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 StateUniversity, 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'sdoctoral dissertation at the University of Michigan, under the direction of the second author, andwith the guidance of Jack Smith, Deborah Loewenberg Ball, and Magdalene Lampert. Thanksas well to Brian My and Angeline Ti for their help in running studies, to Emily Bouk and AndreaKaye for their help coding, and to Valerie Mills and Ellen Hopkins for their assistance in therecruitment of participants. Financial support was provided by a grant to the second author fromthe Office of Naval Research.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 1
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, error-
prone, 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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 2
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; Karmiloff-
Smith, 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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 3
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 4
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 allthe 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 theright 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
2 This solution procedure is "standard" in that it is commonly (and often explicitly) taught in US schools as the bestway 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 allcapital 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 (MOVEVARS), move constant terms to other side (MOVE CONST), and multiple or divide to both sides (DIVIDE).
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 5
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
4 If all (x + 1) terms are temporarily written as z, then equation B becomes 4z + 2z = 3(x + 4). Combing the z termsyields 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 pointwould be 6z = 3z. Subtracting 3z from both sides yields 3z = 0. Substituting back for z gives 3(x + 1) = 0.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 6
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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 7
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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 8
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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 9
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 10
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 pre-
test). 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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 11
(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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 12
"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
6 Although the instruction given to students was "good" (e.g., well-enacted, clearly and legibly presented, well-paced, well-delivered by an experienced teacher), it was certainly not a model of how I believe students should beinstructed in the use of algebra equations. The study was not about evaluating the effectiveness of a particularinstructional program; it was about the knowledge that students developed (largely on their own) about equationsolving procedures.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 13
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 14
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).
