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Learning Trajectory Based Instruction: Toward a Theory of Teaching
Abstract
In this article, we propose a theoretical connection between research on learning and research on teaching through recent research on students’ learning trajectories (LTs). We define learning trajectory based instruction (LTBI) as teaching that uses students’ LTs as the basis for instructional decisions. We use mathematics as the context for our argument, first examining current research on LTs and then examining emerging research on how mathematics teachers use LTs to support their teaching. We consider how LTs provide specificity to four highly used frameworks for examining mathematics teaching, namely mathematical knowledge for teaching, task analysis, discourse facilitation practices, and formative assessment. We contend that by unifying various teaching frameworks around the science of LTs, LTBI begins to define a theory of teaching organized around and grounded in research on student learning. Thus, moving from the accumulation of various frameworks into a reorganization of the frameworks, LTBI provides an integrated explanatory framework for teaching.
... Unfortunately, in fact, students do not understand what they were studying when they study inverse functions. Students encounter learning obstacles when studying inverse functions, according to previous research [7][8][9][10][11][12][13][14][15][16]. The study's findings from Pratamawati [8] indicated that students encountered learning obstacles in the form of difficulties solving inverse function problems because some materials were not included in the textbooks used during the learning process. ...
... Other research, Perbowo and Anjarwati [10] demonstrated an incompatible curriculum in which instructional materials were not supplied completely, resulting in students encountering difficulties solving problems involving the inverse function. Along with obstacles, students encounter various misconceptions when studying inverse functions [11]. The first misunderstanding is that many students believe = ( ) equals = −1 ( ) (Figure 1). ...
... Additionally, Okur [2] discovered inverse function misconceptions, as evidenced by students' application of the same rules to answer various inverse function issues. Students frequently have the misunderstanding [11] that they can derive the graph of an inverse function solely by reflecting the graph ( ) on the line = in Cartesian coordinates ( Figure 2). This error is generated by substituting x and y, which is used to get the inverse of a known function. ...
... Teaching grounded in students' thinking and learning is most likely to develop concepts, skills, and creativity and do so for all children by recognizing and building on their strengths. To do so, research suggests ensuring that learning progresses along researchbased paths [15][16][17]. Learning trajectories are composed of three components: a goal, a developmental progression, and instructional activities [18]. To attain a certain competence in a given domain or topic (the goal), children learn each successive level of thinking (the developmental progression), aided by instruction (activities and teaching practices) designed to build the mental actions-on-objects that enable thinking at each later level. ...
... (the LT instruction) [19][20][21]. In this way, learning trajectories can facilitate developmentally appropriate teaching and learning for all children [17,22], including children with disabilities [23]. ...
Foundational thinking for later use of technology, particularly coding, is necessary for an inclusive and sustainable future. Inclusive practices beginning in early childhood recognize children’s innate development of computational thinking—sequencing, repetition and looping, debugging, decomposing and composing, representation, and causality. This qualitative research describes processes of developing and evaluating hypothesized developmental progressions. Inclusive engagement of children with and without disabilities is described in examples for each level of each developmental progression. Implications for teaching and learning with inclusive practices are described for children with and without disabilities.
... Student outcomes based on measures of higher levels of thinking and reasoning have shown greater success when the cognitive demand of the task is maintained throughout the lesson (Boaler & Staples, 2008;Boston & Smith, 2009;Stein & Lane, 1996;Tarr et al., 2008). Several researchers (e.g., Cai et al., 2011;Grouws et al., 2013;Schoenfeld, 2002;Sztajn et al., 2012) have investigated the effects of mathematics curricula intentionally designed with higher-level tasks and found an increase in student achievement, reasoning, and problem-solving. Tekkumru-Kisa et al. (2015) adapted the mathematics education framework to analyze the cognitive demand of science tasks. ...
... Professional development centered on supporting teachers' implementation of high cognitive demand tasks have demonstrated an increased focus on teaching actions and on student thinking during lesson implementation (Boston & Smith, 2009;Walkoe, 2015). Researchers in mathematics education have looked at the effects of cognitively demanding tasks within the curricula on student achievement (Cai et al., 2011;Grouws et al., 2013;Schoenfeld, 2002;Sztajn et al., 2012), and the TAG-C framework can be used in a similar way to develop and assess curriculum for learning coding. Curriculum developers can use this framework to consider the cognitive demand of their tasks and how they support students' higher-level thinking, and then they can research the effects of such a curriculum on student achievement. ...
... Of the three things, the most important and one of the assessment criteria in teaching practice in schools is lesson planning. In learning planning, it is implied that the activities of preparing Hypothetical Learning trajectory (HLT) and formative assessments [5][6][7]. The activity of preparing lesson plans can also describe the extent to which a student candidate for a mathematics teacher has mastered the knowledge of being a teacher. ...
... In another version, curriculum analysis activities are more about dissecting the curriculum. Understanding the school curriculum is one of the important foundations in the education of prospective mathematics teachers, because the HLT prepared by prospective teachers is the result of this lecture [7]. ...
... It is revealed by Confrey et al., (2017), Ivars et al., (2018),and Nickerson et al., (2017 that lecturers need to interpret and decide how to respond to students' mathematical ideas and analyze students' conceptual understanding, so that lecturers can develop learning activities that can build better ways of thinking, reasoning and communicating. In order to encourage students able to think in a better way, lecturers can use HLT (Hypothetical Learning Trajectory) which serves as a reference for lecturers in identifying learning objectives, interpreting students' mathematical thinking, and responding through appropriate instructions (Sztajn et al., 2012). It also applies in teaching and learning mathematics that uses technology. ...
Mathematical communication skill in a mathematics learning process isa basic skill that must be possessed by students in developing the ability to think, reason, and solve problems. Especially in virtual learning, both online learning and blended learning, teachers must be able to conduct learning that built students' mathematical communication skills. Implementing web-based geometry in space and plane learning can be an alternative for forming mathematical communication skills with the aid of technology in today's virtual learning environment. This research aimed to examine 1) how web-based geometry in space and plane learning is implemented, 2) how students' mathematical communication skills are built through web-based geometry in space and plane learning, and 3) what learning alternatives need improvement to optimize students' mathematical communication skills. This research used a qualitative method with instruments in the form of lecturer activity observation sheet, student activity observation sheet, interview guide, and mathematical communication skills test. Data were collected through observation of learning implementation, mathematical communication skills testing, and interviews based on the test results. The data were analyzed with a descriptive technique. The results showed that 1) web-based geometry in space and field learning can be implemented by exploring with concept discussion on the activity pages, 2) web-based geometry in space and plane learning could help students explore ideas throughout the students' mathematical communication skills formation process, and 3) the learning could be improved by developing an HLT (hypothetical learning trajectory) using technology to develop a higher level of mathematical thinking ability.
