MathRun: An Adaptive Mental Arithmetic Game Using A ... · MathRun: An Adaptive Mental Arithmetic...

http://dx.doi.org/10.14236/ewic/HCI2016.77

MathRun: An Adaptive Mental ArithmeticGame Using A Quantitative Performance

Model

Long ChenDepartment of Creative TechnologyFaculty of Science and Technology

Bournemouth University, Fern BarrowPoole, Dorset, BH12 5BB

[email protected]

Wen TangDepartment of Creative TechnologyFaculty of Science and Technology

Bournemouth University, Fern BarrowPoole, Dorset, BH12 5BB

[email protected]

Pedagogy and the way children learn are changing rapidly with the introduction of widely accessiblecomputer technologies, from mobile apps to interactive educational games. Digital games have the capacityto embed many learning supports using the widely accredited VARK (visual, auditory, reading, andkinaesthetic) learning style. In this paper, we present a mathematics educational game MathRun for childrenage between 7-11 years old to practice mental arithmetic. We build the game as an interactive learningenvironment that fuses game mechanics with learning and uses the popular game genre “infinite runner”as the game mode. The game consists of an automatically generated infinite game map and mathematicalquestions also procedurally generated with varied levels of difficulty and complexity. A novel real-timeperformance evaluation method is developed for quantitative modeling the performance of the player. Themodel evaluates the performance in each primitive map block of the game map and level progression isautomatically carried out based on the result of the evaluation. Therefore, the proposed game-based learningenvironment is adaptive to players with dynamic level progressions based on the combination of not onlymathematics ability, but also gameplay skills of the player to facilitate learning processes through gameplayand appropriate adaptive progression of maths ability.

Educational Games; Incentive Structure in Games; Adaptive Level Progression; Game Design for Children;

1. INTRODUCTION

Games for education have been seen with greatpotential to motivate students. Recently the potentialof games for learning at large scale has beenrecognised (Mayo, M. J., 2009). Maths gameswere designed to teach fundamental mathematicsprincipals through games narratives, feedback andincentive structures (Martin et al, 2012). Inthis paper, we describe our mental arithmeticgame called MathRun, which comprises arithmeticalcalculations currently with the target for Key StageCurriculum (Gov.uk., 2014).

In MathRun student performs arithmetical calcula-tions mentally without writing the figures down orusing a calculator. Mental arithmetic is an importantpart of mathematics. It is also an essential part ofcoping with society’s demands and managing every-day events (Department of Education UK, 2010).Mental arithmetical is an important element of key

stage education, which is proven to be related to themath score of high school (Price et al , 2013).

Experiments have shown that the result of simplearithmetical problems such as single digit addition,subtraction and multiplication are fixed knowledgestored in a semantic memory network and usuallycan be directly retrieved without actual deliberatethinking and computation (Dehaene et al ,1997) (Zago el al , 2001). It has been proposedthat these frequently-used simple questions, calledarithmetic facts, can be learnt and stored asverbal associations mediated by language-basedrepresentations (Dehaene et al , 1992) (Klein,2013).

Recent investigations also indicated that the processunderlying the change from deliberate, consciouscalculation to automatic and direct retrieval ofanswers from memory is actually caused by thelong-term frequent practice. (Bourne et al , 1994)

c© Chen et al. Published byBCS Learning and Development Ltd. 1Proceedings of British HCI 2016 Conference Fusion, Bournemouth, UK

MathRun: An Adaptive Mental Arithmetic Game Using A Quantitative Performance ModelChen • Tang

Figure 1: (a) The sketch of one map block, which consists of one question and three answers, three bonus items and twoobstacles that randomly spawn at corresponding spawn points. (b) One map block in game editor; (c) The stitched map seenin a playing game with three primitive map blocks; (d) A screen capture in player’s perspective.

(Pauli et al , 1994) The experiment results showthat response time speeds up significantly acrosspractice sessions, which follows the power law. Morepractice is completed. the quicker the problem getsprocessed.

