Flexibility in problem solving: The case of equation solving

ARTICLE IN PRESS

+ MODEL

www.elsevier.com/locate/learninstrucLearning and Instruction xx (2007) 1e15

Flexibility in problem solving: The case of equation solving

Jon R. Star a,*, Bethany Rittle-Johnson b

a Harvard University, Graduate School of Education, 442 Gutman Library, 6 Appian Way, Cambridge, MA 02138, USAb Vanderbilt University, TN, USA

Received 4 February 2007; revised 16 August 2007; accepted 26 September 2007

Abstract

A key learning outcome in problem-solving domains is the development of flexible knowledge, where learners know multiplestrategies and adaptively choose efficient strategies. Two interventions hypothesized to improve flexibility in problem solving wereexperimentally evaluated: prompts to discover multiple strategies and direct instruction on multiple strategies. Participants were132 sixth-grade students who solved linear equations for three hours. Both interventions improved students’ flexibility in problemsolving and did not replace, nor interfere with, one another. Overall, the study provides causal evidence that exposure to multiplestrategies leads to improved flexibility in problem solving and that discovery learning and direct instruction are compatible instruc-tional approaches.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Flexibility in problem solving; Equation solving; Strategy use; Discovery learning; Direct instruction

1. Introduction

A key learning outcome in problem-solving domains is the development of flexible knowledge, where learnersknow multiple strategies and apply them adaptively to a range of situations (Baroody & Dowker, 2003; Rittle-Johnson& Star, 2007; Star & Seifert, 2006). For example, expert mathematicians know and use more strategies than novices,even choosing to use different strategies when attempting identical problems on different occasions (Dowker, 1992).Unfortunately, learning is too often plagued by the problem of inflexible knowledge that cannot be used adaptively ortransferred to solve novel problems (National Research Council, 2000). Lack of flexible knowledge has direct links tolow academic achievement in mathematics; students without flexible knowledge have great difficulty on both near-and far-transfer problems across a range of ages and domains (Hiebert & Carpenter, 1992) and in algebra in particular(Kieran, 1992). The problem of inflexible knowledge is not limited to a particular domain or age range, and how tosupport the development of flexible knowledge is a central issue in cognitive science and education.

The research reported here investigated how prompts to discover multiple strategies and direct instruction on mul-tiple strategies support flexible knowledge development. Understanding the impact of these instructional conditionson knowledge flexibility has important implications for learning activities that support strategy change as well as foreducational practice.

* Corresponding author. Tel.: þ1 617 496 2511; fax: þ1 617 496 3095.

E-mail address: [email protected] (J.R. Star).

0959-4752/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.learninstruc.2007.09.018

Please cite this article in press as: Jon R. Star, Rittle-Johnson, B., Flexibility in problem solving: The case of equation solving, Learning and

Instruction (2007), doi:10.1016/j.learninstruc.2007.09.018

mailto:[email protected]

http://www.elsevier.com/locate/learninstruc

ARTICLE IN PRESS

2 J.R. Star, B. Rittle-Johnson / Learning and Instruction xx (2007) 1e15

+ MODEL

1.1. Defining flexibility in problem solving

Based on our previous research, we define flexibility in problem solving as knowledge of (a) multiple strategies and(b) the relative efficiency of these strategies (Rittle-Johnson & Star, 2007; Star & Seifert, 2006). A strategy is definedhere as a step-by-step procedure for solving a problem (e.g., Siegler, 1996).

First, a key feature of flexibility is knowledge of multiple strategies. Flexible problem solvers know more than oneway to complete tasks. For example, young children have a variety of strategies they use to add, ranging from countingobjects to counting up from the larger addend on their fingers to retrieving an answer from memory (Siegler & Jenkins,1989). Variability in strategy use has clear benefits for learning and performance. For example, learners with knowl-edge of multiple strategies at pretest are more likely to learn from instructional interventions (Alibali, 1999; Siegler,1995). More generally, the presence and benefits of multiple strategies among learners have been well documented ina variety of domains, including elementary school mathematics (Alibali, 1999; Carpenter, Franke, Jacobs, Fennema,& Empson, 1998; Resnick & Ford, 1981; Rittle-Johnson & Star, 2007; Star & Seifert, 2006).

Second, flexibility involves knowledge of strategy efficiency. Flexible problem solvers know which strategies aremore efficient than others under particular circumstances. Knowledge of strategy efficiency is a fundamental charac-teristic of problem-solving expertise and is also a prevalent mechanism underlying learning and development (seeSiegler, 1996 for a review). For example, more skilled students know and choose to use mental addition strategiesthat most closely matched the characteristics of the numbers in the problem, because such a matching approach al-lowed for the fewest number of steps to solve the problem (Beishuizen, van Putten, & van Mulken, 1997; Blote, Klein,& Beishuizen, 2000; Blote, Van der Burg, & Klein, 2001).

Developing flexibility is related to transfer and conceptual knowledge growth. Students who develop flexibility inproblem solving are more likely to use or adapt existing strategies when faced with unfamiliar transfer problems and tohave a greater understanding of domain concepts (Blote et al., 2001; Hiebert & Wearne, 1996; Resnick, 1980;Rittle-Johnson & Star, 2007). For example, knowledge of multiple strategies for multi-digit arithmetic calculationswas related to greater success on transfer problems and greater conceptual knowledge of arithmetic (Carpenteret al., 1998).

1.2. Impact of discovery learning and direct instruction

Psychological theories of strategy use have relied largely on descriptive data (e.g., Siegler, 1996). There is surpris-ingly little experimental research designed to identify causal pathways leading to strategy variability and efficiency.The current research evaluates two instructional conditions that may support the development of flexibility: discoverylearningdthat is, prompts to discover multiple strategiesdand a brief amount of direct instruction.