7 Throughout this paper, innovations are indicated by bold and all capital text.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 15
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 pre-
test. 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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 16
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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 17
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
8 It was decided that students should not be penalized on the flexibility measure for solving two identical equationsusing 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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 18
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 post-
instruction 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 post-
instruction test, indicating that instruction was given to both conditions without bias.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 19
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 20
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 post-
test 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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 21
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 re-
solve 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) morefrequently as well, c2(1, N = 32) = 1.88, n.s.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 22
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 23
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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 24
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 25
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 26
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 re-
solving 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
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 27
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 alaboratory 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 theindividual study in this paper, but I make classroom-based recommendations nevertheless.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 28
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 29
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 re-
conceptualize procedural knowledge so as to include a more mindful outcome. If flexibility and
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 30
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.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 31
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Table 1Example 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)
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 37
Table 2Zöe’s solutions
Problem A Problem B Problem C
4(x + 1) + 2(x + 2) = 3(x + 4)4x + 4 + 2x + 4 = 3x + 12
6x + 8 = 3x + 123x = 4x = 4/3
4(x + 1) + 2(x + 1) = 3(x + 4)4x + 4 + 2x + 2 = 3x + 12
6x + 6 = 3x + 123x = 6x = 2
4(x + 1) + 2(x + 1) = 3(x + 1)4x + 4 + 2x + 2 = 3x + 3
6x + 6 = 3x + 33x = -3x = -1
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 38
Table 3Ashley’s solutions
Problem A Problem B Problem C
4(x + 1) + 2(x + 2) = 3(x + 4)4x + 4 + 2x + 4 = 3x + 12
6x + 8 = 3x + 123x = 4x = 4/3
4(x + 1) + 2(x + 1) = 3(x + 4)6(x + 1) = 3(x + 4)
6x + 6 = 3x + 123x = 6x = 2
4(x + 1) + 2(x + 1) = 3(x + 1)6(x + 1) = 3(x + 1)3(x + 1) = 0
x + 1 = 0x = -1
Table 4
Types of innovations in linear equation solving
InnovationT
ransformation
used atypicallyE
xample problem
Solution using standard solutionprocedure
