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This study involving 120 Australian and Chinese teachers introduces a construct of teacher capacity to analyse how teachers help students connect arithmetic learning and emerging algebraic thinking. Four criteria formed the basis of our construct of teacher capacity: knowledge of mathematics, interpretation of the intentions of official curriculum...
Contexts in source publication
Context 1
... between the criteria Table 5 shows the bivariate correlation between any two of the four criteria. Correlations were calculated using SPSS 19.0 (English version) for the 120 Chinese and Australian samples. ...
Context 2
... the means of the four criteria for each sub-category show that each criterion effectively discriminates between the three classifications of high, medium, and low capacity. Table 5 above shows that the correlations between the four criteria are not high enough to suggest that any pair is measuring the same thing, or low enough to suggest that they are completely unrelated. ...
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Citations
... This paper investigates a possible gap between the theoretical framework of knowledge that Chinese mathematics teachers are expected to possess and the practical situations that they encounter in many classrooms. Shulman's pedagogical content knowledge is influenced by the educational background of the United States and may not reflect the actual situation of education and teaching in China because, as Zhang and Stephens (2013) point out, there is no unified national curriculum in the United States of America. However, the theoretical framework of pedagogical content knowledge based on the existing assessment research in China is mostly built on the results of foreign research. ...
The study aimed to establish an assessment model for mathematics teachers’ knowledge of students’ misconceptions in the Space and Shape domain, develop the testing tool, investigate and analyse the overall and differences in performance, and propose suggestions for improvement. The assessment model included content knowledge and performance standards. The content knowledge standard included cognition and content dimensions. The cognition dimension was subdivided into predicting the misconceptions, interpreting the misconceptions, analysing the reasons and teaching strategies to correct misconceptions. According to the division of Space and Shape in official curriculum standard, the content dimension was subdivided into Recognition of graphics, Measurement, Motion, and Location. 701 Chinese mathematics teachers from 8 provinces were involved to verify the assessment model. Item response theory was used to estimate teachers’ performance, and
combinations of response categories was used to divide the teachers’ performance into three response levels: Low, Medium and High. 81% of teachers were performing at Medium and High response levels. Teachers performed the worst in Recognition of
graphics and the best in Measurement. The worst performance occurred in analysing the reasons, the best in interpreting the misconceptions. These results have provided evidences for adjusting teacher activities, improving teacher education courses and teachers training.
... In this study, our attention is directed towards the micro-level curriculum, which is influenced by the central curriculum paradigm. Besides, Shulman's (1986) conflation of curriculum knowledge with pedagogical and subject matter knowledge can be better explained by the nature of the decentralized educational system in the US (Zhang & Stephens, 2013). However, as mentioned by Petrou and Goulding (2010), teachers must be competent in both the required aspects of the curriculum framework they are working within and the materials and resources available to them. ...
Curriculum adaptation is unavoidable, according to both Turkish and international research. Deliberative adaptations, however, are not as simple as they appear, and teachers with low pedagogical design capacity (PDC) may make haphazard adaptations or fail to make the necessary adaptations. Therefore, professional development (PD) is crucial in making teachers’ adaptations more deliberatively and systematically. In this case study, drawing on the interconnected model of professional growth, the change of pathways of four mathematics teachers’ PDC was explored within the context of a PD. The findings revealed that the capacity to use curricular resources, the capacity to use personal resources, and beliefs towards the use of curriculum emerged as three change indicators of PDC. While teachers’ interactions with the facilitator were effective in fostering knowledge change, getting and reflecting on students’ learning outcomes was effective in fostering belief change. The results are discussed in the context of curricular and PD studies.
... 156). This might be explained by the nature of decentralized education system in the USA (Zhang & Stephens, 2013), despite efforts to nationalize it . Further efforts have been made to elaborate on curriculum reasoning (Breyfogle et al., 2010;Roth McDuffie & Mather, 2009) and curriculum noticing (Dietiker et al., 2018) focusing on decentralized education system in the USA. ...
