Summary: ChanceMaker represents probabilities through a range of gadgets. These gadgets calculate the probability of outcomes wrongly. The task of the students is to "fix" the gadgets so that they calculate probabilities correctly.
Focus: Games as meaningful contexts
Metaphor: User centred
Game or Gaming: Playing with toys, relationship with mathematics embedded in toy
Relationship with mathematics: Explicit;
Genre: Puzzle
media: Computer Game;
Holistic components: ?
Temporal components: actions
Mediation: face to face
Players: multiple at one PC
Game Facilitator: Human;
Game Time: Turn Based;
Intended Use: Edutainment.
Hardware: PC ;
User Number: multiple persons around 1 PC
Role: Co-operative
Userinterface: Controls: Mouse and keyboard;
Feedback: GUI;
Content: result (correct answer given);
Accessibility: Skill Level: Novice.
Context of intervention: Research group
Play Context: Local: Players to 1 PC;
Systemic/organisational: 1 classroom, border of curriculum, teacher initiative
Cultural: informative learning
Design and Development: Academic edu & Tech;
Production: Research prototype
Target audience: 11-14, academically-orientated, both genders, all ethnicity, non-gamer
Teacher: Researcher entering classroom
Instructions given to students on how to play game
When you first load up ChanceMaker, you will see 7 gadgets (Figure 1), representing a coin, spinner, dice, two spinners, two dice, a Frisbee and roll-a-penny.
Figure 1: The ChanceMaker gadgets
Which gadgets are working properly?
Drag the strength control of the coin in any direction. The number inside the strength control represents the strength with which you intend to throw the coin (0% to 100% strength).
You will see the coin spinning and it will land on “head” or “tail”
Repeat this several times.
You can also repeat a throw with exactly the same strength as last time by clicking directly on the coin.
Does the coin seem to be working as you would expect?
Repeat for each of the gadgets. Which gadgets seem to be working properly and which do not?
Try to mend the broken gadgets
Choose a gadget that seems not be working properly.
For example, if you think the spinner is not working properly, click on the button marked “open”.
You will see a range of tools to help you mend the spinner gadget (see Figure 2).
You can still use the strength control to make the gadget work.
Results will be collected in the results box.
You can graph the results, say as a pie chart.
You may wish to switch off the graphics and repeat many trials to see what happens.
You may wish to change the workings box, which controls how the spinner works.
Which gadgets do YOU want to mend?
Figure 2: The tools inside the spinner gadget
Students' Work
Exploring and mending the Coin, Spinner and Dice gadgets
Learners bring to statisitical situations expectations and understandings associated with everyday used of items likes coins, spinners and dice. In particular, such tools are usually connected with notions of fairness. The ways in which the tools of the Chancemaker microworld helped one pair of students connect these experiences with the notion of distribution are described here.
Exploring and mending the Coin, Spinner and Dice gadgets
In their first interactions with the Chancemaker microworld, Anne and Rebecca were asked to play with the different gadgets so they might gain a feel for them, but also to form some initial views as to which seemed to be working properly and which might be broken.
As they began these investigations, they tended to seek deterministic explanations of the phenomena they observed. For example, as they tried to make sense of the Coin Gadget, their first idea was to see if they could effect the tool’s outcomes, for example, by altering the strength control they. Because the results of such experiments were available in the gadgets’ Results box, Anne and Rebecca were able to see that altering the strength was not a determining factor in the outcomes. As their investigations continued, the availability of this list of results and especially its representation as a pie chart led to the students to begin to consider aggregated patterns. The speed with which the computer generated results also made it easy for them to consider large-scale experiments. Even so, there was some reluctance to do so. Rather the pair gradually accumulated of results, only increasing the number of trials each time as their confidence grew. After each coin tossing experiment, the resulting charts were consulted and comments were made about the evenness or otherwise of the pie chart or pictogram. This process resulted in a recognition that the pie chart became “more even” as more trials were included in the experiment, which might be interpreted as a situated the first articulation of the law of large numbers. Figure 5 shows how, in the pie chart produced after 200 trials, the number of heads and the number of tails are almost the same.
