Coordinating vector field equations and diagrams with a serious game in introductory physics
Pascal Klein, Nicole Burkard, Larissa Hahn, Merten Nicolay Dahlkemper, Kevin Eberle, Tina Jaeger, Jochen Kuhn, Marc Herrlich
CCoordinating vector field equations and diagramswith a serious game in introductory physics
P Klein , N Burkard , L Hahn , M N Dahlkemper , KEberle , T Jaeger , J Kuhn and M Herrlich Department of Physics, Physics Education Research, Georg-August-UniversityG¨ottingen, Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany Department of Electrical and Computer Engineering, Serious GamesEngineering, Technische Universit¨at Kaiserslautern, Paul-Ehrlich-Strae 11,67663 Kaiserslautern, Germany Department of Physics, Physics Education Research, Technische Universit¨atKaiserslautern, Erwin-Schr¨odinger-Str. 46, 67663 Kaiserslautern, GermanyE-mail: [email protected]
Abstract.
Mathematical reasoning with algebraic and graphical representationsis essential for success in physics courses. Many problems require students tofluently move between algebraic and graphical representations. We developeda freely available serious game to challenge the representational fluency ofintroductory students regarding vector fields. Within the game, interactivepuzzles are solved using different types of vector fields that must be configuredwith the correct mathematical parameters. A reward system implemented in thegame prevents from using trial-and-error approaches and instead encourages theplayer to establish a mental connection between the graphical representation ofthe vector field and the (algebraic) equation before taking any action. For correctsolutions, the player receives points and can unlock further levels. We report aboutthe aim of the game from an educational perspective, describe potential learningscenarios and reflect about a first attempt to use the game in the classroom.
Submitted to:
Eur. J. Phys. a r X i v : . [ phy s i c s . e d - ph ] A ug erious Game
1. Introduction
Vector fields are mathematical objects that assign a vector to every point in thespace or in a subset of space (e.g., the two-dimensional plane). Vector fields areimportant in many physics subjects, some of which are part of the introductory physicscurriculum. Examples include Newton’s gravitational field, velocity fields of fluids, andelectromagnetic fields. Two illustrative examples of vector fields are given in Fig. 1.There are decades of research about students’ difficulties associated with vectors[1, 2, 3, 4, 5, 6, 8]. Numerous studies have shown that first-year university studentshave substantial difficulties regarding basic vector concepts, e.g., interpreting graphicalproperties such as direction, length and component decomposition, vector additionand subtraction, or scalar and cross products [1, 2, 3]. Based on the broad bodyof research, a reliable instrument to assess students’ understanding of vectors wasdeveloped, validated, and used by independent research groups [5, 6, 7]. Students’difficulties occur both with and without physics contexts [4, 8], i.e., when vectorconcepts are treated purely mathematically. A comprehensive knowledge of theseconcepts is a prerequisite for understanding vector fields, thus, students’ difficultieswith single vectors have a direct impact on their understanding of vector fields.In addition, students might understand basic concepts for single vectors but showdifficulties applying the concepts to vector fields [10]. For instance, students mightsucceed to superimpose two force vectors but they might fail to superimpose two forcevector fields.Physics concepts, such as conservation laws or cause-effect relationships, areusually expressed using (one or more) external representations [9]. Commonrepresentations of vector fields are both graphical representations, e.g. vector fielddiagrams (see Fig. 1), and algebraic expressions using unit vectors in a previouslydefined coordinate system (see caption of Fig. 1) [10]. The coordination of both formsof representations (diagram and algebraic expression) represents a further challengefor students in general [11] and, particularly, for vector fields [10]. Among others,the study of Bollen et al. suggest that a confident and flexible handling of multiplerepresentations can have a positive impact on learning and problem solving and todevelop domain-specific expertise (see Sect. 3 for more details). Indeed, several of the
Figure 1.
