Understanding of Patterns, Streaks and Independence by Grade School Children

Understanding of Patterns, Streaks and Independence byGrade School Children

KEYWORDS:Teaching;Representativeness heuristic;Law of small numbers;Coin tossing.

Carolyn RidgwayMerced, California, USA.e-mail: [email protected]

Derry RidgwaySRA International, Research Triangle Park,North Carolina, USA.

SummaryUsing sequences of coin flips as a model of serialindependent events, we asked schoolchildren ingrades 1 through 5 to estimate the likelihood of thenext flip. Most children in each grade expected shortpatterns to continue.

� INTRODUCTION �

Adults and older students may invoke error-prone heuristics to arrive at incorrect estimates

for the likelihood of chance events (Kahnemanet al. 1982). In this article, we use a simple coin-flipping experiment to examine an error-prone heu-ristic related to the independence of serial events inschoolchildren from grade 1 through grade 5 (USages 6–11). We characterize the error-prone reason-ing as follows:

The subject reviews a short sequence of independent randomevents with the goal of predicting the likelihood of the next event.Subjects well grounded in the notion of independence will notattempt to extrapolate any pattern evident in the short sequence tothe next outcome. Subjects less well grounded in independencemay assign likelihoods by erroneously reasoning from the resultsof the immediately preceding outcomes. The expectation that briefsamples will closely reflect the theoretical overall set of outcomesfor longer sequences was called the “representativeness heuristic”by Kahneman and Tversky (1979) and has also been referred to asthe “law of small numbers” (Burns 2004; Burns and Corpus 2004).

The concept of independence is integral to under-standing probabilities associated with more than asingle event (Barnes 1998; Meletiou-Mavrotherisand Lee 2002; Metz 1998a; Polaki 2002; Rubel2006; Shaughnessy 1992; Tarr and Jones 1997).However, independence does not appear in theNational Council of Teachers of Mathematics(NCTM) curriculum standards until secondaryschool (NCTM 2000), and the teaching of indepen-dence in school children is given little attention

(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 forrandom events, we asked approximately 100 school-children to estimate the likelihood (heads morelikely, tails more likely, heads and tails equallylikely) for the seventh flip in a series after reviewingthe results of the first six flips. We hypothesized thatin our experimental model, young students wouldexhibit the representativeness heuristic and, in theabsence of curricular attention, older studentswould continue to exhibit this error even as othernondeterministic reasoning skills increased.

� METHODS �

The subjects were ninety-eight students from grades1 to 5 in a 400-student public elementary school inNorth Carolina. At the time of the study, mathemat-ics teaching followed the 2003 K-12 MathematicsStandard Course of Study for the Public Schools ofNorth Carolina from the State Board of Education(current standards at http://www.ncpublicschools.org/curriculum/mathematics), based on the NCTMPrinciples and Standards for the Teaching ofMathematics (2000) (http://standards.nctm.org).Both standards include five strands: Number andOperation, Measurement, Geometry, Data Analysisand Probability, and Algebra. For this study, the

© 2010 The AuthorsJournal compilation © 2010 Teaching Statistics Trust

34 • Teaching Statistics. Volume 32, Number 2, Summer 2010

Data Analysis and Probability and Algebra strandsare relevant.

Teachers reported that students in all grades hadbeen exposed to lessons involving manipulatives(dice, coins or spinners), data recording and prob-ability of individual events. The manipulatives usedand the concepts developed varied by grade level.The North Carolina first grade standards call fordescriptions of events as “certain”, “more likely”,“less likely” or “impossible” (section 4.02); the firstgrade class had experience with spinners but hadnot studied coin flips. The second grade standardscall for probability experiments, description ofresults and making predictions; the second gradeclass performed coin-flipping experiments. Thethird grade standards add no new objectives; thethird grade classes had performed experiments withboth spinners and dice. The fourth grade standardscall for experiment design and lists of all possibleoutcomes for events; the fourth grade class basedtheir designs and experiments on coin flipping. Thefifth grade standards added no new objectives; the

fifth grade class based their designs and experimentson dice. These objectives and class curricula areconsistent with the recommendations in the Guide-lines for Assessment and Instruction in StatisticsEducation endorsed by the American StatisticalAssociation (Franklin et al. 2005). In each class,students recorded the outcome of single events andsummarized the aggregate results for themselvesand for the class. Sequences of events and, hence,the concept of independence are not part of thestandards for grades 1 through 5. However, curricu-lar standards for the first through fifth grades docall for the creation and extension of patterns, bothnumeric (from grade 1) and geometric (from grade3), within the algebra strand, which will be seen asrelevant in our investigation below.

