A PROBABILISTIC MODEL FOR STUDENTS’ ERRORS AND MISCONCEPTIONS ON THE STRUCTURE OF MATTER IN RELATION TO THREE COGNITIVE VARIABLES

GEORGIOS TSITSIPIS, DIMITRIOS STAMOVLASIS and GEORGE PAPAGEORGIOU

A PROBABILISTIC MODEL FOR STUDENTS’ ERRORSAND MISCONCEPTIONS ON THE STRUCTURE OF MATTER

IN RELATION TO THREE COGNITIVE VARIABLES

Received: 28 May 2010; Accepted: 8 February 2011

ABSTRACT. In this study, the effect of 3 cognitive variables such as logical thinking,field dependence/field independence, and convergent/divergent thinking on some specificstudents’ answers related to the particulate nature of matter was investigated by means ofprobabilistic models. Besides recording and tabulating the students’ responses, acombination of binomial and multinomial logistic regression techniques was used toanalyze the data. Thus, students’ misconceptions as well as the compatible-with-the-scientific-view student’s answers were explored one by one in relation to the above 3cognitive variables. The study took place with the participation of 329 ninth-grade juniorhigh school pupils (aged 14–15). The results showed that mostly logical thinking andsporadically the other 2 cognitive variables were significantly associated with students’answers. Interpretation of the results and implications for science education are discussed.

KEY WORDS: binomial logistic regression, multinomial logistic regression, cognitivevariables, convergence/divergence, field dependence/independence, logical thinking,misconceptions, structure of matter

INTRODUCTION

Understanding the material world in terms of atoms and molecules is ofparamount importance for the contemporary person as a scientist and/orcitizen. The importance of this premise was characteristically stressed byFeynman (1963), who stated that if all scientific knowledge was to bedestroyed and only one sentence has to be passed to the next generation,this would be the statement of the atomic hypothesis. Moreover, he stated:“In that one sentence, you will see, there is an enormous amount ofinformation about the world, if just a little imagination and thinking areapplied” (1963, chapter: Atoms in motion, p. 4).

The above justifies the large amount of work carried out by researchersin order to investigate the teaching and learning difficulties on this topic,which was proved to be particularly challenging. A wide range ofpersisting students’ errors and misconceptions on the structure of matterhas been reported in science education literature. In short, all thesefindings concern the following: Students often consider matter as

International Journal of Science and Mathematics Education (2012) 10: 777Y802# National Science Council, Taiwan (2011)

continuous. Those, however, who adopt a particulate model of matterfrequently show understanding of difficulties in important aspects of theparticle theory, which are mostly related to the space between particles,the intrinsic motion of the particles, the relative spacing between theparticles in the three states, the attractions between particles, and thenature of the particles themselves (Johnson, 1998a). In the main, studentshave problems in conceiving the notion of ‘empty space’ among particles.Thus, many students think that the space among particles is filled withvarious kinds of ‘stuff,’ such as air or dust or with other particles orparticles of the same substance (Novick & Nussbaum, 1978; Lee,Eichinger, Anderson, Berkheimer & Blakeslee, 1993; Johnson, 1998a).Frequently, molecules are thought to be in substances like ‘blueberries ina muffin’ rather than substances to be composed of molecules (Lee et al.,1993). According to this view, “particles are additional to the substance”(Johnson, 1998a, p. 399). The intrinsic motion of particles (structuralunits—usually molecules) was also found to be a difficult concept(Novick & Nussbaum, 1978; Lee et al., 1993; Johnson, 1998a). In case ofgases, that was attributed to various causes, e.g., to low specific gravity orto the action of air (Novick & Nussbaum, 1978; Lee et al., 1993). On theother hand, in solids, where no motion of the substance is visible,molecules are often thought to be still (Lee et al., 1993). In accordancewith the above, Dow, Auld & Wilson (1978) found out that although themajority of the students in their investigation indicated particle motion inthe liquid and gas state, about a third indicated that there was no particlemovement in the solid state.

Significant conceptual difficulties are reflected in the way that studentsmake a distinction between states, which are not always clear orcomplete. Secondary students’ misconceptions about liquids are due tothe fact that they consider liquids to be merely in an intermediate statebetween solids and gases. In this context, students overestimate themolecular spacing in liquids (Dow et al., 1978). In another study,molecular spacing in gases has been found underestimated (Pereira &Pestana, 1991).

In relation to the nature of the particles themselves, students haveshown a great deal of difficulty to understand that the properties of thestates of matter are due to the collective behavior of particles. They oftenregard a particle or a molecule as a little quantity of a substance having allthe macroscopic properties of the substance. That is, ice molecules orparticles are regarded as frozen or ‘solid molecules,’ water molecules as‘liquid molecules,’ and so forth (Lee et al., 1993; Johnson, 1998a).Furthermore, molecules are described to undergo “the same changes as

GEORGIOS TSITSIPIS ET AL.778

the visible changes in the substances. Thus, molecules start to move whenice melts into water, molecules of water are heated up and make waterboil, or molecules expand, contract, melt, evaporate, condense, and soforth” (Lee et al., 1993, p. 268).

Undoubtedly, being aware of pupils’ particle models constitutes asignificant teaching asset (Johnson, 1998b, 1998c; Papageorgiou & Johnson,2005). However, it would be useful for teaching as well, to be aware about thefactors that theremight be behind pupils’models and probably influence them.

