RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE: FLEXIBILITY AND INNOVATION IN EQUATION SOLVING

JON R. STAR MICHIGAN STATE UNIVERSITY COLLEEN SEIFERT UNIVERSITY OF MICHIGAN

Paper presented at the annual meeting of the American Educational Research Association (AERA), New Orleans, LA., April, 2002. Correspondence should be addressed to: Jon R. Star, 242 Erickson Hall, Michigan State University, East Lansing, Michigan, 48824, [email protected], 517-353-2958, www.umich.edu/~jonstar Acknowledgements: The research described here was conducted as part of the first author's doctoral dissertation at the University of Michigan, under the direction of the second author, and with the guidance of Jack Smith, Deborah Loewenberg Ball, and Magdalene Lampert. Thanks as well to Brian My and Angeline Ti for their help in running studies, to Emily Bouk and Andrea Kaye for their help coding, and to Valerie Mills and Ellen Hopkins for their assistance in the recruitment of participants. Financial support was provided by a grant to the second author from the Office of Naval Research.

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Re-conceptualizing procedural knowledge: Flexibility and innovation in equation solving

Introduction For much of this century, mathematics educators have sought to address students’ tendency to view school mathematics as a series of procedures to be memorized. Researchers in mathematics education concur that (a) procedures learned by rote are easily forgotten, errorprone, and resistant to transfer; and (b) the learning of procedures must be connected with conceptual knowledge in order to foster the development of understanding (e.g., Hiebert & Carpenter, 1992). The National Council of Teachers of Mathematics (NCTM) has articulated this emphasis on conceptual learning by calling for decreased attention to “memorizing rules and algorithms; practicing tedious paper-and-pencil computations; memorizing procedures ... without understanding“ (NCTM, 1989, p. 71); and “rote memorization of facts and procedures” (NCTM, 1989, p. 129). There is little doubt that the rote execution of memorized procedures does NOT constitute "mathematical understanding." However, there are other ways in which a procedure can be executed other than by rote, some of which could be characterized as “intelligent” or even as indicative of “procedural understanding” (Greeno, 1978). But few prior studies have considered procedural outcomes other than rote knowledge, much less explored its development. This paper attempts to map out this terrain: I examine the development of students’ knowledge of mathematical procedures, with particular emphasis on learning outcomes other than rote execution.

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Rationale Fundamentally, executing a procedural skill requires that one have knowledge of its component steps and the order in which these steps should be applied. But not all performances of a skill are the same. In particular, skillful execution in mathematics can mean two very different things. On the one hand, skillful execution involves being able to use procedures rapidly, efficiently, with minimal error, and with minimal conscious attention; in other words, to execute a procedure automatically or by rote (Anderson & Lebiere, 1998). On the other hand, being “skilled” means being able to select appropriate procedures for particular problems, modify procedures when conditions warrant, and explain or justify one’s steps to others; that is, to execute a procedure thoughtfully or deliberately (Ericsson & Charness, 1994; KarmiloffSmith, 1992), "relationally" (Skemp, 1976), "intelligently" (Ryle, 1949), or "mindfully" (Brown & Langer, 1990; Langer, 1993). Although acknowledging that both notions of “skilled” are important and necessary (National Research Council, 2001), mathematics educators have had difficulty integrating these two competing visions of mathematical proficiency. The tension between these two visions is a foundational issue in mathematics education: it not only pertains to our educational goals for students, but also speaks directly to what it means to know and to do mathematics. While the first outcome for successful skill execution (automaticity) has been frequently examined by cognitive scientists (Anderson, 1982; Anderson & Fincham, 1994; Anderson & Lebiere, 1998), "mindful" execution of procedures has been less widely studied and is thus the focus of this paper. I begin by articulating what I mean by "mindful" execution of procedures. I then describe a study which explored the development of this capacity.

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Background What does it mean to "mindfully" execute procedures? Consider the work by two hypothetical students, Ashley and Zöe. These two cases, which are an amalgam of the work of many different students that I have worked with in my years as an algebra instructor, are intended to represent two different kinds of successful execution of procedural skills. Ashley’s solutions are designed to exemplify what I mean by mindful execution, while Zöe’s solutions indicate automatic execution. The procedural domain I choose to focus on is linear equation solving. Equation solving is believed by many in the mathematics education community to be a “basic skill” (Ballheim, 1999), and this domain has been widely studied in the mathematics education (e.g., Adi, 1978; Kieran, 1989, 1990; Whitman, 1975) and the cognitive science communities (Bernard & Cohen, 1988; Bundy & Silver, 1981; Lewis, 1981; Matz, 1980, 1982; Sleeman, 1982, 1984). When introduced, the procedures used to solve equations are among the longest and most complex that students have been exposed to, which makes this domain particularly suitable for illustrating the distinctions that I will make below. Consider the 6 linear equations in Table 1.1 All are representative of the kinds of problems that would appear in a typical Algebra I or Algebra II textbook (e.g., Foerster, 1990; Smith et al., 1988). Tables 2 and 3 show solutions from the hypothetical students Ashley and Zöe. (Insert Tables 1, 2, and 3 about here)

1

The reader is encouraged to stop at this point and solve the problems in Table 1.

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Zöe’s solutions The first thing to notice about Zöe’s solutions (Table 2) is that she solves all 3 problems correctly. Her ability to do so indicates that she is adept in the use of the procedures in this domain. However, it is apparent from her work that she knows a limited range of solution procedures. She applies the exact same procedure for each problem. Looking more closely at her solutions, the consistency in which she solves the three problems is apparent. Zöe rigidly adheres to a solution that I refer to as the "standard" solution procedure2, which has the following steps: 1. Use the distributive property to “expand” the parentheses (EXPAND3). 2. Transform the equation to a standard form (ax + b = cx + d) by combining all the variable terms (COMBINE VARS) and constant terms (COMBINE CONST) on each side. 3. Get the variable terms to the left side (MOVE VARS) and the constants to the right side (MOVE CONST). 4. Divide by the coefficient of the variable term on the left side (DIVIDE). Note that there are subtle differences between problems A, B, and C. Zöe either fails to recognize these differences or she chooses to use the same procedure despite the differences. And note again that Zöe is able to successfully solve all 3 problems.

