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... Researchers argued that, for educational research significantly to impact classroom teaching and drive systemic improvement, teachers must develop a robust professional knowledge base (Cai et al., 2018, Cai et al., 2020Hiebert & Morris, 2012;Morris & Hiebert, 2009). This knowledge base is often embedded within artifacts such as sharable and revisable teaching cases (Cai et al., 2023) or instructional products (Morris & Hiebert, 2011). ...
... Improving the quality of classroom instruction has long been a central goal of mathematics education (Stigler & Hiebert, 2009) and lasting improvements in classroom teaching require a professional knowledge base (Cai et al., 2018, Cai et al., 2020Hiebert & Morris, 2012;Hiebert et al., 2002). Although teachers' practical knowledge is detailed, concrete, specific, and integrated, professional teaching knowledge must be widely sharable, storable, and easily accessible to teachers' use (Cai et al., 2018;Morris & Hiebert, 2009). ...
... Moreover, professional teaching knowledge should be embedded in storable and sharable teaching artifacts (Cai et al., 2020, Cai et al., 2023 or instructional products that, "both guide practice and provide a repository for the continuously accumulating knowledge about practice" (Morris & Hiebert, 2011, p. 5). We adopted Morris and Hiebert (2011) definition of instructional products as: (a) annotated instructional plans with specific learning goals that outline all aspects of classroom instruction believed to affect student learning and include all necessary information for successful implementation; (b) assessments that measure whether students achieved these goals. ...
Researchers have argued that sharable and revisable instructional products are crucial for having research impact on teaching practice and improving teaching at scale. This study examined how a tech-supported instructional product developed through lesson study could be adapted and refined by a group of instructors with different expertise. Two theoretical frameworks of instructional orchestrations (Drijvers et al., 2010 Educational Studies in Mathematics 75(2) 213–214) and cognitive engagement (Chi, 2021, Topics in Cognitive Science 13(3) 441–463) were used to examine classroom interactions and its associated learning opportunities. Videos of three lessons using the same instructional product were analyzed through systematic coding. Additionally, instructors’ reflection essays on their experience and perceptions of using the product were analyzed qualitatively. Results show that all instructors created various deep learning opportunities through different instrumental orchestrations. We found that student-centered instrumental orchestrations are most strongly associated with deep learning opportunities. However, the design features of instructional products and the instructors’ pedagogical practices in leveraging specific technology features also contributed to students’ cognitive engagement. With five illustrative episodes, we offer descriptive empirical evidence to unpack the complexity between instrumental orchestrations and associated learning opportunities. While the participating instructors appreciated the usefulness of the instructional product and benefits from adapting it, some challenges are noticed.
... In many international studies, empirical evidence has been provided that students' learning gains significantly depend on the quality of instruction (Brophy, 2000;Cai et al., 2020;Bostic et al., 2021). This applies to more generic coding protocols (e.g., Pianta & Hamre, 2009) and to more subject-related coding protocols (e.g., Blömeke et al., 2022;Hill et al., 2008). ...
... To accomplish this, the next subsection outlines the argument for why differential learning milieus are crucial to consider. Cai et al. (2020) emphasized the research need of "defining and measuring learning opportunities precisely enough" and elaborated that the "urgency of extending and refining the research on learning opportunities comes, in part, from the fact that high-quality learning opportunities are unequally distributed" (p. 13) in relation to various social background factors. ...
... Differential interaction qualities (in conceptual, discursive, and lexical richness) could be identified through a research design that deliberately excluded simple institutional effects (of curriculum, task, and teacher preparation) and controlled for prerequisite effects. In this way, we extend the list of documented (class composition, prerequisite, and institutional) effects to include classroom cultural effects, beyond teachers' often documented low expectations for at-risk students (DIME, 2007): While Cai et al. (2020) called for asking "How does teaching contribute to creating and realizing learning opportunities?" (p. 19), we provide quantitative evidence that learning opportunities in classrooms are not only led by teaching, but are interactively established with what students bring into the discussion. ...
Cognitive demand is a crucial dimension of instructional quality. Its heterogenous operationalizations call for refined investigations, with respect to discursive richness (generic conceptualizations) and conceptual richness (subject-related conceptualizations). Considering not only teachers’ intended cognitive activation (operationalized, e.g., by tasks), but also the enacted activation and individual students’ participation as realized in the interaction, raises the question of how far the interaction quality is associated with students’ prerequisites, school context, and class composition. In this paper, we present a video study of leader-led small-group instruction (in 49 groups of 3–6 middle school students each) with the same fraction tasks, so that differences in interaction quality can be scrutinized in generic and subject-related conceptualizations. In spite of equal task quality, large differences occurred in interaction quality across heterogenous class compositions. The regression analyses revealed that the enacted activation and individual participation were significantly associated with the school context (of higher-tracked and lower-tracked schools), but much less with individual learning prerequisites. These findings reveal the need to capture students’ collective and individual engagement in cognitive demands in the interaction and in generic and subject-related conceptualizations and to systematically investigate their association with class composition.
... International research on subject-matter teaching has repeatedly shown that students' learning gains are substantially influenced by the quality of instruction (Brophy, 2000;Cai et al., 2020;Hiebert & Grouws, 2007). Two recent research reviews on instructional quality frameworks revealed that in particular the quality dimensions of cognitive demands and instructional support had strong effects on students' learning gains, two dimensions with strong overlaps, and multiple conceptualizations and operationalizations (Praetorius & Charalambous, 2018;Spreitzer et al., 2022). ...
... Kunter et al., 2013;Lipowsky et al., 2009;Ni et al., 2018;. These findings have been robust for heterogeneous conceptualizations, but have seemed to vary between students at risk and successful students (Bostic et al., 2021;Cai et al., 2020). ...
... Many case studies, in turn, have specified conditions of mathematical richness and discursive richness of the talk (Lampert & Cobb, 2003;Walshaw & Anthony, 2008), in other words, the conceptual depth of the initiated mental processes and the complexity of elicited and supported discourse practices such as explaining or arguing. Each of these aspects of richness has also seemed to depend on the school contexts, with higher-tracked schools or schools in privileged areas providing richer learning opportunities than lower-tracked schools (Bostic et al., 2021;Cai et al., 2020;Pauli & Reusser, 2015), so school contexts seem to matter. ...
Instructional quality dimensions of cognitive demands and instructional support have been shown to have an impact on students’ learning gains. Existing operationalizations of these dimensions have mostly used comprehensive ratings that combine various subdimensions of task quality and interaction quality. The current study disentangles interaction quality in a video data corpus study (of 49 middle school classrooms sharing the same tasks) to identify those quality features that predict students’ learning gains in conceptual understanding. The regression analysis reveals that quality features of students’ individual engagement do not predict individual student learning, whereas teachers’ support of learning content-relevant vocabulary predicts the small groups’ learning. For at-risk students, the collective time spent on conceptual practices (i.e. explaining meanings of concepts) on students’ learning is significantly predictive. The observation that different operationalizations (for similar aspects of interaction quality) lead to different impacts on the learning gains contributes to ongoing research efforts to refine and increase insight into aspects of interaction quality.
