Twenty-five educative computer games for math education in elementary school ... motivating elements that are so typical for games may get lost. In this paper I ...
Hester A. Glasbeek Faculty of Arts/ Utrecht University Utrecht, the Netherlands Abstract Twenty-five educative computer games for math education in elementary school were analysed on the use of educational (RME) principles, instructional measures and motivational elements. The main objective was to obtain a deeper understanding of the genre of educational games. What are the distinctive features, how are these features realized in specific games and which features and realizations seem successful? A second objective was to explore how designers deal with the dual function of educative games: motivating and stimulating learning. Fifteen of the games in this study come from commercial applications, ten come from a website of the Freudenthal Institute, a scientific institute for math education in the Netherlands. I expected that the two commercial applications would accentuate the fun-aspects of games, while the Freudenthal-games would put more emphasis on the educative and instructional aspects. Some promising candidates for further research are identified. They include the integration of math problems into relevant real tasks, the use of means that can be gained or lost (like money), the presentation form of feedback, and the use of model progression. The expected difference between the commercial applications and the games from the Freudenthal Institute was only partly found. Keywords
Educative games, design, realistic mathematics education, corpus study 1 Introduction Interactive educative tools, like computer games, seem to offer many advantages. They can motivate learners by giving them all sorts of rewards: marks, records, certificates, compliments, applause. They can stimulate learners’ involvement in the learning process, by pulling them into a fictive world. They can also support the construction of adequate mental models, for instance by offering learners explanations “on the spot” and by encouraging them to make elaborations. A problem for designers is that the two main functions that are attributed to educative games –motivating learners and improving their learning- might be in conflict [1]. For players who were strongly involved in the game, it appears to be hard to reflect on concepts, principles or structures the game capitalizes on. Typical motivating elements of games competition, action/reaction, audiovisual elements- could lead to ‘unselective’ learning. Adding instructional measures to the game can prevent this for. As Knotts and Keys put it: “Students must be guided, prompted, motivated, and sometimes forced to learn from experiences” [2]. Although the need of such measures is beyond dispute, they also carry a danger: the motivating elements that are so typical for games may get lost. In this paper I will discuss a small corpus study into Dutch math games. In this study I explored how game-designers make use of motivating and instructional measures. The main objective
was to obtain a more elaborated view of the genre of educative games. What are the distinctive features of educational games, how are these features realized in concrete games and which features and realizations seem successful? A second objective was to explore how designers deal with the dilemma I sketched above. Because of the explorative character of the study I aimed at variety in the corpus. Ten of the games were from Math Web, an educative website for children in elementary school, designed by the Dutch Freudenthal Institute. These games are explicitly based on the theory about mathematics education that is currently dominant in the Netherlands: Realistic Mathematics Education. In the next paragraph I will describe this view shortly. The other games are from commercial publishers. I supposed that the Math Web-games might put a stronger emphasis on the instructional component (support learning), while on the other hand the commercial games might lay more emphasis on the motivational aspects (games should be fun). 2 Educational background: Realistic mathematics education Realistic Mathematics Education (RME) is a teaching and learning theory on mathematics education that was first introduced and developed by the Freudenthal Institute in the Netherlands. It fits into a constructivist view on education, although the development of the theory started before constructivism became popular. Realistic Mathematics Education relies on two key ideas about mathematics [3]. The first idea is that mathematics should be connected to reality. The word realistic does not just imply that mathematics should be related to the real world, it also means offering students problems they can imagine [4]. The second key idea is that mathematics is a human activity, which can be learned best by doing. Students should get the opportunity to develop all kinds of mathematical tools and insights themselves, instead of receiving ready-made mathematics [4]. Freudenthal called this the reinvention principle: students should experience a process that is in a sense comparable to the process by which mathematics was invented. This principle also implies that children work with selfdeveloped models, in order to bridge the gap between informal knowledge and formal mathematics. 3 What is a game? In this study computer games are explored as possible tools for supporting children’s process of “mathematizing”. Leemkuil & De Jong define games as follows [1]: Games are competitive, situated, interactive (learning) environments, based on a set of rules and/or a underlying model, in which, taking into account a number of constraints, under uncertain circumstances a challenging goal is aimed at. [my translation –HG] According to this definition games have the following characteristics [1, 6] • there is a challenging goal state that must be reached • there are constraints and rules involved • there is some form of competition • games are interactive; players get immediate feedback to actions • there is an element of insecurity: although the goal is clear from the start, players are not sure if they will reach it
•
games are situated in a context that is detached from the real world.
