Analyzing student teachers’ use of theory in their reflections on mathematics teaching practice
Abstract
This study was conducted among 269 student teachers at 11 primary teacher training colleges in the Netherlands. To investigate their competence in integrating theory and practice in their reflections on mathematics teaching, a learning environment was designed to evoke theory use in reflections on practice. To be able to systematically describe the use of theory, we distinguished two dimensions, which we called the nature and level of theory use. A Reflection Analysis Instrument was used to univocally code the nature and level of the student teachers’ theory use in the reflective notes of their final assessments into 1740 meaningful units. We found that nearly all student teachers used theory. However, they differed markedly in the way they linked theory and practice and with which depth they used theoretical concepts in their reflections. A remarkable finding of the study was the important influence of prior mathematics education on the nature and level of theory use, especially the low results of the third-year student teachers in their level of theory use. The outcome may have consequences for the design of the teacher education curricula and for the intake of first-year student teachers.
Keywords
Mathematics teacher education Multimedia learning environment Nature and level of theory use Theory and practice Theory-enriched practical knowledgeIntroduction
Interest in theory and theory building of research on and with mathematics teachers has been increasing in recent years, especially in the last decade (Da Ponte 2013; Lerman 2013; Sriraman and Kaiser 2006; Skott et al. 2013). Theory appears in many guises and at many levels (Silver and Herbst 2007). An important question for teacher education is how theories, especially local instruction theories (Gravemeijer 2004), can really be used by student teachers (STs) to deepen their thinking and reasoning about practice. For every theory, it is possible to select a set of coherent concepts that represent that theory. Such a set could provide participants in the discourse of practice with a vital vocabulary (Sfard 2008) to help them to learn to use theoretical concepts.^{1} Supporting STs in appropriating this vocabulary might enable them to integrate theory and existing practical knowledge into “Theory-enriched Practical Knowledge” (Oonk et al. 2004, 2015a). In this process, reflective thought is “a forceful motor” of mathematical discourse and invention (Freudenthal 1991, p.100). It is plausible that such a “motor” is effective and efficient if STs deepen their reflective thoughts with the use of adequate theory and that it will strengthen their own development. This resonates with Schön’s notion of reflective practitioners (Schön 1983). Preparing STs to actually use theory, for example, “to move beyond a superficial ‘right or wrong’ analysis of tasks to a focus on how STs are thinking about the tasks” (National Council of Teachers of Mathematics 2000, p.24), is an important issue for teacher education. In their attempts to spark discussion about the question of how to prepare STs for future society, which means that STs are better equipped to understand the later thinking of students when completing tasks, Gravemeijer et al. (2017) emphasize that attention will have to be given in mathematics education to mathematics-specific forms of argumentation and communication. Their statements endorse the afore-mentioned suggestions about future teachers’ professional development toward integrating theory and practice. In this study, we investigate STs’ use of theoretical concepts in their written reflections on teaching practice.^{2}
The central problem is how the nature and the level of their theory use differs, also with regard to the relationship with their prior education and their year of study. To carry out this research, we created a practice-based multimedia learning environment that invites the use of theory. The research project is characteristic of the practice-based approach to elementary mathematics teacher education in the Netherlands (Goffree and Oonk 1999; Oonk et al. 2019).
Theoretical considerations
Integrating theory and practice: bridging a “gap”?
The failure of teacher education to influence teachers’ practices has been described by a number of researchers (e.g., Ball 2000; Clark and Lampert 1986; Korthagen 2001, 2010; Vescio et al. 2008). For example, the approach in teacher education whereby STs learn theory during lectures and are then expected to apply it in practice still has not disappeared. Changing existing cultures remains difficult (Cohen 2011; Jaworski 2006; Lave 1996; Morris et al. 2009; Stigler and Hiebert 1999).
Conversely, the realization of well-considered teacher training goals can be impeded by the conformist influence that practical training can have on preservice teachers (Kaiser et al. 2007; Zeichner et al. 1987). Since the beginning of the last century, the research community has been working on “bridging the gap between theory and practice” (Cochran-Smith and Lytle 1999; Dewey 1904; Leikin and Levav-Waynberg 2007; Korthagen 2010) and “crossing boundaries” (Akkerman and Bakker 2011; Goos and Bennison 2018). This especially occurred in mathematics teaching, for example, by searching for practice-based theories of instruction intended to support STs in learning to teach students how to solve mathematical tasks that need deep, relational understanding (Cobb 1988; Cobb et al. 2003). Others have emphasized the development of concepts about what kind of knowledge is the key for future teachers (Ball et al. 2008; Nunes et al. 2016; Ruthven 2011). In the view of Freudenthal, “the Gap” between theory and practice “should not have to exist” (1987, p. 14). He considered the interaction between theory and practice as a process in which theory continuously influences practice. It supports understanding practice, improves practice, and grows itself in this way. In this sense, the aim of theory is not to be something “to apply” but a tool for understanding one’s own and others’ practice and to raise practice to a higher level. With this idea of level raising, Freudenthal (1991) saw a relationship between the theory and practice of learning and teaching and the level theory of Van Hiele (1973, 1986). In the original theory, Van Hiele (1973) described three levels in mathematical thinking. At level 0, the base level, elements are judged only by their appearance. At level 1, elements are recognized by their properties, so a relation network exists. At level 2, relations between properties play an important part. Van Hiele’s theory has been broadened to mathematics education (Gravemeijer 2004; Treffers 1987) and to teacher education in general (Korthagen 2010). Freudenthal saw reflection as having an important role within level theory, as a level-raising function, in the course of which reflection happens on a higher level on one’s actions at the lower level (Freudenthal 1991, p. 101). In this way, reinvention of level raising leads to a natural merging of theory and practice. We agree with Freudenthal that a “gap” between theory and practice should not have to exist. It is not helpful to keep returning to the idea that theory and practice are totally different entities and “that in the organization of learning teaching, there is a ‘gap’ to be ‘bridged’” (Lampert 2010, p. 31). Below, we will elaborate the implications of this view for the student teachers’ process of integrating theory and practice.
Theory-enriched practical knowledge
In 1981, Elbaz introduced the concept of “teacher practical knowledge,” “practical knowledge” for short, that can be defined as both the knowledge and the insights that underlie teachers’ actions in practice. Within the term “practical knowledge,” the word “knowledge” is used as an overarching, inclusive concept that summarizes a variety of cognitions from conscious and well-balanced opinions to subconscious and unreflected intuitions (Verloop 2001; Verloop et al. 2001; Oonk et al. 2015a). It does not mean that in this view, practical knowledge is in opposition to theoretical or scientific knowledge. Knowledge gained from lectures, self-instruction, and other sources of teacher education may be integrated into practical knowledge as well as the notion of pedagogical content knowledge, as developed by Shulman (1986). We consider the mathematical knowledge for teaching, distinguished by Ball et al. 2005, 2008; Nunes et al. 2016 or Ruthven 2011, used meanwhile by many other researchers (e.g., Keijzer and Kool 2012; Oonk et al. 2007), as the core of practical knowledge for mathematics teaching. Furthermore, practical knowledge is considered to have a narrative character (Clandinin and Connelly 1996; Lin 2002; Pendlebury 1995), because it often develops from stories of teaching practice. In a sense, this resonates with Freudenthal’s didactical design idea that mathematical concepts can be derived by learners as they experience a need for those concepts, which asks the designer to generate situations in which phenomena need to be organized (Freudenthal 1983; Mor and Noss 2008). At the level of mathematics teacher education, such situations may evoke STs’ use of theory. Oonk et al. (2004, 2015a) found that STs can acquire “theory-enriched practical knowledge” (TEPK), especially if it has to be constructed in appropriate situations. In this way, theoretical concepts (e.g. words, ideas) are meaningfully integrated within practical knowledge, in order to strengthen the STs process of theory use.
