Report - Jan Borg

Transcription

Report - Jan Borg

Master’s Thesis in Software Development and Technology
From Street to Screen: Developing Realistic Math Games
Author:
Jan Borg
221082
[email protected] / [email protected]
Supervisor:
Rilla Khaled
[email protected]
May 2011
Abstract
Realistic Mathematics Education was used as one of the primary inspiration sources in understanding
how mathematics education could increase students’ focus on meaning, motivation, purpose and
importance of doing math. Research concerning out-of-school mathematics was another major
contributor to this process. Both these theoretic foundations, along with various learning game
research, were relevant in supporting the rethinking process of math learning game designs. This gave
rise to a theoretical model consisting of 10 design principles that dictate how such realistic digital
math games could be designed. These design principles address topics like realism, identification,
fantasy, representation, interactivity, feedback and balance. Tests of a game developed in concordance
to these design principles on 37 fifth graders, suggested that the game was capable of improving
students’ motivation and understanding of purpose and importance of doing math. Long term
motivation and retention was not estimated. Low attaining students with game experience are
possibly the best match for a realistic math game of this type, but further research is recommended in
defining the exact, optimal target group and game design. Furthermore, the exact formulation of the
design principles and each principle’s weight and contribution to the whole could be explored deeper.
About the project
The problem formulation and method description has been changed from the original project
agreement. This was necessary since the project scope was narrowed as the project progressed.
This report is available digitally at CD and web: www.host-a.net/u/janborg as thesis.docx
Game available at CD and web: www.host-a.net/u/janborg as Ruby Hunt 1.1.exe
Game source code available at CD.
Game installation instructions
The game consists of a single 37mb .exe file. No installation is required. The game is not recommended
to be played on small screen computers, like laptops. Instead, use a 17” or larger screen on a PC.
It is recommended that readers play the entire game, and that it is done before reading the report. The
game should take 30-60 minutes to finish and can be chosen to be in Danish or English.
To read the source code: Download Game Maker from ‘www.yoyogames.com/gamemaker’. This will
enable reading of the entire source code and view the exact game design.
Acknowledgements
Thanks to my supervisor Rilla Khaled. Thanks to the people who tested and gave feedback on early
testing. Thanks to Julia Hunt for proof reading. Thanks to Keith Devlin for inspiring talks. A special
thanks to students and teachers of 5thA at Hedegårdenes skole and 5th grade of Atheneskolen.
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TABLE OF CONTENTS
Introduction ............................................................................................................................... 4
Problem field ..................................................................................................................................... 5
Method .............................................................................................................................................. 6
Delimitation ....................................................................................................................................... 7
Theory ....................................................................................................................................... 9
Game use in the classroom ................................................................................................................. 9
Learning games ................................................................................................................................ 10
Realistic Mathematics Education ...................................................................................................... 18
Context-relations ............................................................................................................................. 20
Out-of-school mathematics .............................................................................................................. 21
Attaching meaning to results ............................................................................................................ 23
Plausibility ....................................................................................................................................... 26
Model discussion ...................................................................................................................... 29
Endogenous and Exogenous games ................................................................................................... 29
Purposeful activities ......................................................................................................................... 31
Knowledge transfer and fantasy ....................................................................................................... 33
Identity and character ...................................................................................................................... 36
Multiplayer ...................................................................................................................................... 39
Narrative.......................................................................................................................................... 41
Example: MathRider ......................................................................................................................... 44
Game genre ..................................................................................................................................... 46
Math problems as quests.................................................................................................................. 47
Quest design details ......................................................................................................................... 48
Stealth learning ................................................................................................................................ 52
Representation ................................................................................................................................ 53
Interactivity ..................................................................................................................................... 58
Feedback and instructional support .................................................................................................. 61
Difficulty balancing ........................................................................................................................... 63
Sub-conclusion ................................................................................................................................. 66
2
Game design ............................................................................................................................ 71
Technology, resources and process ................................................................................................... 71
Math within the game ...................................................................................................................... 72
Design Principle integration .............................................................................................................. 72
Quest details .................................................................................................................................... 74
Implementation details .................................................................................................................... 76
Game test ................................................................................................................................ 77
Introduction ..................................................................................................................................... 77
Subjects ........................................................................................................................................... 79
Procedure ........................................................................................................................................ 79
Results and discussion ...................................................................................................................... 80
Conclusion................................................................................................................................ 88
References ............................................................................................................................... 90
Appendix A: Credits of resources used in the game ................................................................... 93
Appendix B: Interview guide and transcription ......................................................................... 95
Transcription .................................................................................................................................... 96
3
INTRODUCTION
“When ten or more years instruction fails to leave people having even the faintest idea what something is,
why it is done, or what it is used for, then something is seriously wrong.”1
Keith Devlin, Mathematics Professor.
In 2010 a Danish high school tested the math performance of newly enrolled students. Test results
showed that 80% only managed to do mathematics on a 3rd to 4th grade level, even though they had
received the entire elementary school education math program.2
The interactive math training website Intmath.com has made a series of surveys of their visitors. 74%
of visitors to the website identify themselves as math students. The survey found that 70 % of visitors
report that they rarely get to find out why they study mathematics, and 17% report that they never feel
that they obtain a good understanding of how math is applied to real world problems. Furthermore,
22% of visitors hate math and 11% have no idea if math is useful for their future job. Other studies
have found that students in general have difficulties using real world knowledge when engaging school
context math problems.3
These examples highlight the gap between out-of-school mathematics and school mathematics. Not
only are the students underperforming in math, they are also confused about the very purpose of
mathematics education.
However, studies have found that mathematics practice in out-of-school contexts works very
differently than school math. In many cases children and adults with little or no education are capable
of consistently performing well in math through their daily activities.4 These differences might be
explained by the fact that
a) “Problems in everyday situations are embedded in real contexts that are meaningful to the problem
solver and this motivates and sustains problem-solving activity.
b) The mathematics used outside school is a tool in the service of some broader goal, and not an aim in
itself as it is in school.”5
The gap between school mathematics and out-of-school mathematics could be addressed in various
ways. This thesis project investigates how digital learning games can be designed as a gap minimizer
between the advantages of out-of-school mathematics with the demands and limitations of school
mathematics. Digital learning games have for more than 30 years been hailed as a powerful learning
tool capable of making learning more efficient, more interesting, and more enjoyable.6 Furthermore,
1
Devlin 2010
2
Berlingske 2011.
3Nunes
et al 1993. Verschaffel and Corte 1997. Gravemeijer 1994, Palm
4
Nunes et al 1993
5
Masingila 1994, p. 3
6
Malone 1980
4
scores of more recent studies have reported significant positive effects of instructional computer
based games 7. However, it will not be trivial to create a learning game based on informal out-of-school
mathematics. The Dutch instructional theory of Realistic Mathematics Education (RME) is successful in
grasping some of the advantages of out-of-school mathematics and is systematic in its approach in
making mathematics more real in the students’ minds. Consequently RME can act as a bridge between
out-of-school math and instructional game design. This thesis project will investigate if and how digital
math games can be designed to support a mathematics education inspired by RME and informal
mathematics performance. The goal is to develop a model of design principles and use this to produce
and test a math game sporting effective game based learning while emphasizing purpose and meaning
of mathematics education.
Problem field
In the 60s and 70s audio and video were hyped as technologies that would revolutionize learning8.
However, studies soon found that no significant difference was observed, meaning that overall, media
alone made no difference to learning. The issue was found to lie in the difference between use and
integration of media, as Eck declares: “Using media requires only that the media be present during
instruction. Integrating media, on the other hand, requires a careful analysis of the strengths and
weaknesses of the media, as well as its alignment with instructional strategies, methods, and learning
outcomes.”9 Integrating the video game medium with learning has been the objective of decades of
research. However the approach has often been to adapt the video game creation and usage to
dominating curricular or didactical positions10. Instead of looking at mainstream curricula and game
design ideas and turning them into games, the attempted point of departure for this project will be
where some original ideas of mathematics education dwell. I have found some original math ideas in
what is known as RME and support for expansion of these original ideas in what is known as street
mathematics or out-of-school mathematics. I hypothesize that designing learning games with
inspiration from RME and out-of-school mathematics will be one way to better exploit the strengths of
the video game medium. However, Eck also argues that “instruction is modified to take advantage of the
strengths of the media”11. Consequently the instructional findings of RME and out-of-school math
should be modified by findings from learning game researchers to take advantage of the video game
medium in figuring out how the desired components of the instruction can be transformed into game
form. Such a transformation will possibly be able to tackle the problems outlined in the introduction.
The final result should be a model consisting of design principles combining the best of all above, a
model that can and will be implemented into an actual game and subsequently tested.
7
Ke 2009, p. 20. Egenfeldt-Nielsen 2005
8
Eck
9
Eck, p. 30
10
E.g. Egenfeldt-Nielsen
11
Eck, p. 30
5
Problem Definition:
How could a digital learning game based on RME and out-of-school math be designed to
effectively and meaningfully teach math to primary education students?
Sub-questions:
o
How can a digital game based on such design principles improve students’ motivation and
understanding of purpose and importance of doing math?
o
What type of students and what type of games will be the best match for this purpose?
Method
To outline the exact approach, a concrete explanation of the method intended is presented below.
First a theoretical model of what a modern mathematics learning game should consist of is built up. To
construct this model all kinds of relevant material will be utilized, possibly including (but not limited
to): Research concerning earlier/current learning games, test and analysis of earlier/current learning
games, material regarding Realistic Mathematics Education, informal mathematics, didactics,
psychology and general research concerning video games and simulations. This model will enable the
definition of a set of design philosophies which will be used as heuristics to realize a game design
document. Subsequently the game design will be implemented in into an actual game, using an
appropriate development environment for the development process. This step will apply relevant
academic software- and game design materials. Small scale tests will be performed to correct minor
flaws in coding and design, using state of the art game testing techniques. Lastly the corrected, final
game version will be tested on a larger scale. This test will attempt to asses if the game influenced
factors such as student motivation towards the game, students’ own estimations of the utility and
meaning of math in general, the success ratio of the students’ performance in solving the math and
their teacher’s attitude and estimations. The test results will enable returning to the beginning and
asses the validity of decisions concerning the theoretical model, the set of design philosophies and the
actual game design.
It is on purpose that the design will not be driven by looking at existing games and game theory and
from there try to develop the model. The wish is to let RME and street math control and guide this
process instead, and only use other games and game theory as a form of secondary navigation system.
In other words, this process will let game theory and game examples align to the model that RME and
street math suggests, instead of letting game theory and game examples suggest the model followed by
an attachment of RME and street math to the model. Nevertheless, game theory will be presented early
in order to guide the direction of RME and street math exploration. This will aid in steering clear of
irrelevant sub-topics. Most parts of this project; analysis, discussion/model creation, game design,
game development and game test will to some extent occur in parallel in allowing these parts to
influence each other. The parallelism will however only be partial since a chronological approach will
be necessary to base the game on the design which again is based on theory.
6
Learning model:
The point of entry towards learning and learning in video games will primarily be constructivist
oriented. This is because RME is founded on a constructivist approach and because the domain of
video game researchers is dominated by constructivist learning subscribers. Marc Prensky argues that
according to constructivist learning; ”we learn best when we actively construct ideas and relationships
in our own minds based on experiments we do, rather than being told; and second, that we learn with
particular effectiveness when we are engaged in constructing personally meaningful physical artifacts.”12
The interpretation of constructivist learning takes different forms, e.g. in constructionist learning
where the creation of artifacts is central in the meaning and learning. Situated learning and experiental
learning, close relatives of constructivism, stresses that social and physical environmental factors are
detrimental to learning. This learning approach in video games is dominant in what video game
researcher Simon Egenfeldt-Nielsen calls third generation video games.13 RMEs focus on the
individuals’ meaning- and knowledge creation taking place in social interactions and in nontransmission forms is also fundamentally constructivist and therefore goes hand in hand with
Prensky’s and others’14 constructivist learning approach. Consequently, constructivist learning will be
dominating throughout the theoretical model but other learning paradigms might shine through now
and then, in example the behaviorist approach of Operant Conditioning is relevant in understanding
rewards and reinforcement. Note that the use of constructivism is only as a learning/instruction
theory, not as an epistemology.
Delimitation
Only parts of theories that are found relevant will be included. In example, not all parts of RME
will be included in the theoretic model and not all ideas of video game researchers will be
adopted. However, care will be taken in picking out parts such that they are not too
disconnected from their origin.
Focus on research investigation and game development will be on math students similar to
students attending the middle of the Danish compulsory school system. Adult teaching will
generally be ignored.
The topic regarding assessment for grading or long term retention of learning will be
overlooked, since it would over-expand the scope of this project to analyze or implement ways
of assessing the long term learning outcomes of the students.
12
Prensky 2005, p. 19
13
Egenfeldt-Nielsen 2005
14
Squire, Bogost 2007
7
I assume games are capable of teaching and that players can learn from playing games.
Potential negative or unintentional effects, e.g. aggression from game violence, of games and
learning games will be ignored.
The model will not be designed with non-digital games like board games in mind. Theory
regarding non-digital games might however be included if relevant.
During the development of the model a thorough treatment of non-learning game design
criteria, where a general consensus in the scientific and game development communities exists
concerning their validity, will be refrained from. There is no reason to challenge or
reinvestigate design principles that are found in dominating game design books15.
Consequently it will be assumed that such principles are familiar to game designers pursuing
creation of good games, and the absence of such principles within the model are therefore
intentional. An example of this is flow; that a game’s challenge level must balance itself
between frustration and boredom in order to motivate its players.16 This will give more time
and space to focus on relevant but possibly ambiguous or disputed learning factors.
15
Fullerton 2008, Gee 2003, Malone 1980
16
Fullerton 2008, p. 87
8
THEORY
In order to guide the project, and identify possibilities and limitations, the state of relevant topics will
be outlined. First an introduction to the state of digital learning games (henceforth called learning
games), with a focus on primary school learning games, is presented. Afterwards, an introduction to
general game theory and learning in games is presented. Finally an introduction to RME, out-of-school
mathematics and related theory and studies is presented. Theory and studies below are only
presented on an introductory level and is expanded in the model/discussion chapter when necessary.
Game use in the classroom
In a meta-analysis from 2009, effects of instructional gaming were charted:
“34 out of 65 game effectiveness studies reported significant positive effects of computer based game, 17
reported mixed results (instructional games facilitated certain learning outcomes but not the others), 12
reported no difference between computer games and conventional instruction, and only one study
reported conventional instruction as more effective than computer games.”17
Assuming that this meta-analysis reflects the reality of the effects of learning games, then it is relevant
to consider how games are used in the classroom of the primary education.
Recently made studies and interviews in the UK shows that serious games are not prevalent18. This
might sound as conflicting with the study described before, but this means that games are not used
prevalently in schools, but the games are nevertheless used in classrooms as part of research, and with
good effect. So learning game use in classrooms is still on a somewhat experimental level. This is
despite the fact that teachers perceive advantages of using video games in the class room: It gives a
higher student motivation and provides an alternative presentation.19
A Danish study from 2005 revealed the state of computer game usage in the Danish primary
education20. There seems to be a lack of experience among teachers in using computer games in an
educational setting. The experiences of using learning games in classrooms are often related to quite
simple use of computer games or few limited experimental attempts at using computer games for a
specific topic. According to the survey there is a lack of successful teaching experiences, although all
teachers remain fairly open and positive towards computer games as an educational tool. In the study
teachers were also asked about what they perceived as disadvantages of using computer games in
education. According to the teachers the primary disadvantage is the quality of available titles on the
market. This must be seen in the light that on that the time (and to some degree today as well21) it was
edutainment products that dominated among school games. More about edutainment games will come
17
Ke 2009, p. 20
18
Ulicsak 2010
19
Egenfeldt-Nielsen 2005, p. 74
20
Egenfeldt-Nielsen 2005, p. 65-74
21
Ulicsak 2010
9
below. The other significant barriers were: Access to computers, technical barriers, knowledge of
games, learning to play the game and games that cannot cover curriculum. Teachers can also in some
cases be afraid of the loss of control or the emergence of chaos that might occur in learning
scenarios22. The teacher might feel as an outsider in situations where he might not be competent in the
video game medium, and furthermore he stands outside of the game as well, he will not participate
directly in the video game learning process as is the case in the classroom education.
The above issues should not be interpreted as the teachers are without importance, since teacher’s
facilitation plays an important role in an effective use of learning games in the classroom23.
Furthermore, the above barriers should not be interpreted as an attempt at presenting the teachers as
unqualified or incapable of effectively utilizing learning games in the classroom. The intention is
merely to show that there are some serious challenges that might exhaust even the most skilled and
devoted teacher. Danish video game researcher Thorkild Hanghøj sums it up by listing all the tasks
ahead of a teacher wishing to effectively utilize learning games in the classroom. Teachers must:
“Evaluate and dissect the games, take time to play and understand the games, map the games with
curricular aims, find the time for the game sessions, ensure that the technology is working, facilitate the
games, promote the students’ reflective thinking, and assess the students’ learning outcomes.”24 This
makes the teachers the crucial gate keepers, or ‘change agents’ for using learning games in classrooms.
Nevertheless, learning games are shown a lot of interest in these days. One example of this is the
ambitious STEM-challenge project25. This is a twofold project to use video games to promote Science,
Technology, Engineering and Math (STEM). One part of it is to let students create and submit games
for competitions. The process the students undergo while creating these games is the goal – to e.g.
learn and use math by developing games. The other part consists of a competition open for anyone
within United States to create and submit games that teach or practice science, technology,
engineering or math.
Learning games
This chapter introduces learning game theory and research. The chapter will also help in focusing
definitions and scope regarding such theory.
First, the term learning game will be used interchangeably with other terms, like Game Based Learning
(GBL), learning video games, instructional video games, instructional computer games and educational
games. No matter what term is used, the intention is just to describe a game whose purpose it is to
digitally teach or let a player learn certain curricular contents. The intention is to not subscribe to any
specific definitions or interpretations made by certain game research branches.
Learning games comes in many different forms. Here are some categories of learning games:
22
Meyer, p. 56
23
Ke 2009, p. 21
24
Hanghøj
25
http://www.stemchallenge.org/
10
Commercial games: Also known as COTS, Commercial-Of-The-Shelf games. These are games that are
produced as entertainment products with commercial interests in mind. These games are sometimes
used for educational purposes despite that they were not intended for education. Examples of this
within this category are games like Civilization and Europa Universalis which have been successfully
employed to teach history26
Edutainment: Games with some learning agenda based on the assumption that it must be fun to
learn27.
These are typically somewhat simple in design and often based on a behaviorist learning approach28.
These games usually portray an extrinsic motivation, where “you get a reward for engaging in an
activity, and are not motivated by the activity itself”.29
Simulations: A simulation video game closely simulates aspects of a real or fictional reality.30 These
games usually focus on motivation as intrinsic where mastery of the activity itself is the guiding light,
and no points, high scores or similar extrinsic rewards are central. An example of this is Microsoft
Flight Simulator
Serious games: These have also been called 3rd generation educational games. These apply a
constructivist approach to learning.
One could try placing the above categories on a continuum between a learning focus and a
fun/entertainment focus. However, there is more to understanding educational games than just
balancing learning and entertainment, since there are other factors involved than just those two. This
field is multidimensional, and also includes dimensions of e.g. fantasy, motivation and identification.
One challenge and purpose of this project is to identify these extra dimensions and grasp how the
balancing of these could create an optimal learning experience.
One early approach to identifying effective games and effective learning games can be found in
Thomas W. Malone’s 1980 paper.31 He describes a general taxonomy of intrinsic motivation consisting
of three categories; challenge, curiosity, and fantasy.
Challenge consists of providing a goal that is uncertain. Not all goals are equally well suited for all
tasks i.e. simple games should provide an obvious and compelling goal while complex environments
without built-in goals should be structured so that users can generate their own goal. Furthermore,
challenge is entailed by uncertain outcome which in turn can be supported by variable difficulty level,
score-keeping and randomness.
Curiosity is the motivation to learn, independent of any goal-seeking or fantasy-fulfillment and
consists of sensory and cognitive curiosity. Computer games can appeal to sensory curiosity by
appealing to audio and visual effects, and in turn apply these as decoration in the game, to enhance
fantasy, as a reward or as a representation system.
Fantasy in games can be divided into intrinsic fantasies and extrinsic fantasies. Malone argues that
26
Egenfeldt-Nielsen 2005
27
Meyer, p. 54
28
29
30
http://en.wikipedia.org/wiki/Simulation_video_game
31
Malone, p. 162+
11
“one relatively easy way to try to increase the fun of learning is to take an existing curriculum and
overlay it with a game in which the player progresses toward some fantasy goal, or avoids some fantasy
catastrophe, depending only on whether the player’s answers are right or wrong. To a great degree, the
fantasies used in this kind of game are interchangeable across subject matters.”32
An example of this is that the game Hangman33 can be completely interchanged with any other
curriculum. For example, incorrect algebra responses could trigger the hanging process, instead of
misspelling. Likewise, the curriculum of spelling that is usually attached to the Hangman game could
just as well be attached to another intrinsic game fantasy. Malone states that in extrinsic games, the
fantasy depends on the use of the skill, but not vice versa. However, in intrinsic fantasy games, not only
does the fantasy depend on the skill, but the skill also depends on the fantasy. This means that
problems are presented as elements of the fantasy world, and consequently, the curriculum and the
fantasy is inseparable and therefore not interchangeable like in extrinsic fantasy games. Furthermore,
“In intrinsic fantasies, the events in the fantasy world usually depend not just on whether the skill is used
correctly, but on how its use is different from the correct use”34.
Kurt Squire subscribes to these ideas and expands them to embrace more than only fantasy and
thereby categorizes games as Exogenous and Endogenous. The idea is explained in the table below:
32
Malone p. 164
A spelling game where each incorrect answer increasingly draws a hanged man. The goal is to guess the word
before he is hung.
33
34
Malone p. 164
12
Figure taken from Squire 2006, p. 24
13
An example of a digital mathematics learning game mounting an extrinsic fantasy is Math Olympics of
mathplayground.com35. The screenshot from the game below illustrates the game play.
As seen in the screenshot, a multiple choice question is presented. When the player chooses an answer,
a character appears on the path and attempts to jump the obstacle. If one answers correctly, the
runner succeeds. If one fails (as in the screenshot), the runner crashes into the obstacle and knocks it
down and the player is told to try again. This is an intrinsic fantasy game since the ‘game’ of getting the
runner to jump the bar is interchangeable with any other kind of curriculum. For example, jumping
obstacles and increasing the score might as well depend on success in spelling. Likewise, the multiple
choice math quiz of the game could just as well determine if space invaders are destroyed or not. Math
Olympics also qualifies as an exogenous game since, for example, the knowledge model of the game is
fact based, not tool based and instruction is transmission based since the drill and practice approach is
supposed to train a set of desired responses. This behaviorist agenda qualifies it as a typical
edutainment game.
Squire argues that the very medium of games forces designers and educators to go beyond traditional
notions of education as ‘exposure to content’ and rethink it as enrichment of experience36. He argues
that traditional educational games use context as a motivational wrapper for the game experience
35
http://www.mathplayground.com/olympic_math1.html acquired and tested at 11.10.2010
36
Squire, p. 25
14
whereas contemporary games literally put the players inside game systems. In relation to the table
above, in endogenous games the context is the gameplay whereas in exogenous games, the context is
irrelevant to the game play. This shows the similarities between Malone and Squires ideas.
An example of an intrinsic fantasy game is the game that Malone calls Darts: “A game designed to teach
elementary students about fractions. Three balloons appear at random places on a number line and
students try to guess where the balloons are. They guess by typing in mixed numbers, and each time they
guess an arrow shoots across the screen to the position specified. If the guess was right, the arrow pops
the balloon. If wrong, the arrow remains on the screen and the player gets to keep shooting until all the
balloons are popped.” 37
He states that it is an intrinsic fantasy since the skill of estimating distances is applied to the fantasy
world of balloons and the use of this skill then affects the fantasy world by shooting arrows and
popping balloons.
Figure from Malone 1980, p. 164. “Logical dependencies in extrinsic and intrinsic fantasies”
Consequently Malone’s definition of the difference between intrinsic and extrinsic fantasies is twofold:
Interchangeability of context (fantasy) and content combined with dependencies of skill vs. fantasy.
However, an example of a learning game will be used to show that Malone’s definition is somewhat
vague. The winner of the STEM challenge’s Collegiate and Impact prizes and $50.000 was the game
Numbaland. A sub-game of this is Battleship Numberline which can be compared to Darts. Below is a
screenshot of typical gameplay:
37
Malone, p. 169
15
The purpose of the game is to teach number awareness to students. In this specific scenario, the player
enters a number between 0 and 10 in estimating where the ship is on the interval. In the screenshot,
‘8’ has been entered. Note the tiny missile above the ship that is descending towards the ship. This was
a successful estimation and the immediate moment following the one in the screenshot depicts the
missile impacting with the ship, causing the ship to explode, leading to the next estimation assignment.
This is a game employing an intrinsic fantasy to motivate its players. It would not be possible to
completely detach the fantasy from the game play. For example, it would not make sense that
correct/incorrect spelling would affect if the ship was hit or not. It would directly conflict with the
fantasy since the game content and context is intrinsically connected. However, it is only an intrinsic
fantasy to a certain degree, since it is imaginable that instead of estimating a number between 1 and
10, it could involve calculating the amount of fuel needed for the anti-ship missile to hit the ship.
Actually the designers of Numbaland: Battleship Numberline offer other game modes incorporating
the same game scenario of attempting to hit the battleship, but the curriculum can be freely chosen
from estimating whole numbers between 1-10, whole numbers between 1-100, decimals, fractions and
measurement. Consequently the terms intrinsic and extrinsic fantasies are not distinct, rather a game
can exist on a continuum between these two terms. This makes the terms somewhat vague and
ambiguous, however they are still useful in distinguishing between various approaches to game
design. However, questions still arise concerning how a game can become even more intrinsically
connected to its fantasy. And if this connection is made so strong that the intrinsic fantasy cannot be
replaced with any other task, would that be an advantage?
Video game researcher Ian Bogost’s term procedural rhetoric can help to expand these terms and aid
in understanding how learning games work. Bogost argues that video games cannot be created or
analyzed efficiently with tools used for other media, like television or web pages. He defines
16
procedural rhetoric as a concept for exactly that purpose:
“Procedural rhetoric is the practice of using processes persuasively, just as verbal rhetoric is the practice
of using oratory persuasively and visual rhetoric is the practice of using images persuasively. Procedural
rhetoric is a general name for the practice of authoring arguments through processes.”38
Consequently the word rhetoric is not necessarily literal in meaning persuasion, the term procedural
rhetorics will mostly be used to describe the authoring of arguments, e.g. teaching or practicing
mathematics, through processes. Procedural rhetoric is something more than digital rhetoric which
typically abstracts the computer as a consideration, focusing on the text and image content a machine
might host, like in e-mail, websites, message boards, blogs and wikis. That way, Bogost argues that
verbal, written, and visual rhetorics inadequately account for the unique properties of procedural
expression. Furthermore, Bogost claims that video games are capable of much more than just
supporting existing social and cultural institutional goals, and that this is not just done through the
content of video games39. This can be done by harnessing the power of procedural rhetoric, in both
creating and analyzing games. But since video games are capable of more than just delivering content,
the complexity and barriers revolving around creating the procedural rhetorics are immense.