... In such an experimental scientific activity, it is possible to focus on the understanding of the phenomenon under investigation, but also on the mathematical concepts and procedures needed to describe it. It is not possible to guarantee that students will develop their STEM skills when investigating a complex situation, but the choices students make during the solving process to explain the phenomenon can be transformed into mathematical learning opportunities (Cai et al. [3], Albarrracín and Gorgorió [4]). In this didactic design, it is important to consider not only how to use the available technology to study physical phenomena, but also the teacher's knowledge of it and the teacher's role in carrying out the activity. ...
... Cai et al. [3] consider that any definition of mathematical learning opportunity in the classroom must necessarily consider the interactions between the following three elements: the mathematical tasks, the teacher, and the students. The interactions between the three elements create complexities that are likely to be understood only through multiple iterations of studies based on successive evidence. ...
... The literature on STEM teaching and learning highlights its complexity and the difficulties it poses for both teachers (Diana et al. [34]) and learners (Shekhar et al. [35]). In our study we have focused on identifying the mathematical learning opportunities (Cai et al. [3], Wijaya, van den Heuvel-Panhuizen and Doorman [31]) that sound intensity analysis promotes in a classroom and the catalyst elements that promote the development of the mathematical model that students build. First, we have identified a large number of mathematical learning opportunities. ...
This study aims to describe the design and development of a STEM problem-solving activity involving mathematics and physics content. It is an activity whose starting point is a question based on a real problem: to identify the areas of the classroom where the intensity of sound is higher. In this way, the problem involves the physics of sound. However, in addition, by trying to give an answer to the initial question, students have the opportunity to develop progressively more complex mathematical models from already known mathematical concepts and procedures. The analysis of the results of the implementation based on the identification of the learning opportunities and the catalyst elements that generate them, allows us to state that the activity encourages students to enrich their mathematical models by incorporating new mathematical concepts such as the function of two variables. We conclude that the design of the activity and the choice of data acquisition technology are key, but also the interaction between the students and the teacher.
... Specifically, the value beliefs towards mathematical tasks that students internalize during junior high school affect how they view all future mathematics related tasks (Middleton & Spanias, 1999). The relation among emotions, values, and achievement in mathematics in early adolescence is critical for educators to provide quality mathematics education (Cai et al., 2020). ...
... Pairing mathematics with areas such as sports and extracurricular activities, as well as making tasks authentic and meaningful has the potential to decrease emotional cost and therefore increase performance. Finally, since students' dispositions such as mathematics anxieties and self-efficacy influence whether they make optimal use of their learning opportunities (Cai et al., 2020), researchers may incorporate the dispositional factors into similar studies. ...
Today’s adolescents are digital natives. Information and knowledge are abundant and easily accessible. Learning a subject perceived to be difficult such as mathematics may present a relatively high cost and emotional burden. This study investigated the relations among cost, academic emotion, and performance in mathematics. A longitudinal survey of two waves was conducted among 754 (52.12% males) seventh grade Taiwanese students at both Time 1 and Time 2 with an average age of 12.59 years (SD =0.53) at Time 1. Mathematics academic emotions, cost, and academic performance were significantly correlated. Cross-lagged analysis revealed that mathematics academic emotions (MAE) predicted students’ mathematics performance but indirectly through perceived cost. Academic performance in mathematics was positively associated with subsequent mathematics academic emotions. A loop was detected between MAE and cost as well as cost and academic performance. This paper clarifies the interrelationships among these variables and reveals the crucial role of the perceived cost. Activities that aim to reduce students’ perceived cost of engaging in mathematics tasks will improve their learning emotions and mathematics achievement.
... Having such states of affairs in mind, what is at a stake is access for learners. Cai et al. (2020) referred to learning opportunity or opportunity to learn (OTL) as the best indicator of student learning. They encourage researchers to engage on how to maximize the quality of learning opportunity which is taken as a systemic parameter for boosting equity in mathematics education. ...
... OTL refers to opportunity to learn as defined by Cai et al. (2020). By "structure", we address the issues of organizing students and supporting teachers. ...
In the one hand, the world is not yet secured from COVID-19. On the other hand, educational planning is acontinuous activity. The Federal Democratic Republic of Ethiopia is going to implement new curricula since2021/22academic year. So, what lessons or challenges could be derived from this era? This might be an opportunity for educators and researchers to forward inputs to the decision making bodies. In this article, the construct“opportunity to learning” (OTL) is taken as a parameter for addressing one of the goals of education: equitable access in mathematics education. This was done by adopting two frameworks: NCTM & NCSM (2 020) and Walkowiak, Pinter, & Berry (2017). Finally, we come up with ten discussion points in order to boost the equitableaccess for all learners. Our work may serve as a position paper to inform curriculum implementers and educational material producers in countries like Ethiopia.Keywords: equitable access, mathematics education, opportunity to l earning
... In our January 2020 editorial (Cai, Morris, Hohensee, Hwang, Robison, Cirillo, Kramer, Hiebert, & Bakker, 2020b), we suggested that a useful focus for teacher− researcher partnerships could be the quality of learning opportunities that students experience. We posed an overarching question about the kinds of measures and research designs that could reveal the nature of learning opportunities. ...
... How could technological tools that support communication and sharing of data and resources allow multiple teacher− researcher partnerships to generalize knowledge from multiple cycles of feedback, tweaking artifacts and instructional strategies and testing them with teachers and students in other contexts? Answers to these questions are essential to understanding how technology can be used wisely and effectively to realize the iterative cycle of implementing, testing, assessing, revising, and sharing that is at the heart of our proposals to improve learning opportunities for all students (Cai, Morris, Hohensee, Hwang, Robison, Cirillo, Kramer, Hiebert, & Bakker, 2020b). ...
... But all students, including those students struggling in mathematics for various reasons, need the opposite: Cognitively demanding instruction that focuses on understanding basic mathematical concepts (Blazar & Archer, 2020;Charalambous et al., 2018;Myers et al., 2015). Thus, Cai et al. (2020) and van Mieghem et al. (2020) recently called for more research to maximize the learning opportunities of all students. ...
