Mathematics-related teaching competence of Taiwanese primary future teachers: evidence from TEDS-M

ORIGINAL ARTICLE

Mathematics-related teaching competence of Taiwanese primaryfuture teachers: evidence from TEDS-M

Feng-Jui Hsieh • Pi-Jen Lin • Ting-Ying Wang

Accepted: 21 November 2011 / Published online: 8 December 2011

� FIZ Karlsruhe 2011

Abstract This paper draws on data from the international

TEDS-M study, organized by the IEA, and utilizes a con-

ceptual framework describing the Taiwanese perspective

of mathematics and mathematics teaching competences

(MTCs) with regard to investigating the uniqueness and

patterns of Taiwanese future primary teacher performance

in the international context. The framework includes content-

oriented and thought-oriented categories of mathematics

competence. The latter category contains subcategories adop-

ted and revised from (3rd Mediterranean conference on

mathematical education. Hellenic Mathematical Society,

Athens, 2003) the competence approach by Niss. Hsieh’s

(Research on the development of the professional ability for

teaching mathematics in the secondary school level (3/3).

Taiwan: National Science Council, 2009) model is also

adopted and revised to serve as an analytical framework,

including four categories relating to MTCs, representations,

language, and misconceptions or error procedures. This paper

shows that in thought-oriented mathematics competences

Taiwan and Singapore share a unique pattern of higher

percent correct in competences related to formalization,

abstraction, and operations in mathematics than in those

related to the way of thinking, modelling and reasoning in

and with mathematics. The paper also addresses weak

teaching competences claimed in domestic studies, which

conflict with the TEDS-M results. Namely, in contrary to the

international trend, Taiwanese future primary teachers are

weak at judging mathematics competences required by stu-

dents to learn mathematical concepts or solve problems, and

superior at diagnosing and dealing with student misconcep-

tions and error procedures.

Keywords TEDS-M � MCK � MPCK �Mathematics teaching competence � Teacher education �International comparison

1 Introduction

Previous research has shown that a teacher’s quality and

knowledge are significant school-related factors influenc-

ing students’ performance and learning in the classroom

(Cobb et al., 1991; Rice, 2003), but identifying and mea-

suring the characteristics that constitute a qualified teacher

remains a significant problem (Baumert et al., 2010; Hill

et al., 2007). Many attempts have drawn from theoretical

views, for instance the construction of conceptual frame-

works of qualities in teacher knowledge and skills, and

theory-based frameworks for evaluation (e.g., Ball, Thames

& Phelps, 2008; Baumert et al., 2010; Hill, Schilling &

Ball, 2004; Schmidt et al., 2011). Different domains of

teacher knowledge, such as pedagogical knowledge and

content knowledge have been pointed out as instructional

determinants of student learning achievement and applied

to studies in many fields including mathematics (Ball &

Bass, 2003; Krauss, Baumert, & Blum, 2008; Grossman &

The analysis prepared for this report and the views expressed therein

are those of the authors and do not necessarily reflect the views of the

funding agencies or the IEA.

This article is based on the Taiwan TEDS-M 2008 study conducted by

the National Research Center of Taiwan located at National Taiwan

Normal University.

F.-J. Hsieh (&) � T.-Y. Wang

National Taiwan Normal University, Taipei, Taiwan

e-mail: [email protected]; [email protected]

P.-J. Lin

National Hsinchu University of Education, Hsinchu, Taiwan

123

ZDM Mathematics Education (2012) 44:277–292

DOI 10.1007/s11858-011-0377-7

McDonald, 2008; Hill, Ball, & Schilling, 2008; Shulman,

1986, 1987).

During the past two decades scholarly interest in inter-

national comparison studies about mathematics teachers

has increased (e.g., An, Kulm & Wu, 2004; Ma, 1999).

Studies such as the Mathematics Teaching in the twenty-

first century (MT21) project have shown that different

countries’ future teachers achieved differently in their

teaching knowledge and also had different opportunities to

learn (Blomeke et al., 2008; Schmidt et al., 2011). The

Teacher Education and Development Study in Mathemat-

ics (TEDS-M) was the first data-based international study

about mathematics teacher education with national repre-

sentative samples. It provided participating nations the

opportunity to acquire international perspectives on their

teacher education systems in areas such as future teachers’

knowledge levels (Blomeke, Suhl & Kaiser, 2011; Konig,

Blomeke, Paine, Schmidt & Hsieh, 2011), their opportu-

nities to learn (Schmidt, Cogan & Houang, 2011), and the

quality of mathematics teacher education (Hsieh et al.,

2011).

The TEDS-M study showed that Taiwanese future

teachers’ achievements in mathematics content knowl-

edge (MCK) and mathematics pedagogical content

knowledge (MPCK) ranked at either the first or second

among participating countries at both primary- and

secondary-school levels (Hsieh et al., 2010). However,

conflicting results obtained from domestic studies sug-

gested that Taiwanese pre- or in-service teachers are

limited by a weak understanding of both mathematics

knowledge and students’ mathematical thinking and

learning processes (Hung, 2009; Leu, 1996; Liu, 2002;

Yang, Reys, & Reys, 2009). Other studies focused on the

improvement of the education of pre- or in-service

teachers in mathematics teaching knowledge (Hsieh,

2000; Lin, 2001).

The seemingly conflicting results between Taiwan’s

domestic studies and the international TEDS-M study ini-

tiated an investigation of the essence of Taiwanese future

teacher competences from a domestic perspective, but in an

international context. In light of the TEDS-M data, this

study focuses on the following two research questions:

1. How do Taiwanese future primary teachers perform in

MCK and MPCK in comparison with other countries

and what uniqueness or patterns of performance do

they possess?

2. Under a conceptual framework that expresses Tai-

wan’s views on mathematics and mathematics teach-

ing competences (MTCs), how do Taiwanese future

teachers perform in comparison with other countries’

future teachers and what uniqueness or patterns of

performance do they possess?

2 Conceptual framework

In order to answer the first research question, the authors

adopted the TEDS-M framework. Any further descriptions

of MCK and MPCK may be obtained in Sect. 3.2. This

section illustrates the authors’ framework for addressing

the second research question.

2.1 Framework of mathematical competence in this

study

Many studies have emphasized the importance of mathe-

matical abilities that are not directly related to any specific

mathematical content (Krutetskii, 1976; Niss, 2003;

National Research Council, 2001). Two types of mathe-

matical competence (MC) are discussed in this paper. The

first is content-oriented mathematical competence (CMC),

which is related to specific mathematics topics as the pre-

requisites of advanced-level competence, namely the fac-

tual knowledge and technical skills required to complete a

mathematics teaching task (Niss, 2003). The other type of

mathematical competence, thought-oriented mathematical

competence (TMC), is not bound to any one mathematical

topic, but as Krutetskii (1976) noted, arises from the basic

characteristics of mathematical thought. Niss’s (2003) list

of mathematical competencies is suitable to identify TMC

from the primary school to university levels, and is thus

adopted in this paper.

