No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Using Knowledge of Children's Mathematics Thinking in Classroom Teaching: An Experimental Study
Abstract
This study investigated teachers' use of knowledge from research on children's mathematical thinking and how their students' achievement is influenced as a result. Twenty first grade teachers, assigned randomly to an experimental treatment, participated in a month-long workshop in which they studied a research-based analysis of children's development of problem-solving skills in addition and subtraction. Other first grade teachers (n = 20) were assigned randomly to a control group. Although instructional practices were not prescribed, experimental teachers taught problem solving significantly more and number facts significantly less than did control teachers. Experimental teachers encouraged students to use a variety of problem-solving strategies, and they listened to processes their students used significantly more than did control teachers. Experimental teachers knew more about individual students' problem-solving processes, and they believed that instruction should build on students' existing knowledge more than did control teachers. Students in experimental classes exceeded students in control classes in number fact knowledge, problem solving, reported understanding, and reported confidence in their problem-solving abilities.
... Building mathematics instruction on students' thinking is important for effective mathematics teaching (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989;National Council of Teachers of Mathematics, 2014). Instruction that builds on the students' thinking includes understanding students' existing knowledge and mathematical thinking by assessing their problem-solving processes and reaching appropriate instructional decisions (Carpenter et al., 1989). ...
... Building mathematics instruction on students' thinking is important for effective mathematics teaching (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989;National Council of Teachers of Mathematics, 2014). Instruction that builds on the students' thinking includes understanding students' existing knowledge and mathematical thinking by assessing their problem-solving processes and reaching appropriate instructional decisions (Carpenter et al., 1989). From this point of view, to provide instruction in which students' thinking is at the center, one of the key issues is the teachers' correct analysis of students' thinking (Carpenter et al., 1989;Jacobs, Franke, Carpenter, Levi, & Battey, 2007). ...
... Instruction that builds on the students' thinking includes understanding students' existing knowledge and mathematical thinking by assessing their problem-solving processes and reaching appropriate instructional decisions (Carpenter et al., 1989). From this point of view, to provide instruction in which students' thinking is at the center, one of the key issues is the teachers' correct analysis of students' thinking (Carpenter et al., 1989;Jacobs, Franke, Carpenter, Levi, & Battey, 2007). To be aware of how students think, teachers should be able to identify students' understanding, explore their misconceptions and misunderstandings and make inferences about their mathematical understanding. ...
This qualitative case study aimed to investigate prospective middle school mathematics teachers’ instructional responses on the basis of the students’ correct and incorrect functional thinking within the context of pattern generalization. The data were collected from thirty-two prospective teachers through a written task and semi-structured interviews and analyzed with open coding. The findings of the study revealed that most of the prospective teachers could support the functional thinking of the student who had an incorrect solution. However, they could not extend the student’s functional thinking of those who reached the correct solution. Instead, they asked student to do another similar drill or provided a general response to the student with the correct solution without extending their functional thinking.
... An increasingly prominent line of research in mathematics teacher education has examined teachers' understandings and practices related to children's mathematical thinking (i.e., children's problem-solving strategies, connections between strategies and problem structures, common confusions). This work, which often draws on the Cognitively Guided Instruction (CGI) research program (e.g., Carpenter et al., 1989;Fennema et al., 1996), has linked teachers' knowledge of children's mathematical thinking to productive changes in teachers' classroom practices and student learning. For instance, Fennema and colleagues (1996) found that as teachers learned about the development of children's problem-solving strategies in specific content domains, they began to use this knowledge to inform instructional decisions (e.g., lesson planning, problem selection). ...
... For instance, Fennema and colleagues (1996) found that as teachers learned about the development of children's problem-solving strategies in specific content domains, they began to use this knowledge to inform instructional decisions (e.g., lesson planning, problem selection). In turn, students demonstrated significantly higher levels of achievement on problemsolving tasks (Carpenter et al., 1989;Fennema et al., 1996). Jacobs, Lamb, and Philipp (2010) found that prior to coursework focused on mathematics teaching and learning, PSTs have a limited capacity for attending, interpreting, and responding to children's mathematical thinking. ...
... Journal of Urban Mathematics Education Vol. 9, No. 1 52 matical thinking. Research with practicing teachers has also indicated that connecting to children's mathematical thinking in instruction is a complex teaching practice that takes time to develop (Carpenter et al., 1989;Fennema et al., 1996). ...
In this article, the authors explore prospective elementary teachers' engagement with and reflection on activities they conducted to learn about a single child from their practicum classroom. Through these activities, prospective teachers learned about their child's mathematical thinking and the interests, competencies, and resources she or he brought to the mathematics classroom, and then wrote reports that included instructional suggestions as to next steps to further the child's growth in mathematics. The authors' analyses of these reports indicate that there were a variety of ways which prospective teachers made connections to one or more of their child's knowledge bases. In a high percentage of cases, prospective teachers attended to one of these knowledge bases, indicating that they were attending to particularities about their child and developing the dispositions to continue to do so. Implications for research and practice are discussed.
... One key component of student-centered pedagogy is learning through discussion (Carpenter et al., 1989;Cobb et al., 1997;Delaney, 1991;Helme & Clarke, 2001;Leikin & Zaslavsky, 1997;McKeachie & Kulik, 1975;Wade, 1994). These discussions may serve as instances of formative assessment, as they can expose inconsistencies or gaps in students' reasoning, which teachers can then use to guide future instruction (Gardee & Brodie, 2022;Kazemi & Stipek, 2001;Lampert, 2001;Leinhardt & Steele, 2005). ...
This study uses structural equation modeling to investigate the relationships between pre-service teachers’ (PSTs’) course-taking history, beliefs about mathematics, beliefs about students’ mathematical ability, and opinions about (1) how student errors should be addressed when they occur and (2) how much emphasis should be given to various forms of assessment. The results indicate that some types of courses are strongly associated with beliefs about the nature of mathematics. Specifically, PSTs who have taken more mathematics courses are more likely to see mathematics as a set of rules. PSTs’ views about mathematics strongly predict their beliefs about who can do high-level mathematics, how they think student errors should be addressed, and which forms of assessment they think are most appropriate. Implications for teacher preparation programs are discussed.
... Previous research showed that students perform better if their teachers believe that students' existing knowledge should be built on [64]. Similar research showed that students performed better when their teachers participated in professional development programs where they discussed the importance of building on students' existing knowledge, and where they worked on implementing this approach [80]. Future research can also help find the most efficient ways of incorporating such training in teacher preparation and professional development. ...
Research carried out through the last 20 years gave us undeniable evidence that to learn anything we need to be active participants, not passive observers. One of the important aspects of learning physics is constructing explanations of physical phenomena. To support and guide students toward constructing their explanations, teachers need to be attentive and responsive to students’ explanations. To learn how physics teachers interpret and respond to students’ explanations we investigated pre- and in-service physics teachers’ responses to students’ written explanations of their answers to a complex physics problem. The survey administered to the participants included the problem statement and four authentic student explanations. The participants were asked to identify each student’s strengths and weaknesses and to provide a response to that student. We found that while the participants were successful in identifying productive and problematic aspects of student reasoning, they rarely built on student reasoning when responding to the students, mostly focusing on addressing problematic aspects. The paper discusses why this finding is important for physics teacher preparation programs and professional development programs.