Discovery learning is viewed by many as the ideal learning context for supporting robust learning (e.g., Fuson et al.,1997; von Glasersfeld, 1995; Hiebert et al., 1996; Kamii & Dominick, 1998). For example, Piaget (1973, p. 20) as-serted in his book To Understand is to Invent that ‘‘to understand is to discover, or reconstruct by rediscovery, and suchconditions must be complied with if in the future individuals are to be formed who are capable of production and cre-ativity and not simply repetition’’. Typically, discovery occurs when students are encouraged to work out their ownproblem-solving strategies and to reflect upon multiple strategies. In support of discovery learning, children who dis-cover their own procedures often have better transfer and conceptual knowledge than children who only adopt in-structed procedures (e.g., Carpenter et al., 1998; Hiebert & Wearne, 1996; Kamii & Dominick, 1998).

On the other hand, there is a large literature that suggests that direct instruction is more conducive toward learningthan discovery (e.g., Chen & Klahr, 1999; Klahr & Nigam, 2004; Rittle-Johnson, 2006; Zhu & Simon, 1987). In par-ticular, information-processing theories such as ‘‘cognitive load theory’’ propose that discovery conditions can over-load working-memory capacity (e.g., Kirschner, Sweller, & Clark, 2006; Sweller, 1988). Based on a large number ofempirical studies, Sweller (2003, p. 246) claims that direct instruction, rather than discovery, ‘‘should always be usedif available’’.

The present work builds on existing research by exploring the contributions of discovery learning and direct in-struction for promoting flexibility. In the current study, all students had the opportunity to discover a correct strategy,but we manipulated whether they were prompted to discover multiple strategies (Blote et al., 2000; Rittle-Johnson &Star, 2007; Star, 2001). We refer to our discovery intervention as prompted multiple ways. In addition, some studentswere provided a brief period of direct instruction on three potential strategies; we refer to our direct instruction



ARTICLE IN PRESS

3J.R. Star, B. Rittle-Johnson / Learning and Instruction xx (2007) 1e15

+ MODEL

intervention as strategy demonstration. This label makes clear that we are referring to direct instruction as the way inwhich students were exposed to a strategy, in line with previous research that considers demonstrations of strategies,either by the experimenter or through written worked examples, as direct instruction (Alibali, 1999; Chen & Klahr,1999; Kirschner et al., 2006; Klahr & Nigam, 2004; Sweller & Cooper, 1985). We are not evaluating direct instructionas a larger instructional program with teacher-led demonstrations, guided practice and limited opportunities for stu-dent exploration (e.g., Rosenshine & Stevens, 1986).

In most prior studies, students either learned via discovery or they learned via direct instruction. However, directinstruction and discovery learning do not need to be in opposition to one another. Rather, we hypothesized that the twoinstructional approaches are complementary and should be used in combination. In support of this hypothesis is theevidence that college students transferred their knowledge about human memory to a new task best when they learnedusing a combination of initial discovery followed by direct instruction, rather than only one or the other (Schwartz &Bransford, 1998).

The few existing experiments contrasting instructional methods for improving flexibility also suggest that both di-rect instruction and discovery learning play important roles in the development of flexible knowledge. In a study byBlote et al. (2001), second-graders learned about two-digit addition and subtraction in one of two conditions: (a) gen-erating their own strategies, viewing brief strategy demonstrations by the teacher and frequently discussing the valueof particular strategies for particular problem feature, or (b) receiving direct instruction and extended practice on a re-stricted number of strategies before discussing the value of particular strategies. Students in the first condition, whichemphasized discovery learning with brief bursts of direct instruction, had greater flexibility in problem solving, in thatthey more often chose to use the most efficient strategy on a particular problem, correctly identified the most efficientstrategy to solve a particular problem, and generated two different ways to solve a problem. These findings suggestthat opportunities for discovery learning combined with brief periods of direct instruction are useful for supportingflexibility. However, it is impossible to isolate the key instructional features that contributed to flexibility giventhat the two conditions differed along multiple dimensions. In addition, current best practices in psychometrics sug-gest that the Blote et al. data was analyzed inappropriately, in that the individual was the unit of analysis even thoughrandom assignment and treatment was done at the classroom level. By ignoring the nesting within classroom, differ-ences between conditions are inflated (Bryk & Raudenbush, 1988; Hedges, 2007).

Additional support for the idea that both direct instruction and discovery learning play important roles in the de-velopment of flexible knowledge comes from a recent study by Star and Seifert (2006). Sixth-graders who wereprompted to solve equations in two different ways used a larger variety of strategies and invented more efficient strat-egies than students who solved equations without such prompts (Star & Seifert, 2006). However, Star and Seifert(2006) found that invention and use of multiple strategies were quite low, suggesting that a well-timed and brief dem-onstration of efficient strategies, as was done in Blote et al. (2001), would augment the use of efficient procedures andis important to use in combination with prompts to discover multiple procedures. The purpose of the current study wasto test this hypothesis. Specifically, we manipulated whether students were prompted to solve problems in multipleways and whether they received a brief strategy demonstration, in a 2� 2 design to investigate the relative impactof each of these instructional conditions.