The framework for TMC includes Niss’s two categories:

‘‘the ability to ask and answer questions in and with

mathematics’’—thought in questions (TMC-TQ) and ‘‘the

ability to deal with and manage mathematical language’’—

mathematical language (TMC-ML).1 Though Niss’s list

includes two categories of competencies, each containing

four subcategories, our framework does not cover all of his

subcategories. In TMC-TQ, three subcategories are

emphasized: thinking mathematically, modeling mathe-

matically, and reasoning mathematically. In TMC-ML,

two subcategories are highlighted: representing mathe-

matical entities, and handling mathematical symbols and

formalisms. The framework of MC is shown in Fig. 1.

2.2 Framework of MTC in this study

Hsieh (2009) develops indicators in MTCs, using nation-

wide representative samples of school students and a

variety of samples of elite mathematics teachers in Taiwan.

The term ‘‘competence’’ rather than ‘‘knowledge’’ is used

by Hsieh to properly capture the types of abilities relating

1 Niss’s original second category includes the ability to deal with and

manage tools.

278 F.-J. Hsieh et al.

123

to the operations of thinking, reasoning, judging, or even

executing mathematical tasks.

In this paper, we adopt Hsieh’s (2009) analysis of MTC

which is structured around three objects: element, opera-

tion, and kernel.2 With these, she singles out 20 elements of

mathematics teaching and three operations that engage

those elements: recognizing and understanding (RU),

thinking and reasoning (TR), and conceptual executing

(CE). Additionally, the focus of the competences can be

directed through three kernels of perspective, learning,

teaching and entity. For example, one element is ‘‘mathe-

matics thinking’’. With Hsieh’s framework, this element

may generate many MTC under just the RU operation: with

the kernel ‘‘learning’’ one recovers ‘‘recognizing students’

mathematics thinking’’; with ‘‘entity’’, ‘‘recognize the dif-

ference between the mathematics thinking and other sci-

entific field thinking’’; and with ‘‘teaching’’, ‘‘understand

how to cultivate active mathematics thinking during

classroom teaching’’. Hsieh’s MTC may include the most

oft-mentioned types of MTC in prior studies (Delaney,

Ball, Hill, Schilling, & Zopf, 2008; National Council of

Teachers of Mathematics, 2000).

Due to the limitation of categories available in the

TEDS-M questionnaires, a partial set of four indicator

categories are adopted, closely coinciding with categories

commonly mentioned in MPCK literature. These four

major categories relate to four elements in Hsieh’s frame-

work;3 they are:

1. school students’ mathematical competences pertaining

to concepts, skills, or abilities (MTC-C),4 for instance,

being able to judge what pre-concepts are required

and what mathematical competences to develop in

teaching a concept; the element of the competences

in this category is ‘‘mathematical competence of

students’’,

2. school students’ misconceptions or error procedures

(MTC-M), for instance, being able to diagnose typical

students’ misconceptions, or error procedures and

come up with a way to reduce them; the element is

‘‘misconception of students’’,

3. mathematical representations (MTC-R), for instance,

being able to know the attributes, strengths and

limitations of different mathematical representations

and switch between mathematical representations

adapting to teaching tasks; the element is ‘‘mathemat-

ical representation’’, and

4. mathematical language (MTC-L), for instance, being

able to evaluate the difficulty levels of mathematical

language and properly use mathematics language that

can be understood by students; the element is ‘‘math-

ematical language’’.

3 Research method

3.1 Participants

This paper focuses on future primary teachers in their last

year of training from 15 countries, drawn from the TEDS-M

study. The TEDS-M sampling plan followed a stratified

multistage probability sampling design (Tatto et al., 2009).

A minimum requirement of 75% combined participation

rate was set by IEA as meeting its threshold.5 According to

the IEA’s criterion, samples having a participation rate of

60–75% were also suitable for use, with the IEA advising

an annotation of low participation rates. Therefore, to

ensure additional inclusion of information, we used a

threshold of 60% for this study. Based on this criterion, our

analyses included data from the following countries:

Botswana, Chili, Germany, Georgia, Malaysia, Norway,

Philippines, Poland, Russia, Spain, Switzerland, Singapore,

Taiwan, Thailand, and the United States.6

In Taiwan, there were 30 teacher preparation institutions

at the time of sampling and 11 of them were sampled. A

total of 1,023 future primary teachers were sampled in the

TEDS-M study, with 90.22% of them participating,

Mathematical Competence (MC)

Content-Oriented MC (CMC) Thought-Oriented MC (TMC)

TMC-TQ TMC-ML

Thinking mathematicallyModeling mathematicallyReasoning mathematically

Representing mathematical entitiesHandling mathematical symbols and formalisms

Fig. 1 Conceptual framework of MC

2 This model uses the idea of unary operation in mathematics. An

operator acts on an element in the domain to produce a new element

in the range.3 The classification of items into different operations and kernels in

Hsieh’s framework can be found in Table 2.4 As described in Sect. 2.1, competence includes concept, skills, and

ability.

5 There is another way to meet the IEA’s threshold for participation

rate, namely, when both the institutional and the future teachers’

participation rates are greater than or equal to 85%.6 The combined participation rates of Chile and Poland were between

60 and 75%. Poland limited its participation to institutions with

concurrent programs. Switzerland limited its participation to German-

speaking regions. The United States limited its participation to public

universities. Analyses for Norway were conducted by combining the

two data sets available. The range of the participation rate for Norway

cannot be confirmed yet.

Mathematics-related teaching competence of Taiwanese primary future teachers 279

123

resulting in a total of 923 (un-weighted) participants with a

female to male ratio of 7:3.7 All the future teachers in

Taiwan were prepared to be generalists teaching grades

1–6 in a range of subjects. Across various countries or even

within one country, separate programs existed, resulting in

various definitions of ‘‘teaching grades for the primary

level’’. For example, in Thailand, there were two programs

for teaching grades 1–12, while in Switzerland there was a

program for teaching grades 1–2 exclusively. A thorough

description regarding these grade spans can be found in the

TEDS-M technical report, which will be published soon.

3.2 Measures

3.2.1 MCK and MPCK

The TEDS-M study generated a future primary teacher

questionnaire that included MCK and MPCK tests with a

60-min completion time. According to the TEDS-M

frameworks, there are three cognitive sub-domains of

MCK, knowing, applying and reasoning, and two cognitive

sub-domains of MPCK, curricular knowledge and planning

for teaching (CP), and enacting teaching (ET). A total of

111 knowledge items were included for final analyses and,

after accounting for combinations of items, 105 scores

were utilized. Of the 105 scores, 73 were in MCK and 32 in

MPCK.8 The number of scores in the cognitive sub-

domains of MCK and MPCK, knowing, applying, reason-

ing, CP, and ET was 32, 29, 12, 16, and 16, respectively

(Tatto et al., 2008).

