Understanding of Patterns, Streaks and Independence by Grade School Children by Peter Vargas

Understanding of Patterns, Streaks and Independence by Grade School Children KEYWORDS: Teaching; Representativeness heuristic; Law of small numbers; Coin tossing.

Carolyn Ridgway Merced, California, USA. e-mail: ridgways@mindspring.com

Derry Ridgway SRA International, Research Triangle Park, North Carolina, USA. Summary Using sequences of coin flips as a model of serial independent events, we asked schoolchildren in grades 1 through 5 to estimate the likelihood of the next flip. Most children in each grade expected short patterns to continue.

䉬 INTRODUCTION 䉬

dults and older students may invoke errorprone heuristics to arrive at incorrect estimates for the likelihood of chance events (Kahneman et al. 1982). In this article, we use a simple coinflipping experiment to examine an error-prone heuristic related to the independence of serial events in schoolchildren from grade 1 through grade 5 (US ages 6–11). We characterize the error-prone reasoning as follows: The subject reviews a short sequence of independent random events with the goal of predicting the likelihood of the next event. Subjects well grounded in the notion of independence will not attempt to extrapolate any pattern evident in the short sequence to the next outcome. Subjects less well grounded in independence may assign likelihoods by erroneously reasoning from the results of the immediately preceding outcomes. The expectation that brief samples will closely reflect the theoretical overall set of outcomes for longer sequences was called the “representativeness heuristic” by Kahneman and Tversky (1979) and has also been referred to as the “law of small numbers” (Burns 2004; Burns and Corpus 2004).

The concept of independence is integral to understanding probabilities associated with more than a single event (Barnes 1998; Meletiou-Mavrotheris and Lee 2002; Metz 1998a; Polaki 2002; Rubel 2006; Shaughnessy 1992; Tarr and Jones 1997). However, independence does not appear in the National Council of Teachers of Mathematics (NCTM) curriculum standards until secondary school (NCTM 2000), and the teaching of independence in school children is given little attention 34

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(Barnes 1998; Hawkins and Kapadia 1984; Meletiou-Mavrotheris and Lee 2002; Metz 1998a; Shaughnessy 1992; Tarr and Jones 1997). Using a sequence of coin flips as our model for random events, we asked approximately 100 schoolchildren to estimate the likelihood (heads more likely, tails more likely, heads and tails equally likely) for the seventh flip in a series after reviewing the results of the first six flips. We hypothesized that in our experimental model, young students would exhibit the representativeness heuristic and, in the absence of curricular attention, older students would continue to exhibit this error even as other nondeterministic reasoning skills increased.

䉬 METHODS 䉬 The subjects were ninety-eight students from grades 1 to 5 in a 400-student public elementary school in North Carolina. At the time of the study, mathematics teaching followed the 2003 K-12 Mathematics Standard Course of Study for the Public Schools of North Carolina from the State Board of Education (current standards at http://www.ncpublicschools. org/curriculum/mathematics), based on the NCTM Principles and Standards for the Teaching of Mathematics (2000) (http://standards.nctm.org). Both standards include five strands: Number and Operation, Measurement, Geometry, Data Analysis and Probability, and Algebra. For this study, the © 2010 The Authors Journal compilation © 2010 Teaching Statistics Trust

Fig 1. Worksheet with four sequences of coin flips, completed by 1st through 5th graders

Data Analysis and Probability and Algebra strands are relevant. Teachers reported that students in all grades had been exposed to lessons involving manipulatives (dice, coins or spinners), data recording and probability of individual events. The manipulatives used and the concepts developed varied by grade level. The North Carolina first grade standards call for descriptions of events as “certain”, “more likely”, “less likely” or “impossible” (section 4.02); the first grade class had experience with spinners but had not studied coin flips. The second grade standards call for probability experiments, description of results and making predictions; the second grade class performed coin-flipping experiments. The third grade standards add no new objectives; the third grade classes had performed experiments with both spinners and dice. The fourth grade standards call for experiment design and lists of all possible outcomes for events; the fourth grade class based their designs and experiments on coin flipping. The fifth grade standards added no new objectives; the © 2010 The Authors Journal compilation © 2010 Teaching Statistics Trust

