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The effects and side-effects of statistics education: Psychology students’(mis-)conceptions of probability
Kinga Morsanyi
a,
*
, Caterina Primi
b
, Francesca Chiesi
b
, Simon Handley
a
a
School of Psychology, University of Plymouth, Drake Circus, Plymouth, Devon, PL4 8AA, UK
b
University of Florence, Department of Psychology, Via di San Salvi, 12-50135 Firenze, Italy
a r t i c l e i n f o
Article history:
Available online 10 May 2009
Keywords:
Dual-process theoriesEquiprobability biasHeuristics and biasesIndividual differencesInstruction manipulationProbabilistic reasoningRepresentativeness heuristicStatistics education
a b s t r a c t
Inthree studies welookedat twotypical misconceptions of probability: the representativeness heuristic,and the equiprobability bias. The literature on statistics education predicts that some typical errors andbiases(e.g.,theequiprobabilitybias)increasewitheducation,whereasothersdecrease.Thisisincontrastwithreasoningtheorists’predictionwhoproposethateducationreducesmisconceptionsingeneral.Theyalso predict that students with higher cognitive ability and higher need for cognition are less susceptibleto biases. In Experiments 1 and 2 we found that the equiprobability bias increased with statistics educa-tion, and it was negatively correlated with students’ cognitive abilities. The representativeness heuristicwas mostly unaffected by education, and it was also unrelated to cognitive abilities. In Experiment 3 wedemonstrated through an instruction manipulation (by asking participants to think logically vs. rely ontheir intuitions) that the reason for these differences was that these biases srcinated in different cogni-tive processes.
2009 Elsevier Inc. All rights reserved.
1. Introduction
Probabilistic reasoning consists of drawing conclusions aboutthelikelihoodofeventsbasedonavailableinformation,orpersonalknowledge or beliefs. The notion of probability is notoriously hardto grasp. One reason for this is that the concept of probabilityincorporates two seemingly contradictory ideas: that the individ-ual outcomes of events are unpredictable. However, on the longrun there is a regular pattern of outcomes. A failure to integratethese two aspects of probability (or randomness) leads to eitherinterpreting probabilistic events in an overly deterministic way,or to disregarding the pattern and focusing entirely on the uncer-taintyaspect(Metz,1998).Thatis,peopleeitheroverrateorunder-rate the information given. As a result, when people reason aboutprobabilities they are susceptible to many biases and misconcep-tions. In this study we looked at two reasoning heuristics whichare related to these two aspects of probability: the representative-nessheuristic(whichistheresultofanoverlydeterministicjudge-ment), and the equiprobability bias (which stems fromdisregarding deterministic information).According to Fischbein (1975) intuition plays a very importantrole in the domain of probabilistic reasoning, probably more so,than in other domains of mathematics. He deﬁned the concept of intuition as self-evident, holistic cognitions that appear to be truewithout the need for any formal or empirical proof. Fischbein dis-tinguished between two sources of intuition: primary intuitionswhich are based on individual experiences, especially on interac-tions with, and on adaptations to the environment. Fischbein(1987) also claimed that these intuitions do not disappear witheducation, but theycontinueto inﬂuencejudgment, evenafter for-mal instruction in a particular area. In Fischbein’s view there arealsosecondaryintuitions, whichareformedbyscientiﬁceducationat the school, and which are partly independent from cognitivedevelopment in general.Developing useful primary intuitions about probability is noteasy. Although people have lots of experience with situationsinvolving chance in their everyday lives, these experiences are ingeneral quite ‘‘messy”. In contrast to arithmetic where 2+2 yieldsthe easily testable result of 4, the outcomes of probabilistic eventsare muchharder to evaluate. For example, the lowchances of win-ning at the national lottery are in apparent contrast with the factthat peoplewineveryweek(Borovcnik&Bentz, 1991). At thelevelof personal experiences, ‘‘bad” decisions (in terms of probabilities)are sometimes followed by positive outcomes, or vice versa. Forexample, high risk gambles can occasionally result in big prizes,whereaslowriskgamblescanresultinlosses. Of course, itiscoun-terintuitive (or even culturally unacceptable) to evaluate decisionsirrespectiveoftheirconsequences. Itisevenmoreunusualforpeo-pletoimagineaseriesofequivalenteventshappeningmanytimes,so that they can see some sort of a pattern emerging on the longrun (Borovcnik & Peard, 1996). As a result of probabilistic events
0361-476X/$ - see front matter
2009 Elsevier Inc. All rights reserved.doi:10.1016/j.cedpsych.2009.05.001
*
Corresponding author. Fax: +44 (0)1752 584808.