3.2.2 MC and MTC

All knowledge items from the TEDS-M questionnaire were

re-categorized according to our frameworks of MC and

MTC by five experienced teacher education professors in

Taiwan with an average career length of greater than

20 years. They are either mathematicians or mathematics

educators, and a few also serve as members of working

committees or review committees of the national high

school entrance examinations. We are aware that the sub-

categories of MC and MTC, though distinct, are interwo-

ven and an item may test competences in more than one

subcategory. To lessen the subjectivity of the categoriza-

tion, an operational procedure was developed and admin-

istered. The five professors first worked individually to

classify each item into only one subcategory of MC or

MTC. The classifications were then circulated among this

group of professors and when categorizations mismatched,

several members of the group would negotiate the item into

an appropriate category. If the professors could not reach

an agreement for the categorization of an item, the item

was left out from analyses.

In some cases, if a test item contained many sub-items

(i.e., an item involving four sub-items that required a cer-

tain mathematical concept), we eliminated some of the sub-

items to reduce the weight of the required concept. There

were a total of 14 eliminated items (or sub-items). Among

them, one item did not gain consistent categorization from

professors, eleven required repeated mathematical con-

cepts, and the final two neither gained consistent catego-

rization nor unrepeated. Take one of the items as an

example: the item provided four sub-items; each of them

had a statement regarding the set of non-negative whole

numbers. It asked the future teachers to indicate each was

true or not. Among the statements, two involved the con-

cept of the commutative law and the other two related to

the concept of the associative law. This study eliminated

one item for each concept. Note that this categorization

(and the elimination of some items) is not meant to

establish an extensive consensus across countries; rather, it

is being used to represent a Taiwanese perspective.

Finally, even though all MCK items are classified as

MC, not all MPCK items are classified as MTC—seven

items in MPCK are categorized as MC. In each case a

consensus of the classification among the five professors

was reached, as each of the items possessed the charac-

teristics that the keys to get correct answers required only

mathematical competence, though these items usually

provided a statement including the words: teachers or

students.9 All seven of these items are in the lower sec-

ondary level in Taiwan and they resemble mathematical

problems in the Taiwan senior high school entrance

examination. The complete classification totaled 69 MC

items, with 32 in CMC, and 37 in TMC. The TMC category

was composed of 30 items in TMC-TQ and 7 in

TMC-ML.10 We believe the uneven distribution reflects the

international perspective on the focus of mathematical

competence rather than MTC in the TEDS-M study. In

TMC-TQ, 2 items are characterized as thinking mathe-

matically, 11 as modeling mathematically and 17 as rea-

soning mathematically. In TMC-ML, 2 items may be

classified as representing mathematical entities and 5 as

handling mathematical symbols and formalisms. Note that

the classification to these final subcategories was not7 This ratio also corresponds roughly to the ratio of females to males

of in-service primary teachers in Taiwan in the year of the survey.8 The data sets used in this paper are the TEDS-M released data sets

for national research coordinators: TEDS_MS_NRC-USE_IDB_

20091209_v30. The final TEDS-M data sets include one more score

than used in this paper.

9 The statement might look like ‘‘Indicate whether each of the

following students’ responses is correct or not’’.10 Item examples of TMC-TQ and TMC-ML can be found in Sects.

4.4.1 and 4.4.2 in this paper.


123

intended to contribute substantially to this report, but was

rather used to ensure the items fit into either TMC-TQ or

TMC-ML. The corresponding numbers of TEDS-M

knowledge items in different categories versus the Tai-

wanese classification of MC items is shown in Table 1.

A total of 22 MPCK items were classified into the field

of MTC. Of these 22 items, 5 are in MTC-C, 6 in MTC-M,

8 in MTC-R and 3 in MTC-L.11 As some subcategories

have only a few items, the analyses done for them

may only be exploratory. The corresponding numbers of

TED-M MPCK items in CP and ET versus the Taiwanese

categorization of MTC items is found in Table 2.

3.3 Data processing and analysis

Participants’ responses to the items of MCK and MPCK

were coded and scored according to the Item Scoring

Guide developed by the TEDS-M consortium (Tatto et al.,

2008). The scoring system for each constructed response

item is a two-digit code. The first digit, either a 1 or a 2,

indicates a correct, or partially correct, response and also

signifies the number of score points given to that response.

The second digit captures different approaches used by the

future teachers.12

For the analyses of the measure of a partial or entire MC

or MTC, several variables were either adopted directly, or

derived, from the questionnaire items according to our

conceptual framework and research questions. For each test

item, the percentage of correct answers from each country

was computed (along with its standard error) and this sta-

tistic was called item percent correct for that country. For

any constructed response item scored two points, the item

percent correct is the sum of the percentage of answers

receiving the two points plus half of the percentage of

answers scored one point. For a set of items, for example,

items in a subcategory of MC or MTC, the item percent

corrects were averaged over the set of items to obtain an

average percent correct and this statistic is called percent

correct for that set of items. The international average

percent correct for an item or a category was obtained by

averaging over the percent corrects of all participating

countries. The same process was used to calculate a

country’s percent correct for any item or sub-domain of

MCK and MPCK. The statistics for sub-domains of MCK

and MPCK were not provided by TEDS-M. When there is

a need to express relative strengths and weaknesses rather

than absolute differences of countries, median polish

analyses (Mosteller & Tukey, 1977) were applied. When

comparing two measures of a country or a measure of two

countries, dependent or independent t tests were applied

accordingly.

4 Results and discussions

Throughout the paper, we adopt two approaches to present

or interpret our data, one including the results of all par-

ticipating countries and one including only the ‘‘higher

achieving countries’’—the eight countries that achieved

MCK and MPCK means beyond the international mean of

500.13 The first approach is used when there is a need for

providing a global view and the second is used to make a

more focused interpretation by analyzing countries per-

forming closely with Taiwan.

4.1 Taiwanese future teachers’ achievement

across cognitive domains

With regard to MCK and MPCK, Taiwan’s future primary

teachers achieved the highest score of all TEDS-M coun-

tries (for more information see Blomeke et al., 2011; Hsieh

et al., 2010).14 The following results have not been found

prior to this paper. Taiwan’s future teacher percent correct

in each cognitive sub-domain of MCK was significantly

higher than those in other participating countries. Though

across all countries there was a lower percent correct in

reasoning, there were two patterns that emerged among the

higher achieving countries (see Fig. 2). The first, shared by

Table 1 The TEDS-M knowledge items versus the corresponding

number of MC items under the Taiwanese approach

MC Subtotal Total

CMC TMC

TMC-TQ TMC-ML

MCK

Knowing 17 7 5 29 62

Applying 7 13 2 22

Reasoning 8 3 0 11

MPCK

CP 0 2 0 2 7

ET 0 5 0 5

Subtotal 32 30 7 69 69

Total 32 37 69

11 Item examples of MTC-C and MTC-M can be found in Sects. 4.4.3

and 4.4.4 in this paper.12 For example, a response with a code 20 or 21 was scored 2 points,

whereas a code 10 or 11 was scored 1 point.