fifth grade class based their designs and experiments on dice. These objectives and class curricula are consistent with the recommendations in the Guidelines for Assessment and Instruction in Statistics Education endorsed by the American Statistical Association (Franklin et al. 2005). In each class, students recorded the outcome of single events and summarized the aggregate results for themselves and for the class. Sequences of events and, hence, the concept of independence are not part of the standards for grades 1 through 5. However, curricular standards for the first through fifth grades do call for the creation and extension of patterns, both numeric (from grade 1) and geometric (from grade 3), within the algebra strand, which will be seen as relevant in our investigation below. Near the end of the school year, students from one or two classes at each grade level completed a worksheet (figure 1) as a class lesson under the direction of the first author using a transparency on an overhead projector. Each student had a copy of the worksheet. Students were shown a nickel and told Teaching Statistics.

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that it was a fair coin that was equally likely to land with heads up or tails up (the introductory sentence on the worksheet was read aloud). The nickel was flipped six times. After each flip, the students recorded the outcome in the first row of circles. They were then asked, if the coin were flipped one more time, would heads be more likely, would tails be more likely or would they be equally likely? The students recorded their answers. In the first and second grades, they wrote H, T or = in the circle with the question mark. The third, fourth and fifth grade students recorded their responses in the answer box. The students were then asked to look at the second sequence of flips (THTHTH). The lead-in sentence from the worksheet was read aloud, and the students were told that the same fair coin was used that was equally likely to land heads or tails. The sequence was read aloud and the students were then asked, if the coin were flipped one more time, would heads be more likely, would tails be more likely or would heads and tails be equally likely? The students then recorded their answers. Older students were also asked to write why they chose the answer they did. This process was repeated for sequences 3 (TTTTTT) and 4 (TTTTTH), with a reading of the lead-in sentence on the worksheet for each sequence and the sequence of six flips, but without the reminder about the meaning of a fair coin. At the end of each session, the students were asked if anyone wanted to tell what they answered for each sequence and why. Their oral replies were recorded verbatim by an observer who attended each of the lessons.

䉬 RESULTS 䉬 Student responses are summarized by sequence and grade in table 1. For sequence 1, students witnessed the six coin flips and were asked to respond with the most likely outcome for the seventh flip. Sequences were different in each class; none had an obvious pattern and none terminated with a streak of greater than 2 similar outcomes. The percentage of students correctly responding ‘=’ for sequence 1 increased with grade level (age) and was greater than or equal to 50% for the 3rd, 4th and 5th grades. A similar pattern (generally increasing numbers of correct responses in higher grades) was seen for 36

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responses to sequences 2, 3 and 4, although the overall rates of correct responses were lower, in all grades, than the percentages for sequence 1. Four students (one in 1st grade, one in 4th grade and two in 5th grade) responded ‘=’ for all four sequences. Two students (one in 4th grade and one in 5th grade) responded ‘=’ to three sequences and ‘T’ for sequence 3 (TTTTTT). For sequence 1, students witnessed the flips of a normal-appearing coin and were not distracted by results that called upon a pattern-based decision heuristic. Our hypothesis was that the apparent patterns in sequence 2 (THTHTH), sequence 3 (TTTTTT) and sequence 4 (TTTTTH) led students who understood that heads and tails were equally likely to apply an inappropriate decision algorithm. To examine this hypothesis, we looked at the responses to sequences 2, 3 and 4 for students who responded ‘=’ to sequence 1. The alternating outcomes in sequence 2 and the streak outcome for sequence 3 each might suggest a next outcome of ‘T’. The percentages of ‘T’ responses to sequence 2 and sequence 3 among students who answered ‘=’ for sequence 1 are shown in figures 2 and 3. In figure 4, the percentages of students responding ‘H’ or ‘T’ to sequence 4 are shown for students who responded ‘=’ to sequence 1.