E-mail address:
kinga.morsanyi@plymouth.ac.uk (K. Morsanyi).Contemporary Educational Psychology 34 (2009) 210–220
Contents lists available at ScienceDirect
Contemporary Educational Psychology
journal homepage: www.elsevier.com/locate/cedpsych
being hard to conceptualise, people’s everyday experiences canactually lead to inappropriate intuitions about the nature of probability.
2. Research on the representativeness heuristic
The heuristics and biases literature (Gilovich, Grifﬁn, & Kahn-eman,2002;Kahneman,Slovic,&Tversky,1982)describestherep-resentativeness heuristic as a tendency for people to base their judgement of the probability of a particular event on how muchit represents the essential features of the parent population or of itsgeneratingprocess.Therepresentativenessheuristicoftenman-ifests itself in the belief that small samples will ‘‘look” exactly thesame, and they will also contain the same proportion of outcomes,as the parent population. That is, when relying on the representa-tiveness heuristic, people put too much conﬁdence in small sam-ples. For example, when tossing a fair coin, after a series of headspeople have the feeling that a tails should follow, because this cor-responds more to their expectation of having a mix of heads andtails, rather than a long sequence of just heads. This is called thenegative recency effect (or the gambler’s fallacy). The representa-tiveness heuristic is also at work when we base our probabilistic judgments about people on how much they resemble to a proto-typical member of a certain category. For instance the use of socialstereotypes often leads to the neglect of relevant statistical rulesandbaserates.Onedemonstrationofthisistheconjunctionfallacy(Tversky & Kahneman, 1983). This fallacy violates a fundamentalrule of probability, that the likelihood of two independent eventsoccurring at the same time (in ‘‘conjunction”) should always beless than, or equal to the probability of either one occurring alone(
P
(
A
)
P
P
(
A
and
B
)).Peoplewhocommittheconjunctionfallacyas-sign a higher probability to a conjunction than to one or the otherof its constituents. In the srcinal demonstration of the fallacy(Tversky & Kahneman, 1983) people read a description of Linda,a 31-year-old, smart, outspoken woman who was a philosophymajor, concernedwithdiscriminationand social justice, and a par-ticipant in antinuclear demonstrations. When asked to judge anumber of statements about Linda according to how likely theywere, people usually ranked the statement ‘‘Linda is a bank tellerand is active in the feminist movement” above the statement ‘‘Lin-da is a bank teller,” thus committing the fallacy.
3. Research on the equiprobability bias
In contrast to the representativeness heuristic which is basedonthecontentof problems, andleads tooverlydeterministic judg-ments, the other bias that we looked at in the present study isbased on the structure of probability problems, and it has theopposite effect: people focus entirely on the uncertainty andunpredictability aspect of probabilistic events, and disregard thepatterns in the outcome. The equiprobability bias was describedbyLecoutre(1985)asatendencyforindividualstothinkofrandomeventsas‘‘equiprobable”bynature,andtojudgeoutcomesthatoc-cur with different probabilities as equally likely.This bias emerges as the result of formal education in probabil-ity (thus, in Fischbein’s terms it is a secondary intuition), and it isbasedona misunderstanding of the concept of randomness. Anin-creaseinequiprobabilitybias inthe courseof educationwas foundin the case of both secondary school and university students (seee.g., Batanero, Serrano, & Garﬁeld, 1996; Lecoutre, 1985).Equiprobability responses are usually based on the following(incorrect) argument: the results to compare are equiprobable, be-cause random events are equiprobable ‘‘by nature”. This sort of reasoning is especially common when there is no single effect thatcould account for the outcome, or the effect is hard to identify andconceptualise (Callaert, 2004). For example, when playing at thenational lottery, people do not believethat they have a 50%chanceofwinningthejackpot,althoughthereisrandomnessinvolved,andthere are only two possible outcomes for any person (ie., theyeither win or they do not win). By contrast, when students haveto predict the sum of two dice, they often declare that no total isharderoreasiertoobtainthananyother,becausethediceareindi-vidually fair, and they cannot be controlled (Pratt, 2000).The theory of naïve probability ( Johnson-Laird, Legrenzi, Girot-to, Legrenzi, & Caverni, 1999) offers a different explanation as tothe srcin of incorrect equiprobability responses. According to thismodel, individuals who are unfamiliar with the probability calcu-lusconstructmental modelsof whatistrueinthevariouspossibil-ities,andeachmodelrepresentsanequiprobablealternativeunlessindividuals have knowledge or beliefs to the contrary, in whichcase they will assign different probabilities to different models.Thus, in this approach equiprobability is assumed by default, un-less individuals have beliefs to the contrary, and this effect is inde-pendent of educationin statistics, or shouldactuallydecreasewitheducation.
