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

## Authors

- First Online:

- Accepted:

DOI: 10.1007/s11858-011-0377-7

- Cite this article as:
- Hsieh, F., Lin, P. & Wang, T. ZDM Mathematics Education (2012) 44: 277. doi:10.1007/s11858-011-0377-7

## Abstract

This paper draws on data from the international TEDS-M study, organized by the IEA, and utilizes a conceptual 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 adopted 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 students to learn mathematical concepts or solve problems, and superior at diagnosing and dealing with student misconceptions 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 influencing students’ performance and learning in the classroom (Cobb et al., 1991; Rice, 2003), but identifying and measuring 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 frameworks 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 & McDonald, 2008; Hill, Ball, & Schilling, 2008; Shulman, 1986, 1987).

During the past two decades scholarly interest in international 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 (Blömeke et al., 2008; Schmidt et al., 2011). The Teacher Education and Development Study in Mathematics (TEDS-M) was the first data-based international study about mathematics teacher education with national representative samples. It provided participating nations the opportunity to acquire international perspectives on their teacher education systems in areas such as future teachers’ knowledge levels (Blömeke, Suhl & Kaiser, 2011; König, Blömeke, Paine, Schmidt & Hsieh, 2011), their opportunities 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 knowledge (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 suggested 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).

- 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 Taiwan’s views on mathematics and mathematics teaching 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 mathematical abilities that are not directly related to any specific mathematical content (Krutetskii, 1976; Niss, 2003; National Research Council, 2001). Two types of mathematical competence (MC) are discussed in this paper. The first is *content*-*oriented mathematical competence* (CMC), which is related to specific mathematics topics as the prerequisites of advanced-level competence, namely the factual 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.

*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 mathematically*, and

*reasoning mathematically*. In TMC-ML, two subcategories are highlighted:

*representing mathematical 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 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*, *operation*, 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 “mathematics 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 difference between the mathematics thinking and other scientific 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).

^{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 “mathematical 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 “mathematical 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, 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, reasoning, 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 subcategories of MC and MTC, though distinct, are interwoven and an item may test competences in more than one subcategory. To lessen the subjectivity of the categorization, an *operational procedure* was developed and administered. 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 certain 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 concepts, and the final two neither gained consistent categorization 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 concept 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.

^{9}All seven of these items are in the lower secondary 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 mathematically*, 11 as

*modeling mathematically*and 17 as

*reasoning 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 not 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 Taiwanese classification of MC items is shown in 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}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.

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 |

### 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 statistic 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 participating 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 performing closely with Taiwan.

### 4.1 Taiwanese future teachers’ achievement across cognitive domains

^{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 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 percent 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 classroom by teachers to diagnose problems and respond to students.

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, diagnosing student misconceptions and providing feedback (Tatto et al., 2008).

### 4.2 Future teachers’ mathematical competences (MC)

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 |

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 |

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 subcategories 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 performance in TMC-TQ relatively. These findings demonstrate that Taiwan and Singapore performed relatively better in competences related to formalization, abstraction, and operations in mathematics (TMC-ML) than in competences 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 condition 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

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 |

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 |

### 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 competences of these tested items will be described when applicable.

#### 4.4.1 MC: TMC-TQ

#### 4.4.2 MC: TMC-ML

This item also required translation from natural language 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 numbers 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

*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 problems would be easier for students (see Fig. 8 for two exemplary answers). These future teachers felt that the familiarity 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%.

^{16}the third involved a cultivation 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 judge what mathematical competences one could develop in the given situation.

#### 4.4.4 MTC: misconception

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 orientation 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.

## 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 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 mathematical 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 mathematics. 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 language” and many other subcategories.

With regard to the MTC, in addition to taking a perspective from TEDS-M that describes by kinds—“curricular knowledge and planning for teaching” and “enacting teaching”—this paper adopts a structure emphasizing conceptual components. In the adopted structure from Hsieh (2009), each competence is associated with a certain *mathematics teaching element* (a concept). Different *operations*, for example, recognizing 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, mathematical 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 perspective. 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, Taiwanese and Singaporean future teachers perform relatively better with respect to mathematical language, including representing mathematical entities and handling mathematical 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 promotes 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 competences relating to analyzing students’ mathematical competences. 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 undeveloped 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 formulae, 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 international 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 statements 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 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 international 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 mathematical 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.

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.

The classification of items into different operations and kernels in Hsieh’s framework can be found in Table 2.

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%.

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.

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.

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.

The statement might look like “Indicate whether each of the following students’ responses is correct or not”.

For example, a response with a code 20 or 21 was scored 2 points, whereas a code 10 or 11 was scored 1 point.

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.

This type of answer is incorrect because preservation of length is developed earlier than length measurement.

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.

## 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.