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Developing a Learning Progression for Curriculum, Instruction, and Student Learning: An Example from Mathematics Education
Abstract
Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.
... At the core of learning progressions is the idea of incremental development (Fonger et al., 2017). They emphasize that understanding does not emerge fully formed but grows through a series of stages. ...
... This feedback can then inform subsequent instruction, helping students progress to the next stage of their learning. The integration of learning progressions into curriculum design ensures that instructional content is sequenced in a way that aligns with students' cognitive development (Fonger et al., 2017). This alignment is a core aspect of PCK, as it requires a deep understanding of how to present content logically and coherently. ...
... The progression should be neither too simplistic nor too complex for the target age group, ensuring that students can engage with the material meaningfully and build on their prior knowledge (Goldstone & Landy, 2012). Effective learning progressions are grounded in empirical research on how students learn and develop understanding in science (Fonger et al., 2017). This research includes studies on cognitive development, conceptual change, and common misconceptions. ...
This chapter explores how learning progressions can enhance Pedagogical Content Knowledge (PCK) in science education. By providing a structured sequence of concepts and skills, learning progressions help educators align teaching strategies with students' cognitive development. This alignment offers deeper insights into student understanding and engagement with scientific concepts, improving PCK. The chapter examines the theoretical foundations of learning progressions and PCK, discusses strategies for integrating progressions into classroom practice, and uses case studies to show how they aid in addressing misconceptions, scaffolding learning, and promoting conceptual understanding. It emphasizes the importance of learning progressions in refining PCK and improving science education, ultimately preparing students for advanced inquiry and problem-solving.
... Functional thinking (FT), a key concept of algebraic thinking (Blanton & Kaput, 2011;Blanton et al., 2018;Blanton, Stephens, et al., 2015;Kaput, 2008), has been recommended as an enhanced organizing concept for teaching and learning algebra, compared with the concepts typically used (e.g., expressions and equations) (Stephens, Ellis, et al., 2017). Although studies have shown that elementary school students are capable of engaging in generalizing and representing functional relationships (e.g., Blanton, 2008;Stephens, Ellis, et al., 2017), FT has yet to be addressed systemically at this level Fonger et al., 2018;Stephens, Ellis, et al., 2017). Consequently, methods for systematically and effectively introducing and developing FT along with numeration and arithmetic in elementary schools, as well as preparing students for formal algebraic learning, have yet to be determined in the literature. ...
... However, this may add more content to the "crowded" elementary mathematics curriculum (Hurst, 2015). Building on the standards-based mathematics textbook in Korea, Pang and Sunwoo (2022) explored how to redesign a unit of "pattern and correspondence" in Grade 5 to effectively promote the development of students' FT, but they only focused on one unit without fully considering the progression of students' learning over time (Confrey, 2019;Fonger et al., 2018). Cai et al. (2005) compared early algebra content in six countries and found that in some countries, FT is explored implicitly in elementary school mathematics textbooks. ...
... This finding shows that the Chinese intended curriculum (the standards and standards-based textbooks) provides many opportunities for young children to think in sophisticated and generalized ways about quantitative relationships between variables as suggested by Blanton, Brizuela, et al. (2015). This progression of exploring FT is also in alignment with the curriculum framework and learning progression proposed by Fonger et al. (2018). Different from the effort by Fonger et al. (2018) that aimed to develop a set of separate materials for developing and evaluating algebraic thinking, the examined Chinese textbooks provide an embedded approach to exploring FT-deliberately designed tasks to develop and deepen fluency of the four basic operations and provide OTL about FT as an extension although it is not explicitly stated in standards or teachers' guidebooks. ...
The mathematics education literature indicates a consensus regarding the importance of developing algebraic thinking in elementary school mathematics. However, the approaches used to implement this concept vary around the world. This study examines how a popular standards-based elementary mathematics textbook series in China provides opportunities to learn about functional thinking, which is a key component of algebraic thinking. Building on the literature, an analytical framework was generated to examine the features of function-related tasks in the textbook series across grades. The results of the fine-grained coding analysis show that four categories of function-related tasks in the textbook provide opportunities to learn about multi-modes of functional thinking. These tasks primarily serve to enhance arithmetic learning, while offering opportunities to learn about functional thinking as an embedded component. Elaborated design and arrangement of the function tasks promote opportunities to learn about multiple modes of functional thinking. In addition, two pathways to support the development of functional thinking are identified. Finally, the implications for task design and textbook development which attempt to develop functional thinking are discussed.
... The intention is to enrich the teaching of mathematics during the early stages and contribute to later learning of formal algebra during secondary education [5]. Specifically, it is purported that algebraic thinking should be a cross-functional axis of mathematical thinking throughout all the school stages [6], allowing students to participate in algebraic ways of thinking that are age-appropriate, based on their natural intuition about structure and relations [7]. This view assumes a multidimensional conception of school algebra including generalized arithmetic with a focus on perceiving structure [4], the study of functional relations [8,9], the study of patterns [10] and the representational nature of algebra [11]. ...
... Teaching has a positive effect on relational thinking when it is linked to promoting the understanding of the equal sign as a representation of equivalence [7,23]. The authors cited distinguished between different levels of sophistication in the use of relational thinking according to its lesser or greater connection to calculation. ...
... The context of numerical equalities based on arithmetic properties has been highlighted for its potential to promote this type of thinking [7,15,19]. There is a positive influence in the presence of large numbers and the consideration of equalities with op-erations in both members. ...
Current mathematics curricula have as one of their fundamental objectives the development of number sense. This is understood as a set of skills. Some of them have an algebraic nature such as acquiring an abstract understanding of relations between numbers, developing awareness of properties and of the structure of the decimal number system and using it in a strategic manner. In this framework, the term relational thinking directs attention towards a way of working with arithmetic expressions that promotes relations between their terms and the use of properties. A teaching experiment has allowed to characterize the way in which third grade students use this type of thinking for solving number equalities by distinguishing four profiles of use. These profiles inform about how students employ relations and arithmetic properties to solve the equalities. They also ease the description of the evolution of the use of relational thinking along the sessions in the classroom. Uses of relational thinking of different sophistication are distinguished depending on whether a general known rule is applied, or relations and properties are used in a flexible way. Results contribute to understanding the process of developing the algebraic component of number sense.
... Our work, which was informed by the pioneering work of Carraher et al. (2006) and Carpenter, Franke, and Levi (2003), is a step in that direction. In particular, using a learning progressions approach (described in Fonger et al., 2018), we designed an early algebra intervention for grades 3-5 that focuses on developing students' abilities to generalize, represent, justify and reason with mathematical structure and relationships across a variety of mathematical contexts. We then initiated a series of studies to examine the impact of this intervention in grades 3-5 comparing the performance of students who participated in the intervention with the performance of students who received their district's regularly planned curriculum. ...