Egenfeldt-Nielsen states that: “To see the computer games as ‘merely’ a neutral medium we can use for
delivering content, skills and attitudes is lulling ourselves into dreamland”40. So the task of working with
video game knowledge and persuasion is not to be taken lightly. Video game researcher Suzanne de
Castell of Simon Fraser University, Canada, argued in her keynote speech at the European Conference
of Game Based Learning (ECGBL) in 2010 that there are some basic epistemological issues connected
to this challenge, a perception not unlike the inspiration to the creation of the concept of procedural
rhetorics:
“One of the biggest problems in my opinion is that people want to create and introduce and evaluate
games on the premise that knowledge itself and school knowledge, is the same thing. That, you know, for
example, you have the games where, you know, do such and such, solve a math problem and then you get
a piece of pie, you know. We just changed the medium. So when you change the medium, you change
knowledge representation, so knowledge doesn’t look the same anymore. It won’t come in sentences and
propositions and right answers and multiple choice boxes. It won’t come in mathematical formulas. It
won’t come in lists of dates and facts.”41
Castell seems to agree with the other researchers; that if knowledge embedded in video games is
presented in a way that is intrinsic to the video game medium, then the gains are substantial. The next
chapter will present theory and studies concerning just what Castell describe – alternative knowledge
representation and acquisition.
38
Bogost 2007, p. 28
39
Bogost 2007, p. ix
40
SEN, 2007, Continuum, Educational potential of computer games, p. 157
41
Castell 2010
17
Realistic Mathematics Education
In the Netherlands, mathematics in primary school in is mostly based upon Realistic Mathematics
Education42. RME is largely the brainchild of the Dutch mathematician Hans Freudenthal. Freudenthal
argued that mathematics must be connected to reality, stay close to children’s experience and be
relevant to society in order to be of human value. Furthermore, he viewed math as a human activity
rather than a subject to be transmitted. Basic RME views can be summarized as below:
“Mathematics is viewed as an activity, a way of working. Learning mathematics means doing
mathematics, of which solving everyday life problems is an essential part. A variety of contextual
problems is integrated in the curriculum right from the start.”43
The name realistic math education has been misunderstood at times44. The reason why RME is called
‘realistic’ is not just because of its connection with the real world, but is related to the emphasis that
RME puts on offering the students problem situations which they can imagine (as in realizing
something).
RME is basically a theoretic model that is supposed to be used by teachers to construct better math
curricula. This work can seem outdated since several of the ideas and suggestions explained by RME
are already part of mainstream math education. An example of this is the statement in the official
Danish primary school math curriculum which describes how the real world often is the starting point
in identifying problems that are sought to be solved by the help of mathematics45.
Another example of the influence of RME is the University of Wisconsin project of Mathematics in
Context (MiC), which is a mathematics curriculum, whose development was based on a major
inspiration and cooperation from RME and its protagonists46. They argue that in traditional
mathematics curricula, the sequence of teaching often proceeds from a generalization, to specific
examples, and to applications in context. RME (and MiC) reverses this sequence; mathematics
originates from real problems.
In 2007, fourth-grade students of the Netherlands had the 9th best average mathematics scores47.
Other studies confirm the effectiveness of RME. A study48 shows that 7th grade students get a positive
attitude towards mathematics after receiving RME, and also become aware of usefulness of
mathematics in daily life. Another study49 concludes that students who used realistic math textbooks
scored higher than students who used traditional textbooks.
42
Heuvel-Panhuizen
43
Gravemeijer 1994, p. 91
44
Heuvel-Panhuizen, p.4
45
UVM 2009, p. 35
46
EDC 2001
47
from http://nces.ed.gov/timss/table07_1.asp at 30.3.2011
48
49
Üzel and Uyangör
Romberg, p. 9
18
The basic concepts of RME are elaborated on below and subsequently expanded later in the report. I
want to use this work, not in order to propose or discuss math curricula in general, or to identify the
best way of teaching math in classrooms. On the contrary, this discussion will be used to find academic
sustenance for the previously explained assumptions about how math could be taught in video games.
Selected examples of RME theory that are relevant for this agenda will be presented, while mostly
ignoring the deeper levels of instructional design theory that RME is based on. RME can be considered
as consisting of an array of principles. Several of these principles will not be dealt with in this report,
however extra attention will be given to ‘the reality principle’, since this is what I find most relevant
for video game design.
The Reality Principle
It must be sound to assume that all mathematics curricula aim to enable students to, at the end of the
learning process, apply mathematics to solve problems. In RME, the reality principle is not only
recognizable at the end of the learning process when it is applied; reality is also conceived as a source
for learning mathematics.
“Just as mathematics arose from the mathematization of reality, so must learning mathematics also
originate in mathematizing reality. Even in the early years of RME it was emphasized that if children
learn mathematics in an isolated fashion, divorced from their experiences, it will quickly be forgotten and
the children will not be able to apply it. Rather than beginning with certain abstractions or definitions to
be applied later, one must start with rich contexts demanding mathematical organization or, in other
words, contexts that can be mathematized. Thus, while working on context problems, the students can
develop mathematical tools and understanding.”50
However, these context problems must be carefully chosen, not just any realistic problem will work.
One example of a context problem is the ‘bus chain’ which concerns counting passengers entering and
leaving buses. This is used as an example of how a curriculum could contain an introduction for
students to the operation signs (”+” and ”-”).
Figure of buses51
The task of the student is to check ongoing changes in the amount of passengers. The point of this
context-bound mathematical language is to introduce representation for the description of
quantitative changes, whereupon representations that are gradually less context-bound can be
introduced until finally ending up at the standard notation (e.g. 2+4 = 6). I want to emphasize that
even if the authors use this to explain how various calculation methods can be taught, I intentionally
shift focus on the part of the example that focuses on realism. As one of the leading RME protagonists,
50
Heuvel-Panhuizen, p. 5 (Original source references have been edited out)
51
Scanned from Gravemeijer 1994 p. 29
19
Koeno Gravemeijer points out “Passenger entering and leaving a bus provides a situation in which
addition and subtraction emerge as natural activities” 52. Letting math emerge as a natural activity
might help overcome the obstacle that, as described earlier, many students don not know why they
learn math or what it can be used for.
Context-relations
Many math books have lots of examples that can be seen as realistic, i.e. in the form of word problems.
For example, in teaching to count, one could present a picture of 5 candles and then ask the student
how many there are. It would fit within the realism principle, since it makes sense that a real world
scenario would be to count 5 candles, as opposed to counting 5 dots. But the realism can be improved
by considering the context too. Look at the picture below.
Figure: Context-related counting53
This picture would be presented, instead of a picture of only 5 candles, with the question: How old is
the girl? This embeds the realism into a context that makes sense for a child. Heuvel-Panhuizen
explains the example: “In the context-related questions, the context gives meaning to the concept of
number. This context-related counting precedes the level of the object-related counting in which the
children can handle the direct ‘how many’ question in relation to a collection of concrete objects without
any reference to a meaningful context. Later on, the presence of the concrete objects is also no longer
needed to answer ‘how many’ questions.”54
Two things are explained here. First, the context-related question gives meaning to the example. The
context in this case can also be seen as a story – to put the 5 candles into a story involving a girls’
birthday, and asking to her age instead of the amount of candles, embeds the realism (candles) into a
story (context).
Secondly, this context-relation should, according to RME, only be used as an introductory
representation model which can be gradually replaced by more formal representations. The oldest
students should have no need for context-relation and can work with completely abstract assignments.
I wish to discuss this assertion of RME by turning towards the question of how important the context
actually is, and if it is only important for small school children in the following chapter.
52
Gravemeijer 1994 p. 29
53
Copied from Heuvel-Panhuizen, p. 16
54
Heuvel-Panhuizen, p. 16
20
Out-of-school mathematics
The term out-of-school mathematics has several names. Informal mathematics, street mathematics
and real life mathematics are just some of the names used.
In the book ‘Street Mathematics and School Mathematics’ (from now on referred to as street math), the
authors (from now on referred to as Nunes) discuss various studies that they designed to investigate
the importance of context when learning and using mathematics55. In one study, children who were
working at a marketplace in Brazil were approached by researchers acting as normal customers
interested in their marketplace goods. Around a week later the same children were engaged by the
researchers in the homes of the children or at the marketplace to participate in some formal tests. The
results of the children’s math performance were as following:
In the marketplace situation the children engaged various math questions, i.e. the ‘customer’ asks for
13 coconuts that each cost 35 cruzeiros and the child vendor then solved what the total price was. In a
formal test the children engaged problems involving the exact same math operations as on the street
but made into word problems. One example of such a word problem was: “A fisherman caught 50 fish.
The second one caught five times the amount of fish the first fisherman had caught. How many fish did
the lucky fisherman catch?” In one final test, the students engaged in bare arithmetic operations like
“420 + 80 = ?”
The astounding results were that the overall percentage of correct responses was only 36,8% in the
formal bare number test, 73,7% in the formal word problem test and 98,2% in the marketplace
situation!56 The study involved only five subjects, so it could be questioned how generalized these
findings are. However, subsequent studies by the same researchers involving fishermen, carpenters &
their apprentices and construction site foremen backed these findings and extended them. The street
math authors asks when discussing the findings: “How is it possible that children capable of solving a
computational problem in the natural situation will fail to solve the same problem when it is taken out of
its context?”57
Studies by other researchers58 have encountered similar findings. Furthermore, Masingila et al’s
research concerning mathematic practice in and out of school supports RME and street math findings.
Masingila’s findings were based on several studies, wherein one of them compares secondary school
students to professional carpet layers59. This study is here presented to further illustrate the
differences between school math and out-of-school math. Both groups were given an assignment
wherein they had to convert square feet to square yards. The carpet layers were able to correctly solve
this assignment by dividing square feet by nine, while however being somewhat poor at explaining
why this was true. None of the six pairs of students were able to understand the problem completely.
To compare how professionals that practices math like this every day with students might seem unfair.
55
Nunes et al 1993
56
Nunes et al 1993, p. 22
57
Nunes et al 1993, p. 23
58
Esmonde et al
59
Masingila 1994, p. 5-6
21
However, students had previously had exercises similar to this problem in their textbook, so the
differences might not be ascribed to this.
Masingila et al’s research lead to several conclusions on how to transform education to better connect
in-school with out-of-school experiences. These conclusions can be divided into four parts:
a) A problem solving approach to teaching is preferable to the traditional instructional approach
where the teacher is presenting information and then assigning exercises in which students
practice and applies this information. Using an instructional approach of teaching via problem
solving means that mathematical understandings are constructed by students as they seek to
accomplish emerging goals through problematic situations60. These problems must be
embedded in situations that are real and meaningful to students. It also means that math is to
be seen as a tool to be used and that processes and procedures are learned as they are needed
in the midst of accomplishing emerging goals.
b) In working individually and collectively to accomplish emerging goals, mathematical
knowledge is developed within a meaningful context and cognitive development occurs as
students work together with peers and teacher to negotiate shared meanings61.
c) School activities should make use of cultural artifacts and conventions that students can use to
interpret problems and make sense of them.
d) Teachers can build on students’ prior understandings and mathematical knowledge acquired
in other contexts.
60
61
22
Attaching meaning to results
The street math authors conducted another study, in understanding proportions. Results showed that
school students had great difficulty in attaching meaning to the result of their calculations62. A
difficulty not displayed by foremen of construction sites. For example, a student used a proportional
model to calculate the real life size of a wall and ended up interpreting the result as 3 meters and 750
centimeters, which clearly makes no sense. The students made other mistakes in interpreting results.
A wall represented by a length of 3.3 cm on 1:100 scale was read to be 33 cm long. In Nunes’ words:
“These are clearly unreasonable sizes for the width or length of a room in an apartment.”63 The foremen
however did this without any problems. They were able to scale the proportions correctly all the time.
Comparing professional, experienced foremen in task they are used to, with performance of students
entering a new problem field might not be fair. Firstly, the comparison might be unfair in relation to
the groups’ experience within the domain, but in relation to qualifications; the students were 7th
graders from middle-upper class schools while only 1 in 17 foremen had had formal training – the rest
of the foremen were trained through apprenticeship. Secondly, the point here is not to compare who is
the better mathematician, the point is to underline that students were expected to be better at
attaching meaning to results.
School curriculum designers are usually aware that realistic math problems are better than simple
number problems in relation to student motivation and results interpretation. The usual solution to
this is to embed math problems into word problems. One advantage of this is that the students need
engage in problem solving activities and not just drill-and-practice. RME puts big emphasis on realistic
word problems. However, several studies64 have found that the design of word problems is not trivial:
“… students have a tendency not to make proper use of their real-world knowledge and to suspend the
requirement that their solutions must make sense in relation to the ‘real’ situations”65. For example, in a
study composed by Torulf Palm, two groups of a total of 161 students were tested in their
engagements with word problems. Students engaging in simple word problems provided written
solutions that were consistent with the realities of the ‘real’ situations described in the tasks in 33% of
the solutions66. However, students engaging in more authentic versions of word problems involving
the same scenario and math provided 51% reality-consistent solutions. In the diagram below
differences between standard and authentic word problems can be seen in one of the multiple word
problems of the experiment:
62
Nunes et al, p. 100
63
Nunes et al, p. 101
64
Palm 2007, Verschaffel et al. 2000
65
Palm 2007, p. 38
66
Palm 2007.
23
Standard word problem
Authentic word problem
Elin is planning to ride horses each day for 4 days.
Each day she has 45 minutes of free time to do this.
How many 10-minute rides does she have the time to
do during these days?
You are going to a camp for 4 days, but you also want
to ride horses. Your dad sees in the camp brochure
that you have 45 minutes free time each day, and that
horses can be rented for tours on a path in the woods
that takes 10 minutes. To know how much money you
will bring you must calculate how many tours you
have time to ride. How many 10-minute tours do you
have the time to do during these days?
In compiling Palm’s findings, the issues regarding word problems could be dealt with in several ways:
The event described in the word problem must have taken place or must have a fair chance of
taking place.
The question in the word problem must be one that actually might be posed in the real life
event described.
Give students a clear overall picture of the situation - the information should never be
substantially simplified. This will give students enough information about the circumstances of
the situation, including the task context and the purpose of solving the task. The purpose must
be as clear as it would be in a corresponding real life situation. This must be done while
minimizing the amount of text in order not to punish students with reading difficulties67.
Information given should be specific and not general. The task text must describe a specific
situation in which the subjects, objects, and places in the task context are specific.
Furthermore, data like numbers and values must be identical or very close to the
corresponding numbers and values of a real life situation.
These findings seem interesting for understanding reality and knowledge transfer. However it should
be noted that Palm was unable to test which, if any single, of these principles above were contributing
most to the success of his authentic word problems. Nevertheless, many of these findings could be
proven valuable to keep in mind when creating math video games. Elements of these finding will be
revisited later for that exact purpose.
As described earlier, RME is also addressing word problems and how they should be handled68:
Interpreting, negotiation, and common construction of knowledge through dialogue in between
students and between students and teacher. This should help identify what is the problem, what
counts as a solution, what is the best solution and what is the best solution method.
In wrapping up the research concerning word problems, a major part of the problem and solution
67
Palm 2007, p. 41 & 44
68
Gravemeijer 1994 p. 88
24
seems to revolve around math education as a socio-cultural activity – the social context of a school
situation69. One girl’s answer to the question of why she did not include realistic considerations when
solving math word problems was illustrative for this conclusion: “I know all these things, but I would
never think to include them in a maths problem. Maths isn’t about things like that. It’s about getting sums
right and you don’t need to know outside things to get sums right”70. The expectation is not that any
math game would be able to change factors of socio-cultural problem outlined above. This would
possibly require some major restructuring of classroom practice along with curricular refocus. Further
analysis of the socio-cultural problems can be engaged by others. However, I do expect a math game to
be able to circumvent some of the limitations enforced upon education by these socio-cultural
constructions. The social context of school situations could be manipulated by a game and hopefully
show students like the girl quoted above that there is a connection between ‘outside things’ and
‘maths’, but the game cannot change how the root of school culture spawns this social context.
Even if great care is taken in their creation, word problems seem to have innate limitations. However,
word problems still seem better suited for math problems than bare sum notations problems. This was
illustrated by the main street math study and is also supported by RME. Gravemeijer found that
second graders presented with a problem describing a jar containing 47 beads where 43 of which
were necessary to make a necklace, roughly 60% of them figured out that 4 beads would be left over.
When presented with the bare sum 47 – 43, however, less than 40% of the students found the correct
answer.
It is interesting to think why this happens. Gravemeijer explains that “In contrast to bare sums, the
advantage of this kind of context problems is that the designer is not bound to the notations familiar to
the students. One can thereby bypass unknown forms of notation and anticipate the subject matter that
has yet to be handled in class. Then the students can really show their capabilities”71.
Gravemeijer is here extending the explanation behind word problems and context-relations of the
previous pages to include more than realism. He hints that considering notation or representation of
math problems within realistic contexts potentially enable students to engage math problems beyond
their expected qualifications.
“Moreover, these context problems provide the students with more room to choose a solution procedure.
Bare sum notations are often associated with standard solution methods. When this notation is missing,
the tendency to apply such methods is also absent. Instead, the contexts often offer opportunities for
informal solution strategies.” This explanation must be viewed in the light of one of the central points
of RME, which is development of students’ own solution strategies. The context examples will support
these informal strategies that Gravemeijer mentions, but must be gradually replaced by formal
algorithms.
In gathering the threads presented so far it is seen that math students gain significant advantages in
understanding and solving math that is embedded in a realistic, purposeful context. This is grasped in
the concept of context-relation. It makes logical sense that students lacking understanding of the
purpose and realism of math usage will benefit from be exposed to math that focuses on just that.
69
Palm 2007, p. 56
70
Verschaffel et al. 2000, p. 26
71
25
What has been established within this concept is the fact that not only must the math be less abstract
and more realistic it must also be embedded into a context that makes sense, while carefully
considering the exact problem representation. Furthermore, such a context-realistic approach to math
problems seems suited for not only avoiding problems of other learning artifacts, it seems capable of
increase learning strength. These factors will be relevant for a math game design and will be revisited
accordingly. However, exploring some concepts further before venturing into game design could be
valuable and will be the objective of the next chapter.
Plausibility
One of the limitations of classroom education is the very fact that it takes place inside a classroom and
not in the outside world. This poses challenges as how to put context-realistic math examples into the
education. Nunes explains how it can be difficult to think of good problems to work from in the
classroom. An example of this is the following problem that attempts to test children’s knowledge of
the world in solving word problems. The problem:
“John ate 8 Big Macs. It takes John 15 minutes to eat a Big Mac. How long did it take him to eat them
all?”72
Nunes explains that the situation described goes against the idea of what is sensible to do when going
to a fast-food restaurant. Its solution is not meant to be used in reaching any decision, solving the
multiplication question does not help anyone understand a situation that may arise in everyday life.
This is used to show that just because some math example is embedded in a context-realistic scenario,
it is not necessarily useful. For example, it is a realistic example to use consumption of burgers as way
of calculating time. It is also a meaningful context that John goes to a restaurant and eats some burgers
and it takes some time. But it makes little sense that someone eats as much as eight burgers.
The British Mathematician Keith Devlin argues for the same cause: “please don't use unrealistic, fake
scenarios and tell the students they are seeing "How math is really used." Give them realistic examples”73
He exemplifies this by critiquing what he calls a typical text book example of filling a swimming pool:
“finding out how long it takes a swimming pool to fill from a hose that delivers X gallons of water a
minute. (…) what you, and I, and everyone on the planet would do, is turn on the water, watch for a
minute or two to get a sense how fast the water level seems to rise, then do something else nearby,
checking periodically on the progress until it's getting close to being full, and then watching it until it's
done.”
He argues that we barely use math in our daily lives, even if it actually is omnipresent. What people do
when encountering math is for example to use a smart phone app or an automobile dashboard display
that depends on math. This is so common that it is not at all difficult to show students that math is part
of our daily lives, even if it is the devices performing the math, and not the humans. This makes him
argue that instead of asking students to carry out the swimming pool example with an unrealistic
scenario, instead say that their boss wants them to develop a small automatic valve that can be set to
turn off the water when the pool is full. The students will end up doing the same math, but the
72
Nunes et al, p. 149
73
Devlin 2010
26
formulations will seem more relevant to the students by presenting it in a plausible fashion74. This
statement is analogous the findings of some of the word problem researchers’. They found that
students would engage more realistically with word problems, e.g. about carpentry if they were
provided with concrete materials such as planks, a saw, and a meterstick when calculating lengths.75
The only problem with this approach is that the resources of a school might not support that a class
starts building swimming pools and valves and sawing planks. However, the abilities of the commonly
used methods of teaching math, the classroom and the text book, are in their nature limited in their
ability to pick up the recommendations of realism, context and plausibility that have been identified so
far. The context-realistic scenarios that make sense would be most realistic if the students engaged
them as real problems out in the real world. But as Gravemeijer puts it, “One must acknowledge that
one cannot bring the reality into the classroom. Although students will be able to identify with well
chosen contextual problems, these will never become real life problems”76. This might be obvious but it is
nevertheless an annoying barrier to identify, since street math studies proved how beneficial real
world math problems are. This forces one to think outside the box, and this is where video games
become an interesting choice – to see how real math problems from real life can become in a game.
Context-relations in Math Video Games
In the preceding pages it was established that some kind of dual reality connection is in play between
classrooms and reality:
Classroom -> Reality: Students does not know what the real life purpose of math knowledge is, and
they are not utilizing their school math outside school.
Reality -> Classroom: Students ignore constraints of reality in classroom mathematics – they are not
using their real world knowledge in classrooms.
The research goal of the project at hand is more concrete now: To find out how video games can be
designed to help minimize this gap in the dual reality connection. In other words; to test the
hypothesis that games are capable of simulating purposeful math interaction inspired by RME and
street math. The next chapter will investigate and form this hypothesis including all implications this
can have on the game design and user learning.
Within video games, realism can be simulated without the boundaries the classroom is facing. The
game can embed the player in situations similar to the situations that children working in a
marketplace are facing. Furthermore, if a strong procedural rhetoric is obtained, they can make the
math word problems come to life. However it is necessary at some point to face the question whether
the above described benefits of RME and street math will follow along when translating e.g.
marketplace math into a math video game. Such a game will from now on be referred to as a realistic
math game, or a realistic learning game.
74
Devlin 2010
75
Palm 2007, p. 38
76
27
From the above pages it is already evident that if RME and street math are to guide creation of a
realistic math game, then the game must support the following three points:
Realism: Use the real world for inspiration in creating math problems of the game. Neither
bare sum notation nor simple word problems seems adequate for translation into video game
math problems.
Contexts: Embed these realistic math problems in a rich context comprised of a story involving
the actual math problem. This gives context-relation.
Plausibility: Ensure the context-realistic math problems make sense and are plausible
These points reflect the findings of the previous pages. However, these points are somewhat vague and
raise multiple new issues. These points will serve as the basis for the model discussion of the next
chapter.
28
MODEL DISCUSSION
The purpose of this discussion chapter is to identify, clarify and specify in a concrete way how a game
best can simulate the findings of the theory chapter. Theories will be expanded where necessary and
new theory brought in when needed. The model will end up consisting of a set of design principles.
This model must be specific enough to grasp the findings of the theory chapter while at the same time
be wide enough to enable creation of many different kinds of games for the target group. The target
group is compulsory school children, since the theories and studies of the theory chapter mostly
concerns this target groups. Thus the model must end up as a set of design heuristics that can enable
me and other researchers to create math games based on RME and street math.
Endogenous and Exogenous games
The very first thing to be addressed should be to decide if the game should be of an exogenous or
endogenous type and to decide that a look must be cast on school demands, RME and street math’s
position and GBL researchers’’ advices.
The official curriculum guidelines of the Danish school system (Fælles Mål) urges teachers to evaluate
their teaching tools and ask if these are used exclusively to answer questions like “how many”, “how
much” or “how big” or if they also address questions like “why” or “how”.77 It would be difficult to
engage why and how questions within exogenous games. Nevertheless, the situations in the schools
might force the hand in some situations. A FutureLab researcher said in an interview explaining
findings of a huge survey in Great Britain that “You either have to got short, simple games that get over
one point, that you can pick up and learn almost like a casual game. And then use it very quickly in the
classroom. That I think will be picked up. Or you have to have longer games, ones that clearly tie in with
the goals the teachers have. And that can give the teachers feedback because that’s part of their
problem.”78
This statement indicates that the demands of the schools forces a bipolar situation - either make a
simple, exogenous game, or make a longer, endogenous game that tie in with teachers’ goals. However,
as explained in the theory chapter, the understanding of exogenous and endogenous games is not
distinct; their qualities in regard to e.g. intrinsic fantasy can in some cases overlap. However, if there
before existed a bar to be set in balancing a game between endogenous and exogenous qualities, then
it is gone now, according to FutureLab’s survey of school demands.
As Squire’s table of exo- and endogenous games argues, learning in exogenous games is focused on
memorizing. Endogenous games are focused on doing, experimenting and discovery learning. Malone
agrees on this in relation to intrinsic fantasy games: “One advantage of intrinsic fantasies is that they
often indicate how the skill could be used to accomplish some real world goal.”79 Additionally, most
prominent video game researchers agree80 that if games are to do more than simple drill and practice,
then they should follow a constructivist learning approach – which is the approach of the endogenous
77
78
UVM 2009 s. 46
Ulicsak 2010
79
Malone 1980, p. 164
80
Bogost, Gee, Malone, Squire, Egenfeldt-Nielsen, Prensky
29
games.
In the findings regarding RME, word problems and street math, it seems imperative to ignore rote
learning and focus on the ‘whys’ and the ‘hows’, and also to increase realistic immersion. Also, if
students need to know how math is used in real life, they should know how it works in real life.
Bogost’s constructivist approach of procedural rhetoric supports this: “procedural rhetorics afford a
new and promising way to make claims about how things work”81 Consequently it must entail that if a
math game is to engage students and show them the purpose of math and its real world applications
(how it works), then it will not be fruitful to pursuit exogenous games involving extrinsic fantasies,
since these typically engage students in drill and practice activities.
Thus it must follow that a realistic math game must be based on the endogenous approach. How such
endogenous aspects should be understood and created will be investigated throughout the rest of this
report.
DESIGN PRINCIPLE # 1
ENDOGENOUS GAME WITH INTRINSIC FANTASIES
THE GAME MUST BE DESIGNED AS AN ENDOGENOUS, INTRINSIC FANTASY GAME .
81
Bogost 2007, p. 29
30
Purposeful activities
The comparison and choice between endogenous and exogenous games is not trivial, however it is also
not exhaustive in the approach of education. More choices can be made by considering the very
purpose of doing math. What is learned and how much is learned are both influenced by the learner’s
motivation82. Through the chapters so far, several reasons or purposes have been identified of why
individuals engage in math activities:
- Math for the sake of math. This intrinsic motivation is connected to the desire of mastery; to
become better and better at math and engage in math for the sake of the fun in the very activity.
- Math for the sake of extrinsic purposes. This extrinsic motivation contains purposes like pleasing
the teacher or parents, gaining rewards like money or good grades or avoiding punishment.