... Teaching-learning variety As Carpenter et al. (1989) or Lipowsky and Rzejak (2012) have shown, participants should be given sufficient time to acquire or deepen new competencies at different levels and in different settings. ...
This paper presents the results of a quantitative study investigating the development of teachers’ attitudes and self-efficacy expectations for inclusive mathematics instruction in the context of an in-service training that was designed in a blended learning format and compared to an unsupported online offer. In the blended learning format, 101 teachers participated in an in-service training, designed as a combination of six workshops with concrete activities based on materials for inclusive mathematics teaching and intermittent practical phases with collaborative learning environments for children aged 6–12 years. The teachers’ attitudes toward inclusive mathematics instruction and self-efficacy expectations are collected by using published scales (KIESEL) and scales under development. The effects of the blended learning program are analyzed by the t test for dependent samples or the nonparametric Wilcoxon signed-rank test and compared to the effects of an online offer without further guidance and support. On the one hand, the results indicate the importance of the blended learning program in comparison with the unsupported online offer. On the other hand, they show insights into the relevance of the participants’ interaction with the materials during the concrete activities as well as of their try outs in and their reflection on teaching practice during the intermittent practical phases.
... One of the reasons for these difficulties is placing more emphasis on rule memorization and procedural calculations (Misailadou & Williams, 2003;Modestou & Gagatsis, 2007). Mathematics educators are trying to change their teaching practices to facilitate the development of students' proportional reasoning (Carpenter et al., 1989;Jacobson & Lehrer, 2000). In the accessible literature, a limited number of studies (Ben-Chaim et al., 1998;Çetiner, 2022) were found comparing the traditional teaching method with an alternative teaching method on proportion. ...
This study aimed to examine the effect of using the flipped classroom model on students’ proportional reasoning. In this context, the study focused on students’ achievement and solution strategies for proportional reasoning problems. The participants of the study were 56 seventh-grade students, who were determined by convenience sampling method. In line with the purpose of the research, the concept of proportion was taught to the experimental group using the flipped classroom model and to the control group through teaching in accordance with the mathematics curriculum. The data of the study were collected through the Proportional Reasoning Test. The results of the study showed that the flipped classroom model was more effective in terms of mathematics achievement than the teaching method in line with the curriculum. In addition, the experimental group students used more correct solution strategies and fewer incorrect solution strategies than the control group students while solving the problems.
... Teacher noticing of student thinking has been shown to enhance K-12 learning of mathematics (Carpenter et al., 1989;Empson & Jacobs, 2008). In order for teachers to learn to notice, they need access to high quality PD led by well-prepared facilitators. ...
... Providing content-focused professional development (PD) for mathematics teachers has been a federal policy pathway intended to improve teacher effectiveness and ultimately boost student mathematics performance in the United States (e.g., Every Student Succeeds Act, 2015). Prior empirical evidence also suggests that high-quality PD contributes to mathematics teachers' content knowledge for teaching (e.g., Copur-Gencturk, Plowman, & Bai, 2019;Franke et al., 2001;Garet et al., 2016;Jacob et al., 2017), quality of instructional practice (e.g., Garet et al., 2016;Jacobs et al., 2007;Kraft & Blazar, 2017), and student learning outcomes (e.g., Campbell & Malkus, 2011;Carpenter et al., 1989;Jacobs et al., 2007). ...
https://theconversation.com/ai-can-teach-math-teachers-how-to-improve-student-skills-218210
... To illustrate this with a specific example, let us consider a study by Carpenter and colleagues (1989), which features in many systematic literature reviews that seek to identify effective PD design features. Carpenter et al. (1989) investigated how teacher knowledge about children's understanding of mathematical topics improved their teaching practices and student achievement. Teachers were randomly assigned to either participate in an 80-hour PD program about children's mathematical thinking, or to a business-as-usual control group (i.e., no PD at all). ...
Scholarly efforts to identify core design features for effective teacher professional development have grown rapidly in the last 25 years. Many concise lists of design principles have emerged, most of which converge on a consensus of 5-7 presumably "effective" design features (e.g., collaborative tasks, active learning, focus on content). The proliferation and convergence of reviews create the impression that this consensus is based on strong evidence from large-scale, replicated and rigorously controlled research studies. We critique the empirical foundation on which conclusions about evidence-based design features for teacher professional development have been based, by the same evidential standards that have been adopted within this field of scholarly work. We conclude that the empirical foundations for these lists are problematic and that claims to methodological rigor are misleading as they are based on flawed inferences. We further argue that the ambition to identify general features of effective professional development is also problematic, and reflect on why, despite its weaknesses and potentially adverse consequences for research and practice, we as a field continue to herald this consensus. We call for greater focus on the development, testing and refinement of theories about teacher professional learning in order to advance understanding, policy and practice in the field. Policy-makers appreciate simple answers to complex questions. In the field of teacher professional development (PD), at least, the research community has tended to oblige them. Specifically, educational researchers have been producing lists of the core features of effective PD designs for a quarter century at least. Guskey (2003) reviewed 13 such lists published between 1995-2001, noting how claims about a consensus among researchers, professional development specialists, and policymakers already surfaced before the turn of the century (e.g., Hawley and Valli 1999). More recently, Desimone (2009), synthesized findings from the available literature at the time by concluding that "there is enough empirical evidence to suggest that there is in fact a consensus on a core set of features" (p. 183) for effective teacher professional development efforts: (a) a focus on subject matter content and how students learn that content, (b) collaboration and interaction with colleagues,
... 17-18) and attend to their thinking processes. Beyond the growing realization about the importance for teachers to pay attention to students' cognition (e.g., Carpenter et al., 1989), Erickson (2011) notes that there was also a growing realization about the need to attend to teacher thinking. From that realization, education researchers have spent time inquiring whether teachers recognize students' conceptions documented in research (Even, 1993) and associated it with conceptions of mathematical knowledge for teaching (Ball et al., 2008;Simon, 2006). ...
We contribute to the understanding of teacher noticing by focusing on what a teacher may notice in students' mathematical contributions in the context of problem‐based lessons. Complementing approaches to research on noticing that focus on individual teachers' perceptual, cognitive, or situated skills, this conceptual article offers four categories of perception as examples of affordances available in the practice of teaching mathematics through problems. These include (1) the familiar instructional situations available to frame the problem, and the possibility to see student's work as (2) responsive to the problem, (3) serviceable for the knowledge at stake, and (4) normative with respect to the instructional situation used to frame the problem. The article shows examples of how teachers recognize responsiveness, serviceability, and normativity of student contributions and calls for research that can further uncover how such recognition may matter in the practice of teaching.
... These taxonomies were based upon decades of research on how young children learn to perform operations on whole number (Carpenter, 1985;Carpenter et al., 1988Carpenter et al., , 1999. The stated aim of the CGI program in the 1980s was to incorporate scientific knowledge of how children learn mathematics into instructional practice by focusing teachers' attention on student thinking and providing them with principled frameworks for mathematics problem solving and student thinking (Carpenter et al., 1989;Carpenter and Fennema, 1992). The initial study of the CGI program in the 1980s reported that CGI was beneficial to both teachers and students and over the last thirty years teacher PD based on CGI has taken many different forms. ...