Innovative (atypical) solution
1.C
HA
NG
E IN
VA
RIA
BL
E
CO
MB
INE
VA
RS
4(x + 1) + 2(x + 1) = 04(x + 1) + 2(x + 1)
=0
4x + 4 + 2x + 2=
06x + 6
=0
6x=
-6x
=-1
4(x + 1) + 2(x + 1)=
06(x + 1)
=0
x + 1=
0x
=-1
MO
VE
VA
RS
7(x + 2) = 2(x + 2) + 57(x + 2)
=2(x + 2) + 5
7x + 14=
2x + 4 + 57x + 14
=2x + 9
5x=
-5x
=-1
7(x + 2)=
2(x + 2) + 55(x + 2)
=5
x + 2=
1x
=-1
2.C
AN
CE
L A
TE
RM
MO
VE
VA
RS
6x + 5x + 2x = 10 + 6x + 2x6x + 5x + 2x
=10 + 6x + 2x
13x=
10 + 8x5x
=10
x=
2
6x + 5x + 2x=
10 + 6x + 2x5x
=10
x=
2
3.M
UL
TIP
LY
FIR
ST
DIV
IDE
0.2x + 0.7 = 1.30.2x + 0.7
=1.3
0.2x=
0.6x
=3
0.2x + 0.7=
1.32x + 7
=13
2x=
6x
=3
4.D
IVID
EF
IRST
DIV
IDE
3(x + 1) = 123(x + 1)
=12
3x + 3=
123x
=9
x=
3
3(x + 1)=
12x + 1
=4
x=
3
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 40
Table 5Results, Study 1
All Treatment Control t/c2
Variable n = 32 n = 17 n = 15 p-valueFirst session assessments
Pre-test (% correct) 24% 26% 22%Post-instruction test (% correct) 88% 85% 90%
Session problemsProblems attempted 36 31 42 **% of problems with correct answers 83% 85% 81%% of problems with no transformation errors 78% 74% 84% ~
Post-test problemsProblems attempted 19 19 19% of problems with correct answers 77% 76% 79%% of problems with no transformation errors 91% 90% 92%
All problemsProblems attempted 55 50 61 **% of problems with correct answers 80% 81% 80%% of problems with no transformation errors 83% 80% 86%
Measures of innovationInnovation 44% 59% 27% ~Innovation Type #1 (CHANGE IN VARIABLE) 6% 12% 0%Innovation Type #2 (CANCEL A TERM) 22% 29% 13%Innovation Type #3 (MULTIPLY FIRST) 9% 6% 13%Innovation Type #4 (DIVIDE FIRST) 22% 41% 0% **
Measures of efficiencyEfficiency (High/Moderate) 41%/50% 35%/53% 47%/47%Combine first 84% 82% 87%Move opposite 47% 47% 53%
Measure of flexibilityFlexibility 59% 77% 40% *
** p<.01; * p<.05; ~ p<.10
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 41
Table 6Chris’ solutions to problem 2.3 and 2.4
Problem 2.3: 3(x + 1) = 31. EXPAND 3x + 3 = 152. MOVE CONST 3x = 123. DIVIDE x = 4
Problem 2.4 (repeat of 2.3): 3(x + 1) = 31. DIVIDE FIRST x + 1 = 52. MOVE CONST x = 4
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 42
Table 7Jack’s solutions to problems 4.6, 4.12, and 4.13
Problem 4.6: (2nd repeat of 4.4) 0.1x + 0.8 = 0.2x + 0.3x1. MULTIPLY FIRST x + 8 = 5x2. MOVE VARS 8 = 4x3. DIVIDE 2 = x
Problem 4.12: (repeat of 4.11) 0.5x + 0.2 = 0.3(x + 2)1. MULTIPLY FIRST 5x + 2 = 3(x + 2)2. EXPAND 5x + 2 = 3x + 63. MOVE CONST 5x = 3x + 44. MOVE VARS 2x = 45. DIVIDE x = 2
Problem 4.13: 0.3(x + 1) + 0.1(x + 1) = 0.2(x + 1) + 0.4x + 0.41. EXPAND 0.3x + 0.3 + 0.1x + 0.1 = 0.2x + 0.2 + 0.4x + 0.42. MULTIPLY FIRST 3x + 3 + x + 1 = 2x + 2 + 4x + 43. MOVE VARS 3 + 1 = 2x + 2 + 44. MOVE CONST 0 = 2x + 25. MOVE CONST -2 = 2x6. DIVIDE -1 = x
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 43
Table 8John’s solutions to problems 4.1 and 4.3
Problem 4.1: 2(x + 3) + 4x = 4x + 6x + 21. CANCEL A TERM 2(x + 3) = 6x + 22. EXPAND 2x + 6 = 6x + 23. MOVE CONST
MOVE VARS 4 = 4x4. DIVIDE 1 = x
Problem 4.3: 3(x + 1) + 6(x + 1) + 6x + 9 = 6x + 91. CANCEL A TERM 3(x + 1) + 6(x + 1) = 02. EXPAND 3x + 3 + 6x + 6 = 03. COMBINE CONST
COMBINE VARS 9 + 9x = 04. MOVE CONST 9x = -95. DIVIDE x = -1
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 44
Table 9Casper’s solutions to post-test problems 14, 15, and 18
Problem 14: 5(x + 1) + 10(x + 1) + 5(x + 1) = 5x + 10x1. EXPAND 5x + 5 + 10x + 10 + 5x + 5 = 5x + 10x2. MOVE VARS 5 + 10x + 10 + 5x + 5 = 10x3. MOVE VARS 5 + 10 + 5x + 5 = 04. MOVE CONST 5 + 5x + 5 = -105. MOVE CONST 5x + 5 = -156. MOVE CONST 5x = -207. DIVIDE x = -4