Two examples of 2D vector field diagrams. The left vector field isdescribed by the equation F ( x, y ) = − x ˆ x , and the right one by F ( x, y ) = y ˆ x + x ˆ y erious Game
2. Serious games in physics education
Games and playing in general as forms of learning pose an important factor in humandevelopment, especially, in early years as a child. Play and games follow rules, atleast given implicit but often fully formalized. There have been early (non-digital)examples of the latter for educational purposes especially in the military domain, e.g.,GO and Chess, to train tactical and strategic thinking and decision making.The term serious games was first described by Abt in 1970 as “games [that] havean explicit and carefully thought-out educational purpose and are not intended to beplayed primarily for amusement”[12] originally not referring to digital games. Theterm itself is a bit contradictory, since it might imply that other games might not beserious or that serious games should not be entertaining. There are many other termsused for specific domains (games for learning, edutainment, educational games, etc.).However, the term serious game is often used as the best available summarizing termthat encompassed many different approaches and domains [13].With the ongoing digitalization and transformation of society, digital games haveentered the scene and with the cultural and economic success of (entertainment)games [14], a whole generation has been socialized by gaming which in turn maycontribute to the success of serious games [13].The research on serious games in physics education dates back to the work of
Figure 2.
The effect of an appropriate vector field on the particles: The particles(yellow dots) move from the sources (left) to the target (right). The vector fieldequation and the parameters are present on the upper part of the screen. erious Game
3. Educational and theoretical background: Handling multiplerepresentations of vector fields
Learning processes with multiple representations are a central subject of physicseducation research. Ainsworth formulated three central functions of multiplerepresentations (MR) that can facilitate learning [26]; (1) MR contain complementaryinformation, i.e. the learner benefits from the advantages from both representations[27]; (2) MR can help the learner to develop a better understanding of the subjectby using one representation to limit possible misinterpretations by another; and (3)MR can help the learners to develop a deeper understanding of a concept. Based onthese functions, many researchers report a positive effect of using MR on knowledgeacquisition and problem-solving skills [28, 29, 11, 30, 31, 32, 33, 34, 35]. For example,Nieminen, Savinainen and Viiri (2012) found a strong correlation between learners’ability to interpret multiple representations consistently (representational consistency)and their learning gain in a study on forces [11]. Their result confirms that judicioususe of multiple representations can contribute to a better understanding of physical erious Game “Psychologically they [representations of physical concepts] are differentbecause they are completely inequivalent when you are trying to guess newlaws”.
Despite the potential benefits described above, there are numerous studiesshowing that the use of multiple representations does not per se lead to higherlearning outcomes [30, 38, 39]. In order to benefit from the advantages of multiplerepresentations in learning and problem solving, a deeper understanding of therepresentations is necessary, which is described with the help of two competences,following De Cook (2012) and Nistal et al (2009, 2012) [9, 40, 41]:(i) representational fluency: This competence enables the interpretation andconstruction of representations and enables the correct and quick switching andtranslation between different forms of representation [42].(ii) representational flexibility: this competence enables the choice of an appropriateform of representation in a given problem or learning situation and involves theability to take into account characteristics of the subjects interacting with therepresentation and the context of the interaction [40, 41].A central aspect of representational fluency is called representational competenceand includes the knowledge of how representations are to be interpreted and howthey represent information about the learning content [9, 43]. If learners do notknow how the visual representation encodes information (also referred to as visualunderstanding [43]), multiple representations can have negative effects on problem-solving ability and learning success [9, 39].Bollen et al (2017) investigated students’ errors when switching representationsbetween vector field diagrams, field line diagrams and algebraic expression [10]. Whenconstructing a vector field diagram using an algebraic formula expression, errors werefound that were mainly due to problems with vector addition, representing the changein length and direction of the vectors with increasing distance from the origin, and,for the reverse direction (i.e., changing from graphical to algebraic representations),the choice of a suitable coordinate system and the use of unit vectors posed aproblem. Gire and Price (2012) who dealt with representation changes in the context ofelectric and magnetic fields also found that students often had problems differentiatingbetween coordinates and components [44]. This was especially the case with “mixed”dependencies, e.g. when the x -component was a function of the y -coordinate. erious Game Figure 3.