It are these characteristics that make games fun. However some of them –the competition element, some forms of feedback like sounds, scores and records- may distract students from explicit learning. To prevent from this, designers can build in instructional measures. Leemkuil et al. present a number of instructional measures that could play a role in learning from games [6]. The following measures are the most relevant to the tools involved in this study: 1. Model progression: introducing complex situations stepby-step. 2. Prompting and assignments. Prompts may take a number of forms. “For example prompts may be given to help the learner respond to a question and take the form of an answer or partial answer. Prompts may be used in a less directive manner, such as providing a rule or mathematical formula. Prompts can also be used to promote the learners’ self awareness or self-monitoring.” [7’] 3. Feedback on specific actions of learners. For example, the learning environment may check if all conditions for performing an action have been met, and if this is not the case report to the learner. 4. Additional information that helps learners to understand the game, for instance information about crucial game variables, or specific rules. Such information appears to be most effective when it is offered Just-in-Time [8]. 5. Monitoring facilities, offering learners insight into their discovery process. Examples are overviews of all actions that have been taken so far, notebook facilities for storing numerical and nominal data from experiments, facilities to ‘replay’ one’s own game. 6. Reflection and debriefing. By introducing a final phase of reflection and debriefing learners may acquire more explicit knowledge on the rules of the game and on the strategy that they have followed. Several authors stress the value of oral debriefing (initiated by an instructor), but learners may also benefit from written debriefing [9]. A third possibility is to build in a debriefing stage into the game. Apart from these measures Leemkuil et al. discuss several problem formats that can be used in games. They distinguish between product-oriented and process oriented problem formats. The conventional (product oriented) format is: confront learners with some givens and a (specific) goal. For instance: “Here you see eight fractions and eight diagrams. Select the right fraction for each diagram” (Digikidz: Fraction Race). However, various more process oriented formats are conceivable, that help learners to reflect on strategies and heuristics [10]. Some instances of these process oriented problem formats are worked-out examples, containing a description of a given situation, a goal and a good solution to reach this goal; completion problems, for which a given state, a goal state and a partial solution are provided to learners and goal-free problems, preventing learners from working backward from a specific goal, but forces them to move forward from the givens. This reduces cognitive load and may facilitate learning because learners can more freely explore the problem space. Like the instructional measures discussed above, the process oriented problem formats may carry the danger to ‘kill the game’. Finding a specific goal from the givens will probably be more challenging than exploring a goal-free problem. An example of the latter format is the first assignment in laying
tiles, a game on Math Web that aims at stimulating reasoning about symmetry: “With this program you can make tiled floors. Put the tile just where you want it. You can turn the tile, and you can also change the tiles´ colours.” It is likely that this problem definition indeed reduces learners´ cognitive load. On the other hand it is hard to combine such an open-ended problem with typical game elements like scores and records, simply because there are no restrictions; everything the child does is right. The challenge for designers is to find a balance between entertaining and encouraging explicit learning. To be able to reach such a balance we need to know more about the effects of motivating and instructional measures in games, both separately and in combination. We also need to know more about the best way to apply these measures. This paper aims to offer a small contribution to the study of these questions, by describing a number of existing applications. 4 Global description of the corpus The corpus consists from twenty-five educative computer games, designed for various levels in elementary school. The games come from three sources. Ten of them are from Math Web (“Rekenweb”), an educative website designed by the main advocator of Realistic Mathematics Education: the Freudenthal Institute. At this moment (04 May 2004) Math Web presents 46 games. Ten of them were randomly selected for this study. The other games are from commercial applications: FractionRom (“BreukenRom”, DigiKidz), and Monkey State (“Apenrijk”, Junior Detectives). The designer of FractionRom describes the aim of this application as getting acquainted with fractions [12]. Children learn what fractions are, how to recognize them, what the notation form means and how fractions are used/how to work with them. There are eight games on FractionRom. Seven of these games serve as exercises; the eighth, “FractionRace” can be seen as a test. Although the fraction skills of the intended users are in an initial stage, some prior knowledge is presupposed. FractionRom does not offer any explanations about concepts or notation forms; “the emphasis in the Digikidz-titles is more on practice than on instruction.” Monkey State is part of an adventure. The junior detectives Kim, Marco, Janet and Han are confronted with a mystery in the Numerian rain forest. The wood is polluted, there is a monster hanging around, and Kim’s uncle Horatius was kidnapped just at the moment he would explain these mysterious events. In order to rescue uncle Horatius the detectives need to find two keys. One of them is hidden in MonkeyState, a part of the wood where monkeys appear to struggle with seven kinds of math problems. Kim and Marco go (with the player) into the wood to search for the keys. The other two detectives stay in uncle Horatius’ home camp. You can contact them on a videophone and ask for help. The objectives of Monkey State, according to the designers, are to increase children’s skills with respect to problem solving and abstract thinking, and to appeal to their strong desire for mystery and adventure. 5 Analysis The games were analysed on the following aspects: • RME-principles • (motivational) game elements • instructional measures.