The class has already come up with 2 × 5 followed by 3 × 5. Because the teacher visualizes the five times table for the children, they can also tell a story to accompany a problem. 1 × 5 can be seen as 1 tube times 5 balls. She also makes a connection between concrete material and a grid model. At one point, Clayton [pupil] is counting 10 × 5. The teacher confirms this for the class. In fact, a transition is being made here from multiplication by counting to structured multiplication.
The ST reasoned in this narrative reflection about the relationship between using manipulatives (the tennis balls in transparent tubes) and the higher level of the grid model. In her argumentation, she referred to concepts and instructional approaches (e.g., five times table, visualization, story to accompany a problem, concrete material, grid model, multiplication by counting, and structured multiplication) in a meaningful way and with mutual connections. In this way, the ST showed her TEPK about the STs’ development of mathematical learning.
The student teachers’ process of enriching practical knowledge with theory
The thesis that STs learning can be considered a process of enriching practical knowledge with theory is grounded in a series of assumptions (Oonk et al. 2004, 2015a).
Firstly, the process of enriching practical knowledge is a process of integrating theory with existing practical knowledge. Secondly, such a process is started and stimulated through discourse (Cobb et al. 2003; Ryve 2011). The STs actually go through a “process of appropriation” (Bakhtin 1981). The communication in this discourse could be considered as a “patterned collective activity” (Sfard 2008, p. 93), characterized by narratives representing practice with a specific vocabulary of specific concepts representing theory. The core in our view is that in such discourses, STs learn to use a suitable terminology in a meaningful manner (Oonk et al. 2015a). That means among other things that it is not easy to fulfill this process, because these words initially are the possessions of others, embodied by their beliefs, norms, and experiences. Thirdly, reflection on one’s own and others’ practices in particular can support STs in developing a coherent network of concepts—compare conceptual fields (Vergnaud 1983)—by using the various discourse elements (e.g., particular words and principles) in different ways. Reflection is a tool for analyzing what has happened and also a tool for anticipating situations. In that sense, reflection can be considered as a catalyst of “professional noticing,” a concept that can be used to express the ability to recognize and act on key indicators significant to one’s profession (Jacobs et al. 2010). STs’ capacity to attend, interpret, and decide—three interrelated skills of professional noticing—could naturally enable them to develop a network of concepts for considering practice expertly. Researchers have suggested that professional noticing can be influenced through reflection on video-recorded representations of practice (e.g., Schack et al. 2013; Van Es and Sherin 2010). Fourthly, the narrative character of enriched practical knowledge will support STs to meaningfully learn to generalize and objectify concepts, ideas, and beliefs about teaching mathematics and, conversely, enable them to recall the meanings of subjects and ideas if necessary. Last but not least, the character of the theory that STs are assumed to use may influence the nature and the level of their theory use. The STs involved in the research that is reported here were expected to understand and apply the theory of Realistic Mathematics Education (RME) (Freudenthal 1973, 1991). At elementary teacher preparation colleges in the Netherlands, STs receive practical training in schools that teach, to varying degrees, in the spirit of RME. In teacher education, they are introduced to local instruction theories (Gravemeijer 2004), which have developed within the framework of the RME theory. A local instruction theory comprises theories about potential learning processes for a given topic and the means of supporting those learning processes. Such a theory offers a description of, and a rationale for, the envisioned learning route and the use of a series of instructional activities for a specific topic (e.g., addition and subtraction up to 100, fractions, area). Natural use of such a theory in practice demands knowledge and skills that exceed the local theory itself (e.g., orchestrating a productive classroom discussion and establishing and maintaining an inquiring classroom culture) and it also demands knowledge of the underlying domain-specific theory (Gravemeijer 2008). In this research project, STs were introduced to a local instruction theory on teaching multiplication (Ter Heege 1985; Van den Heuvel-Panhuizen 2001; Oonk et al. 2015a).
The nature and level of theory use
The main goal of this study was to gain insight into the STs’ use of theoretical concepts in their reflections. We therefore investigated the following research question: In what way and to what extent do STs differ in their use of theory and how is this related to their prior education and their year of study?
- 1.
In what way do STs use theoretical knowledge when they describe practical situations after spending a period in a learning environment that invites the use of theory?
- 2.
What is the theoretical quality of statements made by the STs when they describe practical situations?
- 3.
To what extent is there a meaningful relationship between the nature and level of theory use and two variables: the STs’ prior mathematics education and the STs’ year of study?
Method
Context and participants
Complete overview of the population
Number of STs per study year | |
Year 1 | 42 |
Year 2 | 79 |
Year 3 | 148 |
Total STs | 269 |
Number of STs per type of course | |
Full-time | 146 |
Part-time | 54 |
Three year, dual | 15 |
Three year shortened | 12 |
Three year vwo | 42 |
Total STs | 269 |
Number of STs by prior education | |
Mbo without maths | 62 |
Mbo with maths | 17 |
Havo without maths | 31 |
Havo with maths | 96 |
Vwo without maths | 5 |
Vwo with maths | 51 |
Higher education | 2 |
Other | 5 |
Total STs | 269 |
The five, one-and-a-half hour, course meetings were directed by one of the 11 teacher educators. Part of the first meeting was used for an initial assessment; the fifth and final meeting contained a final assessment, which required an hour and a half extra. In total, the course consisted of 40 study hours, nine of which were contact hours with the teacher educator. One Pabo group offered the course as an option, giving the STs the choice of whether or not to follow it; in other groups, the teacher educator determined, in consultation with the researcher, how the course would best fit into the curriculum.
The teachers teaching the course were experienced teacher educators in at least the subject area Mathematics and Pedagogics; they had taken part in the training course that was developed within the framework of this study and taught by the researcher, i.e., the first author. The next section describes the learning environment of the STs and how the teacher educators were prepared to support them.
The learning environment of the student teachers
We created a learning environment which was expected to optimize the chance for STs to make use of theory and in which we could examine their theory use. The research literature describes a rich history of attempts to design this type of learning environment using the advantages of technology (Borko 2016; Brophy 2004; Gaudin and Chaliès 2015; Goldman et al. 2007; Herbst et al. 2011; Lampert and Ball 1998; Masingila and Doerr 2002; Sherin and Dyer 2017; Stockero 2008). The National Council of Teachers of Mathematics (2015) recommends that technology be used strategically, because this strengthens mathematics teaching and learning (Dick and Hollebrands 2011). Strategic use assumes a directed use of technology, especially within the context of instruction, but this does not mean continuous use of technology.