... Each grade level sequence consisted of 18 one-hour lessons 3 that were designed to engage students in a range of critical algebraic thinking practices (generalizing, representing, justifying and reasoning with mathematical structure and relationships) 4 as they occur in the context of different big ideas regarding algebraic content (e.g.: generalized arithmetic; equivalence, expressions, equations and inequalities; and functional thinking). This approach to teaching and learning algebra, which has guided our prior work (see, e.g., Blanton et al., 2015;Fonger et al., 2018), is informed by Kaput's (2008) content analysis of algebra. Overall, these lessons used novel tasks as well as modified existing tasks from research that showed potential to facilitate students' construction of algebraic ideas (e.g., Brizuela & Earnest, 2008;Carpenter et al., 2003;Carraher, Schliemann, & Schwartz, 2008) and allowed for multiple points of entry so as to engage students at varying levels of understanding. ...
... Overall, emphasis on algebraic thinking practices was the core predictor of improved student performance. This finding is aligned with our early algebra framework and its emphasis on algebraic practices (see Fonger et al., 2018 for an extensive discussion of this progression). However, our findings underscore the importance of further refinement of our observation instruments and our coding practices. ...
Guided by a theoretical framework emphasizing the importance of fidelity of implementation (FOI), this paper explores how teachers of grades 3, 4 and 5 implemented an early algebra intervention, and the relationship between FOI and student learning. The data shared in this paper come from a longitudinal experimental research project in which 3,208 students from 46 schools were followed for three years. Videotaped classroom observations, our primary source of FOI data, were coded to capture teachers’ instructional practices, and an algebra assessment was given to assess student performance in response to the teachers’ implementation of our instructional intervention. Results revealed a significant positive relationship between aspects of teachers’ implementation and their students’ performance.
... Britt & Irwin, 2008;Schliemann, Carraher, & Brizuela, in press). We have likewise taken a longitudinal approach to early algebra research, using the construct of learning progressions as a tool to frame our work ( Fonger et al., 2016). This work has included a focus on functional thinking and the characterization of the development of students' understandings around this concept over time (Stephens, Fonger, Blanton, & Knuth, 2016a). ...
... Blanton et al., 2015). The data described here are situated in the context of an Early Algebra Learning Progression [EALP] that involves the coordination of a curricular framework and progression, an instructional sequence, assessments, and 4-244 PME40-2016 levels of sophistication that characterize student' understandings over time (Fonger et al., 2016;Fonger, Stephens, Blanton & Knuth, 2015). ...
... As Fonger et al. (2016) detail, the EALP's curricular framework guided the development of an early algebra intervention and associated assessments. The early algebra intervention consists of an instructional sequence of lessons for Grades 3-5 (ages 8-11 years). ...
This research is part of a larger study that used a learning progressions approach to characterize students' algebraic thinking over time in terms of levels of sophistication. In this paper, we report on analyses of two students' interviews over a three-year period and focus on one big idea from our learning progression—functional thinking— to demonstrate how the development of the two students' functional thinking varied. The results of this study lead us to hypothesize that such variation may be due to differences in the development of students' understandings of other core algebraic concepts.
... Developmental progressions include sequences of skills and concepts that children acquire as they build math knowledge (Frye et al., 2013). Developing a learning progression for children is a logical approach to advancing targeted, practical supports for learning, and is foundational for preschool children's academic growth (Fonger, et al., 2017;Sarama et al., 2008). ...
... This recommendation fits theme 2, logical sequencing, describing the typical sequence of skills/concepts that children acquire as they develop their mathematical understanding. Early math content areas should be taught according to a developmental progression (Fonger, et al., 2017;Sarama et al., 2008). ...
There is a prominent and steadily growing interest in mathematics learning in early childhood education. Teaching mathematics is dependent not only on the teachers’ competencies and characteristics but also on the quality of the curricular resources they use, including manuals. In the Palestinian education system, the kindergarten teacher’s manual is the sole official written curriculum resource available for kindergarten education. The purpose of this study is to analyze and strengthen the kindergarten teacher’s manual based on the Standards of the National Council of Teachers of Mathematics by proposing developmental amendments to it. To achieve this purpose, two methods were used: quantitative content analysis of the Kindergarten Teacher’s Manual based on the National Council of Teachers of Mathematics Standards and qualitative analysis of semi-structured interviews with Palestinian early childhood professionals. The results showed that although the Manual was a strong first-draft document; the mathematics content is incomplete, as it does not include all the standards’ expectations for kindergarten; and the sequencing is not logically aligned with children’s developmental progression. This study proposes amendments which are emphasizing early childhood development; re-arranging the Manual to reflect a logical sequence of mathematical skills; adding the complete mathematics content; and a restructuring of the activities themselves. The intended amended manual will have clear and compelling implications in the field and will empower kindergarten teachers to become more confident in teaching and developing the early childhood curriculum in the Palestinian education system.
... The study reported here is part of Project LEAP (Learning through an Early Algebra Progression), a suite of projects whose overarching goal has been to use a learning progressions approach (Clements & Sarama, 2004;Maloney et al., 2014;Simon, 1995) to build an effective early algebra intervention across elementary grades (Grades K-5), examine the impact of this intervention on children's algebra-readiness for middle grades (e.g., Blanton et al., 2019), and identify progressions in students' thinking as they advance through early algebra instructional sequences (e.g., Blanton et al., 2017Blanton et al., , 2015Blanton et al., , 2018bFonger et al., 2018;Stephens et al., 2017;Ventura et al., 2021). We take learning progressions to include three essential components (Clements & Sarama, 2004): (a) learning goals constructed from empirical findings about children's thinking (e.g., Gravemeijer, 2004); (b) an instructional sequence developed from these learning goals; and (c) a description of increasingly sophisticated levels of thinking students exhibit as they advance through the instructional sequence (cf., Baroody et al., 2004;Clements & Sarama, 2014;Duschl et al., 2007). ...
... 1. The term "parity" is used here to refer to whether a number (or sum) is even or odd. 2. For more on our underlying approach and the development and effectiveness of our Grades 3-5 instructional sequence, now completed, see Blanton, Brizuela et al. (2018b), Blanton et al. (2019), and Fonger et al. (2018). 3. See the section Characterizing Good Mathematical Arguments. ...
Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K–1 students’ thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K–1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students’ practice of argumentation from the start of formal schooling.