However, I believe one more reason for doing math exists:
-Math for the sake of solving a meaningful problem
To explain what is meant by this distinction, an analogy of math history can provide a logical
argument. One can easily imagine that ancient math practitioners developed their math knowledge
because they needed practical tools to improve their world, e.g. improving building construction
methods or distributing amounts fairly for example in relation to trading. One can also imagine that
after these basic needs were identified and solved by developing novel mathematical methods, new
questions arose that would intrigue the curiosity of these early math experts. This curiosity can be
compared to mastery – that math no longer reflects a need to solve a problem like trade, but that math
is developed further because mathematicians wishes greater understanding of the math concepts and
greater expertise in their field. This is the curiosity driven practice of math for the sake of math. But it
is important to recognize that these math needs are based upon the more primitive math need – to
make math for the sake of a purpose. This distinction is important no matter if great mathematicians
are driven by the adventure of exploring the limits of mathematic tools, formulas and theories or if
they are driven by the motivation to use mathematics as a tool or activity to attain a desired goal.
This way, to see the purpose of math practice for the sake of an important goal, should be reflected in
education. This was already established in the theory chapter. Of course, some students are motivated
just fine by racing for high grades or by having intrinsically motivated fun with math challenges. But
other students will keep on asking why they should learn this, what can this be used for, and what is
the purpose. Students who never attain an understanding of this purpose might never dedicate
themselves fully to mathematics learning. Possibly games based on the game design principles
developed through this model might benefit such students the most, since a focus will be this entry to
motivation. But that does not necessarily infer that others students would not benefit from such a
game. Mathematics is special in this regard, as it is a powerful tool whose mastery much of today’s
society rests upon, while it is pretty invisible in daily life. The case is different in regards to reading.
Most can agree that reading is important in our society since texts are found everywhere in our society
and media, everywhere in our daily lives. Another aspect is that intrinsic and extrinsic motivations
might be well suited or possibly even an important prerequisite for allowing deeper learning and
understanding to occur, but that entails that students are willing to even engage in the very activity
with dedication, which again entails they can see the purpose. The fact that street math practitioners
also have a dedicated purpose to perform their mathematics might be one of possibly several
82
Asgari and Kaufman 2009
31
explanations to street math’s success. A vendor in a marketplace might have an extrinsic focus on
earning money or have an intrinsic focus on being able to be good at math. But the purpose of math
usage is not to make money, that is the purpose of doing business. Math is a tool, a means to an end. It
is the tool used for meaningful customer interaction, to ensure fair trade. If fair trade could be easily
ensured without math, then math would not be a part of street vendors’ repertoire. In short, both
intrinsic and extrinsic motivational factors are not sufficiently explaining purpose. However, some
think of the term intrinsic motivation as including meaningfulness (which is similar to purpose) as
well: “… tasks that are personally relevant, meaningful, interesting, appropriate in complexity and
difficulty to the level of learners’ abilities and skills, provide personal choice and control, and tasks that
learners believe they can succeed, can stimulate learner’s intrinsic motivation.”83 Nevertheless, I
postulate that a realistic math game curriculum should have a stronger focus on purpose and being
meaningful, than on other factors. Consequently this will be the main factor for intrinsic motivation,
while the other factor in the above quote will be investigated as well and get their chance to climb the
ranks of importance. Note the difference between the terms intrinsic and extrinsic motivation and
intrinsic and extrinsic fantasy.
In conclusion, it must shine through all design principles and decisions in the following pages that the
game must show purpose of mathematics. How explicit or implicit this purpose must shine through
will be investigated throughout the report.
PURPOSEFUL ACTIVITY
THE GAME MUST IN ALL ASPECTS AND ESPECIALLY IN ITS MATH PROBLEMS REFLECT THE
PURPOSE OF SOLVING THE TASK AND THE PURPOSE OF UTILIZING MATHEMATICS IN WAYS
THAT ARE IN CONCORDANCE WITH PURPOSE -DIRECTED USE OF MATH IN REAL LIFE
83
Asgari and Kaufman 2009
32
Knowledge transfer and fantasy
This chapter will investigate how realistic math knowledge can be transferred to other domains.
Understanding this might help defining how realistic math could be transferred into the video game
domain, and how video game realistic math can be transferred back to school as school knowledge.
One can start by asking if street math and realistic math can be used by the students in other contexts.
This might be difficult since the knowledge has been acquired in another context and the transfer and
understanding of this knowledge out of the context can be hard. Gee acknowledges this idea in his
Situated Meaning Principle: “The meanings of signs (words, actions, objects, artifacts, symbols, texts, etc.)
are situated in embodied experience. Meanings are not general or decontextualized. Whatever generality
meanings come to have is discovered bottom up via embodied experience“. Embodied experience means
experiences that a person has actually had or ones he or she can imagine, thanks to reading, dialogue
with others, or engagement with various media. In other words, Gee claims that knowledge is not
general or decontextualized. This is challenged in a street math study designed specifically to deal with
the question; is street math restricted to reality? Or phrased differently; is street mathematics
hopelessly intertwined with the situational meaning?84 The study consisted of sub-studies involving
fishermen with 0 – 8 years of schooling. They were interviewed individually on the beach where they
worked. The test was made to illuminate if and how the fishermen could solve math with different
content than they were used to in their daily trade. The researchers gave math questions concerning
their daily fishing, questions concerning imagined fish involving unfamiliar values, and questions
concerning agriculture. One question was phrased: “There is a type of oyster in the south that yields 3
kilos of shelled oyster for every 10 kilos you catch; how many kilos would you have to catch for a customer
who wants 12 kilos of shelled oyster?” An example of agriculture question was that fishermen were told
that at farmer was selling 25 kilos of beans for 75 cruzados (minor currency of Brazil) and were asked
to figure out how much the farmer was getting per kilo. Results showed that their degree of success
was quite high, varying between 75% and 91% correct responses in the different groups of problems.
Nunes concludes that fishermen: “… do not display knowledge that is so content-bound that no transfer
is possible”85. The fishermen showed ability to transfer their model of weight-price relation to other
variables in the fishing context and to similar variables in the new problem context of agriculture.
Thus Nunes thinks this brings confirmation to “the idea that fishermen develop a schema for
isomorphism of measures situations that can be used in the solution of new but similar problems.”
The question is now what enables the street math performers to transfer their knowledge to new but
similar problems. This level of or ability to utilize knowledge in other domains is called ‘theoretical
attitude’. Luria 1976 (through Nunes et al p. 130) made some experiments to investigate the effects of
minimal theoretical attitude in unschooled subjects. It showed when it was not possible to transfer
street math knowledge. Unschooled subjects of the tests refused the premises of math problems
whenever the data contradicted their experience. This was in one test done by calculating the distance
between two real towns, A and C given the distance between an intermediary town – that is, given AB
and BC. However the distances given in the problem were wrong, they placed the towns farther away
from each other than what was the case in reality. Unschooled subjects rejected the problem and
refused to search for a solution. This is a relevant finding since it shows that tasks that challenge ones
84
Nunes et al, p. 102
85
Nunes et al, p. 120
33
experience of reality are less believable, especially if a theoretical attitude towards the knowledge
model is absent.
This is also relevant to the previously established design principle about the importance of contexts
that make sense. This nourishes the argument that subjects (at least unschooled ones) prefer to apply
their knowledge in situations that make sense and are plausible in regards to their experience, and
consequently math problems should abide by this.
But to continue the argument, the question is why these unschooled subjects rejected to solve the
problem. One could propose this rejection is connected to the imagination of the learner. If the
imagination of the learner is carefully nurtured during the learning session, then the learning outcome
can become many-fold stronger. I’m not simply saying that some class letting students replicate the
behavior of a street math vendor/customer transaction will be enough to trigger imaginative learning
boosts. “But the power of fantasy in enabling children to project themselves into the roles of actors in
imaginary stories goes beyond just replicating behaviors, for fantasy can also allow children to
experience what it feels like to be a particular character in a story: to feel success upon learning
mathematics, to feel powerful while doing mathematics, to feel effective after using mathematics to solve
problems so that lives can be saved, or perhaps to feel influential after having taught someone
mathematics. Fantasy plays an important role in mathematics instruction when it allows children to
experiment with feelings of mathematical power and success”86. Schiros argument that the power of
fantasy is more than just replicating behavior is interesting. Some keywords in the paraphrase are
projection, role, actor, experience, character, feel success, feel powerful, feel effective, solve problems
and feel influential. These are powerful words, according to Schiro all connected to the power of
fantasy. This power of fantasy must be connected to the math performer’s identity. This will be dealt
with in just a moment. The question for now is if this power of fantasy is powerful enough to help i.e.
unschooled subjects or potential learners lacking the theoretical attitude. The answer to this might lay
in another study that was set up by Dias to confront this. Nunes explains: “In a series of ingenious
experiments, Dias showed that the theoretical attitude could be produced in an experiment either by
asking subjects to engage in make-believe play and imagine that the given premises are true on another
planet or, more generally, by inducing children to create a mental world that they temporarily consider in
their reasoning.”87
This secondary source reference is short and abstract. However if it is assumed to have some level of
validity then it will work as a wrap-up for this discussion. In short, a fantasy world helped the students
engage in the math problems that would otherwise have been rejected. The same was actually the fact
in the street math experiments involving fishermen, they were told to imagine that some other kind of
oyster existed in the south that yielded ratios different from the fishermen’s personal experiences. No
fishermen rejected the problems, and their solutions were satisfying.
This leads to the conclusion that the effectiveness of and the students acceptance of the contextrealistic math examples are increased if they can be embedded in a game that encourages imagination
or fantasy.
86
Schiro, p. 58
87
Nunes et al p. 132
34
So two things have been settled: Street math or highly context realistic math knowledge is transferable
to other applications if this new application is supported by imagination. It was also learned that this
imagination support goes the other way; the theoretical attitude of the learner is boosted and the
applications of math can seem more meaningful in the open imagined world than in the constricted
reality of e.g. the classroom.
One might see a contradiction between this emphasis on fantasy with the previously identified
importance of emphasis on reality. However I see no contradiction between fantasy and reality, since
it is the imagination that is the key here. Imagination or fantasy can in some cases be an obstacle, but
in this case it is a possibility. In combining the previously settled design principles, realistic math must
occur in fantasy scenarios, as long as they are plausible. I will stick with this principle but it forces a
careful consideration of what realism and plausibility in a game is and how it is defined. RME’s HeuvelPanhuizen illuminates this agenda: “The fantasy world of fairy tales and even the formal world of
mathematics can provide suitable contexts for a problem, as long as they are real in the student’s mind.”
88
What is meant by being real in the student’s mind? I will refer to the previously explained idea that
math problems must be plausible, believable and make sense in order to be real in a student’s mind.
This implies a conclusion that it will actually be advantageous to embed math assignments in video
games in fantasy or science fiction settings. It should be noted that the concept of plausibility can be
redefined within a game narrative, thanks to this finding. In the example with John eating eight Big
Macs it will not make sense if he is a human but it will make sense to eat eight burgers if John is a big
orc in a fantasy role playing game. The fantasy setting lets the learner accept math problems that
otherwise would not make sense and might have been refused by the learner.
FANTASY, REALISM AND PLAUSIBILITY
THE EFFECTIVENESS OF AND THE STUDENTS ’ ACCEPTANCE OF THE CONTEXT-REALISTIC
MATH EXAMPLES ARE INCREASED IF THEY CAN BE EMBEDDED IN A GAME THAT STRESSES
IMAGINATION OR FANTASY. I T IS IMPERATIVE THAT THE GAME CONTENTS ARE REALISTIC
AND PLAUSIBLE WITHIN THIS FANTASY FRAME
88
Heuvel-Panhuizen p. 4
35
Identity and character
The study presented in the previous chapter might not be sufficiently covering in explaining why the
differences in performance between school math and street math practitioners. RME argues that two
types of theory could explain these results. One stresses the social-interaction aspects of the situations
and the other stresses the social-cognitive aspects.89 The second explanation will be dealt with in the
chapter named Representation.
The social-interaction approach would stress that the situation in which mathematical calculations are
performed determines the role participants have in respect to each other, thereby influencing the level
of success. If this is a dominant explanation, then the good results of street math performers might be
explained by the social interaction.
“Arithmetic problems that arise in the marketplace are an indispensable part of a commercial
transaction between vendor and customer. It may be that the social relationship established between
vendor and customer is such that children feel more confident of their own ability and perform better.
They trust themselves as vendors but not as students, and this lack of confidence makes their performance
deteriorate when they have to act as students.”90
If this ‘vendor confidence’ can be transferred to other scenarios, then we might deal with students that
are more inclined to solve math correctly. Gee is treating this topic in his chapter about learning and
identity. He uses this chapter to argue that understanding how players’ identities work in video games
will increase the understanding and use of how identities function in other knowledge domains. Gees
argument can be reversed to show how real world identities can teach how video game identities can
be understood.
Gee describes that identity is closely connected to active, critical learning since it requires
commitment of time, effort and active engagement91. “Such a commitment requires that they are willing
to see themselves in terms of a new identity, that is, to see themselves as the kind of person who can learn,
use, and value the new semiotic domain.”92 The presence of this commitment could explain why street
math practitioners outperform themselves in school settings. Understanding how this commitment
and confidence can be put into education might be crucial for effective realistic math. Gee define
virtual identity as one’s identity as a virtual character in the virtual world. However, the real world
person that is playing the game can have several real world identities and the interactions between
the real world person (and his identities) and the virtual identity is called a projective identities.93
The key to the success in street math and RME is the interplay between several real world identities,
i.e. the success of the child’s vendor identity and the failure of the school identity. Gee might suggest
that the very fact that a child enters school makes the child engage in a virtual identity as ‘me in
school’. Likewise with the street market situation, the child would engage in a sort of virtual identity as
89
Nunes et al p. 28
90
Nunes et al p. 28
91
Gee, p. 59
92
Gee, p. 60
93
Gee p. 54-55
36
‘me as a vendor’. In this case, the vendor identity, the one resting in the social interactions and
limitations of the real world, is acting as a confidence and commitment booster.
Researchers investigating stories of how math practice works in the home found examples of
mathematics being used to support one’s sense of personal and social responsibility. Family members
used their stories to illustrate their sense of fiscal responsibility, caring for others, and desire for
precise and thoughtful answers in the context of family values. However, in the school stories, people’s
identities were generally summed up as either ‘good’ and ‘fast’ at math, or ‘bad.’94
Hence, the school identity might be, at least for some students, a catalyst for low confidence, low
commitment and low entertainment value. Furthermore, as established earlier, imagination plays a
powerful role in how such a commitment can occur. Putting these things together suggests that
imagination of the learner must be nurtured not only to boost the learning outcome, but also to boost
the confidence of the learner. The question then stands how the player confidence can be boosted in a
game, and how the projective identity of a player/character can be linked to ‘I can’ real world
identities while losing the ‘I’m not a math type of child’ identities, or whatever other negative
identities that might influence the learner commitment and confidence. Gee argues that many of his
real world identities are in play when controlling the fate of his game character. If this is the case then
it might not be possible to try to welcome some types of player identities into the game scenario while
shooing away others. However, as the research of the previous chapters suggest, fantasy can act as a
highway for injecting theoretical attitude to otherwise refusing learners. This suggests that a realistic
math game must to some degree revolve around a character wherein the student can project his
identity. The idea is that the character should engage in a fantasy world where the virtual identity of
the player has a purpose in doing the math, the commitment in engaging in this math and the
correctness of the results must somehow be important for this character. The fantasy world of the
game must be designed specifically to let this meaningful interaction take place. Depending on the
significance of the level of social-interaction taking place in the game, this interaction and role play
might be sufficiently covered through interaction with in-game environments and non-player
characters (NPCs). Heavier emphasis on the social-interaction significance would require the game to
allow several players to interact with each other in order to increase the commitment. An investigation
into that is relevant and will be dealt with in the chapter named Multiplayer.
This identity talk would be useful for targeting specific target groups. For example, one might apply a
certain character design if the game is meant to be played by unschooled people learning math.
Likewise, a different approach would be relevant if the target audience is girls in middle school. For
instance, math examples and the character’s interaction with these could be designed along certain
guidelines for girls; “Offering players to choose their gender and to control some physical aspects of the
player-character can help to address desires to create a feminine or masculine fantasy persona in a
game.”95 The term fantasy persona in a game is similar to Gee’s definition of a virtual identity. And as
Gee establishes, harnessing this would be detrimental in immersing the player even deeper. Gee’s
Committed Learning Principles says that “Learners participate in an extended engagement (lots of effort
94
Esmonde et al, p. 18
95
Isbister 2006, p. 117
37
and practice) as an extension of their real-world identities in relation to a virtual identity to which they
feel some commitment and a virtual world that they find compelling”96.
Consequently, when engaging identity questions it is worthwhile to use some time on how targetgroup identity could influence the effectiveness of the game design and in particular in relation to
realism. However, in adjusting a game to a specific target group, not only character design is
important. In continuing the example of gender differences, immersiveness can also be improved by
designing content for specific target groups. For example, Isbister says that “Girls would rather spend
their time creating things instead of destroying things.”97 However, investigating how to design games
for specific target groups, e.g. mid-school girls, is not a simple area to enter since it opens up a new
host of complexities whose thorough treatment would be out of scope of this project. An example of
the complexity of gender care is that meta-research shows that some researchers have found gender
differences regarding game-based learning performance and game design preference, while other
researchers have not.98 Nevertheless, if designers wish to address their game at a certain delimited
target group then this step should not be omitted. Characters in video games, if designed properly, will
mediate the connection of the real world identity and the projective identity. In other words, since it
was established that identity play might be important for understanding and give meaning to math,
then it can be concluded that a math video game must employ some kind of character focus. The
question of how strong a character focus is necessary is unknown and this is probably for the best
since this will give the game designer some liberties in their design. Making a math role playing game
seems like a good idea with this in mind. Making a math version of Pong will definitely not be a good
idea with this in mind, but perhaps a Mario clone with some math involved might have a sufficient
degree of charactership. The Mario character is similar to what Isbister calls a Puppet99, which is a
character design with less focus on fantasy and social qualities, and more focus on the physical
manipulation of the character. Super Mario does not have an advanced story engaging the character or
even developing the characters much, but the Mario figure is distinguishable and archetypical enough
to give him a unique character feeling, and that level of character immersion might be sufficient for a
realistic math game. Conclusively, the final choice of character depth and design should be determined
by the target audience and the identity angle of these that the designer wishes to employ, but the game
cannot be characterless, like e.g. some simulation games are.
IDENTITY AND CHARACTER
ANALYSIS OF THE GAME ’S TARGET GROUP MUST BE THOROUGH ENOUGH TO ENABLE
CATERING FOR THE PARTS OF THE TARGET GROUP ’ S IDENTITY THAT CAN INCREASE
MOTIVATION AND IMMERSION. SUCH CONSIDERATIONS MUST BE INCLUDED IN EARLY GAME
DESIGN, ESPECIALLY IN THE DESIGN OF PLAYER CHARACTER BACKGROUND AND ABILITIES .
FAILURE TO ADDRESS THIS SATISFYINGLY WILL IMPERIL LEARNING OUTCOME QUALITY .
96
Gee 2003
97
98
Ke 2009, p. 21
99
38
Multiplayer
Early serious game researcher Abt100 argues that a game is not true before it has interaction with other
players. This is corresponding to the idea that the explanation behind street math performance might
be due to the social-interaction taking place. In the previous chapter it was investigated how this is
connected to identity. Here it will be investigated how the relationships and social interactions
between multiple math performers, be they street math performers or game players, impacts math
performance.
One could say that a vendor and a customer at a marketplace are co-constructing the meaning of their
interaction. The very situation of one being a customer and the other being a vendor justifies and
legitimizes the use of math as a meaningful tool. Since school students have trouble finding meaning
and purpose in math, this might be an interesting angle. One fast conclusion to this would be to focus
on multi player game design. However, a single player math game and a multiplayer math game vary
hugely in terms of implementation and test complexity for game developers. The complexity would
rise especially if the purpose of including multiplayer in the game is to set up situations where
meaningful math interactions can be co-constructed, thereby simulating street math interactions. One
could imagine that it might not be that hard to implement a game where one or more players act as
vendors and others act as customers. For example, this is happening all the time in a game like World
of Warcraft where players often engage in trade with each other. However, World of Warcraft is
designed to be an entertainment game and even if there might be beneficial learning from a game like
that and such vendor/customer interactions, such learning would be limited to understanding how
trade works, i.e. negotiating prices, delivery and payment methods and avoiding trickery or fraud.
What this does not teach or practice would be actual math, and certainly not math of the RME kind.
Furthermore, multiplayer games open up an array of new complexities that should be dealt with if
they are not to negatively influence the previously established design principles. A qualitative
investigation summarizes some of these emerging complexities of multiple player scenarios, here
investigated where small groups of players were present in the same room, in the home of one of the
players:
“1: Small-group game-play interactions are a dynamic blend of different stages, ranging from teaching
(in which the children’s main aim is to collaborative in order to minimize the knowledge gap between
them) to cheating (in which the children’s main aim is to win by maximizing the knowledge gap
between them,).
2: Dynamic states of collaboration, competition, negotiation, and mentoring were detected during gameplay of peers. Several game-plays were accompanied by ad hoc conflicts.
3: Players spontaneously take different roles as leaders, managers, bankers, engineers, observers, and
reporters, to name a few.”101
These findings illustrate the complexity of endeavoring into multi player scenarios. If these
complexities change for better or worse in networked multiplayer games or massively multiplayer
online games is not clear. A further pursuit of the question in relation to what serves a RME math game
best will not be made, as even if it was possible to find substantial positive effects of including
multiplayer interaction in the games, the final decision whether it would be implemented or not would
100
Abt 1970, p. 8
101
Gazit 2009 p. 133-136 (Sentences in parentheses are original)
39
most likely depend on other factors. The technology behind the game might not be suitable for
multiplayer, development time might not allow it, testing complexity might blow up, possibilities of
plagiarism or students over-helping each other, and difficulties of assessing the learning outcome and
performance of individual students (i.e.: Who solved what?) could also set a stop for multiplayer
functionalities in a realistic math game. For instance, the design for i.e. massively multiplayer online
role-playing games (MMORPGs) requires a deep understanding of social play and game economies in
addition to classic role-playing mechanics102.
However, RME is actually stressing that the interactions and negotiations in the classroom in
interpreting problem definitions, determining what a satisfiable or valid solution is and analyzing and
comparing different solution procedures and their effectiveness are vital, and can obviously only occur
if multiple students and teacher(s) are present.103 However, this could possibly be facilitated in aftergame sessions, in the classroom. This would give the benefits of common knowledge creation even if
the game used is a single player game. However, there is the promising possibility of letting stronger
students act as in-game teaching assistants for the weaker students, thereby relieving the teacher. But
this form of mentoring might also be possible if players sit next to each other in the schools computer
room, playing the same game, thereby not requiring multiplayer. The question whether the potential
benefits outweigh the potential disadvantages is still not answered. Therefore this chapter will be
ended by stating that it is likely desirable for a realistic math game to include multiplayer, but it opens
up so many unknowns that either further research could decide if it should be included, or otherwise
the decision could be entailed by availability of game development resources. It is also unknown
whether if the game ends up being a multiplayer game, that it should be a networked multiplayer
game, massively multiplayer game, same-room group multiplayer (several students at each computer),
or same-room single player (multiple students in the same room but only one at each computer).
POSSIBLY MULTIPLAYER
POSSIBILITY OF MEANINGFUL IN-GAME OR OUT -GAME INTERACTION , COLLABORATION AND
COMPETITION BETWEEN PLAYERS IS DESIRABLE , BUT NOT DETRIMENTAL .
Note that out-game interaction in this sense refers to students sitting in the same room and
communicating while playing. There is no doubt that out-game preparation and evaluation taking
place in the classroom before and after learning game engagement is detrimental, not just desirable,
especially for RME.
102
Fullerton, p. 417
103
40
Narrative
In summary of some of the preceding pages; If street math users are to engage in math examples that
are exceeding their existing real world knowledge, then the math examples should preferably be set in
a fantasy or imagined setting, in order to let the students accept the alternative reality.
In other words, the story is possibly important in the delivery of math problems.
Without a story, a game like this will just be a series of various unconnected mini-games, e.g. like a
series of word problems transformed into procedural rhetoric, that might make sense on their own
premises but loose the strengths of contextuality and story binding them together. A game consisting
of an unconnected sequence of contextual realistic examples will seem more like a simulation or
random gathering of edutainment games, and much motivation of the players will be lost. One can
wonder what type of story should be employed – stories about people, objects, animals or gods?
Michael Schiro is in his book ‘Oral Storytelling & Teaching Mathematics’ explaining that “the type of
story used to carry a mathematical message is not as important as the story’s ability to capture children’s
interest, its ability to contextually locate mathematics in children’s lives in ways that are relevant to
them, and its ability to encourage children to project themselves into its characters’ roles so that they can
engage in the mathematical endeavors of those characters.”104
This is comparable to the discussion of the preceding pages of using the story and character-play to
make math real in the student’s mind. What Schiro does not answer is the degree of how a story
should be integrated into the game play, or how a story should develop during the storytelling (or
game). Tracy Fullerton’s ‘Game Design Workshop’ is treating story and premise in relation to game
design. A premise is the setting of the game, e.g. in Monopoly the premise is that the players are
landlords, buying, selling, and developing real estate. Players enjoy the fantasy of being powerful, landgrabbing landlords with plenty of money to spend. The base-level effect of the premise is to make it
easier for players to contextualize their choices, but it’s also a powerful tool for involving players
emotionally in the interaction of formal elements105.
A story differs from premise in its narrative qualities. A premise does not have to go anywhere from
where it begins, while stories unfold with the game.
Since all that was established in the previous investigation was that math must be taught within a
context and somehow engage the imagination of the learner, then it is not easy to conclude whether a
realistic math game should be premise or story based. Furthermore, the question regarding how
deeply the story should be integrated into the game is still unanswered. We do not know how much
story is too much, how little is too little, if gameplay should change the story or if story should dictate
the gameplay106. Fullerton’s approach seems sensible in this case: “There is no one answer to these
questions, but it’s clear from the interest of both players and designers that story integrated with play can
create powerful emotional results”107. Consequently it must be up to the individual game designers to
decide what works best, as long as some kind of imaginative play is involved, relevant and plausible
enough for students to project themselves onto the roles of the characters. Furthermore, a strong story
104
Schiro, p. 56
105
Fullerton, p. 40
106
Fullerton, p. 41
107
Fullerton, p. 41
41
can act as a curiosity booster and thereby motivate the player to continue. A student might be stopped
in his advance through the game and its story if he fails some math problem, but he might be enticed to
try again if the story is compelling enough to awaken his curiosity of how the story will unfold, or what
will happen to character X, if he manages to pass this math problem. Consequently he will end up
doing math, not for the purpose of grades, not for the sake of the teacher and not for the sake of
mathematics mastery. Instead, his motivation will be to use math as a tool to unveil more of the story.
This is what Malone calls completeness in relation to cognitive curiosity, that, i.e. you just read all but
the last chapter of a murder mystery you wish to bring completeness to your knowledge structure by
finding out who the murderer was108. This is similar to the earlier hypothesized concept of math
practice for the sake of solving a meaningful problem, the problem of advancing through the story.
However, one could also say that the story works as an extrinsic motivation to learn, a reward for
doing math correctly.