The design of the learning environment for the STs who participated in this large-scale study was adapted from initial studies of multimedia interactive learning environments (Dolk et al. 1996; Goffree and Oonk 2001; Oonk et al. 2004). It was largely similar to the STs’ learning environment in the earlier exploratory case study (Oonk et al. 2015a), which roughly means an environment centered on a set of teaching narratives suitable for multiplication instruction in grade 2. The narratives were organized in a digital format, referred to as “the Guide” (Goffree et al. 2003), encompassing 25 video clips (e.g., excerpts of lessons, interviews with children, teacher interviews) with related text (e.g., protocols of class discussions and diagnostic talks, teachers’ reflections on lesson preparation and evaluation, STs’ worksheets). The narratives were organized into 12 themes, which were formatted as questions (e.g., How do I start teaching multiplication? How can I use materials? How do STs differ? How do I organize a class discussion?). Each of the 25 narratives was accompanied by an expert reflective note, representing the theoretical background of the local instruction theory for multiplication. These notes provided STs with a written analysis of what they could observe in the video and of what the teacher or the pupils might have considered doing. Relevant theoretical concepts, schemes, and perspectives for teaching ideas were incorporated meaningfully into the text. An important integral component of the guide was a vocabulary presented as a collection of 59 concepts covering the local instruction theory on teaching multiplication in grades 1 and 2. The concepts (e.g., manipulatives, exercizing, memorizing, commutative property, diagnosis, explaining, model, core objectives, learning strand) originated from the 25 expert reflections on the video clips. They were formatted in the reflective notes as interactive links to provide STs with additional information. The concepts were developed by the team of experts who wrote the reflections, including the first author of this article. The criterion for selection was conformity with the concepts used in the Dutch Learning-Teaching Trajectory for mathematics teaching (Van den Heuvel-Panhuizen 2001) and in the standards for mathematics teacher education (Goffree and Dolk 1995).
A few components of the learning environment were adapted or added based on experiences from the small-scale study (Oonk et al. 2015a). This involved, for example, the initial assessment (see “The instruments” section) and a “logbook activity”—entitled “What (else) did you learn in this meeting?”—designed to make STs even more aware of their own learning or increase in learning. The teacher educators were provided with a detailed manual and were given a day of training (see next section). The stated assumption was that, with these adaptations and additions to the learning environment as used in the small-scale study, the STs’ use of theory could be optimized.
Training the teacher educators
Primary mathematics teacher educators in the Netherlands have formed a close network since the 1970s, resulting, among other things, in a considerable consensus of opinion about learning to teach mathematics (Goffree and Dolk 1995; Goffree and Oonk 1999, 2001; Oonk et al. 2019). At the annual conference of the network, the first author of this article provided information about the content of this research project as well as information about the conditions for participation in the study.These conditions were the mandatory and conscientious use of the course materials offered, including the course in the regular curriculum; mandatory training for teacher educators; certification; at least 2-year experience as a mathematics teacher educator; the size of ST groups; the number of meetings; and the number of contact hours. These conditions were mentioned again in the flyer that was handed out at the conference and distributed electronically over the national network. The 12 Pabos that participated showed a geographic spread across the Netherlands. One Pabo dropped out during the study due to organizational problems.
A mandatory day of training for the teacher educator in advance of the study was organized under supervision of the first author. The goal of the training was to optimize the analogy with working with STs by the various teacher educators. Results from the earlier development and research were used to inform teacher educators about how to introduce the Guide as a tool for the discourse and how to create an appropriate investigation context for student teachers. The information was described in a teacher educator’s manual. The manual contained detailed guidelines for each meeting. These guidelines concerned the goals of the meeting, the organization, the subject-specific and course-pedagogical content, suggestions for the STs and aspects that were vital for obtaining valid research data, such as the exact instruction for filling in the lists of concepts and handing out the assignments for the assessments.
The instruments
Initial assessment
The reflective note at the start of the course was intended to test the level at which the STs used theory within a specific category of the nature of theory use (factual description, interpretation, explanation and response to situations) (see the section above on the nature and level of theory use). The four assignments for four different video-recorded teaching situations had been phrased so that they would evoke these four types of theory use in turn. For instance, in the first assignment, the STs were asked to observe student Chantal and then, in their own words, give a factual description of what occurred in that situation. This part of the initial assessment yielded two types of data: primarily the number of theoretical concepts that each ST used in doing the assignments and also statements by STs in which theoretical concepts were used.
Final assessment
The four practical situations presented in the video material and the situation for the final assessment were selected from the lessons about learning the multiplication tables in grade 2, with the same students and teachers for all situations. The situation that was selected for the final assessment was new to the STs. They were given a short explanation about the context of the situation and where the video clip of the situation could be found; there was also some advice on writing the reflection. The large scale of the study necessitated limiting the use of tools and data to those of the written reflections in the initial and final assessments.
Procedure and data collection
The initial assessment was done during the first meeting of the course offered to the STs, with the purpose of determining the number of theoretical concepts used, as well as the level per category for the nature of theory use (factual description, interpretation, explanation, response to situations). For the number of concepts, a distinction was made into the total amount of concepts, the number of different concepts, the number of pedagogical content concepts, and the number of general pedagogical concepts.
The final assessment was performed in the final meeting of the course. This established the nature and level of theory use at the end of the course, as well as the number of theoretical concepts used. Just as for the initial assessment, subcategories were created for the number of concepts, the number of different concepts, the number of pedagogical content concepts, and the number of general pedagogical concepts.
The following variables served as background variables: the institute (the Pabo) at which the ST studied; the ST’s prior education; the kind of course the ST was taking (full-time, part-time, shortened); the study year; the group (class) the ST was in; small or large group; gender; and the primary school group in which the STs did their teaching practice.
Analysis
Reflection analysis instrument
The constructs of nature of theory use and level of theory use (Figs. 1 and 2) formed the Reflection Analysis Instrument (Oonk et al. 2015a) with which the theory use of the 269 STs was analyzed. The reflective notes from the initial assessment and, especially, from the final assessment functioned as the sources of data.
Where possible, the structure imposed on the text by the ST was taken into account. Sometimes, the units to be distinguished were already visible through white space or paragraph demarcations. Syntax also offered support for separating text into meaningful units. For example, words such as “furthermore” or “also” were often indications that a sentence should be part of a preceding sentence or paragraph. When there was doubt about unitizing a text, we chose to keep the text as one single unit.
The theoretical concepts were not only identified in the literal senses (the 59 indicated concepts), but also conceptually, as synonyms or descriptions with the same meaning as the “mother concept,” depending on how the STs used these “derived concepts” meaningfully within the given context.
The discussions about validating the identification and coding of the meaningful units occurred during two sessions between the first author and a second expert.
