did not provide them with explicit motivation to do so. Such motivation is important (Booth, 1988), especially considering that the students were strongly encouraged to use substitutions in the preceding lessons in the teaching sequence when investigating algebraic expressions.


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In contrast, work on investigating algebraic expressions basically involved inductive reasoning. This form of reasoning is often the one used during problem-solving activity in everyday life (Nisbett et al., 1983). It is also known that students often tend to use inductive reasoning to form and justify mathematical conjectures (Harel & Sowder, 2007). Consequently, the transformational-related work associated with developing meaning for expressions, such as using substitutions of numerical values into expressions to learn about the behavior of expressions, was well suited to the students’ preferences. Unlike Rebecca, in Sarah’s classes the opportunities to engage in transformational algebraic activity related to both forming and investigating algebraic expressions and equivalence of algebraic expressions were found to be similar. A possible explanation for these similarities can be related to Sarah’s teaching approach of taking a total responsibility for offering the mathematical ideas in the classroom. Sarah’s approach apparently prevented the mathematical topic and the class from playing a dominant role. Recent research suggests that factors such as the textbook, the teacher, and the class, contribute to shaping students’ opportunities to engage in mathematics (e.g. Eisenmann & Even, 2009, 2011). Our study identified the mathematical topic, an under-researched factor, as a factor that seemed to contribute greatly to the variations in the students’ opportunities to engage in mathematics, with the topic involving deductive reasoning generating high variation. These findings highlight the need for further research into the role of the mathematical topic, and in particular, inductive- and deductive-related topics, in shaping students’ learning opportunities. These findings also have implications for curriculum designers and teacher educators in attempts to support teachers in maintaining the level of cognitive demands of tasks, addressing a problem reported in Stein, Grover & Henningsen (1996), rather than taking over certain task components and doing them for the students (as in Sarah’s case). Teachers however also need tools and guidance to deal with varied scenarios, including different ideas—sometimes invalid ones—suggested by students that can develop in classes in which students take active part in mathematical activities (as in Rebecca’s case). In our case, for example, it is possible that presenting a counter-example to the students’ “inductive proof” would have provoked the need for another way of proving. In terms of Kieran’s (2007) useful model for conceptualizing school algebra, this study unpacked the nature of one of the three types of algebraic activity, i.e. transformational activity, by focusing on its roles, purposes, and characteristics during class work related to two beginning



algebra topics. The two conceptually different uses of transformational activity—developing meaning for algebraic expressions and understanding the notion of equivalence of algebraic expressions—along with the various categories identified in this study, can serve as a basis for providing teachers, curriculum developers, and researchers with a tool for analyzing, comparing, and assessing algebra teaching and learning, written lessons, and students’ opportunities to engage in mathematics in different classes.

ACKNOWLEDGMENTS The authors would like to thank Miriam Carmeli and Naomi Robinson for their assistance with this study.

NOTES 1

Actually two substitutions, each resulting in a value equals to the number of matches in the corresponding train, are sufficient to prove that a linear expression represents a generalization of a given pattern—information that was unknown to the students at this stage. 2 Similarly, two substitutions, each resulting in equal values, are sufficient to prove equivalence of a pair of linear expressions—information that was unknown to the students at this stage.

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