Near the end of the school year, students from oneor two classes at each grade level completed a work-sheet (figure 1) as a class lesson under the directionof the first author using a transparency on an over-head projector. Each student had a copy of theworksheet. Students were shown a nickel and told

Fig 1. Worksheet with four sequencesof coin flips, completed by 1st through5th graders


Teaching Statistics. Volume 32, Number 2, Summer 2010 • 35

that it was a fair coin that was equally likely to landwith heads up or tails up (the introductory sentenceon the worksheet was read aloud). The nickel wasflipped six times. After each flip, the studentsrecorded the outcome in the first row of circles.They were then asked, if the coin were flipped onemore time, would heads be more likely, would tailsbe more likely or would they be equally likely? Thestudents recorded their answers. In the first andsecond grades, they wrote H, T or = in the circlewith the question mark. The third, fourth and fifthgrade students recorded their responses in theanswer box.

The students were then asked to look at the secondsequence of flips (THTHTH). The lead-in sentencefrom the worksheet was read aloud, and the studentswere told that the same fair coin was used that wasequally likely to land heads or tails. The sequencewas read aloud and the students were then asked, ifthe coin were flipped one more time, would heads bemore likely, would tails be more likely or wouldheads and tails be equally likely? The students thenrecorded their answers. Older students were alsoasked to write why they chose the answer they did.This process was repeated for sequences 3(TTTTTT) and 4 (TTTTTH), with a reading of thelead-in sentence on the worksheet for each sequenceand the sequence of six flips, but without thereminder about the meaning of a fair coin. At theend of each session, the students were asked ifanyone wanted to tell what they answered for eachsequence and why. Their oral replies were recordedverbatim by an observer who attended each of thelessons.

� RESULTS �

Student responses are summarized by sequence andgrade in table 1.

For sequence 1, students witnessed the six coin flipsand were asked to respond with the most likelyoutcome for the seventh flip. Sequences were differ-ent in each class; none had an obvious pattern andnone terminated with a streak of greater than 2similar outcomes. The percentage of students cor-rectly responding ‘=’ for sequence 1 increased withgrade level (age) and was greater than or equal to50% for the 3rd, 4th and 5th grades.

A similar pattern (generally increasing numbers ofcorrect responses in higher grades) was seen for

responses to sequences 2, 3 and 4, although theoverall rates of correct responses were lower, in allgrades, than the percentages for sequence 1.

Four students (one in 1st grade, one in 4th gradeand two in 5th grade) responded ‘=’ for all foursequences. Two students (one in 4th grade and onein 5th grade) responded ‘=’ to three sequences and‘T’ for sequence 3 (TTTTTT).

For sequence 1, students witnessed the flips of anormal-appearing coin and were not distracted byresults that called upon a pattern-based decisionheuristic. Our hypothesis was that the apparent pat-terns in sequence 2 (THTHTH), sequence 3(TTTTTT) and sequence 4 (TTTTTH) led studentswho understood that heads and tails were equallylikely to apply an inappropriate decision algorithm.To examine this hypothesis, we looked at theresponses to sequences 2, 3 and 4 for students whoresponded ‘=’ to sequence 1. The alternating out-comes in sequence 2 and the streak outcome forsequence 3 each might suggest a next outcome of‘T’. The percentages of ‘T’ responses to sequence 2and 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 whoresponded ‘=’ to sequence 1.

� STUDENT REASONING �

The most common explanations involved the word‘pattern’ (“In the pattern, tails is next”), followed byexplanations based on counting the number ofheads and tails already in the sequence (“because 1is heads out of 6” or “heads because it had too manytails”). If the number of heads and tails in thesequence were equal (sequence 2), then some stu-dents selected equally likely for the next toss, basedon a counting analysis. A 5th grader explained,“they both have 3 and that’s equal”.

There were other instances of choosing equallylikely using incorrect reasoning, e.g. if both headsand tails have already come up, it is possible thateither may again. This reasoning allowed studentsto respond tails to the sequence of 6 tails (sequence3), 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 foursequences. Their explanations were similar. A 1st



Tab

le1.