Information about program implementation provides critical information for the interpretation of results from randomized trials. The present study provides an evaluation of the implementation of a Cognitively Guided Instruction mathematics teacher professional development as part of a large scale randomized controlled trial with teachers in first- and second- grade in eleven elementary schools in two adjacent school districts. We developed a measure of fidelity of implementation and used it during the in-person training sessions to determine the extent to which the training program was implemented as intended. The results from this study suggest that the program was implemented with high fidelity providing context for interpretation of overall program outcomes on teachers and students.
... Mathematics tasks are related to the teacher's mathematical and pedagogical knowledge. According to "the ingredient Furthermore, teacher's knowledge about effective mathematical pedagogy influences their instructional practices (e.g., Simon & Shifter, 1991;Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). ...
... Standardized test scores are a fairly common measure of intervention effectiveness. In fact, almost 35 years ago, Carpenter et al. (1989) used mathematics standardized test scores as an outcome measure to determine the effects of a teacher professional development intervention. To this day, the effects on mathematics standardized measures are still commonly explored (e.g., Goldhaber et al., 2020;Heppen et al., 2017;Murphy et al., 2020). ...
Recent work has demonstrated that having students study worked examples and answer self-explanation prompts as part of their problem-solving practice improves learning on researcher-developed measures of mathematical proficiency. However, little work has been done to date to investigate whether these benefits translate to improvements on the types of standardized assessments typically used by school districts to assess proficiency and make curricular decisions. In the present study, we examined the impact of using worked-example and self-explanation prompt assignments on the performance of 5th-grade students in one urban district on two district-administered standardized tests: i-Ready and PARCC. Results indicate the approach effectively improves student mathematic scores on both standardized measures. This study provides policy-meaningful outcomes. The findings from the current study show that standardized assessment scores can be improved by merely changing the type of practice problems that students engage with. Interventions like this provide a pathway for underperforming schools and districts to improve student achievement through easy access points, which are low-cost and can be added to the curricula districts currently use.
... Studies have also emphasized that the more information teachers have about students' thoughts, the more the quality of teaching increases (Ball, 1997;Ball et al., 2008;Fennema et al., 1996). There is also evidence that teachers teach better in parallel with their knowledge of students' thoughts, and students achieve higher success (Carpenter et al., 1989;Fennema et al., 1993). Jacobs et al. (2010) state that teachers' professional noticing includes attending to students' strategies, interpreting students' understanding, and deciding how to respond based on students' understanding. ...
In studies over recent years, there has been an increasing interest in teachers’ predicting middle school students’ thinking processes. However, as far as we are aware, there are no studies examining students’ thinking in terms of mathematical thinking components. This study primarily aimed to determine the mathematical thinking of middle school students. Therefore, the study examined how six mathematics teachers and 24 preservice mathematics teachers (from first to fourth grade) predicted the mathematical thinking of 96 middle school students. In this context, the predictions were categorized according to the sub-components of mathematical thinking: conjecturing, specializing, justifying and convincing, and generalizing. Regarding the conjecturing, the teachers explained students’ prediction of their mathematical thinking in more detail than preservice teachers. Regarding the specializing, the study, both groups of teachers could not predict that the students could express different situations in their problem solutions. Within the scope of the justifying and convincing, the preservice teachers had different perspectives on problem solving compared to the teachers. In regard to the generalizing, teachers and preservice teachers made similar predictions but all groups from first to fourth grade lack experience for this component. It can be stated that preservice teachers’ interaction with more students will be effective in predicting students’ mathematical thinking. The same is true for teachers, as it is believed that greater experience will be beneficial.
... Since the beginning of presage-process-product research, and based on theoretical reflections on a subject-specific characterization of teacher cognitions in teaching, which were initiated in the U.S. in the late 1980s, the question of the theoretical conceptualization and empirically examination of teachers' professional knowledge has become increasingly important (e.g. Carpenter & Fennema, 1992;Carpenter et al., 1988Carpenter et al., , 1989Fennema et al., 1996;Neubrand, 2018;Petrou & Goulding, 2011;Rowland, 2014). The research initially sought to identify and isolate more general variables of successful teaching, but has since taken somewhat different forms. ...
Mathematics teaching is subject to cultural and temporal conditions. Not only do
school and societal conditions shift, and with them the composition of the student
body, but also curricular regulations and new mathematical and pedagogical insights
determine the content to be taught and the approach to learning used in mathematics classes. To reflect on mathematics teaching in a changing world, there is a need for continuous scientific research into this process of teaching mathematics. Results of this research also have a retrospective impact on mathematics teacher education insofar as the conditions of education need to be continuously adapted to the professional requirements of teachers in practice. Research on teaching mathematics thus bears a great responsibility and is a constantly evolving field of research for scholars around the globe.
This book comes at the time when the world is facing an ongoing global pandemic
and experiencing violence and unrest due to active war. This publication symbolizes a professional commitment and international collaboration par excellence apropos
teaching mathematics. The editors from three different continents and researchers
who represent sixteen institutions and eight countries worked constructively and
collaboratively with utmost respect for each other, with intentions to reflect on
existing research knowledge and to create new knowledge that can be shared and
used by other educators and researchers across the world.
In preparation for this book, our international group of researchers shared current
issues related to the evolution of research on teaching mathematics. We examined
the present state of research on mathematics teaching and discussed the theoretical and methodological challenges associated with it, including issues related to conceptualization, instrumentation, and design. Additionally, we explored the likely direction of future research developments. In our literature review and discussions on this project, it became evident that studies on teaching frequently establish direct relationships between units of analysis that, at first glance, cannot be assumed to be directly related in a chain of effects. There are examples of studies presented in this book that directly relate teacher competencies to student achievements using empirical measurement models in a causal or relational way. Without criticizing these studies across the board, however, it seems reasonable to consider moderating or intermediate variables in this chain of effects (Baron & Kenny, 1986), such as the initiated student learning activities observable by teachers in the classroom, aspects of instructional quality (e.g., classroom management or cognitive activation), or corresponding student variables such as attention and cooperation in class or students’ prior knowledge (e.g., Fig. 1).
Although there are researchers who do indeed study mediating variables (e.g.,
Blömeke et al., 2022), it became clear to us that there is a lack of a systematic
scientific overview of the complete chain of effects between teacher characteristics, activities, and students’ learning processes. Overviews of precisely these aspects of research on teaching and respective studies are scarce, which inspired this book.
... Since the beginning of presage-process-product research, and based on theoretical reflections on a subject-specific characterization of teacher cognitions in teaching, which were initiated in the U.S. in the late 1980s, the question of the theoretical conceptualization and empirically examination of teachers' professional knowledge has become increasingly important (e.g. Carpenter & Fennema, 1992;Carpenter et al., 1988Carpenter et al., , 1989Fennema et al., 1996;Neubrand, 2018;Petrou & Goulding, 2011;Rowland, 2014). The research initially sought to identify and isolate more general variables of successful teaching, but has since taken somewhat different forms. ...
To assess the effectiveness of teachers and teaching, it is necessary to develop an appropriate understanding of what makes a “good” teacher. According to the framework by Medley, this includes amongst others focusing on the knowledge, skills, and values that a teacher possesses. To appropriately describe these competencies, current research departs from a broad conceptualization of competence that includes dispositional aspects, such as mathematical content knowledge, pedagogical content knowledge, and general pedagogical knowledge. Furthermore, situation-specific skills that are related to school practice such as the perception of instructional quality, interpreting, and decision-making are considered. The chapter gives an overview of different conceptualizations of teachers’ professional competence used in mathematics education studies and describes the evolution of research on mathematics teachers’ competence over the last three decades. It concludes with theoretical and methodological challenges that research in this field focuses on today.