4. The effects of individual differences and statistics educationon probabilistic reasoning
FischbeinandSchnarch(1997)investigatedtherelationshipbe-tween education, age and probabilistic reasoning ability using se-ven different tasks. Participants were secondary school childrenbetween the age of 10 and 17, and college students specialised inmathematics. The results were mixed, with some misconceptionsincreasing, some decreasing, and some remaining stable acrossage groups. In general, representativeness-based responses (therepresentativeness heuristic in a lottery scenario, the negative re-cencyeffect,andtheconjunctionfallacy)decreasedwithage.How-ever, othermisconceptions, suchastheneglectofsamplesizes, theavailabilityheuristicandthetime-axisfallacy(theerroneousbelief that theknowledgeof anevent’soutcomecannotbe usedtodeter-mine the probability of a previous event, because later events can-notretrospectivelyaffectearlierevents)increasedwithage, exceptin the case of college students who generally gave the correct re-sponse. The equiprobability bias was stable across ages in one sce-nario (rolling two dice and getting 5 and 6 vs. getting two 6s) andincreased in the case of another (neglecting sample size when pre-dicting the probability of an unusual ratio of girls and boys beingborn the same day in a small versus in a big hospital).According to Fischbein and Schnarch (1997) these results arebased on the interplay between students’ intuitions and the struc-ture of the problems. In general, the impact of intuitions increaseswithageandwitheducation.Whenaproblemiseasytoconceptu-alise, andtherelevantruleis readilyavailable, misconceptionswilldiminish with age. However, if the task is hard to conceptualise,the effects of misconceptions activated by some irrelevant aspectof the task will increase. Thus, according to Fischbein, the contextof the tasks, personal experiences and education all have an effecton reasoning about probabilistic events.Dual-process theories of reasoning (e.g., Evans & Over, 1996;Kahneman & Frederick, 2002; Stanovich, 1999) also discriminatebetween the effect of heuristics (i.e., intuitions) and the ability toconceptualise problems. They presuppose two types of processesworking on problems. Type 1 processes are fast, automatic, effort-less, independent of cognitive abilities, and contextually cued,whereas Type 2 processes are slow, effortful, related to cognitiveabilitiesanddispositions,andcontext-independent.Anideasharedby all dual-process theories is that heuristics are automaticallyactivated by certain aspects of the problem content (the represen-tativeness heuristic is activated by a stereotypical description of a
K. Morsanyi et al./Contemporary Educational Psychology 34 (2009) 210–220
211
person, for example) and then it depends on a person’s cognitivecapacity and personal motivation, whether they override this ini-tial intuition by conscious, effortful (i.e., Type 2) reasoning, andgive a response based on the logical structure of the problem,rather than its content, or if they go with their initial intuitions(cued by Type 1 processes).Anumberofstudieshavefoundevidencethatpeoplewithhigh-ercognitivecapacityandhigherneedforcognitionarelessinclinedto use reasoning heuristics inappropriately (e.g., Handley, Capon,Beveridge, Dennis, & Evans, 2004; Kokis, Macpherson, Toplak,West, & Stanovich, 2002; Stanovich & West, 1999; however, seee.g., Morsanyi & Handley, 2008). Recently, Stanovich and West
(2008) proposed that people with higher cognitive ability mightnot always reason better than lower ability people. This is usuallybecause they do not possess the necessary ‘‘mindware” (i.e., rele-vant knowledge) to solve the problem. It is also possible that theyknowtherelevantrule,buttheydonotrecognisetheneedtoapplyit. People with higher need for cognition (i.e., people how are in-clinedtothinkharderaboutproblems)aremorelikelytorecognisethe need to apply the appropriate rule. Moreover, instructing peo-ple to think hard/logically about problems also sensitises them topotential conﬂicts between logic and intuitions (e.g., Epstein, Lip-son, Holstein, & Huh, 1992; Ferreira, Garcia-Marques, Sherman, &Sherman, 2006; Klaczynski, 2001), andcanleadtomorenormativeperformance, given that people possess the relevant knowledge tosolve the task.
