In pursuit of knowledge: Comparing self-explanations, concepts, and procedures as pedagogical tools

Abstract

Explaining new ideas to oneself can promote learning and transfer, but questions remain about how to maximize the pedagogical value of self-explanations. This study investigated how type of instruction affected self-explanation quality and subsequent learning outcomes for second- through fifth-grade children learning to solve mathematical equivalence problems (e.g., 7 + 3 + 9 = 7 + _). Experiment 1 varied whether instruction was conceptual or procedural in nature (n = 40), and Experiment 2 varied whether children were prompted to self-explain after conceptual instruction (n = 48). Conceptual instruction led to higher quality explanations, greater conceptual knowledge, and similar procedural knowledge compared with procedural instruction. No effect was found for self-explanation prompts. Conceptual instruction can be more efficient than procedural instruction and may make self-explanation prompts unnecessary.

Introduction

Learning is often plagued both by a lack of connected understanding and by the inability to transfer knowledge to novel problems. Understanding the processes that affect knowledge change is central both to theories of learning and to the development of effective pedagogies for overcoming these problems. Prompting students to generate explicit explanations of the material they study has emerged as one potentially effective tool for promoting learning and transfer in numerous domains (e.g., Chi, DeLeeuw, Chiu, & LaVancher, 1994). Although prompting for such “self-explanations” has been shown to facilitate learning, little is known about how these prompts interact with different types of instruction. Explicating these relations is essential to unlocking the full potential of self-explanation as a tool for supporting learning. Toward this end, the current experiments examined (a) whether procedural or conceptual instruction combined with self-explanation prompts differentially affected learning of conceptual and procedural knowledge for children solving math equivalence problems (e.g., 7 + 3 + 9 = 7 + _), (b) whether the type of instruction affected the quality of self-explanations generated, and (c) whether self-explanation prompts were effective over and above conceptual instruction alone.

In their seminal study on the self-explanation effect, Chi, Bassok, Lewis, Reimann, and Glaser (1989) found that, when studying example exercises in a physics text, the best learners spontaneously explained the material to themselves, providing justifications for each action in a solution sequence. Subsequent studies have shown that prompting for such self-explanations can lead to improved learning outcomes in numerous domains, including arithmetic (Calin-Jageman and Ratner, 2005, Rittle-Johnson, 2006, Siegler, 2002), geometry (Aleven and Koedinger, 2002, Wong et al., 2002), interest calculations (Renkl, Stark, Gruber, & Mandl, 1998), argumentation (Schworm & Renkl, 2007), Piagetian number conservation (Siegler, 1995), biology text comprehension (Chi et al., 1994), and balancing beam problems (Pine & Messer, 2000). Moreover, these self-explanation effects have been demonstrated across a wide range of age cohorts, from 4-year-olds (Rittle-Johnson, Saylor, & Swygert, 2007) to adult bank apprentices (Renkl et al., 1998). Perhaps most impressive is that prompting for self-explanation also promotes transfer in many of these domains even though participants rarely receive feedback on the quality of their explanations (e.g., Renkl et al., 1998, Rittle-Johnson, 2006, Siegler, 2002).

There are, however, substantial differences in the quality of explanations generated among individuals. Importantly, these differences are associated with divergent learning outcomes (Chi et al., 1989, Chi et al., 1994, Pirolli and Recker, 1994, Renkl, 1997). Successful learners tend to give more principle-based explanations, to consider the goals of operators and procedures more frequently, and to show illusions of understanding less frequently (for an effective summary, see Renkl, 2002). Less successful learners, however, offer fewer explanations, anticipate steps less frequently, examine fewer examples, and tend to focus less on the goals and principles governing operators and procedures. Hence, self-explanation prompts are not equally successful across learners at encouraging the types of self-explanations most highly correlated with learning gains. Indeed, a careful review of the literature reveals that prompting learners to self-explain sometimes fails to improve learning over and above other instructional scaffolds (Conati and VanLehn, 2000, Didierjean and Cauzinille Marmeche, 1997, Grobe and Renkl, 2003, Mwangi and Sweller, 1998).