... Since the beginning of presage-process-product research, and based on theoretical reflections on a subject-specific characterization of teacher cognitions in teaching, which were initiated in the U.S. in the late 1980s, the question of the theoretical conceptualization and empirically examination of teachers' professional knowledge has become increasingly important (e.g. Carpenter & Fennema, 1992;Carpenter et al., 1988Carpenter et al., , 1989Fennema et al., 1996;Neubrand, 2018;Petrou & Goulding, 2011;Rowland, 2014). The research initially sought to identify and isolate more general variables of successful teaching, but has since taken somewhat different forms. ...
In this chapter we investigate the evolution of research in mathematics education related to digital resources as an essential element of the external context for mathematics teachers’ professional activity. In the relevant research literature, we identified different themes and different kinds of evolution. We investigate the evolution of research with respect to educational policies related to digital resources, and to teacher integration of digital resources, including digital assessment. We also analyze the evolution of research concerning the quality of digital curriculum resources, and discuss emerging research questions related to mathematics and programming; to collective dimensions of teachers’ work with digital resources; and about the COVID-19 pandemic consequences. The different kinds of research developments are a result of evolution in the external context, or from more general trends in the research in mathematics education. We finally discuss possible directions for future research.
... Since the beginning of presage-process-product research, and based on theoretical reflections on a subject-specific characterization of teacher cognitions in teaching, which were initiated in the U.S. in the late 1980s, the question of the theoretical conceptualization and empirically examination of teachers' professional knowledge has become increasingly important (e.g. Carpenter & Fennema, 1992;Carpenter et al., 1988Carpenter et al., , 1989Fennema et al., 1996;Neubrand, 2018;Petrou & Goulding, 2011;Rowland, 2014). The research initially sought to identify and isolate more general variables of successful teaching, but has since taken somewhat different forms. ...
Lesson planning, assessment, and reflection constitute the key actions that teachers perform when students are not present in the classroom (henceforth, “Type D” variable). These “pre- and post-”actions are the most direct ways through which teachers shape their observable teaching work as mediated by their goals for their teaching. These goals are representations of teachers’ epistemological commitments apropos of teaching mathematics, whether those commitments be consciously espoused or unconsciously reproduced due to constraints within which they work. In this chapter, we survey the literature on lesson planning, assessment, and reflection according to eight epistemological paradigms that are widely known in the field of mathematics teaching. These epistemological paradigms are: Situated Learning Theory, Behaviorism, Cognitive Learning Theory, Social Constructivism, Structuralism, Problem Solving, Culturally Relevant Pedagogy, and Project- and Problem-Based Learning. We situate other perspectives on learning theory, which are derivatives of these prevailing paradigms, within this overarching frame. Our literature search revealed that some of the theoretical perspectives are well-reported in the literature whilst others have not received the same amount of attention from researchers. We detail each perspective, providing a definition, goals for teaching, pros and cons, and examples from the literature. We posit that, with the advent of the digital era of mathematics education, researchers must engage more explicitly with the theoretical perspectives we identified as underserved and must reckon with their own epistemological commitments more intentionally when reporting on studies regarding Type D.
... Traditionally, mathematics education research has been focused on teaching and learning, cognition, and individuals and their approaches to mathematics (i.e., Carpenter et al., 1989;Erlwanger, 1973;Steffe & Kieren, 1994;Vergnaud, 1988). This approach has failed to attend to the social aspects of schooling and learning mathematics; however, more recent work has acknowledged that learning mathematics is a social experience and has moved away from a focus on individuals' thought processes (Lerman, 2000). ...
... Im Bereich Mathematik wird das fachdidaktische Wissen als mathematikdidaktisches Wissen (in der scientific community mit MPCK für mathematics pedagogical content knowledge abgekürzt) bezeichnet . Dieses umfasst u. a. das Lehren von Mathematik, das als gemeinsame Basis aller Inhaltsbereiche verstanden wird und im Gegensatz zum fachlichen Wissen, das mathematisches Wissen in Breite und Tiefe umfasst (MCK als Abkürzung für mathematics content knowledge: Ball et al. 2008) und eher interaktionsbezogene Aspekte beinhaltet (Carpenter et al. 1989). ...
Zusammenfassung
Mathematikdidaktisches Wissen stellt eine zentrale Komponente der professionellen Kompetenz von (angehenden) Primarstufenlehrkräften im Bereich Mathematik dar, die mit situationsspezifischen Fertigkeiten zusammenhängt und, vermittelt über das professionelle Handeln, Effekte auf die mathematischen Leistungen von Schüler:innen haben kann. Um mathematikdidaktisches Wissen im Zusammenhang mit anderen Komponenten professioneller Kompetenz (z. B. Emotionen, Überzeugungen, Handlungsplanung, Instruktionsqualität) untersuchen zu können, bedarf es eines ökonomischen und frei verfügbaren Tests, welcher gängige Gütekriterien hinreichend erfüllt. Der vorliegende Beitrag stellt die Validierung eines solchen Tests vor. Entsprechend der Teststandards werden Grundnahmen in den Evidenzkategorien Inhalt, Struktur und Beziehungen zu anderen Variablen untersucht, um zu prüfen, ob die Testscores Schlussfolgerungen zum Konstrukt mathematikdidaktisches Wissen von Lehramtsstudierenden der Primarstufe zulassen. Die Ergebnisse liefern Validitätsargumente, die auf eine hinreichend hohe Reliabilität des Tests hinweisen und für theoriekonforme Schlussfolgerungen basierend auf den Testwerten sprechen. Der Beitrag schließt ab mit einer integrativen Betrachtung der Validierungsevidenzen, die für den Test bisher vorliegen.
... Based on the assumption that students have a wealth of productive prior knowledge with which to construct new knowledge [19], a responsive teacher seeks to 1) understand the substance of student ideas, 2) identify connections between student ideas and the knowledge and practices of the discipline, and 3) adjust instruction in order to pursue student ideas [20]. Research on responsive teaching has found that it can help develop conceptual understanding [21], engage students in disciplinary practices [22], promote student agency [23], and foster equitable participation [24]. Taking a responsive approach is beneficial, yet it presents the teacher with certain łinstructional tensionsž [25] or łdilemmas of practicež [26]. ...
... Erst unter Berücksichtigung dieser Perspektive wird nicht nur das fachliche Durcharbeiten der Inhalte während der Fortbildung gefördert, sondern auch die inhaltsbezogene Auseinandersetzung der Fortbildungsteilnehmenden mit den jeweiligen fachbezogenen Lern-, Denk-und Verstehensprozessen der Schüler*innen. Leitend für wirksame Fortbildungen ist deshalb nicht nur eine fachliche Perspektive auf die Fortbildungsinhalte, sondern immer auch die fachdidaktische und pädagogisch-psychologische Perspektive des Schüler*innenlernens, aus der heraus die Fortbildungsinhalte betrachtet werden und sich gemeinsam mit den Teilnehmenden gefragt wird, wie das inhaltsbezogene Schüler*innenlernen gefördert werden kann (Carpenter et al., 1989;Kleickmann et al., 2016;Roth et al., 2011). (Posner et al., 1982) zu fördern, sollten kognitive Dissonanzen in Fortbildungen aber nicht nur hervorgerufen, sondern die Teilnehmenden auch dazu angeregt werden, diese Dissonanzen durch eigene forschende Aktivitäten in Auseinandersetzung mit den Fortbildungsinhalten aufzulösen (Lipowsky, 2004;Lipowsky & Rzejak, 2017). ...
