KAI-LIN YANG
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICSTEACHERS' CONCEPTIONS OF SCHOOL MATHEMATICS,
SCHOOL STATISTICS, AND THEIR DIFFERENCES
Received: 11 December 2012; Accepted: 29 January 2014
ABSTRACT. This study used phenomenography, a qualitative method, to investigateTaiwanese mathematics teachers’ conceptions of school mathematics, school statistics,and their differences. To collect data, we interviewed five mathematics teachers by openquestions. They also responded to statements drawn on mathematical/statisticalconceptions and epistemological beliefs in the literature. We also conducted a survey of22 mathematics teachers who responded to open-ended questions for comparing andjustifying the findings from the interview data. We found that the characteristic features ofthe categories and dimensions of these teachers’ conceptions were different from those inthe relevant literature. All of the mathematics teachers’ conceptions of schoolmathematics, school statistics, and their differences could be identified by thesecategories and dimensions as reflected by their characteristic features. We discuss ourfindings and their implications for further research and teacher education programs.
KEY WORDS: conception, epistemology, mathematics teacher, philosophy
INTRODUCTION
Teachers' conceptions of mathematics and mathematical pedagogy werefound to affect their pedagogical decisions and then their student learning(e.g., Cross, 2009; Ernest, 1989; Staub & Stern, 2002). Conceptions are,as Thompson (1992) put it, “a more general mental structure,encompassing beliefs” (p. 130), and beliefs can have meanings connectedto experience and mental constructions (Sigel, 1985). In this study, weadopted the broader interpretation of conceptions and acknowledge itsintegration of cognitive and affective components.
We found relatively little research literature on exploring teachers’conceptions of statistics and the relationship to those of mathematics(Pierce & Chick, 2011). According to statistics educators’ arguments,Burrill & Biehler (2011)outlined that mathematics and statistics differ intheir essential defining characteristics: roles of context, methods ofreasoning, precision, roles of data, and data collection. Statistics isincluded at the school level because of its usefulness in daily life, itsimportant role in developing critical reasoning, and its instrumental
International Journal of Science and Mathematics Education 2014# National Science Council, Taiwan 2014
function in other disciplines and in many professions (Chick & Pierce,2008; Hancock, Kaput & Goldsmith, 1992).
In line with Schoenfeld’s (1998) ideas, conceptions of mathematics andstatistics can be understood as culturally shaped mental constructs.International comparative studies have found that mathematics teachers’beliefs vary between countries (e.g., Felbrich, Kaiser & Schmotz, 2012).Therefore, investigating teachers’ epistemological beliefs in diversecultures can make the greatest contributions to teacher education (e.g.,Chan & Elliott, 2000).
Our review of the literature indicated that teachers’ conceptions ofknowledge and knowing (e.g., Thompson, 1992), as well as thedifferences in their conceptions—between knowledge domains (e.g.,Hofer, 2000), cultures (e.g., Chan & Elliott, 2000), and the particularity ofTaiwanese mathematics teachers (e.g., Hsieh, Wu & Wang, 2014)—haveeffects on their classroom practice (e.g., Cross, 2009). Therefore, thepurpose of our study was to investigate Taiwanese mathematics teachers’conceptions of school mathematics and school statistics as well as thedifferences in their conceptions.
THEORETICAL FRAMEWORK
Teachers’ conceptions about the nature of mathematics have been thesubject of extensive study in mathematics education research. Thesestudies can be conceptualized from two different perspectives: on thefeatures of mathematical foundations and development (see Thompson,1992) and on personal epistemology with the focus on what knowledgeis, how knowledge is constructed, and how it is evaluated (see Muis,2004). Although there is some correspondence between the twoperspectives, studies drawing on each perspective can suggest differentapproaches to teachers’ conceptions and then reveal distinct meanings oftheir conceptions.
From the philosophy of mathematics perspective, Lerman (1983)identified two alternative conceptions of the nature of mathematics,absolutist conception (mathematics is an accumulated body of hierarchi-cal and objective knowledge) and fallibilist conception (mathematics ispresented as investigation of problems and solutions), and then contrastedthe knowledge-centered and problem-solving conceptions of mathematics.Petocz, Reid, Wood, Smith, Mather, Harding & Perrett (2007) researchinto undergraduate students’ conceptions of mathematics identified threelevels of conceptions, including the broadest life view (mathematics is an
KAI-LIN YANG
approach to life and a way of thinking), two intermediate conceptions:modeling view (mathematics is about building and using models, andtranslating some aspect of reality into mathematical form) and abstractview (mathematics is a logical system or structure); and the narrowestnumber view (mathematics as a toolbox of individual components andprocedures). There are correspondences between Petocz et al.’s views andLerman’s conceptions in that number and abstract views can beconsidered as knowledge-centered (or static) conceptions, and modelingand life views as problem-solving (or dynamic) conceptions.