Enhancing understanding is crucial for all students. Instructional approaches have been developed to achieve this goal, yet little is known about the kind of support that teachers need for their effective implementation. We compare two support conditions designed to enable teachers to enhance students' understanding of the mathematical topic percentages. 51 teachers and their grade 7 classes participated in a quasi-experimental pre-post-test control group design, comparing three groups. (1) Teachers of the material condition received the curriculum material of the MATILDA teaching unit on percentages, with a comprehensive teacher manual on the main principles of the approach. (2) Teachers in the workshop condition received the same materials and additionally participated in professional development (PD) workshops covering key ideas of the approach. (3) The waiting control group used their regular textbooks. Hierarchical regressions were used to compare the effects on students' learning outcome. The analysis reveals that students in both support conditions achieved significantly higher learning outcomes than the control group, i.e., curriculum materials and teacher manuals seem to provide effective support. No additional effect on student outcomes was found for the workshop condition, but a substantially lower teacher dropout than in the material condition. This finding indicates that PD workshops may not be necessary for highly motivated volunteer teachers but can be completed by a wider range of teachers. The findings can inform implementation projects for educational innovations and inspire further research to determine for which teachers a written PD is sufficient and for which face-to-face workshops are more promising.
... To achieve academic excellence and ensure equitable educational outcomes in China, it is crucial to address factors that affect student mathematical performance. Opportunity to Learn (OTL), emphasizing equality in educational provision (Cai et al., 2020) and providing students with adequate learning resources, is widely considered as one of these critical factors (McDonnell, 1995;Rodrigues et al., 2024). ...
This study investigates the learning opportunities pertaining to geometric transformations within the New Century Mathematics (NCM) textbooks used in China’s primary schools. To evaluate these opportunities, we developed a specialized analytical model, the SIC model (Situation Representativeness, Information Type, Cognitive Level), providing a nuanced understanding of the educational landscape in relation to geometric transformations. The findings reveal that the NCM textbooks offer a comprehensive and well-balanced curriculum on geometric transformations, closely adhering to the specified guidelines of the national curriculum standards. Specifically, the analysis shows that the NCM textbooks achieve an average learning opportunity score of 0.90 out of a possible 1.00—a high score that highlights the curriculum’s effectiveness in providing quality learning opportunities in this area. Notably, among the three types of geometric transformations examined—reflection (axial symmetry), translation, and rotation—the content related to rotation exhibits slightly higher learning opportunity scores compared to the others. By examining the current state of learning opportunities within Chinese teaching materials, particularly in the realm of geometric transformations, this research provides valuable insights into the strengths of the existing curriculum and lays a foundation for further exploration and enhancement in the field of mathematics education research.
... Some researchers have argued that classroom learning opportunities should rightfully involve interactions among three factors; namely, tasks, teaching, and students although without being able to pinpoint the precise nature of their measurement (see Cai et al., 2020). Indeed, since the 1960s, OTL measures were conceptualized in terms of three broad dimensions at the classroom level, namely the: i) amount of time engaged in learning, ii) coverage of relevant content, and iii) quality of instruction that has been flexibly defined thus far (Eliott & Bartlett, 2016;Kurz, 2011). ...
... According to opportunity-use models, the extent to which teaching promotes and supports students' subject-related understanding and learning processes does not result directly and necessarily from the quality of the learning opportunities in terms of teacher actions and tasks but rather from an interaction between these learning opportunities and the quality of how these are used. Therefore, from a theoretical viewpoint, teaching quality should be seen as a co-production of teachers and students (Cai et al., 2020;Fend, 1998;Vieluf & Klieme, 2023). From a methodological viewpoint, the question arises as to how the quality of use can be adequately measured, especially for characteristics that relate to students' understanding and learning processes. ...
Based on an opportunity-use model of instructional quality, this study investigates the extent to which subject-specific instructional quality rated by experts is reflected in students’ assessments of their own learning and understanding, and how students’ perceptions predict their achievement. The analyses used data from a German-Swiss sample of 36 classes with around 900 lower secondary students, obtained as part of the so-called “Pythagoras study” in the school year 2002/2003. The teachers were instructed to introduce the Pythagorean theorem in three lessons, which were videotaped. Using the videos, the experts assessed the instruction quality with respect to the goal of promoting a deep understanding of the theorem. The students completed the questionnaires assessing their understanding of the content, their learning process, and the general comprehension orientation of the teacher. The results showed significant and moderate correlations on the class level between expert-rated subject-specific teaching quality and students’ perceptions of their own learning and understanding, as well as of the teacher’s general comprehension orientation. Multilevel models revealed that subject-specific expert ratings are reflected in individual students’ perceptions of their own learning and understanding. Student perceptions were also associated with achievement gains. The results suggest that the assessment of quality by students and experts is more closely linked if a distinction is made between the quality of the learning opportunities offered and their use and if subject-specific criteria are used instead of generic criteria. This study contributes to a more nuanced understanding of the validity of student perspective in assessing instructional quality.
... The construction of learning opportunities should be thoughtfully designed to connect the teaching of mathematical concepts to the exhibition of students' learning (Cai et al., 2020). This chapter, therefore, explores strategies that will assist and allow students to exhibit their acquired knowledge through their writing skills in mathematics and enable the building of their confidence when explaining their logic and reasoning -especially those with diverse learning needs. ...
Because mathematics and language are interconnected, often students are asked to “explain” their answers or “justify” their choices. It is critical that students are able to verbally express their thoughts, as this is considered an “authentic practice of the discipline.” The Standards for Mathematical Practice require that all students be able to make sense of problems, and also communicate both in writing and orally; skills which the communication standard has promoted for over two decades. Moreover, in the case of SWDs, those who are dually identified often show more confidence when asked to write their logic, as opposed to speaking. This chapter, therefore, explores strategies that will assist and allow students to exhibit their acquired knowledge through their writing skills in mathematics and enable the building of their confidence when explaining their logic and reasoning – especially those with diverse learning needs.
... It turns out that researchers have reported compelling evidence that links types of SLOs and particular learning outcomes (Bjork & Bjork, 2011;Cai et al., 2020;Hiebert & Grouws, 2007;Richland et al., 2012). Consider the case of mathematics. ...
In this chapter we propose a way to create theories of teaching that are useful for teachers as well as researchers. Key to our proposal is a new model of teaching that treats sustained learning opportunities (SLOs) as a mediating construct that lies between teaching, on the one hand, and learning, on the other. SLOs become the proximal goal of classroom teaching. Rather than making instructional decisions based on desired learning outcomes, teachers could focus on the kinds of SLOs students need. Because learning research has established reliable links between specific types of learning opportunities and specific learning outcomes, theories of teaching no longer must connect teaching directly with learning. Instead, theories of teaching can become theories of creating SLOs linked to the outcomes teachers want their students to achieve. After presenting our rationale for moving from theories of teaching to theories of creating SLOs, we describe the benefits of such theories for researchers and teachers, explain the work needed to build such theories, and describe the conditions under which this work could be conducted. We conclude by peering into the future and acknowledging the challenges researchers would face as they develop these theories.