... Learning progress is an empirically testable framework that defines learning pathways on which students move from simple to more sophisticated concepts or practices in a given domain (Alonzo & Steedle, 2009). A learning progress model often connects to a curricular structure that specifies coherent and substantive content linked in an underlying continuum (Alonzo & Steedle, 2009;Covitt et al., 2018;Fonger et al., 2018;Neumann et al., 2013;Schwarz et al., 2009;Wilson, 2009;Yang & Li, 2018). For example, Schwarz et al. (2009) developed a learning progress model that defined paths for understanding and participating in scientific modeling. ...
... Learning progress assessment (LPA) draws on the curriculumbased measurement method (Deno, 1985) to diagnose learner progress over a long-term period (e.g., semester or year) (Fonger et al., 2018;Förster & Souvignier, 2014;Osborne et al., 2016;Zeuch et al., 2017). LPA outcomes provide information about student progress to teachers, helping teachers determine instructional sequences and strategies. ...
Our literature review has revealed that the definition of learning progress as a function of formative feedback remains unclear, which continues to the lack of well‐reasoned learning progress feedback (LPF) strategies. Also, few scholars have empirically examined student perceptions of informed learning progress in solving a complex problem. The current study aims (a) to generate insights that elaborate LPF in a technology‐enhanced learning environment and (b) to gain better knowledge about LPF design for technology‐enhanced formative assessment and feedback. In this design experiment, we proposed LPF design strategies in five categories. Then, we experimented with two LPF prototypes to see how students reacted to the LPF features and how students perceived the value of LPF feedback. In the context of a graduate‐level online course with 35 student participants, we performed a mixed‐data analysis. Results indicate that students benefited from the two LPF designs to navigate their learning progress. Students' LPF preference appeared to relate to their feedback literacy regarding revision behavior, cognitive progress, and emotional reaction. The current findings lay the groundwork for elaborate LPF models that are pedagogically effective and adaptive to diverse student needs. The proposed LPF design typology and further design suggestions can help advance LPF design experiments and improve technology‐based formative feedback that enables students to pursue mastery.
... In designing our intervention, we were interested in the occurrence of these practices in two of Kaput's (2008, p. 11) three content strands (''the study of structures and systems abstracted from computations and relations'' and the ''study of functions, relations, and joint variation'') because of their close alignment with empirical research on children's algebraic thinking. As reported elsewhere (Fonger et al., 2018), we organized key early algebraic concepts and practices relative to these strands under the ''Big Ideas'' (Shin, Stevens, Short, & Krajcik, 2009) of generalized arithmetic; equivalence, expressions, equations, and inequalities; and functional thinking (see Blanton, Brizuela et al., 2018, for an elaboration of these Big Ideas). ...
... Using this conceptual approach to algebra organized around essential algebraic thinking practices within content-based Big Ideas, we drew from learning progressions research (e.g., Battista, 2004;Clements & Sarama, 2004;Maloney, Confrey, & Nguyen, 2011;Shin et al., 2009;Simon, 1995) to develop an early algebra learning progression for Grades 3 to 5 that includes the following four components (Clements & Sarama, 2004): (1) a curricular framework and associated learning goals that identify core algebraic concepts within the Big Ideas and that are organized around the four algebraic thinking practices, (2) a Grades 3 to 5 instructional sequence (referred to here as the intervention) designed to address the learning goals, (3) validated assessments to measure student learning in response to the intervention, and (4) a specification of the increasingly sophisticated levels of algebraic thinking students exhibit about algebraic concepts and practices as they progress through the intervention (see Fonger et al., 2018, for an extensive treatment of the development of these components). Components 1 to 3 are the basis for the effectiveness study reported here. ...
A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population. Forty-six schools in three school districts participated. Students in treatment schools were taught the intervention by classroom teachers during regular mathematics instruction. Students in control schools received only regular mathematics instruction. Using a three-level longitudinal piecewise hierarchical linear model, the study explored the impact of the intervention in terms of both performance (correctness) and strategy use in students’ responses to written algebra assessments. Results show that during Grade 3, treatment students, including those in at-risk settings, improved at a significantly faster rate than control students on both outcome measures and maintained their advantage throughout the intervention.
... As Marzono (2010) posits, "national and state standards often do not provide guidance in regards to the building blocks necessary to reach the designated learning goals (p.3)." Also, aligning to the work of Fonger et al. (2018), progressions of learning in engineering can serve as a "form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning" (p. 30). ...
... The first objective of the ongoing project was to engage experienced teachers and content experts in the development of a taxonomy of engineering concepts and the formation of PLiEs for secondary TEE programs. To establish the curricular framework and PLiEs, the project has pursued a three-phase method, reinforced by Fonger et al. (2018), which includes Phase 1 research and development, Phase 2 identifying implementation environments, and Phase 3 testing and revision. Each of these phases involve iterative cycles of research, design, and experimentation in order to gather the data necessary to develop validated progressions of learning and assess the effectiveness of the instructional sequences. ...
While current initiatives in P-12 engineering education are promising, a clear vision and roadmap eludes educators. Little is known about how children progress through engineering learning. Few curricula have explored and investigated how an articulate P-12 engineering program may contribute the general literacy of our children. The Advancing Excellence in P-12 Engineering Education (AEEE) project represents a mandate-a call to action-to build a community with a shared focus, vision, and research agenda to ensure that every child is given the opportunity to think, learn, and act like an engineer. This paper will present the first year results of the AEEE project which includes a Taxonomy of Concepts for Secondary Engineering and the Progression of Learning in Engineering framework for articulating a sequence of knowledge and skills that students are desired to learn as they progress toward engineering literacy.
... Secondly, we compared the students' responses across different levels to confirm the unique characteristics of each level (Boeije, 2002). Additionally, we examined the characteristics of each LP level by comparing both (a) students' thinking, including FT modes and layers of algebraic generality, and (b) mathematical sophistication, including the number of variables and the level of complexity in functional relations (Battista, 2004;Fonger et al., 2018). ...
Functional thinking has long been recognized as a crucial entry point into algebraic thinking in elementary school. This mixed-method study investigates the learning progression for elementary students’ functional thinking within the context of routine classroom instruction. Drawing on the existing research, a theoretical framework was constructed to assess the functional thinking of 649 students across grades 3 to 6. The framework includes three modes of functional thinking: recursive patterning, covariational thinking, and correspondence relations, each with particular and general layers of algebraic generality. Psychometric analysis was conducted to validate the assessment instrument. The study identifies a five-level learning progression of functional thinking: Pre-Structure, Pre-Functional Thinking, Emergent Functional Thinking, Specific Functional Thinking, and Condensed Functional Thinking. The distinctive characteristics of each level are identified and illustrated. Furthermore, the developmental sequence of different functional thinking modes within this learning progression is analyzed. Finally, the theoretical and practical implications of these findings are examined.