A second contribution of the present work is that we disentangled knowledge of strategies from use of strategies.Most prior research on flexibility assesses students’ use of strategies, such as students’ ability to generate a strategy formental estimation (Dowker, 1997) and students’ ability to implement a strategy for multi-digit sums. However, in thelarger literature on strategy choice, children frequently exhibit utilization deficiencies (Miller & Seier, 1994), whereknowledge of strategies appears to be present but the ability to use these strategies is lacking. Similarly, preference formore efficient strategies generally preceded use of the more efficient strategies (Blote et al., 2001). Thus, in develop-ing a more complete understanding of how and why flexibility develops, it seems critical not only to measure students’use of strategies but also their knowledge of strategies. The present work used an independent measure of flexibilitythat targeted students’ knowledge of multiple strategies and of strategy efficiency, in addition to more standard anddirect measures of strategy use.

2. The present study

We investigated the respective impact of discovery learning and direct instruction on strategy flexibility in an un-der-studied domain: algebra equation solving. Most of the prior work on flexible knowledge has been on mathematics



ARTICLE IN PRESS


+ MODEL

topics in the elementary grades, including mental estimation (Dowker, Flood, Griffiths, Harriss, & Hook, 1996) andmulti-digit addition (Beishuizen et al., 1997; Blote et al., 2001; Torbeyns, Verschaffel, & Ghesquiere, 2006).

Algebra equation solving is a particularly good domain to investigate flexibility because there are multiple ways tosolve equations, some of which are more efficient than others. We consider equations that can be solved using fourbasic transformations: combining like variable or constant terms, using the distributive property, adding or subtractinga constant or variable term to both sides of an equation, and dividing both sides of an equation by a constant. To solvean equation such as 3(xþ 1)¼ 15, a student could (a) choose one of these four transformations at random, determine ifthe transformation can be applied in the given equation, apply it, evaluate whether the resulting equation is closer tothe solution form of x¼ a, and then repeat this process; (b) follow a set order for executing transformationsda ‘stan-dard’ strategy for solving this linear equation is to distribute the 3, subtract 3 from both sides of the equation, and thendivide both sides of the equation by 3; or (c) decide which transformation to apply based on features of the particularproblemdfor this equation, since the right side is evenly divisible by 3, a student may choose to divide both sides by 3as a first step (see Table 1). All three strategies should lead to the correct solution, but they vary in how efficient theyare. The last, selecting a strategy using particular features of the problem, is typically more efficientdit often involvesfewer steps and fewer computations and thus should be faster and less error prone.

We expected our instructional manipulations to impact different aspects of flexibility. With respect to promptedmultiple ways (discovery learning), prompts to discover multiple strategies should increase both knowledge anduse of multiple strategies by preventing students from implementing possible transformation in a random or a fixedorder. However, it may take a long time for students to discover problem features that allow for more efficient strat-egies and, thus, knowledge and use of efficient strategies may not be improved by discovery prompts in a relativelyshort intervention. In contrast, a strategy demonstration (direct instruction) on more efficient strategies should supportknowledge and use of efficient strategies; direct instruction often leads students to adopt the instructed strategies onfamiliar problem types (Alibali, 1999; Perry, 1991; Rittle-Johnson, 2006). However, these studies also indicate thatdirect instruction often leads to lower variability in strategy use and, thus, the strategy demonstration may not supportuse of multiple strategies.

2.1. Hypotheses

In the present study, we tested the following hypotheses: (1) prompts to solve equations in two different ways(prompted multiple ways) will lead to greater knowledge and use of multiple strategies (Hypothesis 1) and (2) briefdirect instruction in the form of a strategy demonstration (strategy demonstration) will lead to greater knowledge anduse of efficient strategies (Hypothesis 2).

Table 1

Example equations solved using two different strategies

Equation type Solution using standard strategy Solution using more efficient strategy

a(xþ b)¼ c 3(xþ 1)¼ 15 3(xþ 1)¼ 15

Divide composite 3xþ 3¼ 15 xþ 1¼ 5

3x¼ 12 x¼ 4

x¼ 4

a(xþ b)þ d(xþ b)¼ c 3(xþ 1)þ 2(xþ 1)¼ 20 3(xþ 1)þ 2(xþ 1)¼ 20

Combine composite 3xþ 3þ 2xþ 2¼ 20 5(xþ 1)¼ 20

5xþ 5¼ 20 xþ 1¼ 4

5x¼ 15 x¼ 3

x¼ 3

a(xþ b)¼ d(xþ b)þ c 7(x� 2)¼ 3(x� 2)þ 16 7(x� 2)¼ 3(x� 2)þ 16

Subtract composite 7x� 14¼ 3x� 6þ 16 4(x� 2)¼ 16

7x� 14¼ 3xþ 10 x� 2¼ 4

4x� 14¼ 10 x¼ 6

4x¼ 24

x¼ 6



ARTICLE IN PRESS


+ MODEL

3. Method

3.1. Participants

Students who had just completed the sixth-grade were recruited for the present study; 132 (82 girls, 50 boys) vol-unteered to participate over the summer. Students were recruited from two large suburban, middle-class public schooldistricts in the USA. District 1 has 8% of students receiving free or reduced lunch, with 82% of students identified asCaucasian. District 2 has 19% of students receiving free or reduced lunch, with 68% of students identified as Cauca-sian. An analysis of each school’s elementary mathematics curriculum indicated that students had not received formalinstruction on equation solving in school.

3.2. Design

The design of this study was 2 (Prompted multiple ways: present vs. absent) by 2 (Strategy demonstration: presentvs. absent). Students were randomly assigned to each condition and formed small groups of 8e15 students who hadbeen assigned to the same condition.