Reid & Petocz’s (2002) phenomenographic study also found sixconceptions of statistics—statistics is/are: (1) individual numericalactivities, (2) using individual statistical techniques, (3) a collection ofstatistical techniques, (4) the analysis and interpretation of data, (5) a wayof understanding real-life situations using different statistical models, and(6) an inclusive tool used to make sense of the world and developpersonal meanings. We consider that the first three conceptions ofstatistics are similar to a knowledge-based view, whereas the last three aresimilar to a problem-solving view. However, we also consider that instatistics the role of context and the learner’s relationship with the contextare also emphasized in the notion of reading behind the data(Shaughnessy, 2007) and as “art and culture” of exploratory data analysis(Ben-Zvi & Friedlander, 1997). Therefore, we formulated the third viewof statistics as contextual investigation conception which relates the socialcontext to the creation and justification of knowledge.
From the epistemological belief perspective, conceptions of schoolmathematics and school statistics can be counted as beliefs about thenature of knowledge and knowing. For example, Muis (2004) pointed outthat personal epistemology is a powerful predictor of one’s learningapproach and academic achievement in mathematics. In teachingmathematics and statistics, teachers draw not only on pedagogical oracademic knowledge of mathematics and statistics, but also on concep-tions about what knowledge is relevant and critical in learning (Pierce &Chick, 2011; Thompson, 1992). Furthermore, “successful teachingdepends on professional knowledge and teacher beliefs (Döhrmann,Kaiser & Blömeke, 2012, p. 327)”. However, mathematics teachers’belief about mathematics and statistics is seldom investigated with thedimensions of epistemological belief (ref. Hofer & Pintrich, 1997).
Hofer & Pintrich (1997) argued that within the nature of knowledge,there are two dimensions of epistemological belief: certainty andsimplicity, and within the nature of knowing, there are two otherdimensions: source and justification. Accordingly, each dimension ranges
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
from a conception at one end of a continuum to another conception at theother end (e.g., for certainty, it ranges from knowledge that is unchangingto knowledge that is evolving; for more information about other threedimensions; see Table 1).
Hofer (2000) constructed an instrument to justify her multidimension-ality of epistemology. Factor analysis supported the existence of a multi-dimensional model of epistemology across disciplines, but the certaintyand simplicity of knowledge dimensions did not emerge as separatefactors. Besides, an additional factor emerged regarding the perceivedattainability of truth which contained items related to the certainty ofknowledge dimension. Therefore, we adapted the Hofer & Pintrich’s(1997) four presumed dimensions (simplicity, certainty, sources, andjustification) and one emerging dimension of Hofer (attainability of truth)for use in our study (see Table 1 and Fig. 1). Epistemological beliefs suchas attainability of truth have both an ontological aspect (certainty/diversity oftruth) and relational aspect (relationship between the knower and the known)(Kang &Wallace, 2005). For more in-depth probing of teachers’ conceptionsof mathematics and statistics, we adopted two contrasting epistemologicalconceptions along a continuum for each dimension (see Table 4).
To compare the differences between statistics and mathematics isconcerned by statistical researchers and educators. For example, Scheaffer(2006) identified one difference between deterministic and probabilisticthinking, and elaborated that the core of mathematics and statistics lies in
TABLE 1
Five dimensions of epistemological beliefs (based on Hofer & Pintrich, 1997;Hofer, 2000)
DimensionsContrasting epistemologicalconceptions along a continuum
Exemplary questionnaireitems (Hofer, 2000, p. 390)
Simplicity Isolated/interrelated knowledge Most of what is true in thissubject is already known.
Certainty Unchanging/evolving knowledge Principles in this field areunchanging.
Attainability of truth Truth can be/cannot be attained Experts in this field canultimately get to the truth.
Authoritative source Source by authority/by activeconstruction
If you read something in atextbook for this subject,you can be sure it’s true.
Biased justification By feeling or authority/byresearch
First-hand experience is thebest way of knowingsomething in this field.
KAI-LIN YANG
proof and data, respectively. Gattuso & Ottaviani (2011) further outlinedthe difference between statistics and mathematics “in the way thatreasoning takes place, in the way they use numbers, in the way thatvariability and variation are taken into account, and in their approach tomeasurement (p. 130)”. These differences can be associated with thecertainty (numbers and variation) and justification (reasoning andapproach to measurement) dimensions of epistemological beliefs.Nonetheless, teachers’ conceptions of the differences and exploring themas to the two perspectives have not been noticed by research on teachers’conceptions about statistics education (see Pierce & Chick, 2011).