Figure 3: A pie chart produced after 200 “tosses” of the coin When they moved on to consider the Spinner Gadget, they did not immediately carry on experimenting with large numbers of trials. What Anne and Rebecca noticed first was the “unfair appearance” of both the workings box and the pictogram on which their first set of results were represented. As they considered the screen shown in Figure 4, the following discussion occurred:
Anne: Oh, look, it’s got to choose from different numbers.
Rebecca: That’s definitely more chance of it landing on one then.
Anne: Yes . . . There’s more chance of it because that’s (pointing to the 1 sector) much bigger than these (pointing to the other four sectors).
Figure 4: The first pie chart produced for the Spinner Gadget
Anne and Rebecca wanted to make the Spinner Gadget fair and this meant that the Workings box became a focus of their activity. They edited the workings to read: choose‑from[1 2 3 4 5]. However, they only chose to carry out only 50 trials and the pie chart did not appear fair, showing a larger 2 sector. At this point their teacher, Dave, intervened asking them how the pie chart might be made to “look more even”. He was expecting the girls would be provoked to think about the number of trials, but their first thoughts were to alter the workings again:
Figure 5: Testing the change to the Workings box Dave: If your aim was to make that pie chart look more even, what would you do?
Anne: I’d make the five a bit smaller. . .
Rebecca: I’d make the others a bit bigger.
In the end, they decide to replace the list [1 2 3 4 5] with [1 1 2 2 3 3 4 4 5 5] and then once again executed only 50 trials. On the pie chart produced this times, there were more 1s than anything else. They decided to remove a 1 from the list and executed another 50 new trials, this time the number 3 came out tops and the girls decided to go back to the original list [1 2 3 4 5]. Finally, Rebecca remembered the coin activity saying: Rebecca: Maybe throw it more times like we did with the coin? For the first time during the work with the Spinner, the number of trials had been cued as an important factor in controlling the results and their subsequent investigations led Rebecca to conclude:
Rebecca: There’s a higher number, so the more chance of it being even, I think . . . The more times you throw it, the evener it seems to get. And I think that’s because there’s more chances for a number to come up than if you do it say fifty times.
It would seem that although they had already noticed the importance of the number of trials during their interactions with the coin gadget, the need to make the Spinner gadget fair had initially encouraged a focus on the Workings box. This was important as it enabled them to fix the gadget, but initially distracted attention from the need to carry out a large number of trials. In the end, however, this emerged as important again because neither the configuration of the modified to workings nor any part of their naïve knowledge could satisfactorily explain the spinner’s unfair pie chart. The next gadget they worked with was the Dice. This time Anne and Rebecca used 1000 trials from the start of their work, but they seemed to be thinking that using a large number of trials will give what they call an “even” set of results, regardless from the configuration of the Workings box. When they obtained a pie chart with a large sector for 6s (Figure 6), they quickly located the problem in the gadgets workings:
Anne: I think sixes is popular because there’s quite a lot of sixes in the choose-from (pointing to the Workings box) . . . There might be a lot more sixes so it’s got more chances of getting more sixes on it.
Figure 6: Exploring the Dice Gadget
To make the Dice gadget fair, Anne and Rebecca changed its workings to choose‑from [1 2 3 4 5 6]. They carried out 1000 trials and the pie chart appeared fair. In the following discussion with their teacher, they showed an understanding of the relationships between the configuration of the Workings box and the pie chart expected after a large number of trials:
Dave: Let’s say we were playing a game, and for some peculiar reason in this game, it would have to be a computer game because we are using the computer dice, we wanted there to be a good chance of getting ones, and a fairly good chance of getting twos but a pretty low chance of getting anything else. It’s a strange game. How would we make this dice behave like that?