3.2.1. Prompted multiple waysAll students needed to discover correct strategies for solving the equations. Students in the prompted multiple ways

conditions were encouraged to discover multiple strategies. In these conditions, students were sometimes given a prob-lem that they had just solved and were asked to re-solve it using a different ordering of steps. Instead of solving thesame problem twice, students in the unprompted conditions completed a different but isomorphic problem. For exam-ple, all students solved the problem, 2(xþ 4)¼ 14. Students in the prompted multiple ways conditions were given thissame problem again and were asked to solve it using a different ordering of steps. The students in the unpromptedconditions were given a structurally equivalent problem, 4(xþ 1)¼ 12, instead.

3.2.2. Strategy demonstrationStudents in the strategy demonstration conditions received a brief, eight-minute period of strategy instruction. The

first author solved three equations on a blackboard, using the strategy that led to the most efficient solution to eachproblem (e.g., see Table 1). Students were not given a justification for why a particular strategy was chosen, to encour-age reflection on the strategies rather than rote adoption. Students in the no strategy demonstration condition had ad-ditional time to practice solving equations.

3.3. Procedure

All students attended one-hour sessions for five consecutive days (Monday to Friday) during the summer. SeeTable 2 for an overview of what occurred in each session.

Table 2

Overview of procedure

Session Activity

1 Pretest (10 min)

Benchmark lesson (30 min)

2 Independent problem solving (60 min)

Prompted multiple ways introduced on last problem pair (w6 min)

3 Strategy demonstration (depending on condition) (8 min)

Independent problem solving with or without prompted multiple ways (52e60 min)

4 Independent problem solving with or without prompted multiple ways (60 min)

5 Posttest (50 min)



ARTICLE IN PRESS


+ MODEL

In the first session, students completed the pretest and received a scripted 30-minute benchmark lesson on linearequation-solving transformations. In this lesson, students were introduced to the four basic transformations used tosolve linear equations. Students were not given any strategic instruction on how transformations could be chained to-gether to solve an equation.

For the next three sessions, students worked individually at their own pace through a problem booklet. The bookletwas divided into three sections, and students completed only the problems in one section during each session. If stu-dents finished the day’s problems before the end of the session, they were instructed to close their booklet and sit qui-etly. If students became stuck while attempting a problem, they raised their hand and were approached by a helper. Thehelper answered the students’ questions in a semi-standardized format. Specifically, the helper corrected the students’arithmetic mistakes or reminded them of the four possible transformations and how each was used. Helpers never gavestrategic advice to students; in addition, helpers did not tell students if their solutions were correct.

At the end of session 2 and continuing in sessions 3 and 4, students in the prompted multiple ways conditions solvedeach problem twice (with a maximum of 11 problems solved twice). At the beginning of session 3, students in thestrategy demonstration conditions received a brief lesson on three problems: 2(xþ 5)þ 4x¼ 4þ 8x, 3(xþ 1)¼ 15,and 3(xþ 1)þ 2(xþ 1)¼ 20. Each problem was solved using a different ordering of steps and in the most efficientmanner. The first problem was solved using the standard strategy; the second and third problems were solved usingthe more efficient strategy shown in Table 1. The experimenter framed his solutions by saying, ‘‘This is the way that Isolve this equation.’’ The solutions were removed from view before students began independent problem solving. Stu-dents who were not in the strategy demonstration conditions began working independently on the booklets at the verybeginning of the session (and thus had more time for independent work).

3.4. Materials

3.4.1. Intervention problemsDuring the three problem-solving sessions, students were given a booklet with 31 linear equations to solve. For

most linear equations, a ‘standard’ order of steps will always lead to a correct solution (see Table 1). All 31 problemscould be solved using this standard strategy, and for a few problems, this strategy was the most efficient (e.g., involvedthe fewest number of steps). For most problems in sessions 3 and 4, using a different ordering of steps was more ef-ficient, such as dividing first or by combining first (see Table 1).

3.4.2. AssessmentsThe pre- and posttests had assessments of procedural knowledge (e.g., equation-solving success, including trans-

fer) and flexibility in problem solving (see Table 3 and Appendix). Measures were adapted from those used in a priorstudy (Star & Seifert, 2006). Because prior research has linked conceptual knowledge and flexibility in problemsolving (Carpenter et al., 1998), our assessment also included a measure of conceptual knowledge. However,due to low inter-item reliability (Cronbach’s alpha¼ 0.42), conceptual knowledge items were dropped from theanalysis.

The procedural knowledge items assessed students’ ability to solve equations. The familiar equations werefour equations isomorphic to equations presented during the intervention. The transfer equations were five equa-tions that each had an unfamiliar feature and/or could be solved using a new atypical solution strategy (e.g., mul-tiply an equation by 10 to eliminate decimals). Nevertheless, all five problems could be solved using the standardstrategy.

Flexibility in problem solving was assessed in several ways (see Table 3). Our assessment reflected the two com-ponents of flexibility in problem solving outlined in Section 1dknowledge of multiple strategies and knowledge ofstrategy efficiencydas well as our emphasis on disentangling knowledge of strategies and use of strategies. First, weevaluated students’ use of multiple strategies. Students were given two equations and asked to solve each in two dif-ferent ways. They were given a point for each equation that they solved in two different, correct ways. Second, weevaluated students’ use of efficient strategies. On three of the four familiar equations, students could use strategiesthat were more efficient (in terms of the number of operations) than the standard strategy. For each student, we cal-culated (a) the proportion of problems where students used the standard strategy; (b) the proportion of problems wherestudents used composite variable shortcuts that were more efficient than the standard strategy; and (c) the proportionof completed problems where students’ strategy was more efficient than the standard strategy (regardless of whether



Table 3

Description of measuresa as well as proportion correct and Cronbach’s alphas on assessment measures at pre- and posttest in all conditions

