Introducing Algebra in a hands-on augmented reality game app with an adventure, to make math engaging for all!
Algebra is key to the understanding of Mathematics beyond arithmetic, but students practicing procedures up until the learning of algebra are not well-equipped to understand the multiple conceptions of algebra.
Math Heroes aims to target the learning of algebra by allowing students to discover the objective-orientated nature variables through the manipulation of concrete manipulatives, that are dynamically linked to representational and symbolic models on the computer screen, by means of augmented reality. Using game-based activities on equations and the balance beam model, we aim to foster discovery and collaboration in the learning of algebra.
As the camera picks up QR codes, this first one transforms into an airplane. Positive values are represented as weights on each wing that can be moved around in the user’s hands. The scale tips towards the heavier side, but stays balanced even if weights are unknown. Combining negative balloons and positive weights allows a pair to cancel each other out and disappear, leading to a solution for finding the unknown.
Scan the objects
Move them around
Equation changes
The benefits of mathematical cognitive tools are most prominent when facilitating understanding across multiple representations, when action on one representation is automatically and dynamically reflected in another (Kaput, 1992). One method of instruction that allows for making connections in multi-representational thinking is the concrete-to-representational-to-abstract sequence (CRA) (Witzel, Mercer & Miller, 2003), which begins with manipulatives as tools to think and reason with in mathematics problems, progressing to pictorial representations such as diagrams to model reasoning, and eventually scaffolding to discover abstracted patterns.
Manipulatives possess the qualities of a good cognitive tool. Connell (2001) suggests that manipulation of mathematical objects is key to object reification, and that it is a cyclical and iterative process of conjectures and problem solving. Virtual manipulatives offer several affordances over concrete manipulatives, such as constraints that “frees the student to focus on the connections between the actions on the two systems [notation and visuals]” or “hot links” between multiple representations (Kaput, 1992). Students of different achievement levels are able to take advantage of different affordances of virtual manipulatives (Knuth, Stephens, McNeil & Alibali, 2006).
Knuth, Stephens, McNeil & Alibali (2006) observe that the equal sign is not given explicit attention in middle school curricula, resulting in misconceptions that would hinder solving equations in middle school and beyond. The concept of equal sign as an expression of a relationship must be grasped, in order to fully comprehend the transformations performed in the equation. As such the balance beam model serves as commonly used model to understand this relationship (equivalence and equality), as evidenced from implementations ranging from concrete manipulatives such as Hands-on equations® learning system (Borenson, 1997) and the EquaBEAM Balance (2017), to applets (National Library of Virtual Manipulatives, 2010), and Leong & Horn’s (2011) digitally enhanced tangible balance beam.
Manipulatives, being objects that are graspable with hands either physically or virtually, support embodied cognition. Embodied cognition is predicated on the fact that human cognition is not solely formed by activity in the brain, but also includes other modes of perception, such as motor movements (Barsalou, 2008). Tran, Smith & Buschkuehl (2017) have summarized the benefits of embodied cognition in mathematics learning, including improved retention of learnt concepts, reduced cognitive load during learning, and the opportunity for learners to discover concepts through their own exploration.
Augmented reality (AR), an instance of a tangible user interface, has numerous affordances that allows for embodied cognition to take place. AR refers to a digital overlay of “virtual objects or superimposed information” (Bacca, Baldiris, Fabregat, Graf & Kinshuk, 2014) over real world objects. Azuma (1997) makes the distinction that “AR supplements reality, rather than completely replacing it”, highlighting the reliance on the real-world environment. Amongst the features of AR, a key affordance of AR is situated learning (Wu, Lee, Chang, Liang, 2013), providing knowledge at the location and time that the student requires it most, “aligning information to objects and locations in the student’s environment” (Bujak, Radu, Catrambone, Macintyre, Zheng & Golubski, 2013).
A game-based learning experience can serve as a motivating factor in learning tasks, improving students’ attitude towards math learning (Ke, 2008), performing “committed and effortful on-task learning” in cases where math learning was crucial to the success of the gameplay, and was at the right challenge level for the learner. Ke (2008) also noted the importance of debriefing or reflection to learning in game-based learning environments, which may take the form of: individualized feedback to in-game performance to help players cope with the mistakes made; an incentivised component promoting reflection explicitly; a tool that focuses the player’s attention on the salient aspects of the game.