䉬 STUDENT REASONING 䉬 The most common explanations involved the word ‘pattern’ (“In the pattern, tails is next”), followed by explanations based on counting the number of heads and tails already in the sequence (“because 1 is heads out of 6” or “heads because it had too many tails”). If the number of heads and tails in the sequence were equal (sequence 2), then some students selected equally likely for the next toss, based on a counting analysis. A 5th grader explained, “they both have 3 and that’s equal”. There were other instances of choosing equally likely using incorrect reasoning, e.g. if both heads and tails have already come up, it is possible that either may again. This reasoning allowed students to respond tails to the sequence of 6 tails (sequence 3), while responding equally likely to sequence 4, TTTTTH, because “this time she got both” and “because she has done both of them”. Four students responded equally likely to all four sequences. Their explanations were similar. A 1st © 2010 The Authors Journal compilation © 2010 Teaching Statistics Trust

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7 (41%) 10 (59%) 7 (41%) 0 10 (59%) 1 (6%) 1 (6%) 15 (88%) 1 (6%) 2 (12%) 14 (82%) 9 (18%) 3 (6%) 39 (76%)

3 (15%) 3 (15%) 14 (70%)

2 (10%) 3 (14%) 16 (76%)

1 (5%) 13 (62%) 7 (33%)

6 (9%) 19 (31%) 37 (60%)

Grade 2 (n = 17)

4 (20%) 16 (80%)

Grade 1 (n = 20, 21)a

Table 1. Summary of student responses

Two of 21 1st grade students omitted one response. Coins were flipped in the presence of the student; results varied by classroom. c Response of either heads or tails. d Percentage calculated from the total of all responses.

Sequence 1 = Not equalc Sequence 2 (THTHTH?) = Heads Tails Sequence 3 (TTTTTT?) = Heads Tails Sequence 4 (TTTTTH?) = Heads Tails Sequences 2, 3 and 4 =d Headsd Tailsd

8 (10%) 4 (5%) 66 (85%)

2 (8%) 3 (12%) 21 (80%)

0 1 (4%) 25 (96%)

6 (23%) 0 20 (77%)

13 (50%) 13 (50%)

Grade 3 (n = 26)

11 (20%) 7 (13%) 36 (67%)

3 (17%) 4 (22%) 11 (42%)

1 (6%) 2 (11%) 15 (83%)

7 (39%) 1 (6%) 10 (55%)

16 (89%) 2 (11%)

Grade 4 (n = 18)

18 (35%) 4 (8%) 29 (57%)

10 (59%) 1 (6%) 6 (35%)

3 (18%) 2 (11%) 12 (71%)

5 (29%) 1 (6%) 11 (65%)

9 (53%) 8 (47%)

Grade 5 (n = 17)

52 (18%) 37 (12%) 207 (70%)

17 (17%) 23 (23%) 59 (60%)

7 (7%) 9 (9%) 83 (84%)

28 (29%) 5 (5%) 65 (66%)

49 (50%) 49 (50%)

Overall (n = 98, 99)a

100 80

3/4 9/13 4/7

29/49

9/16

4/9

䉬 DISCUSSION 䉬

40 20 0 1st

2nd

3rd

4th

5th

Total

Grade

Fig 2. Fraction (percentage) of ‘T’ responses for sequence 2 (THTHTH) among students who responded ‘=’ to sequence 1

7/7 100

12/13

14/16

40/49

80 60

5/9 2/4

40 20 0 1st

2nd

3rd

4th

5th

Total

Grade

Fig 3. Fraction (percentage) of ‘T’ responses for sequence 3 (TTTTTT) among students who responded ‘=’ to sequence 1

100 6/7

Predicting H Predicting T

10/13

11/16 29/49 60 2/4 40 1/4 20

2/16 1/13

1/9 1/9

6/49

0 1st

2nd

3rd

4th

5th

Total

Grade

Fig 4. Fraction (percentage) of ‘H’ and ‘T’ responses for sequence 4 (TTTTTH) among students who responded ‘=’ to sequence 1

grader stated, “it’s possible to get each one because it’s on the same coin”. These four students invoked the concept of independent events, without using the word. One 5th grade student consistently invoked the idea of a fair coin. He explained that “it’s a fair coine” for his responses of equally likely to sequences 1, 2 and 4, and decided for sequence 3 that “it’s probable a nonfair coine” so he chose tails as being more likely. 38

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No student used the word independent in his explanations or mentioned the idea that past events need not affect future events.