... In order to conceptualize the linkages between algebraic curriculum, algebraic instruction, and K-12 students' mathematical learning, we will implement a modified version of the conceptual framework for learning progression Fonger et al., 2018). This conceptual framework for learning progression was initially developed from empirical research coupled with national and state standards. ...
Algebra has long been recognized as a fundamental component of mathematics education for K-12 students and has been identified as a subject with which students continually struggle. Researchers have utilized various methods across contexts and conditions at the classroom level to improve algebra learning. This systematic review and meta-research (i.e., meta-analysis and meta-synthesis) aims to elucidate which of these efforts are effective, along with the conditions and populations for which they are most effective. In this article, we present our framing for the study under a modified version of the conceptual framework for learning progression, justify selected moderators, and detail our anticipated research process. Conducting meta-research on this topic is essential for providing policymakers, instructors, and researchers with an adequate understanding of the historical landscape of effective practices in algebra instruction.
... внимания. Такие общие характеристики учительской деятельности, как составление учебных планов, формирование образовательных программ, оценивание результатов, традиционно регламентируются общими инструментами регулирования системы образования безотносительно к выбранным методическим подходам и стоящей за ними педагогической философии (Nechayev, Fonger, 2018). В этом смысле целесообразно поставить вопрос, насколько необходимы изменения существующих общих правил регулирования содержания образования с учетом распространения идей деятельностного подхода? ...
В статье рассмотрена проблема норм и правил регулирования образовательного процесса, а также вопросы законодательства в образовательной деятельности. С помощью феноменологического анализа сопоставлены методологическая основа институционального подхода с распространением идей развивающего обучения и деятельностного подхода в образовательных практиках. Сделаны выводы о том, что важнейшим условием для эффективного развития образовательных инновационных практик является их постоянное соотнесение с общими нормами федерального законодательства, диалог с институциональными нормами и их регуляторами. Кроме того, авторами обозначены главные направления деятельности в процессе создания технологии развивающего обучения: 1) формирование критериальных эталонов нормативно заданных результатов выполнения предметных и метапредметных учебных заданий; 2) подготовка педагогов к определенной (отличной от традиционной) роли в учебном процессе; 3) внедрение в образовательный процесс анализа больших данных (Big Data Analysis), поскольку именно сейчас различные учительские наработки сконцентрированы в разнообразных электронных средах и сервисах.
The article addresses the problem of norms and rules governing the educational process, as well as issues of legislation in educational activities. Because of phenomenological analysis, the authors compare the methodological basis of the institutional approach with the dissemination of the ideas of developing education and the activity approach in educational practices. It has been concluded that the basic condition for the effective development of educational innovation practices is their constant correlation, dialogue with institutional norms, and their regulators. In addition, the authors have identified the main areas in the creation of developing education technology: 1) the establishment of criteria benchmarks for the normative results of the implementation of subject and metasubject educational tasks; 2) preparing teachers for a certain (different from traditional) role in the teaching process; 3) introduction of «big data» analysis (Big Data Analysis) into the educational process, since now different teaching methods are concentrated in various electronic environments and services.
... Dynamically, the change will affect students' critical thinking and ability to solve mathematical problems [2], [3], [4], [5]. It supports a learning process in which the teachers focus on the student's thinking process [6], [7]. In its development, the remodeling comprised the selection of interactive-learning strategies adopted by teachers in teaching mathematics [8]. ...
The post-pandemic adaptation demands technological innovations in sustainable learning to support mathematics curriculums. This research aims to develop the Web-based Window Shopping (WBWS) in high school mathematics curriculum with ADDIE procedure. The data were collected through interviews, questionnaires, and tests. The stages of research comprised analysis, design, development, implementation, and evaluation. The experts' validation test results showed that the aspects of materials and medium achieved an average score of 82,75%. Meanwhile, the trial test results showed that the developed WBWS is worth implementing, with the average percentage of effectiveness reaching 83,33%. The results suggest that the creation of WBWS as a web-based learning model for mathematics curriculum is not only feasible but also meets the educational needs of high school students. Teachers and students can use this web-based learning model as an alternative for implementing the independent curriculum on mathematics learning.
... Thus, there is tension between the formal curricula and learning frameworks provided by educational bodies and researchers (Duschl et al., 2011;Fonger et al., 2018) and how children develop their conceptual understanding, skills and practices in their everyday worlds. ...
Learning progressions have become increasingly prevalent in mathematics education as they offer a fine-grain map of possible learning pathways a child may take within a particular domain. However, there is an opportunity to build upon this research in ways that consider learning from multiple perspectives. Many current forms of learning progressions describe learning pathways without explicit consideration of how related skills and contexts directly or indirectly enhance or influence learning. That is, the structured and unstructured learning contexts that can help children develop conceptual understanding in a range of STEM contexts. We consider learning progressions from multiple perspectives, which will be particularly important for supporting learning in early years, play-based contexts. We propose a novel theoretical perspective, termed Bounded Learning Progressions (BLP), which demonstrates the connection and influence ways of reasoning have on the progression of learning in specific domains, bounded by the context in which learning develops. We suggest that this approach provides a broader perspective of children’s learning capabilities and the possible connections between such abilities, acknowledging the critical role context plays in the development of learning.
... A significant body of work documenting young children's productive engagement with this broader set of algebraic competences has convinced the field that "algebra" instruction need not be postponed until middle or secondary grades (Bodanskii, 1991). This work has led researchers to study the impact of early algebraic experiences in elementary grades (e.g., Ayala-Altamirano & Molina, 2021;Blanton & Kaput, 2005;Blanton et al., 2019aBlanton et al., , 2019bCai & Knuth, 2011;Fonger et al., 2018;Kaput et al., 2017;Mulligan et al., 2020;Stephens et al., 2017, and many others), demonstrating that these experiences can support children's future learning of formal, symbolic algebra and thus ease the historically difficult transition from arithmetic to algebra. ...
In this paper, we elaborate the seeds of algebraic thinking perspective, drawing upon Knowledge in Pieces as a heuristic epistemological framework. We argue that students’ pre-instructional experiences in early childhood lay the foundation for algebraic thinking and are a largely untapped resource in developing students’ algebraic thinking in the classroom. We theorize that seeds of algebraic thinking are cognitive resources abstracted over many interactions with the world in children’s pre-instructional experience. Further, we provide examples to demonstrate how the same seeds of algebraic thinking present in early childhood can be invoked in reasoning across contexts, grade levels, and different levels of formality of algebraic instruction. The examples demonstrate how the seeds perspective differs from other accounts of the relationship between children’s early activity and their engagement in algebraic reasoning processes. We anticipate this new theoretical direction for characterizing the nature and development of algebraic thinking will lay the foundation for a robust agenda that sheds light on the development of algebraic thinking and informs algebra instruction, particularly how teachers notice and respond to children’s developing algebraic thinking.