In-game coordination between equation and vector field. The diagramchanges based on the parameter of the field
Against the background of these particular findings, the authors call for specificinstructions to train representational fluency in dealing with vector fields [10, 44], towhich the Vector Field Game can contribute by playfully practicing the transitionbetween vector field diagram and formula.
4. The Vector field game: game concept and design
The Vector Field Game was designed to complement teaching of vector fields and toteach intuition about the abstract concept in an interactive fashion. Therefore, wechose a casual style game to improve accessibility and to visualize the direct effectof different vector fields on particles. The game was developed in an iterative designprocess as proposed by Fullerton [45]. This enabled us to focus on player experience(“playcentric design process”) in every development phase.The game itself consists of recurring elements (see Fig. 4): In every level, there isat least one source of particles (drop shaped), one goal (circle), and one but usuallymultiple boxes representing vector fields. The player needs to select the appropriatevector fields in order to guide the particles to the goal and solve the level. In the designframework of Eccheverra et al. [23], the vector field and the equation correspondto the low-level game atoms (independent variable that can be controlled by theuser), and the track of the particle is the dependent variable. The challenge is toestablish a connection between the source of the particle and the target which reflectsan additional game atom.The scaffolds that Eccheverra et al. describes are also present in the game astwo types of boxes. Boxes with light background indicate a fixed vector field typewhose parameters must be chosen by clicking on the box and subsequently selectingthem on the top bar (see Fig. 4, right). Boxes with grey background indicate nota specific field type, so the player has to choose the correct type by dragging anddropping the respective icon from the button bar in the desktop version. The mobileversion opens an extra menu in which the type can be selected by tapping. The topbar is only present when a box is selected (see Fig. 2) and always displays the vectorfield formula (as link to the mathematical representation) and possible parametersfor it. The game includes four kinds of vector fields: constant vector fields, hookfields, radial fields and rotational fields. When changing the parameters of the vectorfield, the arrow visualization in the vector field box is instantly updated so that theplayer can visually explore the link between the vector field formula and the arrow erious Game
In order to obtain acceptable usability, which we identified as a key qualityrequirement, we included a tutorial and focused on the development of an intuitiveinterface. Our goal is to facilitate the acquisition of intuition complementingthe theoretical knowledge on vector fields. Therefore, we refrained from usingunconstrained mathematics as it would offer too many degrees of freedom withoutan additional benefit in terms of intuition as minimal changes of parameters wouldalmost be indiscernible but might lead to unintended results in some cases. However,we also wanted to avoid pure trial and error type play, which was a difficult balanceto achieve, and partly, this is still open for evaluation.We followed an iterative design and development process to find a good balancebetween offering enough degrees of freedom in the formulas and different types ofvector fields – to facilitate actually making errors to learn from – but also provideguidance and scaffolding in the form of different design elements. First of all, the leveldesign itself and the progression of levels is set up to start simple and raise complexityslowly: the game introduces simpler vector field types first, with fewer parameters tochoose from, and the early levels include overall fewer vector fields. In addition, wedesigned a graphical indicator as a direct reaction to player testing providing feedbackin different tiers: red for wrong field type, amber for correct field type but wrongparameters, and green for correct field type with correct parameters (see Fig. 4). Ascountermeasure for blindly trying to solve the level by random choices, the numberof parameter selections and the overall level time influence the resulting score. Wedecided to put a mild time pressure on players to provide a motivating challenge andto avoid an overly slow and boring pace of the game. However, every level is stillsolvable even if the time runs out to reduce player frustration in challenging levels.In general, the reward design to motivate the players is twofold: on one hand,intrinsic motivation is boosted by good usability and stream-lined, non-distractinginteraction as well as graphical aesthetic design. On the other hand, extrinsicmotivation is given in the form of points and star ratings for each level. The later alsofacilitates replayability because the star ratings are also shown in an overview screen erious Game Figure 4.