... The results in this study highlight the need to reconsider responsive teaching with regard to preparing teachers who can maximize quality of opportunities for all students. While the idea of modifying and accommodating instruction has been discussed in the field of teacher education, the topic of what it means to respond to students with various needs is beginning to attract attention [73]. We describe responsive teaching as accommodations and modifications in mathematics strategies and lesson design that PTs intend to provide to Jose and Liam and interpret the meaning from the perspective of preservice teacher education. ...
This qualitative study investigated 17 preservice teachers’ lesson design for teaching multiplication to an average performing student and a student with mathematical learning disabilities (MLD). Findings reveal how preservice teachers differentiate mathematics instruction to meet the needs of students. They modified mathematical strategies by providing diverse multiplicative concepts and fitting the form of representations. They accommodated lesson design by setting their expectations based on individual needs, managing instructional structure and progress, and adjusting the cognitive demand of tasks. Some formative assessment skills demonstrated how they understood students’ mathematical thinking and responded to it. The needs for further attention and support in lesson differentiation, including content-oriented alternation for equitable responsive teaching and moving away from short-term solutions to sustainable support, were discussed.
... Similarly, Max & Welder (2020) justifying shortcuts stated that they should make generalizations about the operations performed to verify computational shortcuts. For this reason, in studies in the literature (Simon & Blume, 1996;Herbst & Chazan, 2011;Cai, Morris, Hohensee, Hwang, Robison, Cirillo, Kramer, Hiebert & Bakker; similar to the findings. In addition, when the answers given by the students to the problems in the context of the calculating without computing feature, which is included in the abstracting from computation habit, are examined, there is very little finding about calculating without computing. ...
In this study, it is aimed to determine the algebraic thinking habits of two eighth grade school students through the answers they gave in the process of solving mathematical problems. The algebraic habits of mind (ZCA) theoretical framework developed by Driscoll (1999) was used to reveal these thinking habits. The research design of this study is a case study and the participants consist of two eighth grade students. The data were analyzed using Driscoll (1999)'s ZCA framework, which is algebraic habits of mind. Descriptive analysis was used in the analysis of the data. When we look at the findings obtained from the research; It is seen that both students can do describing a rule and justifying a rule in the solutions of the problems. In addition, it is seen that computational shortcuts, equivalent expressions and symbolic expressions come to the fore in the solutions of students' problems. On the other hand, the habit of undoing in solving problems was not encountered very rarely in both students. In the light of the findings obtained, the reasons for the existing and non-existent algebraic habits of mind are discussed. As a result of this discussion, it is thought that it is effective to include guide questions to create and develop algebraic thinking habits in students in classroom teaching practices of teachers.
... Since data triangulation can provide a more detailed and completed picture of the situation (Altrichter et al., 2008), repeated comparisons were conducted to validate the findings. Since the instructional triangle is composed of the three factors -mathematical tasks, teaching, and students (Cai et al., 2020;Chhen et al., 2003), the analysis was conducted through the systematic verification process in the instructional triangle. For example, the author found that the IBL approach constantly created opportunities for the instructor to gain on-going formative assessment through their completed pre-class assignments and in-class activities such as students writing process of reaching a solution, student orally presenting their thinking, group discussions, questions asked when solving a problem. ...
p style="text-align: justify;">This study investigated how implementing inquiry-based learning (IBL) can be an effective tool for an instructor to conduct rich formative assessment. Many researchers have documented that IBL promotes active learning from students’ learning perspective. However, little research examines how IBL affects instructors’ teaching practice from teaching perspective. Based on the data collected from a Calculus II class, the author discussed how the structure of IBL class produced rigorous on-going formative assessment during classroom teaching from the three aspects: helping the instructor “see” student thinking; helping the instructor “see” the level of student understanding; helping the instructor catch teachable moments. The rigorous on-going formative assessment, in turn, helped change student classroom behaviors in terms of asking more questions, showing deep thinking, and gaining confidence.</p
... Student learning and achievement are crucial measures of an education programme (Cai et al., 2020;OCED 2016), and are affected by many factors. To provide a broader picture, Sammons et al. (2009) situate student outcomes by considering the complexity of educational systems. ...
This paper examines mathematical enculturation in Ethiopia, based on Alan Bishop's Framework of Mathematical Activities, which he created after studying a range of indigenous cultures. The aim is to assist a culturally responsive mathematics education in Ethiopia with the intention of improving student learning. Data was collected using an ethnographic approach. The findings demonstrate the existence of rich mathematical activities aligned to Bishop's six fundamental mathematical activities. Our work has implications for curriculum and instruction in mathematics education ranging from early childhood to higher education. It may also inform mathematics educators and policymakers in other countries.
... The second set of research questions that we proposed earlier addresses the spread of an innovation to more students in more contexts. In our January editorial (Cai et al., 2020b), we identified the first of our set of overarching problems for mathematics education: defining and measuring learning opportunities so as to maximize them for every student. In this editorial, we have built on that idea by suggesting that understanding the ways in which an innovation interacts with local contexts will enable scaling up the innovation and spreading ambitious learning opportunities to more students in more contexts. ...
We investigated teachers’ perspectives about what opportunities for learning and teaching could be created using WhatsApp as a social network to help students prepare for the final secondary-school Bagrut (matriculation) exam in mathematics. Launched by the Ministry of Education and the Center for Educational Technology three months before the Bagrut examination, the “WhatsApp Bagroup” project was initiated to serve as an additional environment for learning mathematics. The formation of these WhatsApp groups was meant to provide an online review project during which teachers integrated blended learning, and students presented problems with which they were having difficulties. During this initiative, we applied a quantitative and qualitative research model to analyze the teachers’ points of view about what learning and teaching opportunities were created. The study used a mixed method, sequential explanatory procedure to acquire a complete understanding of the factors that constitute teachers’ perceptions of learning and teaching via the Bagroup project. Quantitative and qualitative data were collected using three tools: a questionnaire with Likert-type statements and open questions, informal semi-structured interviews, and observations of four Bagroup study groups conducted during the three-month period. Factor analysis revealed three categories regarding the Bagroup environment: factors that contribute to learner’s emotional needs, factors that promote learning, and factors that inhibit learning. The findings may have implications for distance and remote learning and teaching opportunities.
Cet article présente les résultats des analyses d’entretiens auprès de quatre enseignantes montréalaises du primaire, à qui nous avons demandé comment elles prévoyaient l’enseignement-apprentissage d’une situation-problème mathématique. À travers le modèle de résolution de problèmes de Verschaffel et coll. (2000) et les indicateurs de la différenciation pédagogique de Tomlinson (1999), nous décrivons les adaptations prévues selon les élèves. Globalement, les résultats indiquent que les enseignantes tiennent compte des besoins des élèves et des obstacles propres aux situations-problèmes et s’y adaptent. Cependant, les explications abondantes lors du moment de la découverte de la situation semblent nuire à l’autonomie des élèves.