When using these five dimensions (Hofer & Pintrich, 1997; Hofer, 2000)as critical lenses to characterize the three types of conceptions of mathematicsand statistics based on the literature (Lerman, 1983; Ben-Zvi & Friedlander,1997; Shaughnessy, 2007), we found that there are some correspondencesbetween these dimensions and conceptions. As for certainty and authoritativesource, the knowledge-centered conception seems to be consistent with theunchanging characteristics of knowledge and authority of knowing, and theother two conceptions, problem-solving conception and contextual investiga-tion conception, seem to be consistent with the evolving characteristics ofknowledge and active construction of knowing. Nonetheless, there is nodeterminate correspondence between types of conceptions and dimensions ofepistemological beliefs. Due to their differences and complement, it would bemore exhaustive to investigate teachers’ conceptions of mathematics andstatistics from the two perspectives.
METHOD
This study used phenomenography, a qualitative research approach,which is “a research method adapted for mapping the qualitative different
Epistemology
Knowledge Knowing
Simplicity Certainty Attainability Authoritative
Source
Biased
Justification
Figure 1. Dimensions of epistemological beliefs (based on Hofer, 2000)
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
ways in which people experience, conceptualize, perceive, and understandvarious aspects of, and phenomena in, the world around them” (Marton,1986, p. 31). The goal of the phenomenographic method in this study wasto reveal how mathematics teachers perceived the nature of schoolmathematics, school statistics, and their differences. Phenomenographytakes a non-dualistic ontological perspective that human reality is notdivided into the objective and subjective worlds but rather simultaneouslyrelated to both objects and subjects (Marton, 2000). There are some recentphenomenographic studies in mathematical education (e.g., Chiu, 2012;Petocz et al., 2007).
For data collection, we used a semi-structured individualinterview—based on our theoretical framework drawing on the literature asdescribed in the preceding section—in order to interpret the participants’responses and to generate categories for qualitatively describing theparticipants’ different ways of experiencing a phenomenon (Marton &Booth, 1997). Next, we conducted a survey using three open-endedquestions to compare and justify the findings from the interview data.
Participants
Five secondary mathematics teachers teaching in the top three senior highschools in the capital city of Taiwan were interviewed in depth (see Table 2).They were selected due to their participation in another study on investigatingthe teaching of secondary school statistics; and they also agreed to participatein this study. Yi and Anna were studying in graduate summer school when theinterviews were conducted. They obtained the master’s degree after thisinterview. Hue, Dong, and Lee obtained their master’s degrees from thebeginning of his teaching career (see Table 2). They had learned the advancedsecondary mathematics topics including number theory, axiomatic geometry,probability, set theory, and analytic geometry.
TABLE 2
Backgrounds of the five interviewees
Pseudonyms GenderTeaching experience(years)
Master degree studyprogram
Yi M 10 Mathematics educationHue M 18 StatisticsDong M 13 Applied mathematicsAnna F 6 Mathematics educationLee M 19 Mathematics
KAI-LIN YANG
About 10 months after the interview, another 22 secondary mathematicsteachers (see Table 3) who were taking a research methodology course in agraduate summer school of mathematics education were surveyed throughopen-ended questions. This research methodology course provided these in-service mathematics teachers with qualitative and quantitative methods.Although this sample could not be representative of secondary mathematicsteachers, their survey responses could be compared to the revealed featuresof teachers’ conceptions in the interview and to justify these findings.
Data Collection
Each of the five teachers was interviewed individually by a trainedresearcher in Chinese. The interview process was organized as “three-interview series” (Seidman, 1991) that lasted from 90 min to 2 h for eachinterview. The first interview focused on the participants’ perceivedconceptions of school mathematics and statistics, and was begun withquestions: What is school mathematics? What is school statistics? Whatare the differences? Based on their responses, the interviewers encouragedthe participants to give examples or explain their perceived conceptions.The second interview asked participants to talk about each statementabout school mathematics and statistics as shown in Table 4. The lastinterview encouraged individual participants to reflect on the beginningquestions and their responses in the first and second interviews.
We designed the statements regarding types of mathematical concep-tions and dimensions of epistemological beliefs in the literature (seeTable 4) for eliciting the interviewees’ conceptions of school mathematicsand statistics in the second interviews. After reading each statement, thefive interviewees were asked to answer the main question: What is youropinion about this? In order to achieve an in-depth understanding of theteachers’ conceptions, thinking about these statements were followed bysupplementary questions. For instance, teachers were asked to explaintheir opinions when they disagreed with one statement or to elaborate thestatement with examples from their teaching experience when they agreed
TABLE 3
Demographics of the 22 survey teachers
Female Male Junior high school Senior high school
Number of teachers 7 15 17 5Median of teaching years 15 8 8 9
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
with it. In addition, the interviewer tried to ask teachers to clarify theirinterpretations of terms used in the statements during the last interview.The interviews were tape-recorded and subsequently transcribed.
TABLE 4
Theoretical framework of conceptions/beliefs and corresponding statements
Phases Corresponding statements
Conception categories (Lerman, 1983; Ben-Zvi &Friedlander, 1997; Shaughnessy, 2007)
Knowledge-centered Some state that mathematics/statisticsa is an accumulatedbody of hierarchical and objective knowledge.