Rebecca: You have to put more of the numbers on here.
She begins to edit the workings until they read: choose‑from [1 1 2 2 3 4 5 6]. They test this out by repeating 1000 trials.
Dave: What will the pie chart look like?
Rebecca: More twos, more ones and less of the others.
Dave: How will the ones and twos compare?
Anne: Roughly even. Reflecting upon this sequence of activities, Anne and Rebecca gradually managed to connect two resources provided by the Chancemaker microworld, the possibility to execute a large number of trials and the distribution of choices as shown in the Workings box – and by doing so they developed a sense of a n invariant relation, invariant across several gadgets, that connected the number of trials, the configuration of the workings box and the appearance of the pie chart.
Working with compound gadgets
Two of the gadgets available in the Chancemaker microworld are compound gadgets. The Two Spinners gadget outputs the sum of the outcomes of two independent spinners (both of which have three sections) and the Two Dice gadget can be used to investigate the sum of the “throws” associated with two dice. Rather differently from the single tool gadgets like the Coin, Spinner or Dice, the outcomes of these two gadgets are not equiprobable. This tends not to be immediately obvious to learners, perhaps because of the strong emphasis on “fairness” associated with such tools – an emphasis that might initially be reinforced during investigations of the single tool gadgets. In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the Workings box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the Workings box (they did not notice the missing 1+3).
Figure 7: Exploring the Two Spinners gadget Having added these two possible outcomes to the Workings box, they repeated another 1000 trials, now expecting a more even distribution in the results. However, as Figure 8 shows, the pie chart they chose to display still showed unequal sectors, with less 2s than anything else.
Figure 8: The pie chart produced after the first modification to the Workings box
Anne wanted to add another 1+1 in order to make the pie chart fair. Rebecca was torn. On the one hand, she did think that the different totals should be equally likely, but on the other, when she thought about actually using the spinners, she could not see the justification for repeating the doubles in the Workings box. At this point the teacher, Dave, intervened, asking Anne to explain why she felt some of the possibilities in the Workings box were the same as each other. As Anne answers, Rebecca tries to clarify her thinking about the relationship between the choices in the Workings box and the spinners: Anne: Well, 1 and 2 and 2 and 1 are the same ... they come to the same number. Dave: They come to the same total, but are they the same as far as the spinners are concerned? Rebecca: No they are not. Because, the second one down, that number (pointing to the 1 of 1+2 in the Workings box) refers to that spinner (pointing to the first spinner), and that number (pointing to the 2 of 1+2) refers to that spinner (pointing to the second spinner). So, say, if that one (the first spinner) lands on 1 and that one (the second spinner) lands on 2, it would be three. And if that one (the second spinner) lands on 1 and that one (the first spinner) lands on 2, it would be three as well. However, even after this exchange, Anne still thinks they need to add 1+1 to the workings, “because 2 doesn’t come up as much, does it?” Rebecca seems to have been convinced by this and they did add another 1+1 and 3+3 although not 2+2 to the Workings box. The inconsistency of not adding an extra 2+2 to the workings was not a concern for Anne since her aim was simply to equalise the likelihoods of the different sectors, and hence make the gadget work properly in her view. After 1000 trials, the sectors for 2, 3 and 4 were fairly even (see Figure 9) and the teacher felt a need to re-problematise the situation so that Anne and Rebecca might question their solution.