Quality of interaction can enhance or constrain students’ mathematical learning opportunities. However, quantitative video studies have measured the quality of interaction with very heterogeneous conceptualizations and operationalizations. This project sought to disentangle typical methodological choices to assess interaction quality in six quality dimensions, each of them in task-based, move-based, and practice-based operationalizations. The empirical part of the study compared different conceptualizations with their corresponding operationalizations and used them to code video data from middle school students (n = 210) organized into 49 small groups who worked on the same curriculum materials. The analysis revealed that different conceptualizations and operationalizations led to substantially different findings, so their distinction turned out to be of high methodological relevance. These results highlight the importance of making methodological choices explicit and call for a stronger academic discourse on how to conceptualize and operationalize interaction quality in video studies.
The field of mathematics education research has been promoting problem-solving-based mathematics instruction (PS-based MI) to afford opportunities to develop students’ conceptual understanding and problem-solving abilities in mathematics. Given its usefulness, there is still little knowledge in the field about how it can afford such opportunities in real classrooms. In this study, an attempt was made to make in-depth observations of such classrooms from the perspective of variation. We examined the differences in the space of learning provided by two lessons of the same teacher in two Ethiopian primary school classrooms. Based on the literature, we identified three key aspects for analysis: mathematical tasks, lesson structure and classroom interaction patterns. Our analysis showed that, even though both lessons focused on the same topic of solving linear inequalities, they were enacted differently. The lesson that employed a PS-based MI approach constituted a wider space of learning than the lesson employing a conventional approach. This study demonstrates the usefulness of our analytical approach for describing and documenting PS-based MI practice, and for qualitatively interpreting the differences in what is mathematically made available to learn. We suggest that it can provide guidelines for mathematics teachers to reflect upon and to enhance learning spaces in their own classrooms.
The problem of formation of practice-and professionally-oriented skills by means of general subject areas in the conditions of constant modernization of educational standards of higher professional education and realization of the concept of development of mathematical education is investigated in the work. The problem of convergence in the educational process of theoretical and applied mathematics is studied, which, in fact, is solved by means of effective use of ideas and methods of mathematical modeling. There is a cross-cutting content and methodologi-cal line of mathematical models, in the implementation of which the greatest potential for increasing the motivation of students to mathematical activities. It is shown that this approach is a carrier of innovation, novelty in the content of mathematical courses and methods of their teaching and practice-oriented orientation. The inseparable connection between the methodology of cognition and the methodology of practical integrity of the study of applied problems of natural sciences, which are studied in various mathematical courses of higher education institutions for the further formation of professional competencies in the course of mathematical modeling.
The quality of curriculum resources and teaching practices can constrain or promote students’ opportunities for mathematics learning, in particular, students with diverse language proficiency. The video study investigates 18 classes that all used the same curriculum resources aimed at developing 367 seventh graders’ conceptual understanding of percentages to identify the interaction of quality dimensions, the enactment of given curriculum resources, and students’ mathematical achievement (when controlling for mathematical preknowledge and language proficiency). Multilevel regression analysis revealed that three quality dimensions that can easily be supported by the curriculum resources ( Mathematical Richness , Cognitive Demand , and Connecting Registers ) were on a high level, and their variance had no additional interaction with students’ achievement. In contrast, the 4 quality dimensions that were enacted in the teacher-student interaction with more variance ( Agency , Equitable Access , Discursive Demand , and, in particular, Use of Student Contributions ) had a significant additional impact on student achievement. These findings reveal important insights into the implementability of equitable instructional approaches.
This paper presents a case study carried out at an elementary school that led to a characterization of mathematical modeling projects aimed at generating social impact. It shows their potential as generators of mathematical learning opportunities. In the school project, upper-grade students (sixth grade, 11-year-olds) studied the way in which the rest of the students at the institution traveled from their homes to school. Its purpose was to identify risk points from the standpoint of road safety and to develop a set of recommendations so that all the children could walk safely to school. In our study, we identified, on the one hand, the mathematical learning opportunities that emerged during the development of the project and, on the other, the mathematical models created by the students. We discuss the impact of the project on the different groups in the school community (other students, parents, and teachers). We conclude with a characterization of the mathematical modeling projects oriented towards social impact and affirm that they can be generators of mathematical learning opportunities.
Background
Within mathematics education research, policy, and practice, race remains undertheorized in relation to mathematics learning and participation. Although race is characterized in the sociological and critical theory literatures as socially and politically constructed with structural expressions, most studies of differential outcomes in mathematics education begin and end their analyses of race with static racial categories and group labels used for the sole purpose of disaggregating data. This inadequate framing is, itself, reflective of a racialization process that continues to legitimize the social devaluing and stigmatization of many students of color. I draw from my own research with African American adults and adolescents, as well as recent research on the mathematical experiences of African American students conducted by other scholars. I also draw from the sociological and critical theory literatures to examine the ways that race and racism are conceptualized in the larger social context and in ways that are informative for mathematics education researchers, policy makers, and practitioners.
Purpose
To review and critically analyze how the construct of race has been conceptualized in mathematics education research, policy, and practice.
Research Design
Narrative synthesis.
Conclusion
Future research and policy efforts in mathematics education should examine racialized inequalities by considering the socially constructed nature of race.
Key findings and theoretical trends that have shaped research on gender and mathematics education are described in context. A brief historical note precedes the overview of the foundational work conducted in the 1970s. The assimilationist and deficit models that framed the early intervention programs designed to promote females’ participation and learning of mathematics are discussed, as are the subsequent challenges and reassessments provided by broader feminist perspectives. The interactive influence on mathematics learning of relevant personal and contextual variables and the move towards more complex models of equity embedded in broader social justice concerns are highlighted. Given its enabling role in educational and career pursuits, and that gender equity concerns will thus remain a significant item on the research agenda of (mathematics) educators in many countries, guidelines for future work are offered.
Feelings of apprehension and fear brought on by mathematical performance can affect correct mathematical application and can influence the achievement and future paths of individuals affected by it. In recent years, mathematics anxiety has become a subject of increasing interest both in educational and clinical settings. This ground-breaking collection presents theoretical, educational and psychophysiological perspectives on the widespread phenomenon of mathematics anxiety.
Featuring contributions from leading international researchers, Mathematics Anxiety challenges preconceptions and clarifies several crucial areas of research, such as the distinction between mathematics anxiety from other forms of anxiety (i.e., general or test anxiety); the ways in which mathematics anxiety has been assessed (e.g. throughout self-report questionnaires or psychophysiological measures); the need to clarify the direction of the relationship between math anxiety and mathematics achievement (which causes which).