The conversations were transcribed. Using a random sample of 15 STs out of 269—i.e., amply 5% for statistical reasons with regard to weighing requirements for practicability versus representativeness—the interrater reliability was determined at 81%. The discussion on the remaining differences led to full agreement between the experts.
The Reflection Analysis Instrument (Figs. 1 and 2) for coding and categorizing the nature and level of theory use (see examples in Fig. 4) was tested to determine the interrater reliability. The Cohen’s Kappa coefficients (Cohen 1960)^{3} for the nature and the level of theory use were .80 and .86, respectively. For the combination of nature and level, the outcome was κ = .77.
The data
Statistics units in final assessment
Number valid STs | 246 |
Number missing STs | 23 |
Number meaningful units | 1740 |
Mean | 7.07 |
Median | 7.00 |
Mode | 6 |
Std. deviation | 1.79 |
Minimum | 4 |
Maximum | 13 |
Results
The general research question was: In what way and to what extent do STs differ in their use of theory and how is this related to their prior education and their year of study?
In the following sections, we discuss the results of the three sub-questions.
The first research question
Number of meaningful units in the final assessments of the STs that could be allocated to the four categories for the nature of theory use. Number of STs: 246. Total number of meaningful units: 1740
Factual description (Cat. A) | Interpretation (Cat. B) | Explanation (Cat. C) | Response to situations (Cat. D) | |
---|---|---|---|---|
Number of meaningful units | 440 | 202 | 730 | 368 |
Mean per ST (St. Dev.) | 1.8 (1.8) | 0.8 (1.3) | 3.0 (2.0) | 1.5 (1.5) |
Mean percentage per ST (St. Dev.) | 25 (25) | 12 (18) | 42 (28) | 21 (21) |
Number of STs in the final assessments that mainly used (≥ 50%) one of the four categories for nature of theory use. Number of STs: 239. Total number of meaningful units: 1740
Factual description Cat. A ≥ 50% | Interpretation Cat. B ≥ 50% | Explanation Cat. C ≥ 50% | Response to situations Cat. D ≥ 50% | |
---|---|---|---|---|
Number of STs | 42 | 16 | 104 | 28 |
Percentage of total number STs | 18 | 7 | 44 | 12 |
We found that about 81% of the STs mainly used (> 50%) one of the four categories. Furthermore, the ranking of the categories (Table 4) matched that of the categories in Table 3. Here too, the relatively high percentage of the category explanation (Cat. C 44%) stands out.
The second research question
The second research question concerned the level of theory use: What was the theoretical quality of statements made by the STs when they were describing practical situations?
Number of meaningful units in the final assessments of the STs that could be allocated to the four categories for level of theory use. Number of STs: 246. Total number of meaningful units: 1740
Level 1 | Level 2 | Level 3 | Level 4 | |
---|---|---|---|---|
Number of meaningful units | 603 | 512 | 625 | 0 |
Mean per ST (St. Dev.) | 2.4 (1.8) | 2.1 (1.3) | 2.5 (1.8) | 0 |
Mean percentage per ST | 35 | 2 | 36 | 0 |
Number of STs in the final assessments that mainly used (≥ 50%) one of the four categories for level of theory use. Total number of STs: 246
Percentage level 1 ≥ 50% | Percentage level 2 ≥ 50% | Percentage level 3 ≥ 50% | Percentage level 4 ≥ 50% | |
---|---|---|---|---|
Number of STs | 73 | 38 | 73 | 0 |
Percentage of total number STs (239) | 30 | 16 | 30 | 0 |
The third research question
The third research question was: To what extent is there a meaningful relationship between the components of theory use and the variables STs’ prior mathematics education or STs’ year of study?
Correlation between nature of theory use and prior education
Beta | Sig. | |
---|---|---|
Correlation factual description (A) and prior education | ||
Percentage factual description and prior education | − 0.131 | 0.041 |
Percentage factual description ≥ 50 and prior education | − 0.195 | 0.003 |
Percentage factual description and mbo without mathematics | 0.130 | 0.041 |
Percentage factual description ≥ 50 and mbo without math | 0.160 | 0.013 |
Percentage factual description and vwo with mathematics | − 0.099 | 0.127 |
Percentage factual description ≥ 50 and vwo with mathematics | − 0.106 | 0.103 |
Correlation interpretation (B) and prior education | ||
Percentage interpretation and prior education | − 0.129 | 0.043 |
Percentage interpretation ≥ 50 and prior education | − 0.042 | 0.514 |
Percentage interpretation and mbo without mathematics | 0.092 | 0.151 |
Percentage interpretation and vwo with mathematics | − 0.043 | 0.498 |
Correlation explanation (C) and prior education | ||
Percentage explanation and prior education | 0.243 | 0.000 |
Percentage explanation ≥ 50 and prior education | 0.275 | 0.000 |
Percentage explanation and mbo without mathematics | − 0.202 | 0.001 |
Percentage explanation ≥ 50 and mbo without mathematics | − 0.246 | 0.000 |
Percentage explanation and vwo with mathematics | 0.138 | 0.031 |
Percentage explanation ≥ 50 and vwo with mathematics | 0.149 | 0.021 |
Percentage responding (D) and prior education | ||
Percentage responding and prior education | − 0.051 | 0.426 |
Percentage responding ≥ 50 and prior education | − 0.037 | 0.573 |
Realizing these expectations and considering the results of the previous research questions, we assume that STs who have a higher level of prior mathematics education would more often tend to explain, to respond to situations, or to reason at level 3, while, on the other hand, factual description and interpreting or reasoning at levels 1 or 2 would mostly correlate with a lower level of prior mathematics education. The same characteristics applied to STs in the third study year as to STs in the first and second year of their study.
Linear regression analysis confirmed most of the assumptions above. Furthermore, as expected, we found a significant positive correlation between the number of concepts and level three of theory use (p < 0.05), also for the different theoretical concepts used, all of them part of the vocabulary of 59 concepts (see Appendix). A stronger relationship was recorded between the number of concepts and the level of theory use in the final assessment than in the initial one. Notable was the strong positive correlation between explanation and the number of general pedagogical concepts but the absence of any correlation between explanation and the number of pedagogical content concepts used. The content-related differences and the difference in reach between general pedagogical and pedagogical content concepts may have played a part. STs tend to initially respond in general terms to teaching situations. This is understandable, since the general pedagogical vocabulary is aimed more at the whole of the pedagogical-didactical actions of teacher and students, and is also used more frequently in teacher training and practice.
A remarkable exception to the assumption that STs who have a higher level of prior mathematics education would more often tend to respond to situations concerned the category response to situations (D) (Table 7). The possible cause of that deviation may be the relative low number of STs with a higher level of prior mathematics education (Table 1).