Figure 9: The pie chart produced after the second modification to the Workings box He asks them to think about how real spinners behave and hence whether it is really so fair to have 1+1 in the Workings box twice. This seems to have provoked the pair to rethink what they were doing: as Ann put it “we don’t want it to be even, we want it to work like a real spinner”. The extra doubles were removed from the Workings box, and their attention was also drawn to the missing option, 1+3. With this revised Workings box, they once again requested 1000 trials, accepting that what they were expecting to see in the pie chart was less 2s and 6s than other numbers. The pie chart did have smaller sectors for these numbers and the largest sector was for the 4s (Figure 10). Figure 10:The pie chart produced after the second modification to the Workings box
To explain the unpredicted finding related to the number 4, Anne and Rebecca looked directly at the frequency of the total 4 in the Workings box and argued the section on the pie chart is a result of this: Anne: It’s written there three times ... Because it’s got more of the numbers. It’s got like three different numbers so it’s coming up much more. When the teacher asked them whether they still felt that the different totals, 2, 3, 4, 5, 6 were equally likely, they answered together “4 is easier to get” and agreed that the hardest numbers to get are 2 or 6. Their experiments with the gadget had helped them move from a view that all outcomes associated with a gadget should come up more or less the same number of times, to a new situated abstraction: The more often a total is represented in the Workings box, the larger will be its sector in the pie chart. To assess their commitment to this idea, the teacher initiates the following discussion: Dave: So, if we were going to play a game, in which we had two spinners like this, and you are going to bet a pound and I am going to bet a pound. And you’re going to bet that a total of 4 comes up, and I’m going to bet that a total of 6 comes up. Would you take that bet? Anne: No ... well, sort of ... I wouldn’t bet as much as a pound. Dave: How much would you bet? Anne: 20p. Rebecca: 2p. Dave: If we were to bet 2p the other way round that I’m betting on a 4 and you’re betting on a 6, would you take that bet? Rebecca: No. Anne: Definitely not.
The next gadget they considered was the Two Dice gadget, the second compound gadget. They noticed immediately that there were missing data in the Workings box. However, despite the fact that they seemed to know it is important that all the possible combinations of dice throws are represented in this box, they also thought that the chance of getting any particular total should be about the same. This shows the strength of the equiprobability bias; it still appears to be present in their thinking even after all the work with the Two Spinners gadget. They manage to incorporate all of the 36 combinations for the two dice totals into the Workings box and decided to run 1000 trials. As they began to make predictions about the outcomes, they also began to make connections with the results from working with the other compound gadget:
Anne: Fairly even? ....Some of the numbers might not be because there’s not as much as the other number. Rebecca: Maybe roughly even because now that we have got all the sums. I’m not too sure at the moment. Anne: I think some will be a bit less because they haven’t got as much as the others ... because some of the numbers will not be the same, will be less, because we didn’t find enough sums for them ... like 1 add 1. Dave: Can you give me an example of one that had a lot of different ways of getting it. Anne: Seven.
The pie chart showed that 7 was indeed the total most frequently obtained and that the totals 2 and 12 were the least common (see Figure 11). To verify the gadgets working they considered the possibility that they may have missed out some of the ways of obtaining these totals, but, during their inspections of the Workings box, they ascertained that there was indeed only one way to obtain a total of 2 or 12, whereas a 7 could be obtained in a number of different ways.
Figure 11:The pie chart produced after ‘fixing’ the Two Dice gadget
As their work with the compound tools progressed, Anne and Rebecca maintained a commitment to obtaining a ‘fair’ set of results, but were beginning to see that not all result sets involve an equal distribution of the possible outcomes and in particular that the size of each sector of the pie charts produced by the gadgets is determined by the frequencies of the corresponding totals in the workings. While the equiprobability bias has not disappeared, and will quite possibly resurface again, the students now have other resources through which they can correctly interpret the functioning of compound gadgets.
Key Mathematical Ideas
Probability is unusual in many respects. Unlike much of mathematical language, the language of probability is present in many different kinds of out of school activities: sports commentators talk about a 50/50 ball, weather forecasters announce an 80% chance of rain; health is assessed in terms of risk factors based upon probabilistic calculations. Indeed, probability is one of the few areas of mathematics that explicitly informs the way we conduct our everyday lives. Yet, although probability is part of the mathematics curriculum for learners from age 7 onwards, because experiences of randomness and variation in the mathematics classroom are frequently limited to situations for which students already have an intuitive feel, they may not be challenged to recognise the differences between theoretical and experimental probabilities or to understanding the powerful connections between randomness, the law of large numbers and the notion of distribution.