There are a number of factors that children canderive and recall mental calculations, and childrencan learn a range of calculation strategies ormethods that they can draw on. The educationalgame is a good way for children to learn whileplaying. The proposed game MathRun is designed toenable frequent practices through fun and motivationwith a fast pace and infinite automatic game mapmethodology. To engage and motivate students topractice, an effective incentive structure in-game isvital to achieving this game-based learning objective.

Since every player has a different preference forthe pace and style of game playing within a game(Charles et al, 2005), and the mental calculationability between individuals varies even in the sameyear group (Dowker , 2005). Educational gamesusually have very narrow range of target students

to avoid too simple for lower grade children or toohard for higher grade children, which greatly reducethe number of available game for the different levelof children. It is vital to have an educational gamethat could cover a wider range of students withoutthe need to select difficulty but to dynamically tailorthe game to suit the ability of players. In addition,the game genre itself should be easy to control andpopular among youths.

In this paper, we describe the development of a new3D game for mental arithmetic practice based on thepopular “infinite runner” game genre. We present anovel quantitative framework of measuring studentsin-game performance and incorporating learningbehaviour while playing an educational game. Ourquantitative framework is developed by introducinga performance model which dynamically guidesthe automatic generation of adaptive incentivesfor game-based learning of mathematical mentalarithmetic. Our system generates rewards based onproperties of mathematical model and student skills.The main contribution of this paper are:

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• A novel quantitative performance model thatcombines mathematical ability and gamingskills to generate motivation and engagement.

• Adaptive procedural level generation for math-ematical mental arithmetic practice for a widerange of abilities.

• A new incentive system based on thequantitative performance model that cangenerate precise rewards and penalty pointsin-game play.

The MathRun game is designed based on thepopular “infinite runner” game genre such as TempleRun (Imangi Studios, 2016) and Subway Surfers(Kiloo, 2016) which has seen more than 1 billiondownloads (Tech Times, 2014). With simple gamecontrol and attractive gameplay mechanics, this typeof games is very popular amongst young peopleaged between 6-25 years old. Therefore, not onlyemploying such game genre in MathRun could drawinterests to the target players of age between 7-11 years old, more importantly, the adaption ofthe fast pace of infinite runner is to meet thespeed requirement for mental arithmetic game-based learning.

We have developed MathRun as a third person3D game with an essential gameplay that a ’therunner’ character runs on an infinitely extensibleroad. The goal of the player is to survive as longas possible by collecting items to earn scoresand avoid obstacles to reducing energy damagewhile answering mathematics questions correctly asshown in Figure 1 (a)). In contrast to Temple Run andSubway Surfers, the runner is not forced to run, butto control the movement of player character freely,so that the player is in control of the time to answerquestions. Details of the game design methodologyare described below, which include automaticmap generation,core mathematics properties, life& energy system, in-game performance evaluation,adaptive level switching, and the reward system.

1.1. Map Generation

We have designed a procedural game mapgenerator to create infinity game map automatically.In addition, a method for generating random mathquestions and incentive elements in the map isalso developed. A primitive map as a fundamentalbuilding block of the map is used to generate onequestion and three answers as shown in Figure 1 (a)and the corresponding scene in game editor Figure1 (b). Figure 1 (c) illustrates an overview of thecurrent map at a certain moment. The entire mapis procedurally generated by stitching three primitivemap blocks sequentially. The middle map block (withyellow border) is the map the player is currently

Figure 2: Overview of the user interaction and system.

standing on; the right map block will be deleted oncethe player has passed the map block, whilst the leftmap block is created. Figure 1 (d) shows a screencapture of the MathRun. The mythic fantasy world iscreated using special visual effects such as fog andanimated assets.

With this procedural map generation process allowsseveral advantages: the game can be playedinfinitely without manual design of a very largemap; objects are created and destroyed dynamicallyto save computational resources; mathematicalquestions are created automatically and incentiveitems are created and controlled either randomly andprecisely.