Research work on students’ errors and misconceptions has been mainlyexploratory and descriptive, and it has been driven by psychologicaltheories of conceptual change. Two apparently competing theoreticalperspectives were dominated: one which considers students’ knowledgeas coherent or theory-like (Chi, 1992; Vosniadou & Brewer, 1992, 1994)and the other which considers it fragmented (diSessa, 1988; diSessa,Gillespie & Esterly, 2004; Harrison, Grayson & Treagust, 1999). Despitethe progress made in both perspectives, conceptual-change research hasprimarily focused on difficulties arising from the nature of concepts itself,without, however, providing explanations for their origin or correlatingthem with independent variables. Research endeavors did not addressquestions regarding mechanisms and mental recourses involved incognitive processes. On the other hand, psychological theories at theinterface between psychology and brain science, such as informationprocessing models and neo-Piagetian theories, view cognitive processesas driven by mental resources that are individual differences explainingvariation in performance on cognitive tasks. An example is Pascual-Leone’s theory of constructive operators (TCO; Pascual-Leone 1969,1970). According to TCO, cognitive performance is the responsibility of avariety of constructive operators, each of which performs a specificfunction: The M-operator deals primarily with mental capacity, the C-operator with content knowledge, the L-operator with logical operationssuch as conservation and formal logic, the F-operator with fielddependence/independence, and so on. Research has supported theconstruct validity of TCO, since CO corresponding to mental resourcesactivated during cognitive tasks can be operationalized by psychometricvariables measured at the behavioral level. Thus, variables, such as theinformation processing capacity (M-capacity), logical thinking (LTh),field dependence/independence (FDI), or convergent/divergent thinking(CD), have been proved to play a significant role in a wide range of tasks,and they affect students’ performance in science, such as physics(Johnstone, Hogg & Ziane, 1993), chemistry (BouJaoude, Salloum &Abd-El-Khalick, 2004; Chandran, Treagust & Tobin, 1987), organic

A PROBABILISTIC MODEL FOR STUDENTS’ ERRORS 779

synthesis (Stamovlasis & Tsaparlis, 2005), physical chemistry (Tsaparlis,2005), and earth science (Chiappetta & Russell, 1982), to mention a few.Recently, three of the above-mentioned cognitive variables—LTh, FDI,and CD—were found to affect students’ mean achievement score on thetopic of the structure of matter and its changes of state (Tsitsipis,Stamovlasis & Papageorgiou, 2010).

In the present work, specific students’ particle ideas regarding thestructure of matter are being analyzed, and an attempt is made toinvestigate associations of these qualitative features, either as misconcep-tions or scientifically compatible models, with the three of the above-mentioned cognitive variables (LTh, FDI, and CD). This is accomplishedby the implementation of probabilistic statistical models, which aredescribed in a next section. Following, a brief description of the cognitivevariables in question is provided.

COGNITIVE VARIABLES

Logical Thinking

Logical thinking is a Piagetian concept and refers to the ability of thesubject to use concrete and formal operational reasoning (Lawson, 1978,1985, 1993). LTh includes proportional, combinational, and probabilisticreasoning as well as reasoning related to the isolation and control ofvariables, conservation of weight, and displaced volume. Literature is richin studies reporting the correlation between LTh and students’ performancein science: Lawson (1982), Chiappetta & Russell (1982), Chandran et al.(1987), Lawson & Thompson (1988), Zeitoun (1989), Niaz (1996), andBouJaoude et al. (2004), to mention a few.

Field Dependence/Independence

Field dependence/independence or disembedding ability according toWitkin, Moore, Goodenough & Cox (1977) is a cognitive style associatedwith the ability to disembed relevant information from a complex andpotentially confusing context. Disembedding ability has been correlatedto academic performance in various disciplines such as language,mathematics, natural sciences, social sciences, art, and computer sciences(Tinajero & Paramo, 1998). In science education, the FDI cognitive styleis also connected with one’s ability to efficiently separate the ‘signal’from the ‘noise,’ and thus, FDI appears important in learning science:problem-solving and conceptual understanding (Bahar & Hansell, 2000;


Danili & Reid, 2006; Kang, Scharmann, Noh & Koh, 2005; Tsaparlis,2005).

Convergence/Divergence

Convergence/divergence is another cognitive style that was introduced asa special aspect of intelligence. Convergent subjects are those who focuson the one right answer in order to find the solution of a problem.Divergent subjects are able to respond successively in problems requiringthe generation of several solutions and show fluency and flexibility (Child& Smithers, 1973). High ability studies have correlated divergent thinkingwith creativity, while it is stressed that convergent and divergent thinkingare not opposites or mutually exclusive but are rather complementary(Heller, 2007). Discussion about the relevance of the above threecognitive variables to a deeper students’ understanding of the particulatenature of matter is provided in the rationale below.

RATIONALE AND RESEARCH HYPOTHESES

The development of rationale for the present work focuses on twoaspects: first the choice of neo-Piagetian framework and second thechoice of the statistical analysis implemented. The first aspect waspartially analyzed in the “INTRODUCTION” section. Since research onstudents’ understanding of the particle nature of matter has focusedmainly on difficulties originated from the content material itself, the roleof cognitive factors (independent variables) on students’ conceptualunderstanding could be a potential area of interest. The neo-Piagetianframework, a well-consolidated one and frequently fostered in scienceeducation research, was chosen for further deductive endeavors. Thepersistent students’ misconceptions could be considered as products ofcognitive processes where mental resources, such as those operationalizedby neo-Piagetian variables, have an active role. Having empiricalevidences that correlation exists between students’ understanding of thestructure of matter and the three cognitive variables, LTh, FDI, and CD(Tsitsipis et al., 2010), the present study goes further and examinesspecific ideas and misconceptions. The dependent variables are specificresponses, qualitative/categorical data which have been analyzed andassociated with the metric cognitive variables within probabilistic modelssuch as binomial and multinomial logistic regressions.