Sum

mar

yof

stud

ent

resp

onse

s

Gra

de1

(n=

20,2

1)a

Gra

de2

(n=

17)

Gra

de3

(n=

26)

Gra

de4

(n=

18)

Gra

de5

(n=

17)

Ove

rall

(n=

98,9

9)a

Sequ

ence

1b

=4

(20%

)7

(41%

)13

(50%

)16

(89%

)9

(53%

)49

(50%

)N

oteq

ualc

16(8

0%)

10(5

9%)

13(5

0%)

2(1

1%)

8(4

7%)

49(5

0%)

Sequ

ence

2(T

HT

HT

H?)

=3

(15%

)7

(41%

)6

(23%

)7

(39%

)5

(29%

)28

(29%

)H

eads

3(1

5%)

00

1(6

%)

1(6

%)

5(5

%)

Tai

ls14

(70%

)10

(59%

)20

(77%

)10

(55%

)11

(65%

)65

(66%

)Se

quen

ce3

(TT

TT

TT

?)=

2(1

0%)

1(6

%)

01

(6%

)3

(18%

)7

(7%

)H

eads

3(1

4%)

1(6

%)

1(4

%)

2(1

1%)

2(1

1%)

9(9

%)

Tai

ls16

(76%

)15

(88%

)25

(96%

)15

(83%

)12

(71%

)83

(84%

)Se

quen

ce4

(TT

TT

TH

?)=

1(5

%)

1(6

%)

2(8

%)

3(1

7%)

10(5

9%)

17(1

7%)

Hea

ds13

(62%

)2

(12%

)3

(12%

)4

(22%

)1

(6%

)23

(23%

)T

ails

7(3

3%)

14(8

2%)

21(8

0%)

11(4

2%)

6(3

5%)

59(6

0%)

Sequ

ence

s2,

3an

d4

=d6

(9%

)9

(18%

)8

(10%

)11

(20%

)18

(35%

)52

(18%

)H

eads

d19

(31%

)3

(6%

)4

(5%

)7

(13%

)4

(8%

)37

(12%

)T

ails

d37

(60%

)39

(76%

)66

(85%

)36

(67%

)29

(57%

)20

7(7

0%)

aT

wo

of21

1st

grad

est

uden

tsom

itte

don

ere

spon

se.

bC

oins

wer

efli

pped

inth

epr

esen

ceof

the

stud

ent;

resu

lts

vari

edby

clas

sroo

m.

cR

espo

nse

ofei

ther

head

sor

tails

.d

Per

cent

age

calc

ulat

edfr

omth

eto

talo

fal

lres

pons

es.



grader stated, “it’s possible to get each one becauseit’s on the same coin”. These four students invokedthe concept of independent events, without usingthe word.

One 5th grade student consistently invoked the ideaof a fair coin. He explained that “it’s a fair coine”for his responses of equally likely to sequences 1, 2and 4, and decided for sequence 3 that “it’s prob-able a nonfair coine” so he chose tails as being morelikely.

No student used the word independent in his expla-nations or mentioned the idea that past events neednot affect future events.

� DISCUSSION �

In our sample, only a few 1st grade students wereable to use nondeterministic reasoning in the simplecase without a pattern (sequence 1), while 50% ormore of students in grades 3 and above had reachedthis level of understanding. Our subjects, withoutdidactic or empirical grounding in the concept ofindependent events or the likelihood of apparentpatterns in a series of random independent events,reviewed the sequences of flips in sequence 2,sequence 3 and sequence 4 and substituted an alter-native analysis for the correct independence-basedresponse of ‘equally likely’. The error of reasoningwas apparent in the small number of very young (1stand 2nd grade) children capable of probabilisticthinking, and persisted as the prevalent pattern ofthinking through grade 5. Students’ comments sug-gested that three distinct alternative analyses wereused.

1. The most common alternative analysis couldbe called ‘extend the pattern’; students whoextended the pattern were applying a naturalhuman reasoning technique, as suggested byBarbara Tversky and others (Burns and Corpus2004; Freeman 1996; Horvath and Lehrer 1998;Metz 1998b; Rubel 2006; Steen 1988; Tarr andJones 1997), or were employing a pattern-identifying skill that receives explicit and strongreinforcement in the Algebra strand of curricu-lum standards (NCTM 2000; North CarolinaState Board of Education 2003). Reasoning bystreak extension underlies the belief of basketballplayers and observers that players sometimespossess a ‘hot hand’, accounting for a series ofsuccessful basket attempts (Gilovich et al. 1985).