13 These means were computed by TEDS-M. German and Russian

means in MPCK were higher than the international mean, though not

significantly. They are regarded as higher achieving countries in this

paper.14 Singapore achieved the same as Taiwan in MPCK.


123

Taiwan, Germany, Singapore, Switzerland and Thailand,

is denoted by three statistically significant, gradually

decreasing percent corrects from knowing, to applying, to

reasoning. Norway did not strictly adhere to this pattern,

but it was close to it by a non-significant deviation between

the percent corrects of knowing and applying. The second

pattern, shared by Russia and the United States, exhibits a

significant drop from the percent correct of knowing to

applying but with similar difference in percent correct from

applying to reasoning. The drops here could come from the

different difficulty levels for items in different sub-domains

rather than representing worse performance. To examine

this problem, all the items were re-classified by school

level (by Taiwan’s definition of ‘school level’) by three

experts in the Taiwanese mathematics curriculum. Chi-

square tests showed no significant differences between the

distributions of items in any two sub-domains. As a result,

it is probable that the items are at the same difficulty levels

for knowing, applying, and reasoning, at least in terms of

school curriculum. Therefore, the noticeable drop of per-

cent correct from applying to reasoning in the Taiwanese

data could serve as a warning to its mathematics teacher

education system that there may be a lack of emphasis on

reasoning, a vital element utilized frequently in the class-

room by teachers to diagnose problems and respond to

students.

With regard to the cognitive domains of MPCK, Taiwan

ranked first in ET (enacting teaching) and second in CP

(curricular knowledge and planning for teaching) among

all participating countries. Figure 3 shows that Taiwanese

pattern of difference between the percent corrects of ET

and CP is different from those of all other higher achieving

countries. To test if Taiwanese pattern is significantly

different from all other higher achieving countries, the

repeated measures ANOVA, with country as a between-

subjects factor, was used. The procedure was performed

repeatedly and each time Taiwan and another country were

compared. The results showed that when comparing with

Taiwan, every higher achieving country had relatively

higher percent correct of CP than ET.

Since the mathematical concepts in all the MPCK items

were not beyond Taiwanese junior high school level

(considered easy items) and some countries achieved better

in junior high school level MPCK items, it is possible that

the differences of percent corrects in CP and ET do not

come from the item difficulties. Therefore, in contrast to

the other countries in the study, Taiwanese future teachers

may perform better in real-time interaction with students

(ET) than in the mathematical curriculum or plans for

teaching and learning (CP). These real-time interactions

involved analyzing student mathematical responses, diag-

nosing student misconceptions and providing feedback

(Tatto et al., 2008).

4.2 Future teachers’ mathematical competences (MC)

This study found that each participating country’s percent

correct of CMC was significantly higher than that of TMC

(see Table 3). The international average percent correct of

TMC may be interpreted as: on average, a future primary

teacher from the participating countries was able to

correctly answer only less than half items in TMC. A

Chi-square test on the distribution of items classified by

Taiwanese school level for CMC and TMC showed no

significant differences between the distributions. As a

result, it is probable that the items are at the same difficulty

Fig. 2 The percent corrects of cognitive sub-domains of MCK

Table 2 The TEDS-M MPCK items versus the corresponding number of MTC items classified under the Taiwanese approach

HF MTC

MTC-C MTC-M MTC-R MTC-L Total

Operation: TR TR TR CE TR TR TR TR

Kernel: L T L T E L L T

MPCK

CP 3 2 0 1 5 0 1 2 14

ET 0 0 4 1 2 1 0 0 8

Total 5 6 8 3 22

HF Hsieh’s framework of MTC


123

levels for CMC and TMC. Thus, the greater percent correct

of CMC than TMC may reflect the event that, interna-

tionally, future teachers performed better in CMC than

TMC. This result matches the assumption that content

knowledge was a prerequisite for an individual to work on

mathematical thought related activities such as thinking,

reasoning, or representing in and with mathematics.

The relationship among the percent corrects of the

higher achieving countries in CMC, TMC-TQ (thought in

questions), and TMC-ML (mathematical language) was

examined through median polish analyses. The country

effect values showed that higher achieving countries dif-

fered in the performance of MC (see Table 4); Taiwan and

Singapore performed best, while the United States and

Germany performed less well.

Chi-square tests showed no significant differences

between the distributions of items classified by school level

for any two of CMC, TMC-TQ, and TMC-ML. Therefore,

the values in MC effect (see Table 4) may represent a

genuine divergence of competences in various subcatego-

ries of MC for these higher achieving countries. These

countries performed less well in TMC-ML, better in TMC-

TQ and best in CMC. The magnitudes of residuals (see

Table 4), which can be considered as the interaction

between the performance of countries on the subcategories

of MC, project that Taiwan and Singapore share the same

pattern with their best performance coming from

TMC-ML, with CMC as the median and the worst per-

formance in TMC-TQ relatively. These findings demon-

strate that Taiwan and Singapore performed relatively

better in competences related to formalization, abstraction,

and operations in mathematics (TMC-ML) than in com-

petences related to the way of thinking, modelling and

reasoning in and with mathematics (TMC-TQ), in contrary

to most other countries. One can note that Taiwan’s con-

dition does not diverge from the common impressions of its

mathematics education at the school and university levels,

namely Taiwan’s strong emphasis on formalization.

4.3 Future teachers’ MTCs

The MTC percent corrects of Taiwan and Singapore were

the highest among all countries (see Table 5). The percent

corrects of Germany, Russia and Thailand may be inter-

preted as: a future teacher from these countries could

Fig. 3 The percent correct of

cognitive sub-domains of

MPCK. ET enacting teaching,

CP curricular knowledge and

planning for teaching. Asteriskdenotes countries with

significantly different CP and

ET using dependent t test

Table 3 Percent corrects of MC, CMC, and TMC of the higher

achieving countries

Country MC CMC TMC Diff CT

Taiwan 78 (0.6) 85 (0.6) 72 (0.6) 13**

Singapore 73 (0.7) 81 (0.6) 66 (0.7) 15**

Switzerland 65 (0.4) 71 (0.5) 60 (0.5) 11**

Russia 63 (1.9) 72 (1.8) 57 (2.0) 15**

Thailand 63 (0.5) 74 (0.7) 54 (0.6) 19**

Norway 61 (0.6) 67 (0.7) 56 (0.7) 11**

US-Public 60 (0.8) 69 (0.6) 52 (0.9) 18**

Germany 58 (0.6) 66 (0.3) 53 (0.7) 13**

IA 56 64 49

The numbers in the parentheses indicate SE

IA international average of all participating countries, Diff CT CMC–

TMC

** p \ .01

Table 4 Results of median polish for the percent correct of MC

subcategories of the higher achieving countries

Country CMC TMC-TQ TMC-ML Country effect

Taiwan 0 -2 6 15

Singapore 0 -3 4 11

Switzerland 0 2 -1 1

Russia 0 -4 2 2

Thailand 6 0 0 -2

Norway -1 0 0 -1

US-Public 4 0 -1 -5

Germany 0 0 -3 -4

MC effect 12 0 -4 59


123

correctly answer only less than half items relating to MTC.