In game view of two different levels of the Vector Field Game. Left:The simulation started but the particles did not enter the goal. The colored iconson the bottom right of each vector field box denote whether the correct type wasused. Right: A level with “charged” particles. The positive and negative particleshave each have a respective goal. of all levels unlocked so far to motivate players to beat their own high-scores.
The game is free and available on the desktop † for Windows and macOS and on mobileplatforms for Android ‡ and iOS § . Note that the vector fields that were constructed for the serious game and forthe educational studies do not necessarily have a physical meaning. For our aimswe wanted to avoid that lacking physical knowledge hinders the acquisition ofrepresentational fluency. To become knowledgeable physicists, however, studentsdefinitely must supplement their mathematical skills with conceptual knowledge ofthe physical world. For instance, the left vector field in Fig. 1 could be interpretedas the force field of an one-dimensional spring, F ( x ) = − k x , that was extended tothe two-dimensional plane, F ( x, y ) = − k x ˆ x . Even though conceptual knowledge inphysics of this kind might be helpful for solving the puzzles, they are no prerequisite.
5. First distribution among students and research potential
In January 2020, the game was distributed to first-term physics students and aninvestigation of the presumed learning potential of the game was prepared. Forthis purpose, a diagnostic test was designed that addresses the connection betweenthe mathematical representation form (formula and equation) and the correspondinggraphical representation of vector fields - a competence that should be promoted bythe game from an educational point of view, see Sect. 3. The instrument includes8 vector field diagrams in total, and students were asked to select the correspondingvector field equation out of four alternatives (see Fig. 5 for an example). The averageability of the students ( N = 68) to establish coherence between formula and diagram † https://tuk-software.procampus.de/de/vektorfeldspiel/ ‡ https://play.google.com/store/apps/details?id=de.unikl.eit.sge.vectorfieldgame.android § https://apps.apple.com/de/app/vektorfeldspiel/id1517002472 erious Game Figure 5.
One out of eight multiple choice items to assess students’representational fluency determined by the test instrument was 57.3% after correction for guessing, measuredas pre-test, i.e. without playing the game.Initially it turned out that the informal distribution of the game withoutinstructions did not produce the desired response. Of the 68 students to whom thedownload link for the game was issued with the request to play the game duringthe upcoming four weeks, only 26 actually downloaded the game. Of those whodownloaded the game, 13 students were engaged with the game for less than 10minutes, 9 between 10 minutes and 1 hour, and 6 students said they had played thegame to the end. 62% of the students thought that the game was fun and the samenumber said that they learned something from it. Due to the low student participationrate, a post-test was not carried out to reassess the competence, so no conclusions canbe drawn about the change in this competence that the game might cause. In follow-up examinations it is advisable to anchor the game formally and instructionally. Inorder to lower the usage threshold for students in the future and to enable testing ona larger scale, the porting and publication of the game to mobile platforms (Androidand iOS) was carried out.
6. Concluding remarks and outlook
In times of increasing importance of online learning due to the COVID-19 pandemic,educational technology and digital teaching materials gain in value. We reportedabout a freely available serious game that offers a contribution for physics studentsand lectures. The Vector field Game was developed to challenge the representationalfluency of introductory students regarding vector fields. It was discussed that (1) beingfluent to make connections between equations and diagrams is an important skill inmathematics and physics education, particularly in the context of vector fields, and(2) that serious games can have the potential to substantially foster learning by amixture of using intuitions, problem-solving heuristics, and feedback from the game.Combining both lines of research, i.e., learning with multiple representations and erious Game
Acknowledgements
The project was partly funded by the ”LehrePlus”-grant of the TU Kaiserslautern,aiming at sustaining and improving the quality of teaching and learning in universityeducation.
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