... We assume that learning conforms to van Hiele's theory. To develop, anticipate, interpret, and respond to evidence of student conception, we follow the aspects of learning progression, as defined by Fonger et al. (2018). Following Cooper et al. (2020), we use the term instructional sequence to refer to an ordered set of instructional activities and the term learning progression to denote a contextualized instructional sequence, as it is enacted by a teacher aiming to adapt the instruction to the needs of the classroom. ...
We used an assessment platform to study the potential of rich student data obtained online to influence classroom instruction and help teachers respond to students’ needs. The article focuses on technology-based formative assessment in the course of teaching a unit on quadrilaterals. We studied example-eliciting tasks, requiring students to determine the truth of a given claim and to use interactive diagrams to construct examples to support their answer. Our aim was to teach the concept of parallelogram by exploring students’ concept images and responding adequately to their submissions. A group of 11-year-old students studied the unit as part of their geometry curriculum. We used an assessment platform, which provides tools designed to monitor examples produced by students interacting with a dynamic geometry environment and to characterize critical and non-critical attributes of the submitted constructions. We present the teaching decisions made by the teacher (the first author), who used automatic feedback on the characteristics of three assessment tasks.
... This work has involved small-scale, cross-sectional, and longitudinal studies Blanton, Stephens, et al., 2015) and, more recently, a large-scale, longitudinal, cluster randomized trial . In a process fully described in Fonger et al. (2018) and Blanton et al. (2018), this earlier work involved building a curricular framework and progression, instructional intervention, and associated assessments around the algebraic thinking practices of generalizing, representing, justifying, and reasoning with mathematical structure and relationships (see also Blanton et al., 2011). We also characterized three "Big Ideas" (Shin et al., 2009) in which these practices can occur: generalized arithmetic; equivalence, expressions, equations, and inequalities; and functional thinking. ...
This research focuses on the retention of students’ algebraic understandings 1 year following a 3-year early algebra intervention. Participants included 1,455 Grade 6 students who had taken part in a cluster randomized trial in Grades 3–5. The results show that, as was the case at the end of Grades 3, 4, and 5, treatment students significantly outperformed control students at the end of Grade 6 on a written assessment of algebraic understanding. However, treatment students experienced a significant decline and control students a significant increase in performance relative to their respective performance at the end of Grade 5. An item-by-item analysis performed within condition revealed the areas in which students in the two groups experienced a change in performance.
... Penalaran merupakan fenomena yang sangat diperlukan dalam menghubungkan potongan bagian konten pembelajaran (Jescovitch et al., 2021). Dampak peningkatan pada aspek penalaran mahasiswa dapat membantu dosen dalam merencanakan pelajaran, mengantisipasi tanggapan siswa, dan merencanakan pembelajaran selanjutnya (Fonger et al., 2018). ...
Understanding student characteristics and the process of designing learning according to students with special needs are the main weaknesses of lecturers in inclusive classes. Students often complain that lecturers' learning is still monotonous, lack of variation in learning strategies, and the absence of learning media so that students experience difficulties in the process of visualizing the content of teaching materials in class. The research objective was to develop Learning Progression during Modeling-Based Teaching (LP-MBT) to help students with learning difficulties in higher education. The research method used in research and development follows Thiagarajan's (1974) 4-D model but is only applied to 3-D, namely the define, design, and developing stages. This preliminary research study involved three student respondents with special needs and twelve regular students from Elementary School Teacher Education students at Djuanda University. The research was conducted during the odd semester of the 2020/2021 academic year. The process of collecting research data was carried out using questionnaires and semi-structured interviews through online or online applications. The research results are as follows; (1) the define stage, the researcher analyzes the problems of students with special needs in the learning process in the classroom; (2) in the design stage, researchers designed Learning Progression during Modeling-Based Teaching (LP-MBT) in inclusive classroom learning based on theoretical studies and the needs of students with learning difficulties; and (3) the developing stage, the researcher validates the experts and conducts limited trials for 15 students online through the zoom application. Suggestions from this study, the Learning Progression during Modeling-Based Teaching (LP-MBT) design can be used for students with special needs, especially learning difficulties because it can activate the learning content visualization process for students and provide opportunities to express more equitable opinions.
... The mathematics education studies in WoS database were first published at early 1980s and despite the change in the rate of growth, the number of publications continued to increase in each period. According to keyword analysis, there were a great variety of studies in mathematics education to determine standards and principles of teaching and learning mathematics such as reform movements (Gravemeijer et al., 2016;Lundin, 2012;Sengupta-Irving, Redman, & Enyedy, 2013), curriculum (Fonger et al., 2018;Fouze & Amit, 2017;Pepin et al., 2017;Voigt, Fredriksen, & Rasmussen, 2020), educational policy (Dalby & Noyes, 2018;Lin, Wang, & Chang, 2018;Nortvedt & Buchholtz, 2018), equity (Jurdak, 2011(Jurdak, , 2014Nortvedt & Buchholtz, 2018;Tan & Thorius, 2019), assessment (Beumann & Wegner, 2018;Kim & Cho, 2015;Nortvedt & Buchholtz, 2018;Veldhuis & van den Heuvel-Panhuizen, 2014;Veldhuis et al., 2013), to evaluate the cognitive and affective skills such as problem solving (Boonen et al., 2016;Verschaffel et al., 2020), achievement (Ciftci, 2015;Veldhuis & van den Heuvel-Panhuizen, 2020), motivation (Schukajlow, Rakoczy, & Pekrun, 2017), to learn more about and support mathematics teachers such as professional development (Sztajn et al., 2007;Williams & Ryan, 2020), teacher education (Buchholtz, 2017;Healy & Ferreira dos Santos, 2014;Tatto & Senk, 2011), teacher knowledge (Koponen et al., 2017;Olfos & Rodríguez, 2019;Scheiner et al., 2019), teacher beliefs (Kang & Kim, 2016;Cetinkaya & Erbas, 2011), to address the relationships with different aspects such as early childhood (Björklund, van den Heuvel-Panhuizen, & Kullberg, 2020;Ulusoy, 2020), algebra (Dougherty et al., 2015; Gökçe & Güner 528 Simzar, Domina, & Tran, 2016), technology (Cullen, Hertel, & Nickels, 2020;Drijvers, 2015;McCulloch et al., 2018;Thomas & Hong, 2013;Trouche & Drizvers, 2010), science education (King et al., 2020;Maass & Engeln, 2019;Swanson & Coddington, 2016). ...