... A hypothetical learning trajectory is a construct that involves hypotheses about "the order and nature of the steps in the growth of students' mathematical understanding, and about the nature of the instructional experiences that might support them in moving step by step toward the goals of school mathematics" (Daro, Mosher, & Corcoran, 2011, p. 12). Therefore, hypothetical learning trajectories provide prospective teachers with a cognitive model for thinking about (interpreting) and acting (deciding) (Sztajn, Confrey, Wilson, & Edgington, 2012;Thomas et al., 2015) and with a specific language to describe mathematics teaching (Edgington et al., 2016). ...
... For LPs to be useful to educators' instructional decision making, they need to accurately and sufficiently represent the plausible pathways that describe how knowledge, skills, and processes interact in increasingly complex ways and through which students can develop greater sophistication in their understandings (Duschl et al., 2011;Graf & van Rijn, 2015;Graf et al., 2021). While LPs are not meant to be deterministic, they can provide teachers with a sense of what skills students are demonstrating proficiently and the next steps toward their learning goals Daro et al., 2011;Duncan & Rivet, 2018;Duschl et al., 2011;Maloney et al., 2014;NRC, 2007;Sztajn et al., 2012). ...
Spatial reasoning comprises a set of skills used to mentally visualize, orient, and transform objects or spaces. These skills, which develop in humans through interaction with our physical world and direct instruction, are strongly associated with mathematics achievement but are often neglected in early grades mathematics teaching. To conceptualize ways to increase the representation of spatial reasoning skills in the classroom, we examined the outcomes of cognitive interviews with kindergarten through grade two students in which they engaged with one spatial reasoning task. Qualitative analyses of students’ work samples and verbal reasoning responses on a single shape de/composition task revealed evidence of a continuum of sophistication in their responses that supports a previously articulated hypothetical learning progression. Results suggest that teachers may be able to efficiently infer students’ skills in spatial reasoning using a single task and use the results to make instructional decisions that would support students’ mathematical development. The practical implications of this work indicate that additional classroom-based research could support the adoption of such practices that could help teachers efficiently teach spatial reasoning skills through mathematics instruction.
... Furthermore, hands-on activities like experiments and lab work allow students to apply their knowledge and skills in real-world contexts (Duit and Treagust, 2003). Using manipulatives is an effective strategy for exploiting a hands-on approach to mathematics (Sztajn, et al. 2012;Adams and Wiemelt, 2016). Minds-on activities, such as discussions and debates, help students develop critical thinking skills and engage with scientific concepts meaningfully (Windschitl et al., 2008). ...
This paper analyses the most effective teaching strategies in 4th-grade mathematics students, considering estimations for Arab countries participating in the TIMSS 2019 assessment. The paper aims to enable educators to design curricula and implement teaching strategies that align with the specific characteristics of these countries. It contributes to the literature by a detailed analysis of the most effective teaching strategies for students in Arab countries and by developing a methodology for assessing variability in mathematics performance. The method employs multilevel models, which allow us to assess where most of the variability in mathematics achievements is found, comparing the attainment between and within classrooms. The results found that nearly half of the variance was associated with differences in classroom achievement. The results show that mathematics strategies involving teacher-guided instruction are negatively associated with achievements. The opposite is observed for strategies involving independent study. The paper also shows that relying on these strategies does not alter the relationship between socioeconomic background and achievement. In addition, the results show the importance of positive student attitudes towards mathematics and of promoting a culture where students value and enjoy learning the subject. This can be achieved through engaging instructional practices and creating student collaboration and exploration opportunities.
... These pathways illustrate students' thought processes during instruction, representing hypotheses based on instructional designs that aim to develop students' thinking toward achieving learning objectives (Clements & Sarama, 2004). They offer insight into expected learning trajectories, informed by empirical experiences that identify the steps students may take in developing mathematical ideas while acknowledging that each student's path may differ (Sztajn et al., 2012). Learning pathways assist teachers in modeling student thinking, identifying the next steps in learning, and interacting effectively during instruction (Wilson et al., 2014). ...
This research aims to find learning obstacles students experienced during natural number multiplication lessons in third-grade elementary school. Learning obstacles were obtained from the Respondent Ability Test (RAT) analysis on students who had studied the lessons on multiplication of natural numbers. The method used in this research is a qualitative descriptive method. The subjects were respondents who participated in RAT, namely 22 third-grade elementary school students in Bandung, Indonesia. Data collection techniques use tests, interviews, and documentation. Data analysis techniques include data collection, reduction, display, and conclusion. Learning obstacles found in natural number multiplication lessons are categorized into three types, namely ontogenic, epistemological, and didactical obstacles. Ontogenic obstacles occur when there is a gap in the students' thinking when analyzing concrete to abstract concepts. Epistemological obstacles may occur if students need help to apply the concepts they have learned in the context of story problems. Didactical obstacles happen when the learning carried out by teachers is only procedural, so students only memorize each multiplication result. Therefore, it is essential to analyze these learning obstacles and find the causes as a reference in finding solutions to design future learning tools that prevent these learning obstacles from happening.
... Researchers can investigate how various instructional strategies affect students' learning trajectories over time by using longitudinal studies (Sztajn et al., 2012). Through the process of recording student experiences and perspectives across several years, teacher's can pinpoint instructional practices that reliably provide transformative learning results. ...
This article examines the efficacy of the Chemistry Education program offered by the Department of Science and Environment Education, Central Department of Education, Tribhuvan University. It also explores students' perceptions and experiences regarding the program, focusing on its strengths and challenges. Using purposive sampling, ten students enrolled in the Science Education program were selected for data collection through face-to-face interviews and conversations. Verbatim and conversational analyses were employed to analyze the data. The findings indicate that the culture of teaching and learning has shifted from traditional, teacher-centered approaches to student-centered pedagogies, with a strong emphasis on research-based and project-oriented learning. Students appreciated the comprehensive coverage of theoretical concepts in the Chemistry Education program. However, they expressed concerns about the limited opportunities for practical laboratory work, which they felt hindered their ability to fully grasp and apply the concepts learned in class. Additionally, the study found that while the program integrates theoretical constructs with real-world applications through experiential learning modalities, doubts persist regarding its effectiveness in cultivating skills essential for long-term professional success. It is strongly recommended to elevate the Chemistry Education program by integrating more dynamic and interactive pedagogical approaches to more effectively nurture the skills critical for sustained professional success.