No matter what the explanation is, triggering player curiosity seems important for math games. As
stated earlier, Malone argues that sensory curiosity can be triggered by audio and visual effects. “When
sound or graphics displays are used to reward good performance, they can increase the salience of the
goal and thus add to the challenge of the game” 109. However, treating an unfolding video game story as
merely a display of sounds and graphics would be an oversimplification. Procedural rhetoric is
invented for just this purpose; to show that games are something more than just its graphics and
sounds. The procedurality of a game is connected to every layer of a game. In relation to the story or
narration layer, this makes us question how procedural or interactive a story should be.
The question whether a linear or non-linear narrative is stronger for this purpose is unknown. The
Danish math game Math in Moontown110 (original title is ‘Matematik i Måneby’) is using a non-linear
narrative structure. The goal is to construct the moon town, but the player unveils the narrative
elements by actively choosing the order that the parts of the town are constructed, effectively making
the narrative completely flat. Specifically in math it is easy to imagine that teachers and students
would find it convenient to be able to easily choose which sub-system of the game they want to play,
e.g. if they know that one part of the game practices addition and another practices multiplication. This
choice will however be immediately impossible if a linear narrative is to control the game structure.
Such a linearity might not expose the player to the multiplication part of the story before the player is
hours into the game experience. However, as seen in some games, a system that allows the player to
skip or replay certain parts of the game could somewhat circumvent this limitation. This would also
serve other purposes, such as adding to the replay value of the game and enable students to repractice specific math content. For a more extensive treatment of narrative development in relation to
learning games, I refer to the article ‘Narrative Development and Instructional Design’111. Particularly
their treatment of narrative structure and character archetypes could be relevant for a realistic math
game narrative.
Furthermore, as established earlier, the fantasy of a game defines what is realistic to occur in the game.
108
Malone 1980, p. 166
109
Malone 1980, p. 165
110
Matematik i Måneby
111
Williams, Ma, Richard and Prejean 2009
42
This is also important for stories, as a strong story can function as a believable link between various
parts of the game. Everything is possible in a fantasy story. No one will consider it odd or out of
context that you one moment must pick berries, the next moment fight someone, and later construct a
ship. That is, if the game story can link the various math problems and other game contents
meaningfully together. In a scenario attempting to imitate the real world, one would wonder why and
how these transitions must occur – this would conflict with ones expectations to what is plausible,
meaningful and realistic to do. In other words, a story or contexts like the ones popular in many word
problems that e.g. involve students of the class planning a field trip are undesirable. Not only would it
emphasize the players’ real world school identity, which is undesirable according to previous chapter.
It would also limit the story to take place in a simulation of the real world whose convincing portrayal
would force depiction of realistic elements. These realistic elements will come out unrealistic if
attempted to be linked together by a meaningful, realistic story. For example, a realistic game set in the
player’s own school or in the players’ own town would have to be pretty creative to realistically
explain why players go through math engagements in jobs like construction site foremen, marketplace
vendors and farmers. A strong fantasy story would have no such issues as it itself defines what is
realistic.
The findings of this chapter can be used to redefine design principle number 3 into a more clear
formulation:
FANTASY STORY THAT DEFINES REALISM AND PLAUSIBILITY
THE EFFECTIVENESS OF AND THE STUDENTS ’ ACCEPTANCE OF THE CONTEXT-REALISTIC
MATH EXAMPLES ARE INCREASED IF THEY CAN BE EMBEDDED IN A GAME WITH A STRONG
STORY FOCUS THAT STRESSES IMAGINATION OR FANTASY . I T IS IMPERATIVE THAT THE GAME
CONTENT IS REALISTIC AND PLAUSIBLE WITHIN THE FANTASY FRAME THAT THE STORY
DEPICTS . THE STORY IS USED TO CONVINCINGLY CONNECT THE VARIOUS MATH PROBLEMS .
43
Example: MathRider
I will now briefly present and analyze a game to contextualize the findings of the previous pages.
MathRider112 is a sort of a racing math game where the goal is to do arithmetic correctly in order to let
your horse jump over obstacles. The horse crashes into the obstacles if you do not calculate correctly.
Here is a picture of the general game play.
The horse traverses from left to right and the player can use the keyboard to enter numbers into the
field below the horse. Then numbers to be entered should be the result of the forthcoming arithmetic
displayed below the obstacles, followed by ‘Enter’ key to confirm. The horse then jumps over the
obstacle if it is correct.
There are also some story elements within this game, or at least a premise. The introduction tells that
your mother is sick and you need to travel by horse to the secret place where the rare flower that can
cure her grows. You can only travel there by using math skills and only on a horse. After each quest, a
map indicating the progress of travel is shown:
112
Acquired via http://mathrider.com
44
The filled part of the road increases after each successful quest. When the player clicks ‘Continue
Quest’, the game continues to the next quest which is similar to the previous image.
This game has some premise and a story that progresses. The visualization of story progress on the
map is a good idea. However the gameplay and narrative appear problematic upon further analysis i.e.
in regards to design principle #3. The story and context does not explain why it should make sense
that a horse only can jump over obstacles of which the player knows the arithmetic answer. It does not
explain why there even are obstacles with numbers and signs below them. Such an explanation could
be possible to include even in an exogenous game like this. This not only makes the game setting
implausible, it makes it unbelievable and the assertion is that this game will have difficulties, despite
its attempts to present a premise/story, in immersing the player into the fantasy of being a Math Rider.
The consequence of that is that even if the student engages in the game and possibly find it
worthwhile, they might end up having difficulties in transferring their acquired knowledge and
practice out of the domain of the game. The implausible scenario does not tell the students why math
is useful since the math usage is not corresponding to the context and realism boundaries set within
the game. This is similar to the situation of the previously described study where unschooled students
refused to solve math problems that conflicted with their experiences. The theoretical attitude is not
built up since the game itself does not abide the rules of realism of neither the game nor real life, thus
conflicting with the players’ experience of what is right.
In summary; a math game must have some kind of story/premise and let the player engage via an
identifiable character in activities that are purposeful, meaningful, and context-realistic within the
game world set by the premise or story.
45
Game genre
Two themes seem to reoccur throughout this investigation, where one theme is represented by the
keywords role, character and identity. The words of the other theme are story, premise and context.
The words within each theme are somewhat ambiguous and might overlap in their coverage.
There are various game genres that are useful for math games and have been used so far. Though I
have not encountered any math games within the strategy genre, nearly all other genres contain math
games. A qualitative study of 40 adults which investigated a variety of game genres for educational
purposes found that all kinds of game genres had potential for educational use113. However, in a RME
math game not all genres would be useful. Actually, only two genres stand out as being able to
embrace keywords of the two themes above. If a heavy emphasis is put on story and character, then
role playing games (RPGs) stand out.
Fullerton describes that role playing games “revolve around creating and growing characters. They tend
to include rich story lines that are tied into quests.”114 This definition embraces strong identity play in
creating and growing characters. Also the demands about fantasy, imagination and story are satisfied.
It mentions quests as tying up story lines. If the story line consists of realistic mathematic examples,
then quests could be a relevant way of organizing these examples into the story in a sensible way. This
will be dealt with in the next chapter.
If the emphasis on story and character is lessened, then adventure games seem to fit better. Fullerton
describes adventure games as emphasizing exploration, collection, and puzzle solving.115 Although
characters are central in adventure games, unlike role-playing games, they are not a customizable
element and do not usually grow in terms of wealth, status, and experience.116 Depending on the
desired character focus, either genre should work fine for the requirements of RME games.
Game genre hybrids might be relevant as well. An example is the game Deus Ex which is actually
classified as an action game, but contains far less ‘shoot first, ask questions later’ than i.e. Quake or
Half-life and more role playing and story- and character development. Let the conclusion to this then
be that a good math video game needs some RPG or adventure game elements, like story and sufficient
character development to let the player identify with the character or story.
Including the findings of the multiplayer chapter could suggest replacing the appearance of the word
RPG in the above paragraph with the word MMORPG. However, in keeping things realistic it might not
be an option for all game developers to create context-realistic math MMORPGS with plausible quests
and a rich story/character element. Consequently the least important design principles must be
ignored, and one of these might be the multiplayer aspect.
113
Ke 2009, p. 10
114
Fullerton, p. 416
115
Fullerton, p. 417
116
Fullerton, p. 420
46
Math problems as quests
RPGs and adventure games are often quest or puzzle based. The word problems of RME and any math
text book is always presenting a limited problem setup and solution range. This ensures that the
student does not lose focus on practicing or learning the mathematical operations that the problem
designer intended for that problem. By organizing the context-realistic math examples as quests, it will
be easier for game designers and teachers to identify the didactical goal of each quest and the story
connecting them. This proposal will align with the previous talk about genre, since core gameplay in
adventure games and RPGs usually consists of puzzles or quests. Susana Tosca defines a quest as
follows117:
From the designer’s point of view, a quest is a set of parameters in the game world (making use of the
game’s rules and gameplay) that specifies the nature and order of events that make up a challenge for the
player, including its resolution. From the player’s point of view, a quest is a set of specific instructions for
action, they can be as vague as a general goal (overthrow the evil king) or extremely precise (take this
bucket to the well, fill it up and bring it back to me); after the quest has been completed it can be
narrated as a story.
This definition is handy as it seems to align well with the findings of the previous pages. The first part,
that the quest must make use of the game’s rules and gameplay, is equivalent to the previously
identified requirement that the math problems must abide by the limits of realism that the game’s
story imposes. These restrictions and fantasies specify the challenge and its resolution. As found in
Palm’s word problem research, word problems’ instructions must be exhaustive and not vague.
Assuming this is transferable to game math problems, this will conflict with Tosca’s definition since
from the player’s point of view, the quest can have vague instructions. If the game consists of both
quests with learning in mind and also quests solely for entertainment or story purposes, then the
learning quests must abide to the second part of Tosca’s definition, that is to be extremely precise, in
order to satisfy e.g. Palm’s findings. However, entertainment quests can be either vague or specific.
The exact balance between pure entertainment quests and learning quests will be discussed later.
Furthermore it should be noted that according to Tosca’s definition, the quest can be narrated as a
story upon completion. If one holds on to the idea that the story should drive and combine the quests,
then an unsuccessful closure of the quest and/or its math components will arrest the player in her
penetration of the story, making the story act as not only a connector, but also as a gate-keeper
between quests. This topic will be expanded in the Balance chapter.
Let us again revisit design principle number 3 to include these findings, while simplifying it a little.
THE GAME MUST BE A ROLE PLAYING GAME OR ADVENTURE GAME . A FANTASY STORY
DEFINES WHAT IS REALISTIC AND PLAUSIBLE AND CONNECTS A SERIES OF MATH QUESTS
THAT ABIDE TO THIS DEFINITION . QUESTS INSTRUCTIONS HAVE SPECIFIC AND PRECISE
INSTRUCTIONS FOR ACTION.
117
Tosca 2003
47
Quest design details
The question of how to design the inner workings of each quest is still not thoroughly explored. How
does one abide by and combine all the previously set principles? One way to get inspiration of how to
exactly design the context realistic quests is to look at one of the other major principles of RME, the
reinvention principle.
“According to the reinvention principle, a learning route has to be mapped out that allows the students to
find the results by themselves. To do so, the curriculum developer starts with a thought experiment,
imagining a route by which he or she could have arrived at a solution him- or herself. Knowledge of the
history of mathematics may be used as a heuristic device in this process” 118. This looks relevant in
relation to the previously discussed topic regarding purpose – to use the history of mathematics as the
inspiration for math game quest design. This will not conflict with the previously set principles of
realism and fantasy, since “the emphasis is on the nature of the learning process rather than on inventing
as such. The idea is to allow learners to come to regard the knowledge they acquire as their own, private
knowledge; knowledge for which they themselves are responsible.”119
The history of mathematics is not to be taken literally; The suggestion is not (necessarily) to make
math games where the player reenacts the problems and solutions faced by ancient civilizations or
renaissance math thinkers. But these problems and solutions can very well act as inspiration to create
math problems in games that are realistic enough and make sense. This will help well-meaning
designers in avoiding creating math problems like the ones described and criticized earlier, like the
swimming pool problem or other unrealistic, meaningless word problems. But it also shows something
else, that is the limitation of each math example. E.g. Thales of Miletus’ usage of geometry to calculate
the height of pyramids and the distance of ships from the shore are good examples of limited problems
and solutions. Such math problems will automatically convey purpose, which was settled earlier as
extremely desirable to math problems.
As was shown in the studies concerning word problems, one way to increase students’ performance
and inclusion of real world knowledge is to include more information into the word problems that is
strictly necessary for solving the problem. This could be transferable to the math quests of a realistic
math game. Such math quests should let the player reenact or simulate actions similar to actions that
have or are likely to take place in the real world. Such actions can be described as embodied
experiences, using Gee’s term. As seen so far, crafting, trade and other manual professions seem like
excellent examples of meaningful, real life math performance. Consequently, simulating such
professions and their math related activities seems viable and sensible within a realistic math game.
Such a focus on simulating professions and labor while including an abundance of superfluous
knowledge will also improve students’ knowledge of the profession in general for interdisciplinary
purposes of e.g. improving a student’s general knowledge about farming while engaging in math that a
real world farmer would be likely to engage in. This applies even if the game is set in a fantasy world,
since figuring out how to construct a carport with your dad (by e.g. calculating material amounts,
costs, or wood board lengths) can teach some math and carpentry knowledge just as realistically as
figuring out how to construct a wooden siege catapult with your green orc friends. This means that
realistic examples can be embedded in fictional game worlds and thereby teach real world knowledge.
118
119
48
This will also train the students in another problem solving aspect which is looking for relevant data
within the problem – learning what to ignore and what to focus on. This will increase realism, as real
world math problems e.g. multiplying a recipe, contains lots of superfluous information. This is also
supportive for developing and maintaining a story – the superfluous elements could be artifacts of the
narration, or of the non-learning part of the game play.
In designing quests, the designer must also consider the didactical goals concerning how the student is
expected to solve the problem. In other words, what solution procedures and algorithms the intention
is to teach and practice. As stated in the beginning, focus will not be on the subject of how the students
solve the math problems in a math game. For instance, it will not be considered if the teacher bans
pocket calculators for students playing the game. This is not a decision taken out in the blue. In the
above paragraph, RME describes how the learning process is important. I hypothesize that such a
learning process might be imperiled if the math quests are too specific in describing how the math
problems should be solved. This is in line with another of RMEs goals which are that children should
be able to make sense of numbers and numerical operations on their own: “… children should be able to
decide for themselves what calculation procedure is appropriate for solving a particular arithmetic
problem. They should know when a mental calculation is adequate, when to use an estimate, and when it
is better to do column arithmetic on paper or to use a calculator.”120 The street math studies are
proponents of the same view. No one is forcing a street market vendor to solve math by either mental
calculation or calculator, instead, they use what is easiest, safest or fastest. A game imitating this
feature might, as quoted earlier, increase the students’ confidence and allows learners to come to
regard the knowledge they acquire as their own private knowledge for which they themselves are
responsible. This could also affect motivation and the way students’ various identities interplay, for
example in encouraging them not to be afraid to experiment, i.e.: ‘I found out how to solve it’ vs. ‘The
teacher/game showed me how to solve it’. Furthermore, teachers who wish to focus train her students
into better mental calculation will be free to do this within such a game design. Consequently, no ingame calculator can be present. No interface, that might otherwise be helpful, should help students
keep track of numbers or formulas.
Consequently, instructions revolving around quests must, as previously established, be rich and
precise and not vague in order to increase fidelity and authenticity, but the instructions are not to be
so precise as to instruct how the math problem is to be solved. It should just present the problem in a
precise way and let it be up to the student to analyze or simplify the problem by removing superfluous
information, identify an appropriate solution and then solve it using mental calculation, pen & paper
or calculator outside the game.
Nevertheless it can still be necessary to consider what kind of math is expected to be used to solve a
certain quest. Gravemeijer points out121 that different realistic examples can be used to teach various
forms of algorithms and solution procedures. Even if RME stresses that good examples opens up for
several solution strategies, ranging from various personal informal strategies to the more official,
formalized strategies, the examples can still be set up with certain learning goals in mind. Gravemeijer
explains how one math problem presents 3 children dividing 36 sweets equally among them, where
the 3 children are drawn, and 36 dots representing sweets are drawn. Gravemeijer found that children
120
Heuvel-Panhuizen p. 24
121
49
of 3rd grade who had never encountered such problems of division before were able to solve this
problem by engaging in a range of different solution procedures, including distribution division and
ratio division. However, if a ratio division approach is desired above others, it may be suggested by a
problem like this: A net keeps three balls. How many such nets will be needed for 36 balls? What to
take from this description is if certain solution procedures are desired by the teacher/curriculum, then
these should be identified and the corresponding realistic examples should be defined and
implemented in cooperation with the game developer. This step will be important if the game is
intended to cover a specific part of a curriculum. However, this step can be omitted in cases where the
game is intended to be more general in its math coverage or where the teacher is expected to fill in the
gaps left out by the game.
We will now revisit Design Principle # 3 to include this new information. However, since knowledge
revolving both story/fantasy and quests/math problems are involved it seems sensible to divide this
into two distinct principles.
THE GAME MUST BE A ROLE PLAYING GAME OR ADVENTURE GAME . A STRONG FANTASY
STORY STRESSING IMAGINATION DEFINES WHAT IS REALISTIC AND PLAUSIBLE AND
CONNECTS A SERIES OF MATH QUESTS THAT ABIDE TO THIS DEFINITION.
REALISTICALLY GROUNDED QUEST-BASED MATH PROBLEMS
DESIRED MATH PROBLEMS/ACTIVITIES ARE DESIGNED AS QUESTS . QUEST INSTRUCTIONS
ARE SPECIFIC AND PRECISE IN DESCRIBING THE MATH PROBLEM BUT THE QUEST MUST BE
SUFFICIENTLY OPEN TO LET THE STUDENT FIGURE OUT HOW TO SOLVE IT. QUESTS MUST BE
INSPIRED BY THE MATH USAGE OF REAL LIFE MATH PERFORMERS . THE QUEST
ENVIRONMENT AND MATH PROBLEM MUST BE SIMULATED RICHLY AND MUST BE
PURPOSEFUL , PLAUSIBLE AND IMPORTANT IN RELATION TO THE GAME ’ S FANTASY
Note how the discussion shifts between emphasis on fantasy and realism. To clarify, the quests are the
realistic part of the game and their dynamics must be realistic. The story/premise narrative is and
must be the fantastic or super-realistic part of the game. Real life’s definition of what is realistic and
what is implausible is altered by the narrative. Next, an example will be presented to illustrate what is
meant.
Math in Moontown is set on the moon, where the player must aid in the construction of a moon base.
In the main overview of the town, the player can click any unfinished building he wants. Each building
represents a quest containing math problems. One of those math problems are shown below as a
screenshot. To construct a radio tower, the player must gather crystals by controlling a moon car. The
car drives relentlessly ahead and the player must make the car jump over aliens and the rocks that do
not contain the currently desired crystals. The radio tower will only be partially constructed if the
player collects too many incorrect stones/crystals and can then try again until it is properly finished.
50
The text reads: “You must collect 3 rocks that fulfill the following condition: - Stones with at least 2
blue crystals.” In using the design principles it can be argued that within Math in Moontown’s
narrative, it is realistic that there is low gravity that can make the moon car jump. It is also plausible
that a radio tower needs crystals but that only specific ones are applicable for this purpose. And within
the narrative, it is an important task. So this fulfills all of the criteria of Design Principle #6. However,
since there is basically no player character presentation or development in the game, it violates Design
Principle #4. Furthermore, the story works more as a premise, and not as a strong imaginative story.
However, this premise is still strong enough to embed the quests into the game by defining the
boundaries of what is realistic and purposeful to do in this fantasy world so it is doubtful if it violates
design principle #3.
51
Stealth learning
Hidden Agenda was a competition urging students to compete for a $25,000 prize by building video
games that secretly teach middle school subjects.122 The word secret might be a little misleading – the
competition website stated that the game had to be “So fun, in fact, that they don’t notice it‘s also
teaching them something. That’s the “stealth education” aspect.”123
Gee is describing that many children brings a real world identity to science and school learning that is
damaged124. Before active critical learning can take place, some repair work must be done, which can
be achieved by the learner being enticed to try (even if he is afraid), put in lots of effort (even if he
begins with little motivation to do so), and the learner must achieve some meaningful success when
the effort has been expended. This is comparable to the previous chapters’ findings. However, what if
Gee and I will not succeed in such repair work, and the bridge between the learners’ real world school
identities and virtual learning identities cannot be built? If that shows to be more complex than
expected, or if only a partial success is achieved in this aspect, then it might be possible to simply
circumvent the issue. Because, if ‘weak’ school learners are blocked in their learning as a result of their
school identities, then the game possibly should not promote itself too much like a school game or a
learning game. I do not necessarily suggest a stealth learning approach of attempting to manipulate
the children into thinking that they are not learning. But a game named ‘Math Training in Space’ might
sound fun, but also might not appeal to RMEs demand of meeting the learner at their ground.
Especially not if the student’s real world school identity is weakened. The same might be the case for
in-game materials – if one is constantly reminded that now you are learning/practicing math, then that
might also conflict with the design principles concerning reality and fantasy. Mark Prensky came to a
similar conclusion in relation to the development of his 3D learning game ‘Monkey Wrench
Conspiracy’ that should endogenously teach CAD drawing:
“Immediately out went the “You will now learn the following three things,” and in came “Come on Moldy,
you’ve got to do this or we’re doomed.” The words “objective,” learn,” and “know how to” were banned,
replaced with imperative action verbs like “build,” “get through,” repair,” and “rescue.”125
The event that triggered this was that Prensky’s partner on the game tested the game and exclaimed
that it was like fire and ice:
“The mix of learning and entertainment isn’t working, he explained. “The game is fire — it’s fun, fast and
engaging. Then you hit the first learning task. Suddenly you’re back in school. It’s boring.”126
One point in these quotes is that if learning and entertainment is not properly welded together, then
the shifts between each can seem disruptive for immersion – like fire and ice. Awareness of this is of
course a big point of previous chapters, and will be dealt with a little more specifically in a coming
chapter. The other point in relevance is that it feels like coming back to school. This is the essence in
this paragraph, that this precise activation of school identity feeds a feeling of being in school to learn
122
Davis 2003
http://web.archive.org/web/20080617111616/http://www.hiddenagenda.com/index.html acquired at
11.02.2011
123
124
Gee 2003, p. 61
125
Prensky 2001, p. 13 of chapter 1
126
Prensky 2001, p. 13 of chapter 1
52
and can, ironically, be so destructive for the learning process of especially realistic math, that the
students completely rejects the games’ premise.
This does not necessarily mean that a 100% dedication to the stealth learning approach is the way to
go. It is unresolved how students will react if they start playing a game which fools them into thinking
that it is only for fun and then later realizes that it is a learning game. Maybe they will feel deceived or
cheated, which probably is not desirable for active, critical learning. Arguably, the point of stealth
learning is to make the user engage in games while never realizing they have been tricked into
learning. The question is if a realistic math game will be capable of such an illusion, without
overstepping other design principles. On the other hand, a realistic math game might nevertheless for
a student seem like an entertainment game and not like a learning game, even if the realistic game
openly declares itself as a math game. If the game is so successful in leading the player into the role of
e.g. building boats (with this activity’s associated math), then the student’s focus, in pointing to
previous chapters, will not be based on a real world identity of math practice nor a school learning
identity. It will be based on an identity of being a boat-builder.
Based on the above it can be concluded that the game does not need to pretend to not be an
educational game. However, it seems advantageous, like in ‘Monkey Wrench Conspiracy’, to minimize
references to school, learning and student identity.
Representation
It have now been established that a realistic math game must consist of quests with specific
instructions and conveying purpose. But how does one represent such instructions and quest content
in such a way? This will be dealt with in this chapter.
The first relevant question to confront is if text or audio is the best way to represent text in learning
games. According to Ke, a study found that students engaging in GBL scored higher on retention,
transfer and program rating in narration conditions than in text conditions.127
Nunes agrees that oral conditions revolving learning are better (especially in relation to meaningmaking) based on their follow up studies of their market place study: “The children kept the meaning in
mind when solving problems in oral mode and seemed to forget about this (…) in written mode”128.
However, these studies focused on how students solved the problems and that algorithmic approaches
differed in oral and text modes, but they did not focus on how the math problems were represented.
Nevertheless, they conclude that “The closeness of the representation of the situation and the
representation of the arithmetic in the schemas developed in street mathematics simultaneously
preservers meaning and allows for greater flexibility in the routes used to a solution”129. Understanding
what this ‘closeness of the representation of the situation’ is likely most relevant.
Malone argues that graphics and sound is well suited to stir sensory curiosity as a representation
system. “Perhaps the best use of sound and graphics in computer games is to represent and convey
127
Ke p. 14
128
Nunes et al p. 48
129
Nunes et al p. 55
53
information more effectively than with words or numbers. For instance, the Darts game uses a graphic
representation system for fractions and the Breakout game signals bounces and misses of the ball with
different tones.”130
Prensky argues that it’s not only a medium strength, it is a user demand. What he calls the ‘Games
Generation’ has, compared to previous generations, an increased focus on visuals with little or no
accompanying text. “This shift toward graphic primacy in the younger generation does raise some
extremely thorny issues, particularly with regard to textual literacy and depth of information. The
challenge is to design ways to use this shift to enhance comprehension, while still maintain the same or
even greater richness of information in the new visual context.” 131
However, others found that graphics might not be that important for motivation. According to Pivec,
FutureLab found that a common belief is that students require rich 3D graphics to play educational
games, whereas less than 10% of FutureLab’s study participants suggested better graphics as a
motivational factor.
So it have now been developed that audio looks more promising than text and that graphical
representations are powerful and necessary but not necessarily have to consist of rich 3D graphics,
nor act as motivators. I assert that this is still insufficient to cover ‘closeness of the representation of the
situation’. Masingila argues that “For example, students may invent notation to indicate when objects are
the same size and shape, in the course of working in a measurement context, before they have formalized
the concept of congruence.“132 In continuing the example, the students will not be able to invent such
notations if the game is thorough in representing sizes and shapes by formalized representations like
numbers. According to RME, whenever possible context-realistic math examples should decrease the
appearance of formal representations, like numbers and formulas. RME would i.e. represent the
number 5 as e.g. 5 candles in a cake. The way to understand representations as models of reality
instead of representation as models of constructions like numbers is related to the concept of a model.
Using models might engage the challenge but that requires a greater understanding of models. A way
to grasp this challenge could be to follow Bogost’s advice concerning using models to enforce
procedural rhetorics. He compares descriptions to models with an example of attempting to learn the
orbits of planets from a textbook or lecture descriptions versus learning it from an orrery. An orrery is
a mechanical model of the planets on a system of gears that models their rotations and orbits at the
correct relative velocities133. The orrerey represents the solar system by serving as a model of it
instead of describing it. In relation to video games, a model would most likely be more immersive if it
was a rich 3D simulation, and consequently more rhetorically strong in its procedurality. Such an
approach will also be able to satisfy Prensky’s ‘Games Generation’. However, the question still stands
regarding how one exactly model or simulate the ‘closeness of the representation of the situation’.