The purpose of this study is to establish the evolution and expose the trends of research in mathematics education between 1980 and 2019. The bibliometric analysis of the articles in Web of Science database indicated four-clustered structure. The first cluster covers the items related to the theoretical framework of mathematics education whereas the second cluster has the terms defining the methods for effective mathematics instruction. The third cluster includes the concepts interrelated to mathematics education while the fourth cluster encloses the studies about international mathematics assessments. The earlier studies look mathematics education mostly in students’ perspective and investigates generalization, restructuring, interiorization and representation. Between 1995 and 2010, curriculum and teacher-related factors were dominant in mathematics education studies. After 2010, the articles investigated specific topics and carried the traces from all stakeholders in mathematics education. The investigation on the trends of mathematics education would provide gain insight about the areas that need more research, contribute to the researchers, teachers, students and policy makers in this field and light the way for further studies.
... Kindergarten lessons were taught during the second semester of the school year (Spring 2018) while first grade lessons were taught throughout the school year (Fall 2018-Spring 2019). 2 Lessons typically took place once per week, depending on school schedules, and were taught by a member of our research team-a former elementary school teacher who participated in the lesson development process-during students' regular mathematics instruction time. The lessons were designed to engage students in the early algebra concepts and practices developed in our previous Grades 3-5 work, including mathematical equivalence, generalizations about arithmetic properties and the properties of even and odd numbers, the use of variable, and functional thinking (see Fonger et al., 2018 for details regarding the development of the curricular framework used in the previous and current studies). Each lesson began with a "Jumpstart" to review previous concepts or prompt students to think about new concepts to be addressed in the lesson. ...
This research shares progressions in thinking about equations and the equal sign observed in ten students who took part in an early algebra classroom intervention across Kindergarten and first grade. We report on data from task-based interviews conducted prior to the intervention and at the conclusion of each school year that elicited students’ interpretations of the equal sign and equations of various forms. We found at the beginning of the intervention that most students viewed the equal sign as an operational symbol and did not accept many equations forms as valid. By the end of first grade, almost all students described the symbol as indicating the equivalence of two amounts and were much more successful interpreting and working with equations in a variety of forms. The progressions we observed align with those of other researchers and provide evidence that very young students can learn to reason flexibly about equations.
... Pedagogically, GC computing formula can be easily added to model-based automated summary evaluation (ASE) tools such as SMART. GC can serve a feedback index that inform students of their learning progression in writing an expert-like summary on top of the existing feedback information-learning progression feedback (Fonger et al. 2018;Nadolski and Hummel 2017). Given the trends explained by GC values, one can use the 3S indices and similarity measures to further detail the characteristics of individual students' mental representations while they revise summaries. ...
During reading, students construct mental models of what they read. Summaries can be used to evaluate the latent knowledge structure of these mental models. We used indices from Student Mental Model Analyzer for Research and Teaching (SMART) to explore the potential of a global index, Graph Centrality (GC), as a measure to describe mental model structure and its relation to the quality of student summaries (e.g., the amount of content-coverage). Students (n = 73) in an online graduate-level course wrote and revised summaries of their course readings. Data preview left the total count of 32 cases to evaluate how students’ mental representations changed from initial to final version. These summaries were analyzed using indices derived from the 3S model (surface, structure, semantic) as well as a measure of GC. The results of this initial investigation are promising, demonstrating that Graph Centrality captures important differences in students’ summaries, including revision behaviors to the wholistic structure of mental models, modification trajectories toward a cohesive and solid mental representation that is semantically similar to the expert model.
... Despite the integral nature of learning trajectories to some design-based research, the work of communicating such complex tools is challenging. Indeed, the domain of research on learning trajectories has grown and diversified in the approaches and theoretical orientations researchers take (e.g., see reviews by Empson, 2011;Lobato & Walters, 2017;and Fonger, Stephens et al. 2018). ...
... Ketterlin-Geller, Zannou, Sparks, and Perry (2020) have relied on educators (including classroom teachers), responding to survey questionnaires, to "recover" such aspects of learning progressions. Fonger et al. (2018) have developed a comprehensive theory of learning progressions that combines ideas from research on learning trajectories. They "view the instructional sequence as vital [to levels of sophistication in student thinking] and the resulting levels of sophistication in student thinking are contextualized with respect to carefully designed and sequenced learning goals and instructional tasks" (p. ...
In their enactment of the curriculum, teachers have a substantial role as instructional designers. Accordingly, any evaluation of the progression of students’ learning should first be concerned with the pedagogical intentions of the teacher. In this article we present a method for reconstructing teachers’ implicit and tacit considerations in their selection, sequencing and enactment of tasks. Two 11th grade teachers tagged all of the tasks that comprised a 5-week learning progression. Tagging consisted of assigning values to prescribed categories of metadata. Visual representations of the metadata revealed patterns in the tagged progressions, and allowed the teachers to reflect upon these patterns. Both teachers, though guided by very different didactical considerations, validated that many of their explicit and implicit intentions were revealed in the representations of the progressions. Furthermore, both teachers had the opportunity to reflect on tacit aspects of their instructional design that they were not previously aware of.
... These lessons were spaced throughout the school year (October-May), took place during students' regular mathematics instruction time and were taught by a member of our research team. They addressed a range of early algebra concepts identified and developed in our previous Grades 3-5 work, including mathematical equivalence and equations, generalizations about arithmetic properties and the properties of even and odd numbers, the use of variable, and functional thinking (see Blanton et al., 2015;Fonger et al., 2018 for details regarding the development of the curricular framework used in the previous and current studies). ...
This research focuses on ways in which balance scales mediate students’ relational understandings of the equal sign. Participants included 21 Kindergarten–Grade 2 students who took part in an early algebra classroom intervention focused in part on developing a relational understanding of the equal sign through the use of balance scales. Students participated in pre-, mid- and post-intervention interviews in which they were asked to evaluate true-false equations and solve open number sentences. Students often worked with balance scales while solving these tasks. Interview analyses revealed several categories of affordances of these tools for supporting students’ productive thinking about equations.
... Another approach might be to establish a class code that specifies steps that students must take before they can ask the teacher a question. Another strategy There are some attempts at designing such sequences and working with teachers on such designs (Fonger, Stephens, Blanton, Isler, Knuth & Gardiner, 2018) although their published sequences do not seem to challenge students, nor do they focus on important mathematical connections. Part of our goal is to clarify what sequences might look like, how they might be interpreted by teachers and how they support student learning. ...