Scale (number of items) Items (see Appendix) Pretest Posttest

M SD a M SD a

Procedural knowledge

Familiar equations (n¼ 4) 5, 7, 8, 9 0.18 0.26 0.78 0.73 0.32 0.75

Transfer equations (n¼ 5) 6. 10, 11, 12, 13 0.10 0.22 0.79 0.59 0.33 0.74

Flexibility in problem solving

Knowledge of multiple strategies (n¼ 5) e e e 0.56 0.29 0.57

Accept multiple solution strategies (n¼ 2) 14, 15 0.16 0.32 0.65 0.59 0.37 0.41

Identify multiple next steps (n¼ 3) 21, 22, 23 e e e 0.54 0.40 0.76

Knowledge of strategy efficiency (n¼ 7) e e e 0.42 0.31 0.79

Prompted implementation of efficient steps (n¼ 4) 24, 25, 26, 27 e e e 0.36 0.35 0.71

Recognition and evaluation of efficient steps (n¼ 3) 28, 29, 30 e e e 0.54 0.31 0.66

Use of multiple strategies (n¼ 2) 19, 20 e e e 0.54 0.41 0.74

Use of efficient strategies

Use of standard strategy e e e 0.53 0.35 0.66

Use of composite variable shortcut strategy e e e 0.11 0.25 0.75

Strategy more efficient than standard strategy e e e 0.16 0.25 0.47

a Items 1, 2, 3, 4, 16, 17, and 18 assessed conceptual knowledge and were dropped due to low inter-item reliability on this measure.

ARTICLE IN PRESS


+ MODEL

they used the composite variable shortcut). The composite variable shortcut strategy was defined as, rather than dis-tributing first, treating the composite variable as a single term and dividing or subtracting from both sides, as appro-priate (e.g., the more efficient strategies shown in Table 1).

Third, we adapted items from our prior work (Star & Seifert, 2006) to assess students’ knowledge of multiple strat-egies. Items were designed to target students’ acceptance of multiple solution strategies and their ability to identifymultiple next steps. Fourth, and also building on our prior work, we evaluated students’ knowledge of strategy effi-ciency. Items targeted the recognition and evaluation of efficient steps and students’ ability, when prompted, to im-plement efficient steps (see Table 3).

At posttest, students completed the full assessment. At pretest, students completed a subset of the overall asses-smentdall the procedural knowledge items, but only the two easiest flexibility items (items on accepting presenceof multiple solution strategies; see Table 3). At pretest, students were unlikely to be able to solve the equations atall, so assessing their flexibility in problem solving seemed unnecessary and potentially discouraging to students.

Students’ accuracy on the pre- and post-assessments were graded by two independent analysts who then met tocompare their codes. Initial interrater reliability was 96%; all disagreements were subsequently resolved by the an-alysts. Accuracy is reported as proportion correct.

4. Results

A prerequisite for flexibility in a domain is the ability to solve problems in this specific domain. Thus, we first eval-uated whether students learned to solve equations from the intervention and whether prompted multiple ways andstrategy demonstration facilitated learning and transfer. Next, we evaluated whether the two manipulations led togreater flexibility in problem solving. Partial eta squared (h2) is used to report effect sizes, which can be interpretedas the amount of variance accounted for by the target variable. Means and Cronbach’s alpha reliabilities across con-ditions are reported in Table 3.

4.1. Procedural knowledge

At pretest, students had little or no prior knowledge of formal equation-solving strategies. Students’ accuracy waslow (mean proportion correct¼ 0.13, SD¼ 0.23), and most students who solved one or more equations correctly did



Table 4

Proportion correct on measures of procedural knowledge (and SD)

Strategy demonstration All equations Familiar equations Transfer equations

Pretest Posttest Pretest Posttest Pretest Posttest

No strategy demonstration

No prompted multiple ways (n¼ 32) 0.17 (0.24) 0.72 (0.30) 0.19 (0.26) 0.80 (0.31) 0.15 (0.25) 0.65 (0.34)

Prompted multiple ways (n¼ 31) 0.14 (0.25) 0.64 (0.30) 0.18 (0.30) 0.68 (0.31) 0.11 (0.24) 0.61 (0.33)

Strategy demonstration

No prompted multiple ways (n¼ 34) 0.12 (0.22) 0.69 (0.36) 0.17 (0.25) 0.79 (0.28) 0.09 (0.22) 0.61 (0.30)

Prompted multiple ways (n¼ 35) 0.11 (0.20) 0.57 (0.32) 0.17 (0.25) 0.66 (0.36) 0.06 (0.19) 0.51 (0.34)

ARTICLE IN PRESS


+ MODEL

so using informal methods such as guess-and-check. There were no differences between the conditions on pretestscores for procedural knowledge (see Table 4).

To investigate improvements in students’ procedural knowledge, a 2 (test administration: pretest vs. posttest) by 2(prompted multiple ways: present vs. absent) by 2 (strategy demonstration: present vs. absent) repeated measuresANOVA was conducted on performance on familiar and transfer equations.

4.1.1. Familiar equationsAcross conditions, students showed significant improvement in their performance on familiar equations, F(1,

128)¼ 343.57, p< 0.001, partial h2¼ 0.73 (see Table 4). In addition, there was a significant interaction betweentest administration and prompted multiple ways, F(1, 128)¼ 4.086, p< 0.05, partial h2¼ 0.03: While students inboth conditions scored the same at pretest, students in the no prompted multiple ways conditions had higher scoresat posttest (79% vs. 67% correct, respectively). There was no effect of strategy demonstration on accuracy on thefamiliar equations.