130
Malone 1980 p. 166
131
Prensky 2001, p. 56
132
Masingila p. 11/13
133
Bogost p. 257
54
In relation to word problems, it was concluded that students’ solution strategies did not involve a
careful analysis of the task situations but instead they focus on the numbers given in the task134. This
conclusion was not explored further by the author, but one explanation could be could be that they
focus on the numbers in the task because it have numbers! If a math problem in a realistic math game
quest concerns figuring out how many cars each capable of holding X people are needed for
transporting Y persons, the visual cues of the word (or audio) problems within the game (the quests)
might alleviate the problem of attaching meaning to results since visualizing i.e. a half jeep might be
harder when the jeeps are visible on the screen. An example of a game situation where the visual
representation of the situation is lacking is one of the quests of Math in Moontown. In the below
example the player must figure out how much asphalt must be used to cover 60 meters of road where
100 kilos are needed for each meter. The student can then use the + and – buttons on the interface to
input their result. However, the problem of attaching meaning to results could be alleviated if the game
was showing a pile of asphalt that increases in size whenever the student clicks the ‘+’. This will be a
visual aid to the student but will not give the answer. This way, the student might get a feeling that
something is wrong if his calculations was wrong and the pile grows to extreme heights.
134
Palm 2007, p. 50
55
Consequently the game must, whenever possible, use representations imitating natural
representations. A simple way of explaining what is meant by such natural representations would i.e.
be for representing the weight of stones - to show that a stone is bigger than some other stone by
letting the drawing of it be bigger than the smaller stone. If one rock is heavier than another rock then
do not let that be clear by printing the rocks weight as a number on it. There are several other
possibilities, i.e. let the rock be larger, let it have another color or let the player be able to operate a
scale and he can then read the number off the scale and thereby get the rock’s weight in a way that
makes sense and abides the plausibility aspects of design principle #3. Such representation of size is
also in concordance to Bogost’s modeling example. This approach might be able to mimic visual and
auditory ‘closeness of the representation of the situation’ found in street mathematics.
In the above figure, yet another quest of Math in Moontown is shown. In this instance, the player is
supposed to shoot all asteroids that weigh at least 1.5 tons, otherwise they will crash into the
spaceship and damage it. However, the player only has a limited amount of energy for shooting which
56
forces him to not shoot too many of the harmless stones. This example is beautifully obeying the
previous design principles, i.e., it makes mathematical and narratively sense that the player cannot just
shoot away at all the asteroids, but the representation is not making sense – they mimic descriptions
instead of models. If accuracy is desired, as in the example, then it might quickly become impossible
for students to distinguish between a stone of size 1.46 and a stone of 1.51 if the stones are
represented by size instead of numbers – they would look almost the same. In this situation, a modular
construction approach could be suggested. For example, the representation of the weight 1.46 could be
naturalized to instead be a cluster of stones where 1 stone is big and black, 4 are medium and blue,
while 6 are small and green. Another solution would be to identify why one even would want such
accuracy in the game. In the example above, the purpose is to teach students to understand concepts of
larger than and lesser than, while practicing understanding decimal numbers. Teaching this might be
possible without such representation, however the two purposes in the above task might have to be
split into two separate quests. In some situations it might be impossible to find ways to meaningfully
transform the representation of numerical values. In this case, this design principle can be ignored.
Summarizing this chapter suggests that RME games as much as possible should use other depictions of
e.g. quantity, size, and weight than the depictions of the numerical system.
REPRESENTATION
MATH PROBLEM QUANTITIES SHOULD BE MODELED AS IN-GAME ELEMENTS REPRESENTING
PHYSICAL PROPERTIES OF THE PROBLEM CONTENTS WHILE MINIMIZING OCCURRENCE OF
NUMERICAL REPRESENTATIONS.
57
Interactivity
In the previous chapter it was investigated how representation could explain the differences in math
performance in street and school math situations. In example, the previously presented study about
students who found it easier to engage in calculations concerning carpentry when given planks and
saw could be explained by the representation design principle. The physical planks modeled or
represented sizes better than numbers in addressing their senses. However, what if the explanation
could also be found in the way students can interact with these physical objects? This will be the topic
of this chapter.
The study concerning foremen vs. students is perhaps not merely explained by differences in factors
concerning identity, social context or representation. Possibly, since the foremen had sufficient
experiences in construction giving them realistic expectations when interpreting arithmetic results,
and enabling to visualize the resulting walls, they were more correct than students. No foreman in his
right mind would propose building a 33cm long room! One interpretation of this result is; since the
foremen have been working within this domain and received feedback from the real environment of
the construction sites, they can use this feedback in interpreting the results. The students, however,
inexperienced in giving any important or meaningful or consequential interpretation of their results,
fail the task. What is proposed here is not only that the foremen succeed since they are experienced in
real life tasks, but also that the real life tasks gives a special feedback when interacting with physical
objects. The rooms in the construction site where the foreman is responsible will not only look silly
with 33 cm walls, it will also trigger an ‘innate’ feeling that something is wrong as he grabs a 33 cm
board with his hands. Likewise, it can be imagined that a math performer on a street marketplace
might miscalculate the amount of apples to give to a customer who wants as many apples he can get
for 2 US$, but he will know that something is wrong when he starts to put 200 apples into a crate for
the customer. This notion, that not only the situation or representation, but the processes or
procedures of e.g. a construction site or goods trade gives tactile embodied experiences, is important.
The effect actions have on the environment gives a special feedback that helps street math performers.
This concept is comparable to what Isbister calls Visceral Feedback. “Facets of visceral feedback include
what sorts of physical power the character has, how it feels to control them and to move through the
world, and the general effects that actions have on the senses”135 Visceral Feedback is not only a way to
think of how senses are affected, it is also a way of thinking how player interaction takes place – how
does it feel to control the character and how his actions affect the game environment. If the effects
actions have on the environment also are important in street math, then it should be grasped in a
realistic math game
This entails that a realistic math game engaging the player in e.g. building construction must not only
capture the realism of the construction site, but must also provide the realistic interactivity or
manipulability of the physical objects that are involved. Consequently, it would be insufficient in
regards to realism to have a construction yard game where a simple pop-up message asks the player to
enter the number of inches to cut off a board for a wall. This is not the case only because it conflicts
with the representation design principle. A better solution would be to try to simulate the properties
of the board and the saw. Let the player grab a saw and a board, let him go to a saw-bench, let him
measure the board, let him saw and hear the sawing sound and see the saw dust fly around, let the
135
Isbister 2006 p. 204
58
game controller vibrate (if possible) as he saws and finally let the cut off piece drop to the floor with
clear audible results. Doing it this way is both realistic and plausible. An associated approach to this is
the use of manipulatives in the classroom. A mathematical manipulative is by Wikipedia defined as “an
object which is designed so that a student can learn some mathematical concept by manipulating it.” 136
One example of a manipulative that is popular in Danish schools is the Centicube, which are bricks of
1x1x1 cm that can be connected to other bricks on all sides. The tactile interactivity of Centicubes is
capable of not only representing numbers and volumes graphically but can also embody the number
experience since the numbers can be touched, felt, bent and pulled apart. It is this embodiment of
numbers which I propose could be helpful for a realistic math game. Again, not only for the purpose of
alternative representation but for the possibility to handle, grab, move and interact with the objects. It
is these characteristics that I propose should be simulated as best as possible. Making a game not
abiding to this proposition is comparable of making a golf sports game for Nintendo Wii and let the
player only use the buttons on the controller to play golf, instead of letting them swing the NunChuck
frantically around. Such a golf game would possibly convey an acceptable game experience; however it
would give up much of the potential ‘game feel’ of the game and would not exploit the medium to its
fullest. The physical feeling is minimized; that of being a golfer that powerfully and with pin-point
precision can control and swing the driver and hit the ball and knock it far away. The visceral feedback
is amputated. Thus, games exploiting qualities of platforms like Nintendo Wii, Kinect, Augmented
Reality (AR), virtual reality, shaking smartphones or force feedback joysticks might be able to lift this
task better than traditional ways of game control. However, as schools of today are extremely limited
in their computer equipment and that not all students can be expected to have smart phones, it is
necessary to consider how this interactivity principle can be simulated with standard personal
computers.
The game should not help too much
One point of entry into understanding this is to consider the amount of indirect help the computer
provides in gaming. In some games, many in-between actions are performed by the game to let the
player move quickly to the focus gameplay. In other games, the focus gameplay is on the in-between
actions that other games would skip over. Example: In the classic game Sim City the focus gameplay is
not on the actual construction of buildings, instead it is on finding the best plot for the building and
then the construction happens automatically. In other city games there might be no worries about
plots, prices or location in regards to neighborhood etc. Instead the focus gameplay is on actual
building design and construction. While the other factors are present and simulated in the game, they
are taken care of by the game. The point is here that this concept is relevant since if realism must be
topped and if some math assignment that would otherwise be a bit tedious needs to be spiced up, then
a realistic game designer must consider what the focus gameplay of the quest is and how the game
should help or not help. For example let’s consider a math quest where the purpose is to multiply
contents of a food recipe to figure out how much of each ingredient is needed to feed X people. It might
be more fun to use the mouse to click, drag and drop the ingredients into the pot one by one than to
either; A: Click the ingredients one by one making them instantly appear in the pot or B: Numerically
input the amount of each ingredient in a popup window.
136
http://en.wikipedia.org/wiki/Mathematical_manipulative acquired 02.04.2011
59
Such balancing of what could be called game laziness vs. player involvement is possibly an old topic
within game design; however I was unable to find any sources directly dealing with this. Nonetheless,
here it serves a new purpose which is to increase the feeling of realism in benefit of better learning.
Conversely, forcing the player to perform too much of this manual interactive labor might make the
task seem tedious or lengthy.
INTERACTIVITY
PLAYER GETS RICH OPPORTUNITY TO INTERACT WITH SIMULATED PHYSICAL OBJECTS THAT
ARE INVOLVED IN MATH PROBLEMS . MANIPULATING SUCH OBJECTS CORRECTLY CAN BE A
MEAN TO SOLVE THE MATH PROBLEM .
60
Feedback and instructional support
Although the constructivist learning approach suggests learning by doing, it does not make a teacher
superfluous. The teacher acts as a mentor or guide who facilitates learning. Within GBL, the game is
the teacher. Consequently it should be considered how an important teacher tool, feedback, could be
organized within a realistic math game. It has already established that quests and math concepts must
make sense within the story-line. This chapter will treat the question of how that will influence ingame feedback, e.g. in relation to rewards.
Feedback in learning games is sometimes understood as split between game-wise feedback and
learning feedback:
“Most game design studies indicate significant results. A common finding extracted from these design
studies is that instructional support features are a necessary part of instructional computer games. The
studies generally conclude that learners without instructional support in game will learn to play the
game rather than learn domain-specific knowledge embedded in the game.”137
This suggests that if learning feedback is impotent, then the game feedback will dominate the
experience, effectively letting the student only learn to play the game. That is the reason why internal
instructional support features are a necessary part of instructional games and should be embedded
within a game through elaborative feedback, pedagogical agent and multimodal information
presentation138.
Malone argues that feedback is twofold – in relation to the game-wise feedback it must be surprising,
and in relation to learning it must be constructive139. However, I will suggest that these two forms of
feedback can be integrated: That the instructional feedback can be integrated directly into the
gameplay and narrative. The first assertion is that that the consequences of calculating correctly or
incorrectly in the game should have some consequences similar to the consequences of calculations in
the real world.
Gee presents an imaginary game where students of architecture have to learn a complicated 3D
architectural drafting system. If this game operated like a good video game then the player’s
understandings of the system’s words, symbols and procedures would have to be embodied in
materials, images and actions in the game’s virtual world. The effect would be that “the player would
have to actively assemble these understandings on the spot and face real consequences in the virtual
world for these assemblies. In fact, it is these consequences that allow the player to test whether the
situation- and action-specific meanings he or she has constructed are viable or not.”140
This idea explains why it is important with feedback that makes sense and demonstrates
consequences of the actions. Otherwise the player will not start to experiment and test the strength of
this new situated meaning. In transforming this idea to a realistic math game, attention must be turned
to street math performers again. There is consequence and responsibility involved when children
work in the marketplace. Assume that if they are good at calculating, the family business prospers and
the child will see the effects of that. It means that it is more than just feedback like correct or incorrect,
137
Ke 2009, p. 21
138
Ke 2009, p. 23
139
Malone 1980, p. 166
140
Gee, p. 86
61
the result manifests directly in their surrounding environment. Likewise, the feedback or
consequences for a marketplace child that had developed a sloppy approach to calculating and giving
change to customers would not just face the question whether the calculation was correct or faulty, the
child and family would in worst case see the consequences manifest in their immediate world when
they are forced to sell their market booth or house because of continuous losses over time. The
example might be exaggerated but it is to show the point which is that many games forget to simulate
this richness of feedback provided by realistic scenarios. Many games simply let a popup message or a
voice utter ‘Correct – good job’ if the player calculated the correct answer. I propose the term
procedural feedback to describe the concept of emulating the rich consequential feedback of out-ofschool math. If a game is about designing an automatic valve that controls the water level of a
swimming pool, then it would be a poor procedural feedback solution to just let the player know
whether they calculated correctly or not. A higher degree of procedural feedback would be, in case of
miscalculation, to let the water flood the yard of the customer resulting in the player character to get
fired from his job. This approach might sound rough and could conflict with Gees design principle
called ‘Psychosocial Moratorium’. “Learners can take risks in a space where real-world consequences are
lowered”. If the in-game consequences of failure rise, then player risk-taking frequency might decrease.
Nonetheless, this is often a central design fact in many ordinary video games – if you mess up, you die
and have to start over from the beginning or the last saved game.
I do not suggest that simple feedback is not effective. In fact, a great deal of research has gone into
motivating and molding humans by somewhat simple feedback. The thought in mind is of course
operant conditioning, which is the major or sole motivator in many games, especially exogenous,
extrinsic fantasy games. The discussion whether one kind of motivational feedback/reward is better
than the other is irrelevant and will be skipped elegantly since it already have been established that a
realistic math game must be endogenous and motivate its player’s intrinsically to learn, and not by
extrinsic reinforcement like grades or points. However, extrinsic rewards can be useful for nonlearning purposes of a realistic math game. Within the story it might prove relevant to include
extrinsic rewards like collection of artifacts to motivate the player in advancing the story and to act as
a gate-keeping device to prevent player’s advance before certain tasks have been completed. Likewise,
in RPGs, character development often revolves around rewarding the player with experience points
that occasionally increases ‘character stats’, which can be explained as minor improvements of the
character’s attributes that can surmount in creating increasingly more potent characters over time.
Usually in RPGs, such rewards are given upon successful completion of quests. Consequently, extrinsic
rewards like scoring, experience or artifact collection should not be an integrated motivator of quests;
the purposeful completion of the math within the quest should be sufficient, intrinsic reward. But
completion of the quest can still reward an extrinsic reward if that is meaningful in carrying the story
or narrative ahead, but should not be included just for the sake of having a reward. An example of this
is the score-keeping of the Math Rider game that was previously explained. The scoring in Math Rider
serves no purpose to the game and is not intrinsically connected to its story.
Instead, rewards, or feedback, should be procedural in relation to the quest and show what
consequences your math performance have on the game environment and narrative. If a quest
involves calculating how to properly construct a boat, then the negative procedural feedback in the
case of miscalculation could be that the boat is sinking when you test it, making you lose precious time
in the pressing story. In the case of correct calculation, the positive procedural feedback will surmount
in a boat that simply allows the player to sail her merry way to new areas and continue exploring the
62
game. A similar application of this design principle is in several of Math in Moontown’s quests where
intolerable quest solution, of e.g. failing to protect from asteroids the spaceship carrying building
materials for a house, entails incorrect building constructions. However it is important to recognize
that the player still needs ‘traditional’ detailed feedback if he/she wishes to know why the calculation
was wrong. In other words, it is necessary to provide elaborative feedback to the player so she is able
to progress mathematically and learn from her mistakes that lead to the boat sinking.
PROCEDURAL FEEDBACK
FEEDBACK ON MATH PERFORMANCE IS INTRINSICALLY INTEGRATED INTO THE NARRATION
AS REALISTIC CONSEQUENCES THAT MANIFESTS INTO THE GAME WORLD , COMPARABLE TO
THE REAL LIFE CONSEQUENCES OF THE TASK THAT THE QUEST WAS INSPIRED FROM .
FURTHER INSTRUCTIONAL SUPPORT SHOULD BE ELABORATIVE, PEDAGOGICAL, MULTIMODAL
AND CONSTRUCTIVE .
Difficulty balancing
This chapter will deal with balancing the difficulty level of game-wise challenges like defeating
monsters or controlling the character precisely. Likewise, learning challenges like difficulty and
complexity of math will be treated, as well as the interplay and balancing between the two.
One of Gee’s design principles is addressing this: The ‘Regime of Competence’ principle: ”The learner
gets ample opportunity to operate within, but at the outer edge of, his or her resources, so that at those
points things are felt as challenging but not “undoable”141.This is a basic constructivist learning
principle, and in example central in Vygotsky’s theory of the zone of proximal development. Parts of
this are also comparable with Csikszentmihalyi’s idea of motivation flow. As stated in delimitations, I
will not enter a thorough treatment of flow – the art of balancing game difficulty so the challenges of
the player neither makes her frustrated nor bored. I refer to Tracy Fullerton’s ‘Game Design
Workshop’ for general game design criteria and tips regarding creating good gameplay and game flow.
Likewise I refer to literature concerning teaching differentiation in the question of adapting teaching
material to various types of learners.
However, the difference between difficulty in learning and difficulty of the game must be addressed.
Prensky’s solution is as follows. ”Personally, I think it would be great if every learning program in the
world had a little slider control, always sitting in the same place like a car’s temperature controls, which
let each user (or trainer, or teacher, or parent) set the mix of edu and tainment for themselves.” (…)
“Perhaps separate controls for game difficulty and learning difficulty (kind of like bass and treble) might
work even better.“142
Another solution could be to let the game automatically adapt to the player – if it registers that a
player is struggling, it will automatically decrease the difficulty of the game challenge or the learning
141
Gee 2003
142
Prensky 2001, p. 381
63
challenge. No matter if it is by ‘slider controls’ or automatic adaption, it must be guaranteed that the
difficulty of game parts does not constrict the learning process. What is meant is that in balancing the
difficulty level of the game, it must be absolutely guaranteed that if i.e. little Bob is unable to complete
or progress in the learning game, that it is not because he was unable to defeat some monster or failed
to control his character to perform some difficult jump over a trap. In other words, game difficulty
must either be automatically balanced or the student must be able to turn down the difficulty of NPC
enemies etc. if they are struggling. Otherwise, some students who might be capable math performers
will underperform as a consequence of their inexperience with gaming in general and not as a
consequence of their underperformance in math. Thus, unless the game’s target group is extremely
narrow and i.e. only focus on teaching math to Uzbek 5th grade boys with high gaming experience, the
game must have adaptive content or the player must be able to choose his/her skill level or a mix of
both approaches.
In a way, finding how to balance the difficulty of learning and finding the difficulty of game-play is the
same task since, as Bogost puts it, “Learning is inseparable from the learner’s interaction with the
environment.”143 This view is in concordance to constructionist views; however in this case it might
only cover learning in relation to learning to play the game, i.e. mastering the game controls or figuring
out how to defeat opponents. But some game and learning elements will definitely be mixed up,
especially when taking a quest approach as the quest must both lift and connect the gameplay and
narrative elements while exposing meaningful math content. Consequently it can be hard to use
Prensky’s ‘slider control’ to affect the difficulty of the game as it will also affect the difficulty of the
curriculum and vice versa.
In relation to this, another question is relevant to revisit: Should it only be possible to progress
through the game if the player correctly answers the math challenges? In other words, should there be
‘doors’ in the game that cannot be passed before you give the correct answer and prove that you have
learned? The Danish video game researcher of the Danish university of pedagogics Morten Misfeldt
said in his keynote of a 2010 GBL conference that
”… Og så skal man altid være opmærksom på at have nogle ret klare læringsmål og have en rigtig god
sammenhæng mellem læringsmål og spilmål eller hvis du spiller godt, hvis du klarer dig godt og spiller
fornuftigt, så skal det være et tegn på at du faktisk har lært det som man skal lære af spillet. Så de to ting
skal spille sammen ret tæt, ellers kan man få sådan en dum situation hvor man bare sidder og trykker
enter for at komme igennem bogen (sic) og ikke rigtig lærer noget.”144
My translation is:
”… And then one must always be careful to have some pretty clear learning goals and have a really good
connection between learning goals and game goals or if you play well, if you perform well and play
reasonably, then it must be an indication that you actually have learnt what one should learn from the
game. So the two things must play together really tight, otherwise you can get such a stupid situation
where one just sits and pushes enter to get through the book (sic) and not really learn anything.”
143
Bogost, p. 234
144
Misfeldt
64
If one adopts Misfeldt’s idea then it will be imperative to consider this in relation to the
Representation principle. If the math is represented too simply, then a game can quickly become too
easy, in example if the arithmetics can be solved by counting objects instead of finding the total by
multiplication or addition. Testing of the game should address if the balance of the quests opens up for
‘cheating’ – to test if the game can be completed by just playing the game and not engaging the
embedded curriculum.
Summarily, it could look like a mixed approach, a compromise is most beneficial.
Some quests will have math and game difficulty so closely integrated that separating them will not
only be hard, it will be counterproductive as it could promote ‘cheating’. Nevertheless, other parts of
the game where math is not central in acting as a gate-keeper or quests involving no math at all should
automatically or manually be adaptable to the learner. An example of this could be that the player can
change the toughness of monsters (like in many games, e.g. action games where the player can choose
easy, medium or hard). This should only affect ‘bridge’ monsters that as the part of the story and to
plug entertainment acts as connections between different quests. If a specific quest involves utilizing
mathematical tools optimally to defeat some special monster, then this monster should not be affected
by the difficulty setting, otherwise the player can possibly skip over the learning goals of the quest to
easily.
Thus, in quests that are intrinsically combining hard math challenges with demanding game
interaction should pace down the game interaction challenge as to not punish non-gamers.
BALANCE
QUESTS THAT ARE INTRINSICALLY COMBINING HARD MATH CHALLENGES WITH DEMANDING
GAME INTERACTION SHOULD PACE DOWN THE GAME INTERACTION CHALLENGE LEVEL AS TO
NOT PUNISH NON -GAMERS . ADAPTION OF DIFFICULTY OF SUCH QUEST TYPES SHOULD NOT
BE POSSIBLE TO PREVENT CHEATING , BUT OTHER QUEST TYPES AND GAME CONTENT MUST
BE ADAPTABLE TO ADDRESS DIFFERENT PLAYER TYPES.
65
Sub-conclusion
The findings of Lisa Galarneau can be used to sum up many parts of the theory chapter. “We can start
by building on a fundamental component of constructivist learning approaches: the idea that a learner is
challenged to construct their own knowledge via an ‘authentic’ learning experience.” An authentic
learning experience is to “Place the learner firmly at the centre of the learning experience, encourage
him or her to take an active role, and make sure that the learning situation is not abstracted from reality,
but is placed directly in a real-world context, either physically or virtually”145
The theory chapter was used to conclude that, in realistic digital math games, math should be taught
like street math and decrease focus on how math is taught in classrooms and textbooks today. This is
done by attempting to identify the positive factors of street math and RME and investigating how these
could be translated into gameplay. A RME game must immerse the player, not through focus on fun or
spectacularly immersive graphics and sounds, but through a focus on realism, purpose, meaning and
consequential feedback. The support for this postulate is resting on two pillars: The performance of
informal street math practitioners and the Dutch theory of Realistic Mathematics Education. The
discussion surmounted in the below heuristic design principles to be used in creating games that
grasps such authentic learning experiences.
THE GAME MUST BE DESIGNED AS AN ENDOGENOUS , INTRINSIC FANTASY GAME.
PURPOSEFUL ACTIVITY
THE GAME MUST IN ALL ASPECTS AND ESPECIALLY IN ITS MATH PROBLEMS REFLECT THE
PURPOSE OF SOLVING THE TASK AND THE PURPOSE OF UTILIZING MATHEMATICS IN WAYS
THAT ARE IN CONCORDANCE WITH PURPOSE -DIRECTED USE OF MATH IN REAL LIFE
THE GAME MUST BE A ROLE PLAYING GAME OR ADVENTURE GAME . A STRONG FANTASY
STORY STRESSING IMAGINATION DEFINES WHAT IS REALISTIC AND PLAUSIBLE AND
CONNECTS A SERIES OF MATH QUESTS THAT ABIDE TO THIS DEFINITION.
ANALYSIS OF THE GAME ’S TARGET GROUP MUST BE THOROUGH ENOUGH TO ENABLE
CATERING FOR THE PARTS OF THE TARGET GROUP ’ S IDENTITY THAT CAN INCREASE
MOTIVATION AND IMMERSION. SUCH CONSIDERATIONS MUST BE INCLUDED IN EARLY GAME
DESIGN, ESPECIALLY IN THE DESIGN OF PLAYER CHARACTER BACKGROUND AND ABILITIES .
FAILURE TO ADDRESS THIS SATISFYINGLY WILL IMPERIL LEARNING OUTCOME QUALITY .
POSSIBILITY OF MEANINGFUL IN-GAME OR OUT -GAME INTERACTION , COLLABORATION AND
COMPETITION BETWEEN PLAYERS IS DESIRABLE , BUT NOT DETRIMENTAL .
145
Galarneau 2005, p. 3
66
DESIRED MATH PROBLEMS /ACTIVITIES ARE DESIGNED AS QUESTS . QUEST INSTRUCTIONS
ARE SPECIFIC AND PRECISE IN DESCRIBING THE MATH PROBLEM BUT THE QUEST MUST BE
SUFFICIENTLY OPEN TO LET THE STUDENT FIGURE OUT HOW TO SOLVE IT. QUESTS MUST BE
INSPIRED BY THE MATH USAGE OF REAL LIFE MATH PERFORMERS . THE QUEST
ENVIRONMENT AND MATH PROBLEM MUST BE SIMULATED RICHLY AND MUST BE
PURPOSEFUL , PLAUSIBLE AND IMPORTANT IN RELATION TO THE GAME ’ S FANTASY
REPRESENTATION
MATH PROBLEM QUANTITIES SHOULD BE MODELED AS IN-GAME ELEMENTS REPRESENTING
PHYSICAL PROPERTIES OF THE PROBLEM CONTENTS WHILE MINIMIZING OCCURRENCE OF
NUMERICAL REPRESENTATIONS.
INTERACTIVITY
PLAYER GETS RICH OPPORTUNITY TO INTERACT WITH SIMULATED PHYSICAL OBJECTS THAT
ARE INVOLVED IN MATH PROBLEMS . MANIPULATING SUCH OBJECTS CORRECTLY CAN BE A
MEAN TO SOLVE THE MATH PROBLEM .
PROCEDURAL FEEDBACK
FEEDBACK ON MATH PERFORMANCE IS INTRINSICALLY INTEGRATED INTO THE NARRATION
AS REALISTIC CONSEQUENCES THAT MANIFESTS INTO THE GAME WORLD , COMPARABLE TO
THE REAL LIFE CONSEQUENCES OF THE TASK THAT THE QUEST WAS INSPIRED FROM .
FURTHER INSTRUCTIONAL SUPPORT SHOULD BE ELABORATIVE , PEDAGOGICAL, MULTIMODAL
AND CONSTRUCTIVE .