Binomial and/or multinomial logistic regressions constitute the suitablestatistical approach for processing nominal data such as misconceptions inrelation to the independent three cognitive variables. From statistical pointof view, these differ substantially from linear regression in the underlyingmathematics and the computational method. The latter can be solvedexplicitly, while the equations of the former are solved iteratively.Moreover, the logistic regression does not presuppose three assumptionson which the general linear model (GLM) is based. These are thefollowing: (1) There are linear relations among variables, (2) the errorterm in the stochastic equation has to follow normal distribution, and (3)the independent variables have to be uncorrelated with the error term(identically and independently normally distributed errors). In addition tothe above, in logistic regression, the homogeneity of variance is notassumed, either. Therefore, the logistic regression analysis constitutes adistinct approach for exploring data and establishing possible effects, andthe results would be informative in a complementary manner.

There are some theoretical, a priori arguments for a possible effect ofthe three cognitive variables on students’ answers. It can be reasonablyassumed that understanding of the invisible microscopic structures of thesubstances needs the subjects’ imagination as well as the “development ofthose formal operations which allow construction of a postulatory-deductive system within which the concepts derive their broadermeaning” (Lawson & Renner, 1975, p. 348) which correspond to formaloperational reasoning. A significantly positive effect of LTh on the ‘right’students’ responses related to the particular nature of matter may be foundfor the above reasons. Besides, a significant effect of CD on students’answers might be found due to the major role that linguistic skills play incomprehending and interpreting a scientific text. Divergent thinkers arethought to show superiority in language (Hudson, 1966; Runco, 1986;Danili & Reid, 2006) and a linguistic advantage is considered to be ofparamount importance for reasoning in science (Byrne, Johnstone &Pope, 1994). Due to the above, divergent thinking might favor the ‘right’students’ answers. Finally, the chances are that various misleading andconfusing contexts prevent field-dependent pupils from locating thesignificant information. Among else, some of the various misconceptionsassociated with the particulate nature of matter could operate as confusingcontexts in this study. If so, field dependence would favor some ‘wrong’pupils’ responses.

Ergo, the hypothesis in this research study is that the following threecognitive variables—(a) LTh, (b) FDI, and (c) CD—have an effect on thespecific students’ ideas concerning the structure of matter, which constitutes


either misconceptions or models compatible with the scientific view. In otherwords, it is hypothesized that each of the students’ ideas or misconceptionscould be associated with the cognitive variables within a probabilistic model.

METHOD

Subjects

This study was conducted with the participation of 329 ninth-grade juniorhigh school Greek pupils (age 14–15, 51% females). The subjects were ofdifferent socioeconomic status and belonged to 18 different junior highschools in the prefecture of Fthiotida, in central Greece; seven of them arein the capital while the other 11 are dispersed in the municipalities of theprefecture. Each school contributed with one class section. All of thejunior high schools of the capital and almost half of the remaining juniorhigh schools in the prefecture took part in the research.

Instruments

Data were collected during one school year through paper-and-penciltests. The instruments used are briefly described below.

Logical Thinking. Pupils’ logical thinking abilities were measured usingthe Lawson paper-and-pencil test of formal reasoning (Lawson, 1978).The 45-min test consists of 15 items involving the following: conservation ofweight (one item), displaced volume (one item), control of variables (fouritems), proportional reasoning (four items), combinational reasoning (twoitems), and probabilistic reasoning (three items). The students had to justifytheir answers. A Cronbach’s alpha reliability coefficient of 0.79 wasobtained for the present sample.

Field Dependence/Independence. FDI ability of the subjects was assessedby a version of theWitkin, Oltman, Raskin &Karp (1971) Group EmbeddedFigures Test. This is a timed test (20 min) in which the subject’s task was tolocate and outline simple figures concealed in complex ones. In this study, aCronbach’s alpha reliability coefficient of 0.84 was obtained.

Convergence/Divergence. A six-item test was used to measure the extentof divergency of the subjects. Each item substantially constituted a mini-test itself lasting for 2–5 min. The whole test along with the directionsgiven lasted 45 min. Pupils were asked to generate words, pictures,


sentences, or to list ideas (Bahar, 1999). For the measures in this study,the Cronbach’s alpha reliability coefficient of the instrument is 0.76.

Pupils’ Achievement, Concerning Their Understanding of the ParticulateNature of Matter. The instrument was synthesized by selected itemsutilized in a number of related research studies (Johnson, 1998a, b, c;Papageorgiou, Stamovlasis & Johnson, 2010), and it forms part of the fullinstrument used by Tsitsipis et al. (2010). A pilot study followed byinterviews was carried out in order to correct possible communicationdeficiencies of the test, and thus, enhanced validity is expected. Thecalculated Cronbach’s alpha reliability coefficient was 0.78.