This chapter uses the notion of relentless consistency, elaborated by Laurinda Brown, to describe an approach to teaching mathematics we are researching. A fundamental assumption is that students learn better and engage actively when tasks on which they are working on are more rather than less challenging. The article describes how challenge is connected to the structure of lessons, our interpretation of ways sequences might support learning, justification for proposing specific approaches, and the importance of positive classroom and teacher learning cultures. An extract from one sequence is presented to exemplify our overall approach along with implications for teacher education.
... Quasi-experimental, small-scale study to gather efficacy data within grades Cross-secƟonal study at each of Grades 3 -5 (e.g., Blanton, Stephens et al., 2015Fonger et al., 2018Stephens et al., 2013 ImplementaƟon taught by member of research team ...
We share here results from a quasi-experimental study that examines growth in students’ algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3–5. Analyses showed that, while there were no significant differences between experimental and control students on a grade 3 pre-assessment measuring students’ capacity for generalizing and representing generalizations, experimental students significantly outperformed control students on post-assessments at each of grades 3–5. Moreover, experimental students were able to more flexibly interpret variable in different roles and were better able to use variable notation in meaningful ways to represent arithmetic properties, expressions and equations, and functional relationships. This study provides important evidence that young children can learn to think algebraically in powerful ways and suggests that the earlier introduction of algebraic concepts and practices is beneficial to students.
... Our focus on evidence-based learning progressions is intended to assist teachers and students to improve the quality of their algebraic reasoning and performance. Early in this chapter, there is a brief review of hypothetical learning trajectories and progressions as reported by researchers in the field such as Clements and Sarama (2014), Corley (2014), Fonger, Stephens, Blanton, Isler, Knuth, andGardiner (2018), Ronda (2004) and Simon and Tzur (2004). The work reported here on algebraic reasoning comes from a larger research project on mathematical reasoning, the Reframing Mathematical Futures II (rmfii) project, which was conducted between August 2014 and December 2017 (see Siemon, Day, Stephens, Horne, Seah, Watson, & Callingham, 2017). ...
Mathematical reasoning is an important component of any mathematics
curriculum. This chapter focuses on algebraic reasoning in the middle
years of schooling, often termed as a predictor of later success in school
mathematics. It describes the design research process used to develop an
evidence-based Learning Progression for Algebraic Reasoning using three
key ideas of Equivalence, Pattern and Function, and Generalisation. The
Learning Progression for Algebraic Reasoning may be used to identify where
students are in their learning journey and where they need to go next. Once
the Learning Progression for Algebraic Reasoning was developed it was used
to design Teaching Advice to help teachers to provide appropriate activities
and challenges to support student learning. Two implied recommendations of
this chapter are that algebraic reasoning based on these three key ideas should
precede symbol use; and, that algebraic reasoning as described here needs to
be cultivated in the primary school years.
... The study, from which the data described in this chapter were drawn, involves a quasiexperimental comparison of Grades 3-5 (ages 8-11) student performance from two classroom contexts: an intervention classroom (implementing an intervention based on an Early Algebra Learning Progression [EALP]) and a traditional elementary mathematics classroom. The EALP consists of a curricular framework, with learning goals, an instructional sequence of lessons to achieve those goals, assessments, and levels of sophistication that characterize students' understandings over time (Fonger et al. 2015(Fonger et al. , 2017. Drawing upon Kaput's (2008) analysis of algebra in terms of content strands and thinking practices, we designed the EALP, which then guided the development and implementation of our longitudinal early algebra intervention, as well as the development of accompanying assessments. ...
This study considers classroom situations in which students and the teacher co-contribute to promoting generalization. It specifically focuses on the ways in which students and a teacher in one classroom engage in generalizing arithmetic. Generalized arithmetic is an important route into early algebra (Kaput in Algebra in the Early Grades. Routledge, New York, 2008); its potential as a way to deepen students’ understandings of concepts of school arithmetic makes it an important focus of early algebra research. In the analysis we identified generalizations around properties of arithmetic and the actions that promoted these types of generalizations, and then considered the relationship between these actions. Analysis revealed that generalizations became platforms for further generalization.
The teaching needs expressed by new generations require teachers to have various skills: pedagogical, didactic, communicative and design. The work is the result of an experimental project conducted in the academic field, which involved two teachers with extensive professional experience both as teachers in schools and universities. The interest of the research was to demonstrate the validity of Shulman's PCK method through classroom instruction and, at the same time, to promote in future teachers the ability to assess, through direct experience, the importance of the design and implementation of teaching and the significant role of the choice of didactic method.
Nelson Mandela (1990) stated “[e]ducation is the most powerful weapon which you can use to change the world.” This has never been truer as we live in a world in need of strong teachers. We live in a world that underappreciates teachers and their role in educating the next generation often goes unnoticed. Teaching children to become literate in math and science is a concern as these disciplines can lead to increased employment opportunities, higher pay, and a higher quality of life. Having a scientifically literate citizenship is critical to fostering innovation, improving the quality of life, and solving crises such as pandemics and climate change. Unfortunately, interest and motivation in STEM disciplines often decline after early grades which illustrates the important role that these teachers have in math and science education. This chapter combines research-based practices and activities that the author has used to promote STEM learning in preservice teachers. It combines these with narratives from former students who add evidence for their value.
A growing body of research in early algebra has illustrated that young children have the capacity to reason algebraically and has recommended introducing algebraic ways of thinking in early-elementary grades. Over the past two decades, there have been great strides in exploring children’s capacity for algebraic thinking and this work has opened the door to designing early algebra curricula and classroom experiences. What is less explored, however, are ways to support teachers in implementing early algebra lessons and supporting students’ thinking about early algebra concepts. In this paper, we explore early algebra teacher education. We introduce an approach to teacher education that leverages a novel algebraic thinking perspective. We demonstrate how using this perspective in video-based professional development can support teachers in identifying and reasoning about students’ nascent algebra thinking, which, we claim, is an important skill in teaching early algebra.
