Equations Squared: Combining Educational Game Design Theory With an Equations Learning Progression to Design a Mathematics Assessment Game

• In Event: Designing Game-Based Assessment Around Learning Progressions

Abstract

Students construct mental models of abstract concepts, basing their representations on their concrete experiences. Learning progressions provide a taxonomy through which subjective student understanding can be classified; this is done by matching student assessment data to levels of the learning progression. This evidence is conventionally gathered via written homework and quizzes, and we posit that well-designed games provide an excellent opportunity for both gathering assessment data and encouraging meaningful practice. A game is itself a formal abstract system that is defined by its rules (Salen & Zimmerman, 2003). By playing the game, players discover, learn, and potentially master the formal rules that govern it. According to Koster (2004), this process is a major source of "fun" in gameplay, "Fun from games arises out of mastery. It arises out of comprehension" (p. 40).

Equations Squared is a single-player educational video game that is designed to be fun while also providing mathematics assessment data. In this game, the player is given a sequence of 36 virtual tiles, each containing either a digit, an operator, or a variable. The player also has a limitless number of equal sign tiles. Each turn, the player must arrange tiles on an 11-by-11 board to make mathematically true equations. As in Scrabble (Butts, 1938), tiles played in subsequent turns must touch existing tiles, and they can be added so as to extend existing equations. The mathematical operations consist of addition, subtraction, multiplication, and division, each of which is worth more points than the previous. Using multiple operators will follow conventional order of operations and will earn multiplicative bonuses. Variables follow single-assignment semantics: once one is played, the player must provide a value that makes the equation true, and that value is bound to the variable for the duration of the game.

Equations Squared features a system of badges and demerits that provide direct evidence of student achievement. Whereas the player's score is a coarse-grained approximation of the player's relative achievement, the badges and demerits correspond directly to the equations and variables learning progression proposed by Arieli-Attali, Wylie, and Bauer (2012). Badges are earned by taking actions in the game that represent high levels in the learning progression. For example, the "Operators on Either Side" badge can be earned by playing 1+1=2 followed by 4=2+2; this shows evidence of a relational understanding of equality, which corresponds to level 3 of the progression (Arieli-Attali et al.). Similarly, playing 1+1=2-0 earns the "Operators on Both Sides" badge, which corresponds to the fourth level. Demerits represent mathematical errors and correspond to the lowest level of the progression. A player earns the "Unbalanced Equation" demerit for playing a sequence of tiles that form a mathematically untrue statement. By tracking the frequency of a player's badges and demerits, a teacher gains insight into their level within the learning progression. This achievement system is also designed to increase players' motivation to continue playing and learning (Blair, 2012).

Author

• Paul Gestwicki, Ball State University