... These activities can be discussions, experiments, assignments, or other interactions that allow students to be actively involved in the learning process; (3) Prediction of Student Response: Considers how students might respond or react to the designed learning activity, including possible difficulties or confusion they may face; and (4) Learning Adjustments: Planning strategies to overcome challenges or difficulties that students may face during the learning process. This could include providing additional assistance, simplifying materials, or providing additional resources (Baroody et al., 2022;Simon, 2020;Clements & Sarama, 2020;Sztajn et al., 2012). By using the HLT concept, educators can more effectively plan and manage the learning process, as well as be more responsive to students' individual needs by helping to ensure that the learning experience is more directed and meaningful for each student, thus enabling them to achieve their learning potential optimally (Lantakay et al., 2023), but there is a lack of data specific to microbiology. ...
The application of effective teaching methods needs to be implemented to improve the quality of microbiology education which can empower important competencies in the current era. This study aims to analyze the application and trajectory of argumentation-based inquiry learning through the application of microbiology lectures. This research uses research design methods consisting of preliminary design steps, teaching experiments and retrospective analysis. The source of the data comes from student learning activities in argumentation-based inquiry learning implemented in microbiology courses. The results showed that the learning trajectory of Hypothetical Learning Trajectory (HLT) in microbiology lectures with an argumentation-based inquiry model was in accordance with the stages of student research ranging from determining research themes, compiling proposals, designing and implementing data collection, analyzing data, discussing research results, writing research reports to conducting scientific publications in journals. Students who carry out microbiology lectures using argumentation-based inquiry learning through the implementation of different research in the field of microbiology experience a similar learning trajectory so that a specific and distinctive set of Hypothetical Learning Trajectory (HLT) can be formulated.
... HLT is carried out in order to be able to anticipate various strategies and correct teacher misunderstandings with rich and accurate explanations (Yilmaz, 2015). HLT can assist researchers in improving the quality of activity development design (CCSSI, 2010) and is a first step toward a teaching theory centered around research on learning (Sztajn et al, 2012). HLT also provides opportunities for teachers to be able to collaborate in learning (Wawro et al., 2012) so they can generate productive ideas (McClelland, Acock, & Morrison, 2006). ...
This study aims to gain an in-depth understanding of how 15 elementary school teachers in a professional learning community demonstrate how to build their technological pedagogical content knowledge. The qualitative method used is Didactical Design Research. Data collection uses triangulation techniques, namely document studies, observations and interviews. The results of the study show that studying in a professional learning community will make it easier for teachers to master technology so that it can make it easier for them to plan, implement and assess learning. It was also revealed that the teacher's teaching practice was in line with the plan that had been prepared beforehand. Based on the implementation of professional teacher learning activities, it can be concluded that the teacher's knowledge and skills in teaching develop very well, can further develop meaningful learning activities, and can better master student characteristics. In addition, students can be more independent and really help students in higher order thinking.
... For a similarly complex question in the posttest, students' answers reflected a more structured answer ( Figure 2). Sztajn, Confrey, Wilson, and Edgington (2012) summarize that learning trajectory became a unifying element for instructions when teachers organize teaching from a learning trajectory perspective. In this current study, designing Hypothetical Learning Theory for teaching linear equations in which learning goals (PISA and national curriculum consideration) embodied in teaching activities and presumed obstacles in learning becomes a sound and effective basis for preparing assistance efforts blended within linear equations learning instruction. ...
Prediction of how learning might proceed can be used as a basis for designing and achieving successful learning. A Hypothetical Learning Trajectory (HLT) consisted of learning goals, a set of learning activities, and a hypothesized learning process was developed for learning linear equations in which PISA and the scientific approach model also became points of consideration. HLT implementation in three learning schemes suggested the importance of scaffolding in learning linear equations. Sufficient and strategic scaffolding can improve student’s understanding and facilitate the students in overcoming obstacles when learning linear equations.
... Methodologically, while many studies assess the effectiveness of automated formative feedback through posttest-to-pretest comparisons (Chew et al., 2019) or via case studies (Kim & McCarthy, 2021;Zhang, 2020), the researchers' approach involves evaluating change over multiple time points, employing learning trajectories. These trajectories depict students' probable cognitive developments as they progress in their task (Sztajn et al., 2012). This approach enables teachers to make diagnostic inferences and offer tailored feedback based on the data supplied, even with large sample sizes (Beese, 2019;Plass & Pawar, 2020). ...
We conducted a study with the aim to investigate the effectiveness of automated formative feedback in improving students' ability to summarize. One-hundred and thirty-eight undergraduate students in an elementary education program were asked to summarize six scientific texts, with the experimental group (N=87) receiving automated formative feedback in a computer-based learning environment (FALB). FALB provides automated feedback about content coverage, copying words avoidance, redundancy avoidance, relevance, and length. Comparing the experimental group to a control group (N=51), results implied that summarizing skills could be fostered when interacting with FALB. In particular, the automated formative feedback promoted the adherence to the predefined length and the avoidance of copying words while maintaining a high content coverage, fostering cognitive processes essential for constructing a mental model of a text. In addition, students in the experimental group were able to maintain high quality summaries in their final session when not scaffolded. In conclusion, FALB supports the alignment of internal standards with external standards and provides an incentive to revise and engage with texts.
... Learning trajectories 6 can be defined as a representation of the expected paths a student may follow as they gain successively more sophisticated ways of thinking about an idea, concept or topic (Simon, 1995;Sztajn et al., 2012;Groff, 2017). If a student is considered to be 'lagging' behind the standards set out in the curriculum, there is the risk of the system losing track of them and that they fall behind even further. ...
In this chapter, we present an overview of the main messages of the OECD Learning Compass 2030 and their implications for the mathematics curricula. We introduce the key concepts of this learning framework, which describes the kind of competencies today’s students will need for the future, and then look at how it and the OECD Future of Education and Skills project can influence mathematics learning and teaching. We present common challenges that OECD countries face when redesigning their mathematics curriculum and how these challenges can be addressed. Finally, we discuss implications for the design of mathematics curricula, including how learning progressions and design principles can support more effective and equitable mathematics learning.