5. The effects of studying different disciplines on reasoning about general and discipline-speciﬁc problems
There is some evidence that studying different disciplines atuniversity level affects certain aspects of reasoning abilities in dis-tinguishable ways. For example, Lehman, Lempert, and Nisbett(1988) found that psychology and medical training increased per-formance on statistical and conditional reasoning problems (forexample, there was a marked increase in students’ relying on thelaw of large numbers when reasoning about everyday situationsinvolving uncertainty, and their ability to recognise the effect of confounding variables also improved), whereas law students gotbetter only on conditional reasoning during their undergraduateyears. On the other hand, chemistry training had no effect on anyof the types of reasoning studied. Although there is clearly a self-selectioneffect (i.e., studentswithdifferentinterestsandstrengthsenrol in different courses) whichmayexplainsome of these differ-ences, because of the longitudinal nature of the study (the samestudents were tested in the ﬁrst and third year of their course) itwas clear that education in a speciﬁc discipline also had a distin-guishableeffect onstudents’ reasoning, dependingontheir subjectof study.Lehmanetal.(1988)proposedthatpsychologyandmedicalstu-dents’ statistical reasoning ability increased because these areprobabilistic sciences, where statistical reasoning is a very impor-tant aspect of education. By contrast, chemistry is a non-probabi-listic (i.e., deterministic) science and law is a non-scientiﬁcdiscipline. They also proposed in line with an earlier study (Fong,Krantz, & Nisbett, 1986) that these changes in statistical reasoningability not only affect the way students think about their own dis-cipline, but it also has an effect on how students of different disci-plines reason about everyday-life events.Heller, Salzstein, and Caspe (1992) compared ﬁrst, second andthird year paediatric residents’ reasoning about medical and non-medical problems. Some effects that they studied were consistentacross training level and problem content. However, ﬁrst year res-identsmadeuseofbase-rateinformationmorethanthirdyearres-idents, whereas, ﬁrst year residents were inﬂuenced by the waymedicaldatawaspresentedtothem,butsecondandthirdyearres-idents were not. On the non-medical problems there was no effectof training level. By contrast, higher year residents relied more onrepresentativeness, which was considered to be the result of med-ical training consisting of drilling in disease conditions and syn-dromes. Thus, just like the students in Fischbein and Schnarch’s(1997) study, they got worse in one aspect of probabilistic reason-ing, and got better in another. Unlike the changes identiﬁed byFong et al. (1986) and Lehman et al. (1988) these effects were
not general, but speciﬁc to medical problems.