This conceptual paper puts forward the construct termed teacher curriculum competence, which is an amalgamation of theoretical or formal and personal practical teacher knowledge and orientations in relation to curriculum. We situate the competence in institutional, political, and philosophical contexts. Drawing on research related to mathematics curriculum at different stages (i.e. the official, intended, and enacted curriculum) and teacher competence frameworks, we elaborate on how a teacher interacts with curriculum informed by their knowledge and orientations. When working with curriculum, teacher curriculum competence refers to what teachers attend to and how they interpret curriculum, as well as why they make decisions, including when to introduce particular concepts or skills, based on their local students' needs. In addition, we argue that the process of interacting with curriculum helps shape teacher knowledge and orientations. This article is significant for the contribution it makes to conceptualizing teacher curriculum competence in a centralized curriculum system, that is a nationally mandated. Although we use the mathematics discipline as an example, implications for research in promising areas for future studies in this space are also discussed. ARTICLE HISTORY
... We designed the teacher instrument items based on previous measures of teachers' knowledge for teaching equivalence. Teacher items have previously focused on teachers' knowledge of students' understanding of mathematical equivalence (Hunter & Jones, 2016;Jacobs et al., 2007;Stephens, 2006;Zhang & Stephens, 2013). For example, Jacobs et al. (2007) asked teachers to generate strategies students might use to solve five open number sentences such as 8 þ 5 = _ þ 6 (grade 1). ...
Many primary school students have difficulties understanding mathematical equivalence with considerably poorer performance in some countries than in others. However, students’ formal understanding of equivalence has significant and long-lasting effects, as it predicts arithmetic and algebra achievement throughout school years. Currently, little is known about the factors influencing students’ understanding of mathematical equivalence particularly across different countries. We have conducted the first large-scale study to explore the factors associated with primary school students’ understanding of mathematical equivalence across six countries (China, England, New Zealand, South Korea, Turkiye, and the United States). Participants were 2,760 primary school students and their teachers (N = 108). Using multilevel structural equation modeling, we found that students’ knowledge of definitions of the equals sign relates to their equation-solving performance. We also found that while teachers’ knowledge of students’ relational strategies does relate to students’ understanding of equivalence, teachers’ knowledge of students’ operational strategies, and the format of arithmetic practice presented in the students’ current year textbooks do not. Using England as the reference country, we found that this pattern was similar across the samples from all the participating countries. Taken together, our findings have important theoretical and practical implications, providing a more complete picture of the individual and classroom-level factors associated with students’ understanding of equivalence.
... De la misma forma, los materiales curriculares pueden comunicar un tipo de mensaje, pero su implementación final dependerá de cómo son interpretados por los docentes. La comprensión que alcanza el profesorado acerca de lo dispuesto en los materiales curriculares impacta en la forma en que estos recursos son utilizados y, por lo tanto, en la implementación efectiva de cualquier reforma (Choppin, 2011;Zhang y Stephens, 2013). ...
... Una implementación efectiva del currículo de cualquier reforma depende de la delicada interpretación que hagan los profesores de los documentos curriculares oficiales y su disposición profesional a esas ideas (Zhang y Stephens, 2013). Cuando los profesores se enfrentan a los diferentes materiales, estos influyen en su actuación, transformándola. ...
... La implementación de cambios curriculares no es sencilla, especialmente cuando los cambios se producen verticalmente (Zhang y Stephens, 2013). Dado que la puesta en práctica del currículo implica atravesar por un proceso complejo de interpretación y adaptación a un contexto local, es importante que los materiales curriculares transmitan de forma adecuada lo que se espera de la instrucción Stephens, 2013). ...
... Curriculum knowledge was considered to be an understanding of how mathematical topics were arranged and connected across school experiences. Many scholars have subsequently used Shulman's categories of knowledge and refined them to various degrees, adding further descriptions or coining new anachronisms and insights (e.g., Ball and Bass 2000;Delaney et al. 2008;Ball et al. 2008;Beswisk and Goos 2012;Chapman 2015;Jacobson and Kilpatrick 2015;Zhang and Stephens 2013). In considering mathematics teaching and learning, adding M to PCK refers to pedagogical knowledge specific to mathematics; thus, mathematics pedagogical content knowledge (MPCK) (Dohrmann et al. 2012). ...
In Australia, there is increasing scrutiny of teacher education processes in learning to teach primary mathematics. In this study, mixed methods, including linear regression, are used to examine the entry mathematical content knowledge of third-year Bachelor of Education Pre-service Students and to relate this to their graduating level of content and a measure of pedagogical content knowledge based upon capacity to describe student errors and provide learning support. The data indicate low levels of mathematical content knowledge at the beginning of the course, prompting questioning of the focus of earlier mathematics curriculum courses. Over the life of the study, there was improvement in some domains, which is to be expected where knowledge of mathematics was an intended outcome. Mathematical content knowledge at the start, but particularly at the end of the study, was highly predictive of expressions of mathematical pedagogical content knowledge. The stronger predictive value of mathematical content knowledge at the end of the study for mathematical pedagogical content knowledge suggests merit in developing the two aspects of teacher knowledge in tandem, rather than in different courses. The relevance of the data to teacher preparation in the institution and more broadly is discussed.
