Elsevier

Learning and Instruction

Volume 18, Issue 6, December 2008, Pages 565-579
Learning and Instruction

Flexibility in problem solving: The case of equation solving

https://doi.org/10.1016/j.learninstruc.2007.09.018Get rights and content

Abstract

A key learning outcome in problem-solving domains is the development of flexible knowledge, where learners know multiple strategies and adaptively choose efficient strategies. Two interventions hypothesized to improve flexibility in problem solving were experimentally evaluated: prompts to discover multiple strategies and direct instruction on multiple strategies. Participants were 132 sixth-grade students who solved linear equations for three hours. Both interventions improved students' flexibility in problem solving and did not replace, nor interfere with, one another. Overall, the study provides causal evidence that exposure to multiple strategies leads to improved flexibility in problem solving and that discovery learning and direct instruction are compatible instructional approaches.

Introduction

A key learning outcome in problem-solving domains is the development of flexible knowledge, where learners know multiple strategies and apply them adaptively to a range of situations (Baroody and Dowker, 2003, Rittle-Johnson and Star, 2007, Star and 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 or transferred to solve novel problems (National Research Council, 2000). Lack of flexible knowledge has direct links to low 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 to support 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 multiple strategies support flexible knowledge development. Understanding the impact of these instructional conditions on knowledge flexibility has important implications for learning activities that support strategy change as well as for educational practice.

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 and Star, 2007, Star and Seifert, 2006). A strategy is defined here 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 one way to complete tasks. For example, young children have a variety of strategies they use to add, ranging from counting objects 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 knowledge 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 in a variety of domains, including elementary school mathematics (Alibali, 1999, Carpenter et al., 1998, Resnick and Ford, 1981, Rittle-Johnson and Star, 2007, Star and Seifert, 2006).

Second, flexibility involves knowledge of strategy efficiency. Flexible problem solvers know which strategies are more efficient than others under particular circumstances. Knowledge of strategy efficiency is a fundamental characteristic of problem-solving expertise and is also a prevalent mechanism underlying learning and development (see Siegler, 1996 for a review). For example, more skilled students know and choose to use mental addition strategies that most closely matched the characteristics of the numbers in the problem, because such a matching approach allowed for the fewest number of steps to solve the problem (Beishuizen et al., 1997, Blöte et al., 2000, Blöte et al., 2001).

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

Psychological theories of strategy use have relied largely on descriptive data (e.g., Siegler, 1996). There is surprisingly 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: discovery learning—that is, prompts to discover multiple strategies—and 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 and Dominick, 1998). For example, Piaget (1973, p. 20) asserted in his book To Understand is to Invent that “to understand is to discover, or reconstruct by rediscovery, and such conditions must be complied with if in the future individuals are to be formed who are capable of production and creativity and not simply repetition”. Typically, discovery occurs when students are encouraged to work out their own problem-solving strategies and to reflect upon multiple strategies. In support of discovery learning, children who discover their own procedures often have better transfer and conceptual knowledge than children who only adopt instructed procedures (e.g., Carpenter et al., 1998, Hiebert and Wearne, 1996, Kamii and Dominick, 1998).

On the other hand, there is a large literature that suggests that direct instruction is more conducive toward learning than discovery (e.g., Chen and Klahr, 1999, Klahr and Nigam, 2004, Rittle-Johnson, 2006, Zhu and Simon, 1987). In particular, information-processing theories such as “cognitive load theory” propose that discovery conditions can overload working-memory capacity (e.g., Kirschner et al., 2006, Sweller, 1988). Based on a large number of empirical studies, Sweller (2003, p. 246) claims that direct instruction, rather than discovery, “should always be used if available”.

The present work builds on existing research by exploring the contributions of discovery learning and direct instruction 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 (Blöte et al., 2000, Rittle-Johnson and Star, 2007, Star, 2001). We refer to our discovery intervention as prompted multiple ways. In addition, some students were provided a brief period of direct instruction on three potential strategies; we refer to our direct instruction intervention as strategy demonstration. This label makes clear that we are referring to direct instruction as the way in which 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 and Klahr, 1999, Kirschner et al., 2006, Klahr and Nigam, 2004, Sweller and Cooper, 1985). We are not evaluating direct instruction as a larger instructional program with teacher-led demonstrations, guided practice and limited opportunities for student exploration (e.g., Rosenshine & Stevens, 1986).