A description of the items used is given in Table 1. Pupils’ responseswere recorded as categorical data and were tabulated for the needs of thepresent study. It is important to emphasize that the pupils were asked tofill the research questionnaire without any notification almost 1 year afterthe time they had been taught the relevant matter. Thus, the instrument isconsidered to measure the residual knowledge on this matter.

Statistical Analysis

In this study, the logistic regression analysis (Agresti, 2007) wasimplemented, where the loge[odds] of receiving each ‘right’ answer ormisconception is modeled as a function of the three cognitive variables.

logep

1� p

� �¼ b0 þ bL � LThþ bF � FDIþ bc � CD

Odds express the relative probability of receiving the ‘right’ answeragainst the ‘wrong’ one; as ‘right’ are characterized the answers that wereconsidered compatible with the scientific view incorporating the basicaspects of the particulate theory taught at the eighth-grade level of theGreek curriculum, while all the others are characterized as ‘wrong.’

For the ‘right’ answer, a binary logistic regression is used with thedichotomous ‘right’/‘wrong’ response as dependent variable. For thevarious misconceptions or alternative ideas, a multinomial logisticregression is used with dependent variable the polytomous categoricalresponse. In this case, the odds of a misconception were examined againstthe ‘right’ answer. The above enable the exploration of the effect of thethree cognitive variables on each student’s response. The coefficients bi’sin the logistic equation are determined stochastically from the data. Fortheir significance, the Wald test is used, which tests the null hypothesis(Ho) that b=0. The larger the absolute value of bi, the greater the effect of


TABLE 1

Description of the questionnaire items

Part 1: the particulate nature of matter

Concern the solid state 1.A Pupils are asked to choose among five alternatives(see “APPENDIX”), the figure that best represents whatthey would ‘see’ if they observed a sugar grain using ahypothetical magnifying glass enabling the view of thegrain structure.1.B Pupils are asked to explain what they think existsbetween molecules, in case they chose a figuredepicting molecules. Otherwise, they do not have toanswer this question.1.C Pupils are asked to answer whether or not theythink that the view of the sugar structure through thehypothetical magnifying glass would remain ‘frozen’as the time is passing. They are also asked to explainor justify their answers.

Concern the solid state 2.A Pupils are asked to choose among five alternatives(see “APPENDIX”), the figure that best represents whatthey would ‘see’ if they observed a drop of pure(liquid) water using a hypothetical magnifying glassenabling the view of the structure of the drop.2.B Pupils are asked to explain what they think existsbetween molecules, in case they chose a figuredepicting molecules. Otherwise, they do not have toanswer this question.2.C Pupils are asked to answer whether or not theythink that the view of the water structure through thehypothetical magnifying glass would remain ‘frozen’as the time is passing. They are also asked to explainor justify their answers.

Concern the gas state 3.A Pupils are asked to choose among five alternatives(see “APPENDIX”), the figure that best represents whatthey would ‘see’ if they observed a very small quantityof oxygen, found inside a vase containing pureoxygen, using a hypothetical magnifying glassenabling the view of the structure of the oxygen.3.B Pupils are asked to explain what they think thatexists between molecules, in case they chose a figuredepicting molecules. Otherwise, they do not have toanswer this question.3.C Pupils are asked to answer whether or not theythink that the view of the oxygen structure through thehypothetical magnifying glass would remain ‘frozen’as the time is passing. They are also asked to explainor justify their answers.


the cognitive variables on the odds of receiving a certain answer. Animportant interpretation of the logistic regression uses the odds. Theestimated odds for a given misconception (or answer) are multiplied byexp (b) as the cognitive variable increases by 1 unit, ceteris paribus. If exp(b)91, that is, b9 0, then the odds of a given answer is positivelycorrelated with a cognitive variable, while it is negatively correlated ifexp (b) G 1, that is, bG 0 (Agresti, 2007).

RESULTS

The results of logistic regressions are shown in Tables 2, 3, 4, and 5. Eachtable contains remarkable students’ responses, which comprise well-known misconceptions and are examined along with the ‘right’ answers.The responses or categories shown in Tables 2, 3, 4, and 5 were actually

Part 2: the properties of a state as a result of the collective behavior of particles

Concern the same substancein three different temperatures

4.A Pupils are prompted to make the assumption thatthey have separated one single molecule from one ofthe following: a block of ice, some pure (liquid) water,or some pure water in the gas state. They are askedwhether or not they could understand if the separatedmolecule has come from ice, liquid water, or water inthe gas state, respectively. Then, they are also asked toexplain or justify their answers.4.B Pupils are prompted to make the assumption thatthey have separated one single molecule from a blockof ice, another single molecule from a quantity of pureliquid water, and a third single molecule from aquantity of water in the gas state. They are askedwhether or not they could determine a physical statefor each of the three molecules and if yes, what thisstate is. Then, they are also asked to justify theiranswers.4.C Pupils are prompted to make the assumption thatthey have separated one single molecule from a blockof ice, another single molecule from a quantity of pureliquid water, and a third single molecule from aquantity of water in the gas state. They are asked tocompare the shape and the magnitude of the threemolecules.