... and engineering students' learning trajectories evolve over time, from 1 st year to senior year, along a novice to expert spectrum. We borrow the idea of "learning trajectories" from mathematics education that can paint the evolution of students' knowledge and skills over time over a set of learning experiences (Clements & Sarama, 2004;Simon, 1995;Sztajn et. al., 2012;Corcoran, Mosher & Rogat, 2009;Maloney and Confrey, 2010). We use a theoretical framework based on adaptive expertise and design thinking adaptive expertise to further advance a design learning continuum (Hatano and Inagaki, 1986;Schwartz, Bransford & Sears, 2005;McKenna, 2007;Neeley, 2007). ...
... Of the notable potentials of LPs, Corcoran et al. (2009) proposed and others reinforced (e.g. Krajcik, 2011;Sztajn et al., 2012) that LPs can play a critical role in supporting adaptive instruction. Without understanding student thinking, teachers are unlikely to respond to student learning needs central to their instruction. ...
Learning progressions (LPs) are considered to have great potential to improve pedagogical practices. However, even with LPs, teachers may still be unaware of the barriers that keep students from progressing; many are struggling with essential pedagogical strategies to support students' progression. This study thus proposed an educative LP (ELP), a framework that informs teachers about students' cognitive development and provides pedagogical strategies to facilitate learning. Specifically, we developed an ELP of scientific modelling competence (ELPoSMC), integrating LP levels, learning challenges, and model-based inquiry, to assist teachers in lesson plan critique. We implemented the ELPoSMC with novice physics teachers (n = 32) and compared teachers' lesson plan critiques with a control group (n = 40). Results indicate that the experimental group significantly outperformed the control group, especially on seven out of 12 functions of ELP. Teachers' critiques addressed relationships among four elements in lesson planning: student characteristics, learning goals, teaching strategies, and curriculum materials. Experimental group were more likely to tap relationships between 'student characteristics' or 'learning goals' with other elements, while the control group tended to use 'teaching strategies' or 'curriculum materials' as a proxy for critiques. The study indicates that the ELP, compared to general LP, facilitated teachers' attention to supporting students with varying characteristics to achieve learning goals. ARTICLE HISTORY
... In many cases, they have taken multiple years of graduate-level coursework (e.g., Harrington et al., 2017;Goodman et al., 2017), often in programs that are based on the 2013 Association of Mathematics Teacher Educators' (AMTE) Standards for Elementary Mathematics Specialists or 2012 National Council of Mathematics Teachers' (NCTM) Standards-Elementary Mathematics Specialist. These programs seek to develop strong knowledge of elementary mathematics, including the ways that student knowledge develops in particular domains (Sztajn et al., 2012), what representations can effectively portray and connect mathematical concepts , and how school policies and structures (including the use of textbooks, testing practices, teacher autonomy, leadership practices, etc.) can influence mathematics teaching and learning (de Araujo et al. 2017). Research suggests that EMS programs can have a substantial impact on teachers' knowledge, as well as their beliefs about mathematics (e.g., Campbell & Malkus, 2014;Swars et al., 2018), and that EMSs can support improved learning outcomes for children (Kutaka et al., 2017;Rigelman & Lewis, 2023). ...
We report on the differences in mathematics learning environments in classes taught by certified Elementary Math Specialists (EMSs) (n = 28) and their peers (n = 33) as determined by observations of instruction. We used path analysis to examine how variables such as mathematical knowledge for teaching, beliefs, and background characteristics were related to the learning environment. We used the Classroom Learning Environment Measure (CLEM) observation protocol, which attends to aspects of mathematics lessons such as opportunities for students to justify their reasoning and attend to mathematical concepts. Our analysis revealed that learning environments incorporating such elements were significantly more prevalent in classes taught by EMSs, and that there were two paths indicating mediation effects on the relationship between EMS status and learning environment. One path was related to teachers’ beliefs about the primacy of computation in learning mathematics; the other path was related to teachers’ mathematical knowledge for teaching and their beliefs about the extent to which mathematical knowledge is constructed by the learner. We share implications for EMS programs and recommendations for future research on the impact of EMSs in elementary schools.
... membantu proses pemahaman siswa. Simon mengekspresikan hipotesis LT (hypothetical learning trajectory atau HLT) sebagai gambaran proses pembelajaran ketika siswa mengalami proses pembelajaran mulai dari awal sampai tercapainya tujuan pembelajaran (Sztajn, 2012;Fuadiah, 2015). Sejalan dengan itu, Rezky (2019) menyatakan bahwa melalui HLT, terciptanya dugaan guru tentang bagaimana siswa belajar, sehingga bukan hanya materi yang menjadi bahan pertimbangan tetapi juga melihat paham atau tidaknya siswa selama proses pembelajaran berlangsung. ...
In today's modern era, students are required to become learning centers. Thus, students must have prior experience and knowledge in order to be able to connect with the next concept (material). However, because the differences between the nature of elementary schooler and mathematics, teacher needs to structure learning that can be a bridge to neutralize these differences. One way to structure the learning is make a learning design such as Hypothetical Learning Trajectory (HLT). HLT is an alleged learning process as an anticipation of things that might happen, both students' thinking processes and other things in the learning process. The term HLT refers to the teacher's lesson plan which is designed not only to consider the material, but also the level of understanding, knowledge and student characteristics of the material being studied, so that students experience the learning process from the beginning to achieving the expected mathematics learning goals for students. This article is a literature study that aims to discuss HLT in elementary mathematics learning.
... To accomplish our objectives, we conjectured that Learning Trajectories (LT) could guide teachers' instruction. They offer structured developmental pathways for teaching spatial orientation skills and incorporate activity ideas to facilitate children's progression through each stage (Clements & Sarama, 2014;Maloney et al., 2014;Sztajn et al., 2012). LT is a road map for teachers to determine children's baseline level, where to start training, and what path to follow. ...