6. The aims of the present investigation
Thepurposeofthepresentstudywastoinvestigatethechangesin the representativeness heuristic (a primary intuition whichleads to a deterministic viewof probability), and the equiprobabil-itybias(asecondaryintuitionwhichresultsintheoverratingoftheuncertainty aspect of probability) with statistic education.Reasoning theorists (e.g., Stanovich & West, 2008) predict thatbiasesingeneralshoulddecreasewitheducation.Bycontrast,edu-cational research (e.g., Fischbein & Schnarch, 1997) showed thatdifferent heuristics can follow different developmental patterns,and the overall impact of heuristics on reasoning increases withexperience. However, the use of the representativeness heuristicwas found to decrease with education.According to Batanero et al. (1996) and Lecoutre (1985) the
equiprobability bias should increase with statistics education. Ontheotherhand, Johnson-Lairdetal.(1999)andStanovichandWest
(2008)wouldpredict that theequiprobabilitybiasshoulddecreasewith statistics education. Moreover, dual-process theories (e.g.,Stanovich & West, 2008) claim that students with higher needfor cognition and higher cognitive ability will be less biased, atleast when they possess the necessary ‘‘mindware” to solve thetasks (Stanovich & West, 2008).Inordertotestthispredictionweincludedneedforcognitionasa control measure. Cacioppo and Petty (1982) described the needfor cognition as individual differences in the tendency to engagein and enjoy effortful cognitive activity. The need for cognition ispositively related to academic performance and course grades(Leone & Dalton, 1988; Sadowski & Gulgoz, 1996). Students highon the need for cognition are able to comprehend material requir-ingcognitiveeffortbetter(Leone&Dalton,1988),andtheyarealsomore effective information processors (Sadowski & Gulgoz, 1996).As a measure of cognitive abilities we included a short form(Arthur & Day, 1994) of the Raven Advanced Progressive Matrices(APM). The APM is a nonverbal measure of ﬂuid intelligence witha low level of culture-loading which made it appropriate to usewith groups of students from different countries.We wanted to address the question that in the case we ﬁnd achange in statistical reasoning ability with education, whether itis a general effect (as in Fong et al., 1986, and Lehman et al.,
1988) or if it is speciﬁc to the subject of study (see Heller et al.,1992). To do this, we investigated the effects of problem content.We used problems with both people content (based on the ideathateducationinpsychologymightaffectthewaywereasonaboutpeople), and we compared students’ performance on these taskswith their reasoning about tasks with the same underlying logic,butwithalessindividualisedcontentwherethetasksweremostlyabout objects and natural phenomena.The mainaimof the present investigationwas to test the (oftenconﬂicting)predictionsofreasoningandeducationaltheoristwith-in one study, and to potentially integrate these approaches. In or-der to separate out the effects of individual differences, theeducationalsystem,andeducationinacertaindiscipline,weinves-tigated the performance of different groups of students on the
212
K. Morsanyi et al./Contemporary Educational Psychology 34 (2009) 210–220
same problems. In Experiment 1 we compared the performance of undergraduate psychology (probabilistic science) and marine biol-ogy (deterministic science) students from a UK university. Thesamples included both ﬁrst year students who have had no educa-tioninstatistics,andhigheryearstudentsatdifferentstagesofsta-tisticseducation.InExperiment2weexaminedtheperformanceof Italian psychology students (ie, students studying the same disci-pline at a different educational setting) on the same problems. Fi-nally, in Experiment 3 we manipulated the amount of cognitiveeffort that students invested in solving the problems, by instruct-ing them to reason logically vs. intuitively. The aim of this manip-ulation was to distinguish between reasoning errors that stemmedfroma shallowprocessing of information, and errors that were theresult of the lack of relevant knowledge.
7. Experiment 1
This study investigated the following questions. Is there any ef-fect of psychology education on the representativeness heuristicand on the equiprobability bias? If so, do these biases increase ordecrease with education? Are these changes in psychology stu-dents’probabilityjudgmentsonlypresentwhentheyarereasoningabout people, or do they also affect their reasoning about objectsandnatural phenomena?Are theseeffects present inboththe psy-chology and the biology student groups, or only in the psychologystudents (as educationinprobabilisticreasoningis morecentral totheir education)? Finally, in addition to education-related changes,do we also ﬁnd any effect of individual differences on probabilisticreasoning?In order to address these questions we used 12 tasks to assessundergraduate psychology and marine biology students’ probabi-listic reasoning ability.
