# Designing Pedagogically Effective Haptic Systems for Learning: A Review
## 3 Haptics Applications for Learning
### Stem
#### 3.1.3 Mathematics  
A close relative of the theory of Embodied Cognition, [PERSON] and [PERSON] [22] propose a theory of Grounded and Embodied Mathematical Cognition (GEMC). GEMC asserts that a person's actions serve as inputs capable of altering cognitive states and views mathematical communication as a \"multi-modal discourse practice, rather than a formal, written, propositional form\" [22]. Studies supporting their theory have found that gesture and directed action can significantly improve learning outcomes. They define directed actions as: \"physical movements that learners are instructed to formulate by some kind of pedagogical agent\" and define gestures as: \"spontaneously generated movements, often of the hand, that accompany speech and thought\". According to [PERSON] et al. [23], a network of conceptual metaphors is the basis of mathematical knowledge. [PERSON] et al. define conceptual metaphors as: \"Cognitive mechanisms [that are] used automatically and unconsciously, having abstracted patterns of inferences from physical experiences\". On the premises of Embodied Cognition and Cognitive Load Theory, it is thought that mathematical connections are made when a physical action is connected to an abstract mathematical concept and that physical actions can facilitate the learning of biologically secondary information (mathematics). According to these principles, mathematical pedagogy should be developed to incorporate physical ways for students to interact with abstract mathematical concepts, which emphasizes a demand for haptic intervention in mathematics education. Capitalizing on embodied cognition research, [PERSON] et al. [24] designed a haptic-supported learning environment and investigated the ability of this system to teach trigonometry concepts to four high school students in a preliminary study. Their research objective was to determine how haptic feedback might impact learning and they propose a theoretical design framework for educational habits. [PERSON] et al. collected student data concerning three metrics: (1) student interactions with the software, (2) student interactions with the haptic device, and (3) student drawings of the learning material, graphs of sine and cosine, which served as a measurement of conceptual understanding. The haptic device used was created from an open-source kit, called Hapkit. Their haptic device was programmed to be used with a program called Trigonometry Explorer, where the student uses the Hapkit as a joystick to move across the graph of a trigonometric function. The users \"feel\" the graph by receiving force feedback from the Hapkit joystick depending on the position the user is at in the graph. Results analysis indicated that the haptic feedback helped the students make connections between the different representations for sine and cosine, and that the effectiveness of haptic feedback on learning was tied to the students' ability to interpret it [24]. For example, one student named [PERSON] systemically manipulated the Hapkit and proposed a hypothesis for how the haptic feedback was related to the sine graph: \"So when it's doing this and it gets harder then you're going up higher and higher; then you start getting here and it starts getting easier as it's going down then you start getting around here again and it starts getting a little harder; at that point you're going up again\" [24]. [PERSON]'s personal hypothesis was in line with the force feedback presented by the haptic joystick, the minima and maxima of the graph were located at the two points where the force generated by the Hapkit was highest. When [PERSON] was asked to draw the graph, he returned to the Hapkit several times before completing the task. [PERSON] drew the graph of sine correctly on his first try. One of the other student participants in the study, [PERSON], had a harder time interpreting the haptic feedback. Unlike [PERSON], he did not appear to be making the connection between the haptic feedback and the graph of cosine, saying \"I don't know.