Some of the principles and measures needed further operationalisation. An example is the RME-principle Math problems should be connected to reality. To this principle two aspects were distinguished: (1) Does the game include realistic elements? and (2) Are the real context and task presented in this game relevant for the math problem to be solved? A talking frog for instance (Math Web) that asks you to come up with multiplying problems can be viewed as realistic, since it relates to children’s imagination. On the other hand interacting with a frog is not in itself a “multiplying generating” task, like treating cookies. So Math Frog, as the game is called, does meet the first, but does not meet the second aspect of the ‘reality’ principle. Another distinction I made is between motivational and instructional feedback. Leemkuil and de Jong mention feedback both as a game feature and as an instructional measure. In my analysis I viewed scores, records, rewards and disapprovals as motivational feedback. Of course children may also learn from knowing that their answer was wrong, but within the context of games rewards and disapprovals will predominantly intensify their desire to win. Other types of feedback like solutions, explanations and formulas are more directed at slowing down and stimulating reflection. These types are analysed as instructional feedback. (Partial) solutions, explanations and rules or formulas can also be presented before actions are performed. In that case, they are analysed as prompting. The analyses were based on the explanations and descriptions provided by the designers and on my own observations of the games. As a first step I examined to which learning objectives each game contributes, and how these objectives are distributed over the three sources I focus on: Math Web, FractionRom and Monkey State. Secondly I investigated which of the principles and measures that were discussed above can be recognized in these sources. 6 Results Learning objectives I found three types of learning objectives: reasoning (translating real problems into math problems), exploring new math problems or operations and practicing familiar operations. Table 1 shows the distribution of these objectives over the three sources. Table 1 Learning objectives MathWeb (n= 10) Reasoning 4 Exploring 4 Practicing 6 Testing 0
FractionRom (n=8) 0 7 8 1
Monkey State (n=7) 6 1 7 0
The two commercial applications, FractionRom and Monkey State, both are quite uniform with respect to learning objectives. The games on Math Web on the other hand show a wider variation. Six out of seven games in Monkey State present a reasoning problem. A real problem (a monkey wants to fence off a well with a rank. Which rank will fit?) must be translated into a math problem (calculate the perimeter of the well, by adding up the lengths of its four sides). Within the context of these real problems, children practice several mathematical operations. The only exception to the rule is “belt of volcano’s”,
the end game of MonkeyState where the access to the lost key can be gained. This game does not present any real problem; it consists of a series of math problems. Exploring and practicing seem to be its main objectives. On Fraction Rom all games are directed at practicing with more or less familiar operations. According to the designer most games (except for ‘FractionRace' , which is meant as a test) also aim at exploring fractions. Usability research is needed to find out if the games are appropriate for this aim; in my view they may be too difficult. One of the analysed games from Math Web is merely meant for practicing. This is Falling figures (“vallende sommen”), an educative variant on the well-known computer game Tetris. The other games seem to aim at reasoning/mathematizing or at exploring math problems, sometimes in combination with practicing. Use of principles and measures The opposition uniformity versus variety also applies to the use of motivational and instructional measures. The games from the commercial applications, Fraction Rom and Monkey State, are very uniform. There is one concept behind all games; all problems are introduced in a similar way, in every game you get the same type of feedback on the same type of actions (with small variations), the same prompts pop up in the same contexts et cetera. When you open a new game on Math Web it is far less predictable what you will meet. To a certain extent the variety on Math Web can be explained from the learning objectives of the games; various objectives ask for various principles and measures. On the other hand differences between the games do not always seem to be there on purpose; sometimes they make an arbitrary impression. RME principles. Table 2 shows to what extent RME principles play a role in the corpus. For each source is indicated in how many of the games the principle or measure described in the left column can be recognized. Table 2 RME-principles MathWeb (n= 10) Realistic 9 elements Relevant real task 5 Stimulate 4 reinvention
FractionRom (n=8) 2
Monkey State (n=7) 7
0 0
4 0