Ashley’s solutions Ashley is also able to solve all 3 problems correctly (see Table 3). However, she shows considerably more variation in her solution procedures. Ashley solves problem A using the

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This solution procedure is "standard" in that it is commonly (and often explicitly) taught in US schools as the best way to solve most linear equations. 3

For the remainder of this paper, I refer to each type of equation solving step with a word or phrase written in all capital letters. The possible steps are: use the distributive property (EXPAND), combine like variable terms (COMBINE VARS), combine like constant terms (COMBINE CONST), move variable terms to other side (MOVE VARS), move constant terms to other side (MOVE CONST), and multiple or divide to both sides (DIVIDE).

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standard solution procedure. However, in problem B, she uses a different procedure. In her first step, she temporarily changes the variable, from x to (x + 1), to combine the (x + 1) terms.4 After this initial “change in variable” step, the remainder of Ashley’s solution on problem B follows the standard solution procedure. Ashley’s solution in C is different from ones she used in either A or B. In her first step, she combines 4(x + 1) and 2(x + 1) to yield 6(x + 1), as she did in problem B. In her second step, she subtracts 3(x + 1) from both sides of the equation.5 At this point, note that the standard solution procedure would suggest that Ashley use the distributive property to “expand” parenthesis, as she did when she reached a similar place on problems A and B (and as Zöe did on all three problems). Ashley chooses again to deviate from the standard solution procedure by dividing both sides of the equation by 3. Ashley uses 3 different solution procedures on these 3 problems. It is clear that she recognizes subtle variations in the initial conditions of each problem (its constants and coefficients), and she takes these variations into account when selecting and executing her solution procedures.

Comparing Ashley and Zöe Both Ashley and Zöe know a set of mathematical procedures that can be used to solve a variety of equations. Each is able to apply these procedures accurately to reach a correct solution. Both can be considered to be adept in the use of procedures in this domain: in fact, on

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If all (x + 1) terms are temporarily written as z, then equation B becomes 4z + 2z = 3(x + 4). Combing the z terms yields 6z = 3(x + 4). Substituting back for z gives 6(x + 1) = 3(x + 4). I refer to this strategy as a change in variable, since instead of combining x terms, it involves combining (x + 1) terms. 5

This is also a change in variable. If the (x + 1) terms were again written temporarily as z, the equation at this point would be 6z = 3z. Subtracting 3z from both sides yields 3z = 0. Substituting back for z gives 3(x + 1) = 0.

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many traditional tests of equation solving that emphasize only a final answer, Ashley and Zöe would earn equal and perfect scores. However, the solutions in Tables 2 and 3 suggest that Ashley and Zöe are not equivalent in their knowledge of how to solve equations. Ashley’s solution procedures are, on the whole, more efficient (fewer steps and/or less difficult calculations) than Zöe’s. Ashley’s greater efficiency may result in fewer errors; her work may also be considered more innovative or even more elegant. Ashley also exhibits greater flexibility in the ability to use procedures than Zöe. One can also imagine classes of unfamiliar problems where Ashley’s flexibility would enable her to solve problems that Zöe could not. In addition, when Ashley learns more advanced symbol manipulation procedures (e.g., in a Calculus course), it may be the case that her greater flexibility and her ability to use multiple procedural approaches to solving will be to her advantage. Ashley and Zöe exemplify two types of “skillful” performance. I consider Ashley to be an example of a more mindful executor. She has knowledge of a number of mathematical procedures, and she is able to use these procedures to solve many types of problems in efficient, creative, and innovative ways. Zöe is an example of an automatic executor. She knows a much smaller set of procedures and is able to use them fluently. Although each can be considered successful, their performances on these example problems point to real differences in what Ashley and Zöe know about equation solving.

Mindful execution My interest is in solvers such as Ashley. What does Ashley know that enables her to use procedures "mindfully"? How does such a capacity develop? A first step in considering these questions is to think more carefully about what distinguishes Ashley's solutions from Zöe's

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solutions on these three problems. With these examples as inspiration, I propose that two central features of "mindful" execution of procedures are (a) flexibility and (b) innovation. Flexibility refers to the ability to use a wide range of mathematical procedures in order to generate the "best" solution for particular problems (Beishuizen, van Putten, & van Mulken, 1997; Feltovich, Spiro, & Coulson, 1997). Flexible solvers have knowledge of "standard" solution procedures, but they also choose to use alternative or non-standard procedures on certain problems, when doing so results in a better or more efficient solution. An inflexible solver always relies on the same procedure, without considering whether there might be an alternative and better route to the solution. Metaphorically, flexible solvers have more "tools" in their procedural "toolbox." Flexibility can be seen by looking at a collection of problems and determining whether a student uses several, or only a few, solution procedures on these problems. Using the examples of Ashley and Zöe above, on the collection of 3 problems, Ashley uses 3 solution procedures (including the standard solution procedure), while Zöe only uses one. Thus Ashley is a more flexible solver. A second feature of mindful execution is innovation (Gick, 1986; Ryle, 1949; Simon & Reed, 1976). Innovation refers to the ability to use steps within a procedure in atypical ways in order to produce a more efficient solution. An innovative solver is able to use the individual steps of a procedure in ways other than that suggested by a standard solution. Metaphorically, innovation refers to the ability to use the "tools" in one's toolbox in non-standard ways that do a better job of performing certain kinds of tasks. Innovation can be seen by looking at students' work in a single problem. Consider how Zöe uses the COMBINE steps in her solution to problem B (see Table 2). In this solution (which follows the standard solution procedure), Zöe combines like variable and constant terms in the

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second step of her solution: she combines 4x and 2x to yield 6x, and 4 and 2 to yield 6. Note that this is how the COMBINE steps are typically used in the standard solution procedure. In Ashley's solution to the same problem, she uses the COMBINE steps differently. Ashley combines as a first step, and she makes a change in variable to combine 4(x + 1) and 2(x + 1) to yield 6(x + 1). Thus, Ashley's use of the COMBINE steps in her solution to problem B is atypical, as compared to the standard solution procedure. Such an atypical use of the COMBINE steps results in a more efficient solution on this problem, and this is example of what I mean by innovation. Thus, in the current conceptualization, mindful solvers both have more tools in their procedural toolboxes (flexibility) and can use their tools in both standard and non-standard ways (innovation).