Some state that mathematics/statistics is acquired first andapplied afterwards.
Problem-solving Some state that mathematics/statistics is a subject of problemsolving.
Some state that mathematics/statistics is investigation ofproblems and their solutions.
Contextual investigation Some state that context is important for mathematics/statistics.
Some state that mathematics/statistics is investigativeprocesses in the context of societal activity.
Dimensions of epistemological beliefs (Hofer & Pintrich,1997; Hofer, 2000)
Simplicity Some state that mathematics/statistics is organized as highlyinterrelated concepts. (R)b
Some state that mathematics/statistics is organized asisolated pieces.
Certainty Some state that mathematics/statistics is unchanging.Some state that mathematics/statistics is evolving. (R)
Authoritative sources Some state that mathematics/statistics is handed down byauthority.
Some state that mathematics/statistics is acquired throughactive construction. (R)
Biased justification Some state that mathematics/statistics is justified throughpersonal experience.
Some state that mathematics/statistics is justified by inquiryand integration of experts’ views. (R)
Attainability of truth Some state that the truth in mathematics/statistics can beattained.
Some state that there is no absolute truth in mathematics/statistics. (R)
aMathematics/statistics referred to school mathematics and school statisticsbR denotes reverse statements
KAI-LIN YANG
Then, 22 other mathematics teachers were surveyed by a questionnaireusing open-ended questions in Chinese for collecting data moreefficiently but this method is limited by its written responses. The open-ended questions were: (1) “If students ask you, ‘what is schoolmathematics?’, what will you tell them?” “What are the characteristicsof school mathematics regarding its certainty, truth and validity?”; (2) “Ifstudents ask you ‘what is school statistics?’, “what will you tell them?”“What are the characteristics of school statistics regarding its certainty,truth and validity?”; and (3) “What are the differences between the twosubject areas: mathematics and statistics?” “What do you think about thedifferences?”
In both the interview and survey situations, we told teachers thatschool mathematics and school statistics respectively refers to “algebraand geometry” and “data presentation and analysis” taught in junior andsenior high school.
DATA ANALYSIS AND FINDINGS
In the analysis, five secondary teachers were designated by pseudonyms(see Table 2) and 22 secondary teachers by T6 to T27. The interview andsurvey data were analyzed in four stages. First, we compared andcontrasted each interviewee’s responses with those of the other four. Keywords and common properties were found through constant comparativeanalysis (Merriam & Associates, 2002; Strauss & Corbin, 1998). Second,we connected the common properties with phases of conceptions typesand dimensions from the epistemological beliefs (see Table 4). From theconnections and analyses, we then modified the three phases as anotherthree new categories (see Table 5), and the original five dimensions asanother new five dimensions (see Table 6). Interview and survey open-ended question responses were used to illustrate these common propertiesof the categories and dimensions that emerged from the study (see somesample quotes in Tables 5 and 6). Third, we coded the written responsesof the open-ended questionnaire survey using the new categories and thedimensions specified in stage two. All of the survey responses could becoded by the categories of conceptions, but some could not be coded bythe dimensions of conceptions due to insufficient responses. Finally, weadopted categories and dimensions to classify and contrast eachparticipant’s conceptions of school mathematics and school statistics.
In this phenomenographic study, we tried to ensure its trustworthinessand credibility using multiple procedures (Strauss & Corbin, 1998). For
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
TABLE5
Three
catego
ries
ofteachers’conceptio
ns
Categories
Key
words
identified
Com
mon
prop
erties
Samplequ
otes
from
interviewsan
d/or
survey
respon
ses
Instrumental
application
Too
ls;prerequisite;
structure;
logic
Alogicalstructureor
system
ofuseful
tools
requ
ired
forsolving
prob
lems
Statistical
topics
intextbo
oksareindepend
ent…
morefor
application…
Alth
ough
it(m
athematics)
isno
tnecessarily
dedu
ctive,itexistsob
jectively.
Itisun
derlainby
asystem
anda
setof
logic(A
nna).
Formulas
instatisticsarehard
togrou
ndon
solid
theories.You
(I)
feel
itisalittle
independ
ent.…Statisticsisuseful
andvaluable
butno
twellrelated(H
ue).
…Wecanusecompu
terto
calculatestatisticsor
draw
graphs
(Yi).
Statistical
form
ulas
areeasy
tousebu
thard
toun
derstand
(Lee).
Relational
application
Multip
lerepresentatio
ns;
concepts;relatio
nsMultip
leandrelatio
nal
representatio
nsof
conceptsrequ
ired
for
solvingprob
lems
…The
dedu
cedconcepts(m
athematics)
arerelated(H
ue).
Interpretin
gdata
instatisticsisthemostim
portant.…ho
wto
interpretdepend
son
ourdecision
s(Y
i).