TABLE 1

(continued)


TABLE2

Abo

utsubstances’structure

Questions

Answers

Frequency

Valid

percent

Cognitivevariables

bWald

1AAbout

sugarstructure

Multin

omialregression

analysis

1.Nothing

exists

113.7

LTh

−0.148

7.264**

2.Uniform

-contin

uous

substance

3010.0

LTh

−0.094

10.951**

3.Gas

structure

268.7

LTh

−0.105

8.924**

4.Liquidstructure

131

43.7

LTh

−0.046

7.647**

FDI

−0.102

6.065*

Binom

ialregression

analysis

5.Solid

structure(‘right’answ

er)

9832.7

LTh

0.064

16.613***

FDI

0.084

4.915*

2AAbout

water

structure

Multin

omialregression

analysis

1.Nothing

exists

258.3

LTh

−0.088

7.554**

2.Uniform

-contin

uous

substance

4314.3

LTh

−0.091

11.584**

3.Gas

structure

6120.3

LTh

−0.047

5.856*

Binom

ialregression

analysis

4.Liquidstructure(‘right’answ

er)

112

37.3

LTh

0.057

14.652***

3AAbout

oxygen

structure

Multin

omialregression

analysis

1.Nothing

exists

7625.3

LTh

−0.082

14.242***

2.Uniform

-contin

uous

substance

4414.7

LTh

−0.053

4.791*

3.Liquidstructure

3511.7

LTh

−0.047

3.883*

Binom

ialregression

analysis

4.Gas

structure(‘right’answ

er)

113

37.7

LTh

0.058

14.697***

*pG0.05;**pG0.01;**

*pG0.001


TABLE3

Whatexistsbetweenmolecules

Questions

Answers

Frequ

ency

Valid

percent

Cog

nitive

variab

les

bWald

1BWhatexistsbetween

sugarmolecules

Multin

omialregression

analysis

1.Sug

ar/other

substances/other

molecules/air/oxy

gen

113

37.7

LTh

−0.046

5.38

5*Binom

ialregression

analysis

2.Nothing

/vacuu

m(‘righ

t’answ

er)

5418

.0LTh

0.04

66.97

9**

2BWhatexistsbetween

water

molecules

Multin

omialregression

analysis

1.Water/other

substances/other

molecules/air/oxy

gen

118

39.3

LTh

−0.085

14.009

***

Binom

ialregression

analysis

2.Nothing

/vacuu

m(‘righ

t’answ

er)

3812

.7LTh

0.08

115

.744

***

3BWhatexistsbetween

oxyg

enmolecules

Multin

omialregression

analysis

1.Other

substances/other

molecules/air/oxy

gen

7826

.0LTh

−0.051

6.96

2**

Binom

ialregression

analysis

2.Nothing

/vacuu

m(‘righ

t’answ

er)

5317

.7LTh

0.05

39.64

4**

FDI

0.08

94.30

1*

*pG0.05

;**pG0.01;**

*pG0.00

1


TABLE4

Abo

utmolecules’motionin

solid

s,liq

uids,andgases

Questions

Answers

Frequ

ency

Valid

percent

Cog

nitive

variab

les

bWald

1CAbo

utsugar

molecules’motion

Multin

omialregression

analysis

1.Nomotion.

Sug

arissolid

.Molecules,structurearestable.

4615

.3–

––

Binom

ialregression

analysis

2.Molecules

aremov

ing/aremov

ingrand

omly

arou

ndconstant

positio

ns(2)(‘righ

t’answ

er)

4414

.7LTh

0.04

45.76

4*

2CAbo

utwater

molecules’motion

Multin

omialregression

analysis

1.Molecules

aremov

ing(nomoredescription)

5819

.3LTh

0.06

14.23

2*Binom

ialregression

analysis

2.Molecules

aremov

ingrand

omly

‘slid

ing’

oneach

other,

keepingconstant

distances(‘righ

t’answ

er)

186.0

CD

0.07

55.96

1*

3CAbo

utox

ygen

molecules’motion

Binom

ialregression

analysis

1.Molecules

aremov

ingrand

omly

andfreely

(‘righ

t’answ

er)

134.3

LTh

0.07

86.44

7*

*pG0.05


TABLE5

The

prop

ertiesof

stateas

aresultof

thecollectivebehavior

oftheparticles

Questions

Answers

Frequ

ency

Valid

percent

Cog

nitive

variab

les

bWald

4.A

Abo

utthepo

ssibility

fora

water

molecule’sdistinctionon

thestreng

thof

thestateof

matteritcomes

from

Multin

omialregression

analysis

1.Moleculeisdistingu

ished.

Different

state

ofmatter/moleculedistances/moleculearray.

7023

.3LTh

−0.060

7.96

2**

2.Moleculeisdistingu

ished.

Different

motion.

217.0

––

–Binom

ialregression

analysis

3.Moleculecann

otbe

distingu

ished.

The

3arethesame(‘righ

t’answ

er)

4916

.3LTh

0.07

417

.149

***

4.B(I)Abo

utan

icemolecule’s

hypo

thesized

stateof

matter

Multin

omialregression

analysis

1.Moleculeissolid

.Iceis

solid

/macroscop

icsolid

features

119

39.7

LTh

−0.111

12.867

***

2.Moleculeissolid

.Specificmoleculemotion.

124.0

––

–3.

Moleculeisno

tsolid

orliq

uidor

gas.

No/no

n-remarkableexplanation.

113.7

LTh

−0.113

5.77

7*

4.Molecules

arealwayssolid

.7

2.3

––

–Binom

ialregression

analysis

5.Wecann

otspecifyastateof

matter

foron

esing

lemolecule(‘righ

t’answ

er)

165.3

LTh

0.10

914

.342

***


4.C(I)Com

parisonbetweenan

iceandawater

molecule

Multin

omialregression

analysis

1.Different

shapes

ormagnitudesor

both.