Although earlier studies have documented the importance of spatial orientation skills in early childhood development, teachers often need more experience and guidance in teaching these skills. This research investigates how two kindergarten teachers, guided by two teacher education researchers, utilized Learning Trajectories (LT) in a Professional Development (PD) program to enhance their instructional practices in teaching spatial orientation skills. Employing a qualitative research approach, researchers collected data through semi-structured interviews, field notes, and documents prepared by the teachers and analyzed it using content analysis. Findings reveal that the program provided teachers with a framework for making instructional decisions and facilitated children’s engagement in spatial orientation tasks. However, there is a need for further in-service programs on child assessment using LT and instructional activities to improve target skills. This study highlights the potential of LT-based PD programs in fostering effective teaching of spatial orientation skills while identifying areas for further improvement in teacher education.
... PfBT-theories, like learning trajectories (Simon, 1995;Sztajn et al., 2012) and local instructional theories (Gravemeijer, 2004;Stephan & Akyuz, 2012), describe a possible learning route for a given topic and guide teaching through a prior analysis of the concepts and procedures to be learned. All three support student learning in certain directions, but do not claim to determine that learning. ...
This article presents a case study of two Grade 5 boys’ argumentation concerning addition and subtraction of negative numbers while using an interactive tablet-based application simulating positive and negative tiles. We examine the properties of integers they conjectured, and the kinds of evidence and arguments they used to support their conjectures. The proof-based teaching theory used to develop the tasks, and the features of the virtual manipulative environment, are described. The results show that the tasks, in combination with the virtual manipulative environment, allowed the boys to perform calculations that they had not been able to perform previously, that in one instance one boy used a deductive argument to explain a conjecture they had made, and that a known weakness of the counterbalance model of integers that was used was not a problem in this case.
... Over the last decade, learning progressions, or learning trajectories as they are often called in mathematics education research, have influenced 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 been found to impact teacher learning, instructional practice, and student learning Clements et al., 2011;Clements et al., 2013;Edgington, 2012;Supovitz et al., 2021). ...
... Thus, providing high cognitive demand tasks that will contribute to the students' understanding of elementary school mathematics is essential (Huinker & Bill, 2017;Van de Walle et al., 2019). Studies have also indicated that using high cognitive demand tasks in the classroom positively affects both students' understanding of mathematics and their attitudes toward mathematics (Ni et al., 2018;Schoenfeld, 2002;Sztajn et al., 2012). ...
As a result of the COVID-19 pandemic, distance education started at K-12 levels in the spring semester of the 2019-2020 school year. The Ministry of National Education had also published instructional tasks to be used in distance education at all grade levels in order to create mathematics learning opportunities for students and to provide resources for teachers. Well-structured and high-quality instructional tasks play an important role in students' learning mathematics. The aim of this study is to examine the quality of the elementary school mathematics tasks recommended for distance remedial education from multiple perspectives, in particular their cognitive demand levels. A total of 85 tasks focusing on 79 critical objectives in grades 1-4 mathematics were examined using document analysis. Results of this study showed that the majority of the tasks were at low cognitive demand level, cognitive demand levels did not show a balanced distribution, and some tasks had mathematical errors.
... The preparation of HLT according to Simon [29] explained that the presumptive learning path is an activity plan prepared by the teacher in anticipating possible student learning paths by considering knowledge acquisition, level of understanding, and selection of learning activities with mathematics learning objectives. The preparation of the presumptive learning path is based on the learning objectives to be achieved in the form of learning stages in the form of a series of didactical situations that are mutually sustainable [30], [31]. ...
This research aims to obtain a comprehensive picture of the hypothetical learning trajectory (HLT) that should be developed based on students' learning trajectory (LT) in learning fraction division arithmetic operations in grade V elementary school. The HLT was developed based on the analysis of the Learning Implementation Plan document made by the teacher by considering the students' learning trajectory on the material of fraction division arithmetic operation, including the learning carried out by the teacher in the classroom, examining aspects of learning obstacles that occur in the learning process, and examining what didactical situations will be built, predicting student responses that may occur to the situation created, and determining didactical and pedagogical anticipation of these responses. The HLT in this study started with the context of generating the idea of number division, learning the concept of number division such as natural numbers divided by natural numbers, and recalling the concept of fractions and the concept of division of fractions. The learning objective designed in the HLT is that students can solve problems with at least two ways related to division of fractions.
... The main purpose of the program was to enhance teachers' understanding and implementation of mathematics instruction for children between 3 -6 years old. Among other frameworks (Confrey & Maloney, 2010;Simon & Tzur, 2009;Sztajn et al., 2012;Wickstrom & Langrall, 2020), The Learning and Teaching with Learning Trajectories (LT) approach for early mathematics guided the activities of the PD. The LT consists of three related components: a) a mathematical goal, b) a hypothetical developmental progression through which the child will move forward to reach the goal and c) a set of instructional activities that are supposed to help the child to move from one level of thinking to a more complex level (Sarama et al., 2016). ...
Executive functions are all mental skills that enable and
direct behaviors for various purposes. Executive functions begin
to appear in infancy, show rapid development from early
childhood, grow mature in late adolescence, and regress in late
adulthood. Early childhood, which includes critical periods for
the development of executive functions, is considered important.
In this study, it is aimed to share current studies on executive
functions, which is a relatively new concept. In this study, data
were collected through document analysis based on the survey
model, one of the qualitative research methods. In this study, the
definition of executive functions and their neurological basis,
components, and development are discussed in the context of
early childhood. In addition, executive functions in early
childhood are explained in detail as they have an effect that
predicts executive function potential in later life.
... The main purpose of the program was to enhance teachers' understanding and implementation of mathematics instruction for children between 3 -6 years old. Among other frameworks (Confrey & Maloney, 2010;Simon & Tzur, 2009;Sztajn et al., 2012;Wickstrom & Langrall, 2020), The Learning and Teaching with Learning Trajectories (LT) approach for early mathematics guided the activities of the PD. The LT consists of three related components: a) a mathematical goal, b) a hypothetical developmental progression through which the child will move forward to reach the goal and c) a set of instructional activities that are supposed to help the child to move from one level of thinking to a more complex level (Sarama et al., 2016). ...
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Project: Algorithmic Thinking Skills through Play-Based Learning for Future’s Code
Literates
Project number: 2020-1-TR01-KA203-092333
Project financier: EU Comission, Erasmus+
... The main purpose of the program was to enhance teachers' understanding and implementation of mathematics instruction for children between 3 -6 years old. Among other frameworks (Confrey & Maloney, 2010;Simon & Tzur, 2009;Sztajn et al., 2012;Wickstrom & Langrall, 2020), The Learning and Teaching with Learning Trajectories (LT) approach for early mathematics guided the activities of the PD. The LT consists of three related components: a) a mathematical goal, b) a hypothetical developmental progression through which the child will move forward to reach the goal and c) a set of instructional activities that are supposed to help the child to move from one level of thinking to a more complex level (Sarama et al., 2016). ...