Problem 15: 3(x + 1) + 6(x + 1) + 6x + 9 = 6x + 91. EXPAND 3x + 3 + 6x + 6 + 6x + 9 = 6x + 92. MOVE CONST 3x + 3 + 6x + 6 + 6x = 6x3. MOVE VARS 3x + 3 + 6x + 6 = 04. MOVE CONST 3x + 3 + 6x = -65. MOVE CONST 3x + 6x = -96. MOVE VARS 3x = -157. DIVIDE x = -5
Problem 18: 3(x + 2) + 9(x + 2) = 6(x + 2)1. EXPAND 3x + 6 + 9x + 18 = 6x + 122. MOVE VARS 6 + 9x + 18 = 3x + 123. MOVE VARS 6 + 6x + 18 = 124. MOVE CONST 6x + 18 = 65. MOVE CONST 6x + 12 = 06. MOVE CONST 6x = -127. DIVIDE x = -2
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 45
Table 10Tess’ solutions to post-test problems 14, 15, and 18
Problem 14: 5(x + 1) + 10(x + 1) + 5(x + 1) = 5x + 10x1. EXPAND
COMBINE VARS 5x + 5 + 10x + 10 + 5x + 5 = 15x2. COMBINE CONST
COMBINE VARS 20x + 20 = 15x3. MOVE VARS 20 = -5x4. DIVIDE -4 = x
Problem 15: 3(x + 1) + 6(x + 1) + 6x + 9 = 6x + 91. EXPAND 3x + 3 + 6x + 6 + 6x + 9 = 6x + 92. COMBINE CONST
COMBINE VARS 15x + 18 = 6x + 93. MOVE CONST 15x + 9 = 6x4. MOVE VARS 9 = -9x5. DIVIDE -1 = x
Problem 18: 3(x + 2) + 9(x + 2) = 6(x + 2)1. EXPAND 3x + 6 + 9x + 18 = 6x + 122. COMBINE CONST
COMBINE VARS 12x + 24 = 6x + 123. MOVE CONST
MOVE VARS 12 = -6x4. DIVIDE -2 = x
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 46
Table 11Minnie’s solutions on the 6 eligible problems on the post-test
Problem 14: 5(x + 1) + 10(x + 1) + 5(x + 1) = 5x + 10x1. CHANGE IN VARIABLE 20(x + 1) = 5x + 10x2. COMBINE VARS 20(x + 1) = 15x3. EXPAND 20x + 20 = 15x4. MOVE VARS 5x + 20 = 05. MOVE CONST 5x = -206. DIVIDE x = -4
Problem 15: 3(x + 1) + 6(x + 1) + 6x + 9 = 6x + 91. CHANGE IN VARIABLE 9(x + 1) + 6x + 9 = 6x + 92. CANCEL A TERM 9(x + 1) = 03. EXPAND 9x + 9 = 04. MOVE CONST 9x = -95. DIVIDE x = -1
Problem 16: 4(x + 2) + 6x + 10 = 2(x + 2) + 8(x + 2) + 6x + 4x + 81. CANCEL A TERM 4(x + 2) + 10 = 2(x + 2) + 8(x + 2) + 4x + 82. CHANGE IN VARIABLE 2(x + 2) + 10 = 8(x + 2) + 4x + 83. MOVE CONST 2(x + 2) + 2 = 8(x + 2) + 4x4. EXPAND 2x + 4 + 2 = 8x + 16 + 4x5. COMBINE VARS 2x + 4 + 2 = 12x + 166. COMBINE CONST 2x + 6 = 12x + 167. MOVE CONST 2x = 12x + 108. MOVE VARS 0 = 10x + 109. MOVE CONST -10 = 10x10. DIVIDE -1 = x
Problem 17: 4x + 3x + 5x + 4 = 3x + 5x + 161. COMBINE VARS 12x + 4 = 8x + 162. MOVE CONST 12x = 8x + 123. MOVE VARS 4x = 124. DIVIDE x = 3
Problem 18: 3(x + 2) + 9(x + 2) = 6(x + 2)1. CHANGE IN VARIABLE 12(x + 2) = 6(x + 2)2. CHANGE IN VARIABLE 6(x + 2) = 03. EXPAND 6x + 12 = 04. MOVE CONST 6x = -125. DIVIDE x = -2
Problem 19: 2(x + 3) + 4x + 8 = 4(x + 2) + 6x + 2x1. EXPAND 2x + 6 + 4x + 8 = 4x + 8 + 6x + 2x2. CANCEL A TERM 2x + 6 = 6x + 2x3. MOVE VARS 6 = 6x4. DIVIDE 1 = x
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 47
Table 12David’s solutions on the 6 eligible problems on the post-test
Problem 14: 5(x + 1) + 10(x + 1) + 5(x + 1) = 5x + 10x1. EXPAND 5x + 5 + 10x + 10 + 5x + 5 = 5x + 10x2. COMBINE CONST
COMBINE VARS 20x + 20 = 15x3. MOVE VARS 20 = -5x4. DIVIDE -4 = x
Problem 15: 3(x + 1) + 6(x + 1) + 6x + 9 = 6x + 91. EXPAND 3x + 3 + 6x + 6 + 6x + 9 = 6x + 92. COMBINE CONST
COMBINE VARS 15x + 18 = 6x + 93. MOVE CONST
MOVE VARS 9x = -94. DIVIDE x = -1
Problem 16: 4(x + 2) + 6x + 10 = 2(x + 2) + 8(x + 2) + 6x + 4x + 81. EXPAND 4x + 8 + 6x + 10 = 2x + 4 + 8x + 16 + 6x + 4x + 82. COMBINE CONST
COMBINE VARS 10x + 18 = 20x + 283. MOVE CONST
MOVE VARS -10x = 104. DIVIDE x = -1