Alternative ordering task Framing mindful equation solving in this manner raises the question of how innovation and flexibility develop. This question is largely unexplored. Basic skill practice has been linked to the development of rote knowledge (Anderson, 1982; Fitts, 1964), but the development of more expert knowledge appears to require a different kind of practice, which has been referred to as "deliberate" (Ericsson, Krampe, & Tesch-Romer, 1993). One hypothesis for what such deliberate practice looks like comes from studies where participants were asked to solve a problem repeatedly in order to observe changes in their solutions that emerged with practice. There is ample evidence that solving a problem multiple times can lead to more automatic execution (e.g., Simon & Reed, 1976; Anzai & Simon, 1979; Blessing & Anderson, 1996; Koedinger & Anderson, 1990). However, there is also reason to hypothesize that, under certain

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conditions, re-solving previously completed problems can lead to more mindful solving (Blöte, Klein, & Beishuizen, 2000; Krutetskii, 1976). In the present study, I test this hypothesis by utilizing a task I refer to as the "alternative ordering task." Participants are asked to re-solve previously completed problems but using a different ordering of steps. In this task, students are not merely practicing the same solution over and over again, but instead are generating, comparing, and evaluating the effectiveness and efficiency of different solution procedures. There is reason to speculate that such a task may lead to more mindful solving, in the form of greater innovation and flexibility.

Goals The study described below has the following goals. First, I seek to demonstrate that mindful execution of procedures, as described above, exists. This capacity has been infrequently studied among K-12 students and, in fact, has been shown to be relatively rare even among expert mathematicians (Lewis, 1981). Second, I seek to determine how mindful execution develops, particularly the capacities for innovation and flexibility. I am particularly interested in the effect of alternative ordering tasks on the development of mindful solving.

Method

Participants The 36 participants (20 female, 16 male) in this study were recruited on a first-come, first-served basis using flyers that were distributed late in the school year to all 6th grade students in the public school district of a medium-sized city in the midwestern United States.

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Participants were paid $50, in the form of a gift certificate to a local bookstore, to participate in this research. The participants attended schools with an integrated middle school mathematics curriculum -- Connected Mathematics Project [CMP] (Lappan, Fey, Friel, Fitzgerald, & Phillips, 1995). Symbolic algebra is not introduced in CMP until after the 6th grade, and it is not covered at all in the K-5 curriculum. Thus, it is unlikely that, at the time of the study, any of the participants had received any formal instruction in the use of symbolic methods of equation solving. However, a pre-test was administered to assess students’ prior knowledge on the first day of the study. The problems on the pre-test were selected in order to determine whether students had any knowledge of formal equation solving procedures.

General procedure Participants attended one-hour experimental sessions for 5 consecutive days (Monday through Friday), at the same time each day. Students met in groups of 6; on each day, there were 6 sessions of 6 students. The first experimental session on Monday was devoted to administering the pre-test, providing instruction (described below), and administering the post-instruction test. The final session on Friday was used to administer the post-test (which was the same as the pretest). During the 3 remaining sessions (Tuesday through Thursday), students solved algebra equations for the entire hour (described in depth below). All sessions took place in a seminar room on the campus of the University of Michigan. In the first session, I gave students a scripted, 30-minute lesson on the transformations of equation solving. I began the lesson by defining what an equation was and also giving examples and non-examples of equations. I then introduced students to the six basic transformations (referred to, in the lesson, as “steps”) used in solving linear equations: combining constants

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(COMBINE CONST), combining like variable terms (COMBINE VARS), using the distribute property (EXPAND), adding/subtracting a constant from both sides (MOVE CONST), adding/subtracting a variable term from both sides (MOVE VARS), and multiplying/dividing to both sides (DIVIDE). For each transformation, I showed students a symbolic pattern to look for (“When you see this....”) and a transformation that could be applied to that pattern (“you can use this step to rewrite it as ......”). For example, for the “combine like terms” transformation, students were told, “When you see 2x + 3x, you can use the COMBINE step to rewrite it as 5x.” For each transformation, I showed students 4 or 5 examples of how to apply the transformation, and then gave 3 or 4 additional examples that the group did together, in recitation-style format. After all six transformations had been presented, I gave students 5 additional practice problems to complete on their own, and then went over these problems as a group. Note that in this initial lesson, students were never given any strategic instruction as to how the transformations could be used together to solve equations. The focus on instruction was strictly on pattern recognition: identifying which transformation could be used for particular patterns of symbols, and how that transformation was correctly applied. On all example equations presented to the participants during instruction, only one transformation was applied. Students never saw any examples where a transformation was applied and then the resulting pattern of symbols was evaluated as to what might be a good next step. My intent in this brief period of instruction was to provide novice algebra learners with sufficient knowledge to enable them to begin to solve very straightforward equations. However, I wanted students to discover on their own how to solve more complex equations. As a result, the instruction was purposely