1.2. Integration of Mathematical Questions

To create a mathematical game as an infinite runnergame, questions are designed as obstacles in theway of the player. Three answers are placed beloweach question so that the player needs to choose thecorrect answer to continue the game. Questions arederived from the National mathematics curriculum inEngland (Gov.uk., 2014), but the difficulty is slightlyreduced with the expectation of mental calculationin this game and randomly generated based on therange designed for each current level; operators arealso randomly generated with addition, subtraction,multiplication, and division; and correct or wronganswers are randomly selected from the positions ofthree pumpkins.

The current version of MathRun consists of 7levels of difficulties with mathematical questionsrange from very simple 0 ∼ 5 addition/subtractionand 1 ∼ 3 multiplication/division to the difficult 3numbers addition/subtraction ranges from 0 ∼ 30

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Table 1: Level Design

and multiplication/division ranges from 2 ∼ 5. Thedetailed level design can be seen from Table 1 withexample questions.

1.3. Energy & Life System

In a normal infinite runner game, the playerusually only controls the runner left, right and jumpmovements, and the forward movement of the playeris controlled by the game. Instead of forcing theplayer to keep moving forward, in the MathRun, theplayer has the full freedom to control the pace ofthe movement so that there is sufficient time tocomplete the mathematical challenges. Therefore,the game flow in MathRun has been redesignedby adding an energy system to facilitate the game-based education. The energy bar indicates thecurrent energy decays along with time and If theenergy is reduced to zero or the player selected awrong answer, a life will be lost, and the energywill be refilled to full (see Figure 4(c) and (d)).Correct answers and picking up bonus items willboost the energy by 20% and 50% respectively (seethe energy bar in Figure 4(a) and (b)) . If the energyexceeds 100%, a life would be rewarded.

2. QUANTITATIVE PERFORMANCE MODEL

In order to apply adaptive difficulty in the MathRungame, for each map, we have designed aquantitative model to analyse the performance ofplayer faced with the current difficulty. The modelallows dynamic modifications of the difficulty ofquestions according to the player performancecalculated by a performance score within each of theith map block (see Figure 1 (a) for one map block):

si = ci + a · b(ti−T ) − (d − D) (1)

Equation 1 describes the performance model, whereci denotes the number of incentive items obtained inthe ith map block, ti is the total time spent on theith map block with T as the minimal time required tocomplete the map block from the beginning to theend of the block. An estimated average value forT is about 6 seconds, therefore, ti − T representsthe delay time of the player completing the block,which means the longer it takes to finish the map, thesmaller score of the player to be awarded. We designb as a constant value between 0 to 1, thus mappingthe effect of the time delay using the exponent ofb to 0 ∼ 1 and to avoid the performance scorebecoming negative. In our experiments, we set b =0.8 for a smoother mapping. in order to determine theeffect of the time delay on the overall performance,a coefficient a is used and based on our test, we seta = 5. The last term in Equation 1 is the numberof obstacles that the player runs into d, while Dstands for the number of obstacles in the map block,which is 2 in the current configuration. The finalrange of our performance score is between 0 to 15(8 maximum number of coins in a map block, 5 forcorrect answers, and 3 for bonus items).

The performance score is awarded by taking intoaccount of correct answers to the respond time toquantitatively assess the performance of the playerin one map block, thus, offering guidance to adaptivelevel progressions and incentive elements delivery.

3. ADAPTIVE LEVEL PROGRESSIONFRAMEWORK

The MathRun game is designed with UK 7-11years old students as target players. The mentalcalculation ability between individuals varies even inthe same year group (Dowker , 2005). Therefore,fixed level progression method will not be able to

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Figure 3: The adaptive level progression framework.

accommodate students with a variety of mentalcalculation abilities. More capable students shouldbe charged and motivated while less able studentsshould be encouraged to continuous their practice.The adaptive level progression is designed to makethis game playable for a wider range of studentsability.