BALANCE
QUESTS THAT ARE INTRINSICALLY COMBINING HARD MATH CHALLENGES WITH DEMANDING
GAME INTERACTION SHOULD PACE DOWN THE GAME INTERACTION CHALLENGE LEVEL AS TO
NOT PUNISH NON -GAMERS . ADAPTION OF DIFFICULTY OF SUCH QUEST TYPES SHOULD NOT
BE POSSIBLE TO PREVENT CHEATING , BUT OTHER QUEST TYPES AND GAME CONTENT MUST
BE ADAPTABLE TO ADDRESS DIFFERENT PLAYER TYPES.
Below, the design principle has been concretized into an example that revolves around the subject of
calculation in relation to cutting boards for building a boat. The list describes how individual parts of
such a math quest can be either in low or high concordance to the various design principles. Some
principles have been left out of this quest example as they are on a more abstract meta-level, like
design principle #1, while others have been left out since they require target group analysis (design
principle 4). This example is merely meant as a demonstration appetizer that also functions as a
summary in exemplifying some of the design principles. The game design presented in the next
chapter will demonstrate a full implementation of all design principles.
# 2: Purposeful Activity Principle
o
High: You must calculate correctly to build the boat; otherwise you won’t be able to
pursue the evil pirate ship that kidnapped your family!
67
o
Low: You must calculate correctly to build the boat, otherwise you will lose a point
#3: Fantasy Story Principle
o
High: Cut wood for building a boat that can carry one person across an ocean. Make
sure to cut all pieces correctly, or the boat will be too weak.
o
Low: Cut some wooden boards now
#3: Fantasy Story Principle - Make sense
o
High: Wood for a boat of lengths 10cm – 10 meters.
o
Low: Wood for a boat of lengths 782364 meter or -1 meter
#7: Representation Principle
o
High: Wooden board length can be estimated by observing the boards. Perhaps a
measurement device is given to the player, or perhaps a reference object is near the
board
o
Low: Each wooden board has its length printed on it
#8: Interactivity Principle
o
High: Player moves avatar to pile of wood, picks up wood, uses measurement device
on board to find its length, moves it to a saw and positions board on saw feeder to cut
the desired amount, cuts the board by starting the saw, moves the two pieces of boards
away,
o
Low: A popup box appears: “How much will you saw off this board? Enter a number”.
#9: Procedural Feedback Principle
o
High: Initially player will not know if she cut the boards correctly, but the player’s boat
sinks if the boat’s boards have been miscalculated.
o
Low: Player gets immediate feedback after sawing a board with a popup on screen that
tells if board was cut correctly or incorrectly.
68
Deviating from the model
In some cases it will be necessary to deviate from some of the points suggested in this model. This
could for example be that if you make a quest where the intention is to train the conversion between
fractions and decimals, then it would be counterproductive to abide too seriously to the
representation principle. Unless you can come up with some clever way of minimizing the amount of
formal representations and still teach formal representations! Other cases can be to deviate from the
procedural feedback principle and the interactivity principle in cases where you want a quest to be
quick and simple. This is expected to be acceptable as long as the majority of the game abides by the
majority of the principles.
Discussion
This model is based upon theory presented in the theory chapter. Much of this theory has been created
for very different purposes than digital GBL. In the model it is asserted that many finds, for instance
the research findings concerning word problems, are valid when transferred to the video game
domain. However, I have chosen to focus on the aspects of theories that are relevant for realistic
games. For instance, word problems do usually not have graphics and cannot be considered as
procedural rhetoric even if they sometimes require a somewhat interactive or algorithmic approach.
But understanding the pitfalls and silver bullets of this domain might improve understanding of
pitfalls and silver bullets in related domains, like GBL.
Developing games in concordance to this model is not as cheap as developing standard math
edutainment games, since in this model; nearly nothing can be reused or interchanged within different
games and curricula. However, this seems to be the price to pay to reach the goal of efficiently teaching
via authentic experiences.
The model treats so many different topics that several of them are only treated superficially. This
makes the model wide in its grasp and coverage, but not especially deep. For instance, it would have
been interesting to drill deeper into motivation and rewards in GBL. Unfortunately, due to limited time
for a project of this type it would have been impossible to both gain width and depth of the model.
Several of the street math studies focused on working class, low-schooled individuals from Brazil.
Consequently their findings might not be valid in a western country like Denmark or other EU
countries or USA. However, other studies extended and supported their findings to other population
groups. Nevertheless, it might be relevant to consider if the realistic math games based on the design
model proposed here will be more beneficial or relevant for children who are underperforming in
school, whereas elite students might think of the game as waste of their time. Possibly normal or elite
students are more used to abstract thinking and prefer numbers and representations over real life and
contextual examples. Possibly only the low attainers need the focus on purpose and meaning. This
hypothesis will be addressed when the game design is tested.
Furthermore, this model’s focus on contextualizing and concretizing concepts might make it
unsuitable for teaching abstract mathematics. This is however intentional as the primary and early
69
secondary school curricula of RME focus on concrete and representational concepts while gradually
introducing more and more abstract representations.
One could argue that this model is a sort of self-fulfilling prophecy since its theoretical and
interpretational components are used to confirm soundness of the inclusion and validity of other such
components, while these other components returns the same reassurance back to the original
components. The risk is that this process constructs a bubble of artificial knowledge that has nothing
to do with the very reality that the model attempts to address. This is why it is imperative to test the
model – to find out if the assumptions and decisions made in the model were sound. Furthermore, as
shown in some of the examples concerning Math in Moontown, the model can be utilized as a game
analysis tool as well as a game design tool. However, the intended purpose of the model is not as an
analysis tool, the analysis examples presented earlier were intended to act as demonstrators of
particular design principles.
There is one possible disadvantage to RME in relation to its constructivist approach to meaning
creation. RME146 suggests that meaning of math problems should be negotiated in the classroom.
However, there can be issues regarding this approach. One example illustrates this: 12 year old low
attainers where presented with a problem concerning a school party:
“There are 18 bottle of cola for 24 students and the bottles must be distributed over the tables fairly,
taking into account the different numbers of students at each table (tables with 1, 2, 4, … students). What
was intended as the task was the production of equivalent ratios (bottles per students). Some students,
however, did not want to interpret the task in this way. They thought equivalent ratios of the bottles per
students was inadequate because ‘Some students don’t drink cola’ and also ‘They don’t drink the same
amount’.”147
This shows that the chosen contextual example was successful in enabling the students to accept the
context as relevant, and thereby engage them in solving it. But perhaps their interpretation was a bit
too realistic when proposing solutions and interpretations as above. This is of course somewhat
changed for math context problems within video games, since solutions will be limited by the design.
But it shows the importance of identifying the various interpretations that students can create when
encountering the quests. This must then occur within the testing phase of the game, for example by
asking children to explain their interpretation of the math problems and solutions and then adjust the
game problems accordingly. This is yet another reason why the design principles of the model should
be implemented into an actual game and then test the game. This will be dealt with in the next chapter.
Still, due to the extent of theoretical analysis and discussion with this chapter, this model could also be
viewed as a stand-alone model, valid in its own right to the degree of limitations imposed by its
associated methodology.
146
147
70
GAME DESIGN
The design principles specified in the previous chapter were central in the development of the game.
How they manifest through the game play and design decisions of the produced game are described in
this chapter. In addition to the design principles, Fullerton’s ‘Game Design Workshop’148 as well as
observation of countless other games was consulted in many game design decisions.
Technology, resources and process
Game Maker 8149 was utilized as development tool and framework. This is an excellent game
development tool for simpler games e.g. since resources are easily imported and coding like collision
detection is available and easily launched. Audacity150 was used to edit sounds.
Game Maker’s built in graphic editor was used for graphics editing.
A limited approach to paper prototyping was utilized to work out quest scopes and mechanics.
The built in debugger of Game Maker 8 was used for testing and no dedicated test tools were used.
At various stages of the game development, a series of mini-tests were performed to address issues
not related to the model and administered to various volunteer testers. Some testers were individuals
within the target group range while others were outside of it. These tests were conducted already
from early development and until final version release. These tests were set in informal settings and
focused primary on usability and understanding. Several problems of control were identified and
fixed, bugs were found and fixed, spelling mistakes corrected and logical or formulation ambiguities
within the math problems were straightened out. Finally, a stable version was achieved and ready for
testing with the design principles in mind.
The entire game idea and programming was made by Jan Borg. However, some frameworks and code
fragments were adopted from other projects to gain extra functionalities while decreasing my own
personal development burden, like the menu system and the inventory system. Furthermore, nearly
all graphics and sounds used were created by others. See full credits in the appendix of this report or
inside the game. Development of the game and inclusion/exclusion of certain game ideas was to some
extent driven by the availability of certain resources like graphics.
148
Fullerton 2008
149
http://www.yoyogames.com/make
150
http://audacity.sourceforge.net/
71
Math within the game
The math book MatematikTak was used to ensure the math problems were within the scope of the 5th
grade Danish curriculum. Also, the official ‘Fælles Mål’, which is the national curriculum guidelines,
was consulted to ensure relevance and backing of the game’s math topics and difficulty level. Some
math was included that might be in the lower end of 5th graders (like somewhat simple subtraction
tasks) and other tasks was in the upper end and might be beyond 5th graders normal capabilities (e.g.
area of circle). This was intentional and the lower end tasks were included to ensure a common
ground and an easy introduction for all students no matter how skilled they were. The higher end
tasks were included to test the hypothesis that well-made realistic math is capable of letting students
engage in math that they otherwise would not be able understand or solve. Furthermore, the hardest
math was placed at the end of the game to see if the story based approach was capable of motivating
students to keep on playing even if they were encountering increasingly difficult challenges.
Design Principle integration
The design principles were obeyed and their influences on various game design aspects are explained
below.
The game was designed as being intrinsically connecting curricular aims with its fantasy. An
endogenous approach to knowledge and learning was pervasive during development.
PURPOSEFUL ACTIVITY
All quests were designed to portray gamified real-life activities where math is employed as a tool to
solve purposeful problems. More about this can be read in the quest section below.
The game is inspired from both RPGs and adventure games. The narrative is linear and connects
various quests that the player must solve along the way, in order to progress. The story is as follows:
Life is peaceful and happy on the island of Mathagaskar where you live with your parents. The daily
life consists of taking care of the family’s berry orchard. But one day a pirate ship kidnaps the parents
which catapults the player into an epic and exciting journey through near and distant lands… This
fantasy is super-realistic in the sense that pirates, trolls, and medieval technology/society is mixed up.
However its definition of reality in sense of natural laws is defining a realism model similar to the real
world. This enables quests to function like tasks of historical medieval times.
A careful target group analysis was performed to understand character integration in the game. The
target group was chosen to be 5th grade (circa 11 year old) Danish or English speaking students.
Consequently the game was made in both English and Danish. To get further knowledge about the
target group informal talks with teachers and acquaintances near the target group’s age group were
performed. Furthermore, the math book Matematiktak151 and national curricular recommendations
151
See MatematikTak in references
72
(Fælles mål) 152 were confronted to get an understanding of how this target group could be addressed.
In extending understanding of statistical target group preferences, an attempt was made to gather
some statistics but not much seems available. Finally, successful learning games and non-learning
games addressing the target group were evaluated for purposes of inspiration in relation to charactercreation and development. This resulted in a character model which was not particularly deep, but
hopefully sufficiently complex to engage player identities. The player can e.g. choose if their avatar is
female or male, but they cannot customize face, hair, etc. as often is possible in RPGs. The player can
move their character around using keyboard and look around, fight and interact with NPCs and objects
using the mouse. In creation of the control scheme it was the intention to portray a feeling of being a
capable individual, who can act as the student desires.
The game is designed to only be a single player experience, to decrease development time.
Each quest was organized around specific NPCs. Through in-game dialogue system, these quest NPCs
introduces the quest at hand, sends the player out to act out the quest objectives and math primarily
using in game-objects (in concordance to Design Principle #8) and possibly guides the player along the
way. These NPCs also help the player if stuck or failing by giving tips and aid in continuing the
narrative upon quest completion. The typical engagement is that the NPC represents a professional in
the natural environment of a person having such a profession, and that he/she needs help performing
some purposeful task related to the profession. See quest descriptions below for further examples of
how this principle was obeyed.
REPRESENTATION
There are no audio narrated quests or character dialogue. No audio was recorded for the game, it is
purely text based and it is expected this will be a drawback in relation to the game design principles,
however, as there are graphics supporting all quest tasks, the text dominance should be limited to
possibly only impact the textually weak students. All game text is available in both Danish and English.
The reason for no audio was to cut down development time, which would have been extraordinarily
extended if all text should have been narrated to both Danish and English.
In regards to minimizing numerical representations, see the quest details.
INTERACTIVITY
See the quest details below for a description of how this principle was implemented.
PROCEDURAL FEEDBACK
See the quest details below for a description of how this principle was implemented as consequential
feedback. See the subchapter below named Implementation details for a description of the message
system which handles further instructional support.
152
UVM
73
BALANCE
Difficulty level of monsters can be changed by the player anytime. In designing quest mechanics,
attempts were made to guarantee that players who e.g. were bad at controlling the character, would
still be able to progress to and through the mathematical parts. However, such inexperienced gamers
might be slower in progress than others.
Quest details
Here is a list of all quests in the game. The basic design ideas and purpose behind each quest is
explained. A more thorough experience of the quests can obviously be acquired by playing the game.
Intro and rat
o
Description: This first quest introduces the game’s controls and lets the player train
this in chasing a rat away.
Pick berries
o
The first math engagement where the player must pick and put berries into boxes. The
representation and interactivity principles are not strictly followed here to in order to
let the player get comfortable with the graphical user interface.
Trees and trolls
o
The first ‘real’ quest. The player must practice math in figuring out how many trees will
be needed for a boat, while fighting some trolls at the same time. The Procedural
Feedback principle was intended to be implemented but had to be left out as it was
hard to find a way to realistically incorporate it into the story in a way that makes
sense and is not too time-demanding in implementation time. Consequently, if the
player fails, the trees will just reappear and the player must try again.
Sawing and sailing
o
The player must now cut boards for a boat, by subtraction exercises. He will not know
if he did it right or wrong, besides the very first board which acts as an introduction to
the quest mechanics. In regards to the Interactivity Principle; the player moves the
boards to and from the action, they don’t just appear there. In regards to the
Procedural rhetorics principle; if just one of the boards besides the first is
miscalculated, then the finished boat will sink and the player must try again. If he is
correct he can move on over sea to the next area.
‘Chasing’ pirates and ruby hunt
o
Not a real quest, more like an intermezzo giving impetus to the story. The player
follows the pirates’ trail and ends up at a guy who can help if he gets 10 rubies
74
Animal feeding jobs
o
This quest mathematizes feeding farm animals with correct amount of food and/or
other animals. One idea was to let the player engage problems like this: If 4 dinosaurs
needs to eat 3 giant worms each, how many giant worms would have to be put into the
dinosaur compound? In regards to Representation Principle; the player measures in
animals instead of numbers. In regards to the Interactivity Principle; the player moves
and herds the animals from compound to compound, like some life-size animal abacus.
In regards to procedural feedback; if the player provides too few giant worms, then the
dinosaurs would end up eating each other from famine. If the player provides too
many, the worms would triumph over the dinosaurs and eat them. Either fail would
result in the player, after watching how the animals eat each other, would somehow be
forced to try again or try in another compound. Complexity could be increased by
adding several types of animals that had to be combined correctly to find the correct
summed combination. This quest is not playable as development time was redelegated
to other purposes, but the semi-implemented, non-functional, non-playable quest
assets remain in the game world. The intention was that each of these compounds
would reward 1 ruby each.
Farmer field jobs
o
Geo the farmer provides 1 ruby per field you can help him with. Here the player
engages in calculation of area of various geometric shapes and has to find the correct
amount of seeds to throw onto fields. The Interactivity Principle is implemented as
throwing of seeds onto the field, however, the amount of seeds to be thrown is
determined in a pop-up window, which violates the principle. The Procedural
Feedback principle manifests as dark, plagued grain if the player throws too much or
too little seeds. The idea is that the field is weak and sick if overfed or too thin in
growing density. Since the animal feeding quest was never implemented, a lack of
rubies would have prevented game progress, so the final farm field quest rewards 3
rubies.
Pirate showdown
o
A non-math quest. This is the grand finale of the game, empowering the player while
showing off some of the game engine’s capabilities.
Other game design considerations
The name of the game; ‘Ruby Hunt’, was chosen in lack of better names. However, the main reason
behind this name to exclude any school or math references in the game, in concordance to the model.
Nevertheless the name might give associations to simple puzzle or edutainment games and is actually
not centered on hunting for rubies. The ruby hunt does not occur before end-game and were
implemented both as an extrinsic story motivator, but also as a method of gate-keeping while
providing difficulty adaption and player freedom of choice. If the feeding quest had been implemented,
75
a total of 14 rubies could have been gathered from engaging in either increasingly difficult geometry or
multiplication/addition. However, since only 10 rubies would be needed in total, this would have
enabled the students or teacher to actively choose which type of math to pursue deepest, while still
not being able to skip either topic completely, thereby forcing exposure to both math types.
There is another implementation of adaptability within the game, and was more implemented as a
demonstration of simple automatic adaption. If the player messes up a math response up once in the
berry picking quest or in the trees-for-boat quest, then the player will have to only cut three planks for
the boat. Otherwise the player must cut 5 planks.
Implementation details
This chapter presents various details of game-contents in relation to implementation. It is not fully
covering the development process and implementation choices, but is focused on specific examples.
The source code is heavily commented at most non-trivial places, but not all code is commented. A
basic understanding of Game Maker is beneficial for anyone wanting to find specific code snippets.
Most of the game, including quests, was mostly coded without using magic numbers, but instead tidily
arranged variables. This increases development time but gives two advantages: Balancing of the game
is faster while also enabling possibilities of difficulty adaption.
Some expenditure was spent on integrating an inventory and item system into the game. This is
functional and monsters dropping loot could be toggled on, but it is severely underused. Only few
quests involves using the inventory/item system since development time constraints prevented
planned quests revolving around gathering and spending items and involving this gathering and
spending in math problems had to be abandoned.
An AI engine consisting of a Finite State Machine was made to create an interesting AI for both friendly
and enemy NPCs.
An external graphics library was used. However, this graphics library consisted only of a series of
pictures. In example, a troll walking north-east was represented by 8 different pictures representing
gradual animation increases. As such, for all directions and for all character states, i.e. walking, fighting
and dying, pictures were acquired from that graphics library. A non-trivial task was to code a system
that modularly and in a scalable way could animate these many different pictures for different
purposes.
A message system was made to take care of various ways of communicating from game to player and
gathering player input. The message system is composed of three parts supporting both Danish and
English text:
1: Short simple messages in appearing in the top right corner like, ‘you cannot do that’ or ‘you are too
far away’. This is used to give smaller meta-messages.
2: Dialogue system: This is the system that carries most NPC communication and interaction. It
supports 1-8 replies and requires only few lines of code to be activated. Intricate dialogues can be set
using two states. State of NPC (and quest) and state of messages.
3: Popup-messages appear for communication tasks whose information density is in between simple
messages and complex dialogues.
76
GAME TEST
This chapter will present a field test of
the game described in the previous
chapter
Introduction
The test has two foci. One is to estimate
if the game succeeds in showing the
importance of math, how math makes
sense, and how math is relevant for solving Students playing the game.
real life problems. The other purpose is to
test if the design principles found in the model are somewhat radiating through the game design.
However, these two focal points overlap. This is the case because if the model is assumed to be valid
and the game manifests the design principles, then it can be argued that the game reflects the model
and is thereby as valid as the model.
Hypotheses
In expanding the test focal points, a series of hypotheses have emerged throughout the scientific
investigation that could be relevant to be put to a test. The model chapter should be read for a deeper
understanding of these hypotheses. It might be incorrect to term the points below as hypotheses, as
some of them are not exactly asserting anything in regards to outcome but are instead describing
where and why special focus in the interpretation effort could be laid.
A. It will not be easy to distinguish which design principles contribute more to potential success
than others. Consequently, the overall theme of this project, which is explained as the first focal
point which could be boiled down into intrinsic motivation, will be the main device for
measuring the success of the game design. This could be done by asking to students’ beliefs
regarding math’s purpose and importance before and after playing and then observe if a
change in response occurs. Such change could possibly be credited to the game use. However,
some Design Principles effect could be probed individually, as will be explained further below.
B. Mathematically weak students are expected to gain more from the game than strong students.
It is not expected that the stronger students do not get anything out of the game, it is simply
expected that the weak students will fare comparatively better in the math game session than
they usually do in their math class sessions, compared to the difference between how the
strong students fare in the game and in the class. This is expected since the model upon which
the game is based takes lots of inspiration from how unschooled or poorly schooled individuals
learn and perform math. It is assumed that such individuals are comparable to weak school
77
students and therefore the success of street math performers might rest on factors that are
also important for weak school students.
C. It is expected that the game will be well received by the students during the testing. If it is
assumed that most of their normal math education is traditional book and blackboard based
learning, then it must be a thrill to get to play a game during the class hours that normally
contain these entirely different activities. Even if it should prove to be a terrible and boring
math game, the students are expected to be somewhat enthusiastic about it since it simply
represents a different activity than what they are used to.
D. The game is expected to be received with an enthusiasm above the level that might be
expected from the above hypothesis. This is expected since it is assumed that the model and
design principles of the theory chapter is a sound scientific product, then the model’s
assertions concerning purpose, entertainment and motivation should shine through the game
and thereby through to the players. Distinguishing between enthusiasm stemming from the
activity represented by the game and the enthusiasm stemming from the game contents will
not be easy but might be approached by probing students if they liked it and why.
E. In relation to the Balance Principle, it must be addressed if potential differences in
performance of the various players can be ascribed to high game-playing experience, high
math qualifications or a combination. If the Balance Principle is well integrated, then there
should not be any difference in performance between game-experienced students and nongamers.
F. As proposed in the model summary/discussion, the test should address how the children
interpret the mathematical parts of the quests, i.e. as what is the problem and what counts as a
solution to the problem.
G. Math in Moontown has many similarities with Ruby Hunt. One of the few key differences is the
narrative structure. I hypothesize that a linear story is better for an RME game than an unlinear story. Though a linear story is not strictly required by Design Principle 3, I expect the
benefits will be greater since the linear story will be better at adding structure and purpose
than an open-ended story. Furthermore, it can act as a gate keeping device in a way that semiun-linear stories might be able to as well, but completely open ended stories or worlds cannot.
This is not a central task to investigate, and it will not be easy since no direct comparison
between games with different story types will be made in the test. Nevertheless, this topic will
be probed by asking questions to the students regarding their view on Math in Moontown (it is
fairly popular so they might know it) and how they think it compares to Ruby Hunt.
78
Subjects
Subjects tested were within the target group of the game. The study was split up over three different
sessions, all taking place in Denmark in April 2011. Two sessions were conducted at Atheneskolen153
in Copenhagen within the same day. Each session involved 10 students of 5th grade. The final session
was conducted some days later at Hedegårdenes Skole154 in Roskilde, involving 17 students of 5th
grade.
The math teacher of each class was present during the test to introduce the researcher and keep order.
During the students’ playing, the teachers discretely informed the researcher about students’ math
qualifications by pointing out the students who he considered to be at a low, medium and high math
skill level.
Procedure
One school gave permission to record the play test with video camera while the other did not. It was
the intention to use Fraps155 to capture on-screen action. However, both schools’ hardware proved
unable to run both the game and Fraps at the same time without considerable lag.
Since the computer rooms are hard to schedule, the teachers preferred filling the computer room with
students. Thus it was not possible to perform small group or solo tests of the game. Thus the intention
was to let 10 to 17 students play at the same time while only focusing observations and interviews
with a few, in order to gain narrow but detailed observations. Consequently a low, medium and high
math performing student was placed next to each other to be the focused group. However, it was soon
found that many interesting things happened all over the classroom which could be observed while
tracking progress, understanding and interaction of the students in focus. Thus, all students were in
focus, giving a wider but less detailed view. Nevertheless, knowledge about most students’ math level
and general game experience level was obtained from teacher (math level) and directly asking the
students about their game playing habits. In one school, all observations were written while in the
other they were video-recorded and supported by written observations.
In probing for evidence regarding to the above hypotheses and general test interest, a pre-interview
before game play and a post-interview was performed on all students in plenum. Effort was put into
formulating these questions open and non-leading, and some general guidelines concerning interviews
of game participants were followed156. Each test session lasted nearly an hour with around 5 minutes
dedicated to pre-interview and 10 minutes to the post-interview. Both the pre- and post-interviews’
audio was recorded and subsequently transcribed. These Danish transcriptions can be found in the
appendix.
153
http://www.athene-skolen.dk/
154
http://www.hedegaardenesskole.skoleintra.dk
155
http://www.fraps.com/
156
79
In the pre-interview, questions ranged from various topics like probing the students’ feelings towards
math learning and math’s importance. In the post-interview, questions revolved around the how the
game was experienced (i.e. character, story, quests) and how they understood the math of the game.
Questions asked were not exactly the same in each of the three test sessions as some student
responses were pursued with further questioning.
Results and discussion
Interpretations of observations, interviews
and video analysis produced the findings
outlined and discussed below. First, the key
findings are presented. These are followed by
attending to each of the hypotheses and test
focal points.
Key findings
Nearly all of 37 test subjects were positive
about the game experience. Only one
student thought that a book may be better at teaching math since “the game doesn’t have as much
information as an ordinary book”.
Nearly all students had obvious fun and were entertained.
The vast majority of the students reported they would prefer Ruby Hunt over other math games
and traditional text-book math.
If performance is measured as depth of progress into the game, then the majority of students who
were pointed out by the teachers as weak math performers did as good as the general crowd! Only
the very top math students were better/faster than the crowd. More about this below.
Hypothesis A
As expected, it was not easy to identify which design principles were strongest, most liked by students
or best manifested into the game design. However, indicators of effects related to specific design
principles appeared now and then during the observations and interviews. Such appearances are
discussed throughout the rest of this chapter. The second part of this hypothesis concerning
motivation and subjects’ views upon purpose of math seemed hard to isolate from the context. Instead
students’ relative progress into the game is deployed as a measure of effectiveness of the game as a
math motivator and single qualities regarding differences in this relative progress are attempted to be
singled out. This is also done in below discussions. However, one observation regarding motivation
can be presented here without considering relative progress to other students.
In one school, the class had not yet learned how to calculate the area of a circle. When one of the
mathematically strong students reached the sub-quest of the farmer jobs quest asking for the area of a
circle, an interesting thing took place. The student sprang eagerly from his computer through the room
towards the teacher and impatiently asked him how one can find the area of a circle while jumping up
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and down. This can be interpreted as a positive finding for several reasons: The boy was eager and
motivated to learn a new math formula, which must be expected to otherwise be hard to intrinsically
motivate students to do. Secondly, he took initiative himself to acquire this knowledge and was eager
to acquire it. This could possibly be explained by Design Principle 3 (fantasy story) as that might have
made him curious to progress into the game and find out how the story ended. Consequently it could
be an example of Malone’s idea about desire to bring ‘completeness to your knowledge structure’, as
described in the model. This pressing desire to learn could also have been explained by other design
principles, e.g. that the purposeful application of math was intrinsically motivating him. This is not
confirming the hypothesis that well-made math quests can let students engage and solve math beyond
their expected qualifications, but it is confirming that well-made math quests can motivate students to
engage in math beyond their expected qualifications.
