International perspectives on frameworks for mathematics teachers’ knowing and quality of mathematics instruction

In this Special Issue (SI), an international group of researchers presents current issues related to the design and development of measures of mathematics teacher knowledge and quality of instruction in the mathematics classroom in the context of mathematics teachers’ change. The goal of the SI is to examine current strategies used by researchers to design measures of the different types of knowledge needed for teaching mathematics. Researchers use a wide range of theoretical frameworks to conceptualize and develop measures of different types of teachers’ knowledge. The discussion of theoretical and methodological challenges associated with the design of measures from a diverse group of researchers is included in this Special Issue.

Over the past decade, many conceptual frameworks for describing mathematical knowledge have been put forth by scholars (e.g. Hill, Ball, & Schilling, 2008; Kaarstein, 2014; Kaiser, Blömeke, Busse, Döhrmann, & König, 2014; Lindmeier, 2011; Manizade & Martinovic, 2016; Manizade & Mason, 2011; Rowland & Ruthven, 2011; Shulman, 1987; Silverman & Thompson, 2008; Tirosh, 2000). These frameworks help both to define the domain and provide potential foundations upon which assessments can be built; by using them as the basis for assessment, the field can critically examine the frameworks to offer refinements. At the same time, there has been growth in the number and variety of assessment instruments that have been developed to measure the quality of mathematics instruction, teachers’ knowledge for teaching mathematics, and its related constructs (e.g. Herbst & Kosko, 2014; Hill et al., 2008; Kaiser et al., 2014; Knievel, Lindmeier, & Heinze, 2015; Krauss, Baumert, Blum, 2008; Lindmeier, 2011; Manizade & Mason, 2011; Martinovic & Manizade, 2018; Silverman & Thompson, 2008; Thompson, 2016; Tirosh, 2000). This proliferation of instruments suggests that there is both interest in operationalizing the conceptual frameworks, and that there is tremendous opportunity to explore teachers’ knowledge and actions in a wide variety of ways.

The articles in this special issue offer further illumination of existing theoretical frameworks for mathematics teacher knowledge that can benefit the field. Although there are existing important frameworks based on the aforementioned conceptual foundations, the domain of mathematics teacher knowledge remains under-conceptualized and understudied. Providing researchers with access to a wide array of conceptual frameworks can promote innovation in ways that more readily translate across cultures. It is essential that teacher knowledge is appropriately conceptualized when designing measures of teacher knowledge and quality of mathematics instruction. There are pros and cons related to the conceptual and methodological aspects of any framework, which account for strengths and limitations of the measures developed from the corresponding frameworks as well as the validity and reliability of such measures.

This aligns with Rowland who argues that “the distinction between different kinds of mathematical knowledge is of lesser significance than the classification of situations in which mathematical knowledge surfaces in teaching” (Even, Yang, Buchholtz, Charalambous, & Rowland, 2017). As relevant researchers state, the conceptualization of the constructs of different types of mathematics teacher knowledge needs to be focused on usable knowledge for teaching that only includes knowledge as it is implemented by a mathematics teacher (Kersting, Givvin, Sotelo, & Stigler, 2010; Kersting, Givvin, Thompson, Santagata, & Stigler, 2012). They assume that some teacher knowledge may be inactive except in the process of teaching.

Therefore, the designs of their measures of mathematics teachers’ knowledge, or of the quality of mathematics instruction, are based on teacher actions that can be observed by a researcher and this affects the form of the assessment and the format of the designed instrumentation. What a researcher observes, however, depends on their theoretical perspective, as demonstrated below.

There are multiple ways to conceptualize the knowledge teachers need to use in a mathematics classroom, as well as conceptualize the quality of mathematics instruction, which affects the way we measure these constructs in a research setting. Some of these are captured in Figure 1.

Figure 1. Conceptualizing “good” mathematics teaching from different theoretical perspectives.

The quality of mathematics instruction, and any measures designed to evaluate such quality, is content-specific (e.g. content knowledge, CK, or pedagogical content knowledge, PCK, for fractions is different from CK and PCK for area). In addition, when a researcher evaluates the quality of mathematics instruction or type of mathematics teacher’s knowledge, they base it on their concept of “good” mathematics teaching. The challenge in mathematics education research is that researchers do not have one common definition of “good” mathematics teaching or a “good” mathematics teacher (Medley, 1987). Based on the mathematics education researchers’ conceptual and theoretical framework, the definition of “good” mathematics teaching varies (Medley, 1987). In some cases, mathematics teaching is considered good quality if particular learning outcomes are achieved by students. In other cases, quality of mathematics instruction could be tied to student activities, particularly the types of learning experiences in which students engage within a mathematics classroom.

In this special issue, Makonye (2020) discusses the importance of learners’ cultural knowledge systems when teaching financial mathematics. In his study, he conceptualizes financial mathematics pedagogical content knowledge (fmPCK). He also speaks of the great importance of culturally relevant pedagogy by demonstrating that learners from South Africa do not hold a “time-value-of-money” construct, therefore making the concept of banking and investment more challenging for students to understand. This internal context variable, associated with South African students, affects responses to mathematics teachers’ activities by attenuating the relationship between interactive teacher activities in the presence of students (i.e. the process of teaching) and the type of student activities that occur in the classroom.