Median polish analysis was again performed to the percent

corrects in the four subcategories of MTC for the higher

achieving countries.

Based on the magnitudes of country effect values (see

Table 6), Taiwan and Singapore were in the best MTC

performance group, followed by the median group,

including Norway, US-Public and Switzerland. One can

also note that across all higher achieving countries the

MTC effect values decreased from MTC-C (mathematical

competences of students), MTC-L (misconceptions of

students), MTC-R (mathematical representations), to

MTC-M (mathematical language) (see Table 6). If one

compares the two countries in the best MTC performance

group, their patterns are inconsistent. Taiwan performed

better in MTC-R compared to Singapore, but worse in

MTC-C and MTC-L. Among the four subcategories of

MTC, compared to future teachers in most other countries,

Taiwanese future teachers had relatively better compe-

tences on diagnosing and dealing with student miscon-

ceptions and error procedures (MTC-M) than competences

such as identifying prerequisites for learning new concepts

or solving problems (MTC-C).

4.4 In-depth analysis of Taiwanese future teachers’

MC and MTC

Items for in-depth analyses were chosen to exemplify a

category if Taiwan’s performance in that item followed one

or more of the following criteria: typical levels or patterns

were revealed, the performances of Taiwanese future

teachers were substantially better or worse than other

countries, and unique patterns deviated from international

norms. This organization creates a set of six items, two

categories in MC and four categories in MTC. The com-

petences of these tested items will be described when

applicable.

4.4.1 MC: TMC-TQ

Item #509 was chosen to exhibit Taiwanese future teach-

ers’ competence in reasoning mathematically, including

devising formal and informal mathematical arguments, and

transforming heuristic arguments to valid proofs (see

Fig. 4, also for a partial rubric). Item #509 displays three

types of correct answers, but two of them are particularly

worthy of note, namely, Type A (code 20 and 10) and Type

B (code 22 and 12). Taiwanese future teachers provided a

greater number of correct or partially correct Type A

solutions than Type B ones. In contrast, those countries

whose percent corrects differed from Taiwan’s by 5%, such

as Singapore and Norway, had more Type B responses than

Type A.

One could argue that no Type B response should be

awarded full credit because it lacks generalization to rig-

orously validate the reasoning; however, Type B responses

do successfully show the responders’ chain of reasoning

and thus, if reasoning is valued over rigor proof, the value

of Type B responses can be seen. The Taiwanese lower

percentage of Type B responses may indicate that the

Taiwanese system is one that values formalism and closely

associates it with the explicit expression of one’s reasoning

processes. This conclusion gains support when one exam-

ines the percentage of attempts to, with at least partial

success, answer this item. While Taiwan ranked first in

overall percent corrects, Taiwanese future teachers had the

fewest attempts (53.7%), where other countries such as

Norway and Singapore had more (57.8 and 66.5%,

respectively). In other words, when incapable of providing

formal proofs, Taiwanese future teachers tended not to try

a more natural heuristic approach to show their reasoning.

Figure 5 provides four examples of Taiwanese future

teachers’ answers to show their Type A (Example 1: code

Table 5 Percent correct of MTC of the higher achieving countries

Country MTC

Taiwan 65 (0.7)

Singapore 65 (0.8)

Norway 54 (0.7)

US-Public 54 (0.6)

Switzerland 52 (0.5)

Germany 43 (0.8)

Russia 41 (1.8)

Thailand 40 (0.5)

IA 42

The numbers in the parentheses indicate SE

IA international average of all participating countries

Table 6 Results of median polish for the percent correct of MTC

subcategories of the higher achieving countries

Country MTC-C MTC-M MTC-R MTC-L Country

effect

Taiwan -17 8 0 0 14

Singapore -6 7 -7 5 14

Norway -1 13 0 -1 0

US-Public 5 0 -1 0 2

Switzerland -2 0 5 0 0

Germany 1 0 2 -3 -9

Russia 1 0 2 -10 -11

Thailand 7 -1 -1 1 -14

MTC effect 2 -12 -1 1 56


123

20; Example 4: code 10) and Type B answers (Example 2:

code 22; Example 3: code 12).

4.4.2 MC: TMC-ML

In order to exemplify Taiwanese future teacher handling

and manipulation of statements or expressions containing

symbols and formulae, the authors chose item #207 (see

Fig. 6).

This item also required translation from natural lan-

guage to symbols; however, this knowledge is not required

to successfully solve the problem. There were two keys

needed to successfully solve the problem. First, problem

solvers have to transform between quantities to correctly

express the quantitative relationship of objects. Second,

x and y should be viewed as variables representing num-

bers rather than the labels of objects A and B. If a future

teacher fails to do this, she might make a ‘‘reversal error’’

(Clement, 1982), which entails seeing x and y as labels and

the quantities as adjectives to describe the unknowns.

Taiwanese future teachers performed significantly better

than all participating countries (see Fig. 6). However,

there were still 29.7% of future teachers that made a

‘‘reversal error’’ (A2 and A3). Though this percentage was

high, it was still the lowest among all the participating

countries—all other countries ranged between 44.9 and

74.0% and the international average was large at 53.2%.

An examination of the wordings of this item in English and

Chinese versions revealed syntax structure dissimilarities

which changed the relative orders of the quantities and

variables. Whether this kind of variances affects the

solutions of this type of problem may require further

investigation.

4.4.3 MTC: students’ mathematical competence

This study identified item #206B as a measure of future

teachers’ knowledge about primary students’ difficulty in

dealing with uneven ratios and multiples to solve problems

(see Fig. 7).This item regards the uneven ratio of 2.4 (l) to

30 (h) and multiple of 100 (h) to 30 (h) as the elements to

be altered when creating an easier problem for primary

students to solve. In contrary to its high standing in the rest

of the scoring, Taiwan ranked 12th on this item, with a

percent correct (44.3%) far below the international average

(55.1%). Through the responses to this item, we once again

see that high mathematical competence (96.5% correct in

Taiwan in #206A) alone is not sufficient for high caliber

teaching of mathematics.