Offering a revaluation of the negative connotations usually associated with mathematics anxiety and prompting avenues for future research, this book will be invaluable to academics and students in the field psychological and educational sciences, as well as teachers working with students who are struggling with mathematics anxiety
We began our editorials in 2017 seeking answers to one complex but important question: How can we improve the impact of research on practice? In our first editorial, we suggested that a first step would be to better define the problem by developing a better understanding of the fundamental reasons for the divide between research and practice (Cai et al., 2017). This sparked subsequent editorials in which we delved deeper into some of the many complicated facets of this issue. In our March (Cai et al., 2017b) editorial, we argued that impact needs to be defined more broadly than it often has been, notably, to include cognitive and noncognitive outcomes in both the near term and longitudinally. This led us to focus our May (Cai et al., 2017a) editorial on the ways that research might have a greater impact on the learning opportunities that help students reach broader learning goals. We argued that it is not enough to identify learning goals–it is also necessary to conduct research that breaks those learning goals into subgoals that can be appropriately sequenced. We highlighted research on learning trajectories as an example of this sort of work but also emphasized the need to work at a grain size that is compatible with teachers' classroom practice. Finally, in our July (Cai et al., 2017c) editorial, we argued that the implementation of learning opportunities in the classroom is an integral element of research that has an impact on practice.
In our May editorial (Cai et al., 2017), we argued that a promising way of closing the gap between research and practice is for researchers to develop and test sequences of learning opportunities, at a grain size useful to teachers, that help students move toward well-defined learning goals. We wish to take this argument one step further. If researchers choose to focus on learning opportunities as a way to produce usable knowledge for teachers, we argue that they could increase their impact on practice even further by integrating the implementation of these learning opportunities into their research. That is, researchers who aim to impact practice by studying the specification of learning goals and productively aligned learning opportunities could add significant practical value by including implementation as an integral part of their work.
The variation theory of learning emphasizes variation as a necessary condition for learners to be able to discern new aspects of an object of learning. In a substantial number of studies, the theory has been used to analyze teaching and students’ learning in classrooms. In mathematics education, variation theory has also been used to explore variation in sets of instructional examples. For example, it has been reported how teachers, by using variation and invariance within and between examples, can help learners to engage with mathematical structure. In this paper, we describe the variation theory of learning, its underlying principles, and how it might be appropriated by teachers. We illustrate this by an analysis of one teacher’s teaching before and after he participated in three lesson studies based on variation theory. Both the theory and the empirical illustration focus on ‘what is made possible to learn’ in different learning situations. We show that in the two analyzed lessons, different things were made possible to learn.
In our first editorial (Cai et al., 2017), we highlighted the long-standing, critical issue of improving the impact of educational research on practice. We took a broad view of impact, defining it as research having an effect on how students learn mathematics by informing how practitioners, policymakers, other researchers, and the public think about what mathematics education is and what it should be. As we begin to dig more deeply into the issue of impact, it would be useful to be more precise about what impact means in this context. In this editorial, we focus our attention on defining and elaborating exactly what we mean by “the impact of educational research on students' learning.”
Mathematics education has increasingly in the last 30 years acknowledged the crucial role that language plays in learning. However, this has mainly been the roleof language in cognition, such as students’ understanding of mathematics concepts and relationships, and not necessarily its impact on social, cultural, and political issues and learning in mathematics. Furthermore, multilingualism in mathematics classrooms has been underpinned by a deficit discourse that viewed languages other than the language of instruction as a “problem”.
This chapter describes the interplay between task design and student learning that informs teachers’ decisions about goals and pedagogies. Based on their mathematical goals for students, teachers face many choices: choosing or designing tasks and sequences of tasks; selecting media for presenting tasks to students and for students to communicate results; planning pedagogies to best realize opportunities afforded by tasks; determining the level of complexity of tasks for their students, including ways of adapting for them; and anticipating processes for assessing student learning. Each of these choices is influenced by teachers’ understanding of the relevant mathematics, by assessments of the preparedness of their students, by the teacher’s experience or creativity or access to resources, by their expectations for student engagement, and by their commitment to connecting learning with students’ lives. Decisions are also informed by teachers’ awareness and willingness to enact the relevant pedagogies.
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The Third International Mathematics and Science Study (TIMSS) 1999 Video Study examined eighth-grade mathematics teaching in the United States and six higher-achieving countries. A range of teaching systems were found across higher-achieving countries that balanced attention to challenging content, procedural skill, and conceptual understanding in different ways. The United States displayed a unique system of teaching, not because of any particular feature but because of a constellation of features that reinforced attention to lower-level mathematics skills. The authors argue that these results are relevant for policy (mathematics) debates in the United States because they provide a current account of what actually is happening inside U.S. classrooms and because they demonstrate that current debates often pose overly simple choices. The authors suggest ways to learn from examining teaching systems that are not alien to U.S. teachers but that balance a skill emphasis with attention to challenging mathematics and conceptual development.
This case study explores the contextual factors involved in one high school mathematics department that is successful in getting African-American students to take advanced levels of mathematics. Examination of the school site, teacher and staff interviews, and school documents highlight the particular manner in which student success may be aided by five characteristics of the department: a rigorous curriculum and the support to maneuver through it; active commitment to students; commitment to a collective enterprise; a resourceful and empowering chairperson, and standards-based instructional practices. This article also explores the role of race and racism on student learning. Implications for policy and research are discussed.
This study analyses student advancement through us mathematics curricula as an outcome of departmental organization and patterns of collective teacher beliefs and practices. Evidence from a study of eight high school mathematics departments supports the argument that four key components facilitate students taking additional mathematics courses ‐‐ especially those at advanced levels ‐‐ and scoring higher on mathematics achievement tests. These components are: (1) a rigorous and common curriculum, (2) active commitment to students, (3) commitment to a collective enterprise, and (4) innovative instructional practices. The interaction between these components is discussed.
Mathematics education is linked to modern technological advancement, citizenship, and matters of political economy. Access to quality mathematics teachers can have a profound influence on African American males. This article focuses on several salient issues related to the question “What is a qualified teacher?” In particular, the article describes common concerns and challenges associated with teacher quality and the advancement of African American males and other traditionally under served demographic groups in mathematics.
The research program described in this article has focused on the work students do in classrooms and how that work influences students' thinking about content. The research is based on the premise that the tasks teachers assign determines how students come to understand a curriculum domain. Tasks serve, in other words, as a context for students' thinking during and after instruction. The first section of this article contains an overview of the task model that guided research. The second section provides a summary of findings concerning the properties of students' work in classrooms, with special attention to work in mathematics classes. I conclude with a brief discussion of implications of this research for understanding classroom processes and their effects.
The purpose of this article is to identify some necessary conditions of learning. To learn something, the learner must discern what is to be learned (the object of learning). Discerning the object of learning amounts to discerning its critical aspects. To discern an aspect, the learner must experience potential alternatives, that is, variation in a dimension corresponding to that aspect, against the background of invariance in other aspects of the same object of learning. (One could not discern the color of things, for instance, if there was only one color.) The study results illustrate that what students learn in a sequence of lessons is indeed a function of the pattern of variation and invariance constituted in that sequence. All teachers make use of variation and invariance in their teaching, but this study shows that teachers informed by a systematic framework do it more systematically, with striking effects on their students' learning.