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"procedural" and minimal; I wanted to investigate what students came to know about procedures in this domain on their own.6 Immediately following the lesson, students were given a post-instruction test consisting of 12 problems. On the first 6 problems, students were given an expression or equation and asked to re-write it in a different way, using one of the transformations that were just learned. On the remaining 6 problems, students were asked to perform a specific transformation; for example, students were given the equation, 2x = 12, and asked to apply the DIVIDE transformation. This post-instruction test was intended to evaluate whether students learned the material provided during instruction. At the conclusion of the first instructional session, the groups of 6 in each study were randomly assigned to either a control group or a treatment group for the remainder of the week. The treatment and control groups differed only in that the treatment group completed alternative ordering tasks, while the control group did not. On alternative ordering tasks, students were given problems that they had previously solved and asked to re-solve them, but using a different ordering of steps. On problems where the treatment group was asked to provide an alternative solution, the control group completed a different but isomorphic problem. For example, both treatment and control groups were asked to solve the problem, 4x + 10 = 2x + 16. Students in the treatment group were given this same problem again and were asked to solve it using a different ordering of steps. The students in the control group were given a structurally equivalent

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Although the instruction given to students was "good" (e.g., well-enacted, clearly and legibly presented, wellpaced, well-delivered by an experienced teacher), it was certainly not a model of how I believe students should be instructed in the use of algebra equations. The study was not about evaluating the effectiveness of a particular instructional program; it was about the knowledge that students developed (largely on their own) about equation solving procedures.

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problem, 6x + 9 = 3x + 12, instead. Treatment group students regularly engaged in the alternative ordering task, as shown in Appendix A.

Problem-solving sessions During the three problem-solving sessions (Tuesday, Wednesday, and Thursday) students worked individually on all problems. Students sat alone at tables and were positioned far enough from each other so that it was impossible for any participant to see the work of another. The collected problems for the 3 sessions were bound into an 8.5” by 11” spiral notebook, and students worked through a pre-determined number of problems for each session, at their own pace, using pencils. Students were instructed not to erase any of their mistakes, because the research team was interested in looking at their mistakes as well as their correct answers. If participants completed all of the problems allocated for a particular day’s session, they were instructed to remain seated for the rest of the time. There was no student-to-student interaction during any of the sessions of this study. If a student became stuck while attempting a problem, s/he raised his/her hand and was approached by a helper -- either the experimenter or a research assistant. The helper answered the student’s questions in a semi-standardized format. Specifically, the helper corrected the student’s arithmetic mistakes (e.g., if the student multiplied 2 by 3 and got 5, the helper pointed out this error) or reminded him/her of the 6 possible transformations and how each was used. Helpers never gave strategic advice to students, such as suggesting which transformation to apply next or whether one method of solution was any better than another method. Students independently came up with their own choices for which transformations to apply.

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Materials The problems used in the problem-solving sessions of this study are shown in Appendix A. Problems 1.1-1.6 and 2.1-2.13 (a total of 19 problems) were assigned on Tuesday. Problems 3.1-3.13 (13 problems) were assigned on Wednesday, and the remaining 13 problems (4.1-4.13) were assigned on Thursday. Thus, there were a maximum of 45 problems that students solved during the 3 problem-solving sessions. Students who did not finish a particular day’s problems returned the following session and began where they had left off. For example, if a participant completed up to problem 2.6 on Tuesday, she began working on problem 2.7 on Wednesday. As students worked at different paces, not all students had time to complete all 45 problems. The problems shown in Appendix A were carefully designed according to the following principles. Whenever possible, problems had integral coefficients and constants, in order to minimize any cognitive load issues and also to avoid biasing students, because of load issues, toward choosing one strategy over another. In addition and for similar reasons, all problems had integral solutions. The problems gradually increase in complexity, although more straightforward problems are presented in all 3 sessions in order to evaluate changes in students’ approaches to them. In addition, the problems used in this study were designed to give students maximal opportunities to show certain kinds of innovation in their solution strategies. Recall that innovation is the use of a step in an atypical way, in a single problem, that results in a more efficient solution. Problems were created to give students the chance to use 4 types of innovation: CHANGE IN VARIABLE7, CANCEL A TERM, MULTIPLY FIRST, and DIVIDE FIRST (see Table 4).

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Throughout this paper, innovations are indicated by bold and all capital text.

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Innovation Type #1 (CHANGE IN VARIABLE) is the “change in variable” strategy that has been previously discussed. It results from an atypical use of the COMBINE VARS transformation (e.g., adding 4(x + 1) + 2(x + 1) to get 6(x + 1)) or the MOVE VARS transformation (e.g., subtracting 2(x + 2) from both sides of an equation). Innovation Type #2 (CANCEL A TERM) allows solvers to “cancel” a term, and it results from an atypical use of the MOVE CONST or MOVE VARS transformations. Innovation Type #3 (MULTIPLY FIRST) results when solvers multiply an entire equation by 10 in order to eliminate decimal coefficients, using the DIVIDE transformation. Similarly, Innovation Type #4 (DIVIDE FIRST) give students the opportunity to divide an entire equation by a constant. (See Table 4 for examples of each of these 4 innovations.) (Insert Table 4 about here)

Post-test Students took a post-test during the final session. The post-test was identical to the pretest. Students were given 30 minutes to complete the post-test, and all students finished in the allotted time.

Analysis

Coding for transformations Students’ solutions on all problems other than the pre-test (in other words, all problems attempted during the solving sessions and on the post-test) were given to coders. Three coders (2 independent coders and the experimenter) generated a list of the transformations that were observed in each solution and in what order. A total of 1,966 problems were attempted by

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participants; solvers used between 1 and 13 transformations on each problem. On 89% of all problems attempted, at least 2 of the 3 coders generated transformations lists that were identical. On the remaining 11% of the problems, all 3 coders had slightly different codes, so the experimenter examined each of these problems and selected the most accurate code.

Scoring of study instruments The three study instruments (the pre-test, the post-instruction test, and the post-test) were graded by 2 independent coders. For each instrument, two scores were calculated: one for the percentage of problems completed correctly (e.g., those on which the students arrived at the correct numerical answer) and one for the percentage of problems completed without any errors in how transformations were applied, though arithmetic errors were allowed.