Further analysis revealed a particular pattern in Taiwan,

which may demonstrate a unique focus of Taiwanese

mathematics teacher education. When creating a simpler

version of the original problem, 29.8% of Taiwanese future

teachers concentrated on diverse problem situations, which

are not regarded as correct by the TEDS-M study, rather

than the relationships of the numerals in the problem. The

Taiwanese future teachers tended to provide problems with

situations closer to the students’ daily life experiences or

with fewer scientific concepts,15 as they believed such

Item Content Domain CompetenceSubcategory Competence Country Percentage

#509 Algebra and Function MC:TMC-TQ Reason mathematically PC 20 22

Students who had been studying algebra were asked the following question:

Give the answer and show your reasoning or working.

Taiwan 42.3 29.3 0.1Singapore 42.0 15.1 2.5Norway 37.3 10.9 5.4Switzerland 35.1 20.4 1.3Germany 30.3 16.9 0.3Thailand 27.6 13.3 0.7Russia 26.8 15.4 0.4US-Public 24.4 4.8 0.8IA 23.4 10.6 1.2

Rough rubric (partial) 10 12

Type ACode 20: General arguments either with words or inequalities.Code 10: On the right track of 20, but incomplete or limited.

Type BCode 22: Correct, ordered, specific-value checking and making general conclusions.Code 12: On the right track of 22, but incomplete or limited.

Taiwan 7.6 15.2Singapore 9.8 39.0Norway 13.0 27.8Switzerland 16.6 10.1Germany 20.3 5.9Thailand 14.0 13.1Russia 10.5 11.3US-Public 8.8 28.6IA 8.2 14.3

Fig. 4 Future teachers’

performance in #509. PCpercent correct, IA international

average of all participating

countries

15 The situation and units of the test problem #206(a) employed a

sense of a ‘‘speed’’ concept.


123

problems would be easier for students (see Fig. 8 for two

exemplary answers). These future teachers felt that the famil-

iarity of situations in a problem is a key to success for students.

In order to evaluate future teachers on their competence

in judging what students’ competences will be developed

in a teaching activity, item #513 (see Fig. 7) required

future teachers to give at least two reasons why a teacher

would begin an exercise in a particular way. Three

accepted reasons given in the TEDS-M coding rubrics

were: (a) enabling student understanding of the meaning of

measurement as comparing unknown to known entities,

(b) showing the need for standard units, and (c) helping

students learn to choose appropriate units. Only 16.2% of

the Taiwanese future teachers provided two appropriate

reasons and 36.2% of the future teachers could only come

up with one accepted reason among the three in an almost

balanced distribution: reason (a) 14.3%, reason (b) 10.5%,

and reason (c) 11.4%.

A total of 34.2% of Taiwanese future teachers provided

responses that were either too vague or improper to explain

why the teacher in the problem used paper clips and pencils

instead of rulers. Their responses could be divided into

three basic categories. The first pertained to the intentions

of enhancing students’ concrete sense of length (too vague,

see Example 1 in Fig. 9); the second related to the pres-

ervation of the notion of length under different measuring

units (see Example 2 in Fig. 9);16 the third involved a culti-

vation of other mathematical abilities or concepts (see

Examples 3 and 4 in Fig. 9, respectively).17 These responses

showed a weakness in the Taiwanese future teachers’ ability to

Example 1 Response:

Translation:

When n is less than 2, n+2 > 2n

n+2-2n=2-n n has to be less than 2 Positive number

When n is equal to 2, 2n = n+2

2 2=2+2=4

When n is greater than 2, 2n > n+2

2n-(n+2)=n-2 n has to be more than 2 Positive number

Example 2 Response:

Translation: n = -1, 2n=-2, n+2=1, then n+2 >2nWhen n = 1, 2n=2, n+2=3, then n+2 >2nWhen n = 2, 2n=4, n+2=4, then 2n = n+2 n = 3, 2n=6, n+2=5, then 2n > n+2 n = -4, 2n=8, n+2=6, then 2n > n+2

If n < 2, then n+2 > 2n n = 2, then n+2 = 2n n > 2, then n+2 < 2n

Example 3 Response:

Translation: 2n n+2 When n=0, 2n=0 < 2 n=1, 2n=2 < 3 n=2, 2n=4 = 4 n=3, 2n=6 > 5 n=4, 2n=8 > 6

2n n+2 If when n=-1, 2(-1)=-2 < 1 n=-2, 2(-2)=-4 < 0 n=-3, 2(-3)=-6 < -1

When n is negative integers, n+2 is greater than 2n n is positive , it is uncertain

Example 4 Response:

Translation:

When n > 2, 2n > n+2 n 0

After exceeding 2 itself, one side will increase by multiple.

Fig. 5 Taiwanese future

teachers’ original responses to

#509 and the translations

16 This type of answer is incorrect because preservation of length is

developed earlier than length measurement.17 This type of answer is incorrect because teachers will usually not

teach advanced concepts or develop abilities in other fields at the time

they teach length measurement.


123

judge what mathematical competences one could develop

in the given situation.

4.4.4 MTC: misconception

Across all participating countries in the TEDS-M study,

future teachers had more difficulties developing teaching

practices to reduce student misconception than under-

standing student misconception. Item #105A was meant to

assess future teachers’ understanding of student miscon-

ceptions in mathematics, and item #105B, future teachers’

practical competence in reducing student misconceptions.

Taiwan ranked second in both of these items (see Fig. 10).

ItemContentDomain

CompetenceSubcategory

CompetenceCountry

Percentage

#207 Algebra MC:TMC-MLHandle mathematical symbols and formalisms

A1 A2

Description of the item:

A quantity relationship of two objects, say A and B, is given with a certain percentage in the stem. The symbols x and y are assigned to represent the numbers of A and B respectively. Future teachers were asked to choose a correct algebraic equation from four options to represent the quantity relationship.

Taiwan 63.7 11.2

Russia 40.5 12.3

Singapore 39.6 31.1

Switzerland 28.7 35.7

Norway 24.5 30.7

Germany 21.0 30.2

US-Public 18.6 16.6

Thailand 15.7 9.7

IA 23.8 20.5

Descriptions of algebraic equations in the four options A3 A4

A1: Correctly transform the quantity in the stem, and correctly see x and y as variables not labels.

A2: Correctly transform the quantity in the stem, but see xand y as labels not variables.

A3: See x and y as labels not variables, and directly use the quantity in the stem without transforming it.

A4: See x and y as variables not labels, but directly use the quantity in the stem without transforming it.