Benefiting from our performance score model andautomatic map generation system, the adaptivedifficulty is very effectively implemented. The leveldifficulty will be increased as well as decreasedautomatically based on the player score analysis.As can be seen from the flow chart in Figure 3,the adaptive level progression makes the use of alevel buffer for a smooth transition between levels.Table 2 summarises the level transaction process,in which the performance score of the current mapblock is mapped between −1 to +1 with the levelbuffer as 2. When the level buffer reaches 1 or -1,the game will be one level up or down and the levelbuffer is reset to 0. The extreme performance score15 or 0 will directly increase or reduce the gamelevel. The middle range performance score will begiven small value in the level buffer until it reachesthe level switch threshold (+1 or -1), minimizing thedisturbance of performance score and to ensure asmooth adaptive level switching process.

4. THE REWARD SYSTEM

A multilevel reward system is designed in theMathRun to give reward, praise or penalty to theplayer based on the performance evaluation model.

On each map block level, different events can triggerreward or penalty points. Incentives take in differentforms ranging from playing fancy graphics, sound toincrease or decrease the energy and number of lives

Table 2: Performance Score to Level Buffer Mapping Table

of the player. For example, obtaining a golden coinand bonus items will trigger crispy coin sound, thewrong answer will cause an explosion of pumpkinswith the painful sound of player, etc.

The performance of the player is also taken intoaccount by comparing of the current map blockwith that performed on the previous map block toevaluate the performance in real-time taking intoaccount of the mathematics ability and game skills.More importantly, based on long term gameplayperformance, the system praises the player forpersistence and an extra life is rewarded when theenergy is full or when player collected a certainnumbers of reward coins.

• Coin reward: One correct answer gains fivecoins. A wrong answer causes the reward coinand bonus items disappear (see Figure 4(d)).

• Lose of life: A wrong answer will make a lifelost.

• Bonus items: Three bonus items are placedfor the player to collect, if the question wasanswered correctly (see Figure 4(a) and (b) forpicking a mushroom bonus item). Otherwise,the bonus items will disappear (see Figure 4(c)choose the wrong answer and (d) the pumpkinexplodes along with the disappearance ofbonus items).

• Energy Boost: On chosen of the correct answerand pick up bonus items, the energy willincrease 20% and 50% respectively (see theenergy bar in Figure 4(a) and (b)).

• Life Bonus (Energy): When player continuouslypicks the correct answers and bonus items with

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Figure 4: Game screen captures: (a) The player is going topick a bonus item. (b) The player gets the bonus coin andgains 20% energy boots after collecting the bonus item.(c) The player is going to select a wrong answer. (d) Thepumpkins exploded and the player lost one life after thewrong answer, along with the disappearance of the bonusitem (mushroom).

low respond time, the energy will exceed 100%which can trigger the obtainment of an extralife.

• Life Bonus (Coin) : When player collects 100coins, an extra life will be rewarded to prize thepersistence.

• Level Switching Notice: Based on the levelswitching rule in Section 4, when the levelprogresses, the words “Congratulations, youstep into Level XX !” will be displayed on thescreen. Whereas when the level regresses, thewords “Cheer up, you step back into Level XX!” will be shown on the screen.

These elaborate-designed incentive methods allowthe user play the game in an encouragingenvironment, guiding the user to try their best toselect the correct answer to practice the math mentalcalculation skills. While in the meantime, in theinterval of two questions, collecting the randombonus items is an entertaining and encouragingprocess, which guarantee the user an interest in thisgame.