Hypothesis B
As vaguely clear from the
photo, four different
students are at the same
time engaging the same
quest. From observations,
this was generally the case
for most students. The
students on the photo were
pointed out by the teacher
as ranging between both
high and low math
Four students at the same quest.
performers. That could be
an indication that either the game slows down the strong students so the weak ones can catch up, or,
more likely and hopefully, the game presents the math in such a fashion which enables otherwise weak
students to perform as well as the students who usually are strong. This seems like promising support
for declaring the game design successful in implementing the design principles of the model. However,
other factors might contribute to the difference as explained in the Balance chapter of the model.
Differences were observed regarding performance of students who identified themselves as ‘gamers’
or non-gamers. Gamers, people with high and regular computer gaming experiences, generally
performed better than the other students. This was expected but unfortunately students who
identified themselves as gamers were also identified by the teacher as strong math performers.
Consequently, since it was impossible to observe any non-gamer students strong in math, it was hard
to know if these top performers were performing well as a consequence of their math performance,
gaming experience or a combination. A few girls identified themselves as medium gamers, while being
strong in math. These girls performed very well too, and were nearly as quick in progression as the top
performing strong math gamer boys. Only one student was identified as a gamer and low math
performer. Let us call that student LG.
In the pre-interview, LG said “To me math is very very hard. But I think it is - I know it is very important
to (unintelligible). But anyway I think I never will become very good at it”. LG apparently agreed with
the teacher that he was a low math performer. During play he quickly picked up the game mechanics
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and raced ahead, leaving the other students behind. However he slowed down significantly and was
left behind the other students as soon as he encountered the mathematical parts of the game. But he
was able to progress steadily while only being a little slower than the flock. In the post-game interview
he was nevertheless positive about the experience: “I thought it was a really funny game (…) It is really
good especially for people who want to learn math because you mix a funny story with jokes and you
know a little of this and that into mathematics and that is really good I think”. This quote demonstrates
that he understood the basic game focus on mixing entertainment and education. However he later
demonstrated that he also understood the factors beyond the basics:
LG: “… If you can make math to a funny game.”
Researcher: “Did the game succeed in that?”
LG: “Yes it did that is the worst part of it! I am pretty sure that you can release it like a school game. And
if you can get things like cosinus and nasinus and all that and can make become funny, then anyone can
learn it”
The game is not teaching/practicing cosine etc., but he is enthusiastic and ambitious in his envisioning
of using games to learn difficult topics while having fun. In other words, his agenda is not just to have
fun, but also to learn. However, if LG was not really challenged mathematically then that could also
have explained his enthusiasm. If he was never really mathematically challenged, and being used to
playing games, then his expectations of a clash between something comfortable (games) and
uncomfortable (math) might have given him low expectations and he would have been positively
surprised when the game did not pull him out of his comfort zone. No matter what, he seemed to have
a positive experience. Perhaps, as the teacher of the other school said, “it is important that they get a
success experience by sitting and solving an assignment”. If the game can do this, then this might be a
new focal point of such realistic math games – and it might be a job that can be fulfilled easier or better
by a realistic math game than e.g. an edutainment game. This is expected since an intrinsic game like
Ruby Hunt can possibly portray a feeling of success while actually being easy. Contrarily, if an
edutainment game is too easy, then it might be boring as well.
Hypothesis C and D
Despite small helpings now and then, during the actual game playing, most students were quiet and
focused on the game. Every now and then a ‘DIEEEE’ or ‘YES’ filled the room.
This could indicate the children were immersed and felt related to their character’s adventures,
despite the rather shallow character design. Other students commented their progress or state and
one was even humming some kind of hero theme song while throwing out seeds on the fields. During
post-interview, a student asked why it was not possible to swim in the water. Another student replied
“don’t you wear too much armor?” This indicates the student is accepting and believing in the story and
extends it to explain inconsistencies like not being able to swim in water.
Several students were noticeably amused by the fact that the boat sinks when failing the sawing and
sailing quest, and gave loud exclamations when that happened. They generally seemed to think that it
made sense. One girl said “I think it was a good that the boat, like, sank. I mean it was obviously built
unstably so I think it was good that it sank so one could try again. Even if my boat didn’t sink“. This
student demonstrated that she understood that the boat sinking is not just a part of the story and is
not just a punishment for miscalculating. She acknowledges the causal flow that when someone fails
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math performance then the boat gets unstable which results in its demise. This was also demonstrated
by another student who, when asked why the boat sank, replied “because you didn’t cut them (the
boards) properly. Or subtracted wrong”. When asked if they thought it made sense, several students
replied yes. In relation to the farmer field quest, the subjects were not as expressive or positive about
the fact that the field turns dark if one miscalculates the amount of seeds. One answers that it does not
make sense. But no one expressed anger or frustration at this design and in the observations they
looked as understanding and accepting the process of the seed-to-fields quest and its ways of
communicating success, failure and progress through the Procedural Feedback principle.
These quotes emphasize the positive reception of the game among the students.
Student: “I thought it was very good, it is also very funny. Erhmm yes I like it.”
Another student: “Very funny, but also a bit hard”
Yet another student: “It is okay. Maybe a little easy in the beginning. Pretty comical actually. For example
there is a dinosaur in a fenced area up there near the desert a little to the right of it. And then there where
the boat sinks and all that and then crocodiles with weapons.”
One other student: “I think the game gives a good insight that math is important”
Whether these utterances demonstrate a positive attitude towards the game beyond the fact that it is a
game representing a different activity than normal class education, is unknown. However, some of
these and other comments, e.g. the student who explained the inability to swim was due to heavy
armor, were specific in describing the game’s qualities beyond simple entertainment. This could be
coupled with the generally observed appreciation of the specific design principles (whenever they
shone through individually), e.g. the Procedural feedback principle. This coupling indicates that the
entertainment value and positive atmosphere to a greater degree could be ascribed to the exact game
design based on the model than the game medium, i.e. simple game mechanics.
Hypothesis E
Near the end of one of the game sessions, a girl was still at
the tree-painting quest (early game) while some fast
students were at the farmer quest (late game), while the
rest was in between. This suggests that even if most
students at times were ‘on the same page’; some
differences shone through near the end of this session. The
same was observed in the other sessions as well. The girl
playing on the computer shown in photo was designated as
a ‘weak’ math student by the teacher. The correct solution
of this quest requires marking 2 big trees and 3 small trees
with paint. However, this girl got stuck in a cycle of fail
Screen of the girl who misunderstood the tree
marking quest. All the trees have been marked
while her peers continued their progress through the
game. She repeatedly marked all the trees she could find,
went to the NPC who told her she did it wrong, and then she tried the exact same thing again. It would
have been another situation if she had painted only some trees, but the wrong amount, since that
would show she understood the assignment but failed to perform it. This indicates that she had not
understood the quest objective, regardless of her ability to solve the quest correctly or incorrectly. If
she failed because of lack of understanding of the quest, then it must be attributed to the description of
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the quest. Consequently, since the description was textual, this student might have fared better at this
task had the quest description been audio-based. However, it could also be attributed to other factors
than medium; she might be one of the students that Gee describes as having a ‘broken’ school identity,
thereby consciously or subconsciously refusing or blocking herself from engaging in math. If this is the
case, then the game failed in the hoped circumvention of such broken identities. Regardless of the
reason, this was a complete failure of the game to bring a student forward, even if it only occurred with
one of 37 students. After spending nearly all the allotted game time on the quest, she received help
from a classmate sitting next to her.
Hypothesis F
Pre-interviews partially confirmed the study’s premise; that students do not know why they should
learn math. I.e. when asked if math is important, students generally agreed it was important but were
not really able to explain why. Only one student was able to formulate her understanding: “You need to
use plus and minus and multiplication and division all the time no matter what kind of person you are.
And then it depends on what job you will get if you will need geography (sic) or something.” The girl is
later stating that she meant geometry, not geography.
Students could not quite agree if the difficulty and balancing of the game was appropriate, as some
were arguing whether there was too much, just right, or too little math in the game.
Instance of classmates helping each other was observed. These did not consist of just giving the
answer, instead they tried to let their comrades understand the problem and figure out the result for
themselves. This was only observed in Hedegaardenes school, and could possibly be owing to, as the
teacher explained later, that students were usually told to work in pairs and help each other by not
giving the answer. However it could possibly or partially be due to the game being able to
communicate math in a fashion that encourages problem solving, communication and understanding.
However the indications for this hypothesis are too weak to allow firm conclusions in this regard.
In the Athene school, a boy had obvious troubles but his peers did not help him despite being utterly
stuck for around 15 minutes. Finally the researcher assisted him. The problem he faced was that he
was continuously trying to attack the rat using right mouse button instead left, which resulted in
nothing. This might indicate the control scheme is too advanced or the explanation of it was bad. This
was not picked up in the mini-tests preceding these tests as the subjects had no problems with the
character controls. Still they should possibly be changed, especially since it does not seem that
necessary to use both mouse and keyboard e.g. in fighting.
Hypothesis G
During post-interviews, students were asked about Math in Moontown. In one session they seemed to
not know it. In another session the game was known and a general consensus was observed/recorded
that subjects did not like Math in Moontown:
Researcher: “Why do you think Math in Moontown is bad?”
Student: ”It is because there are the same tasks all the time. That you just have to complete and then
choose yourself. So it is one of their musts, since then you can decide for yourself when you want to solve it
and what kind you want to solve. But there are not really different assignments for each time. It is the
same geometry assignments”.
This girl made a fine demonstration that she understood the differences between the games in relation
to their narrative structures. However, her critique ends up falling on an argument that repetition of
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quests leads to boring gameplay. Further attempts to investigate the hypothesis were fruitless or
observations were too weak to merit deduction. Thus it is hard to conclude if Math in Moontown’s
open-ended narrative or Ruby Hunts linear story is the better choice. However, the gate-keeping
function of the story in Ruby Hunt was not identified as troublesome. However, further research could
address this topic.
Other findings
Some interpretations of findings are unrelated to the hypotheses/focal points and will be presented
here.
Female students of one session claimed their girl character was not looking like a girl. They
asked if their character could look more girly and for the possibility to be able to customize it
more.
Several complained that the player character moved too slowly.
One asks if the game would be available for purchase in stores and was disappointed when
learning that was unlikely. However, both schools were ‘rewarded’ with a game copy to keep,
as a reward for participating in the study.
One student of a session where no one had reached the end of the game yet, made some
interesting comments about the story. He started to talk about what would happen later in the
game, as if he fully believed his anticipations about the rest of the game would be true. He
talked about he liked the story but it was pretty classic; rescue family members from bad guys,
but he assumed that the player would actually end up saving the parents, even if he could not
know that yet. That might indicate that the story, even if it was classic, simple and predictable,
was sufficiently rich to motivate the players to keep going and to think ahead about what to
expect. It would be interesting to investigate how a less predictable story-line or an un-linear,
flat structure narration would fare with such students.
Teacher interview
After the final test session was completed the teacher was interviewed. He had been present but
mostly silent during the session:
“We are very happy to get the opportunity to try this. I think it is exciting for the children to see, like I told
you, that we work a lot on telling them that mathematics is not just for the sake of the mathematics or for
my sake, but it is a question that they in the daily life enters problem solving where they need the
mathematics. “
This really demonstrates, despite the teacher’s self-description of being a game-illiterate, that he
acknowledges the purpose, need, and effectiveness of the game. Furthermore:
“You can see how interested they actually are, that they just approach it – on that level the game fits very,
very well. They think it is exciting and they engage it vigorously and they try and talk – and there is also
quite a part of that they seek each other and talk about how do you solve that task, how did you do that.”
This utterance is very relevant to the RME approach of making students’ knowledge their own and to
use the peers in meaning interpretation and problem solving. It was decided to not interview the
teacher of the other class in a formal interview as he is a personal acquaintance of the researcher, and
as such the objectivity of interviewer, interviewee and the following interpretations could be disputed.
Still, he was openly positive about the game, the design philosophies behind it and game-based
learning in general.
Early in the report some barriers considering using learning games in the classrooms were explained.
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These barriers were somewhat present here, i.e. as the first school only had a poorly equipped
computer room, barely capable of supporting this game. Furthermore the teacher of the second school
was aware of curricular ideas similar to RMEs proposals; however he confessed time constraints and
insecurities as a major barrier for implementing alternative teaching approaches. Such school barriers
could possibly be circumvented by proposing the application of the game as a homework artifact.
However this opens a host of new complexities to consider but the benefit (and possible pitfall) would
be that this approach would push the pressure towards students’ homes in regards to installation
expertise, hardware performance and socio-interactional changes. Another argument revolves around
how the teacher of the other school handled a special situation. Two students were forced to sit at the
same computer due to lack of available computers. Soon, one took complete control of the game and
the other was obviously not immersed or involved in the game while even physically withdrawing
from the screen. However, the teacher noticed this and forced them to swap places. This drastically
improved their teamwork as both were involved in solving the quests. This seems to support
Hanghøj’s and Egenfeldt-Nielsen’s findings that competences of teachers are directly proportional to
the students’ educational utility of the game. However, none of the findings within this test are
substantial enough to convincingly claim that student conflicts and teacher resource constraints could
be guaranteed to be evaded if the game would be networked multiplayer, group multiplayer or single
player. It is also uncertain how a homework approach to the game would end. Nevertheless, an
approach of realistic math games as homework artifacts and/or multiplayer game would be a
welcomed research direction.
Methodological considerations of the study
Most of the time the children were disciplined in not interrupting each other when talking, but various
exclamations and quick comments sometimes flew in the air, especially at Atheneskolen. This
disrupted the recording quality, making many audio recordings unintelligible. Moreover, no
recordings were made concerning who was speaking on the audio recordings, so I had no obvious way
to link the observations of the players with their utterances in the interview, besides memorizing
names, places and faces and voices. Furthermore, the sound recording equipment was so sensible to
noise so several parts of the recordings became unintelligible, even when students were by and large
silent. Furthermore, several students kept on playing at the post interview even if told to stop. That
distracted them from the interview questions and sometimes produced sounds of the game which
overrode the voice recordings. Both these factors contributed to obfuscating many minutes of
recording beyond recognition. Better sound recording equipment would have been desirable, or
smaller groups of students should have been tested.
Since Fraps could not be used, no direct dedicated record of on-screen activity was acquired. This
could have enabled analysis of where they clicked, how they performed and so on. This did not show
to be a failure in assessing the model, but it would have given better usability feedback. However, such
data might be hard to interpret if it is not accompanied by e.g. a webcam simultaneously recording the
face and body of the player. This would give a complete picture of what went on inside and outside the
game and could have enriched the understanding of how the game was used and experienced.
However, such a setup would have required a larger setup phase which was not possible due to the
high traffic in the schools’ computer rooms. Only few minutes were available at each school for
installing the game on all computers.
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Within all three test sessions, as soon as the test sessions began, nearly all subjects pressed enter or
clicked OK to the very first message without reading it. This message is however crucial as it does not
allow the player to move before they click the house, as is explained in the message. But since most
missed this, they started asking each other, the teacher and researcher what to do. In the second
school they found out a way to exploit a bug to bypass this intended design by entering and leaving the
menu system. Nevertheless, it still shows that something was missed in the mini-test sessions.
Possibly, players sitting alone with a researcher next to them would be too careful, concentrated and
attentive to miss messages like these. The situation might be very different in a big test session with a
computer room full of classmates. The conclusion is: Test in a class full of students if it is meant to be
played that way.
The teacher of Atheneskolen expected his students to be on above average level in math performance,
but no general, visible difference in performance between students of the two different schools were
observed, despite the fact that only Atheneskolen had previously trained calculating areas of
geometrical figures. Consequently it is asserted that the students of the test generally were
representative of average Danish students. However, grading the generalizability of the findings of this
study is hard as many assertions and choices regarding model, game design and test methodology are
present.
No retention test was performed. It is not only unknown if the students learnt anything new, it is also
unknown if the practice of math in the game have given them better math performance or retention of
math facts. However, this was never the purpose of the test. The purpose was to test motivation, if
students could see the purpose and if students enjoyed this way of engaging math. Actual effectiveness
of learning in computer games is not easy to estimate.157
Some design principles utilized in the game design overlaps in scope. Others make others obsolescent,
e.g. the Interactivity Principle is only possible if the Representation Principle is present, making a
complete one-way overlap. Further scientific treatment of the model could specify the model further
and avoid such overlaps. However, it is not desired to make it too specific either since effort was put
into making the model wide enough for several different game ideas and types while narrowing it
sufficiently for qualifying as a realistic math games. Nevertheless, a more specified and properly
divided model could have been easier to test. In example, the model seemed able to accomplish its
goals but it is not known which subparts of the model accounts for this accomplishment. More
rigorous testing would be necessary to understand this. This could be done by testing and comparing
games that are based on differing design principles.
Test summary
Despite technical obstacles in relation to gathering the empirical material, the study seems sufficiently
methodologically sound to warrant interpretation of the test results. The beginning of the chapter
presents key findings. These can be boiled down to a conclusion that the test indicates the game was
primarily successful in motivating the students to engage in mathematics. Further summaries and
conclusions in comparing the game test, game design and model are presented in the next chapter
157
Hanghøj 2010
87
CONCLUSION
It appears that math performing school students lack purpose, motivation and proper success when
engaging in traditional school math. Other math artifacts, like word problems, do not seem viable for
addressing such issues. Realistic math, defined by Realistic Mathematics Education and out-of-school
math seem viable in solving these issues. This project has investigated how a digital learning game can
be designed to teach realistic math. This investigation has produced a model consisting of 10 design
principles that such a realistic math game should be based on. The summary of the model chapter
functions as a conclusion for the entire first part of the report, and basically consists of the 10 design
principles. These 10 design principles were used to govern the creation of a realistic, digital math
game named Ruby Hunt. The game was tested on 37 fifth grade students from Denmark. In the
previous chapter key findings were presented and discussed.
The purpose of this project was to address some research questions. Concluding answers for each of
these questions are offered below.
How could a digital learning game based on RME and out-of-school math be designed to
effectively and meaningfully teach math to primary education students?
This question is two-fold. One is regarding how a game could be designed with inspiration from RME
and out-of-school math. The theory chapter was used to conclude that, in realistic digital math games,
math should be taught like street math and decrease focus on how math is taught in classrooms and
textbooks today. A RME game must immerse the player, not through focus on fun or spectacularly
immersive graphics and sounds, but through a focus on realism, purpose, meaning and consequential
feedback. The support for this postulate is resting on two pillars: The performance of informal street
math practitioners and the Dutch theory of Realistic Mathematics Education.
The discussion manifested as a model consisting of 10 heuristic design principles. This would be
beneficial to defining the design principles more clearly, as the current design principles contain some
ambiguities and overlap. It is asserted that the scientific process followed in the creation of the model
resulted in a game that is at least equally capable of teaching math effectively and meaningfully as RME
and out-of-school math procedures. However, the model might have been made tighter by including
more thorough and deep research and analysis of the various topics that the model touches upon.
It cannot be claimed that this model is the only answer to how an RME and out-of-school math game
could look like since many choices of inclusion and exclusion of scientific elements in the model took
place during its development. However, the approach seems to be viable according to the theoretic
model as no greater theoretical or methodological obstructions or inconsistencies were identified that
could have prevented a meaningful construction of the model.
How can a digital game based on such design principles improve students’ motivation and
understanding of purpose and importance of doing math?
The game that was designed on the basis of the model was created to improve students’ motivation
and understanding of purpose and importance of doing math. Consequently the field test of this game
can address the above question. The test confirmed findings of other studies, that game based learning
seems like a viable learning tool. It also hinted a confirmation to the assertion that students in general
have troubles finding purpose in doing math.
88
However, one student became during the game play test eager to quickly learn new math. Other
students declared that the game was successful in linking math and entertainment. In all, the test
indicated that most of 37 students found motivation, purpose and importance of playing a game that
practiced math. Yet it is not possible to conclude if this proves that the game improved students’
motivation and understanding of purpose and importance of doing math, beyond the time spent on the
game. It is only possible to conclude that this was the case during the game interaction, and as such it
is hard to say if their motivation and understanding was improved by the game. This is the case since
long term motivation towards math was not measured. Also, the question still stands if it actually
increases math performance of children who played it, compared to children who did not. It is to be
assumed that the act of repeatedly engaging in math content in games will be beneficial for math
performance in the long run, but further research could determine how beneficial this game is in
comparison to other teaching methods, like other games or other media.
What type of students and what type of games will be the best match for this purpose?
The model represents a set of choices in regards to fulfilling the agenda, and these choices are
narrowing the scope of possibilities. Likewise, the game design represents a further narrowing of the
scope of possibilities and choices. The question is how choices resulting in narrowing of scope, e.g.
creating another model or game, would manifest in relation to the target group of a game. Likewise,
several possibilities exist in combining an optimal target group with each of these potential models
and games. This vast problem field cannot be grasped in this project, but the results of the chosen
game design and chosen target group can be discussed.
It was hypothesized that mathematically weak students was the ideal target group for the chosen
game design as this type of student, more than others, would need to see purpose and meaning in
math. However, the game test found that both mathematically strong and weak students received the
game well. Generally the game seemed appropriate for the target group, while possibly slightly
favoring boys over girls, and gamers over non-gamers as optimal target audience. Whether changing
the model or the game design based on the current model will address other target groups is
unanswered. Further game implementations based on the current model could address how design
liberties within the boundaries of the model affect its effectiveness if compared to this study. Likewise,
testing the same game on groups related to the target group, e.g. 4th or 6th grade children, could
identify if the game fits better or worse for these as it was not cemented if the exact target group
choice was optimal.
Furthermore, one principle (Multiplayer principle) was defined theoretically but not implemented and
tested in the game. This, and the fact that some of the design principles’ overlap, makes it non-trivial to
conclude what elements of the model contributed more or less to the findings of the test. An
understanding of this could be approached by testing and comparing games that are based on differing
design principles.
In summary, a game based on the theoretical model has via tests indicated that 37 fifth grade students
in Denmark received it well and that the game was capable of conveying purpose of math while
motivating students to practice math. Realistic math games like Ruby Hunt seems viable to deserve a
place within todays’ mathematics education.
Jan Borg, May 2011
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Isbister, K. : “Game Usability – advice from the experts for advancing the player experience”.
Elsevier Inc. 2008.
Ke, F. : “A Qualitative Meta-Analysis of Computer Games as Learning Tools”. Handbook of
Research on Effective Electronic Gaming in Education Vol. 1. Richard E. Ferdig (ed). IGI Global
2009.
90
Malone, T. W. : “What Makes Games Fun to Learn? Heuristics For Designing Instructional
Computer Games“. SIGSMALL '80 Proceedings of the 3rd ACM SIGSMALL symposium and the
first SIGPC symposium on Small systems 1980.
Masingila, J. O., Davidenko, S., Prus-Wisniowska, E., Agwu, N. : “Mathematics Learning and
Practice In and Out of School: A Framework for Making These Experiences Complementary”.
Paper presented at the Annual Meeting of the National Council of Teachers of Mathematics
(72nd, Indianapolis, IN, April 1994). Acquired from
http://eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb=true&_&ERICExtSearch_Searc
hValue_0=ED373960&ERICExtSearch_SearchType_0=no&accno=ED373960 at 11.12.2010.
MatematikTak 5. Klasse. Contributors: Høegh, J., Benedict-Møller E. M., Andersen, C., Esbensen,
E. Alinea 2nd edition 2007.
Matematik i Måneby. Math game - only available in Danish. Made by BitFrost Interactive. As of
23.04.2011 a demo is available from http://www.bitfrost.dk/files/maaneby_demo.exe and
information is available at http://www.bitfrost.dk/mim_about.htm . The full version was
tested by the researcher.
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volume 4, number 9, 2009. Pages 52-57. Forening bag Dansk Pædgogisk Tidsskrift 2009.
Misfeldt, M. : Videotaped interview. Acquired from
mms://stream.dpu.dk/public/lgh/Morten_Misfeldt.wmv at 23.12.2010
Nunes, T., Schliemann, A. D., Carraher, D. W. : “Street Mathematics and School Mathematics”.
Cambridge University Press 1993.
Palm, T. : “Impact of authenticity on sense making in word problem solving”. Acquired from
http://www.springerlink.com/content/v443203614213512/ at 29 June 2007.
Pivec, P. : “Game-based Learning or Game-based Teaching?” acquired from
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ecta.org.uk//uploaddir/downloads/page_documents/research/emerging_technologies/game_based_learning.pdf
at 23.01.2011.
Prensky, M. : “Computer games and learning: Digital Game-Based Learning”. Handbook of
Computer Game Studies, MIT press 2005.
Prensky, M. : “Digital Game Based Learning”. McGraw-Hill 2001.
Romberg, T. A. : “Designing middle-school mathematics materials using problems set in context
to help students progress from informal to formal mathematical reasoning”. Acquired from
http://www.scribd.com/doc/22375957/Designing-Middle-School-Math-Materials-UsingProblems-in-Context at 15.03.2011.
Schiro, M. S. : “Oral storytelling & teaching mathematics”. Sage Publications, Inc. 2004.
Squire, K. : ”From Content to Context: Videogames as Designed Experience”. Educational
Researcher, Vol. 35, No. 8, pp. 19–29. November 2006.
Tosca, S. : “The Quest Problem in Computer Games”. Paper presented at the Technologies for
Interactive Digital Storytelling and Entertainment (TIDSE) conference, in Darmstadt, 2003, and
printed in its proceedings, published by Fraunhofer IRB Verlag). Acquired from http://www.itc.dk/people/tosca/quest.htm at 24.02.2011.
Ulicsak, M. : Videotaped interview at ECGBL 2010. Acquired from
mms://stream.dpu.dk/public/konf10/ECGBL/ECGBL101022sq-05.wmv at 21.02.2011.
UVM 2009 : Acquired from
http://www.uvm.dk/~/media/Publikationer/2009/Folke/Faelles%20Maal/Filer/Faghaefter/
matematik_31.ashx at 13.11.2010.
Üzel, D., Uyangör, S. M. : “Attitudes of 7th Class Students Toward Mathematics in Realistic
Mathematics Education”. Acquired from http://www.m-hikari.com/imf-password/37-402006/uzelIMF37-40-2006.pdf at 13.01.2011.
91
Verschaffel, L., Corte, E. : ”Teaching Realistic Mathematical Modeling in the Elementary School:
A Teaching Experiment With Fifth Graders”. Journal for Research in Mathematics Education,
Vol. 28, No. 5, 577-601. 1997.
Verschaffel, L., Greer, B., Corte, E. : “Making sense of word problems”. Swets & Zeitlinger B.V.
2000. Partial example acquired via Google Books at 20.04.2011.
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Handbook of Research on Effective Electronic Gaming in Education Vol. 3. Richard E. Ferdig
(ed). IGI Global 2009.
92
APPENDIX A: CREDITS OF RESOURCES USED IN THE GAME
Ruby Hunt
A math game made by Jan Borg as a part of a thesis investigation.
[email protected]
Game design and programming by Jan Borg
Additional credits:
Most graphics:
A very big thanks to Reiner "Tiles" Prokein of http://reinerstileset.4players.de/englisch.html for all his wonderful graphics.
Acquired from 6th December 2010 and onward.