... These participants aimed to elicit students' ideas, which they then used to learn about student thought processes or to incorporate students' ideas into class discussions. Instructors who regularly leverage student thinking can improve student learning, create more equitable participation, and support the development of specialized teaching knowledge, even in large classes (100 to 300 students) (19)(20)(21)(22)(23). ...
Instructor discourse, defined as verbal interactions with students in the classroom, can play an important role in student learning. Instructors who use dialogic discourse invite students to develop their own ideas, and both students and the instructor share ideas in back-and-forth exchanges. This type of discourse is well-suited to facilitate deep learning for students but is rare in undergraduate biology classrooms. Understanding the reasoning that underlies the use of dialogic discourse can inform teaching professional development for instructors who are learning to use discourse to support student learning. Through classroom video recordings to identify dialogic discourse and stimulated recall interviews to elicit instructor reasoning, we investigated why undergraduate biology instructors used dialogic discourse in active-learning lessons. Using inductive and deductive qualitative analysis of interview transcripts, we identified and characterized seven reasons that instructors used dialogic discourse, including three aligned with a theoretical framework of student cognitive engagement and four that emerged from our data set. In addition to aiming to prompt generative cognitive engagement in 34% of instances of dialogic discourse, instructors used dialogic discourse to prompt activity, supply information, provide feedback, decipher student thinking, leverage student thinking, and cue students to make connections. Reasoning varied across different types of dialogic discourse. These findings provide valuable insights that can inform research, teaching professional development, and individual instructors’ reflections.
... Such knowledge could allow teachers to create an effective classroom environment including performing appropriate instructional activities, designing instruction to overcome students' misconceptions/difficulties, evaluating students' understanding, addressing students' needs, and designing tasks to further student understanding (An et al., 2004;Ball et al., 2008;Johnson & Larsen, 2012;. In order to create this kind of classroom environment, one of the requirements is teachers' knowledge of students, that is, knowing students' misconceptions, knowing students' prior knowledge and knowing the strategies invented by the students (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). Among these issues, students invented strategies which allowed teachers to enter the students' mind and understand students' thinking which is considered as a basis for effective mathematics instruction (Franke, Kazemi, & Battey, 2007;NCTM, 2014). ...
Although students' nonstandard strategies have great importance in understanding students' thinking and creating effective mathematics classrooms, much remains unexplored in the literature. This study investigated 22 middle school teachers' reasoning about a student's nonstandard strategy for the division of fractions. The data were collected through semi-structured interviews and a task consisting of a student's nonstandard strategy within a classroom excerpt which simulates how mathematical work emerges in the context of teaching. Six categories of layers were formed based on their reasoning about the validity, generalizability, and efficiency of the nonstandard strategy. These layers were categorized as a surface, intermediate, and deep level of reasoning. It was found that while half of the teachers had a surface level of reasoning, only one-third of teachers are at a deep layer of reasoning. On the other hand, teachers' reasoning approaches of how and when the nonstandard strategy works for all problems were determined as equating the answer, equating the process, being multiples of each other, and equating the denominators. The results and implications are discussed, and recommendations are presented in accordance with the findings of the study.
... We can interpret students' understandings through student actions reflecting student thoughts (Leatham, Peterson, Stockero, & Van Zoest, 2015). For this reason, to conduct effective mathematics teaching, it is necessary to build mathematics instruction based on student thinking (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989;National Council of Teachers of Mathematics, 2014). When teachers focus on student thinking, they can "interpret what the student does and says, and try to build a 'model' of the student's conceptual structures" (von Glasersfeld, 1995, p.14). ...
This study aimed to investigate the specialized content knowledge of pre-service primary school teachers about multiplication through problem posing and the justification for the accuracy of the multiplication of a two-digit number and a single-digit number. The research questions were formulated based on the theoretical framework of Ball and her colleagues’ teacher knowledge. The data were collected from 2nd, 3rd and 4th year pre-service teachers studying in the Faculty of Education of a state university using two open-ended questions. The data were analyzed with the content analysis method to provide a detailed perspective on the specialized content knowledge of pre-service teachers about multiplication. The findings of the study showed that although the pre-service teachers had difficulties in writing a problem related to multiplication operations, in which one of the multipliers was zero, the number of correct problem statements for the given operation increased as the pre-service teachers proceeded in their education. Another finding of the study is that the majority of the pre-service teachers made correct interpretations about the correctness or incorrectness of the solutions to the multiplication of a two-digit number and a single-digit number. However, it was observed that the pre-service teachers based their justifications for the student solutions given to them on students’ operational knowledge rather than their conceptual knowledge of the multiplication operation. The findings of the study were discussed within the framework of the relevant literature and some recommendations were made.
... Research on the connection between knowledge and culture is grounded in the perspective that students come to understand phenomena as they are related to what they already know (Foster, Lewis, & Onafowora, 2003;Ladson-Billings, 1994). Studies have shown that drawing on informal mathematics knowledge leads to increased understanding and attitudes towards the relevance that school mathematics has in students' lives (e.g., Carpenter, Fennema, Peterson, Chiang, & Loef, 1989;Gutstein, 2003;Gutstein, Lipman, Hernandez, & de los Reyes, 1997; ...
In this article, the authors report on a 3-year professional development research project. The project focused in general on early mathematics teaching and learning in urban schools and in particular on promoting teachers’ awareness of the importance of making connections between students’ out-of-school experiences to promote deep understanding of K–2 school mathematics. Children cross into school spaces bringing with them a wide variety of out-of-school experiences; this is especially true in the early elementary grades when they have spent more of their lives out of school than in school. Effective teaching at this level requires that teachers put forth concerted efforts to make connections between these out-of-school experiences and formal curricular content. The authors present the strategies that participating teachers (n = 49) employed in their attempts to make such connections as well as implications for future professional development research.
... Das Ausmaß professionellen Wissens von Lehrkräften wurde auch über den Besuch von Fortbildungen erfasst (Lipowsky, 2006). Das Projekt Cognitively Guided Instruction (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989) (Staub & Stern, 2002). ...
... A focus on content is critical because the CCSS require moresophisticated mathematical content knowledge to support student work and to help students develop mathematical proficiency (Wu, 2011;Ball and Forzani, 2011). Studies of cognitively guided instruction (Carpenter et al., 1989), the problem-centered mathematics project (Cobb et al., 1991), and the educational leaders in mathematics project (Simon and Schifter, 1991) all demonstrated that PD programs that were content focused could have positive impacts on instructional practice and student achievement by improving teachers' understanding of mathematics and student mathematical thinking. ...