Different

states

ofmatter/macroscop

icstatefeatures.

6822

.7LTh

−0.057

6.92

9**

FDI

−0.118

4.47

7*Binom

ialregression

analysis

2.Sam

eshapeandmagnitude.Theyare

thesamesubstance/molecules

(‘righ

t’answ

er)

5217

.3LTh

0.07

517

.280

***

*pG0.05;**pG0.01

;**

*pG0.001


emerged from students’ answers. Unremarkable or ambiguous responsesor categories were excluded from any table presented; however, they weretaken into account by the statistical procedure. In each table, frequencies,valid percents, and the estimated b’s with the Wald test statistic for theirstatistical significance are provided. Since the computational method inlogistic regression works iteratively, the convergence of the procedure hasbeen confirmed in all cases. The direction of the effects—positive ornegative—was judged from the sign of b coefficient as described above.

The misconceptions located in students’ answers and examined in thispaper are the following:

M1: The matter is continuous (Johnson, 1998a; Adadan, Irving &Trundle, 2009; Ayas, Ozmen & Calic, 2010).

M2: Between molecules of a substance, the same substance or moleculesor other substance—usually air—or molecules exist (Novick &Nussbaum, 1978; Lee et al., 1993; Johnson, 1998a; Adadan et al.,2009).

M3: There is no intrinsic motion of molecules especially in the solid statewhere no motion of the substance is visible (Lee et al., 1993;Adadan et al., 2009).

M4: Students give macroscopic characteristics to a single molecule atmicroscopic level (Lee et al., 1993; Johnson, 1998a; Adadan et al.,2009).

The misconception M1, i.e., ‘matter is continuous,’ was detected instudents’ responses to questions 1A, 2A, and 3A analyzed in Table 2.LTh was found to have a significantly negative effect on the odds ofgetting the M1 in questions (1A, 2A, and 3A) concerning the threephysical states. That is, a higher LTh ability does not favor the specificstudents’ misconception. Moreover, LTh is negatively correlated with theodds of getting all of the remaining ‘wrong’ answers that are significantlyaffected, while it is positively correlated with the odds of receiving the‘right’ students’ responses. FDI was found to affect only the answersabout sugar structure (question 1A). In particular, FDI has a positiveeffect on the odds of getting the ‘right’ students’ response, while it has asignificantly negative effect on the odds of receiving the (‘wrong’)response that asserts ‘liquid’ structure for sugar (answer 1A.4).

The misconception ‘between molecules of a substance, the samesubstance or molecules or other substance—usually air—or moleculesexist’ was detected in students’ responses to question 1B, 2B, and 3B, andit is analyzed in Table 3. LTh has a significantly negative effect on the


odds of getting all of the three answers (1B.1, 2B.1, and 3B.1) thatdescribe the above misconception. In other words, this specificmisconception is not favored by a higher LTh ability. On the contrary,LTh is positively correlated with the odds of getting the ‘right’ students’answers that support the idea of empty space or vacuum between sugar,water, or oxygen molecules. FDI affects positively the odds of receivingthe compatible with the scientific view answer, which supports the notionof empty space between oxygen molecules (3B.2).

In Table 4, the alternative idea that there is no intrinsic motion ofmolecules especially in the solid state where no motion of the substance isvisible is confirmed by students’ answers (answer 1C.1). No directsignificant effects of the three cognitive variables on the odds of gettingthe above students’ idea were found; however, for solids (1C) and gases(3C), a positive correlation of LTh with the ‘right’ answers was detected.A similar effect was shown for the answer 2C.1 for the cases of liquids,which is close to the correct one.

In Table 5, the prevalent misconception is that students give macro-scopic characteristics to a single molecule at microscopic level as it canbe deduced from students’ justifications for the answers 4A.1, 4B(I).1,and 4C(I).1. The odds of receiving each of these three answers are liableto a significantly negative effect of LTh. In addition, FDI has asignificantly negative effect on the odds of getting the answer 4C(I).1,which claim different shapes or/and magnitudes between an ice and awater molecule due to an asserted different state of matter or to differentmacroscopic state features such as the shape or the volume of themacroscopic substance. Two other misconceptions in Table 4 are thefollowing:

(a) Molecules are always solid (4B(I).4). Similarly, in Griffiths &Preston’s (1992) study, students believed that water molecules arecomposed of two or more solid spheres.

(b) A single molecule’s characteristic motion can lead to its distinctionfrom another molecule of the same substance that comes from adifferent state of matter (4A.2). In accordance with the above, inJohnson’s (1998a) study students held that a single moleculemaintains a characteristic motion according to the state of matter itcomes from.

Logistic regression showed no effect of the LTh, FDI, and CD on theodds of getting the above two misconceptions, indicating that these mightbe deep-rooted misconceptions and apparently common to all cognitive


styles or LTh abilities. Interestingly, Griffiths & Preston (1992) hadattributed the students’ belief that water molecules are composed of twoor more solid spheres to academic instructional treatment and particularlyto the use of three-dimensional diagrams or to the use of ball-and-stickmodels.

It is worth mentioning that students’ responses in question 4B, whichconcern a water and/or a water-vapor molecule (not showed in Table 5),are similarly affected by the cognitive variables as the answers for an icemolecule’s hypothesized state of matter (4B(I)). The same holds forstudents’ responses in question 4C, which compare a water-vapor moleculewith an ice or a water molecule. These are not presented in Table 5.