Thus, although the relation between self-explanation prompts and improved learning has been documented and replicated, not all learners generate effective self-explanation even when prompted. One unexplored possibility is that the type of instruction preceding self-explanation prompts may influence subsequent explanation quality and learning. Although method of instruction has been varied between experiments, the type of instruction used within an experiment has rarely been manipulated. In this study, we contrast the effects of conceptual and procedural instruction on self-explanation quality and learning.

Debate over the comparative merits of procedural and conceptual instruction has a rich history spanning the 20th century (for an overview, see Baroody & Dowker, 2003), yet the relations between the types of instruction employed and the types of mathematical understandings generated remain largely unresolved. Does instruction focusing on procedures primarily build procedural knowledge, or does it effectively promote conceptual knowledge as well? Likewise, what types of knowledge does instruction on concepts promote? In line with our current concern for getting the most out of self-explanations, we add another question: Which type of instruction best supports the types of explanations associated with the best learning gains?

First, we offer some functional definitions. We define conceptual knowledge as explicit or implicit knowledge of the principles that govern a domain and their interrelations. In contrast, we define procedural knowledge as the ability to execute action sequences to solve problems (see Baroody et al., 2007, Greeno et al., 1984, Rittle-Johnson et al., 2001: Star, 2005). Similarly, we define conceptual instruction as instruction that focuses on domain principles and procedural instruction as instruction that focuses on step-by-step problem-solving procedures. Although mathematics education researchers sometimes emphasize conceptual knowledge at the expense of procedural knowledge, we recognize that both procedural and conceptual knowledge are critically important (National Mathematics Advisory Panel, 2008, Star, 2005). Greater conceptual and procedural knowledge both are associated with better performance on a variety of problem types (e.g., Blöt et al., 2001, Byrnes, 1992, LeFevre et al., 1993, Rittle-Johnson et al., 2001), and both are key characteristics of expertise (Koedinger and Anderson, 1990, Peskin, 1998, Schoenfeld and Herrmann, 1982). Hence, we are ultimately interested in instruction that can maximally promote both types of knowledge.

Several classroom researchers have argued that, compared with procedural instruction, conceptual instruction supports more general knowledge gains. Hiebert and Wearne’s (1996) study of place value and multidigit arithmetic is one widely cited case. Procedural instruction that focused on standard algorithms could quickly move students’ procedural knowledge ahead of their conceptual knowledge. In contrast, conceptual instruction that focused on the base-10 system and inventing procedures improved both procedural and conceptual knowledge simultaneously. Others also have found evidence that, compared with procedural instruction, an emphasis on conceptual instruction leads to greater conceptual knowledge and to comparable procedural knowledge (Bednarz and Janvier, 1988, Blöt et al., 2001, Cobb et al., 1991, Fuson and Briars, 1990, Hiebert and Grouws, 2007, Kamii and Dominick, 1997).

Randomized experimental studies have provided some corroboration of these classroom findings. All of these studies have used math equivalence problems as the target task. In one early precursor to the current experiment, providing students with conceptual instruction led many children to generate accurate solution procedures that they could appropriately adapt to solve transfer problems (Perry, 1991). In contrast, procedural instruction improved performance on problems specifically targeted by instruction but was less effective in promoting procedural transfer. Similarly, Rittle-Johnson and Alibali (1999) found that procedural instruction was less effective than conceptual instruction at promoting conceptual knowledge. Interestingly, Perry (1991) also found that procedural instruction could actually impede learning when students who received hybrid instruction on both concepts and procedures performed worse on procedural transfer items than did those who received instruction on concepts alone.