The idea behind the Chancemaker microworld is that students will be motivated to go beyond their intuitive feelings about probabilistic situations and, by examining and modifying different gadgets, mini-computational devices that simulate everyday random generators (a coin, a spinner, a dice and so on), they will connect what they already believe about the behaviour of such devices with an emerging sense of distribution as mathematical structure. The gadgets provided within the microworld are modelled upon devices usually seen as unbiased or fair, associated with the production of a set of equiprobable outcomes. In the microworld activities, however, these gadgets are often encountered in a ‘broken’ state – some sort of bias has been inserted into their operation and microworld users are challenged to identify what is wrong with their workings and to use the tools of the microworld to mend them.
When experimenting with any one of the gadgets, the user can control the numbers of trials to be performed, so they can see how patterns in outcomes become more stable when experiments are repeated a large number of times. The results can be displayed as lists, or as pictograms or pie charts. To mend a gadget, the user has access to its ‘workings’ – the list of the choices from which the values outputted are chosen. By changing this list, the user can change the list of possible outcomes as well as the number of times each possible outcome appears. This enables them to attend to possible biases in the way the gadget works. Each gadget also has its own strength control, which permits the user to control how hard the coin, spinner or dice is thrown or tossed. The strength tool is included so learners can experience how increasing the strength only makes the simulation last for longer, it does not effect the distribution of the outcomes generated.
Mending Gadgets
In order to inspect a particular gadget, it needs to be opened. Figure 1 shows the microworld screen after the dice gadget has been opened. The dice is “thrown” by clicking on its image and the result of each throw is listed in the Results box. The list in the Results box can also be displayed as a pictogram (Pictogram button) or as a pie chart (Pie button). Using the Controls box, the number of trials in an experiment can be altered (in Figure 12, it is prepared for an experiment of only 10 trials). The results will be accumulated until a new experiment is begun (New button).
Figure 12. The tools in the dice gadget (Chancemaker dice.jpg) The Workings box shows the computational core of the gadget. In this particular case, the workings have been configured so that the dice “chooses” from the list 1, 2, 3, 4, 5, 6, 6, 6. The choose-from command executes a process in which a member on the given list is chosen at random: in the configuration of the tool shown in Figure 1, the dice will have a bias towards producing sixes – this is because of the eight items in the list, three are 6s as opposed to only 1 of each of the other numbers. Figure 13 shows the pie chart produced after an experiment involving 800 trials with the biased dice (we would expect the number 6 to occurred 300 times and the other numbers 100 times). In order to make a fair dice, the microworld user needs to think how its Workings box should be edited. Figure 14 shows the pie chart produced in an experiment with the dice after the Workings box has been changed so that the dice “chooses” from the list 1, 2, 3, 4, 5, 6.
Insert Figure 13. The results of an experiment involving 800 “throws” (Chancemaker dice2.jpg)
Insert Figure 14. The results of an experiment involving 800 “throws”with revising the Workings box (Chancemaker dice2.jpg)
In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3). In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3). In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3). In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3).
In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3). In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3). In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3). In Anne and Rebecca’s initial work with the compound tools, their first interpretations of the results from working with the Two Spinners gadget suggested they were guided by an equiprobability bias. As they watched the generation of results for their first 1000 trials of the gadget, they were expecting that 3s and 4s would appear more frequently than other numbers, not just because of the emerging empirical results, but also because of the choices available in the box and they clearly felt that the overabundance of 3s and 4s was something that needed correcting. The pictogram they constructed at the end of the experiment (Figure 7) confirmed their prediction about the numbers 3 and 4 and also indicated a complete absence of the number 5. This led Rebecca to notice that the possibilities for 2+3 and 3+2 were missing from the box (they did not notice the missing 1+3).
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