Taiwan 18.5 5.3

Russia 32.6 13.2

Singapore 18.0 11.3

Switzerland 15.6 17.6

Norway 27.0 15.3

Germany 18.5 22.7

US-Public 43.8 20.3

Thailand 43.5 25.9

IA 32.7 16.6


performance in #207. A1 is the

correct answer. IA international

average of all participating

countries

ItemContentDomain


CompetenceCountry

PC

#206B Number MTC-CJudge competences required to learn concepts/solve problems

#513

#206B

(a) A machine uses 2.4 litres of fuel for every 30 hours of operation.How many litres of fuel will the machine use in 100 hours if it continues to use fuel at the same rate?

(b) Create a different problem of the same type as the problem in (a) (same processes/operations) that is EAS IER for <primary> children to solve.

Singapore 82.0

Switzerland 73.9

Norway 72.4

US-Public 71.6

Germany 64.9

Thailand 53.9

Russia 52.5

Taiwan 44.3

IM 55.1

#513 Geometry MTC-CJudge what competences to develop in teaching

#206B

#513

When teaching children about length measurement for the first time, Mrs. [Ho] prefers to begin by having the children measure the width of their book using paper clips, then again using pencils.Give TWO reasons she could have for preferring to do this rather than simply teaching the children how to use a ruler?

Singapore 50.7

US-Public 48.7

Russia 38.7

Norway 34.7

Taiwan 34.3

Germany 26.2

Switzerland 22.6

Thailand 22.5

IA 29.0


performance in #206B and

#513. PC percent correct,

IA international average of all

participating countries


123

In these two items, two possible student misconceptions

were classified with two codes: considering the hypotenuse

as the base (code 21) and the unfamiliarity with the ori-

entation of the triangle (code 20). The Taiwanese future

teachers tended to consider the specific case more, namely,

that the misconceptions concerned the hypotenuse (code

21). In contrast, all other higher achieving countries except

Singapore and Thailand had more code 20 counts than code

21. Although both codes are considered correct, the authors

regard the code relating to ‘‘orientation and position’’ (code

20) to be more advanced as it explicitly describes not only

the specific structures given in the problem, such as the

hypotenuse, but also a more general, abstract orientation

and position concept of spatial ability.

The concentration on specific cases in Taiwan was fur-

ther emphasized by the results of item #105B. A total of

66% of Taiwanese future teachers provided teaching

practices applicable only to the given specific right trian-

gle, or at best, to right triangles generally (code 10,

partially correct). Only 23.1% described a general teaching

practice applicable to all triangles (code 20). Among the

higher achieving countries, greater quantities of code 10

responses for item #105B were observed, except in Norway

and Singapore, which had about equal percentages. These

results may lead one to ask whether having a more general,

abstract view of student misconceptions can also generate a

more generalized and universally applicable solution that

develops concept images. The results showed that among

all participating countries only Norway, Singapore and

Thailand maintained any consistency. In Taiwan, only

28.5% of future teachers who possessed a general per-

spective in student misconception gave universally appli-

cable solutions and 62.4% of them still provided solutions

dealing with limited, specific cases. Further, nearly 4.6% of

the future teachers from Taiwan provided responses to item

#105A that indicated they had a problem with recognizing

the pictorial representations of right triangles (see exam-

ples in Fig. 11).

Responses:

Translation:

There are thirty tubs of ice cream. Ming needs 2.4 hours to eat them all. How many hours will Ming need to eat up all if there are 100 tubs of ice cream?

Responses:

Translation:

Ming can get 2.4 pieces of small cakes for jumping 30 times. How many pieces of small cakes will Ming get if he jumps 100 times?



#206B and the translations

Example 1 Responses:

Translation:

Let students have the concrete sense of length via the articles in daily life.


Translation:

The length of desk will remain the same even if the units or instruments used for measurement are different.


Translation:

Can let students convert between [length of] objects, not obtain the answer directly, and also can train students’ ability of calculating and converting units of instruments.


Translation:

Can understand the concepts of factors and multiples. Ex. The width is composed of [the length] several paper clips, or [that of] several pencils.



#513 and the translations


123

5 Conclusion

International comparisons have provided a useful way to

examine the emerging influences and the relationship of

these influences, shaped by globalization, on teaching and

teacher education to test relevant theoretical assumptions

about globalization; however, more data are necessary to

verify these assumptions (Wang, Lin, Spalding, Odell,

Klecka, 2011). The conflicting results obtained between the

international TEDS-M study and Taiwanese domestic

studies further confirmed a need to investigate TEDS-M

data from different perspectives. Consequently, this study

attempts to investigate the essence of future Taiwanese

primary teacher mathematics-related competences from a

domestic perspective in an international context.

5.1 Structure for measuring mathematics-related

competence for teaching

The types of knowledge that should be included in testing

mathematics teachers, as well as how much each type of

knowledge is necessary for mathematics teaching is a

common topic of discussion in mathematics education. The

TEDS-M study has many more items testing MCK than

ItemContentDomain


Competence

Country

Percentage

#105A Geometry MTC-MDiagnose students’ misconceptions or error procedures #105A

#105B Geometry MTC-MFind a way to reduce students’misconceptions/error procedures

PC 20 21

Description of the item:

The stem shows a graph of a right triangle with a horizontal hypotenuse and lengths of three sides, and states that a sixth grader claims no way to find the area for lacking the height.

#105AThe item asked why the student claims that.#105BThe item asked responders to come up with a good teaching practice to reduce the student’s misconception.

Singapore 90.1 39.3 44.8

Taiwan 85.2 21.7 48.1

Norway 71.2 49.2 9.6

Thailand 59.7 13.8 26.2

US-Public 59.4 28.0 15.7

Russia 51.2 28.5 12.3

Switzerland 37.1 14.1 7.9

Germany 17.4 8.7 4.6

IA 47.0 20.5 14.8

#105B

Descriptions of rubric (partial)PC 20 10

#105ACode 20 relates to students’ unfamiliarity with the orientations/positions of the triangle.Code 21 relates to regarding the hypotenuse as the base and thus unable to find the height.

#105BCode 20 refers to teachings applicable to general triangles. Code 10 refers to teachings applicable only to the specific right triangle in the question or to right triangles only.

Singapore 58.8 40.1 37.5

Taiwan 56.1 23.1 66.0

Norway 55.1 37.5 35.2

Thailand 32.1 15.6 32.9

US-Public 32.0 12.2 39.6

Switzerland 26.3 5.3 42.1

Russia 24.0 4.9 38.2

Germany 9.9 0.7 18.3

IA 27.6 12.8 27.6


performance in #105A and

#105B. Some of the codes for

correct or partially correct

answers of #105A are not

presented. PC percent correct,

IA international average of all

participating countries

Responses:

Translation:

He does not know that the triangle is a right triangle, one of the two sides of the right angle can be the base and the other can be the height to calculate the area.