Correlation level of theory use and study year
Beta | Sig. | |
---|---|---|
Correlation level 1 and study year | ||
Percentage level 1 and study year 1 | 0.153 | 0.017 |
Percentage level 1 ≥ 50 and study year 1 | 0.170 | 0.008 |
Percentage level 1 and study year 2 | − 0.187 | 0.003 |
Percentage level 1 ≥ 50 and study year 2 | − 0.159 | 0.013 |
Percentage level 1 and study year 3 | 0.062 | 0.329 |
Percentage level 1 ≥ 50 and study year 3 | 0.034 | 0.601 |
Correlation level 3 and study year | ||
Percentage level 3 and study year 1 | − 0.124 | 0.053 |
Percentage level 3 ≥ 50 and study year 1 | − 0.078 | 0.220 |
Percentage level 3 and study year 2 | 0.221 | 0.000 |
Percentage level 3 ≥ 50 and study year 2 | 0.253 | 0.000 |
Percentage level 3 and study year 3 | − 0.115 | 0.072 |
Percentage level 3 ≥ 50 and study year 3 | − 0.141 | 0.027 |
Conclusion and discussion
To investigate STs’ competence in integrating theory and practice of mathematics teaching, a learning environment was designed to evoke theory use in their reflections on practice. We distinguished two dimensions of theory use: the nature and level of theory use.
Theory and practice: enriching practical knowledge
First of all, the study showed that each meaningful unit in the STs’ final assessment, in total 1740 and on average seven per ST, could be interpreted using one of the characteristics for nature, and one of the characteristics for level of theory use. Nearly all STs, 98% of the population, used theory in their final assessment, on average 12 theoretical concepts per ST, which means that most STs were able to relate theory and practice in the context of the learning environment offered.
What does this result mean for the STs’ competence in integrating theory and practice, that is, their competence in acquiring theory-enriched practical knowledge (TEPK)? Considering the process of enriching practical knowledge as a process of increasingly integrating theory with existing practical knowledge, we found a spectrum of theory use from STs. This ranged from STs who described zero meaningful units with a theoretical concept in the reflective note of their final assessment (3 STs out of 246), to STs for whom we categorized ≥ 50% of the meaningful units in their assessment as level 3 of theory use (73 STs out of 246). This level of theory use can be understood as practical theorizing (Ruthven 2001) or what Simon (1995) considers as the beginning of developing a hypothetical learning trajectory.
What stood out was the absence of any meaningful units that could be adjudged to be level 4 of theory use. This means that none of the 246 STs’ reflective notes showed developing a new relation between relations within the structure of a network of relations between concepts. This result is consistent with earlier findings. However, Oonk et al. (2015a) found that while theory use did not occur at level four in the written reflections, it really did happen during video stimulated recall interviews. Expressing thoughts that integrate theoretical concepts in writing appears to be something that requires more or other specific skills than is the case for thoughts that are expressed orally during interaction and interviews. The yield of oral reflections is often higher than that of written ones (Jaworski 2006). Theory use may be particularly evoked by activities where oral input is natural. This argues in favor of a variety of written and oral activities, also in assessments.
Another striking finding of this study was the important influence of prior mathematics education on the nature and level of theory use, especially the low results of the third year STs in their use of theory at level 3. This is all the more remarkable because the local instruction theory offered on teaching multiplication was relatively easy compared with other theories, for example the theory on teaching fractions. The outcome may have consequences for the design of the teacher education curricula and for the intake of first-year STs (see next section).
This study was limited to a certain extent by choices that were made. One example of a limitation was the context in which the study was conducted. It was not the STs’ own teaching practice that was at the center of the study, but “practice” for the student teachers consisted mainly of practice situations that were represented in multimedia form. Despite all the advantages of the multimedia practice, for example professional development (Gaudin and Chaliès 2015), theoretical enrichment (Oonk et al. 2004), opportunities for learning (Sherin and Dyer 2017), or the development of a reflective stance (Stockero 2008), the question remains whether situations from the student teachers’ own practice as an object of discussion and reflection would not have led to a better insight into making connections between theory and practice. It is possible that focusing on their own practice may have helped the STs to successfully go through the “process of appropriation” (Bakhtin 1981) of TEPK. Many student teachers forget the theory presented at university and revert to the “norm” of behavior in that school, or department. However, it is just in the real practice of teaching that STs can become particularly aware of theory as a necessary instrument for reflection on their thinking and actions, with as its goal understanding and responding adequately to situations.
Another example of the limitations of this study was in the collection of data. The nature of the data collection, mainly consisting of reflective notes, may have limited the STs’ insight into some aspects of the use of theory.
Implications for teacher education
We found that most of the STs in this study integrate theory and practice in a natural way by acquiring “theory-enriched practical knowledge” (TEPK) if they are in an adequately equipped multimedia learning environment, aimed at integrating theory and practice. The learning environment in this study was, for a variety of reasons, a vital component of the teacher education curriculum offered: vital because it enabled STs to become conscious of phenomena of real teaching practice and to acquire TEKP in a way they could rarely experience in their own teaching practice. Some characteristics of this multimedia learning environment (MLE) could qualify for general application in MLEs for teacher education. First, STs could freely surf in a conveniently arranged, “rich” collection of video-recorded real teaching practices around the course theme, together with expert reflections on these practices. Former studies (e.g., Brophy 2004; Goldman et al. 2007; Lampert and Ball 1998) have shown that STs need a clear view to survey the environment; the recorded teaching practices alone will not naturally prompt them to use theory. New visions on using digital tools may provide ideas about the integration of digital technology in teaching and learning (Drijvers 2019; Kang and Van Es 2018). Second, the vocabulary of 59 concepts covering the local instruction theory of the theme in this study (teaching multiplication) was a tool that supported STs at different levels. It was a dictionary in all parts of the course and an advance organizer (Ausubel 1968). It also served as a tool to gauge their learning process when they indicated in a special format (see Appendix) which concepts were known or unknown to them and which were meaningful in the context of a practice story and the source of that story (own practice, literature, videoclips, lectures, and workshops). Third, the study showed that the role of the teacher educator is crucial to stimulating improvements in level. The teacher educator has the expertise to theorize, to evoke, and to stimulate theory use, by for instance selecting adequate video fragments, asking challenging questions, making use of differences in argumentations, presenting confronting situations (Piaget 1974) and inspiring pedagogical conflicts, sharpening the discourse with theory-laden summaries, or by stimulating hypothetical thinking (Simon 1995). It is precisely the combination of these ingredients that can lead STs to expand their own repertoire through assimilation of the experts’ TEPK and through “adaptation and accommodation” (Oonk et al. 2004, p.152) enlarging their own repertoire by modifying the experts’ repertoire.
The training for the teacher educators provided them with knowledge and skills to enable them to perform the activities discussed, for example to serve as a model to “orchestrate” (Cobb et al. 2003) the discourse in class meetings (Kilpatrick et al. 2001). This included supporting the STs in using the set of 59 theoretical concepts as a vital vocabulary (Sfard 2008) for their thinking and reasoning about the teaching practice presented on the video. In doing so, they made hidden practical knowledge explicit and enriched it with theory aimed at acquiring a network of relations among concepts, i.e., acquiring TEPK. Such networks may be comparable with what Vergnaud (1983) describes as the context in which students learn in terms of “mastering situations” to produce a mastered collection of situations which he calls a “conceptual field.” The learner (e.g., maths student) “masters” a conceptual field if he or she masters several concepts of a different nature. Lampert (2001) applies the idea of “conceptual field” not just to learning but also to teaching. She considers her teaching approach as facing students with conceptual fields. In this study, STs had to acquire a rather complex “field,” a kind of “conceptual web,” as a web of relations among relations between 59 concepts.