4.1.2. Transfer equationsStudents also showed improvement from pretest to posttest in their ability to solve transfer equations, F(1,

128)¼ 339.39, p< 0.001, partial h2¼ 0.73 (see Table 4). There were no other significant main effects or interactions.Overall, all treatments were effective at improving students’ ability to solve both familiar and transfer equationscorrectly.

4.2. Flexibility in problem solving

All students improved in their ability to solve equations; however, the question was whether discovering multiplestrategies or strategy demonstration leads to greater flexibility in problem solving.

4.2.1. Pretest flexibilityRecall that the two easiest flexibility items (items on accepting presence of multiple solution strategies) were in-

cluded on the pretest. Overall performance on these two items was very low, M¼ 0.16, SD¼ 0.32, confirming thatstudents did not endorse use of multiple strategies at pretest. There were no differences between the conditions onthis measure at pretest.

4.2.2. Effects of condition on knowledge of strategiesBoth prompts to discover multiple strategies and strategy demonstration led to greater knowledge of multiple

strategies and knowledge of strategy efficiency. An MANCOVA was conducted, with the two scales of strategyknowledge as the dependent variables, prompted multiple ways and strategy demonstration as between-subjectfactors, and pretest equation-solving score as a covariate. There were significant main effects for prompted mul-tiple ways, F(1, 128)¼ 4.792, p< 0.05, partial h2¼ 0.04, and strategy demonstration, F(1, 128)¼ 6.388,



Table 5

Proportion correct on measures of knowledge of strategies scale at posttest (and SD)

Strategy demonstration Overall knowledge

of strategies

Knowledge of

multiple strategies

Knowledge of

strategy efficiency


No prompted multiple ways (n¼ 32) 0.39 (0.18) 0.46 (0.30) 0.33 (0.27)

Prompted multiple ways (n¼ 31) 0.51 (0.17) 0.55 (0.25) 0.43 (0.26)




ARTICLE IN PRESS


+ MODEL

p< 0.05, partial h2¼ 0.05. There was no interaction between prompted multiple ways and strategy demonstrationon students’ knowledge. Univariate F-tests on each subscale confirmed that prompted multiple ways and strategydemonstration led to higher posttest flexibility on the knowledge of multiple strategies scale, F(1, 127)¼ 3.867,p¼ 0.051, partial h2¼ 0.03, and F(1, 127)¼ 4.653, p< 0.05, partial h2¼ 0.04, respectively, as well as on theknowledge of efficient strategies scale, F(1, 127)¼ 3.174, p¼ 0.07, partial h2¼ 0.02, and F(1, 127)¼ 5.696,p< 0.05, partial h2¼ 0.04, respectively, although the effect of prompted multiple ways was marginal (seeTable 5).

4.2.3. Effect of condition on use of strategiesAs predicted, the two manipulations had different effects on use of multiple strategies vs. the use of efficient strat-

egies. An ANCOVA was conducted for each dependent variable, with prompted multiple ways and strategy demon-stration as between-subject factors, and pretest equation-solving score as a covariate. Specifically, for the use ofmultiple strategies scale, students who were prompted to solve equations in multiple ways during the interventionwere better able to solve equations in more than one way at posttest than students who were not prompted(M¼ 0.63, SD¼ 0.44 vs. M¼ 0.46, SD¼ 0.36), F(1, 132)¼ 5.645, p< 0.05, partial h2¼ 0.04. The strategy demon-stration did not impact success on this measure. On the use of efficient strategies scale, the strategy demonstrationimpacted students’ use of efficient strategies on the familiar equations. As shown in Table 6, strategy demonstrationled to increased use of the composite variable shortcuts, F(1, 128)¼ 24.112, p< 0.001, partial h2¼ 0.16, a marginalreduction in use of the standard strategy, F(1, 128)¼ 3.156, p¼ 0.078, partial h2¼ 0.02, and use of more efficientstrategies in general, F(1, 128)¼ 13.475, p< 0.001, partial h2¼ 0.12. In contrast, prompted multiple ways did notimprove strategy efficiency, although it did lead students to use the standard strategy less often, F(1, 128)¼ 7.028,p< 0.01, partial h2¼ 0.05.

Overall, prompting for the discovery of multiple ways and providing direct instruction in the form of a strategydemonstration both led to increased knowledge of multiple strategies and efficient strategies. However, for themore difficult use measures, the effects of condition varied by outcome, as predicted. Prompting for discovery of mul-tiple ways promoted use of multiple, non-standard strategies that were correct, but not more efficient than the standardstrategy. A strategy demonstration increased use of more efficient strategies, but did not increase use of multiplestrategies.

Table 6

Frequency of using each strategy on the familiar equation-solving items at posttest (and SD)

Strategy demonstration Standard strategy Composite variable

shortcut strategies

Strategy more efficient

than standard strategy









ARTICLE IN PRESS


+ MODEL

5. Discussion

The present research explored the development of students’ flexibility in knowledge and use of mathematicalstrategies. All students improved in their equation-solving ability after three hours of problem-solving experi-ence, but the prompts to discover multiple strategies and direct instruction on multiple strategies promotedgreater flexibility in different ways. In particular, both of our hypotheses were confirmed: prompts to solve equa-tions in two different ways led to greater knowledge and use of multiple strategies, while direct instruction led togreater knowledge and use of efficient strategies. We discuss the implications of the current work for improvingflexibility, for theories of strategy use, and for the complimentary benefits of discovery learning and strategydemonstration.