Coding for innovations, efficiency, and flexibility A variable was created for each of the 4 types of innovations, indicating whether a student used each innovation at least once on at least one eligible problem attempted during the problem-solving sessions or the post-test. In addition, an overall innovation variable was created from the collection of the 4 innovation variables, indicating whether any innovation was used on at least one eligible problem. In addition to coding for innovations, students' solutions were also coded for whether or not they incorporated an efficient solution method for solving equations. A maximally efficient solution was defined as one having the following two features. First, terms on each side of the equation were combined first, rather than being moved individually from one side to the other side. I refer to this feature as "combine first," and it indicates whether students were able to efficiently transform an equation from its initial state to the form ax + b = cx + b. The second

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feature, which I refer to as "move opposite", indicates whether students were able to efficiently transform an equation from the form ax + b = cx + d to the solution state. Variables were created for each of these two features, and these measures indicated whether a student made the particular efficiency modification on at least one eligible post-test problem. Also, an overall measure of efficiency was created by combining the “move opposite” and “combine first” measures. This variable had 3 levels: high (or “2”), moderate (or “1”), and none (or “0”). Flexibility was operationalized as follows. A subset of the problems on the post-test (the final 6 problems) was identified which met the following criteria: they were similar but not identical8, and they were sufficiently complex so as to allow to many different orderings of solution steps. Looking only at this set of 6 post-test problems, the number that each student attempted that were solved with the exact same sequence of solution steps was calculated. If the student used the exact same solution step sequence on no more than 2 of the 6 problems (e.g., at least 4 of the problems had unique step sequences), he/she was coded as having a high degree of flexibility in his/her solution strategies. If 3 or more problems were solved with the exact same sequence of steps (e.g., fewer than 4 problems with unique step sequences), the solver was coded with as lacking flexibility.

Results Four students were omitted from the analysis. One student withdrew from the study in the middle of the week and one student showed knowledge of formal equation solving techniques on the pre-test. In addition, two students who did not show knowledge of formal

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It was decided that students should not be penalized on the flexibility measure for solving two identical equations using the same sequence of steps. If a solver has discovered a maximally efficient solution for a particular problem, one would hope that she would use this same solution if the exact same problem were encountered again.

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equation solving techniques on the pre-test nevertheless did show such knowledge in the first problem-solving session and were dropped. Thus, of the 36 students initially enrolled in this study, the work of 32 (15 in the control group and 17 in the treatment group) was used in further analysis. The results are shown in Table 5 below. I review the results in the following categories: scoring instrument measures, measures of innovation, measures of efficiency, and measures of flexibility. (Insert Table 5 about here)

Pre-test Students did poorly on the pre-test (M = 24%), indicating lack of prior knowledge of formal equation solving techniques. The few problems that students were able to complete correctly were the very straightforward ones from the beginning of the pre-test. On these initial problems, correct solutions were arrived at using various informal methods, including unwinding and guess-and-check. There were no significant treatment effects on the pre-test, indicating that random assignment of students to condition was done without bias.

Post-instruction test Recall that students were given a short test immediately following instruction in order to assess learning of the 6 equation solving transformations. Students did very well on the postinstruction test (M = 88%), indicating that the short period of instruction led to mastery of the equation solving transformations. There were also no significant treatment effects on the postinstruction test, indicating that instruction was given to both conditions without bias.

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Problem-solving sessions Students attempted an average of 36 problems during the 3 problem-solving sessions. However, there were significant differences in how many problems each group attempted. Over the 3 sessions, the control group solved approximately 11 more problems than did the treatment group: for the control group, M = 42, and for the treatment group, M = 31, t(30) = 3.43, p < .01. Since the conditions were otherwise identical, the most likely explanation for why the control group worked so much faster is that it took treatment students more time to complete a previously-solved problem using a different ordering of steps, as the alternative ordering task requested, then it took the control group to solve an isomorphic problem. Students in this study were quite successful on equations that were solved in the 3 problem-solving sessions. On average, students solved 78% of attempted problems without transformation errors and arrived at the correct solution on 83% of attempted problems. The fact that such a high percentage of attempted problems were solved correctly is noteworthy, especially considering that students had no prior knowledge of formal solving procedures and did not receive any instruction in how to chain together transformations to solve equations. While there were no significant treatment differences in the percentage of attempted problems solved correctly in the problem-solving sessions, there was a marginally significant treatment difference in terms of the frequency of transformation errors in the problem-solving sessions. The control group made fewer mistakes than the treatment group: the treatment group solved only 74% of problems without transformation error, while the control group solved 83% without error, t(30) = 1.90, p < .07. Transformation use errors most likely emerged while treatment students attempted to use transformations in a different way to complete previously solved equations, as the alternative ordering tasks required.

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Post-test Students attempted an average of 18.7 of the 19 problems on the post-test. The rate in which students correctly used transformations on the post-test (M = 91%) was higher than during the problem solving sessions; however, students were less likely to arrive at a correct answer on the post-test (M = 77%). There were no significant treatment differences in the number of posttest problems attempted, the percentage of attempted problem solved without transformation error, or the percentage of problems solved with a correct numerical answer (see Table 5).

All problems Combining the problems solved during the sessions and the post-test, the overall rate in which students arrived at a correct answer (M = 80%) and used transformations without error (M = 83%) were quite high. There were no treatment differences in either of these rates; however, by virtue of the difference in how many session problems each group solved, there was a significant difference in the number of total problems solved by the treatment group (M = 50) and the control group (M = 61), t(30) = 3.28, p < .01.

Measures of innovation 44% of all students in this study used at least one innovation on at least one problem. Innovation Types #2 (CANCEL A TERM) and #4 (DIVIDE FIRST) were the most likely to be discovered by students; each was used at least once by 22% of solvers. Innovation Types #1 (CHANGE IN VARIABLE) and #3 (MULTIPLY FIRST) were not as likely to be discovered by students; only 6% (Type #1) and 9% (Type #3) of students used each of these two innovations at least once.