The ST’s own teaching practice was part of the ST’s course activity, but the focus in this study was on analyzing and discussing representations of practice. However, these activities can have a positive influence on STs’ attitude to mathematics (Oonk and De Goeij 2006) and their teaching practice. They may support STs to elicit and respond to students’ ideas and to learn how to enact aspects of practice in complex situations, activities that can be improved through theoretical reflection on their practice. Pedagogies of investigation and of enactment are necessary if teachers are to develop classroom practices that focus on student reasoning (Grossman et al. 2009), especially if they are to scaffold students’ language required for mathematical learning (Smit et al. 2016).
The approach of integrating theory and practice, as demonstrated in this study, has been incorporated into a series of five books for primary mathematics teacher education in the Netherlands. In these books, teaching practices are the starting point for theoretical reflections and tools to recall in the discourse, lists of concepts are means for teacher educators and STs to support and to judge the processes of learning to integrate theory and practice and, an accompanying website provides STs and teacher educators with access to video episodes, tasks, and guides (e.g., Oonk et al. 2015b, 2017). The STs and teacher educators involved appreciate learning and teaching in this way. However, it demands a high level of “doing and understanding”.
To really support the integration of theory and practice, we have to rethink our vision on learning and teaching mathematics, for example to discuss the need for a theory of reflective practice that enriches practice with theoretical knowledge. But this is not the only effort we have to make. Gravemeijer et al. (2017) answered the question of how mathematics education might prepare students for the society of the future with a list of propositions. Their proposition that choosing to aim for twenty-first century skills and high-level conceptual understanding requires a significant effort in teacher professionalization, curriculum design, and test design serves as an incentive. It may encourage the design of learning environments that students, student teachers, and teacher educators evoke to invent “conceptual fields” (Vergnaud 1983) at different levels of learning and teaching mathematics.
Footnotes
Notes
References
- Akkerman, S., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research, 81, 132–169. https://doi.org/10.3102/0034654311404435.Google Scholar
- Ausubel, D. P. (1968). Educational psychology: a cognitive view. New York: Holt, Rinehart & Winston.Google Scholar
- Bakhtin, M. M. (1981). The dialogic imagination: four essays (C. Emerson & M. Holquist, Trans.). M. Holquist (Ed.). Austin: University of Texas Press.Google Scholar
- Bales, R. F. (1951). Interaction process analysis: a method for the study of small groups. Cambridge: Addison-Wesley Press.Google Scholar
- Ball, D. L. (2000). Bridging practices: intertwining content and pedagogy in teaching and learning to teach. Journal of Teacher Education, 51(3), 241–247. https://doi.org/10.1177/0022487100051003013.Google Scholar
- Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29, 14–22 43–46.Google Scholar
- Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554.Google Scholar
- Borko, H. (2016). Methodological contributions to video-based studies of classroom teaching and learning: a commentary. ZDM Mathematics Education, 48(1–2), 213–218. https://doi.org/10.1007/s11858-016-0776-x.Google Scholar
- Brophy, J. (Ed.). (2004). Advances in research on teaching: using video in teacher education (Vol. 10). New York: Elsevier Science.Google Scholar
- Clandinin, D. J., & Connelly, F. M. (1996). Teachers’ professional knowledge landscapes: teacher stories—stories of teachers—school stories—stories of schools. Educational Researcher, 25(3), 24–30. https://doi.org/10.3102/0013189X025003024.Google Scholar
- Clark, C. M., & Lampert, M. (1986). The study of teacher thinking: implications for teacher education. Journal of Teacher Education, 37(5), 27–31. https://doi.org/10.1177/002248718603700506.Google Scholar
- Cobb, P. (1988). The tension between theories of learning and instruction in mathematics education. Educational Psychologist, 23(2), 87–103. https://doi.org/10.1207/s15326985ep2302_2.Google Scholar
- Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. https://doi.org/10.3102/0013189X032001009.Google Scholar
- Cochran-Smith, M., & Lytle, S. L. (1999). Relationships of knowledge and practice: teacher learning in communities. Review of Research in Education, 24, 249–306. https://doi.org/10.2307/1167272.Google Scholar
- Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20(1), 37–46. https://doi.org/10.1177/001316446002000104.Google Scholar
- Cohen, D. K. (2011). Teaching and its predicaments. Cambridge: Harvard University Press. https://doi.org/10.4159/harvard.9780674062788.Google Scholar
- Da Ponte, J. P. (2013). Theoretical frameworks in researching mathematics teacher knowledge, practice, and development. Journal of Mathematics Teacher Education, 16, 319–322. https://doi.org/10.1007/s10857-013-9249-0.Google Scholar
- Dewey, J. (1904). The relation of theory to practice in education. In C. A. McMurry (Ed.), The third yearbook of the National Society for the Scientific Study of Education. Part I: The relation of theory to practice in the education of teachers (pp. 9–30). Chicago: University of Chicago Press https://archive.org/details/r00elationoftheorynatirich/.Google Scholar
- Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: technology to support reasoning and sense making. Reston: NCTM.Google Scholar
- Dolk, M., Faes, W., Goffree, F., Hermsen, H., & Oonk, W. (1996). A multimedia interactive learning environment for (future) primary school teachers with content for primary mathematics teacher education programs. Utrecht, the Netherlands: Freudenthal Institute/National Association for the Development of Mathematics Education.Google Scholar
- Drijvers, P. (2019). Embodied instrumentation: combining different views on using digital technology in mathematics education. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.Google Scholar
- Elbaz, F. (1981). The teacher’s “practical knowledge”: report of a case study. Curriculum Inquiry, 11(1), 43–71. https://doi.org/10.2307/1179510.Google Scholar
- Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, the Netherlands: Reidel.Google Scholar
- Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.Google Scholar
- Freudenthal, H. (1987). Theorievorming bij het wiskundeonderwijs. Geraamte en gereedschap [Formation of theory in mathematics education. Framework and tools]. Tijdschrift voor nascholing en onderzoek van het reken-wiskundeonderwijs, 5(3), 4–15.Google Scholar
- Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, the Netherlands: Kluwer Academic Publishers.Google Scholar
- Gaudin, C., & Chaliès, S. (2015). Video viewing in teacher education and professional development: a literature review. Educational Research Review, 16, 41–67. https://doi.org/10.1016/j.edurev.2015.06.001.Google Scholar