... Learning trajectories provide prospective teachers with a cognitive model for thinking about (interpreting) and acting (deciding) (Sztajn, Confrey, Wilson, & Edgington, 2012;Thomas et al., 2015) and a specific mathematical language to describe students' thinking (Edgington, Wilson, Sztajn, & Webb, 2016). In this context, learning trajectories can support prospective teachers in identifying learning goals, in interpreting students' mathematical thinking and in responding with appropriate instruction. ...
... Because research on learning largely developed separately from research on teaching, our work used LTs to link these two bodies of research. We theorized the concept of Learning Trajectories Based Instruction (LTBI) as a model of teaching where instructional decisions are grounded in research on student learning in the form of trajectories and we interpreted several highly developed domains of research on mathematics teaching in relation to these trajectories (Sztajn, Confrey, Wilson, & Edgington, 2012). Since that time, we have worked to share this model with teachers in professional development settings, and our research has empirically examined and elaborated the affordances of LTBI. ...
... Taking this perspective is also one of Buitink's (2009) criteria for welldeveloped practical theories. It is supported additionally by a core philosophical idea presented in the context of the construction of a scientific theory of teaching: how a pupil learns should be the starting point for a theory of teaching (Sztajn, Confrey, Wilson, & Edgington, 2012). This is one possible content-based criterion in judging how welldeveloped a practical theory is. ...
In this review of studies an answer is looked for the questions of the title, and an attempt is made to construct a system of integrative knowledge which can be utilized to benefit student teachers in their studies as well as teachers in their life-long learning. The teacher's practical theory is a scattered and non-uniform concept in educational studies. The teacher's beliefs is a wide construct which, in addition to the practical theory, contains self-efficacy, i.e., a motivational aspect. So far, the research has analyzed the sources of the teacher’s self-efficacy but also its connection to the teacher’s way of thinking, emotions, quality of the teaching, and the pupil's acting and learning. Emotions are an essential power in improving the quality of teaching and the student-teacher interaction. In summary, the teacher's practical theory, self-efficacy and emotions are essential research themes in educational research, but usually they have been studied separately from each other. However, in the past few years there have appeared studies which have managed to incorporate emotion and self-efficacy in the same research. In practice, a teacher has to function as a whole person and has to take into consideration the challenges of the context, so the need for more integrative research designs will be growing.
... These core content areas build on each other to create mathematical proficiency. 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). ...
Responsive teaching requires teachers to have knowledge of children’s thinking. Prospective teachers need support in developing such knowledge because they often enter teacher preparation programs with fragile mathematical knowledge and anxiety about teaching mathematics to children. We designed a learning trajectory–based intervention to support prospective teachers’ development of their knowledge of content and students in the area of children’s multiplicative thinking. The intervention was a four-lesson unit that scaffolded prospective teachers’ knowledge development according to a trajectory of children’s multiplicative reasoning. Results indicate improvements in prospective teachers’ anticipation of children’s multiplicative strategies and knowledge of the range of multiplication strategies to which children have access.
For the past two decades, the development of preservice elementary teachers’ mathematical knowledge and skills has been central to mathematics education research. Two frameworks that researchers have drawn upon to examine such development are mathematical knowledge for teaching and professional noticing (of children’s mathematical thinking). We have identified shared theoretical space between these two frameworks, and we hypothesize that effective professional noticing occurs at the intersection of developed mathematical knowledge for teaching and a high level of responsiveness with respect to the mathematical activities of students.
The study aims to explore the effect of STEM education on the hypothetical-creative reasoning skills of the pre-school pre-service teachers. The pre-school pre-service teachers were educated with STEM education for 14 weeks within the scope of the study. A mixed-method was employed in the study. The quantitative data were collected with the "Hypothetical-creative Reasoning Skills Inventory” in the study. The hypothetical-creative reasoning Inventory was applied to the pre-school pre-service teachers as a pre-test before the STEM education and a post-test after the STEM education. The qualitative data were obtained with a semi-structured interview form applied to the pre-school pre-service teachers. At the end of the study, it was found that STEM education developed the pre-school pre-service teachers' hypothetical-creative reasoning skills. The qualitative data also supported this result. The pre-school pre-service teachers claimed that they used the steps of determining the situation, collecting necessary data, suggesting solutions, doing research, analysing data, evaluating, that is, scientific problem solving during STEM education.
In educational systems where high-stakes assessment, standardized testing, normative grading, and inter-student competition are prevalent, there have been ongoing concerns that too many students are denied educational success experiences such as personal accomplishment, improvement, and progress. In the past decade, there has been increasing interest in growth goals as a means to optimize students’ access to these forms of educational success. Growth goals (comprising growth goal setting and growth goal striving) involve students seeking to outperform their past best efforts and performance, encouraging them to consider educational success in terms of personal benchmarks, more than in terms of external benchmarks. Growth goals have typically been operationalized in theory and research by way of personal best goals and self-approach goals. Given the uptick in research into growth goals, this review is a timely opportunity to synthesize definitional, theoretical, and measurement considerations that have underpinned and directed much growth goal research. It examines the environmental (e.g., teachers, classroom climate), personal, and background attributes (e.g., age, gender, SES, at-risk learning status) that are implicated in students’ growth goal setting and striving, as well as the impact of growth goals on students’ academic outcomes (e.g., engagement, achievement). Evidence from causal modelling and intervention growth goal studies is also reviewed, concluding with suggestions for practice-oriented strategy aimed at supporting students’ growth goal efforts, as well as critical issues and directions for growth goals research going forward.
This study aims to explore how a research-informed product, a hypothetical learning trajectory (HLT), was implemented by a group of Shanghai teachers in mathematics classrooms and promoted by a researcher through a Chinese lesson study. The instruction informed by the HLT served as a boundary object to promote conversations between the researcher and the teachers. The researcher played an indispensable role as a broker to facilitate the transformation of the boundary object. A fine-grained analysis of the data set showed that the team of teachers and the researcher formed a boundary encounter between teaching and research communities in which the participants negotiated and agreed on HLT-informed instruction. The study identified four turning points that are crucial for transforming HLT-informed instruction from an abstract object to a concrete form. Based on the turning points, we revealed the role played by the researcher as a broker in this lesson study. Finally, the challenges of bringing HLT into the classroom are discussed.