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In our sample, only a few 1st grade students were able to use nondeterministic reasoning in the simple case without a pattern (sequence 1), while 50% or more of students in grades 3 and above had reached this level of understanding. Our subjects, without didactic or empirical grounding in the concept of independent events or the likelihood of apparent patterns in a series of random independent events, reviewed the sequences of flips in sequence 2, sequence 3 and sequence 4 and substituted an alternative analysis for the correct independence-based response of ‘equally likely’. The error of reasoning was apparent in the small number of very young (1st and 2nd grade) children capable of probabilistic thinking, and persisted as the prevalent pattern of thinking through grade 5. Students’ comments suggested that three distinct alternative analyses were used. 1. The most common alternative analysis could be called ‘extend the pattern’; students who extended the pattern were applying a natural human reasoning technique, as suggested by Barbara Tversky and others (Burns and Corpus 2004; Freeman 1996; Horvath and Lehrer 1998; Metz 1998b; Rubel 2006; Steen 1988; Tarr and Jones 1997), or were employing a patternidentifying skill that receives explicit and strong reinforcement in the Algebra strand of curriculum standards (NCTM 2000; North Carolina State Board of Education 2003). Reasoning by streak extension underlies the belief of basketball players and observers that players sometimes possess a ‘hot hand’, accounting for a series of successful basket attempts (Gilovich et al. 1985). 2. Less often, our subjects used a counting analysis to conclude that the less frequently appearing outcome will be more likely to appear in the next flip (the ‘gambler’s fallacy’) or that because equal or nearly equal numbers of Hs and Ts have appeared so far, heads and tails are equally likely on the next flip (a correct response wrongly arrived at through the application of the law of small numbers). 3. Finally, one student used a counting method to make an explicit determination that the coin was not fair, contradicting the declared worksheet conditions, and leading to a response of T after TTTTTT. © 2010 The Authors Journal compilation © 2010 Teaching Statistics Trust

Topics in probability and nondeterministic mathematical thinking are included in the recommended curricula from the earliest grade (NCTM 2000; North Carolina State Board of Education 2003). In response to the questions on our instrument, a few children in 1st grade and half or more of children in grades 3 and higher were able to answer ‘equally likely’ for the next toss in the first sequence. The percentage of students able to give this response generally increased from grades 1 to 5, consistent with the pattern observed for older students by other researchers (Rubel 2006). In Rubel’s interviews with junior and senior high school students about the outcome of serial coin flips, independence of successive flips did not appear in students’ reasoning until the 9th grade (Rubel 2006). In an analysis of responses to questions about probability concepts given to nearly 3000 high school students, and follow-up interviews of 44, Green (1983) found that while other aspects of statistical reasoning improved with age, the ability to reason correctly about events in sequence was uncommon at all ages. Only 4 of the 98 students in our study appeared to reason from the concept of independence in all four series. In our study, the invariable correct response was ‘=’, reflecting both the fairness of the coin and the independence of each flip. Students in all grades showed a willingness to change their probability estimates based on the outcomes of short preceding sequences – extending patterns and streaks or using counts of recent results to make future judgements. Our data suggest that the reasoning error is present from the earliest time that students are able to reason nondeterministically. There is little published research related to earlygrade curricula that could address questions of serial independent events. Kjartan Poskitt provides a discussion that might assist teachers constructing relevant lessons (“coins have no memory”; Poskitt 2001). Focused lessons in the early grades might allow young elementary-school children to resist development of the error-prone heuristic. Directed hands-on lessons could be designed around the concept of independent events, the likelihood of patterns and streaks in a series of independent chance outcomes and the extension in reasoning required for events that do not have equally expected outcomes. Both the occurrence of patterns and reasoning associated with unequal probabilities could be taught with explicit didactic lessons and with experience-based, hands-on lessons. Focused lessons with student-generated data could include the following: © 2010 The Authors Journal compilation © 2010 Teaching Statistics Trust

• a hunt for patterns in a long sequence of out• • • •

comes from coin flips, generated over several days by members of the class a review of unlikely results from sequences of outcomes from an unbalanced spinner finding patterns in sequences of outcomes from ‘real-world’ events based on student skills (such as basketball free throws) forecasting games for sequences of coin flips or spinner outcomes (balanced or unbalanced) individual or class discussions to expose the potential for both correct and incorrect reasoning to provide correctly reasoned forecasts.