- Goffree, F., & Dolk, M. (Eds.). (1995). Standards for mathematics education (R. Rainero, Trans.)[translation of a portion of F. Goffree & M. Dolk (Eds.). (1995). Proeve van een nationaal programma voor rekenen-wiskunde & didactiek op de Pabo]. Utrecht, the Netherlands: NVORWO/ SLO. Retrieved from http://www.fi.uu.nl/publicaties/literatuur/1063.pdf
- Goffree, F., & Oonk, W. (1999). Teacher education around the world. Educating primary school mathematics teachers in the Netherlands: back to the classroom. Journal of Mathematics Teacher Education, 2(2), 207–214. https://doi.org/10.1023/A:1009903205316.Google Scholar
- Goffree, F., & Oonk, W. (2001). Digitizing real practice for teacher education programmes: the MILE approach. In F.-L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 111–145). Dordrecht, the Netherlands: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-010-0828-0_6.Google Scholar
- Goffree, F., Markusse, A., Munk, F., & Olofsen, K. (2003). Gids voor rekenen/wiskunde. Verhalen uit groep 4, versie 2003 [Guide for mathematics. Stories from grade 2, version 2003]. Groningen, the Netherlands: Wolters-Noordhoff.Google Scholar
- Goldman, R. P., Barron, B., & Derry, S. (Eds.). (2007). Video research in the learning sciences. Mahwah: Lawrence Erlbaum.Google Scholar
- Goos, M., & Bennison, A. (2018). Boundary crossing and brokering between disciplines in pre-service mathematics teacher education. Mathematics Education Research Journal, 30, 255–275. https://doi.org/10.1007/s13394-017-0232-4.Google Scholar
- Gravemeijer, K. P. E. (2004). Local instruction theories as a means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. https://doi.org/10.1207/s15327833mtl0602_3.Google Scholar
- Gravemeijer, K. (2008). RME theory and mathematics teacher education. In D. Tirosh & T. Wood (Eds.), Tools and processes in mathematics teacher education (pp. 283–302).Google Scholar
- Gravemeijer, K., Stephan, M., Julie, C., Lin, F., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education, 15(Suppl 1), S105–S123. https://doi.org/10.1007/s10763-017-9814-6.Google Scholar
- Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. W. (2009). Teaching practice: a cross-professional perspective. Teachers College Record, 111(9), 2055–2100.Google Scholar
- Herbst, P., Chazan, D., Chen, C.-L., Chieu, V.-M., & Weiss, M. (2011). Using comics-based representations of teaching, and technology, to bring practice to teacher education courses. ZDM–The International Journal on Mathematics Education, 43(1), 91–103. https://doi.org/10.1007/s11858-010-0290-5.Google Scholar
- Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.Google Scholar
- Jaworski, B. (2006). Theory and practice in mathematics teaching development: critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), 187–211. https://doi.org/10.1007/s10857-005-1223-z.Google Scholar
- Kaiser, B., Schwarz, B., & Krackowitz, S. (2007). The role of beliefs on future teacher’s professional knowledge [Monograph]. The Montana Mathematics Enthusiast, 3, 99–116.Google Scholar
- Kang, H., & van Es, E. A. (2018). Articulating design principles for productive use of video in preservice education. Journal of Teacher Education. https://doi.org/10.1177/0022487118778549.
- Keijzer, R., & Kool, M. (2012). Mathematical knowledge for teaching in the Netherlands. Paper presented at the 12th International Conference on Mathematics Education, Seoul, South Korea. Retrieved from http://dspace.library.uu.nl/handle/1874/272294
- Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
- Korthagen, F. A. J. (2001). Linking theory and practice: the pedagogy of realistic teacher education. Mahwah: Lawrence Erlbaum Publishers.Google Scholar
- Korthagen, F. A. J. (2010). How teacher education can make a difference. Journal of Education for Teaching: International Research and Pedagogy, 36(4), 407–423. https://doi.org/10.1080/02607476.2010.513854.Google Scholar
- Krippendorff, K. (2004). Content analysis: an introduction to its methodology (2nd ed.). Thousand Oaks: Sage.Google Scholar
- Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press. https://doi.org/10.1023/A:1015870009117.Google Scholar
- Lampert, M. (2010). Learning teaching in, from, and for practice: what do we mean? Journal of Teacher Education, 61(1–2), 21–34. https://doi.org/10.1177/0022487109347321.Google Scholar
- Lampert, M., & Ball, D. L. (1998). Teaching, multimedia, and mathematics: investigations of real practice. New York: Teachers College Press.Google Scholar
- Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture, and Activity, 3, 149–164. https://doi.org/10.1207/s15327884mca0303_2.Google Scholar
- Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommend dations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349–371. https://doi.org/10.1007/s10649-006-9071-z.Google Scholar
- Lerman, S. (2013). Theories in practice: mathematics teaching and mathematics teacher education. ZDM–The International Journal on Mathematics Education, 45(4), 623–631. https://doi.org/10.1007/s11858-013-0510-x.Google Scholar
- Lin, P.-J. (2002). On enhancing teachers’ knowledge by constructing cases in classrooms. Journal of Mathematics Teacher Education, 5(4), 317–349. https://doi.org/10.1023/A:1021282918124.Google Scholar
- Masingila, J. O., & Doerr, H. M. (2002). Understanding pre-service teachers’ emerging practices through their analyses of a multimedia case study of practice. Journal of Mathematics Teacher Education, 5(3), 235–263. https://doi.org/10.1023/A:1019847825912.Google Scholar
- Mor, Y., & Noss, R. (2008). Programming as mathematical narrative. International Journal of Continuing Engineering Education and Life-Long Learning (IJCEELL), 18(2), 214–233. https://doi.org/10.1504/IJCEELL.2008.017377.Google Scholar
- Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: what can pre-service teachers learn. Journal for Research in Mathematics Education, 40(5), 491–529.Google Scholar
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: Author.Google Scholar
- National Council of Teachers of Mathematics. (2015). Strategic use of technology in teaching and learning mathematics. A position of the National Council of Teachers of Mathematics. Reston: Author.Google Scholar
- Nunes, T., Vargas Dorneles, B., Lin, P., & Rathgeb-Schnierer, E. (2016). Teaching and learning about whole numbers in primary school, ICME-13 Topical Surveys. https://doi.org/10.1007/978-3-319-45113-8_1
- Oonk, W., & De Goeij, E. T. J. (2006). Wiskundige attitudevorming [Educating a Mathematics Attitude]. Tijdschrift voor nascholing en onderzoek van het reken-wiskundeonderwijs, jaargang, 25(4), 37–39.Google Scholar
- Oonk, W., Goffree, F., & Verloop, N. (2004). For the enrichment of practical knowledge: good practice and useful theory for future primary teachers. In J. Brophy (Ed.), Advances in research on teaching: using video in teacher education (Vol. 10, pp. 131–168). New York: Elsevier Science. https://doi.org/10.1016/S1479-3687(03)10006-5.Google Scholar