... In agreement with Pretorius (2014) the results reflected in Table 3 show the majority (91.5%) of respondents agreed that teachers' professional development is a useful strategy that teachers can use to improve learners' understanding. Pretorius (2014) indicates that every school should have a plan for the continuous professional development of its teaching staff in place.The findings by Carpenter et al. (1989) show that professional development programs which are focused on student thinking can help teachers increase their understanding of children's mathematical understandings and misconceptions enhance their ability to build on children's understanding in their teaching, and improve students' problem-solving skills. ...
... Additive reasoning lays the foundation for multiplicative reasoning; multiplicative reasoning lays the foundation for fractions; fractional reasoning lays the foundation for proportional reasoning; and all of these are important for algebraic reasoning in the secondary school years Preface • xi Over the last decade, learning progressions, or learning trajectories as they are often called in mathematics education research, have had an influence on current mathematics standards, instructional models, and curriculum and assessment materials (Daro et al., 2011;Lobato & Walters, 2017;Sztajn et al., 2012). The use of learning progressions in professional development and classroom instruction has also been found to impact teacher learning, instructional practice, and student learning (Carpenter et al., 1989;Clements et al., 2011;Clements et al., 2013;Supovitz et al., 2021). A recent study of the implementation of the OGAP formative assessment system in multiplication and fractions in grades 3-5 showed significant impacts on teacher knowledge and student performance in a large urban school district (Supovitz et al., 2018(Supovitz et al., , 2021. ...
Secondary geometry teachers from several urban school districts participated in a two-year professional development focused on integrating dynamic geometry into teaching. The chapter documents the positive impact of the professional development for teachers' Technological Pedagogical Content Knowledge (TPACK) development and their students' achievement in geometry through the use of the dynamic geometry approach. Instruments used to develop and assess teachers' TPACK included a Conjecturing-Proving Test, interviews and observation protocols. Participants' TPACK levels were identified using a TPACK Development Levels Assessment Rubric. Findings show that teachers' TPACK tended to remain within the three middle TPACK levels (accepting, adapting, and exploring). Recommendations and suggestions for future research are offered to those who implement school-based, mixed methods research studies involving technology.
In this study, we examine two U.S. elementary teachers’ use of worked examples, representations, and deep questions in elementary mathematics lessons both before and after a project intervention designed to promote teaching improvement. This intervention was guided by a cognitive construct containing these three aspects, which were demonstrated via cross-cultural video lessons of both U.S. and Chinese teachers. We specifically analyze two teachers (Ann and Bea) who taught first-grade lessons on the topic of inverse relations. Although their lessons were rated similarly in year 1, the rating of their lessons in year 4, post-intervention, demonstrated different patterns of change. While Ann's use of worked examples and representations underwent no significant improvement, Bea spent more time unpacking a single worked example with a representational sequence fading from concrete to abstract (also called “concreteness fading”). Neither teacher showed improvement in asking deep questions. An analysis of the teachers’ responses to the project intervention revealed that their video noticing skills and personal takeaways were related to the areas of teaching where they improved. Other factors such as teachers’ prior knowledge, beliefs, and textbook resources, also played a role in teaching changes. These findings shed light on a recent endeavor called the “science of improvement” in mathematics teaching, which seeks to document teaching changes and the underlying mechanisms that might induce these changes. Implications for both research and practice are discussed.
Bu çalışmanın amacı, araştırma temelli bir hizmet öncesi eğitim programına katılan öğretmen adaylarının oran kavramıyla ilgili öğrenci düşünme modelleri geliştirme sürecini araştırmaktır. Nitel araştırma yöntemlerinin kullanıldığı çalışmada katılımcılar bir devlet üniversitesinde son sınıfta okuyan dört ortaokul matematik öğretmeni adayıdır. Eğitim programının tasarımı, Bilişsel Yönlendirmeli Öğretim (CGI) ilkelerine dayanmakta olup, öğretmen adaylarının öğrencilerin oran kavramı gelişimine dair araştırma temelli bilgiyle etkili bir şekilde etkileşimde bulunmalarını sağlayan görevleri içermektedir. Çalışmanın verileri, beş oturumdan elde edilen görüşmelerin kayıtlarından ve öğretmen adaylarının görevlere verdiği yazılı cevaplardan oluşmaktadır. Verilerin analizinde öğrenci düşünme modellerini dört süreç (tanımlama, karşılaştırma, çıkarım ve yeniden yapılandırma) ile tanımlayan bir çerçeve kullanılmıştır. Öğrenci düşünmesini açıklarken, öğretmen adayları öğrenci ifadelerini tekrarlamış, öğrenci çözümlerindeki önemli noktalara vurgu yapmış, öğrenci çözüm yöntemlerini ayrıntılı bir şekilde açıklamış veya genel çözüm özelliklerine odaklanmıştır. Karşılaştırma sürecinde öğretmen adayları, öğrenci çözümlerini kendi çözümleriyle açık veya üstü kapalı olarak karşılaştırmış, genellikle kendi veya diğer öğretmen adaylarının düşüncelerine odaklanmışlardır. Çıkarım süreci, öğrenci çalışmalarından gelen kanıtları yorumlamayı içermiştir, bazen gerekçe sunmadan çıkarımda bulunmuşlardır. Öğretmen adayları, öğrenci düşünmesini tahmin ederek ve öğrenci düşüncesini dikkate alan problemler oluşturarak yeniden yapılandırma sürecinde bulunmuşlardır. Oluşturdukları modellerle öğrenci düşünmesini tahmin edebilmişlerdir.
Bu çalışmanın amacı, ilköğretim matematik öğretmen adaylarının aritmetik ortalamaya yönelik mesleki fark etme becerilerinin belirlenmesi ve öğretim deneyi yöntemiyle geliştirilmesidir. Çalışmada, istatistik öğretimini kapsayan 8 haftalık öğretim dizisinde yaklaşık 4 hafta süren aritmetik ortalama öğretimine odaklanılmıştır. Çalışmanın verileri 3. sınıfta öğrenim gören 35 öğretmen adayından ön test, son test, yarı yapılandırılmış görüşmeler ve öğretmen adaylarının öğretim deneyi sürecinde tutmuş oldukları günlükler vasıtasıyla toplanmıştır. Öğrencilerin Matematiksel Düşünmelerine Yönelik Mesleki Fark Etme kuramsal çerçevesinin üç bileşenine yönelik soruların olduğu ön-test ve son-test, aritmetik ortalamaya yönelik bir problem ve bu probleme ait üç öğrenci çözümü içermektedir. Veriler, aynı kuramsal çerçeve temel alınarak hazırlanan kodlama tablosu kullanılarak analiz edilmiştir. Ön-test sonuçları öğretmen adaylarının öğrenci düşünüşü odaklı istatistik öğretimine katılmadan önce öğrencilerin aritmetik ortalamaya yönelik problemlerdeki çözüm stratejilerini dikkate alma, öğrencilerin matematiksel kavrayışlarını yorumlama ve öğrencilere karşılık verme becerilerinin düşük olduğunu göstermektedir. Öğrenci düşünüşü odaklı istatistik öğretiminin sonunda, öğretmen adaylarının, dikkate alma, yorumlama ve karşılık verme becerilerini önemli derecede geliştirdikleri görülmektedir. Bu bulgular doğrultusunda, ,amaca yönelik planlanan ve sistemli bir şekilde yürütülen matematik eğitimi dersleri ile öğretmen adaylarının fark etme becerilerinin geliştirilebileceği söylenebilir.