Responses:

Translation:

1. Because he doesn’t know the meaning of “right angle” and what its sign means.

2. Because he doesn’t know two sides of a right triangle could be used as base and height when calculating.



#105A and the translations


123

MPCK, which perhaps demonstrates an unbalanced focus

on mathematics knowledge for primary teachers in

TEDS-M. In light of the fact that it is impossible to include

enough test items to cover all school mathematical topics

in a large-scale test like the TEDS-M study, perhaps one

could inquire into future teachers’ abilities or competences

rather than their particular knowledge of specific mathe-

matical concepts or domains.

Although the types of mathematical competence or

mathematics teaching (pedagogical) competence identified

in different theoretical approaches are not identical, there is

overlap though with different emphases. This gave us the

chance to analyze future teachers’ knowledge or ability by

utilizing structures with different focuses to present various

perspectives and results. Based on the assumption that a

mathematics teacher must be equipped with competences

that enhance the understanding of students’ mathematical

reasoning, argumentation or representation, we believe that

CMC is not sufficient. Thus, this paper adopts Niss’s

(2003) structure of mathematical competence, which

emphasizes the basic characteristics of mathematical

thought embedded across almost all domains of mathe-

matics. This structure includes categories of ‘‘the ability to

ask and answer questions in and with mathematics’’ and

‘‘the ability to deal with and manage mathematical lan-

guage’’ and many other subcategories.

With regard to the MTC, in addition to taking a perspective

from TEDS-M that describes by kinds—‘‘curricular knowl-

edge and planning for teaching’’ and ‘‘enacting teaching’’—

this paper adopts a structure emphasizing conceptual com-

ponents. In the adopted structure from Hsieh (2009), each

competence is associated with a certain mathematics teaching

element (a concept). Different operations, for example, rec-

ognizing or reasoning, act on the concept with a kernel of

teacher, student or entity (i.e., the concept itself) to generate

different instantiations of competences. Hsieh’s structure can

thus detail 20 mathematical teaching elements, while the

TEDS-M MPCK only allowed to analyze four categories:

competences of students, misconception of students, mathe-

matical representation and mathematical language. This

shortage of dimensions concerning competences relating to

pedagogy may be regarded as an indicator of the TEDS-M

study’s lack of interest in pedagogical types of competences.

Note that both of the mathematical and pedagogical structures

for competences in this study provide a Taiwanese perspec-

tive. Other conceptual structures expressing other countries’

perspectives are required in order to gather information for

generating or testing any globalized assumptions.

5.2 Patterns of performance

Focusing on TMC, this study found that with respect to

different subcategories of mathematical competence,

although Taiwanese competence was higher than that of

Singapore, only these two East Asian countries shared the

same structural pattern in their responses. The pattern

suggests that, when compared with other countries, Tai-

wanese and Singaporean future teachers perform relatively

better with respect to mathematical language, including

representing mathematical entities and handling mathe-

matical symbols or formalisms, than in other mathematical

competences. This result points to one possible hypothesis:

competence in mathematical language may be an important

element in primary future teacher education, one that pro-

motes general mathematical competence. However, this

requires further research before any conclusion may be drawn.

Further results based on our structure showed that

countries, even those having similar country competences,

did not usually share the same patterns of performance

across subcategories of MTC. When compared with other

countries, Taiwan performed better in competences relating

to students’ misconceptions, but much worse in compe-

tences relating to analyzing students’ mathematical com-

petences. Singapore performed at the same level as Taiwan

in the MTC, but its pattern similarity with Taiwan was

restricted to its performance in competences relating to

students’ misconceptions. An area of research still undev-

eloped is whether there is a shared atmosphere in some

East Asian countries that focuses on competences relating

to mathematical language in a teaching context.

5.3 Insights from the in-depth analyses

Previous studies with in-depth analyses have often been

confined to a domestic scope. The in-depth analysis with a

complement of international comparisons conducted in this

paper provides the researcher with the opportunity to

uncover some unique insights. An analysis on item #207,

examining the ability of handling and manipulation of

statements or expressions containing symbols and formu-

lae, revealed that internationally more than half of future

primary mathematics teachers made ‘‘reversal error’’

(Clement, 1982). Even Singapore had about half of its

participating future primary teachers make this error. This

percentage is 20% higher than that of Taiwan. This result

raises a question concerning the construction of interna-

tional tests of the TEDS-M-type: Did the wording of this

item in English (as administered to future teachers in

Singapore and many other countries) and in Chinese (as

administered in Taiwan) change the syntax of the state-

ments and thus result in different types of potential errors

for responders? Further research on this question is

required if more international tests of this scope are to be

employed by mathematics pedagogy researchers.

Our analyses also uncovered possible explanations for

conflicting results between the limited knowledge shown in


123

Taiwanese domestic studies and the stronger knowledge

shown in the TEDS-M study. Though Taiwanese future

primary teachers had strong competences in mathematics

and mathematics teaching compared to other countries, for

some of the items the future teachers’ performance did not

achieve the expectations of Taiwanese teacher educators or

researchers. A 63.7% correct percent performance for item

#207 is not satisfactory for the Taiwanese criteria for

teachers. There were even items with a correct percent in

the 30–50% range (item #206B and #513); these items fell

into the subcategories relating to the judging of students’

competence in a teaching context. The poor performance of

Taiwanese future primary teachers in this subcategory of

this study confirmed the claims made by domestic studies

that Taiwanese teachers demonstrate an unsatisfactory

understanding of student learning and this is a weak link in

their education of future teachers.

As evidenced by the results of item #206B, an excellent

understanding of a mathematical concept engaged in a

teaching episode is not sufficient to successfully teach that

concept. This result gives evidence to teaching competence

not dependent on mathematical knowledge alone and lends

weight to our claim that MPCK should not be undervalued

in favor of MCK in teacher training.

This study also shows that when Taiwanese future

teachers were incapable of providing a formal proof, they

tended to not try a more heuristic approach. In an inter-

national context, this appeared to limit the Taiwanese

participants. Future research could investigate whether a

heuristic method of proof for teaching is more valuable

than a more formal method when the tracing of mathe-

matical reasoning is desired.

An era of globalization demands international perspectives

on the problems of characterizing components of teacher

competence, balancing different components, and detailing

information for each component. Due to the limitations of the

TEDS-M data set, we have had to be careful with drawing final

conclusions. However, our data have provided us with an

initial approach. More data from all countries, including

Taiwan, are needed to further investigate these problems.

Acknowledgments We gratefully acknowledge the following: the

IEA, the International Study Center at Michigan State University,

the Data Processing Center, the ACER, the U.S. NSF, the Taiwan

TEDS-M team, and all TEDS-M national research coordinators for

sponsoring the international study and providing information and

data. We also acknowledge Sarah-Jane Patterson for her assistance

with editing the paper. Taiwan TEDS-M 2008 was supported by the

National Science Council and Ministry of Education.

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Mathematics-related teaching competence of Taiwanese primary future teachers: evidence from TEDS-M