Statisticsisno
tmathematics.Itiswho
llyuseful
inreal
life.Ifyo
udo
notkn
owho
wto
useit,
youcann
otkn
owits
meaning
…(Lee).
It(m
athematics)
needsapplications
andtransformations
(Don
g).
Relational
metho
dology
Con
nection;
integration;
waysof
thinking
Waysof
thinking
for
prop
osingprob
lemsand
developing
concepts
…mathematicsislearnt
noto
nlyfortherigidof
structures
butalso
forthinking
andsolvingprob
lems…
(Ann
a).
Mathematicsrepresentsasystem
ofthinking
andthen
isused
tosolveprob
lem…(H
ue).
It(m
athematics)
teachesus
how
tothink(Y
i).
Mathematicsissimultaneou
slyrelatedto
itselfandothersubjects
fordeveloping
new
know
ledg
e(D
ong).
KAI-LIN YANG
example, two researchers condensed the transcripts of teacher interviews,reduced them to about 20 keywords which were then merged, compared,and clustered for formulating the categories and dimensions of teachers’conceptions. Lastly, one researcher and one transcriber translated quotesfrom Chinese to English. The five interviewees were invited for member-checking of the transcripts and codings (Seidman, 1991) which weresubsequently modified after their discussion with us. Furthermore, thesurvey data were used to examine the generality of categories, someof which were subsequently modified to code some previouslyunassignable data. In the following sections, we report the in-depthmeaning of each category and dimension by describing instances ofinterview or the written responses with sample quotes as shown inTables 5 and 6.
Teachers’ Conceptions of School Mathematics and School Statistics
As shown in Tables 5 and 6, analysis of the interview transcripts and thewritten responses provided evidence for generating three qualitativelydistinct categories of conceptions and for five distinct dimensions ofconceptions drawing from the literature on various aspects of thephilosophy of mathematics, which could represent the variation withinall participants. Although the categories and the dimensions wereillustrated by some quotations from either the interview transcripts orthe written responses to open-ended questions, they did not represent anobjective experience of a single individual.
Three Categories of Teachers’ Conceptions. We found three categoriesof Taiwanese teachers’ conceptions in this study: instrumental applica-tion, relational application, and relational methodology. Table 5 sum-marizes the key words and common properties of these three conceptions.Instrumental and relational were summarized according to the focus on‘knowledge as prerequisite tools’ and ‘knowledge as meaningfulconnection’. Application and methodology were summarized accordingto their focus on “what are required to solve problems” and “therelationships among what are related to problem solving”.
For instrumental application and relational application categories,teachers perceived school mathematics/statistics respectively as “a logicalstructure or system of useful tools required for solving problems” and“multiple and relational representations of concepts required for solvingproblems” (see Table 5). The latter category emphasized that the basis ofapplication is more on relational understanding than on the structure ofknowledge. As for relational methodology category, teachers perceived
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
TABLE6
Fivedimension
sof
teachers’conceptio
ns
Dimension
sKey
words
identified
Com
mon
prop
erties
Samplequ
otes
from
interviewsan
d/or
survey
respon
ses
Degreeof
certainty
Chang
ing;
unsure;
unvarying;
accurate
Alm
ostor
hardly
sure
ofinvariance
ofkn
owledg
eIagreemathematicsandstatisticsarebo
thevolving
,bu
ttheinfluence
ofchangesin
scho
olstatisticsislarger…The
curriculum
contentof
scho
olmathematicsmay
change,bu
t…its
definitio
nsandrelated
prop
ertiesareno
tvaried
(Ann
a).
…mathematicsisalmostun
changing
,scho
olmathematicsespecially;
butstatistical
form
ulas
forestim
atingthesameparameter
may
bedifferentin
differentedition
sof
textbo
oks(T13
).Sou
rces
ofun
certainty
Chaos;person
alun
derstand
ing;
reality
;intuitive;
artificial
Uncertainty
comingfrom
subjectiv
eor
objective
factors
Yi:The
power
ofscho
olmathematicsdepend
son
theidentity.
Why
teachers
need
tosummarizewhatiscorrector
incorrect…
?Teachers
have
moreauthority
than
stud
ents.Schoo
lstatisticsissimilar(Y
i).
Statisticsdealswith
uncertainty,
andmakinginferencefrom
uncertain
chaosisprob
abilistic.According
ly,itisim
possible
tobe
100%
certainof
thecorrectnessof
statistical
inference(H
ue).
Autho
rity
andactiv
econstructio
nrepresentsenseandsensibility.
Mostly
,ou
rsensecomes
from
outsideauthority
,andou
rsensibility
comes
from
innerconstructio
n(D
ong).
Sou
rces
ofcoherence
Coh
erentsystem
;local;incoherent
Coh
erence
comingfrom
definite
orindefinite
sense
Wekn
owthatthedevelopm
entof
mathematicsisregu
latedby
form
ing
acoherent
system
(e.g.,Euclid
eangeom
etry)andcoherencein
statisticsislocalandrelatedto
perspectives
onun
certainty(e.g.,
Classical
orBayesianperspective)
(Yi).