The short-term in-service EFL teacher education programs are assumed to be of crucial importance in upgrading teachers methodologies and gearing their teaching more closely to the students needs. Therefore, a dynamic in-service program for EFL teachers is needed to keep abreast of the time. The present study aims to investigate the role of experience in EFL teachers satisfaction of the in-service teacher education programs in Zanjan city. 200 EFL teachers from Zanjan province (Districts 1 & 2) participated in this study. The data were collected through the Course-evaluation questionnaire with five-level Likert scale. The results were analyzed through both descriptive and inferential statistics. The findings showed that there is a significant difference between teachers experience and their satisfaction of in-service programs. Thus, novice teachers and experienced teachers have different expectations of the in-service teacher education programs. This production of significant difference about the relationship between the EFL teachers experience and their satisfaction of the in-service teacher education programs can contribute to different perceptions of the teachers on the relationship between the EFL teachers experience and their satisfaction of the in-service teacher education programs.
Posing purposeful questions is one of the most effective teaching practices (NCTM in Principles to actions: Ensuring mathematics success for all. National Council of Teachers of Mathematics, 2014). Although the types and functions of teacher questioning have been abundantly studied, research on the role of teacher questioning in students’ contextualization process as they solve word problems is rather scarce. This study was conducted to investigate the function of six elementary preservice teachers’ questioning, its impact on students’ contextualization, as well as the successes and difficulties of enacting questioning. The collected data were analyzed using thematic analysis. The findings indicated that the implementation of task clarification (TC) moves effectively enhanced contextualization only when students possessed a relatively strong sense of agency while solving word problems. Furthermore, when students’ attentional focus was not appropriately redirected by the functional moves, including procedural understanding (PU), making connections (MC), the rationale behind a strategy (RA), and an alternative strategy (AS), their understanding of the contextual features and construction of mathematical relationships in word problem solving could not be refined. Implications for field experience design and future research on the quality of teacher questioning in mathematics teacher education programs are discussed.
The purpose of this study was to examine mathematics preservice teacher (PST) learning to notice for equity as they participated in an experimental, practice‐based methods course. Activities intended to support PST learning included the use of equity‐based lenses in video analysis, live lesson observations, and case study analysis. We analyzed PSTs' noticing for equity throughout the course and examined both the foci and depth of their noticing. We focused on three case study participants for an in‐depth investigation of trends in their noticing. We present how PSTs attended to and made sense of equity at the beginning and end of the course and describe their focus as primarily dominant (equity framed as access) or critical (equity framed as affirming identity). Findings indicated that activities with equity‐based lenses supported PSTs in noticing equity along the dominant and critical axes. We conclude with implications for mathematics teacher educators who are interested in supporting PSTs' learning about and development of equitable pedagogies.
Proof facilitates conceptual and meaningful learning in mathematics education rather than rote memorization. In this study, incorrect theorems and proofs are used to assess secondary school pre-service mathematics teachers’ proof assessing skills. Using the case study method, the study is conducted on pre-service mathematics teachers studying at the Department of Mathematics Education. There were eight pre-service mathematics teachers selected from each grade, resulting in 32 participants in total. A semi-structured proof form containing 13 questions was used to collect data, which was analyzed using content analysis. As the analysis reveals, pre-service mathematics teachers are highly likely to make incorrect decisions regarding theorems and proofs, and the margin of error is unaffected by grade level. Moreover, pre-service mathematics teachers tend to use proving terms incorrectly and, at times, are unable to differentiate between terms that are commonly used in proving. The pre-service mathematics teachers are believed to have learned proofs by rote rather than understanding how proofs work. With the help of interviews and tests created for different proof methods, it has been suggested that pre-service mathematics teachers should be tested on their proof evaluation skills in more detail.
More than 35 years have passed since Shulman introduced the term “pedagogical content knowledge” [PCK] to the field of teacher education. As is the case with any new construct in a field, Shulman's advocacy for PCK in the mid-1980s was almost certainly a reaction to the priorities of the field at that time. A primary goal of this paper is to refresh our collective memories of the origins of and development of PCK as a research domain, with a particular focus on what may have prompted scholars such as Shulman to propose the construct of PCK in 1986. With this historical lens as background, I also seek to consider what might be fruitful future directions in the field's thinking about teaching and teacher knowledge.
Although early-life adversity can undermine healthy development, an evolutionary-developmental perspective implies that children growing up in harsh environments will develop intact, or even enhanced, skills for solving problems in high‐adversity contexts (i.e., 'hidden talents'). This Element situates the hidden talents model within a larger interdisciplinary framework. Summarizing theory and research on hidden talents, it proposes that stress-adapted skills represent a form of adaptive intelligence enabling individuals to function within the constraints of harsh environments. It discusses potential applications of this perspective to multiple sectors concerned with youth from harsh environments, including education, social services, and juvenile justice, and compares the hidden talents model with contemporary developmental resilience models. The hidden talents approach, it concludes, offers exciting directions for research on childhood adversity, with translational implications for leveraging stress-adapted skills to more effectively tailor education, jobs, and interventions to fit the needs of individuals from a diverse range of life circumstances.
The development of 21st century abilities necessitates satisfying students' needs. Preservice teachers may find it difficult to meet the criteria. Responsive teaching needs to take precedence to effectively meet student needs. As a result, it is essential to consider how preservice teachers view this issue. This chapter first sets to highlight some of the issues preservice teachers face during microteaching in a mathematics classroom. Next, it looks at possible ways to promote responsiveness by using 21st century skills. It further discusses some solutions, suggestions, and recommendations based on the highlighted issues. Specifically, this chapter aims to identify ways for preservice teachers to contribute to teaching mathematics in a more creative way based on their responsiveness in microteaching. Finally, input is provided to educators on how to meet the output of responsive teaching by applying classroom microteaching strategies.
Technological Pedagogical Content Knowledge (TPACK) is one of the important models describing teachers’ competencies for teacher development model with technology. This study examines the reliability and validity of a Mongolian version of the short questionnaire for measuring Technology Pedagogical Content Knowledge (TPACK.xs) in the Mongolian secondary education. The instrument consists of 28 items used to evaluate seven components. 450 mathematics teachers participated in this study. All components had high reliabilities (all Cronbach’s alpha .86 or higher). Results show that TPACK.xs instrument, consisting 28 items, can be considered a reliable and valid instrument for assessing mathematics teachers TPACK.
Professional Development is an integral part of a teacher's professional career and it is a topic most discussed in the teaching profession since the last decade. Lastly, it was called in-service training but now it is known as "Professional Development". The main aim of the present study was to analyze the impact of teachers' professional development on the students' achievement in Elementary teachers' training institutes of Punjab, Pakistan. Government Colleges for Elementary Teachers (GCET) of Punjab was the target population. From 30 colleges of Punjab, data was collected using a self-developed tool from the sample of 70 teachers, and 100 students of the same colleges were used in the study. This study was cross-sectional. The data regarding the professional development of teachers was primary data collected from both teachers and students. The data about achievement scores of students was secondary. The statistical methodology correlation and regression analysis were used to analyze the impact. Further to test the significance of correlation and regression results the t-test and ANOVA were applied respectively. Data were analyzed with the help of Statistical software SPSS. Study findings showed that the correlation between 'Professional Development' and 'Students achievement' is high i.e. 0.314 for students and 0.343 for teachers along with significance values 0.004 and 0.001 respectively. The regression results are also significant having an F-statistic of 10.699 for students and 9.070 for teachers along with significant values 0.001 and 0.004 respectively. Analysis of the results showed that 'Professional Development of teachers have a great positive impact on 'Students achievement'.