This paper problematizes the issue of how decisions about the content of mathematics education can be made. After starting with two examples where research in mathematics education resulted in different choices on the content of primary school teaching, I explore where and how, in the scientific enterprise within the domain of education, issues of the what of mathematics education are addressed. My conclusion is that, although it is often thought that scientific research cannot resolve issues related to decisions about subject matter content and goals, there is a powerful movement, with its roots in the European tradition, that pleads for the further development of didactics as a scientific discipline.
The Third International Mathematics and Science Study (TIMSS) 1999 Video Study sampled eighth grade mathematics lessons in seven countries including Australia. As well as describing teaching in these countries the study aimed to : develop objective, observational measures of classroom instruction to serve as appropriate quantitative indicators of teaching practices in each country; compare teaching practices among countries and identify similar or different lesson features across countries; describe patterns of teaching within each country; and develop methods for communicating the results of the study, through written reports and video cases, for both research and professional development purposes. The results in this report are presented from an international perspective.
Students’ attitudes toward mathematics and the strength of their mathematics preparation typically go hand in hand such that their specific effects are difficult to disentangle. Employing the method of propensity weighting of a continuous variable, we built hierarchical linear models in which mathematics attitudes and preparation are uncorrelated. Data used came from a national survey of U.S. college students taking introductory calculus ( N = 5,676). A 1-standard-deviation increase in mathematics preparation predicted a 4.72-point higher college calculus grade, whereas a 1-standard-deviation increase in mathematics attitudes resulted in a 3.15-point gain. Thus, the effect of mathematics preparation was about 1.5 times that of mathematics attitudes. The two variables did not interact, nor was there any interaction between gender and these variables.
Students' attitudes toward mathematics and the strength of their mathematics preparation typically go hand in hand such that their specific effects are difficult to disentangle. Employing the method of propensity weighting of a continuous variable, we built hierarchical linear models in which mathematics attitudes and preparation are uncorrelated. Data used came from a national survey of U.S. college students taking introductory calculus (N = 5,676). A 1-standard-deviation increase in mathematics preparation predicted a 4.72-point higher college calculus grade, whereas a 1-standard-deviation increase in mathematics attitudes resulted in a 3.15-point gain. Thus, the effect of mathematics preparation was about 1.5 times that of mathematics attitudes. The two variables did not interact, nor was there any interaction between gender and these variables.
In this paper, we argue that although mathematics educators are concerned about social issues, minimal attention has been paid to student-student interactions outside the classroom. We discuss social network analysis (SNA) as a methodology for studying such interactions in the context of an undergraduate course. We present results on the questions: Who studies with whom? What are students' study habits, and are these systematically related to the habits of those with whom they interact? Do individual and collaborative study habits predict attainment? We discuss the implications of these findings for research on undergraduate learning and on social issues in mathematics education, suggesting that SNA may provide a bridge between mathematics education researchers who focus on cognitive and on social issues.
In our March editorial (Cai et al., 2018), we considered the problem of isolation in the work of teachers and researchers. In particular, we proposed ways to take advantage of emerging technological resources, such as online archives of student data linked to instructional activities and indexed by learning goals, to produce a professional knowledge base (Cai et al., 2017b, 2018). This proposal would refashion our conceptions of the nature and collection of data so that teachers, researchers, and teacher-researcher partnerships could benefit from the accumulated learning of ordinarily isolated groups. Although we have discussed the general parameters for such a system in previous editorials, in this editorial, we present a potential mechanism for accumulating learning into a professional knowledge base, a mechanism that involves collaboration between multiple teacher-researcher partnerships. To illustrate our ideas, we return once again to the collaboration between fourth-grade teacher Mr. Lovemath and mathematics education researcher Ms. Research, who are mentioned in our previous editorials(Cai et al., 2017a, 2017b).
Mathematics educators have been publishing their work in international research journals for nearly 5 decades. How has the field developed over this period? We analyzed the full text of all articles published in Educational Studies 1n Mathematics and the Journalfor Research in Mathematics Education since their foundation. Using Lakatos's (1978) notion of a research programme, we focus on the field's changing theoretical orientations and pay particular attention to the relative prominence of the experimental psychology, constructivist, and sociocultural programmes. We quantitatively assess the extent of the "social turn," observe that the field is currently experiencing a period of theoretical diversity, and identify and discuss the "experimental cliff," a period during which experimental investigations migrated away from mathematics education journals. © 2018 National Council of Teachers of Mathematics. All rights reserved.
THE REAL WORLD OF MATHEMATICS, SCIENCE, AND TECHNOLOGY EDUCATION In this Preface, I would like to focus on what I mean by “education” and speak about the models and metaphors that are used when people talk, write, and act in the domain of education. We need to look at the assu- tions and processes that the models and metaphors implicitly and explicitly contain. I feel we should explore whether there is a specific thrust to mat- matics education in the here and now, and be very practical about it. For me education is the enhancement of knowledge and understanding, and there is a strong and unbreakable link between the two. There seems l- tle point in acquiring knowledge without understanding its meaning. Nor is it enough to gain a deep understanding of problems without gaining the appropriate knowledge to work for their solution. Thus knowledge and understanding are each necessary conditions for the process of education, but only when they are linked will the process bear fruit. Only in the b- anced interplay of knowledge and understanding can we expect to achieve genuine education.
Educational research communities bear responsibility for establishing a substantial body of evidence to support claims that drive the field. For example, one commonly accepted claim is that there is a relationship between the cognitive demand of mathematical task enactments and students’ learning. One study that is often cited in association with this claim is Stein and Lane (1996), and in 44% of those citations, Stein and Lane (1996) is the sole reference. Citation analysis reveals that many of these claims go beyond the warrants provided by the Stein and Lane study, either by granting more confidence in the relationship than the study design allows or by phrasing the claim as a causal relationship between cognitive demand and student learning. A few other studies are occasionally cited in conjunction with Stein and Lane (1996) and are summarized in this article, but there remains a need for replication studies to provide better empirical support for claims about cognitive demand and student learning and to refine our shared understanding.
Free Access Link: http://www.sciencedirect.com/science/article/pii/S0732312316301109
In a previous study of 2 schools in England that taught mathematics very differently, the first author found that a project-based mathematics approach resulted in higher achievement, greater understanding, and more appreciation of mathematics than a traditional approach. This follow-up showed that the young adults who had experienced the 2 mathematics teaching approaches developed profoundly different relationships with mathematics knowledge that contributed toward the shaping of different identities as learners and users of mathematics.