2. Less often, our subjects used a counting analysisto conclude that the less frequently appearingoutcome will be more likely to appear in the nextflip (the ‘gambler’s fallacy’) or that because equalor nearly equal numbers of Hs and Ts haveappeared so far, heads and tails are equally likelyon the next flip (a correct response wronglyarrived at through the application of the law ofsmall numbers).

3. Finally, one student used a counting method tomake an explicit determination that the coin wasnot fair, contradicting the declared worksheetconditions, and leading to a response of T afterTTTTTT.

Grade

1st 2nd 3rd 4th 5th Total0

20

40

60

80

100

3/4

4/79/13

9/16

4/9

29/49

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

Grade


20

40

60

80

100

2/4

7/7

12/13 14/16

5/9

40/49

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

Grade


20

40

60

80

100

Predicting H

Predicting T

2/4

1/4

0

6/7

1/13

10/13

2/16

11/16

1/9 1/9 6/49

29/49

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



Topics in probability and nondeterministic math-ematical thinking are included in the recommendedcurricula from the earliest grade (NCTM 2000;North Carolina State Board of Education 2003). Inresponse to the questions on our instrument, a fewchildren in 1st grade and half or more of children ingrades 3 and higher were able to answer ‘equallylikely’ for the next toss in the first sequence. Thepercentage of students able to give this responsegenerally increased from grades 1 to 5, consistentwith the pattern observed for older students by otherresearchers (Rubel 2006). In Rubel’s interviews withjunior and senior high school students about theoutcome of serial coin flips, independence of succes-sive flips did not appear in students’ reasoning untilthe 9th grade (Rubel 2006). In an analysis ofresponses to questions about probability conceptsgiven to nearly 3000 high school students, andfollow-up interviews of 44, Green (1983) found thatwhile other aspects of statistical reasoning improvedwith age, the ability to reason correctly about eventsin sequence was uncommon at all ages. Only 4 of the98 students in our study appeared to reason from theconcept of independence in all four series.

In our study, the invariable correct response was ‘=’,reflecting both the fairness of the coin and the inde-pendence of each flip. Students in all grades showeda willingness to change their probability estimatesbased on the outcomes of short preceding sequences– extending patterns and streaks or using counts ofrecent results to make future judgements. Ourdata suggest that the reasoning error is present fromthe earliest time that students are able to reasonnondeterministically.

There is little published research related to early-grade curricula that could address questions ofserial independent events. Kjartan Poskitt providesa discussion that might assist teachers constructingrelevant lessons (“coins have no memory”; Poskitt2001). Focused lessons in the early grades mightallow young elementary-school children to resistdevelopment of the error-prone heuristic. Directedhands-on lessons could be designed around theconcept of independent events, the likelihood ofpatterns and streaks in a series of independentchance outcomes and the extension in reasoningrequired for events that do not have equallyexpected outcomes. Both the occurrence of patternsand reasoning associated with unequal probabilitiescould be taught with explicit didactic lessons andwith experience-based, hands-on lessons. Focusedlessons with student-generated data could includethe following:

• a hunt for patterns in a long sequence of out-comes from coin flips, generated over severaldays by members of the class

• a review of unlikely results from sequences ofoutcomes from an unbalanced spinner

• finding patterns in sequences of outcomes from‘real-world’ events based on student skills (suchas basketball free throws)

• forecasting games for sequences of coin flips orspinner outcomes (balanced or unbalanced)

• individual or class discussions to expose thepotential for both correct and incorrect reason-ing to provide correctly reasoned forecasts.

� CONCLUSION �

The number of students able to perform nondeter-ministic mathematical thinking increases with age.On our 4-item instrument, a majority of students in3rd grade and above were able to respond ‘equallylikely’ for flips of a fair coin performed in theirpresence. But when sequences of only six flipslooked unlike the students’ expectations for chance,students in grades 1 through 5 failed to invoke theconcept of independence, favouring instead anerror-prone heuristic, to predict the outcome of aseventh flip. Students extended patterns andstreaks, perhaps misapplying skills taught as part ofthe algebra curriculum; a few students based theirresponses on counts of recent outcomes.

Acknowledgements

We are grateful for the assistance of Ms AerinRidgway in the collection and processing ofresponses, and for the encouragement of Dr MarionWalter in the completion of this manuscript.

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Understanding of Patterns, Streaks and Independence by Grade School Children

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