This chapter is concerned with the development of an important aspect of children's problem-solving skiU in arithmetic-the ability to solve arithmetic word problems. There are several factors that might enable older children to perform better in problem-solving tasks than younger children, including the complexity of conceptual knowledge about the problem domain and the sophis tication of problem-solving procedures . The studies reviewed here suggest that , with age, children's improved ability to solve word problems primarily involves an increase in the complexity of conceptual knowledge required to understand the situations described in those problems . We will describe these findings in this chapter and consider some general issues about the development of problem solving skill .
72 5th- and 6th-grade students were randomly assigned to 1 of 6 classes using a factorial assignment of ability level crossed with attitude. Each class was taught a 2-day unit on probability, using the direct instruction model. Ss were videotaped during the lesson and afterward were interviewed about their thought processes using a stimulated-recall procedure. Independent of ability, Ss' reports of their understanding of the lesson were significantly related to achievement. Also, Ss who reported using specific cognitive strategies, such as relating the information being taught to prior knowledge, did better on the achievement test than those who did not report such strategies. (26 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The study concentrated on derived facts strategies in which the child uses known number facts to find the solution to unknown number facts. The study documented the children's spontaneous derived facts strategies and the influence of instruction in derived facts on children's solution strategies. The use of derived facts strategies more than doubled during the instruction and accounted for half the answers to addition problems. The increase in derived facts was accompanied by a decrease in counting. Counting on did not appear to be a prerequisite for learning derived facts strategies.
This study examined how six experienced teachers acquired information about students' knowledge and used that information to adjust their instruction while tutoring. Each teacher tutored five simulated students and one live student in the algorithm for whole number addition. A diagnostic/remedial perspective in which the teacher forms a detailed model of the individual student's knowledge and misconceptions was assumed in the early stages of the study, but did not describe adequately the tutoring of the teachers. Diagnosis was not their primary goal. Rather, each teacher appeared to move through a curriculum script-a loosely ordered but well defined set of skills and concepts students were expected to learn, along with the activities and strategies for teaching this material.
This study investigated students’ reports of attention, understanding, cognitive processes and affect during mathematics instruction. Two classes of fifth grade students (N = 38) were taught a 9-day mathematics unit on measurement by one of their teachers. Students were videotaped during instruction and interviewed subsequently using a stimulated-recall procedure. Students completed an achievement test and questionnaires about their attention, cognitive processes, motivational self-thoughts, and attitudes toward mathematics. Results suggest that students’ reports of attention, understanding, and cognitive processes were more valid indicators of classroom learning than observers’ judgments of students’ time on task. Findings also indicate that students’ reported affect as well as cognitions mediated the relationship between instructional stimuli and student achievement and attitudes.
This research identified the classroom activities that were related to the low level and high level mathematics achievement of boys and girls. In December and in May, students in 36 fourth grade mathematics classes completed a mathematics test containing low level and high level items from the National Assessment of Educational Progress. During January through April, the engagement/nonengagement in mathematics activities was observed for six randomly selected students of each sex in each class. Results showed that girls and boys did not differ significantly in either mathematics achievement or in observed engagement/nonengagement in mathematics activities. However, engagement in the following four types of activities was consistently and differentially related to girls' versus boys' low level and high level mathematics achievement: competitive mathematics activities, cooperative mathematics activities, social activities, and off-task behavior.
Student learning in elementary mathematics can be increased by some key teacher behaviors identified by research. (Author)
This study examined the teacher-student interaction patterns that were related to the low-level (LL) and high-level (HL) mathematics achievement of girls and boys. In December and in May, students in 36 fourth-grade mathematics classes completed a mathematics test containing LL and HL items from the National Assessment of Educational Progress. During January through April, observers coded the teacher-student interactions for six randomly selected students of each sex in each class. Many teacher-student interaction patterns that were related significantly to girls' mathematics achievement were unrelated to boys' mathematics achievement or were related significantly in the opposite direction. Further, teacher-student interaction patterns that predicted girls' and boys' LL achievement were not the same as those that predicted their HL achievement.
Curriculum scripts and adjustment of content to lessons. Paper presented at the annual meeting of the
- R T Putnam
- G Leinhardt
Putnam, R. T., & Leinhardt, G. (1986, April). Curriculum scripts and adjustment of content to lessons. Paper presented at the annual meeting of the American Educational Research Association, San Francisco.
Teachers' pedagogical content knowledge in mathematics
- T P Carpenter
- E Fennema
- P L Peterson
- D Carey
Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. (1988). Teachers' pedagogical content knowledge in mathematics. Journal for Research in Mathe-matics Education, 19, 345-357.
Observation manual Using Knowledge of Children's Mathematics Thinking Studies of the application of cognitive and instruction science to mathematics instruction
- P L E Peterson
- T P Fennema
- Carpenter
Peterson, P. L. (1987). Observation manual. In E. Fennema, T. P. Carpenter, & P. Using Knowledge of Children's Mathematics Thinking L. Peterson (Eds.), Studies of the application of cognitive and instruction science to mathematics instruction (Technical Progress Report, August 1, 1986 to July 31, 1987).
Specialization: teachers' knowledge. CHI-PANG CHIANG, Assistant Professor of Psychology, National Chengchi Uni-versity, Taipai, Taiwan, Republic of China. Specialization: research design. MEGAN LOEF, Graduate Student in Educational Psychology
- Erickson Hall
- East Lansing
- Mi
Erickson Hall, East Lansing, MI 48824. Specialization: teachers' knowledge. CHI-PANG CHIANG, Assistant Professor of Psychology, National Chengchi Uni-versity, Taipai, Taiwan, Republic of China. Specialization: research design. MEGAN LOEF, Graduate Student in Educational Psychology, University of Wisconsin—Madison, 225 N. Mills St., Madison, WI 53706. Specialization: teachers' knowledge.
Learning to add and subtract: An exercise in problem solving Teaching and learning mathematical problem solving: Multiple research perspectives
- T P Carpenter
Carpenter, T. P. (1985). Learning to add and subtract: An exercise in problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 17-40). Hillsdale, NJ: Lawrence Erlbaum.
Specialization: mathematics education. ELIZABETH FENNEMA, Associate Professor of Education, University of Wis-consin—Madison, 225 N. Mills St., Madison, WI 53706. Specialization: teachers' knowledge and beliefs
- Thomas P Authors
- Carpenter
- Professor
- University
- N Wisconsin—madison
- Mills
- St
- Madison
- Wi
Authors THOMAS P. CARPENTER, Professor, University of Wisconsin—Madison, 225 N. Mills St., Madison, WI 53706. Specialization: mathematics education. ELIZABETH FENNEMA, Associate Professor of Education, University of Wis-consin—Madison, 225 N. Mills St., Madison, WI 53706. Specialization: teachers' knowledge and beliefs. PENELOPE L. PETERSON, Associate Professor, Michigan State University, 510