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Recent work physics education research has included a substantial focus on the importance of emphasizing a small set of core physics ideas over time and across contexts. Accordingly, standards documents increasingly emphasize the importance of core ideas such as energy, interactions, and waves. This emphasis on core ideas has been influenced by research into learning progressions which describe how learners build increasingly sophisticated understanding over extended time. Much of the existing empirical learning progressions research relies largely on very large sample sizes which are (by design) not dependent upon particular instructional interventions. As a result, alternative pathways for developing competence in physics using non-traditional instructional interventions may be under-identified. Coherent instruction, which emphasizes a small set of core ideas as students collaborate to make sense of meaningful phenomena over time, shows particular promise for supporting student competence in physics and identifying new pathways toward competence. In this chapter, we discuss the interplay between coherent instruction and learning progressions. Using the example of energy teaching and learning, we discuss how existing learning progressions research can serve as a basis for instructional design and how coherent instruction may present novel pathways toward the development of competence in physics over time.
The chapters in this section illustrate one of the ‘challenges’ of considering the challenge in mathematics: the terminology of ‘challenge’ itself. Challenge is conceived of as not only presenting a challenging curriculum for learners, but also in terms of the challenges faced by teachers in teaching a ‘challenging’ curriculum, the affective challenges faced by learners in terms of resilience and perseverance, and also the systemic and research challenges of curriculum change in mathematics. In this commentary, I discuss the challenge in terms of curriculum materials and tasks, the teacher’s role and the curriculum.KeywordsMathematical challengeCurriculum materialsTeachingCurriculum content
This chapter begins by outlining the key ideas and problems of the theorizing of teaching as discussed in selected English-language literature published over the past six decades. The focus is on the value of theories of teaching and the ways theories of teaching and related terms have been defined. After creating a synthesis of the various attributes which researchers have suggested can be used for assessing the quality of theories of teaching, we discuss the process and difficulties of generating theories, and present a summary of theories of teaching found in the literature. The second part of this chapter clarifies the aims of this book, describes the sampling criteria for the selection of contributors, provides an overview of the structure of the book, and lists the questions that the contributors were asked to address.
This study employed a cluster randomized trial design to evaluate the effectiveness of a research-based intervention for improving the mathematics education of very young children. This intervention includes the Building Blocks mathematics curriculum, which is structured in research-based learning trajectories, and congruous professional development emphasizing teaching for understanding via learning trajectories and technology. A total of 42 schools serving low-resource communities were randomly selected and randomly assigned to 3 treatment groups using a randomized block design involving 1,375 preschoolers in 106 classrooms. Teachers implemented the intervention with adequate fidelity. Pre- to posttest scores revealed that the children in the Building Blocks group learned more mathematics than the children in the control group (effect size, g = 0.72). Specific components of a measure of the quantity and quality of classroom mathematics environments and teaching partially mediated the treatment effect.
While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
My aim in this article is to explore 3 perspectives on bilingual mathematics learners and to consider how a situated and sociocultural perspective can inform work in this area. The 1st perspective focuses on acquisition of vocabulary, the 2nd focuses on the construction of multiple meanings across registers, and the 3rd focuses on participation in mathematical practices. The 3rd perspective is based on sociocultural and situated views of both language and mathematics learning. In 2 mathematical discussions, I illustrate how a situated and sociocultural perspective can complicate our understanding of bilingual mathematics learners and expand our view of what counts as competence in mathematical communication.
Teachers who attempt to use inquiry-based, student-centered instructional tasks face challenges that go beyond identifying well-designed tasks and setting them up appropriately in the classroom. Because solution paths are usually not specified for these kinds of tasks, students tend to approach them in unique and sometimes unanticipated ways. Teachers must not only strive to understand how students are making sense of the task but also begin to align students' disparate ideas and approaches with canonical understandings about the nature of mathematics. Research suggests that this is difficult for most teachers (Ball, 1993, 2001; Leinhardt & Steele, 2005; Schoenfeld, 1998; Sherin, 2002). In this article, we present a pedagogical model that specifies five key practices teachers can learn to use student responses to such tasks more effectively in discussions: anticipating, monitoring, selecting, sequencing, and making connections between student responses. We first define each practice, showing how a typical discussion based on a cognitively challenging task could be improved through their use. We then explain how the five practices embody current theory about how to support students' productive disciplinary engagement. Finally, we close by discussing how these practices can make discussion-based pedagogy manageable for more teachers.
Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.
This article is a review of the literature on classroom formative assessment. Several studies show firm evidence that innovations designed to strengthen the frequent feedback that students receive about their learning yield substantial learning gains. The perceptions of students and their role in self‐assessment are considered alongside analysis of the strategies used by teachers and the formative strategies incorporated in such systemic approaches as mastery learning. There follows a more detailed and theoretical analysis of the nature of feedback, which provides a basis for a discussion of the development of theoretical models for formative assessment and of the prospects for the improvement of practice.
Describes assessment tasks and a conceptual framework for understanding elementary students' thinking about the concept of length. Teachers will learn about student difficulties with length and how to differentiate instruction to reach these learners.
This article focuses on mathematical tasks as important vehicles for building student capacity for mathematical thinking and reasoning. A stratified random sample of 144 mathematical tasks used during reform-oriented instruction was analyzed in terms of (a) task features (number of solution strategies, number and kind of representations, and communication requirements) and (b) cognitive demands (e.g., memorization, the use of procedures with [and without] connections to concepts, the “doing of mathematics”). The findings suggest that teachers were selecting and setting up the kinds of tasks that reformers argue should lead to the development of students’ thinking capacities. During task implementation, the task features tended to remain consistent with how they were set up, but the cognitive demands of high-level tasks had a tendency to decline. The ways in which high-level tasks declined as well as factors associated with task changes from the set-up to implementation phase were explored.