䉬 CONCLUSION 䉬 The number of students able to perform nondeterministic mathematical thinking increases with age. On our 4-item instrument, a majority of students in 3rd grade and above were able to respond ‘equally likely’ for flips of a fair coin performed in their presence. But when sequences of only six flips looked unlike the students’ expectations for chance, students in grades 1 through 5 failed to invoke the concept of independence, favouring instead an error-prone heuristic, to predict the outcome of a seventh flip. Students extended patterns and streaks, perhaps misapplying skills taught as part of the algebra curriculum; a few students based their responses on counts of recent outcomes. Acknowledgements We are grateful for the assistance of Ms Aerin Ridgway in the collection and processing of responses, and for the encouragement of Dr Marion Walter in the completion of this manuscript.

References Barnes, M. (1998). Dealing with misperceptions about probability. Australian Mathematics Teacher, 54(1), 17–20. Burns, B.D. (2004). Heuristics as beliefs and as behaviors: The adaptiveness of the “hot hand”. Cognitive Psychology, 48(3), 295–331. Burns, B.D. and Corpus, B. (2004). Randomness and inductions from streaks: “Gambler’s fallacy” versus “hot hand”. Psychonomic Bulletin and Review, 11(1), 179–184. Teaching Statistics.

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Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M. and Schaeffer, R. (2005). Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report. Alexandria, VA: American Statistical Association. http://www.amstat.org/ education/gaise/ (accessed 8 October 2007). Freeman, K. (1996). At home in a random universe: Amos Tversky. In: M. Siegel and R. Baker (eds) The Last Word, (pp. 144–146). New York: William Morrow. Gilovich, T., Vallone, R. and Tversky, A. (1985). The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 17, 295–314. Green, D.R. (1983). A survey of probability concepts in 3000 pupils aged 11–16 years. In: D.R. Grey, P. Nolmes, V. Barnett and G.M. Constable (eds) Proceedings of the First International Conference on Teaching Statistics, pp. 766–783 Sheffield: Teaching Statistics Trust. Hawkins, A.S. and Kapadia, R. (1984). Children’s conceptions of probability – A psychological and pedagogical review. Educational Studies in Mathematics, 15, 349–377. Horvath, J.K. and Lehrer, R. (1998). A modelbased perspective on the development of children’s understanding of chance and uncertainty. In: S.P. Lajoie (ed.) Reflections on Statistics: Learning, Teaching, and Assessment in Grades K–12, pp. 121–148. Mahwah, NJ: L. Erlbaum. Kahneman, D. and Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–292. Kahneman, D., Slovic, P. and Tversky, A. (1982). Judgment under Uncertainty: Heuristics and Biases. New York: Cambridge University Press. Meletiou-Mavrotheris, M. and Lee, C. (2002). Teaching students the stochastic nature of statistical concepts in an introductory statistics course. Statistics Education Research Journal, 1(2), 22–37. Metz, K.E. (1998a). Emergent understanding and attribution of randomness: Comparative analysis of the reasoning of primary grade children and undergraduates. Cognition and Instruction, 16(3), 285–365.

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Metz, K.E. (1998b). Emergent ideas of chance and probability in primary-grade children. In: S.P. Lajoie (ed.) Reflections on Statistics: Learning, Teaching, and Assessment in Grades K–12, pp. 149–174). Mahwah, NJ: L. Erlbaum. National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. North Carolina State Board of Education. (2003). K–12 Mathematics Standard Course of Study for Public Schools of North Carolina. Raleigh, NC: Public Schools of North Carolina, Department of Public Instruction. Polaki, M.V. (2002). Using instruction to identify key features of Bosotho elementary students’ growth in probabilistic thinking. Mathematical Thinking and Learning, 4(4), 285–313. Poskitt, K. (2001). Do You Feel Lucky? London: Scholastic Children’s Books. Rubel, L.H. (2006). Students’ probabilistic thinking revealed. The case of coin tosses. In: G.F. Burrill and P.C. Elliott (eds) Thinking and Reasoning with Data and Chance. Sixty-eighth Yearbook, pp. 49–59 Reston, VA: NCTM. Shaughnessy, J.M. (1992). Research in probability and statistics: Reflections and directions. In: D.A. Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning. A Project of the National Council of Teachers of Mathematics, pp. 465–494. New York: Simon & Schuster Macmillan. Steen, L.A. (1988). The science of patterns. Science, 240, 611–616. Tarr, J.E. and Jones, G.A. (1997). A framework for assessing middle school students’ thinking in conditional probability and independence. Mathematics Education Research Journal, 9(1), 39–59.

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