5.1. Improving and assessing flexibility in problem solving

Prior research has focused exclusively on students’ use of multiple strategies and/or use of efficient strategies.The present study also included independent measures of strategy knowledge, including knowledge of multiple so-lution strategies and knowledge of strategy efficiency. With respect to the first, both prompts to discover multipleways and direct instruction via a strategy demonstration helped students accept that there are multiple ways to solveequations and to be able to identify possible next steps. Knowledge of strategy efficiency was also improved by bothinstructional interventions: prompted multiple ways and strategy demonstration on multiple strategies led studentsto recognize more adaptive strategies (e.g., more efficient ones) and to implement efficient strategies whenprompted.

However, our two instructional conditions affected different aspects of strategy use. Prompts to discover mul-tiple strategies increased use of multiple strategies, but not use of more efficient strategies, and direct instructionon strategies increased use of efficient strategies, but not use of multiple strategies. Prompts to discover multiplestrategies should increase both knowledge and use of multiple strategies by preventing students from implement-ing possible transformation in a random or a fixed order. However, it may take a very long time for students todiscover or utilize problem features that allow for more efficient strategies. Our findings suggest that the discov-ery prompts were sufficient to increase knowledge, although not use, of more efficient strategies. Prompting formultiple ways helped students recognize, positively evaluate, and implement efficient steps when prompted to doso, but this knowledge was not sufficient to support choice of these steps during problem solving. In contrast,direct instruction on more efficient strategies supported knowledge and use of efficient strategies, convergingwith prior findings that direct instruction often leads students to adopt the instructed strategies on familiar prob-lem types (Alibali, 1999; Perry, 1991; Rittle-Johnson, 2006). Direct instruction also increased knowledge of mul-tiple strategies; students in these conditions accepted that there were multiple solution strategies and couldidentify multiple next steps to solve a problem; however, they did not use this knowledge to solve equation inmultiple ways when prompted. Prior research indicates that direct instruction decreases variability in strategyuse (Alibali, 1999; Perry, 1991; Rittle-Johnson, 2006), but this research suggests that knowledge of multiple strat-egies may persist.

The present study underscores the importance of disentangling knowledge from use and multiple strategies fromefficient strategies. First, research on metacognitive skills and utilization deficiencies both suggest that learners de-velop knowledge that they do not always use effectively (Miller & Seier, 1994; Veenman, Kok, & Blote, 2005). Strat-egy development often happens in fits and spurts; when new strategies are discovered, it takes time before they becomeregularly used and benefit performance (e.g., Siegler, 1996; Siegler & Jenkins, 1989). Relying exclusively on mea-sures of use can fail to capture initial stages of flexibility. Second, strategy flexibility requires use of multiple strategiesand adaptive choice of efficient strategies and different instructional manipulations can influence the two componentsdifferently.

5.2. Balancing direct instruction and discovery learning

Current findings suggest independent, and non-conflicting, benefits of discovery learning and direct instructionfor flexibility in problem solving. With respect to discovery learning, prompts to solve the same problem in two



ARTICLE IN PRESS


+ MODEL

different ways led to greater gains in flexibility than moving on and solving new problems. This finding providescausal evidence that converges with descriptive evidence on the benefits of students’ generating more than oneway to solve a problem (Carpenter et al., 1998; Resnick, 1980).

At the same time, our results highlight the potential value of direct instruction: direct instruction and discoverylearning are not mutually exclusive. Schwartz and Bransford (1998) argued that there is a ‘‘time for telling’’dthat despite some educators’ concerns, direct instruction can be very effective under certain conditions. Directinstruction may be particularly effective after students have developed a moderate level of differentiated domainknowledge through problem exploration (Schwartz & Bransford, 1998). In the present study, direct instruction onmultiple strategies was provided after students had achieved basic fluency in solving linear equations throughproblem-solving exploration, and this led to reliable gains in problem-solving flexibility. Discovery prompts tosolve a problem in two different ways did not replace this benefit of direct instructiondrather, the two wereadditive.

5.3. Limitations and future directions

The present study is an important first step in identifying instructional conditions that support knowledge anduse of multiple and efficient strategies. However, several limitations need to be considered when interpreting ourresults.

First, the present study did not include a follow-up test to assess whether procedural knowledge and flexibilitygains persisted beyond the week of the study. Second, internal consistencies were only moderate on some scales(e.g., posttest inter-item reliability of 0.57 on the knowledge of multiple strategies subscale). Third, we did notassess whether our results were influenced by the order in which the treatments were administered (discovery,followed by direct instruction). Schwartz and Bransford (1998) argue that the order matters, but this hypothesisremains untested.

In terms of future directions, additional research is needed to evaluate when and for whom different instructionalconditions support flexibility in problem solving. For example, several characteristics of the target learners should beconsidered, such as their motivation and metacognitive skills. Participants in the current study were volunteers, sowere likely motivated to learn. Discovery prompts and direct instruction can promote different levels of motivationin students (Timmermans, Van Lieshout, & Verhoeven, 2007), differences that were likely minimized in our volunteerpopulation. Second, students’ metacognitive skills may influence how much students reflect on characteristics ofa problem that make a particular strategy most efficient for those problem features. We did not measure nor directlysupport metacognitive skills. Supporting metacognitive skills would likely enhance the effects of other instructionalmanipulations.

Additional research is also needed to identify when different instructional conditions support flexibility in problemsolving. By disentangling strategy knowledge from use of strategy and separating flexibility in problem solving intotwo components, we provided initial evidence for how different instructional conditions might impact flexibility inproblem solving. Studies using microgenetic methods and using dual tasks are needed to identify specific mechanismsunderlying these effects, such as whether direct instruction reduces cognitive load and frees resources for reflectionupon when to use particular, more efficient, strategies.