Ido
notdoubtthattheshortestdistance
betweentwopoints(ontheplane)
isthelength
ofastraight
segm
ent…
How
ever,I
donotb
uylottery
tickets…Iknow
thetheoreticalprobability.W
ehave
multip
leview
son
uncertainty.
The
view
smay
beincoherent
with
each
other(A
nna).
KAI-LIN YANG
Waysof
justification
Experience;
logic;
dedu
ctive;
indu
ctive
Log
ical
orem
pirical
valid
ationof
results
Alth
ough
experts’
view
swerecriticalto
judg
ethevalid
ityof
know
ledg
e,em
piricalinferences
sustainthedevelopm
entof
statistics,andlogicalrulesareadop
tedto
judg
ethevalid
ityof
mathematics(D
ong).
Mathematicsfollo
wsdedu
ctionto
prov
eprop
osition
s,bu
tstatistics
follo
wsindu
ctionto
valid
atehy
potheses
(T15
).Generality
oftruth
Con
text;forall;ow
n;specific
Self-containedor
context-
depend
enttruthof
results
The
truthandvalid
ityof
statisticsdepend
onwhether
itismeaning
ful
incontexts(T23
).The
samestandard
answ
er,even
usingmultip
lemetho
ds,canbe
derivedforon
emathematicalprob
lem,b
utitisdifferentto
solveon
estatistical
prob
lem
(T6)
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
school mathematics/statistics as “ways of thinking for proposing problemsand developing concepts” (see Table 5).
The sample quotes from the five interviewees are shown in Table 5 toillustrate some common properties of the three conceptions that emergedin our study. Accordingly, we interpret that Anna perceived schoolstatistics more as isolated knowledge and school mathematics more as astructural and reasonable system. Like Anna, Hue perceived schoolstatistics as instrumental application. But, he perceived school mathemat-ics as multiple conceptions: abstract structures, developed step by stepand systemized thinking. Unlike Hue and Anna, Yi and Lee perceivedschool statistics as meaningful interpretation in addition to instrumentalapplication. Dong inclined to view statistics as prerequisite tools and viewmathematics as meaningful and developmental.
When the five interviewed teachers were asked to think about thecorresponding statements, all of them disagreed that school mathemat-ics and school statistics are just an accumulated body of hierarchicaland objective knowledge. As for school mathematics, all of them re-emphasized the connection of concepts to problems or the integrationof problems and concepts. For example, Dong said that “to findproblems from observing patterns is part of developing mathematicaltheories” (not shown in Table 5). As for school statistics, all of themagreed that context is important, but considered that school statisticsdoes not present itself as investigative processes in the context ofsocietal activity. Yi’s and Hue’s quotations were presented asillustration.
Yi: I … sometimes teach students statistical knowledge by integration of their thinking. …I know context is important. But investigating contextual problems is not suitable for allstudents, maybe only for gifted students, to learn statistics.Hue: … Students who can propose reflective questions can be guided to think forward andsolve more open-ended and realistic problems. But, investigation in real context is notincluded.
When the researcher asked the five interviewees why they did notconsider to engage students in contextual investigation, they said that theywere commonly concerned that the type of investigation problems hadnever been tested in the entrance examinations in Taiwan. This concernmight explain why none of them reported the “contextual investigation”conception (see Table 4) of either school mathematics or school statisticsin their teaching. This concern also indicated that teachers’ conceptions ofschool mathematics and school statistics can vary with nationalassessment methods in addition to students’ backgrounds.
KAI-LIN YANG
Five Dimensions of Teachers’ Conceptions. From the perspective ofepistemological beliefs, teachers’ conceptions in this study werecharacterized with five new dimensions that emerged from our dataanalyses as shown in Table 6. All of the participating teachers did notagree with simplicity of school mathematics and statistics. Thus,simplicity, used in our theoretical framework (see Table 1), was excludedin our new dimensions. The source dimension was further distinguishedas sources of uncertainty and sources of coherence. Sources ofuncertainty were characterized as subjective or objective, and sources ofcoherence were characterized as definite or indefinite. The attainabilitydimension was replaced by generality of truth which was characterized asself-contained or context-dependent. For the degree of certainty, all thefive interview teachers’ conceptions were related to the evolving propertyof knowledge.
For example, in Table 6, Anna referred to the influence of changes inknowledge when the researcher asked her about the evolution ofknowledge. In the written responses, we interpreted that T13 viewedtruth of mathematical knowledge as invariant and then its certainty asunchanging, but statistical knowledge is just contrary to mathematicalknowledge. As for sources of uncertainty, Yi, Hue, and Dong all agreedwith both authority and active construction according to their regard ofauthority as identity, responsibility and sense, and of active constructionas discussion, active thinking, and sensibility.