As for Math Web and Monkey State, almost all games have clear connections to reality. They refer to objects and referents children know: cakes, stars, frogs, clouds, monkeys, queens, snakes, et cetera. The games on FractionRom on the other hand hardly refer to the “real world”. Only abstract objects are used in these games, like blocks and strokes. The most realistic elements are the FractionMachine and the racing sounds that serve as feedback to correct answers in FractionRace. The absence of realistic elements on FractionRom implies that the two other RME-principles I focused on, relevant realistic tasks and stimulation of reinvention are also lacking. As expected, on Math Web these principles can be recognized. Five out of ten games present ‘real’ problems that are relevant for the math problem the game is about. A clear example is fairly dividing, a game for exploring fractions. In this game children are asked to divide a number of (for instance) pancakes among a (higher) number of children. In
other games however the relevance of the real task is not that obvious. Examples are ‘Crack the safe’ and ‘Mail’. Crack the safe aims at exploring divisibility of numbers. The code of the safe can be found by inserting the numbers by which the number in the middle can by divided. This is not a common feature of safe codes. In Mail multiplying problems are represented by a number of stamps on a letter, for instance six stamps of eight cents. If the answer is right, the letter gets stamped. Although on first sight the task seems to be related to the real world and to be “generating multiplying”, it can be doubt if it is meaningful to children. Stamps of eight (or two or four cents) do not exist. Most Dutch children will only be familiar with stamps of the standard tariff of 39 eurocents. So for anchoring mathematics in a real domain, this task does not seem the most appropriate. However, observation of real players is necessary to find out if they bother about these discrepancies from reality. For Monkey State the situation is similar; four of the ‘real’ tasks are apparently relevant, the other two are more artificial. The game Queen´s calendar for instance presents the following situation. Goatknitter, Queen Itchabella´s pet, wants to know on which day he will get feed. Since the indicator is on the floor, the players need to sort out the right day from the instructions of King Vasco. Vasco´s instructions specify a day and some features of the number, like: “it is a Tuesday, the date is an even number the numbers the date consists of add up to three.” Of course this is not a realistic process for determining a pet’s next feed. On the other hand, it is possible that children accept such “made up” problems because of the overall fantasy context (rather than on Math Web, where problems are presented in isolation). The third principle, stimulate reinvention, does not apply for Monkey State. The format of the reasoning problems in this adventure does lend itself for using models of your own, but children are not explicitly encouraged to do so. The designers of Math Web do intend to stimulate independent reasoning. This can be derived from their own descriptions of the games [11] and from instructional measures, like explanations from other children, or invitations to design your own problem and mail it to Math Web. On the other hand some games that are apparently designed for reasoning and exploring impose a ready made and pretty complicated model to the children. The clearest example is the game Cake. The designers describe the aim of Cake as exploring divisibility, in particular divisibility by 60. The task is to decorate a cake. Various decorations are available: cream, chocolates, fruit and sweets. Children can decide to make their own cake, or to replicate an example. The place where the next decoration will turn up is indicated in minutes. The essence of the game is that choosing a number of minutes that is dividable on 60 will result into a ‘neat’ (symmetric) cake, whereas choosing a number that is not dividable on 60 will produce a messy cake. “Playing with cake children should discover how to make neat cakes: be sure that the number of minutes is dividable on 60”[11].
Figure 1: “neat cake” (number of minutes is dividable on 60) Organizing a game around one specific discovery that is build in by the designers does not seem in line with the reinvention principle. Moreover I expect the clock metaphor to cause trouble. Most children will not draw the analogy between cakes and clocks spontaneously. Therefore I assume that the concept of minutes (that is hardly explained in the information about the game) will be confusing in this context. Motivational elements. Table 3 shows the distribution of motivational elements that are typical for games. As expected the two commercial applications, FractionRom and Monkey State, contain more motivational elements than Math Web. Table 3 Motivational (‘typical game’) elements MathWeb FractionRom (n= 10) (n=8) Challenging goal state Constraints & rules? Strict rules Means involved Competition Feedback? Reward/disapproval Scores Records
6
8
Monkey State (n=7) 7
6 1 2
8 1 8
7 7 7
5 1 0
8 8 1
7 0 0
The games from FractionRom and Monkey State all have a well-specified goal state. Besides there are scores involved (FractionRom) or means (Monkey State), which will add to children´s competitive feelings. Math web on the other hand uses more process oriented, goal free problem formats, especially for games with a reasoning or exploring objective. However, some of the games with more emphasis on practicing also lack a well-specified goal. An example is Math Frog: there is no closure to this game; the frog just keeps asking “do you know another multiply problem”? An advantage of this approach is its friendliness: there is no pressure, the frog will not make you feel stupid. On the other hand there is a considerable chance that children will get bored pretty soon.