In conclusion, both direct instruction on multiple strategies and discovery learning improved flexibility in problemsolving. Specifically, discovery prompts facilitated use of multiple strategies whereas direct instruction facilitated useof efficient strategies. This provides causal evidence for both discovery-driven and instruction-driven sources of flex-ible knowledge.

Acknowledgments

Thanks to Kristine Rider, Marcy Wood, Marie Turini, Katie Kawel, Jessica Stewart, and Natalie Marino fortheir assistance in data collection and analysis. Thanks to Laura Novick for her very useful feedback on a draftof this paper.



ARTICLE IN PRESS


+ MODEL

Appendix. Assessment items



ARTICLE IN PRESS


+ MODEL



ARTICLE IN PRESS


+ MODEL

References

Alibali, M. W. (1999). How children change their minds: strategy change can be gradual or abrupt. Developmental Psychology, 35(1), 127e145.

Baroody, A. J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ:

Lawrence Erlbaum.

Beishuizen, M., van Putten, C. M., & van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred.

Learning and Instruction, 7(1), 87e106.

Blote, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10, 221e247.

Blote, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: instruction

effects. Journal of Educational Psychology, 93, 627e638.

Bryk, A. S., & Raudenbush, S. W. (1988). Toward a more appropriate conceptualization of research on school effects: a three-level hierarchical

linear model. American Journal of Education, 97(1), 65e108.

Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in

children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3e20.

Chen, Z., & Klahr, D. (1999). All other things being equal: acquisition and transfer of the control of variables strategy. Child Development, 70(5),

1098e1120.

Dowker, A. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23(1),

45e55.

Dowker, A. (1997). Young children’s addition estimates. Mathematical Cognition, 3, 141e154.

Dowker, A., Flood, A., Griffiths, H., Harriss, L., & Hook, L. (1996). Estimation strategies of four groups. Mathematical Cognition, 2(2), 113e135.

Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., & Olivier, A. I., et al. (1997). Children’s conceptual structures for multidigit

numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130e162.

von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Bristol, PA: Falmer Press, Taylor & Francis.

Hedges, L. V. (2007). Correcting a significance test for clustering. Journal of Educational and Behavioral Statistics, 32(2), 151e179.

Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teach-

ing and learning (pp. 65e97). New York: Simon & Schuster Macmillan.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., & Murray, H., et al. (1996). Problem solving as a basis for reform in curriculum

and instruction: the case for mathematics. Educational Researcher, 25, 12e21.

Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3),

251e283.

Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1e4. In L. J. Morrow, & M. J. Kenney (Eds.), The teaching and

learning of algorithms in school mathematics. 1998 Yearbook (pp. 130e140). Reston, VA: National Council of Teachers of Mathematics.



ARTICLE IN PRESS


+ MODEL

Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning

(pp. 390e419). New York: Simon & Schuster.

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance does not work: an analysis of the failure of constructivist, discovery,

problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75e86.

Klahr, D., & Nigam, M. (2004). The equivalence of learning paths in early science instruction: effects of direct instruction and discovery learning.

Psychological Science, 15(10), 661e667.

Miller, P. H., & Seier, W. L. (1994). Strategy utilization deficiencies in children: when, where, and why. In H. W. Reese (Ed.), Advances in childdevelopment and behavior, Vol. 25 (pp. 107e156). New York: Academic.

National Research Council. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

Perry, M. (1991). Learning and transfer: instructional conditions and conceptual change. Cognitive Development, 6, 449e468.

Piaget, J. (1973). To understand is to invent. New York: Grossman.

Resnick, L. B. (1980). The role of invention in the development of mathematical competence. In R. H. Kluwe, & H. Spada (Eds.), Developmental

models of thinking (pp. 213e244). New York: Academic.

Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Lawrence Erlbaum.

Rittle-Johnson, B. (2006). Promoting transfer: the effects of direct instruction and self-explanation. Child Development, 77(1), 1e15.

Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study

on learning to solve equations. Journal of Educational Psychology, 99(3), 561e574.

Rosenshine, B., & Stevens, R. (1986). Teaching functions. In M. C. Wittrock (Ed.), Handbook of research on teaching (pp. 376e391). New York:

McMillan.

Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475e522.

Siegler, R. S. (1995). How does change occur: a microgenetic study of number conservation. Cognitive Psychology, 28(3), 225e273.

Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press.

Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Lawrence Erlbaum.

Star, J. R. (2001). Re-conceptualizing procedural knowledge: Innovation and flexibility in equation solving. Unpublished doctoral dissertation,

University of Michigan, Ann Arbor.

Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280e300.

Sweller, J. (1988). Cognitive load during problem solving: effects on learning. Cognitive Science, 12, 257e285.

Sweller, J. (2003). Evolution of human cognitive architecture. In B. H. Ross (Ed.), The psychology of learning and motivation, Vol. 43 (pp. 215e

266). San Diego, CA: Academic.

Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and

Instruction, 2(1), 59e89.

Timmermans, R. E., Van Lieshout, E. C. D. M., & Verhoeven, L. (2007). Gender-related effects of contemporary math instruction for low per-

formers on problem-solving behavior. Learning and Instruction, 17(1), 42e54.

Torbeyns, J., Verschaffel, L., & Ghesquiere, P. (2006). The development of children’s adaptive expertise in the number domain 20 to 100. Cog-

nition and Instruction, 24(4), 439e465.

Veenman, M., Kok, R., & Blote, A. W. (2005). The relation between intellectual and meta-cognitive skills in early adolescence. InstructionalScience, 33(3), 193e211.

Zhu, X., & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition and Instruction, 4(3), 137e166.



Documents

Flexibility in problem solving: The case of equation solving