These findings notwithstanding, we should be careful not to conclude prematurely that conceptual instruction is typically more effective than procedural instruction in promoting conceptual knowledge and procedural transfer. First, in the classroom studies considered above, some procedural instruction was typically included in the conceptual instruction and children were not randomly assigned to condition, making it difficult to draw conclusions about the effects of one type of instruction versus the other. Second, the experimental studies considered above offered few examples, little opportunity for practice or feedback, and/or no prompts for reflection, all of which may be important for establishing effects of procedural instruction (e.g., Peled and Segalis, 2005, Siegler, 2002). Third, recent experiments have shown explicitly that either procedural instruction or procedural practice in the absence of instruction can promote both procedural and conceptual knowledge of math equivalence and decimals (Rittle-Johnson and Alibali, 1999, Rittle-Johnson et al., 2001).

All told, prior experimental studies often have not provided opportunities for practicing and reflecting on procedures that should reduce cognitive load, increase problem-solving efficiency, and free cognitive resources for improving procedural transfer and conceptual knowledge (e.g., Kotovsky et al., 1985, Proctor and Dutta, 1995, Sweller, 1988). The procedural instruction intervention in the current study offered both instruction on a procedure and multiple opportunities for practice using that procedure with feedback. Moreover, this study incorporates self-explanation prompts for reflection that may further boost the effects of procedural instruction. Such boosts to the efficacy of procedural instruction may make it equal to or more effective than conceptual instruction.

Self-explanation prompts add a new dimension to consider when choosing between procedural and conceptual instruction. The effects of a given type of instruction might be augmented or weakened when used in combination with self-explanation prompts. Likewise, the effects of self-explanation prompts might vary in response to the type of instruction used prior to prompting.

Self-explanations can promote transfer when used in combination with procedural instruction to teach mathematical equivalence (Rittle-Johnson, 2006). Self-explanation prompts may push learners to consider the conceptual underpinnings of instructed procedures. Similarly, procedural instruction may free cognitive resources that can be dedicated to generating more effective self-explanations than when conceptual instruction is provided. Alternatively, conceptual instruction may boost the benefits of self-explanation prompts by directly augmenting knowledge of domain principles and directing attention to conceptual structure. Hence, self-explanation prompts might help students to fill in knowledge gaps by promoting inferences that can be drawn from knowledge provided by conceptual instruction. To date, direct comparison of self-explanation effects across the two types of instruction remains unexamined.

The current experiments investigated the relations among type of instruction, self-explanation prompts, and the types of self-explanations and knowledge that are promoted. We used math equivalence problems of the type 7 + 3 + 9 = 7 + _ as the primary task. These problems pose a relatively high degree of difficulty for elementary school children (Alibali, 1999, Perry, 1991, Rittle-Johnson, 2006). Importantly, these problems tap children’s understanding of equality, which is a fundamental concept in arithmetic and algebra (Kieran, 1981, Knuth et al., 2006, McNeil and Alibali, 2005). Because equality is such a central concept in mathematics, the current task is potentially fruitful for exploring the relations between conceptual and procedural knowledge in mathematical thinking more generally. Prior research has shown that self-explanation prompts can improve procedural learning and transfer on math equivalence problems (Rittle-Johnson, 2006, Siegler, 2002).

The goals of the current study were threefold. First, because of the previously established relation between quality of self-explanation and learning outcomes, we wanted to evaluate the relations between type of instruction and the quality of children’s subsequent self-explanations. Second, we wanted to evaluate the relations between type of instruction and children’s conceptual and procedural knowledge of mathematical equivalence. Finally, we wanted to determine whether self-explanation prompts used in conjunction with conceptual instruction improve learning over and above conceptual instruction alone when equating time on task. Specifically, Experiment 1 examined the comparative effects of conceptual and procedural instruction when all children were prompted to self-explain. Experiment 2 examined the effects of self-explanation prompts when all children received conceptual instruction and spent approximately the same amount of time on the intervention.