Academic journal article
*Educational Technology & Society*

# Effects of Self-Explanation and Game-Reward on Sixth Graders' Algebra Variable Learning

## Article excerpt

Introduction

Algebra variable learning

Many students struggle with algebra variable concepts (Novotna & Hoch, 2008; Warren, 2003). MacGregor and Stacey (1997) noted that students usually do not understand how to use variables, the basic and essential concepts of algebra, to help them solve problems accurately and efficiently. Thus, Stacey and MacGregor (1997) suggested that students could be better prepared for further algebra learning by frequently using algebra notation and symbolism (i.e., the variable system) in learning contexts. According to the suggestion, Ross and Willson (2012) conducted an instructional experiment and found that engaging students in the given content with practical examples led to the better understanding of algebra variables. However, to date, issues regarding students' algebra variable learning, such as use of certain strategies and design of examples, are still only partially explored (Bush & Karp, 2013).

As to proper understanding of algebra variables (i.e., the symbol system), Lucariello, Tine, and Ganley (2014) pointed out some key concepts. First, a symbol must be interpreted as representing an unknown quantity (Kuchemann, 1978; McNeil et al., 2010), meaning a student must realise that a symbol represents a unit that does not have a certain value. Second, a student must interpret a symbol as representing a varying quantity or range of unspecified values (Kieran, 1992; Philipp, 1992). This concept is named as the multiple values interpretation of literal symbols (Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Lucariello et al., 2014). To explore learners' understanding of these two concepts, Knuth et al. (2005) tested seventh and eighth graders with a problem: "The following question is about the expression '2n+3.' What does the symbol (n) stand for?" The correct responses should represent ideas that (1) the symbol could represent an unknown value (e.g., "the symbol is a variable, and it could stand for any value") and (2) the symbol could represent more than one value (e.g., "it could be 3, 74, or even 123.4567"). The results, however, showed that approximately 39% of seventh graders and 25% of eight graders answered incorrectly. The researchers thus suggested that an appropriate training would be necessary before middle-school education (e.g., when in sixth grade) for students' better understanding of variables.

The third concept involves the relationships between symbols, and in such relationship the values change in a certain approach (e.g., as X increases, Y decreases, Lucariello et al., 2014). For example, Kuchemann (1978) invited and tested 3000 high-school students who had learnt variables with the following problem: "Which is the larger, 2n or n+2? Explain." The results revealed that just only 6% of the students correctly realised the concept that the relation between 2n and n+2 is actually changing with n. Knuth et al. (2005) also explored this understanding of algebra variables by presenting middle-school students with similar problems. Likewise, the results showed that only approximately 18% of sixth graders, 50% of seventh graders, and 60% of eighth graders were aware of the relationship between symbols because their values change systematically.

Self-explanation strategy

Self-explanation, a concept-oriented learning activity, is defined as generating explanations to oneself to make sense of new and known information to be correct, and it has been regarded as an effective strategy in students' algebra learning (Chi, 2000; Chi, de Leeuw, Chiu, & Lavancher, 1994; Neuman, Leibowitz, & Schwarz, 2000). For example, Aleven and Koedinger (2002) facilitated students' self-explanation by using intelligent instructional software for algebra, and they found that students who explained their steps of problem-solving outperformed those who did not. According to Roy and Chi (2005), the process of self-explanation can encourage four forms of cognition: (1) recognise what information is missing while generating inferences, (2) integrate information into what learnt from a lesson, (3) relate new information to learners' prior knowledge, and (4) identify and correct information received. …