This study examined changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period in which the teachers participated in a CGI (Cognitively Guided Instruction) teacher development program that focused on helping the teachers understand the development of children's mathematical thinking by interacting with a specific research-based model. Over the 4 years, there were fundamental changes in the beliefs and instruction of 18 teachers such that the teachers' role evolved from demonstrating procedures to helping children build on their mathematical thinking by engaging them in a variety of problem-solving situations and encouraging them to talk about their mathematical thinking. Changes in the instruction of individual teachers were directly related to changes in their students' achievement. For every teacher, class achievement in concepts and problem solving was higher at the end of the study than at the beginning. In spite of the shift in emphasis from skills to concepts and problem solving, there was no overall change in computational performance. The findings suggest that developing an understanding of children's mathematical thinking can be a productive basis for helping teachers to make the fundamental changes called for in current reform recommendations.
In the NRC report, Scientific Research in Education, (Shavelson & Towne, 2002) three broad types of research were discussed: trends, causal effects, and mechanism. Mechanism was described as research that answers the question, "how or why is it happening"; the authors 2 described "design experiments" as an "analytic approach for examining mechanism that begins with theoretical ideas that are tested through the design, implementation, and systematic study of educational tools (curriculum, teaching methods, computer applets) that embody the initial conjectured mechanism" (p. 120). The Committee identified two products of such work as "theory-driven process of designing" and "data-driven process of refining [instructional strategies]" (p. 121). Both of these products can be viewed as related to a class of research known as design studies, the focus of this chapter. Researchers across the country have recognized the need to strengthen the "instructional core" (Elmore, 1996) and to identify effective "instructional regimes" (Cohen et al., 2003) as critical to the improvement of education. Likewise, Lagemann (2002) focused on the need for more research that produces useable classroom guidance. This review synthesizes the current progress of the methodology and identifies areas for future development. Design studies are defined as "entailing both 'engineering' particular forms of learning and systematically studying those forms of learning with the context defined by the means of supporting them. This designed context is subject to test and revision. Successive iterations that result play a role similar to systematic variation in experiment" (Cobb, Confrey, diSessa, Lehrer and Schauble, 2003, p. 9). A design study is an extended investigation of educational interactions provoked by use of a carefully sequenced and typically novel set of designed curricular tasks studying how some conceptual field, or set of proficiencies and interests, are learned through interactions among learners and with the guidance of an instructor or form of tutor. The 1 The author wishes to acknowledge the assistance in preparation and editing by Dr.
This review focuses on the intrinsic character of academic work in elementary and secondry schools and the way that work is experienced by teachers and students in classrooms. The first section contains a review of recent research in cognitive psychology on the intellectual demands of the tasks contained in the school curriculum, with particular attention to the inherent complexity of most of the tasks students encounter. The findings of this research are brought to bear on the issue of direct versus indirect instruction. The second section is directed to studies of how academic work is accomplished in classroom environments. Classrooms appear to shape the content of the curriculum in fundamental ways for all students and especially those who find academic work difficult. In addition, the processes that are likely to have the greatest long-term consequences are the most difficult to teach in classrooms. The paper concludes with an analysis of issues related to improving instruction and extending current directions in research on teaching.
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.
The purpose of this report is to describe the work that has been done so far on learning progressions in science, examine the challenges to developing usable learning progressions, determine if further investments are warranted, and if so, what investments are needed to realize their promised benefits. The report examines the quality and utility of the work done to date and identifies gaps in the fields that would have to be addressed in order to move the work forward. This report has been informed by discussions and addresses (1) the nature and quality of existing work on learning progressions in science, (2) the essential elements of learning progressions, (3) the outstanding issues, challenges and debates about learning progressions in science, and (4) the research and development that must be done to realize their potential as tools for improving teaching and learning. The recommendations are based on the authors' review of transcripts of the discussions at the panel meetings; review of research reports, papers, and funding awards on learning progressions; and extended conversations with many of the participants in the meetings. Three appendixes are included: (1) CCII (Center on Continuous Instructional Improvement) Science Panel Meeting Participants; (2) Table of Learning Progressions in Science Discussed, Mentioned, and Identified by Panelists; and (3) Examples of Learning Progressions. (Contains 6 footnotes.)
structured around the key areas of research within the field [teachers' beliefs and knowledge] / begins with the history of development of the research area / followed by an account of the research methodology that has grown up around this topic and a review of empirical inquiries into teacher cognitions, including research on teacher planning, interactive thinking, and postactive reflection / research on teachers' knowledge is reviewed, followed by a short discussion of the implications of research in the field for teacher education, curriculum development, and school improvement / review concludes with a look at possible future directions of research
methodology [simulations, commentaries, concept mapping and repertory grid, ethnography and case studies, narratives] / purpose, validity, and reliability (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Based on the results of a generalizability study of measures of teacher knowledge for teaching mathematics developed at the National Center for Research on Evaluation, Standards, and Student Testing at the University of California, Los Angeles, this article provides evidence that teachers are better at drawing reasonable inferences about student levels of understanding from assessment information than they are at deciding the next instructional steps. We discuss the implications of the results for effective formative assessment and end with considerations of how teachers can be supported to know what to teach next.
Teacher knowledge and student achievement on alge-bra Assessment and classroom learning
- E G Begle
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Five practices for orchestrating produc-tive mathematical discourse Toward a working model of constructivist teaching: A reaction to Simon
- M Smith
- M K Stein
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Learning trajectories in mathematics
- P Daro
- F Mosher
- T Corcoran
Daro, P., Mosher, F., & Corcoran, T. (2011). Learning trajectories in mathematics (Research Report No. 68). Madison, WI: Consortium for Policy Research in Education.
Teacher’s uses of a learning trajectory for equipartitioning (Unpublished doctoral dissertation)
- P H Wilson
Wilson, P. H. (2009). Teacher's uses of a learning trajectory for equiparti-tioning (Unpublished doctoral dissertation). North Carolina State University, Raleigh, NC.
Retrieved from http://www.ccsso.org/Resources/Programs/Formative_Assessment_for_ Students_and_Teachers_%28FAST%29 Instructional conversations: Promoting comprehension through discussion. The Reading Teacher
- C Goldenberg
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Rousing minds to life: Teaching, learning, and schooling in social context Content then process: Teacher learning communi-ties in the service of formative assessment Ahead of the curve: The power of assessment to transform teaching and learning
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- R Gallimore
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Preparing pre-service elementary teachers to teach mathematics with learning trajectories (Unpublished doctoral dissertation)
- G F Mojica
Learning progressions: Supporting instruction and formative assessment
- M Heritage