Alternatively, “good” mathematics teaching could be connected to the best practices/activities in which the mathematics teacher engages while in the presence of their students. In this special issue, Friesen and Kuntze (2020) discuss professional knowledge and the quality of mathematics teaching with respect to instructors’ capabilities to analyse a classroom situation through the use of such best practices as multiple representation. In their study of 298 teachers, they investigated different aspects of teachers’ knowledge on using multiple representations in the context of the teaching and learning fractions and their role for teachers’ corresponding competence of analysing. Similarly, Kristinsdóttir, Hreinsdóttir, Lavicza, and Wolff (2020) used “silent” video tasks with Icelandic high school students when teaching trigonometric concepts. In these tasks, students were asked to annotate a silent video with their own mathematical explanations, and teachers’ responses to student productions were analysed to evaluate teachers’ knowledge that influenced their engagement in innovative practices.

Another way to assess the quality of mathematics instruction is to associate it with pre-active mathematics teacher activities such as lesson planning, assessment of student work, and other activities teachers engage in while outside of the classroom. In this Special Issue, Mellone et al. (2020) discuss the development of interpretive knowledge of pre-service mathematics teachers with respect to the sum of powers of ten. They defined a “good” mathematics teacher as one who can assess and interpret students’ work and mathematical reasoning. This aligns to the work of Manizade and Martinovic focused on professionally situated knowledge (based on Shulman’s PCK construct) of in-service mathematics teachers in the context of teachers’ capability to interpret and assess presented students’ challenges in geometry, specifically finding the area of a trapezoid, as well as to plan the response to each conceptual challenge presented by a student (Manizade & Martinovic, 2016; Manizade & Martinovic, 2018; Martinovic & Manizade, 2018).

Other researchers associate the quality of mathematics instruction with mathematics teacher competencies, knowledge, and skill, as well as pre-existing mathematics teacher characteristics, such as attitudes, motivations, and beliefs. In their article, Lindmeier et al. (2020) present a study of 170 early elementary teachers’ development of Action-related Competence (AC) and Reflective Competence (RC) with respect to early number sense. Based on their model of teacher competence (Lindmeier, 2011), researchers designed two professional development programmes with a focus on AC and RC, then implemented them in a randomized, controlled experiment. Overall, they found that AC, RC, and professional knowledge were sensitive to interventions. Their results indicated that the AC intervention group was the only group with a growth in AC, while all groups, including the control group, gained in RC. Depending on the definition of “good” mathematics teaching to which a researcher subscribes, the focus of the instrumentation used to judge the quality of instruction is going to vary.

Additionally, in the post-industrial era, the way that teachers teach mathematics, plan lessons, assess students, and interact with their students in and out of the classroom has changed compared to the way mathematics was taught during the last century. Therefore, the way to measure change in mathematics teacher knowledge, as well as the quality of the mathematics instruction, must also be adjusted with technological and methodological advances. For example, in this special issue, Herbst, Ko, and Milewski (2020) propose a new methodological approach for evaluating change in mathematics teachers’ knowledge for a small group of teachers while affording the validity and reliability of a large-scale study. These researchers used a national distribution of responses from 416 practicing teachers to test items of mathematical knowledge for teaching geometry to estimate changes in knowledge by a group of 11 practicing teachers who participated in a professional development programme. To draw a reliable interpretation of the change, they used multiple measurement models, and they demonstrated how Diagnostic Classification Modeling (DCM) can be used to measure participants’ growth. In their article, using the examples of two of their most recent studies, Orrill, Copur-Gencturk, Cohen, and Templin (2020) explore the importance of: (1) the conceptualizing of the domain of different types of mathematics teacher knowledge; (2) identifying the purpose of measuring the knowledge of mathematics teachers; (3) selecting psychometric models that support meaningful sensemaking of what was being measured. They also discuss current methodological advances in design of measures of mathematics teacher knowledge for various purposes (e.g. IRT, Topic Analysis Models, DCMs).

For this Special Issue of Research in Mathematics Education, the goal was to examine current strategies used by researchers to design measures of different types of knowledge needed for teaching mathematics, including, but not limited to dynamic measures of mathematics teachers’ pedagogical content knowledge; classroom observation protocols; silent video analyses; animations suitable for teacher professional discussions; mathematics teacher certification assessments; analyses of classroom scenarios through cartoons, video enactments and transcripts; analyses of classroom videos; and paper and pencil tests of different types of teacher knowledge based on diverse theoretical frameworks. Researchers used a wide range of theoretical frameworks to conceptualize and develop measures of teachers’ knowledge and their quality of mathematics instruction. The set of articles in the Special Issue illustrates how the field is developing and shows the opportunities for further research.

Disclosure statement

No potential conflict of interest was reported by the author(s).