An investigation into Grade-6 responses to a random generator Paul L Ayres University of Western Sydney (Nepean), Australia. Paper presented at the Conference jointly organised by Educational Research Association Singapore and Australian Association for Research in Education Singapore Polytechnic November 25-29 1996. Paul Ayres Faculty of Education UWS (Nepean) PO BOX 10, Kingswood NSW 2747, Australia. Email p.ayres@uws.edu.au Tel 61 47 360784. Two groups of grade-six students were each shown a sequence of thirty trials where coloured cubes (6 brown, 3 white and 1 yellow) were drawn randomly from a box with replacement. After every five selections students were required to predict the next colour to occur. One group, which observed a high number (73%) of browns occurring, chose brown for most predictions. This group demonstrated probabilistic reasoning consistent with the odds; however the second group, which observed fewer browns (53%), but still a majority, made more random choices. Evidence emerged that the second group was influenced by the results of their predictions rather than the overall pattern of colours which emerged. In recent years there has been a movement in many countries (see Shaughnessy, 1992; Watson, 1995) towards incorporating more chance and data into the school curriculum. Many recommendations have been made to expose younger (grades K-6) children to the concept of probability. Coinciding with this direction, has been an increase in research on probabilistic reasoning (see Konold, Pollatsek, Well, Lohmeier & Lipson, 1993), with a focus on cognitive development. Highly influential to recent studies has been the earlier work of Piaget &Inhelder(1975); Fischbein (1975) and Green (1982). Although, research has so far failed to produce (see Konold et al.) a common model for probabilistic development, a number of significant findings have emerged. In particular, it has been found that primary-aged children have informal knowledge of probability (Carpenter, Corbit, Kepner, Lindquist & Reys, 1981); are influenced by culture (Amir & Williams, 1994); are intuitive (see Watson & Collis, 1994); and employ a number of heuristics (see Konold et al.,1993). The study reported here was designed to add to the literature on probabilistic development It is part of a larger project which is examining the processes employed by K-7 students in making decisions based on chance. In the following two related experiments, students (grade six) were asked to make a number of predictions based on a random generator. The decisions they made and the reasons why they made them were investigated. Experiment 1A Subjects Twenty grade-six girls from a Sydney Metropolitan girls school participated in this experiment. None of the students had been formally taught chance or probability. Method and procedure To prepare for the task ahead and to motivate the students, an introductory demonstration was completed using chocolate bars. In the main experiment, ten 2cm cubes (6 browns, 3 whites and 1 yellow) were placed in an opaque ice-cream container (14cm x 14 cm x 10cm) with no lid. A student was asked to select a cube, show it to the rest of the class, before returning it to the container (box). This procedure was completed a total of thirty times. The whole sequence was recorded on the whiteboard in the classroom. A space was left after every fifth colour. After every five selections, students were asked to complete the following: "I think the next colour will most likely be ........". The wording "most likely" was used to encourage students to make decisions based on their concepts of chance; however, it should be noted that children may interpret words, like "likely", differently to what is expected (Konold, 1991). Each student recorded their prediction on an answer sheet. Then they were asked to give the reason for her choice by ticking one of the following alternatives: a) It is this colour's turn; b) There are more of these colours in the box; c) There are less of these colours in the box; d) It is just a guess; e) More of these colours have already been pulled out; f) Less of these colours have already been pulled out; g) Some other reason .................. Reason (a) was included because people will often exhibit the "gambler's fallacy" (see Shaughnessy, 1992) of switching from a frequent event to a non-frequent event because it is the latter's turn. Reason (b) was included to see if students would at any stage make assumptions about the number of colours in the box. In contrast, Reason (c) was included to see if any student thought that low frequencies were an important factor, in much the same way as Reason a). Reason (e) was included to see if students would be guided by the higher frequencies without making assumptions about the contents. Again, (f) was included in contrast to (e). Reason (d) was included to gauge the extent of guessing, while (g) allowed students to express any other reason they wanted. After each phase of five selections, the above seven reasons were presented in a randomly scrambled