INTERPRETATION OF THE RESULTS AND DISCUSSION

The present study is rather exploratory, searching models in order toprovide clues about the predictors and suggest questions for furtherresearch and theory building. The results have provided support for thehypothesis that students’ ideas or models concerning the structure ofmatter could be probabilistically associated with the three cognitivevariables. As a matter of fact, LTh was found to be the dominatingvariable associated with the overwhelming majority of pupils’ responses,compatible or incompatible with the scientific models, while associationsof FDI and CD with pupils’ answers are few and far between.

Research findings have confirmed the significance of LTh as apredictor variable on science achievement (Chandran et al., 1987; Lawson& Thompson, 1988; Johnson & Lawson, 1998; Kang et al., 2005) and ledto the conclusion that achievement deficiencies in the sciences,mathematics, etc. are probably due to deficiencies in formal reasoning(Lawson, 1985). Besides, full comprehension of the abstract conceptsrelated to the particulate nature of matter cannot be achieved unlesshypotheticodeductive reasoning about the unseen particle entities hasbeen developed. Such reasoning, though, is characteristically formaloperational (Lawson & Renner, 1975; Cantu & Herron, 1978). The abovecould explain the findings of the present study that as a rule show LTh tobe positively correlated with the odds of getting the ‘right’ answers andnegatively correlated with the odds of getting the ‘wrong’ ones. The onlyexception to this rule is the answer 2 of the question 2C (Table 4).However, this students’ answer, i.e., ‘molecules are moving,’ could beaccepted as partially correct, as it shows that molecules’ motions are notignored despite the fact that their detailed features are not described.


FDI cognitive style appeared to have a significant effect merely on afew answers. In general, field-independent students were more likely togive a ‘right’ answer, while field-dependent subjects were more likely togive a ‘wrong’ answer. Although this is a small-scale effect, it isconsistent with the fact that field independents have been found to achievemore in science generally (Lawson, 1983; Johnstone & Al-Naeme, 1995;Niaz, 1996; Tinajero & Paramo, 1998; Bahar & Hansell, 2000; Danili &Reid, 2004, 2006; Kang et al., 2005; Tsaparlis, 2005; Stamovlasis &Tsaparlis, 2005). This serious advantage of field-independent pupilscould be attributed to their ability to separate readily the significantinformation from its context (Witkin & Goodenough, 1981). On theother hand, field-dependent pupils might be proved unable to utilizecritical information. The above characteristics of FDI cognitive stylejustify why it appears to have affected certain answers of the questions1A (Table 2), 3B (Table 3), and 4C(I) (Table 5). In question 1A,which is about sugar structure, the almost same intermoleculardistances in liquid and solid structures may constitute a confusingcontext when it comes to the distinction between the two structuresdespite their different molecule array. The effect of FDI is strikinghere. FDI is positively correlated with the odds of selecting the solidstructure, i.e., field-independent pupils were more likely to choose the‘right’ answer, while it is negatively correlated with the odds ofselecting liquid structure, i.e., field-dependent pupils were more likelyto choose the ‘wrong’ answer. Interestingly, the distinction is easierwhen the ‘wrong’ answer of a gas structure (answer 3) is the casebecause the increased intermolecular distances make this structuremore distinguishable, and therefore, the effect of FDI vanishes.

In question 3B, an explanation for the effect of FDI on the ‘right’students’ response that considers vacuum between molecules (answer 2)may be due to the reported in the literature confusion about the nature ofgases. Johnson (1998b) found out that students did not consider thatthe water particles themselves in a bubble in boiling water are thegas. Thus, they “needed a substance that was ‘a gas’ to form the‘gas’ of the bubble” (p. 581). Broaderly speaking, there is a well-known students’ difficulty regarding the notion of empty spacebetween particles, especially for the gas state (Johnson, 1998a).Considering the above, the field-independent pupils may be morecapable to perceive that a gas is nothing more or less than itsmolecules (important information), and therefore, the intermediatespace is a vacuum. The misleading context here could be themisconception itself for the nature of gases.


Further, the answer 3 of the question 4C(I) (Table 5) that is the well-acknowledged misconception where students give macroscopic charac-teristics to a single particle at microscopic level seems to have aconnection with FDI cognitive style. The odds of demonstrating thismisconception are negatively correlated with FDI. Field-dependentpupils refer to macroscopic features of the states of matter, such as theshape or volume of a macroscopic substance (water or ice), or simply toa different state of matter (e.g. ‘ice is solid,’ ‘water is liquid’) in theireffort to justify their comparison between an ice and a water molecule.This indicates that the macroscopic model of a state of matter mightconstitute the dominating frame of reference, from which field-depend-ent subjects are not capable to escape even when they refer to one singlemolecule.

The effect of CD on students’ answers is also restricted, as is the casewith FDI. An interpretation of CD’s role on the ‘right’ answer of question2C (Table 4), which sufficiently describes water molecules’ motion, couldinvolve the determinative role of language. Linguistic skills, such ascomprehension and interpreting of a scientific text, are considered to be ofparamount importance for reasoning in science (Byrne, Johnstone &Pope, 1994). Students, though, who show superiority in language, arethought to be divergent thinkers (Hudson, 1966; Runco, 1986; Danili &Reid, 2006). Thus, divergent thinkers may understand more of a scientificmaterial that needs increased linguistic abilities in order to be studied ortaught. The above is consistent with the fact that an increased divergentability favors the specific students’ response.