In most prior studies, students either learned via discovery or they learned via direct instruction. However, direct instruction and discovery learning do not need to be in opposition to one another. Rather, we hypothesized that the two instructional approaches are complementary and should be used in combination. In support of this hypothesis is the evidence that college students transferred their knowledge about human memory to a new task best when they learned using 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 direct instruction and discovery learning play important roles in the development of flexible knowledge. In a study by Blöte et al. (2001), second-graders learned about two-digit addition and subtraction in one of two conditions: (a) generating their own strategies, viewing brief strategy demonstrations by the teacher and frequently discussing the value of particular strategies for particular problem feature, or (b) receiving direct instruction and extended practice on a restricted number of strategies before discussing the value of particular strategies. Students in the first condition, which emphasized discovery learning with brief bursts of direct instruction, had greater flexibility in problem solving, in that they more often chose to use the most efficient strategy on a particular problem, correctly identified the most efficient strategy to solve a particular problem, and generated two different ways to solve a problem. These findings suggest that opportunities for discovery learning combined with brief periods of direct instruction are useful for supporting flexibility. However, it is impossible to isolate the key instructional features that contributed to flexibility given that the two conditions differed along multiple dimensions. In addition, current best practices in psychometrics suggest that the Blöte et al. data was analyzed inappropriately, in that the individual was the unit of analysis even though random assignment and treatment was done at the classroom level. By ignoring the nesting within classroom, differences between conditions are inflated (Bryk and Raudenbush, 1988, Hedges, 2007).

Additional support for the idea that both direct instruction and discovery learning play important roles in the development of flexible knowledge comes from a recent study by Star and Seifert (2006). Sixth-graders who were prompted to solve equations in two different ways used a larger variety of strategies and invented more efficient strategies 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 demonstration of efficient strategies, as was done in Blöte et al. (2001), would augment the use of efficient procedures and is important to use in combination with prompts to discover multiple procedures. The purpose of the current study was to test this hypothesis. Specifically, we manipulated whether students were prompted to solve problems in multiple ways and whether they received a brief strategy demonstration, in a 2 × 2 design to investigate the relative impact of 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 for mental estimation (Dowker, 1997) and students' ability to implement a strategy for multi-digit sums. However, in the larger literature on strategy choice, children frequently exhibit utilization deficiencies (Miller & Seier, 1994), where knowledge of strategies appears to be present but the ability to use these strategies is lacking. Similarly, preference for more efficient strategies generally preceded use of the more efficient strategies (Blöte et al., 2001). Thus, in developing 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 flexibility that targeted students' knowledge of multiple strategies and of strategy efficiency, in addition to more standard and direct measures of strategy use.

Section snippets

The present study

We investigated the respective impact of discovery learning and direct instruction on strategy flexibility in an under-studied domain: algebra equation solving. Most of the prior work on flexible knowledge has been on mathematics topics in the elementary grades, including mental estimation (Dowker, Flood, Griffiths, Harriss, & Hook, 1996) and multi-digit addition (Beishuizen et al., 1997, Blöte et al., 2001, Torbeyns et al., 2006).

Algebra equation solving is a particularly good domain to

Participants

Students who had just completed the sixth-grade were recruited for the present study; 132 (82 girls, 50 boys) volunteered to participate over the summer. Students were recruited from two large suburban, middle-class public school districts in the USA. District 1 has 8% of students receiving free or reduced lunch, with 82% of students identified as Caucasian. District 2 has 19% of students receiving free or reduced lunch, with 68% of students identified as Caucasian. An analysis of each school's

Results

A prerequisite for flexibility in a domain is the ability to solve problems in this specific domain. Thus, we first evaluated whether students learned to solve equations from the intervention and whether prompted multiple ways and strategy demonstration facilitated learning and transfer. Next, we evaluated whether the two manipulations led to greater flexibility in problem solving. Partial eta squared (η2) is used to report effect sizes, which can be interpreted as the amount of variance

Discussion

The present research explored the development of students' flexibility in knowledge and use of mathematical strategies. All students improved in their equation-solving ability after three hours of problem-solving experience, but the prompts to discover multiple strategies and direct instruction on multiple strategies promoted greater flexibility in different ways. In particular, both of our hypotheses were confirmed: prompts to solve equations in two different ways led to greater knowledge and

Acknowledgments

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

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