format. This was designed to make students read the list each time and thus reduce the possibility of the same pattern being followed. In total, six phases of five selections were followed by a prediction and a given reason. It was anticipated that all three colours would appear at some stage; however, to incorporate the slight chance that a particular colour would not appear, a further question was asked- "do you think that there could be other colours in the box?". Students were then given a choice of three answers: yes, no or possibly. Even if all three colours did appear, this question would investigate whether students at this age could appreciate that there is a chance, even though small, that a colour may not have been selected. After this selection was made, students were informed that there were only brown, yellow and white cubes in the box. In a pilot study, trialed with Grade-7 students, it was observed that some students gave unrealistic estimates when asked to guess how many cubes were in the box. For example, some students would estimate that there were thirty cubes (the number of trials), even though it appeared improbable that there could be that many. In addition, when asked to proportion their estimate into the respective colours, some students gave answers which did not reflect the observed events; for example: 10 brown, 10 yellow and 10 white. To investigate these factors further, two other tasks were set. Firstly, students were asked to estimate how many cubes were in the box. They were given a choice of 5, 10, 15, 20, 25, 30 or over 30. After they had recorded their prediction they were told that there were ten cubes. Students were then asked to divide these ten cubes into the respective number of colours. It was appreciated that students of this age would have little experience of working with ratios (which may explain the results in the pilot study). Therefore to make the latter task easier, a number of alternatives was given as choices. All six combinations of the colours in the ratio 6:3:1 were listed, as was the possibility that there could be an equal number of colours (approximately). After this stage students were shown exactly what was in the box. It should be noted here that some research suggests that ratios (proportions) can be learnt at an earlier age than is normally expected (see Confrey, 1995; Spinillo, 1995; Watanabe, Reynolds & Lo, 1995). One other task was also set. The cubes were returned to the box and students were once again asked to predict the next colour selected.This trial was designed to explore whether students would change tactics compared with previous attempts when they knew exactly what was in the box. Results and Discussion The following sequence of coloured cubes occurred: BWBBB WBBBB BYBBB BBWBB BBBWW BWBBW W* * Underlined colours indicate when predictions were made. In the first 30 selections there were 22 Browns (73%), 7 Whites (23%) and 1 Yellow (3%) which contrasts with the theoretical percentages of 60:30:10 indicating that more browns occurred than might be expected. The predictions for each student are listed in the Appendix and a summary is shown in Table 1. Table 1 Summary of the number of colour predictions made in Experiment 1A Fifteen students (see appendix) chose brown for each of their six predictions. Two other students (listed as numbers 1 and 8 in the Appendix) made exactly one non-brown selection; whereas students 17 and 18 made two non-brown selections, and student 4 made three non-brown selections. Few white or yellow choices were made These results suggest that this group of students was very much influenced by the outcomes of the experiment. In particular it should be noted that brown occurred in the key 11th, 16th, 21st, 26th positions, and thus brown predictions were rewarded. All twenty students chose brown when they were required to make their last prediction following the revelation about the exact ratio of the colours. From the analysis of reasons given it was revealed that eleven students (Numbers 2-3, 5, 7-9, 13-16, and 20) made their prediction based on the frequency of the browns. Three students (Numbers 6, 10 &19) started out making their predictions based on this same reason before making the assumption that there were more browns in the box. Two students (numbers 1 & 18) made their decisions on more browns occurring except when they chose non-browns, which they attributed to guessing. One student (number 17) chose white as the eleventh selection because she had noticed that a pattern of four browns had emerged (this was the only time that a student offered a reason other than the ones listed). The remaining three students offered a mixture of reasons. For the task concerning an estimate of how many cubes were in the box, seventeen students chose 10 and three students chose 15. All the students estimated correctly the ratio of colours indicating that they believed that the experimental data would coincide with the closest ratios. These last two results were not consistent with the pilot study where many students gave