8. Method
8.1. Participants
Participants were 40 ﬁrst, 40 second and 31 third year psychol-ogystudents, and31ﬁrst, 21secondand22thirdyear biologystu-dents fromthe University of Plymouth. The experiment was run inthe ﬁrst term, so ﬁrst year students were at the beginning of theirstatistics course, second year students had a year’s education instatistics, and third years had two years of statistics education be-hind them.For the purpose of analyses we combined the second and thirdyear groups (group 2 – those students who have had education instatistics), andcompared themwiththe year 1 students (group1–no education in statistics). The mean age for the students in group1 was 19.2years (age range 18–28years) for the psychology stu-dents (
n
=40, 29 females), and 19.7years (age range 18–31years)for the biology students (
n
=31, 21 females). In group 2 the meanage of the psychology students (
n
=70, 64 females) was 22.5years(age range 19–52years). The mean age of the biology students(
n
=42, 32 females) was 21.2years (age range 19–37years). Psy-chology students took part in the experiment for ungraded coursecredits, biology students participated as volunteers.
8.2. Probabilistic reasoning tasks
We used 12 probabilistic reasoning tasks (see Appendix for theactual tasks, and also for the normative and heuristic response foreach). There were 6 different types of task: 3 problems measuringthe representativeness heuristic (problems 1, 2 and 4 in each set),and 3 problems measuring the equiprobability bias (problems 3, 5and 6 in each set).All the problems were presented twice (once in set 1, and onceinset2).Thelogicalstructureofthecorrespondingproblemsinthetwo sets was equivalent, but the content of the problems was dif-ferent. In set 1 the presentation and content of the problems wassimilartoprobabilisticproblemsthatareusuallyincludedinstatis-ticstextbooks(everydaycontext).Inthesetaskstherandomnessof theprocesses involvedwere moresalient (i.e., theywere relatedtoactivities which are known to generate random outcomes, such asthrowing a dice, or taking part in a lottery) and the stories wereusually about objects (e.g., ﬂipping a coin, drawing a marble, etc.)or natural phenomena (e.g., the weather). Set 2 required probabi-listicreasoningaboutpeople.Theseproblemsalsocontainedinfor-mation about the probabilities of the different outcomes, but therandomness of the generating process was less salient than inthe object content tasks (e.g., when we know that most studentswith a learning disability are dyslexic, we still do not think thatthe type of learning disability an individual suffers from is ran-domly assigned to them). These problems resembled to the wayprobabilistic data are usually presented in psychology papers andtextbooks.In each task participants chose from three different responseoptions. One corresponded to the representativeness heuristic ortheequiprobabilitybias,onewasthenormativeresponse(i.e.,are-sponse that corresponds to the statistical rule that is required tosolvethetask),andthethirdoneneithercorrespondedtothemainmisconception nor was normative.An example for an everyday content task that measured therepresentativeness heuristic is the following:
A fair coin is ﬂipped ﬁve times, each time landing with tails up;TTTTT. What is the most likely outcome if a coin is ﬂipped a sixth time?
a.
Tails.
b.
Heads (heuristic response; representativeness).
c.
Tails and a heads are equally likely (normative response).
This task is measuring the negative recency effect; to expectheads after a series of tails in order to make the pattern more rep-resentative to a randomsequence of heads and tails (i.e., the alter-nation of the two outcomes). Students were given arepresentativeness point if they chose (b) and they were given anormative point if they chose (c).An example for a psychology content task measuring the equi-probability bias is the following:
The two most common causes of learning difﬁculties among uni-versity students are dyslexia (speciﬁc problems in learning to read,write and spell) and dyscalculia (speciﬁc problems in learning arith-metical concepts and procedures). Out of 15 university students withlearning difﬁculties approximately 9 are dyslexic, and 6 have dyscalcu-lia. Joe is a student with a learning difﬁculty. Which of the following isthe most likely?
a.
Joe is dyslexic (normative response)
b.
Joe has dyscalculia
c.
Both are equally likely (heuristic response; equiprobability)
Students were given an equiprobability point if they chose re-sponse (c) and they were given a normative point if they chose re-sponse (a).The problems were presented in a ﬁxed random order, and theorder of the problems within both sets was the same. The order of the presentation of the two sets was counterbalanced acrossparticipants.For each problem students were given either a representative-ness or equiprobability, or a normative point. The maximumnum-ber of representativeness and equiprobability responses was 3 per
K. Morsanyi et al./Contemporary Educational Psychology 34 (2009) 210–220
213

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