5. RESULTS AND DISCUSSION

We tested our adaptive level, the result can befound in Figure 5. The horizontal and vertical axesrepresent the round of questions (horizontal axes)and the performance score/level (left vertical axes)as well as the buffer value (right vertical axes). Theblue curve denotes the performance score of eachround that is in very high value in the beginning asthe game level (represented by grey curve) startswith 2. The high score had a direct impact on thebuffer value, which would cause the boost of thegame level (1 to 22 round). The performance scoredropped dramatically (after given several wronganswers) after the game level reaches 6, whichresult in the decrease of the level buffer. After thelevel buffer was lower than -1, the level degradationwas triggered and the game level was dynamicallyadjusted to 5. After that, the performance score wasraised again along with the rise of buffer and thengame level. From this point, the game found thesuitable level for this player and the game level cameinto a steady state of level 5 and level 6.

The experiment results prove the effectivenessof our adaptive level progression model and itsusefulness in the educational game. As we testedthis adaptive level educational game by studentswith high mental calculation abilities, the game levelwas dynamically adjusted to be in a very high level.Therefore, for the students with medium or poormental calculation abilities, the game level will beadaptively adjusted to medium or low levels. In thisversion of MathRun, many variables such as thecoefficients of coins, delay time, and obstacles aswell as the mapping relationship of performancescore and buffer change are selected based on our

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Figure 5: The experiment result shows the effectness of the adaptive level.

test or experience. Further work will involve the testof students with different mental calculation levelsto evaluate our system and find a better mappingrelationship between entertainment and learning.

6. CONCLUSION

A mental arithmetic game is presented based onthe popular “infinite runner” game genre. With thequantitative performance model, real-time dynamicanalysis of the player’s performance of mentalcalculations can be carried out in-game along withperformance evaluation of game skills for adaptivelevel progression. An automatic map generatoris developed to create an efficient learning andgaming environment. The experiment results showthat our proposed progression model can effectivelyswitch levels to a suitable difficulty of mathematicalquestions that match the mental calculation abilityof the player. Further study will involve a large scaleuser tests among a wider range of mental calculationabilities of students to evaluate the efficacy of theproposed game-based mental calculation practice.

REFERENCES

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Charles D, Kerr A, McNeill M, et al. (2005).Player-centred game design: Player modelling andadaptive digital games. Proceedings of the DigitalGames Research Conference. Vol. 285.

Dehaene, S. (1997). The number sense. New York:Oxford University Press.

Dehaene, S. (1992). Varieties of numerical abilities.Cognition, 44(1-2), pp.1-42.

The National Strategies (2010) Primary: Teachingchildren to calculate mentally, Department forEducation.

Dowker, A. (2005). Individual differences in arith-metic. Hove [U.K.]: Psychology Press.

Gov.uk. (2014). National curriculum in England:mathematics programmes of study - GOV.UK. [on-line] Available at: https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study [Accessed 31 May 2016].

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Martin, T., Smith, C. P., Andersen, E., Liu, Y. E.,& Popovic, Z. (2012). Refraction time: Makingsplit decisions in an online fraction game.In American Educational Research AssociationAnnual Meeting, AERA.

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Mayo, M. (2009). Video Games: A Route to Large-Scale STEM Education?. Science, 323(5910),pp.79-82.

Pauli, P., Lutzenberger, W., Rau, H., Birbaumer,N., Rickard, T., Yaroush, R. and Bourne, L.(1994). Brain potentials during mental arithmetic:effects of extensive practice and problem difficulty.Cognitive Brain Research, 2(1), pp.21-29.

Price, G., Mazzocco, M. and Ansari, D. (2013).Why Mental Arithmetic Counts: Brain Activationduring Single Digit Arithmetic Predicts High SchoolMath Scores. Journal of Neuroscience, 33(1),pp.156-163.

Tech Times. (2014). ’Temple Run’ now in thebillion download club. [online] Available at:http://www.techtimes.com/articles/7992/20140604/temple-run-now-in-the-billion-download-club.htm[Accessed 29 May 2016].

Zago, L., Pesenti, M., Mellet, E., Crivello, F.,Mazoyer, B. and Tzourio-Mazoyer, N. (2001).Neural Correlates of Simple and Complex MentalCalculation. NeuroImage, 13(2), pp.314-327.

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