Even though the above analysis attempted to provide a comprehensiveinterpretation of the statistical effects, in order to avoid, at this point,reductionistic flaws, it is imperative to stress that the formation of ideas isa more complex process and cannot be reduced merely to some cognitivevariables. In contemporary psychological theories, knowledge is repre-sented mentally in terms of concepts, categories, and propositions, andontologically is characterized as a hierarchical associative network ofconcepts (Newell, 1995). These organization structures, considered eitheras fragmented or theory-like, is what changes when conceptual changeoccurs. Even though such processes are intractable, an importanttheoretical contribution to science education could be the investigationof variables involved. Conceptual change has been proposed as adynamical cognitive process, where the related control variables interactrather with a nonlinear manner (Stamovlasis, 2006, 2010, 2011). In thepresent study, what has been learnt is that LTh, FDI, and CD are morelikely to be among the potential protagonists.


EDUCATIONAL IMPLICATIONS

The implications of the present findings for the practice of scienceeducation are associated with teaching strategies, which should bedeveloped to overcome the barriers imposed by students’ insufficiencyin formal reasoning. Informed teachers should assist students by utilizingmethods that make abstract concepts more accessible through concreteoperational thought. These methods make use of illustrations, diagrams,and models that constitute perceptible entities or concrete materials tofocus attention on critical and variable attributes of abstract concepts.Research has shown that such methods can enhance the attainment ofabstract concepts (Cantu & Herron, 1978; Howe & Durr, 1982; Zeitoun,1984). Alternatively, training programs that promote development instudents’ formal operational reasoning (Lawson, 1985) might be designedand implemented.

Analogous is the implication that concerns students’ insufficiency indisembedding ability. Even though the field-independent ability may bedeveloped naturally with experience, it is difficult to teach someone to befield independent. Instead, it is advisable to avoid any irrelevant ormisleading information (‘noise’) that could be misinterpreted by thestudents resulting to erratic ideas or misconceptions. Effort should bemade during teaching in order to help students make sense of the materialtaught, when attending lessons in classroom or reading their schooltextbooks, by focusing on central ideas and disembedding only therelevant information.

Finally, circumvention as much as possible of the determinative roleof language for teaching a subject matter is suggested, when divergentthinking is found to favor a ‘right’ answer corresponding to the specificsubject matter. According to the findings of this study, lack of divergentthinking is regarded as a disadvantage connected with restrictedlinguistic skills. If so, illustrations and diagrams can clarify theparticulate structures of the substances in the three physical states, andsoftware simulations can demonstrate the transition from one statemodel to another.

Conclusively, the analysis implemented in this study proved to beuseful in providing a means of association between student’sresponses/categorical data and psychometric/scale variables. From amethodological point of view, it demonstrated that students’ mis-conceptions could be probabilistically correlated with individualdifferences and encourages further investigations with other potentialcandidate variables.


APPENDIX

TABLE 6

Sample items of the tests

Test 1: Logical Thinking (LTh)The three strings in the drawing on the right have metal bobs attached to their ends. The strings A and C are of the same length while the string B is longer. A bob of 100 gr is attached to the end of the string A. A bob of 100 gr is also attached to the end of the string B. A bob of 50 gr is attached to the end of the string C. The strings (and the attached bobs) can be swung back and forth and the time it takes to make a swing can be timed. Suppose you want to find out whether the length of the string has an effect on the time it takes to swing back and forth.

Item

7

Which strings would you use to find out?

1) A and B 2) A and C 3) B and C

Select one of the three choices and then justify your answer. …………………………………………………………………………………………………..

Test 2: Field dependence/independence (FDI)The simple shape (B) on the right is “hidden” in the complex pattern of lines on the left (pattern (A)). It is of the same size, has the same proportions and it faces in the same direction as when it appears alone. Can you pick it out? Using a pen, mark the position of the shape (B) that exists within the pattern (A) by tracing round its outline.

Item

1

Test 3: Divergency (CD)

Item

4

This is a test to see how many things you can think of that alike in some way. For example: What things are always red or that are red more than any other colour? You may use one word or several words to describe each thing. e.g.: tomatoes, bricks, blood.

Go ahead and write as many things as you can think that are ‘round’ or that are round more often than any other shape. ……………… ……………… ……………… ……………… ……………… ……………… ……………… ……………… ……………… ……………… ……………… ………………

2 minutes

Test 4: Pupils’ understanding of the particulate nature of matterSuppose that you could magnify billions of times a very small area of a sugar grain by means of a hypothetical magnifying glass so that you could see the structure of the sugar. 1A: Which of the following five figures do you think that it best represents what you would see?


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Dimitrios Stamovlasis

Faculty of Philosophy, Department of Philosophy and EducationAristotle University of Thessaloniki54 124, Thessaloniki, GreeceE-mail: [email protected]

Georgios Tsitsipis and George Papageorgiou

Department of Primary EducationDemocritus University of Thrace68100Nea Chili, Alexandroupoli, GreeceE-mail: [email protected]: [email protected]


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A PROBABILISTIC MODEL FOR STUDENTS’ ERRORS AND MISCONCEPTIONS ON THE STRUCTURE OF MATTER IN RELATION TO THREE COGNITIVE VARIABLES