... In mathematics education research circles, there is still no widespread consensus on a framework of types of knowledge specifically needed for teaching mathematics (Petrou and Goulding 2011). None of the prevalent models are considered easy to operationalise when analysing teachers' specific actions in classrooms (Wilkie 2016a;Zhang and Stephens 2013), but the model proposed by Ball et al. (2008) does highlight that there is much more to teachers' knowledge for teaching mathematics than merely subject-matter (content) knowledge. Even for that type of knowledge, they distinguished between three types: common content knowledge, specialized content knowledge, and knowledge at the mathematical horizon. ...
Mathematics teaching at secondary levels has proven surprisingly resistant to
change over the past century. This study draws on two theoretical models to investigate
how the process of changing secondary teaching in algebra through school-based professional
learning might occur, and its relationship to different external and internal influences
on teachers and researchers. A cyclic change model is used to discuss three different
change pathways that were found amongst six practising secondary teachers participating
in an algebra teaching experiment, one phase of a larger design-based research project.
Meta-didactical transposition is used to examine the dynamics between teachers and
researchers and the institutional dimension of professional learning. Affordances and
constraints related to the teachers’ internal domains and social contexts in responding to
professional learning opportunities are discussed. The bidirectional nature of brokering
processes between teachers and researchers during professional learning is examined.
... Ma (1999) used the descriptor ''profound understanding of fundamental mathematics'' to describe the forms of mathematical knowledge needed to teach effectively. More recently, Zhang and Stephens (2013) use the term ''teacher capacity'' as a general term that refers to teachers' ability to understand and act on reforms that policy makers are seeking to implement. Zhang and Stephens argued that in order to assist students with mathematical learning, teachers needed to know the subject matter themselves as well as to have curriculum knowledge, knowledge of students, general and specific pedagogy, and appropriate disposition. ...
... The first five standards relate to the practice of managing classroom discourse and include: (1) understand how students learn, (2) know the content and how to teach it, (3) be able to plan for and implement effective teaching, (4) create and maintain supportive and safe learning environments, and (5) assess and provide feedback and report on student learning. It can be argued that effective classroom discourse required by the standards above is dependent upon teachers' knowledge of mathematics and the specific pedagogies to teach it (Poulson 2001;Zhang and Stephens 2013). What research that has been undertaken on mathematical knowledge of teachers in Western and Eastern institutions suggests that Eastern prospective primary school teachers (in China and Japan) had greater mathematical knowledge than those in most Western nations (Burghes 2011). ...
The preparation of teachers to teach mathematics to primary school children differs across nations and cultures. This study used mixed methods to examine the basic content knowledge of trainee teachers in Australia and China. A simple test (30 questions) of content based on an international comparative study in mathematics teacher training found that many of the Australian trainee teachers struggled with material that they might be expected to teach, while the Chinese teachers largely demonstrated mastery. The significance of this finding is examined in the context of the teacher preparation programs in two teacher training institutions. Cultural commentary is added by leading academics in each institution. It was found that in the Australian teacher training institution there was a focus on generic skills and relatively limited opportunity to develop trainee teachers’ content knowledge or specific pedagogy. The relevance of the findings is discussed through the framework of different beliefs in the nature of mathematics and mathematics teaching that have been reported to dominate the different educational systems in China and Australia.
... It is widely accepted that if a teacher does not know the mathematics, then they are unlikely to be able to structure lessons to teach it effectively (Ball, Lubienski, & Mewborn, 2001;Graven, 2002;Masters, 2009;Murphy, Neil, & Beggs, 2007;Rowland, Huckstep, & Thwaites, 2005;Stipek et al., 2001;Zhang & Stephens, 2013) not least because they will not be able to manage effective classroom discourse or use meaningful models to assist students to understand the mathematical concepts. Confidence and depth of mathematical knowledge are considered particularly relevant to teachers' practices in "inquiry-orientated" mathematics teaching. ...