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This figures are quite impressive and much higher than would be suggested by earlier studies of algebra equation solving. In particular, Lewis (1981) found that both experts and undergraduates were unlikely to innovate in their solutions (9-12% and 20-40% of innovation opportunities, respectively). It is reasonable to expect that 6th graders would have a substantially lower rate of innovation than undergraduates and experts. Considering the brief period of instruction (30 minutes) and practice (3 one-hour sessions), the lack of prior knowledge, and the age of the study participants, the fact that any solvers discovered how to innovate is a surprising finding. There were also significant treatment effects in how frequently students discovered and used innovations. Students in the treatment group were somewhat more likely to use at least one innovation on at least one problem, c2(1, N = 32) = 3.35, p < .07. Approximately twice as many treatment students (M = 59%) as control group students (M = 27%) discovered at least one innovation. In terms of the four individual innovations, the treatment and control groups differed significantly only in Innovation Type #4 (DIVIDE FIRST): 41% of treatment group students discovered this innovation, while none of the control group did, c2(1, N = 32) = 7.91, p < .01.9 Thus, the treatment group’s completion of the alternative ordering tasks, which was the only difference between the treatment and the control groups, led to a greater likelihood that innovations would be discovered and used. Below are some examples of students’ solutions that use innovations. Table 6 shows Chris’ (a student in the treatment group) solutions to problems 2.3 and 2.4. Chris’ solution to problem 2.3 follows the standard solution procedure. On problem 2.4, Chris was asked to resolve problem 2.3 but using a different ordering of steps. When asked to complete this problem 9

There was a trend for treatment students to discover Innovation Type #1 (CHANGE IN VARIABLE) more frequently as well, c2(1, N = 32) = 1.88, n.s.

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another way, Chris discovered Innovation Type #4 (DIVIDE FIRST). Prior to problem 2.4, Chris had never used the DIVIDE transformation in any way other than the last step. This example shows how the alternative ordering task led some students to innovate. (Insert Table 6 about here) Table 7 shows Jack’s use of Innovation Type #3 (MULTIPLY FIRST), which involved multiplying an equation by 10 in order to eliminate decimals. Jack used this innovation on a total of 3 problems (all of which are shown in Table 7). He was a student in the treatment group, and the first two times he used this innovation occurred on alternative ordering tasks (problems 4.6 and 4.12). His third use of this innovation, however, occurred on the first time he solved problem 4.13. In each problem below, Jack multiplies both sides of the equation by 10, which has the effect of changing all decimal constants and coefficients into integers and thus simplifies the problem. (Insert Table 7 about here) Table 8 shows John’s use of Innovation Type #2 (CANCEL A TERM), which involved canceling a term common to both sides of an equation as a first step. John was a student in the control group, and looking for the opportunity to cancel common terms as a first step became part of his standard way of approaching equations. In the problems in Table 8, John canceled variable and constant terms before distributing. Doing so is a very quick way of reducing the number of terms in an equation and thus can result in a more efficient solution. (Insert Table 8 about here)

Measures of efficiency Almost all students in this study discovered how to use the standard solution procedure, with 91% of students earning a rating of at least moderate (“1” or “2”) on the efficiency measure.

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41% of students discovered both how to "combine first” and also how to “move opposite,” while 50% of students discovered only one of these features of the standard solution procedure. Discovering how to “combine first” (combine before moving terms) was very common. 84% of solvers used the “combine first” strategy on at least at least one eligible post-test problem. Discovering how to “move opposite” (move variables and constants to opposite sides) was less common: 47% of solvers used a “move opposite” strategy on over at least one eligible post-test problem. These results indicate that an extremely high percentage of students were able to discover the standard solution procedure on their own, despite the lack of strategic guidance about how to solve equations efficiently. Interestingly, there were no significant differences between the treatment and control groups on any measures of efficiency. Recall that the treatment group completed significantly fewer problems than the control group. The fact that both treatment and control students reached near ceiling levels of efficiency indicates that the treatment group was able to learn to solve equations efficiency despite having less practice. In Tables 9 and 10, I give examples to illustrate efficient and inefficient solvers. After an initial EXPAND step, Casper’s solutions on all three problems in Table 9 are an elaborate series of moves (5 in a row, on each of the above problems). Had Casper chosen to combine variable or constant terms before moving, as the standard solution procedure suggests, he would have been able to complete each of the equations in Table 9 in fewer steps. In fact, it appears that Casper does not know how to use the COMBINE CONST or COMBINE VARS transformations at all; in none of his solutions to these three problems does he ever combine variable or constant terms. In essence, Casper is only able to combine terms by moving them. For example, in problem 14, when faced with the line, 5 + 10 + 5x + 5 = 0, rather than combining

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the 5, 10, and 5 on the same side, Casper combines these constants by performing three MOVE CONST transformations, subtracting 5, then 10, then 5 from both sides, resulting in the equation 5x = -20. (Insert Table 9 about here) Casper’s inefficient solutions can be compared with those of Tess (see Table 10, below). Tess follows the standard solution procedure in all 3 problems. After distributing, she combines all like constant and variable terms on each side, and then she moves constant terms to one side and variable terms to the other side. On all three problems, Tess’ solutions require fewer steps and thus are more efficient than Casper’s. (Insert Table 10 about here)

Measures of flexibility Flexibility refers to how often students used the exact same sequence of transformations to solve a set of eligible problems on the post-test. 59% of participants were flexible solvers, meaning that they used the same sequence of solution steps on no more than 2 of the 6 eligible post-test problems. There were significant treatment effects in students’ flexibility -- in particular, treatment students were more flexible solvers than control students. 77% of treatment students used the same solution step sequence on no more than 2 of the eligible post-test problems, while only 40% of control students did, c2(1, N = 32) = 4.39, p < .05. Tables 11 and 12 gives example solutions for two solvers, one from the treatment group who was flexible and the one from the control group who was inflexible. These two solvers show very different patterns of solutions on the 6 post-test problems that assessed flexibility.