Furthermore, they were asked to think about sources of developingmathematics and statistics. We distinguished sources of coherence fromsources of uncertainty based on their responses. All intervieweesperceived that uncertainty is the nature of statistics rather thanmathematics, and hence uncertainty is objective in statistics but subjectivein mathematics. As for sources of coherence, Yi and Anna perceived thatcoherence within school mathematics and school statistics was concernedby experts when developing each knowledge domain, but that coherencemight come from indefinite sense instead of definite sense (see Table 6).Lee also mentioned that “one single truth is sought to construct coherencewithin the mathematical world, but multiple truth is sought in statistics.”Dong and Hue did not actively express any ideas about the sources ofcoherence.
As for ways of justification, all five teachers emphasized that logicalrules instead of experts were critical in mathematical justification, andDong further expressed that the experts’ empirical inference was the basisof statistical justification (see Table 6). Dong also mentioned that“methods to find roots can be justified with deductive derivation,” and
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
“senior high statistics focuses on data analysis” which is part of empiricalinference. Besides expressing how mathematics and statistics are justified(see Table 6), T15’s written response mentioned what is justified:propositions in mathematics and hypothesis in statistics. Finally, forgenerality of truth, school mathematics was perceived by the teachers tobe true forever in its own system (self-contained), but the truth of schoolstatistics was perceived to partly rely on contexts according someteachers’ responses in this study. The features of generality related totruth can also be found in the written responses, for instance, those of T23and T6 (see Table 6).
Differences in Terms of Categories
Teachers’ conceptions of the two knowledge domains—school mathe-matics and school statistics—were observed through the lens of the abovethree categories of conceptions. Table 7 shows the number of teachersclassified into different conceptions of school mathematics and statistics.About 70 % (19 out of 27) of the teachers reported the instrumentalapplication conception of school statistics, and a lower percentage of theteachers, about 40 % (11 out of 27), reported such conception of schoolmathematics. On the other hand, their relational application and relationalmethodology conceptions were less for school statistics than for schoolmathematics. No teacher in this study showed the contextual investigationconception of school mathematics and school statistics we reviewed in theliterature (e.g., Ben-Zvi & Friedlander, 1997; Shaughnessy, 2007). Wefound that two thirds (18 out of 27) of the teachers’ conceptions of schoolmathematics were different from their conceptions of school statistics asto the philosophy of mathematics perspective.
We also found that six teachers reported dual-relation conceptions ofschool mathematics, which means a dual relationship between a staticview (instrumental application or relational application) and a dynamicview (relational methodology).
Differences in Terms of Two Dimensions
Tables 8 and 9 show respectively the differences in teachers’ (N = 27)conceptions of school mathematics and school statistics in terms of two ofthe five dimensions of conceptions—the degree of certainty and ways ofjustification—based on epistemological beliefs. About 56 % (15 out of27) of teachers perceived high certainty of school mathematics but lowcertainty of school statistics (see Table 8). Another 56 % (15 out of 27) ofteachers perceived logical rules to justify mathematics but empirical
KAI-LIN YANG
inference to justify statistics (see Table 9). No teacher showed that both ofschool mathematics and statistics have the property of low certainty andempirical justification.
Based on the teachers’ interview and open-ended question responses,we also found that these mathematics teachers’ conceptions of schoolmathematics were more likely to have properties of high certainty,subjective uncertainty, definite coherence, logical justification of
TABLE 7
Possible combinations of teachers’ (N = 27) conceptions of school mathematics andstatistics in terms of categories
Schoolmathematics School statistics IA RA RM IA+RA Total
Teachers in thesurvey (n = 22)
Instrumental application (IA) 8 3 0 0 11Relational application (RA) 3 1 0 0 4Relational methodology (RM) 1 1 0 0 2IA+RA 4 0 0 0 4IA+RM 0 0 0 1 1RA+RM 0 0 0 0 0IA+RA+RM 0 0 0 0 0Total 16 5 0 1 22
Teachers in theinterviews(n = 5)
IA 0 0 0 0 0RA 0 0 0 0 0RM 0 0 0 0 0IA+RA 0 0 0 0 0IA+RM 1 0 0 0 1RA+RM 1 0 0 2 3IA+RA+RM 1 0 0 0 1
Total 3 0 0 2 5
TABLE 8
Differences in teachers’ conceptions of two domains in terms of degree of certaintydimension (N = 27)
School statisticsSchool mathematics H L I Total
High certainty (H) 8 15 0 23Low certainty (L) 2 0 0 2Insufficient responses (I) 0 1 1 2Total 10 16 1 27
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
propositions, and self-contained truth; whereas, their conceptions ofschool statistics were more likely to have properties of low certainty,objective uncertainty, relative coherence, empirical justification ofhypotheses, and context-dependent truth. Furthermore, our resultsappeared to provide some empirical evidence for our claim that theseTaiwanese teachers’ conceptions were different between these twoclosely related knowledge domains, and that distributions of theproperties of teachers’ conceptions occurred—in terms of the degreeof certainty and ways of justification—in ways similar to Hofer’s(2000) finding that there was the same dimensional structure butdifferent epistemological beliefs between two subjects.