An interesting feature of MonkeyState is the use of means. Each problem the children solve along their way through Monkey State yields a number of (red, blue, yellow..) berries. These berries are needed in the end game, the belt of volcanos. When you get out of berries in this game, you need to go back to one of the problems along the road, and gain some more. Means could have a very motivating effect, maybe even more than scores. Firstly they make the game more real and secondly they play with feelings of control and insecurity (your luck may turn; you can influence your chances by earning or losing means). On the other hand, if too many berries are needed to pass the belt of volcano' s this feature can get pretty frustrating A final observation about the use of motivational elements can be made on the presentation form of feedback. In most games from Math Web and FractionRom rewards and disapprovals have the form of anonymous sounds (a bell, a buzzer, applause from an invisible audience). In Monkey State they are verbalized by a monkey in trouble (“thank you, my tail is free!” or: “oh no, my poor tail”). This personal approach might be more engaging. Instructional measures. Table 4 shows the distribution of instructional measures. My supposition that Math Web would contain more instructional measures than the other two sources is not confirmed. Table 4 Instructional measures MathWeb (n= 10) Model progression Various levels 4 Other forms 4 Feedback solutions 1 Rules/formulas 0 Explanations 1 Prompting Hints 4 Partial answers 1 Possible solutions 2 Rules/formulas 1 2 Promote self monitoring Additional information Description of 8 aim and rules Just in time? 0 0 Monitoring facilities 1 debriefing
FractionRom (n=8)
Monkey State (n=7)
8 0
7 7
4 1 0
0 0 0
0 0 0 0 0
7 0 0 0 7
8
7
8 8
7 7
0
0
Math Web does show the greatest variety in prompting, but it has less instructional feedback than FractionRom. In contrast with the two commercial applications, Math Web does not offer monitoring facilities (the designers explain this choice by referring to the medium; it is hard to design such facilities for internet applications). Moreover the information about aims and rules of the games on Math Web is often less clear and sometimes absent. Some interesting differences were found for the measure model progression. Most of the games involved in this study can be
played on various levels. The underlying models of problems on higher levels are more complex than the models on lower levels. Before the game starts, the learner chooses the desired level. This can be considered as a learner initiated form of model progression; the learner decides if she is ready for the next level. In this respect the games in this study differ from ‘real’ computer games, where model progression is usually determined by the system: players must start at level 1 and deserve access to higher levels. A second distinction that can be made relates to the function of model progression. In the ‘real’ games described above, where learners need to deserve access to the next level, model progression will serve mainly as a motivating factor. In the educative games involved in this study, two functions of model progression can be identified. The first is relating to prior knowledge and skills. This function applies to most of the games; they enable learners to solve problems on their own level, without offering explicit help to reach a next level. The second function is support reasoning. Only a few games (all on Math Web) show this function. In those games model progression aims at a deeper understanding about a problem domain, by presenting various aspects, step by step. Although I expect that the progression in these particular games might be too fast (the problems seem difficult for children in elementary school and the games provide little additional help like solutions or explanations), this does seem an interesting function for educative games. 5 Conclusions and preview to the presentation In this corpus study I explored how designers of educative games make use of educational (RME) principles, instructional measures and motivational elements. The main objective was to obtain a more elaborated view of the genre of educative games. A second objective was to explore how designers deal with the dual function of educative games: motivating and stimulate learning. I expected that the two commercial applications in this study would accentuate the fun-aspects of games, while the applications from the (scientific) Freudenthal Institute would put more emphasis on the educative and instructional aspects. Various interesting features and differences were found. Some promising candidates for further research appear to be the application of the RME-principles (‘connect mathematics to reality’ and ‘stimulate reinvention’), the use of means, the presentation form of feedback, and the use of model progression. I formulated several suppositions about these variables. I assumed that games aiming at reasoning/mathematizing should not only include real elements, but also present a credible real problem, that is relevant for the math problem involved. Furthermore I suggested that the use of means might have a strongly motivating effect; that personal feedback, verbalized by a visible ‘problem owner’ may be more engaging than anonymous feedback and that model progression may be further exploited as a measure to support reasoning. As indicated above, observation of real players from the intended audience is necessary to find out if these assumptions have any foundation. I hope to present the first result of these observations in my presentation on EISTA. The expected difference between the commercial applications and the games from the Freudenthal Institute was only partly found. The two commercial applications do include more motivational elements, but the differences for the educational and instructional aspects are not that clear.
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