- Oonk, W., van Zanten, M., & Keijzer, R. (2007). Gecijferdheid, vier eeuwen ontwikkeling: Perspectieven voor de opleiding [Numeracy, four centuries of development: Perspectives for teacher education]. Reken-wiskundeonderwijs: Onderzoek, Ontwikkeling, Praktijk [Primary Mathematics Education: Research, Development, Practice], 26(3), 3–18.Google Scholar
- Oonk, W., Verloop, N., & Gravemeijer, K. P. E. (2015a). Enriching practical knowledge: exploring student teachers’ competence in integrating theory and practice of mathematics teaching. Journal for Research in Mathematics Education, 46(5), 559–598. https://doi.org/10.5951/jresematheduc.46.5.0559.Google Scholar
- Oonk, W., Keijzer, R., Lit, S., Barth, F., Den Engelsen, J. F. M., Lek, A. T. E., & Van Waveren Hogervorst, C. (2015b). Rekenen-wiskunde in de praktijk-Kerninzichten [Mathematics in practice: Big ideas] (2nd ed.). Groningen, the Netherlands: Noordhoff Uitgevers bv.Google Scholar
- Oonk, W., Keijzer, R., Lit, S., & Barth, F. (2017). Rekenen-wiskunde in de praktijk-Verschillen in de klas. [Mathematics in practice: Differences in class] (2nd ed.). Groningen, the Netherlands: Noordhoff Uitgevers bv.Google Scholar
- Oonk, W., Keijzer, R., & Van Zanten, M. (2019). Mathematics & didactics as a subject in primary school teacher education in the Netherlands. In M. Van den Heuvel-Panhuizen, P. Drijvers, M. Doorman, & M. Van Zanten (Eds.), Reflections from inside on the Netherlands Didactic Tradition in Mathematics Education. Utrecht: Freudenthal Institute, Utrecht University.Google Scholar
- Pendlebury, S. (1995). Reason and story in wise practice. In H. McEwan & K. Egan (Eds.), Narrative in teaching, learning and research (pp. 50–65). New York: Teachers College Press.Google Scholar
- Piaget, J. (1974). La prise de conscience [The grasp of consciousness]. Paris: Presses Universitaires de France.Google Scholar
- Putnam, R. T., & Borko, H. (1997). Teacher learning: implications of new views of cognition. In B. Biddle, T. L. Good, & I. F. Goodson (Eds.), International handbook of teachers and teaching (Vol. II, pp. 1223–1296). Dordrecht: Kluwer Academic Publishers.Google Scholar
- Ruthven, K. (2001). Mathematics teaching, teacher education, and educational research: developing “practical theorizing” in initial teacher education. In F. Lin & T. Cooney (Eds.), Making sense of mathematics teacher education. Dordrecht, the Netherlands: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-010-0828-0_8.Google Scholar
- Ruthven, K. (2011). Conceptualising mathematical knowledge in teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching. Dordrecht, the Netherlands: Springer. https://doi.org/10.1007/978-90-481-9766-8_6.Google Scholar
- Ryve, A. (2011). Discourse research in mathematics education: a critical evaluation of 108 journal articles. Journal for Research in Mathematics Education, 42(2), 167–198.Google Scholar
- Schack, E. O., Fisher, M. H., Thomas, J. N., Eisenhardt, S., Tassell, J., & Yoder, M. (2013). Prospective elementary school teachers’ professional noticing of children’s early numeracy. Journal of Mathematics Teacher Education, 16(5), 379–397. https://doi.org/10.1007/s10857-013-9240-9.Google Scholar
- Schön, D. A. (1983). The reflective practitioner: how professionals think in action. New York: Basic Books.Google Scholar
- Sfard, A. (2008). Thinking as communicating: human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511499944.Google Scholar
- Sherin, M. G., & Dyer, E. B. (2017). Mathematics teachers’ self-captured video and opportunities for learning. Journal of Mathematics Teacher Education, 20, 477. https://doi.org/10.1007/s10857-017-9383-1.Google Scholar
- Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.3102/0013189X015002004.Google Scholar
- Silver, E. A., & Herbst, P. G. (2007). Theory in mathematics education scholarship. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 39–67). Charlotte: Information Age Publishing.Google Scholar
- Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. https://doi.org/10.2307/749205.Google Scholar
- Skott, J., Van Zoest, L., & Gellert, U. (2013). Theoretical frameworks in research on and with mathematics teachers. ZDM–The International Journal on Mathematics Education, 45(4), 501–505. https://doi.org/10.1007/s11858-013-0509-3.Google Scholar
- Smit, J., Bakker, A., Van Eerde, D., & Kuijpers, M. (2016). Using genre pedagogy to promote student proficiency in the language required for interpreting line graphs. Mathematics Education Research Journal, 28, 457. https://link.springer.com/article/10.1007/s13394-016-0174-2.Google Scholar
- Sriraman, B., & Kaiser, G. (2006). Theory usage and theoretical trends in Europe: a survey and preliminary analysis of CERME4 research reports. ZDM–The International Journal on Mathematics Education, 38(1), 22–51. https://doi.org/10.1007/BF02655904.Google Scholar
- Stigler, J. W., & Hiebert, J. (1999). The teaching gap: best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.Google Scholar
- Stockero, S. L. (2008). Using a video-based curriculum to develop a reflective stance in prospective mathematics teachers. Journal of Mathematics Teacher Education, 11(5), 373–394. https://doi.org/10.1007/s10857-008-9079-7.Google Scholar
- Ter Heege, H. (1985). The acquisition of basic multiplication skills. Educational Studies in Mathematics, 16(4), 375–388. https://doi.org/10.1007/BF00417193.Google Scholar
- Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics education—the Wiskobas project. Dordrecht, the Netherlands: Kluwer Academic Publishers.Google Scholar
- Van den Heuvel-Panhuizen, M. (Ed.). (2001). Children learn mathematics: a learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Utrecht, the Netherlands: Freudenthal Institute.Google Scholar
- Van Es, E. A., & Sherin, M. G. (2010). The influence of video clubs on teachers’ thinking and practice. Journal of Mathematics Teacher Education, 13, 155–176. https://doi.org/10.1007/s10857-009-9130-3.Google Scholar
- Van Hiele, P. M. (1973). Begrip en inzicht. Werkboek van de wiskundedidaktiek [Understanding and insight. Workbook of mathematics pedagogy]. Purmerend, the Netherlands: Muusses.Google Scholar
- Van Hiele, P. M. (1986). Structure and insight. A theory of mathematics education. Orlando: Academic Press.Google Scholar
- Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 128–174). New York: Academic Press.Google Scholar
- Verloop, N. (2001). Guest editor’s introduction. International Journal of Educational Research, 35(5), 435–440. https://doi.org/10.1016/S0883-0355(02)00002-2.Google Scholar
- Verloop, N., Van Driel, J., & Meijer, P. (2001). Teacher knowledge and the knowledge base of teaching. International Journal of Educational Research, 35(5), 441–461. https://doi.org/10.1016/S0883-0355(02)00003-4.Google Scholar
- Vescio, V., Ross, D., & Adams, A. (2008). A review of research on the impact of professional learning communities on teaching practice and student learning. Teaching and Teacher Education, 24, 80–91. https://doi.org/10.1016/j.tate.2007.01.004.Google Scholar
- Zeichner, K. M., Tabachnick, B. R., & Densmore, K. (1987). Individual, institutional, and cultural influences on the development of teachers’ craft knowledge. In J. Calderhead (Ed.), Exploring teachers’ thinking (pp. 21–59). London: Cassell.Google Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.