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As shown in Table 11, Minnie, a student in the treatment group, solved 6 problems using 6 different solution methods. In other words, on no two problems does she use the exact same sequence of transformations, and this earns her a “yes” score on the flexibility measure. Note that in addition to being a flexible solver, Minnie is an innovator: on 5 of the 6 problems, she uses at least one innovation. The only problem that Minnie follows the standard solution procedure on problem 17. Clearly Minnie looks at the specific features of each problem and modifies her solution accordingly. However, despite the fact that she is an innovator and is a flexible solver, note that Minnie does not always follow the most efficient solution path. In particular, note that her solution to problem 16 involves a total of 4 MOVE transformations, indicating some redundancy in her solution. (Insert Table 11 about here) Table 12 shows the solutions of David, a student in the control group. David’s solutions are reminiscent of Zöe’s from the beginning of this paper; he uses the exact same sequence of steps on 5 of the 6 problems. David’s solutions would earn him a “no” score on the measure of flexibility, as he repeated the same sequence of steps to solve more than 2 of the 6 problems. It is clear that David rigidly follows the standard solution procedure. (His slight deviation on problem 17 comes only because this problem did not have distributed terms to EXPAND, which is David’s typical first step.) (Insert Table 12 about here) Both Minnie and David solved all 6 problems correctly, yet they represent opposite ends of the flexibility spectrum. Minnie is an extremely flexible solver, while David is a very rigid, inflexible solver.

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Discussion and Conclusions In this study, I found that 6th grade students with minimal prior knowledge were able to figure out how to solve equations in a relatively short period of time, and largely on their own. Study participants developed reliable but sometimes idiosyncratic strategies that they consistently used to arrive at a solution, and they subsequently made modifications to these solution strategies for increased efficiency. More interestingly, I found that engaging in alternative ordering tasks, which involved resolving previously solved equations using a different ordering of steps, led students to believe that equations could be solved in more than one way and that some strategies were better than others. Treatment students’ cognizance of multiple ways that equations can be solved led to an increase in their ability to innovate, where innovation refers to the use of a transformation in an atypical way that generally results in increased efficiency. The ability to innovate was also related to increased flexibility in treatment students’ solutions, where flexibility refers to a reluctance to rigidly adhere to the exact same solution sequence when solving similar problems. The increased flexibility that accompanied participation in the alternative ordering tasks came without a cost to efficiency; indeed, treatment students became more flexible and innovative solvers while remaining as efficient in their solutions as control group students. Students who did not experience this treatment were more likely to develop one solution strategy that was rigidly adhered to on all problems. This finding has not previously been reported in the literature on algebra learning. At the elementary school level, reform documents have long advocated the use of multiple and invented algorithms for solving arithmetic problems (NCTM, 1989, 2000). It has been suspected that allowing children to work with invented arithmetic algorithms, rather than being drilled in the

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use of standard algorithms, is beneficial toward their understanding of number (e.g., Fuson et al., 1997). However, prior to the present study, adapting these methods for post-arithmetic, symbolic mathematics has not been considered. This study suggests that there are significant benefits to having students invent their own symbolic methods of solving equations and subsequently attempt to modify and refine these methods. As almost all students discover the standard solution procedure on their own anyway, there appears to be little cost in efficiency to allowing for discovery as opposed to explicitly teaching the most efficient solution strategy. These studies add to the literature on equation solving by shifting the focus from students’ errors to the capacities that successful performers exhibit. A review of the literature on the use of mathematical procedures (with its emphasis on cataloging the multitude of errors that students make) might suggest that the most important feature of success in this domain is the ability to rapidly execute error-free procedures. The present study suggests that another important feature of a successful solver is the ability to “mindfully” use procedures; that is, to selectively choose to deviate from standard and practiced methods in order to produce even more efficient solutions. Students who have capacities for innovation and flexibility have more sophisticated knowledge of equation solving transformations that only emerges in their application. The fact that the alternative ordering task was effective suggests that the results of this study could be used to inform classroom practice. In particular, three recommendations follow from this study10. First, during instruction on equation solving (and other symbolic mathematical

10

Note that a serious limitation of the study described here is its use of an individualized learning environment in a laboratory setting. As promising as these results may be, it may be the case that they are less applicable in "real" classroom environments. However, it should be noted that I have confirmed these results in a somewhat more "authentic" classroom environment (Star, 2001). Because of space limitations, I have chosen to focus only on the individual study in this paper, but I make classroom-based recommendations nevertheless.

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procedures), teachers should frequently and regularly ask students to re-solve previously completed problems using a different ordering of steps. The multiple solutions that are generated in such a task can then be compared and contrasted. The study described in this paper suggests that the incorporation of such tasks will result in substantial gains in students’ ability to innovate and be flexible, without any cost to their solution efficiency. Implicit within this recommendation is a strong suggestion against direct and explicit instruction on the standard solution method. Especially for novice learners, teachers should avoid labeling any one solution method as being the best way, the right way, or the only way. Benefits that arise from engaging in the alternative ordering tasks come when students think carefully about how to generate additional solution strategies and how to compare multiple solution strategies. Students come to their own conclusions about the features that identify one solution as different from another (e.g., efficiency), and direct instruction on a standard, efficient procedure subverts this process. Teachers need not be concerned that students will fail to discover the standard solution method unless it is shown to them; in fact, the vast majority of students in the study described here came to develop a reliable and efficient solving solution procedure entirely on their own. It follows from these recommendations that the complexity of problems that are typically used during the learning of equation solving techniques needs to be substantially increased. Straightforward problems (e.g., those with only a few terms) are not amenable to being easily solved in multiple ways. Since increasing the problem complexity does not require the learning of any new solving transformations, the present study indicates that students are capable of solving problems that are much more complex than those typically used. Increasing the problem complexity allows students to explore the domain in more depth, via alternative ordering tasks.