... The models that describe the relationship between teachers' content knowledge and how this needs to be supported by knowledge of students, learning theories, curriculum intention and specific pedagogy have evolved over the past few decades (e.g., Ball & Bass, 2000;Ball, Hill, & Bass, 2005;Ball, Thames, & Phelps, 2008;Hattie, 2009;Ma, 1999;Shulman, 1987Shulman, , 1999Zhang & Stephens, 2013). While the models have evolved and become ever more sophisticated, it is apparent that concerns about the depth of teachers' content knowledge (MCK) remain a relatively constant theme in the Western educational research literature (e.g., Ball et al., 2001;Burghes, 2011;Henderson & Rodrigues, 2008;Ingram & Linsell, 2014;Jensen, 2012;Klein, 2005;Livy & Herbert, 2013;Ma, 1999;Ma & Kishor, 1997;Major & Perger, 2014;Tatto, Rodriguez, & Lu, 2015;Tatto et al., 2008). ...
... As noted in the literature review the critical relationship between understanding mathematics (MCK) (to a relevant level) and capacity to teach it is almost universally accepted (e.g., Ball & Bass, 2000;Ball et al., 2008;Hill et al., 2005;Ma, 1999;Shulman, 1987;Zhang & Stephens, 2013). Further, it has been found by earlier authors (e.g., Brady & Bowd, 2005;Henderson & Rodrigues, 2008;Schackow, 2005) that pre-service teacher confidence in understanding mathematics was not robust. ...
: The purpose of this paper is to examine trainee primary school teachers’ confidence in their mathematical content knowledge (MCK) and confidence to teach specific primary mathematics concepts (mathematics pedagogical content knowledge –MPCK) which was correlated to their actual MCK on specific tasks. For this correlational study survey and test data were collected from a cohort of 210 trainee teachers. It was found that confidence to do and to teach mathematics was reasonably strongly correlated with competence. Trainee teachers’ confidence varied greatly depending on the specific mathematics they were attempting. When presented with specific tasks, trainees were well aware of the link between personal numeracy levels and their potential to teach primary mathematics. A further finding was that the trainee teachers tended to over report their confidence. It is unknown if this is a cultural manifestation or a limitation of the scale. The data also add to the body of knowledge with respect to the MCK of about-to-graduate primary teachers.
... It is widely accepted that if a teacher does not know the mathematics, then they are unlikely to be able to structure lessons to teach it effectively (Ball, Lubienski, & Mewborn, 2001;Graven, 2002;Masters, 2009;Murphy, Neil, & Beggs, 2007;Rowland, Huckstep, & Thwaites, 2005;Stipek et al., 2001;Zhang & Stephens, 2013) not least because they will not be able to manage effective classroom discourse or use meaningful models to assist students to understand the mathematical concepts. Confidence and depth of mathematical knowledge are considered particularly relevant to teachers' practices in "inquiry-orientated" mathematics teaching. ...
... The models that describe the relationship between teachers' content knowledge and how this needs to be supported by knowledge of students, learning theories, curriculum intention and specific pedagogy have evolved over the past few decades (e.g., Ball & Bass, 2000;Ball, Hill, & Bass, 2005;Ball, Thames, & Phelps, 2008;Hattie, 2009;Ma, 1999;Shulman, 1987Shulman, , 1999Zhang & Stephens, 2013). While the models have evolved and become ever more sophisticated, it is apparent that concerns about the depth of teachers' content knowledge (MCK) remain a relatively constant theme in the Western educational research literature (e.g., Ball et al., 2001;Burghes, 2011;Henderson & Rodrigues, 2008;Ingram & Linsell, 2014;Jensen, 2012;Klein, 2005;Livy & Herbert, 2013;Ma, 1999;Ma & Kishor, 1997;Major & Perger, 2014;Tatto, Rodriguez, & Lu, 2015;Tatto et al., 2008). ...
... As noted in the literature review the critical relationship between understanding mathematics (MCK) (to a relevant level) and capacity to teach it is almost universally accepted (e.g., Ball & Bass, 2000;Ball et al., 2008;Hill et al., 2005;Ma, 1999;Shulman, 1987;Zhang & Stephens, 2013). Further, it has been found by earlier authors (e.g., Brady & Bowd, 2005;Henderson & Rodrigues, 2008;Schackow, 2005) that pre-service teacher confidence in understanding mathematics was not robust. ...