DISCUSSION AND IMPLICATIONS
Features of Taiwanese Mathematics Teachers’ Conceptions
The findings of the study provided some evidence that Taiwanesemathematics teachers’ conceptions of school mathematics and schoolstatistics can be more comprehensively characterized using the philo-sophical and epistemological perspectives. One may query why somefeatures of conceptions, for example, sources of coherence and ways ofjustification, could be related to school mathematics and school statistics.For this, we conjecture two possible reasons. One could be that themathematics curriculum of senior high school in Taiwan includes someadvanced ideas (e.g., calculus and confidence intervals). The other could bethat these teachers teach in the top three senior high schools where theyextended some advanced ideas in school mathematics for their students.
The features of the categories of conceptions we found in this studywere not all similar to those traditional categories of conceptions, nor to
TABLE 9
Differences in teachers’ conceptions of two domains in terms of ways of justificationdimension (N = 27)
School statisticsSchool mathematics L E I Total
Logical justification (L) 8 15 1 24Empirical justification (E) 0 0 1 1Insufficient responses (I) 0 1 1 2Total 8 16 3 27
KAI-LIN YANG
those dimensions of epistemological beliefs in the literature. Specifically,the instrumentalist view did not involve the idea that the facts, rules, andskills could be related to mathematical structures for solving manyproblems and learning in the future in the instrumental applicationconception. The problem-solving perspective in those of previous studies(e.g., Ernest, 1989; Lerman, 1983; Thompson, 1992) was different fromrelational application conceptions which focused on relational represen-tations instead of a subject of problem solving. We also identified the newrelational methodology conception to reveal teachers’ concern about theintegration of development of thinking into problem solving in addition tothe integration in a vice versa way.
Similarly, on the basis of our data analysis, we found that the featuresof the dimensions of conceptions related to the epistemological beliefswere distinct from the original dimensions in the literature (e.g., Hofer &Pintrich, 1997; Hofer, 2000). We further modified the sources dimensionto distinguish between sources of uncertainty and sources of coherence.We replaced the justification dimension with ways of justifications andthe attainability of truth dimension with generality of truth. Regardingthese participants’ conceptions in this study, school statistics wasperceived as more instrumental, but less certain, less definite, less self-contained than was school mathematics.
Implications
In this study, we uncovered some different conceptions of schoolmathematics and statistics reported by the participating teachers, whichmay be useful for developing a conception that emphasizes therelationship between the two knowledge domains. Our findings couldalso shed light on how teachers become aware of contradictions in theirown reasoning and more open to alternative conceptions as they reflect ontheir conceptions of school mathematics, school statistics, and theirdifferences.
In the past, researchers have discussed the differences betweenmathematics and statistics. Nevertheless, this study contributes atproviding a way to specifically distinguish their differences frommathematics teachers' conceptions of mathematics and statistics basedon the philosophical and epistemological perspectives. Despite a numberof large-scale studies (e.g., Felbrich et al., 2012) on typical teachers’mathematical beliefs or conceptions, our findings may serve as a base todevelop an instrument for measuring teachers’ conceptions of statisticsand its differences from mathematics, and to explore the relationships
AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS
between teachers’ conceptions of the differences in the two domains andtheir teaching.
On the other hand, researchers have acknowledged a complementaryrelationship between statistics and mathematics (delMas, 2004; Gattuso &Ottaviani, 2011). Moreover, Rossman, Chance & Medina (2006) notedthat teachers’ conceptions of the differences between mathematics andstatistics are also significant when they design lessons that accuratelyrepresent the two domains. However, little effort so far has been made todevelop an integrated program that allows students to compare andcontrast thinking mathematically and statistically. The statements weemployed in the interview of this study may provide mathematicseducators with a resource for designing tasks of reflecting on thinkingmathematically and statistically.
Although the participants in this study were categorized to be reportingcertain conceptions of school mathematics and school statistics, it isimportant for researchers to note that these teachers’ conceptions weredynamic and depended on particular contexts in which they were studied.Therefore, we suggest some follow-up studies to explore how teachers’conceptions of school mathematics and statistics differ under differentnational curricula and assessments.
ACKNOWLEDGMENT
The development of this paper was supported by a grant from the NationalScience Council (NSC 101-2511-S-003-010-MY3). We are grateful for theassistance as well as thank reviewers for their helpful suggestions towardimproving the paper.
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Department of MathematicsNational Taiwan Normal University11677, No.88 Sec 4, Ting-Chou Rd., Taipei, TaiwanE-mail: [email protected]
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