unrealistic answers. It may be that as ratios and numbers were given as choices in this experiment, rather than asking the students to produce the numbers themselves, the task was made easier. Finally, nine students believed that there were no other colours in the box, two students believed there were other colours, while nine students believed that there was a possibility. Experiment 1B Subjects Twenty grade-six girls from a second class of the same Sydney Metropolitan girls school described in experiment 1A participated. None of the students had been formally taught chance or probability. Method and procedure The exact procedure described in Experiment 1A was followed. Results and Discussion The following sequence of coloured cubes occurred: WBBYW WYBBB YBBBB WWBBB BBBYW WBWYW W* * Underlined colours indicate when predictions were made. In the first 30 selections there were 16 Browns (53%), 9 Whites (30%) and 5 Yellows (17%) which compared with the theoretical percentages of 60:30:10 indicated that less browns and more yellows occurred than might be expected. This sequence was radically different to the one in Experiment 1A. The predictions for each student are listed in the Appendix and a summary is shown in Table 2. Table 2 Summary of the number of colour predictions made in Experiment 1B In this experiment no student predicted a brown each time. Four students (numbers 5, 7, 9 and 11) began predicting browns for three or more goes, but then chose other colours towards the end of the sequence. Each of these students cited a higher frequency of occurrence as her reason initially before changing to guessing. Although no student indicated that she had guessed for all of her choices, overall, guessing contributed fifty percent of the reasons given. All students selected a brown at least once. When a brown was predicted, 76% of the stated reasons were either made because of the frequency of previous brown selections or assumptions concerning more browns in the box. These reasons were not used in connection with any other choice of colour, except in two cases (white) following the first prediction, indicating that students did realise that more browns were occurring overall; however, students were not convinced enough to select brown more comprehensively. It should be noted that only one brown appeared at a prediction time (21st selection). In contrast three whites appeared at these critical times. It may well be this factor which influenced student selections rather than the overall sequence. More evidence to support this argument came from the prediction following observation of the exact ratios. Thirteen students selected brown, five selected white and two yellow. Even though students knew there was a majority of browns, seven students did not make a selection consistent with the odds. For the estimate of how many cubes were in the box, thirteen students chose 10, three students chose 15 and four students chose 5. Fifteen students estimated correctly the ratio of colours, four students thought the colours were in the same ratio while one student thought that there were more whites. Finally, seventeen students believed that there were no other colours in the box, and three students believed that there could possibly be other colours. Conclusions The two experiments produced radically different colour sequences. In the first experiment, browns were an obvious majority and students gladly predicted their occurrence. There is no doubt that the first group of students was guided by the experimental outcomes when making their decisions. As browns popped up in the critical positions students were rewarded for their selections and were not surprised by less likely outcomes. Little indication of the ÒgamblerÕs fallacy was observedÓ. In contrast, the students in the second experiment saw fewer browns (even though they were in the majority) and were not rewarded for choosing brown in the critical positions. The surveys indicated that students tended to guess more, rather than favour the most likely outcome. There is evidence to suggest that many students in the second group ignored the overall picture and concentrated on the patterns of the critical events. Many students changed their choice of colours frequently. Overall there is evidence to suggest that students at this age can exhibit probabilistic reasoning and link ratio with likelihood. However, it may be that grade-6 students, in a similar fashion to many adults, are highly influenced by their own rewards in situations that involve predictions. Acknowledgments The author wishes to acknowledge the help and advice given by Jenni Way in conducting the study. References Amir, G. & Williams, J. (1994). 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Teaching Secondary School Mathematics.Australia: Hardcourt Brace. Watson, J.M. & Collis, K.F. (1994). Multimodal functioning in umderstanding chance and data concepts. PME 18, Proceedings of the Eighteenth International Conference, Portugal, 369-376. Appendix: Selections made by each student in both Experiments