Problem 17: 4x + 3x + 5x + 4 = 3x + 5x + 161. COMBINE VARS 12x + 4 = 8x + 162. MOVE CONST
MOVE VARS 4x = 123. DIVIDE x = 3
Problem 18: 3(x + 2) + 9(x + 2) = 6(x + 2)1. EXPAND 3x + 6 + 9x + 18 = 6x + 122. COMBINE CONST
COMBINE VARS 12x + 24 = 6x + 123. MOVE CONST
MOVE VARS 12 = -6x4. DIVIDE -2 = x
Problem 19: 2(x + 3) + 4x + 8 = 4(x + 2) + 6x + 2x1. EXPAND 2x + 6 + 4x + 8 = 4x + 8 + 6x + 2x2. COMBINE CONST
COMBINE VARS 6x + 14 = 12x + 83. MOVE CONST
MOVE VARS 6 = 6x4. DIVIDE 1 = x
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 48
APPENDIX A
Equations used in problem-solving sessions
# a Treatment Problem Soln Control Problem Soln1.1 3x = 3 + 12 x = 5 same as treatment1.2 4x + 2 = 10 x = 2 same as treatment1.3 4x = 2x + 12 x = 6 same as treatment1.4 10x - 5x = 20 x = 4 same as treatment1.5 4x + 10 = 2x + 16 x = 3 same as treatment1.6* 4x + 10 = 2x + 16 x = 3 6x + 9 = 3x + 12 x = 12.1 6x + 8 = 2x + 16 x = 2 15x + 10 = 5x + 20 x = 12.2* 6x + 8 = 2x + 16 x = 2 same as treatment2.3 3(x + 1) = 15 x = 4 2(x + 3) = 10 x = 22.4* 3(x + 1) = 15 x = 4 same as treatment2.5* 3(x + 1) = 15 x = 4 5(x + 2) = 20 x = 22.6 4(x + 3) = 8x x = 3 same as treatment2.7 3(x + 2) + 9x = 3x + 15 x = 1 2(x + 5) + 4x = 4 + 8x x = 32.8* 3(x + 2) + 9x = 3x + 15 x = 1 same as treatment2.9* 3(x + 2) + 9x = 3x + 15 x = 1 5(x + 3) + 10x = 35 + 5x x = 22.10 2(x + 4) + 6x = 8x + 4x x = 2 same as treatment2.11 4(x + 3) = 2(x + 3) x = -3 3(x + 2) = 6(x + 2) x = -2
2.12* 4(x + 3) = 2(x + 3) x = -3 same as treatment2.13 9(x + 2) + 3(x + 2) + 3 = 6x + 9 x = -3 same as treatment3.4* 3(x + 2) = 21 x = 5 same as treatment3.5* 3(x + 2) = 21 x = 5 5(x + 3) = 25 x = 23.6 2(x + 1) + 6(x + 1) = 4(x + 1) x = -1 same as treatment3.7 2(x + 1) + 6(x + 1) + 4(x + 1) = 10x
+ 8xx = 2 same as treatment
3.8* 2(x + 1) + 6(x + 1) + 4(x + 1) = 10x+ 8x
x = 2 3(x + 2) + 6(x + 2) + 9(x + 2) = 3x+ 6x
x = -4
3.9* 2(x + 1) + 6(x + 1) + 4(x + 1) = 10x+ 8x
x = 2 5(x + 1) + 10(x + 1) + 5(x + 1) =5x + 10x
x = -4
3.10 2x + 6 = 14 x = 4 same as treatment3.11 2(x + 1) + 4(x + 1) + 6x + 8 = 6(x +
1) + 8x + 2x = 3 same as treatment
3.12* 2(x + 1) + 4(x + 1) + 6x + 8 = 6(x +1) + 8x + 2
x = 3 3(x + 2) + 6(x + 2) + 3 + 9x = 9(x+ 2) + 6 + 6x
x = 1
3.13 2(x + 5) = 4(x + 5) x = -5 same as treatment4.1 2(x + 3) + 4x = 4x + 6x + 2 x = 1 same as treatment4.2* 2(x + 3) + 4x = 4x + 6x + 2 x = 1 3(x + 2) + 6x + 9x = 18 + 6x x = 14.3 3(x + 1) + 6(x + 1) + 6x + 9 = 6x + 9 x = -1 same as treatment4.4 0.1x + 0.8 = 0.2x + 0.3x x = 2 same as treatment4.5* 0.1x + 0.8 = 0.2x + 0.3x x = 2 0.2x + 0.4x = 0.9 + 0.3x x = 3
aDesignates problems for which the treatment group engaged in the alternative ordering task.
RE-CONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 49
# a Treatment Problem Soln Control Problem Soln4.6* 0.1x + 0.8 = 0.2x + 0.3x x = 2 0.5x + 0.3x = 0.6x + 0.8 x = 44.7 4(x + 2) + 6x + 10 = 2(x + 2) + 8(x
+ 2) + 6x + 4x + 8x = -1 6(x + 1) + 9(x + 1) + 9 + 6x = 12(x
+ 1) + 9x + 6x + 6x = 1
4.8* 4(x + 2) + 6x + 10 = 2(x + 2) + 8(x+ 2) + 6x + 4x + 8
x = -1 same as treatment
4.9* 4(x + 2) + 6x + 10 = 2(x + 2) + 8(x+ 2) + 6x + 4x + 8
x = -1 5(x + 1) + 15(x + 1) + 30 + 15x =10(x + 1) + 15x + 20 + 20x
x = 2
4.10 2(x + 3) + 4x + 8 = 4(x + 2) + 6x +2x
x = 1 same as treatment
4.11 0.5x + 0.2 = 0.3(x + 2) x = 2 same as treatment4.12* 0.5x + 0.2 = 0.3(x + 2) x = 2 0.3(x + 5) + 0.1x = 0.3 x = -34.13 0.3(x + 1) + 0.1(x + 1) = 0.2(x + 1)
+ 0.4x + 0.4x = -1 same as treatment
aDesignates problems for which the treatment group engaged in the alternative ordering task.