Subsystem programming:
Thanks to Game Maker forums user FoxInABox for his script that concatenates variable names with strings. Acquired via
http://gmc.yoyogames.com/index.php?showtopic=369769&st=0&p=2643867&#entry2643867 at 06.12.2010
Thanks to Game Maker forums user Davve941018 for his Inventory System and Item engine. Acquired from
http://gmc.yoyogames.com/index.php?showtopic=448094#entry3313327 at around mid December 2010
Thanks to Game Maker forums user Shocker51374 for his achievements engine from which I used a small part for some
messages. Acquired from
http://gmc.yoyogames.com/index.php?showtopic=476353&st=0&p=3532298&hl=achievement&fromsearch=1&#entry353
2298 at 1st of February 2011
Thanks to Game Maker forums user Mordi for his/her 'Professional Menu' scripts. This is the menu engine used in the
game. Acquired from http://gmc.yoyogames.com/index.php?showtopic=318527&st=0 at mid January 2011
Audio:
Samples used from Freesound. All sounds used in the game are in the list below. Some of the sounds have not been
used
Download date, user, and file.
--------------------------------------February 12, 2011
By Benboncan (http://www.freesound.org/usersViewSingle.php?id=634166)
Two Bells,Ship Time.wav (http://www.freesound.org/samplesViewSingle.php?id=77699)
--------------------------------------January 27, 2011
By theta4 (http://www.freesound.org/usersViewSingle.php?id=606715)
ding30603-spedup.wav (http://www.freesound.org/samplesViewSingle.php?id=66136)
By zimbot (http://www.freesound.org/usersViewSingle.php?id=1449999)
WoodDebrisFall5.wav (http://www.freesound.org/samplesViewSingle.php?id=89387)
--------------------------------------January 25, 2011
By SpazTastic (http://www.freesound.org/usersViewSingle.php?id=1767083)
me ouch.wav (http://www.freesound.org/samplesViewSingle.php?id=104170)
By Parking Sun (http://www.freesound.org/usersViewSingle.php?id=5550)
torturer massage.aif (http://www.freesound.org/samplesViewSingle.php?id=74308)
--------------------------------------January 24, 2011
By Snoman (http://www.freesound.org/usersViewSingle.php?id=29481)
grass5.wav (http://www.freesound.org/samplesViewSingle.php?id=9912)
By NoiseCollector (http://www.freesound.org/usersViewSingle.php?id=4948)
mysteryclickloop.wav (http://www.freesound.org/samplesViewSingle.php?id=111167)
--------------------------------------January 17, 2011
By cmusounddesign (http://www.freesound.org/usersViewSingle.php?id=1059930)
Wood, Sawed_Dropped.wav (http://www.freesound.org/samplesViewSingle.php?id=84693)
By zimbot (http://www.freesound.org/usersViewSingle.php?id=1449999)
WoodSmashAndFall_04.wav (http://www.freesound.org/samplesViewSingle.php?id=89391)
93
By MAJ061785 (http://www.freesound.org/usersViewSingle.php?id=1275441)
Chopping wood.aif (http://www.freesound.org/samplesViewSingle.php?id=85534)
--------------------------------------January 14, 2011
By Dynamicell (http://www.freesound.org/usersViewSingle.php?id=47052)
Fire_Forest_Inferno.aif (http://www.freesound.org/samplesViewSingle.php?id=17548)
--------------------------------------January 10, 2011
By m_O_m (http://www.freesound.org/usersViewSingle.php?id=829608)
onion_break_hit_flesh_vegetable_blood_01.wav (http://www.freesound.org/samplesViewSingle.php?id=107642)
--------------------------------------December 10, 2010
By qubodup (http://www.freesound.org/usersViewSingle.php?id=71257)
swosh-08.flac (http://www.freesound.org/samplesViewSingle.php?id=59995)
swosh-43.flac (http://www.freesound.org/samplesViewSingle.php?id=60030)
By jus (http://www.freesound.org/usersViewSingle.php?id=157342)
cellos down down.wav (http://www.freesound.org/samplesViewSingle.php?id=41618)
By FreqMan (http://www.freesound.org/usersViewSingle.php?id=92661)
gypsy_violin_variation.wav (http://www.freesound.org/samplesViewSingle.php?id=42953)
By mumagi (http://www.freesound.org/usersViewSingle.php?id=186050)
bells.mumagi.aiff (http://www.freesound.org/samplesViewSingle.php?id=26396)
By junggle (http://www.freesound.org/usersViewSingle.php?id=128404)
djembe_loop_13.wav (http://www.freesound.org/samplesViewSingle.php?id=27359)
By acclivity (http://www.freesound.org/usersViewSingle.php?id=37876)
GullsByTheSea.wav (http://www.freesound.org/samplesViewSingle.php?id=13564)
By Kyster (http://www.freesound.org/usersViewSingle.php?id=646701)
little forest again.wav (http://www.freesound.org/samplesViewSingle.php?id=98264)
By earthsounds (http://www.freesound.org/usersViewSingle.php?id=1209938)
Twig Snap 1.wav (http://www.freesound.org/samplesViewSingle.php?id=101492)
--------------------------------------December 9, 2010
By Robinhood76 (http://www.freesound.org/usersViewSingle.php?id=321967)
01683 witch groan.wav (http://www.freesound.org/samplesViewSingle.php?id=100896)
01685 harp groan.wav (http://www.freesound.org/samplesViewSingle.php?id=101120)
By tschapajew (http://www.freesound.org/usersViewSingle.php?id=1757489)
pain_scream_hard_2.wav (http://www.freesound.org/samplesViewSingle.php?id=103527)
pain_scream_light_2.wav (http://www.freesound.org/samplesViewSingle.php?id=103530)
pain_scream_light_3.wav (http://www.freesound.org/samplesViewSingle.php?id=103531)
pain_scream_serious_1.wav (http://www.freesound.org/samplesViewSingle.php?id=103532)
random_vocal_1.wav (http://www.freesound.org/samplesViewSingle.php?id=103534)
By ryansnook (http://www.freesound.org/usersViewSingle.php?id=430094)
groan1.wav (http://www.freesound.org/samplesViewSingle.php?id=103569)
growl2.wav (http://www.freesound.org/samplesViewSingle.php?id=103574)
snort.wav (http://www.freesound.org/samplesViewSingle.php?id=103578)
By m_O_m (http://www.freesound.org/usersViewSingle.php?id=829608)
onion_break_hit_flesh_vegetable_blood_07.wav (http://www.freesound.org/samplesViewSingle.php?id=107772)
By Syna-Max (http://www.freesound.org/usersViewSingle.php?id=111920)
punches_and_slaps.wav (http://www.freesound.org/samplesViewSingle.php?id=43586)
By man (http://www.freesound.org/usersViewSingle.php?id=14447)
swosh.aif (http://www.freesound.org/samplesViewSingle.php?id=14609)
By inchadney (http://www.freesound.org/usersViewSingle.php?id=28867)
danish crowd.wav (http://www.freesound.org/samplesViewSingle.php?id=96145)
By fogma (http://www.freesound.org/usersViewSingle.php?id=93683)
Bar atmosphere - Not So Busy.wav (http://www.freesound.org/samplesViewSingle.php?id=19870)
By Luftrum (http://www.freesound.org/usersViewSingle.php?id=553671)
oceanwavescrushing.wav (http://www.freesound.org/samplesViewSingle.php?id=48412)
By ERH (http://www.freesound.org/usersViewSingle.php?id=215874)
wind.wav (http://www.freesound.org/samplesViewSingle.php?id=34338)
By reinsamba (http://www.freesound.org/usersViewSingle.php?id=18799)
evening in the forest.wav (http://www.freesound.org/samplesViewSingle.php?id=18765)
By Kyster (http://www.freesound.org/usersViewSingle.php?id=646701)
Restaurant chatter.wav (http://www.freesound.org/samplesViewSingle.php?id=82479)
---------------------------------------
94
APPENDIX B: INTERVIEW GUIDE AND TRANSCRIPTION
These are the translated questions/observation points that were guiding the game test process. The
questions were not asked literally as they are written here, but acted as a guideline for topics that
should be covered during the interview.
Pre-test Interview
Is math important? Why / why not?
Do you think math is useful in every-day situations?
Is math hard?
Raise your hand if you play lots of computer games all the time.
Raise your hand if you almost never play computer games.
Observations:
What happens if someone doesn’t know what to do?
Do they have fun?
Do they accept the game’s premise/context/world/story as serious?
Do they roleplay, alter their voice or seem to identify with the character? In what situations?
How is their interaction with the NPCs?
Are there exclamations of joy/fear/frustration? In what situations?
How do they solve the assignments? Calculator/pen-paper/head? Do they solve them as
expected in the quest design or with other methods/formulas? With formal school methods or
with own invented methods?
Does it look like the math makes sense to them or do they seem like it is an unserious or
moronic task to play the game / engage in specific quests.
Post-Interview
What did you think about the game?
Can you describe what math assignments there were in the game?
Can you describe how math assignments were solved?
Did you learn anything new?
Did it make sense to do the math in the game?
How did it feel to play the game? Was it like playing or was it like being in classroom hours
95
Was the game hard (math problems)
Was the game hard (opponents, controls, understanding what to do)
How was the story?
Soon a sequel to Ruby Hunt will come that can be played via internet. If you could chose what
your homework should be next time, would you then prefer that it was:
o
To play and finish Ruby Hunt 2
o
Play and finish another math game
o
Make ordinary homework, for example assignments from the math book
Transcription
Direct quotes have only been translated when directly used in the report, so the text below is in Danish.
Questions by the researcher that gave no replies have been left out of the transcription. The students raised
their hands and the researcher picked who should talk. Often there were several hands in the air after a
question. Most of the time the children were disciplined in not interrupting each other, but various
exclamations and quick comments sometimes flew in the air, especially at Atheneskolen. This disrupted the
recording quality, making the responses unintelligible. Completely unintelligible passages have been left out
making some interview sessions appear shorter than they were. Only few of the respondents are marked as
male as it can be hard to determine the gender from the voice from kids at that age. Videotaped recordings
were not transcribed and are not made available for privacy reasons.
R = Researcher
T = teacher
U/S = unknown student
F = Female
M = Male
Test atheneskolen d. 1.04.2011.
Atheneskolen 1:
Pre-interviews:
R: Er matematik vigtigt?
F: Altså plus og minus og gange og dividere. Jeg ved så ikke med geografi (sic) det kommer an på job. Man skal hele tiden
bruge plus og minus og gange og dividere uanset hvilket menneske man er. Og så kommer det så an på hvilket job man får om
man skal bruge geografi eller sådan noget. Fx hvis man er håndværker og skal skære en cirkel ud af en trekant så er det en
god ide at kunne vide hvordan man laver en (unintelligible) cirkel…
U: Geometri altså !
R: Er matematik svært eller nemt?
U: Nogle ting er svære andre ting er nemme.
U: Fifty-fifty
Post interview:
96
T: Nallerne væk fra keyboardet:
R: Hva synes i om spillet?
M: Der er for meget matematik.
R: Der måtte gerne være lidt mere sjov imellem matematikstykkerne?
M: Ja.
R: Hva synes du?
F: Det var nogenlunde sjovt. Jeg synes godt man kunne bevæge sig lidt hurtigere. Og kunne man arbejde lidt mere med
grafikken? Det er jo ikke fordi det skal være sådan totalt wannabe matematik i måneby men der kunne godt være nogle
detaljer som kunne gøres lidt bedre. Men ellers synes jeg det er et sjovt spil og så også teksten den er lidt ulæselig nogle
gange.
R: Hvad tænker du på med grafikken og detaljer?
F: Det ser lidt groft ud.
R: Kan I prøve at beskrive hvad det var for noget matematik man skulle lave i spillet?
M2: plus og minus
Unknown: Unintelligible
M2: Der er utroligt meget korn, sådan et par 100 kilo
R: Så det er lidt urealistisk?
M2: Ja
R: Er de andre ting realistiske? Altså bortset fra det med kornsækken – giver det mening?
Unknown: Ja det giver mening det meste af det. Der kunne godt være lidt der kunne finpudses.
Noise
R: Har i lært noget nyt?
F: Jeg vil sige jeg har brugt mine evner inden for matematik, jeg har ikke rigtigt lært noget nyt.
R: Gav det mening at lave det her matematik
F: Det har en (unintelligible) at så går man ned til KalkuLars og så kommer piraterne og så skal man gå op igen og så siger de
så man skal gå ned igen.
R: Hvordan føltes det at spille spillet? Føltes det sådan som et almindeligt computerspil eller var det som at have almindelig
undervisning?
Unknown: Almindelig undervisning.
R: Var det som en leg eller var det som øhh ja hva siger du?
Unknown: Både og synes jeg. Unintelligible
R: Hvor svært var matematikproblemerne?
Unknown: Det var vildt svært der ved den der kornmark.
F: (Laughs)
U3: Problemet var at du skulle gange 56 med 3.
U4: Skulle man det? Hvorfor skulle man det?
U3: Fordi det er jo kun arealet af det.
U4: Det bliver jo overmeget!
Others: (Laughs)
R: Ville du sige noget?
U5: Jeg synes de fleste af sådan nogle (spil) hvor man lærer noget, det er jo ikke sådan et rigtigt computerspil, altså det er jo
sådan en blanding. Man lærer samtidigt noget (unintelligible …) et computerspil der er lavet til skolen. Som det nu så også er.
R: Hvordan er historien?
U1: Der var meget matematik i det.
U2: Nej der var ikke.
U1: Der var lidt meget.
R: I den der matematik i måneby, er det meget camoufleret der?
U: unintelligible noise
R: Hvorfor er det dårligt? Matematik i måneby
F: Det er fordi der er de samme opgaver hele tiden. Som man bare skal løse og så vælger man selv. Det er så en af deres musts,
fordi så kan man selv bestemme hvornår man vil løse dem og hvilken slags man vil løse. Men der er ikke rigtig forskellige
opgaver til hver gang. Der er de samme geometriopgaver.
UM: Der er noget klassisk i det: Ens forældre bliver taget af piraterne og så skal man sådan redde dem. Jeg ved ikke hvordan
det ender. Jeg ved at piraterne dør. Ej det ved jeg ikke.
R: Men hvad mener du om historien så?
UM: Den er okay men sådan ret klassisk i det. Altså pirater tager forældre og så skal man redde dem.
R: Der kommer snart en 2’er
Several: Jaaa
R: som man kan spille via internettet! Hvad ville I sige til hvis I selv kunne vælge hvad for nogle lektier i skulle have for næste
gang, ville I så vælge den her Ruby hunt, altså 2’eren, eller ville I vælge et andet matematikspil? Eller ville I hellere vælge en
bog altså en almindelig form for lektier. (De bliver bedt om at tage hånden op ved hver kategori. Først spørges der om et
andet matematikspil. Ingen. Så spørges der om ruby hunt 2, alle tager hånden op.
97
F: Et computerspil er sjovere end normal, og man kan ikke rigtig sige et andet matematikspil fordi der skal jo være noget som
er i kontrast. Det eller det.
Slut
Atheneskolen 2:
Pre interview:
R: er matematik vigtigt?
T: Hånden op!
S: Ja selvfølgelig
S: Matematik er ret vigtigt men det er ikke rigtig mit yndlingsfag.
R: Så det er vigtigt selv hvis man ikke kan lide det?
S: Ja.
R: Er matematik nemt eller svært?
S: Matematik kan både være nemt og svært. (…) Men jeg elsker matematik. Jeg synes det er rigtig sjovt og der er så mange
muligheder og forskellige grader af det.
S2000: For mig er matematik meget meget svært. Men jeg tror at det er, jeg ved at det er vigtigt for at (…) Men alligevel, jeg
tror aldrig jeg kommer til at blive særlig god til det.
End.
Post interview:
R: Nu er vi færdige, slip keyboardene. Jeg vil gerne stille nogle spørgsmål nu.
Several: Noise and complaints that its over.
R: Hva synes I om spillet?
U: Jeg synes det var meget godt, det er også meget morsomt. Øhmm ja jeg kan meget godt lidt det.
U: Meget sjovt, men også lidt svært
U1: Det er fint nok. Måske lidt nemt i starten. Ret komisk faktisk. Bl.a. der er en dinosaur I en indhegning op der ved ørkenen,
lidt til højre for, og så der hvor båden synker og alt så noget og så krokodiller med våben.
U: Ja
U: yeah right (laughs)
U1: Det eneste dårlige var faktisk at man ikke kunne holde en af pilene nede der helt ude i starten i menuen. Og så scrolle ned
og ned og ned og ned. Sådan så den hele tiden begyndte forfra. Det er sjovt.
R: Kan I prøve at beskrive hvad for noget matematik var der i det her spil?
U: unintelligible
R: Og hvordan skulle man så løse det her matematik?
S2000: Der var regnestykker, det der med træerne. Jeg nåede at se at der var noget om geometri! Nogle firkanter og
trekanter. Jeg synes det var et rigtig sjovt spil fordi (unintelligible)… Det er rigtig godt, især for folk der gerne vil lære
matematik fordi du blander en morsom historie med vittigheder og sådan lidt forskelligt ind i matematik og det er rigtig godt
synes jeg.
R: gav det mening at lave matematikken inde i spillet? Eller var det meningsløst eller tåbeligt?
U: Jeg synes det var lidt mærkeligt at man var en Viking. (Laughs)
R: Hvordan føltes det at spille spillet? Var det ligesom at have undervisning eller var det ligesom at lege?
U: Altså det var lidt ligesom begge dele og jeg synes det var en god måde at lære på. Det var lidt ligesom hvis man laver det
der lyt og læs. Så det er en sjov ting blandet sammen med en ikke så sjov ting, det giver som regel en (unintelligible)
S2000: Altså hvis du kan gøre matematik til en leg så tror jeg det simpelthen at det kan (unintelligible). Hvis du kan lave
matematik til en sjov leg.
R: Lykkedes spillet med det?
S2000: Ja det gjorde det, og det er det værste ved det! Jeg er ret sikker på du kan sende det ud som et skolespil. Hvis man kan
få sådan noget der med cosinus og nasinus og alt det der og kan få det til at blive sjovt, så kan enhver lære det (changes his
voice to be more childish while blushing) og det synes jeg du skal gøre, Christian (looks at teacher whos name is christian)
U: laughs
R: Hvordan var historien?
U: Kommer spillet ud?
R: Nej. Blablal
U: Jeg syntes historien den var lidt komisk fordi (…) pirater. Den var sådan lidt sjov og lidt lam. Meget sjovt men alligvel lidt
underligt.
R: Kender i det spil der hedder matematik i måneby? Er det sjovere at lave matematik i matematik i måneby end at lave
matematik fra en matematikbog?
U: Hmm, det er det samme
R: Ok. Er det her spil sjovere end at lave matematik fra en matematikbog?
Several: Ja !
R: Tror I man lærer lisså meget?
Several: Ja, hmm. Neej. Der går lidt længere tid mellem opgaverne her.
R: Hvis I fik det her for som en lektie, hvor man skal spille en halv time, tror I så at I ville spille mere end en halv time?
U: Ja
98
U: Ja
U: Jeg har ikke nogen tidsfornemmelse!
R: Er der nogen der synes de hellere ville lære matematik med en bog eller et andet matematikspil?
U: Nope
S2000: Altså jeg tror jeg har lært noget af det her. Jeg er ikke så god til areal, trekanter. Så.. det er ret godt.
U: Det er et socialt spil!
Er der nogen der har nogle kommentarer eller spørgsmål?
U: Hvorfor hedder det ruby hunt? Når man nu kun laver sådan nogle ting som at måle areal. Hvorfor hedder det ikke pirate
hunt?
R: Replies.
S2000: Jeg synes det er sjovt at der også er noget kamp inde I det. Det er lidt mere som et spil og ikke bare som matematik.
Der noget der er mærkeligt, den ø man bor på hedder Matagaskar… Det staves ikke med T!
End
S2000 = Underperforming student. He eagerly engaged the game in the beginning and raced through the tasks, being
obviously experienced in playing games. However he quickly slowed down whenever he engaged the math problems.
Test Hedegårdenes skole. Kurt Lausen 5. A. 6/4-2011
13.05-13.50
Pre:
Unintelligible and noisy
Post:
R: HVa syntes I om spillet?
U: Jeg synes det var et godt spil
U: Det er et godt spil men det lukkede ned lige pludselig
UM: Det er et godt og lærerigt spil og det er godt at man nogle gange skal dræbe ting og sådan noget. Det er lidt irriterende at
man ikke går særlig hurtigt for hvis man skal gå lang tid så bliver det lidt trættende.
R: Hvafornoget matematik var der i spillet?
U: Man skulle finde arealet af en firkant og cirkler og trekanter.
U: Man skulle også minusse og bruge tabellerne og gange
R: Var det svært at forstå hvad man skulle?
U: Jeg synes ik det var svært at forstå hvad man skulle, det var først efter starten det blev sværere.
U: Jeg synes det samme, det var let i starten og lidt svært senere.
R: Var det svært ift modstandere, monstre osv?
U: De var lette. Undtagen rotten
U: Laughs
R: Asks students to raise hands if they would prefer a book assignment as their next homework assignment, another video
game, or more of Ruby Hunt. All except 1 raises hands at the Ruby hunt option.
U: Det ville også være let at snyde. Hvordan kan læreren se at man har gennemført det? Medmindre man kan gemme og gå
ind på den og vise hvor langt man er nået.
UF: Jeg foretrækker stadigvæk bogen.
R: Hvorfor?
UF: Det er et godt spil men der er ikke så mange informationer som i en almindelig bog synes jeg. Det er lidt sværere for
læreren at se hvad man kan og ikke rigtig kan. Det kan man med bogen synes jeg.
R: Tror du man lærer mere med bøgerne end det her?
UF: Ja…
R: Viser spillet om matematik er vigtigt eller ej?
U: Det er det for man bruger det i dagligdagen. Man har næsten altid brug for det. Også i fremtiden.
U: Jeg synes spillet giver et godt indblik på at matematik er vigtigt.
R: Var der nogen af jer hvor jeres båd den sank ned i vandet?
U, several: Ja
R: Hvorfor skete det?
U: Fordi man ikke skar dem ordentligt. Eller minussede forkert.
R: Var der nogen af jer der fik hældt korn ud på marken?
U, several: Ja
U: Kun på nummer 1
R: Ja? Hvordan gik det?
U: Det gik godt
U: Det er let med nummer 1. 2’eren er sværere
R: Hvad skete der hvis man regnede forkert med kornet?
U: Så blev kornet sort.
R: Så blev kornet sort… Giver det mening?
U: Nej
R: OK. Giver det mening at en båd synker hvis man regner forkert?
99
U, several: Ja
R: Er det fedt? Eller ville det være federe hvis man bare fik at vide ’du har regnet forkert, din båd er ikke god nok’. Eller er det
sjovt det der med at båden synker og man så skal prøve igen?
U: Altså nu prøvede jeg det ikke, men jeg synes det var meget sjovt specielt fordi det ikke gik ud over mig
UF: Jeg synes det var godt at båden sådan sank, altså den var jo ustabilt bygget, så synes jeg det var godt den sank så man
kunne prøve igen. Altså selvom min båd ikke synkede.
R: Det var det, jeg har ikke flere spørgsmål. Nogen spørgsmål?
U: Spørger om liv og hvad der sker hvis det løber ud. En anden snakker om at han døde. Andre griner ad ham.
U: En anden spørger hvorfor der er valgmuligheden i dialogen om at sige ’det gider jeg ikke’ eller ’nu går jeg’
U: Bliver det her udgivet så man kan købe det i forretninger senere og bruge det derhjemme for at lære noget nye ting i
matematik
R: Nej
U: Det er ellers synd. Det er ellers godt for born og unge at matematik godt kan være sjovt.
U: Hvorfor kan man ikke svømme?
R: Øhh det ved jeg ikke. Det har du ikke lært. Laughs.
U, someone else: Har du ik for meget rustning på?
U: Kan du ikke gøre sådan så man går lidt hurtigere?
R: Det kunne man godt, jo. Er der flere spørgsmål, kommentarer?
F: Altså jeg var en pige og jeg synes ikke rigtig min figur lignede en pige.
R: Nej
F: Kunne man ik få det til at se mere piget ud?
R: Ville det være…?
F, several: Ja
R: Ville det være godt hvis man også kunne vælge, altså hvad for noget tøj den skal have på eller hvad for en farve hår?
F, several: Ja
R: Ville det gøre det mere troværdig? Og historien bedre?
F, several: Hmm nej
F: Det ville bare være sjovere.
F: Spørger om kø-besked systemet. Og at man ikke kan trykke på dem. Hun mener de ligeså godt kunne komme som beskeder
i dialog systemet i stedet for.
U: Hvad sker der hvis man får alle de der… Hvad sker der hvis man kommer ind i ørkenen?
U: En anden studerende fortæller til forskeren og sine medstuderende om at der findes hjælpebeskeder til mark opgaven i
dialogen. Dette kan hjælpe ham som forsøgte at finde arealet af en cirkel. Forskeren fortæller at det er rigtigt, men dem der
handler om arealet af cirkel og trekant henviser blot til matematikbogen.
Teacher – Hedegårdenes Skole:
R: Du snakkede om successoplevelser?
T: Det er vigtigt de får en successoplevelse med at sidde og løse en opgave.
…
T: Spil har jo også noget med tekster at gøre. Så det er jo også en del af det vi snakkede om at læseforståelsen er også meget
forskellig. Og det er ikke sikkert at den der er dygtig til matematik har den gode læseforståelse.
R: Talks about RME
T: Vi bruger et matematiksystem der hedder Kontekst. Det lægger op til at børnene selv også finder deres algortimer. Det kan
godt være jeg kan vise noget, men alt er tilladt kan man sige. Når de så ganger eller dividere eller plusser eller minusser at de
selv finder den algoritme i første omgang der passer dem. På et tidspunkt kan det så være at de finder ud af at den algoritme
de har er uhensigtsmæssig til den måde stykket er sat sammen på.
R: Hvad siger du til at der ikke er nogen lommeregner eller formelsamling indbygget i spillet?
T: Det kommer an på opgavens art. For hvis opgaven ligesom lægger på at til at det er et spørgsmål om udregning så er det
klart, så skal der ikke ligge den der lommeregner. Men ellers kunne det sagtens være når de fx skal løse en cirkels areal så kan
man så godt sige at hvis de kan huske hvordan cirklens areal er, så er det jo lidt lige meget om de kan hovedregne den, så er
det jo fint nok at der er en lommeregner så de lige kan taste tallene ind, så får de et svar.
…
T: Vi er meget glade for at have fået lov til at prøve det her. Jeg synes det er spændende for børnene at se, som jeg sagde til
dig, vi arbejder jo meget med at fortælle dem at matematik ikke bare er for matematikkens skyld eller for min skyld, men det
er også et spørgsmål om at de i dagligdagen sådan kommer ind i at løse opgaver hvor de skal bruge matematikken
R: …
T: Du kan jo se hvor interesserede de rent faktisk er, altså de går ind i det lige – det der niveau der passer spillet jo rigtig rigtig
godt. De synes det er spændende og de går til den og de prøver og snakker – og der er jo også en god del af det at det at de
søger hinanden og snakker om hvordan løser du den opgave, hvordan gjorde du det.
100

Report - Jan Borg

Transcription

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