AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS OF SCHOOL MATHEMATICS, SCHOOL STATISTICS, AND THEIR DIFFERENCES

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)


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


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


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


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)


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


“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


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


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]

KAI-LIN YANG

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AN EXPLORATORY STUDY OF TAIWANESE MATHEMATICS TEACHERS' CONCEPTIONS OF SCHOOL MATHEMATICS, SCHOOL STATISTICS, AND THEIR DIFFERENCES