In this book an experienced classroom teacher and noted researcher on teaching takes us into her fifth grade math class through the course of a year. Magdalene Lampert shows how classroom dynamics--the complex relationship of teacher, student, and content--are critical in the process of bringing each student to a deeper understanding of mathematics, or any other subject. She offers valuable insights into students and teaching for all who are concerned about improving the learning that happens in the classroom. Lampert considers the teacher's and students' work from many different angles, in views large and small. She analyzes her own practice in a particular classroom, student by student and moment by moment. She also investigates the particular kind of teaching that aims at engaging elementary school students in learning fundamentally important ideas and skills by working on problems. Finally, she looks at the common problems of teaching that occur regardless of the individuals, subject matter, or kinds of practice involved. Lampert arrives at an original model of teaching practice that casts new light on the complexity in teachers' work and on the ways teachers can successfully deal with teaching problems.
In this article, Graham Nuthall critiques four major types of research on teaching effectiveness: studies of best teachers, correlational and experimental studies of teaching-learning relationships, design studies, and teacher action and narrative research. He gathers evidence about the kind of research that is most likely to bridge the teaching-research gap, arguing that such research must provide continuous, detailed data on the experience of individual students, in-depth analyses of the changes that take place in the students' knowledge, beliefs, and skills, and ways of identifying the real-time interactive relationships between these two different kinds of data. Based on his exploration of the literature and his research on teaching effectiveness, Nuthall proposes an explanatory theory for research on teaching that can be directly and transparently linked to classroom realities.
In order to develop students' capacities to "do mathematics," classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.
Two questions of scholarly and public policy interest concerning the well-documented racial difference in scores on achievement tests are How much of the racial difference ("gap") can be attributed to social-class differences between blacks and whites? and How much has the racial gap changed over the past 30 years? To address these questions, the authors analyzed evidence from seven probability samples of national populations of adolescents from 1965 to 1996 and found that black-white differences in achievement are large and are decreasing slowly over time. About a third of the gap in test scores is accounted for by racial differences in social class, and although this gap appears to have narrowed since 1965, the rate at which it is narrowing seems to have decreased since 1972. The two groups are becoming more equal at the bottom of the test-score distribution, but at the top, blacks are hugely underrepresented and are approaching parity with whites slowly, if at all.
This article specifies how the setup,or introduction,of cognitively demanding tasks is a crucial phase of middle-grades mathematics instruction. We report on an empirical study of 165 middle-grades mathematics teachers' instruction that focused on how they introduced tasks and the relationship between how they introduced tasks and the nature of students' opportunities to learn mathematics in the concluding whole-class discussion. Findings suggest that in lessons in which (a) the setup supported students to develop common language to describe contextual features and mathematical relationships specific to the task and (b) the cognitive demand of the task was maintained in the setup,concluding whole-class discussions were characterized by higher quality opportunities to learn.
This article reports on a review of the mathematics education research literature 1989-May 2011 specific to K-12 African American students’ opportunities to learn mathematics. Although we identify important developments in the literature, we conclude that the existing research base generally remains at the level of broad principles or orientations to teaching and is therefore inadequate for specifying forms of instructional practice that support African American students’ participation in rigorous mathematical activity. We suggest a research agenda focused on specifying forms of practice that are empirically shown to support African American students’ learning of mathematics and development of productive mathematical identities.
In the knowledge-based economy that characterizes the 21st century, most previously industrialized countries are making massive investments in education. The United States ranks poorly on many leading indicators, however, primarily because of the great inequality in educational inputs and outcomes between White students and non-Asian “minority” students, who comprise a growing share of the U.S. public school population. Standards-based reforms have been launched throughout the United States with promises of greater equity, but while students are held to common standards—and increasingly experience serious sanctions if they fail to meet them—most states have not equalized funding and access to the key educational resources needed for learning. The result of this collision of new standards with old inequities is less access to education for many students of color, rather than more. This article outlines current disparities in educational access; illustrates the relationships between race, educational resources, and student achievement; and proposes reforms needed to equalize opportunities to learn.
This paper reports on 3-year case studies of 2 schools with alternative mathematical teaching approaches. One school used a traditional, textbook approach; the other used open-ended activities at all times. Using various forms of case study data, including observations, questionnaires, interviews, and quantitative assessments, I will show the ways in which the 2 approaches encouraged different forms of knowledge. Students who followed a traditional approach developed a procedural knowledge that was of limited use to them in unfamiliar situations. Students who learned mathematics in an open, project-based environment developed a conceptual understanding that provided them with advantages in a range of assessments and situations. The project students had been "apprenticed" into a system of thinking and using mathematics that helped them in both school and nonschool settings.
The current debates about the future of mathematics education often lead to confusion about the role that research should play in settling disputes. On the one hand, researchers are called upon to resolve issues that really are about values and priorities, and, on the other hand, research is ignored when empirical evidence is essential. When research is appropriately solicited, expectations often overestimate, or underestimate, what research can provide. In this article, by distinguishing between values and research problems and by calibrating appropriate expectations for research, I address the role that research can and should play in shaping standards. Research contributions to the current debates are illustrated with brief summaries of some findings that are relevant to the standards set by the NCTM.
This article reports on an ethnographic study of a 10-year-old's pursuit of school-based mathematics across school and home to suggest that participating in school-based mathematics is a cross-setting phenomenon in at least 2 ways. First, I illustrate how accomplishing school-based mathematics literally extends into the home and how individuals recruit resources from their histories of participation in alternative settings to accomplish the work of school-based mathematics. Second, I show how a youth's social identification in the classroom is shaped by his teacher's partial accounts of how learning is arranged for in the home. Approaching participation in school-based mathematics as a cross-setting phenomenon illustrates the complexity inherent in participating in schooling and raises questions about how to coordinate schooling across school and home settings.
In the present study the relationship between teaching and learning was examined using a conceptual framework that links dimensions of instructional tasks with gains in student learning outcomes. The greatest student gains on a performance assessment consisting of tasks that require high levels of mathematical thinking and reasoning were related to the use of instructional tasks that engaged students in the “doing of mathematics” or the use of procedures with connections to meaning. In addition, student performance gains were greater for those sites whose tasks were both set up and implemented to encourage the use of multiple solution strategies, multiple representations, and explanations. Whereas, student performance gains were relatively small for those sites whose tasks tended to be both set up and implemented in a procedural manner and that required a single solution strategy, single representations, and little or no mathematical communication.
Many researchers who study the relations between school resources and student achievement have worked from a causal model, which typically is implicit. In this model, some resource or set of resources is the causal variable and student achievement is the outcome. In a few recent, more nuanced versions, resource effects depend on intervening influences on their use. We argue for a model in which the key causal agents are situated in instruction; achievement is their outcome. Conventional resources can enable or constrain the causal agents in instruction, thus moderating their impact on student achievement. Because these causal agents interact in ways that are unlikely to be sorted out by multivariate analysis of naturalistic data, experimental trials of distinctive instructional systems are more likely to offer solid evidence on instructional effects.
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