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Note that increasing the problem complexity does not refer to making equations more arithmetically challenging but instead to increasing the number of terms within each problem. It is the presence of multiple steps that provides opportunities for novices to reap the benefits of engaging in alternative ordering tasks. A challenge that necessarily accompanies these recommendations concerns student motivation. Many teachers find students to be uninterested in learning how to solve equations, and so it might appear that asking students to re-solve previously completed equations would further reduce already low levels of motivation. This is certainly a valid concern. However, there is a great deal of evidence from the elementary grades that such concerns can be addressed. Many examples exist of classrooms where a climate has been created that incorporates the features that are integral to the recommendations detailed above: student collaboration, the sharing of multiple solution strategies, and the group comparison and evaluation of mathematical procedures and reasoning (Ball, 1993; Chazan & Ball, 1999; Lampert, 1990). There are fewer examples of this kind of classroom environment at the high school level, particularly related to the instruction of mathematical procedures. This paper suggests that there is much to be gained from efforts to make such changes at the secondary level. Procedures are an integral component of mathematics. While fluency is certainly one educational outcome, this paper has identified another in the ability to vary the ways that one uses procedures on particular problems in order to arrive at maximally efficient solutions. While students with rote knowledge of procedures are relatively easy to find, mindful solvers present a much more significant challenge. The study described here represents a first attempt to reconceptualize procedural knowledge so as to include a more mindful outcome. If flexibility and

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innovation in the use of procedures are integral to our educational goals for students, further investigation of the development of this kind of procedural knowledge is vital.

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Table 1 Example problems Problem A

Problem B

Problem C

4(x + 1) + 2(x + 2) = 3(x + 4)

4(x + 1) + 2(x + 1) = 3(x + 4)

4(x + 1) + 2(x + 1) = 3(x + 1)

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Table 2 Zöe’s solutions Problem A 4(x + 1) + 2(x + 2) 4x + 4 + 2x + 4 6x + 8 3x x

= = = = =

Problem B 3(x + 4) 3x + 12 3x + 12 4 4/3

4(x + 1) + 2(x + 1) 4x + 4 + 2x + 2 6x + 6 3x x

= = = = =

3(x + 4) 3x + 12 3x + 12 6 2

Problem C 4(x + 1) + 2(x + 1) 4x + 4 + 2x + 2 6x + 6 3x x

= = = = =

3(x + 1) 3x + 3 3x + 3 -3 -1

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Table 3 Ashley’s solutions Problem A 4(x + 1) + 2(x + 2) 4x + 4 + 2x + 4 6x + 8 3x x

= = = = =

3(x + 4) 3x + 12 3x + 12 4 4/3

Problem B 4(x + 1) + 2(x + 1) 6(x + 1) 6x + 6 3x x

= = = = =

3(x + 4) 3(x + 4) 3x + 12 6 2

Problem C 4(x + 1) + 2(x + 1) 6(x + 1) 3(x + 1) x+1 x

= = = = =

3(x + 1) 3(x + 1) 0 0 -1

Innovation 1. CHANGE IN VARIABLE

2. CANCEL A TERM 3. MULTIPLY FIRST 4. DIVIDE FIRST

Transformation used atypically COMBINE VARS

MOVE VARS

MOVE VARS

DIVIDE

DIVIDE

= = = =

12 12 9 3

Solution using standard solution procedure 4(x + 1) + 2(x + 1) = 0 4x + 4 + 2x + 2 = 0 6x + 6 = 0 6x = -6 x = -1 7(x + 2) = 2(x + 2) + 5 7x + 14 = 2x + 4 + 5 7x + 14 = 2x + 9 5x = -5 x = -1 6x + 5x + 2x = 10 + 6x + 2x 13x = 10 + 8x 5x = 10 x = 2 0.2x + 0.7 = 1.3 0.2x = 0.6 x = 3

Table 4 Types of innovations in linear equation solving Example problem 4(x + 1) + 2(x + 1) = 0

7(x + 2) = 2(x + 2) + 5

6x + 5x + 2x = 10 + 6x + 2x

0.2x + 0.7 = 1.3

3(x + 1) = 12

3(x + 1) 3x + 3 3x x

0 0 0 -1

2(x + 2) + 5 5 1 -1

= = = =

Innovative (atypical) solution

= = = =

4(x + 1) + 2(x + 1) 6(x + 1) x+1 x

7(x + 2) 5(x + 2) x+2 x

= = = = = = =

1.3 13 6 3 12 4 3

6x + 5x + 2x = 10 + 6x + 2x 5x = 10 x = 2

0.2x + 0.7 2x + 7 2x x 3(x + 1) x+1 x

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Table 5 Results, Study 1 All n = 32

Variable First session assessments Pre-test (% correct) Post-instruction test (% correct)

Treatment Control t/c2 n = 17 n = 15 p-value

24% 88%

26% 85%

22% 90%

Session problems Problems attempted % of problems with correct answers % of problems with no transformation errors

36 83% 78%

31 85% 74%

42 81% 84%

Post-test problems Problems attempted % of problems with correct answers % of problems with no transformation errors

19 77% 91%

19 76% 90%

19 79% 92%

All problems Problems attempted % of problems with correct answers % of problems with no transformation errors

55 80% 83%

50 81% 80%

61 80% 86%

**

Measures of innovation Innovation Innovation Type #1 (CHANGE IN VARIABLE) Innovation Type #2 (CANCEL A TERM) Innovation Type #3 (MULTIPLY FIRST) Innovation Type #4 (DIVIDE FIRST)

44% 6% 22% 9% 22%

59% 12% 29% 6% 41%

27% 0% 13% 13% 0%

~

Measures of efficiency Efficiency (High/Moderate) Combine first Move opposite Measure of flexibility Flexibility ** p