Knowledge of algebra is known to be critical for not only school mathematics but also higher-level studies in mathematics. However, many students find learning algebra problematic and experience difficulty, particularly in transitioning from arithmetic to algebra. Therefore, the mathematics education research community has endeavored to develop students’ algebraic thinking in the elementary grades to help them learn algebra successfully. Although research on early algebra has been extensively conducted overseas since the 1990s, in Korea, this research began only in the 2000s and is relatively scarce. Therefore, in this study, we aim to compare the elementary mathematics textbooks of Korea and the U.S. based on the five big ideas of early algebra and draw implications for early algebra education in Korea. The results demonstrate that Korean textbooks contain significantly less content related to early algebra than U.S. textbooks. It was also found that the U.S. textbooks than that of Korea more emphasize the relational meaning of the equal sign. Furthermore, there were differences in the textbooks of the two countries in the way of handling the properties of numbers, analyzing patterns in functional relationships, and presenting the meaning of variable. Based on the results, we discussed ways to develop elementary school students’ algebraic thinking through textbooks in Korea.
Although literacy research has traditionally focused on content area and disciplinary literacies; in this study, we argue for the importance of transdisciplinarity. We employed a participatory action research design to examine pre-service teachers’ (PST) planning of a transdisciplinary math and music lesson to understand how conceptual framing of pedagogy and instructional practice can be shifted from content literacies to transdisciplinarity. Our area of inquiry focused on how PSTs’ understanding of transdisciplinarity shifted as they planned, taught, and reflected on instruction and how these shifts were supported by others. We documented evidence of how the lesson progressed from disciplinary thinking to interdisciplinary, multidisciplinary, and transdisciplinary thinking throughout the planning process. We also documented how the process of planning, teaching, coaching, and reflecting provided PSTs with an opportunity to better understand the connections between transdisciplinarity and pedagogy.
In the United States, national and state legislative mandates have forced school districts
to include student growth measures in teacher evaluation systems. However, statistical
models for monitoring student growth on standardized tests have not been found to foster teachers’ reflective practice or pedagogical content knowledge and goal-based models have been found to lack adequate structure for supporting implementation. This basic qualitative inquiry explored how teachers perceive using standards-based rubrics to
monitor student growth for teacher evaluation influences their reflective practice and
pedagogical content knowledge in mathematics. Nine teachers who have used standards�based rubrics to monitor student growth were recruited through snowball sampling. Through semi structured interviews and inductive and deductive coding, six themes were identified to understand teacher perceptions of the experience monitoring growth with standards-based rubrics: (a) fosters collaborative dialogue and descriptive feedback, (b) promotes standards-based focus, (c) supports evidence-based assessment, (d) supports student-centered instruction, (e) encourages students’ reflective practice, and (f) cultivates a positive teacher evaluation experience. This study may inform standards�based growth monitoring practices for formative and summative teacher evaluation in K–8 education systems. Formative teacher evaluation has been found to promote positive social change by improving both teacher practice and student achievement, thereby supporting teachers and students to continuously grow in knowledge, skill, and understanding. These findings indicate that monitoring student growth on standards�based rubrics may provide the necessary structure other models have been lacking.
The aim of this study was to show that some of the errors made by students when responding to mathematics assessment items can indicate progress in the development of conceptual understanding. By granting partial credit for specific incorrect responses by early secondary students, estimates of the difficulty of demonstrating full and partial knowledge of skills associated with the development of proportional reasoning were determined using Rasch analysis. The errors were confirmed as indicators of progress, and hence partial knowledge, when the thresholds of achievement followed a logical order: The greater the proficiency of the students, the more likely they were to receive a higher score. Consideration of this partial knowledge can enhance the descriptions of the likely behaviours of students at the various levels of learning progressions and this can be informative for teachers in their planning of learning activities.
Many calls have been made for more research on social studies teachers' practices and preservice training. Instructional practices employed by teachers are important for encouraging student learning. However, there is a history of social studies teachers focusing much of their time on teacher-centered instructional techniques that have not demonstrated strong learning for students. Therefore it is important to examine not just how teachers chose to teach, but also where they may have learned to teach. This study examined data from the Teaching and Learning International Survey (TALIS) 2018 of 240 secondary social studies teachers to understand what instructional practices they report employing and their feelings about their preparation. Data analysis provided direct empirical evidence of the power of teacher preparation programs to shape social studies teachers’ instructional practices well into their teaching careers.
Purpose
As accountability policies worldwide press for higher student achievement, schools across the globe are enacting a host of reform efforts with varied outcomes. Mounting evidence suggests reforms, which encourage greater collaboration among teachers, may ultimately support increased student learning. Specifically, this study aims to investigate the relationship between human and social and student achievement outcomes.
Design/methodology/approach
In exploring this idea, the authors draw on human and social capital and examine the influence of these forms of capital on student achievement using social network analysis and hierarchical linear modeling.
Findings
The results indicate that teacher human and social capital each have a significant and positive relationship with student achievement. Moreover, both teacher human and social capital together have an even stronger effect on student achievement than either human or social capital alone.
Originality/value
As more schools across the globe adopt structures for teacher collaboration and the development of learning communities, there is a need to better understand how schools may capitalize on these opportunities in ways that yield improved student learning. Our work sheds new light on these critical foundational elements of human and social capital that are individually and collectively associated with student achievement.
Background: This scoping literature review was undertaken by the Science and Engineering Education Research and Innovation Hub at The University of Manchester to enhance the understanding of how teachers can be supported to plan for progression in engineering education in primary and secondary schools in England.
Purpose & Method: The aim of this literature review is to provide insight into Learning Progressions (LPs) published globally for primary and secondary school level. In setting out the context and parameters for the study the paper identifies and compares definitions for LPs. It synthesises emergent themes from 25 academic papers.
Findings: Four main findings were deducted from the data papers. Firstly, UK curricular were not discussed. Secondly, within the dataset near parity between science and engineering-focused papers was revealed. Thirdly, of the data papers reviewed nearly the same number used pre-existing definitions of LPs to those that did not offer any definition or description of LP. Furthermore, around half this number created their own, or used a generalised description of LPs. Finally, the data papers highlighted a lack of common definition for engineering education LPs, unlike science LPs. None of the data papers provided an LP specific to engineering education aligned to the National Curriculum (NC) in England.
Conclusions: Four recommendations emerge: i) engineering education should be recognised as a distinct subject within the NC for England; ii) more academic research and curriculum development is required within the field of engineering education LP specifically aligned with the NC in England; iii) industry and education would benefit from further collaboration to ensure that their respective needs and positions are adequately met through schools; iv) teacher professional development and resources need focused auditing and investment.
This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic growth as phenomenon of constantly-changing rate of change. We elaborate the duality, necessity, repeated reasoning framework, and methods of creating learning trajectories. We report five WoT: Variation, Early Coordinated Change, Explicitly Quantified Coordinated Change, Dependency Relations of Change, and Correspondence. We also articulate instructional supports that engendered transitions across these WoT: teacher moves, norms, and task design features. Our integration of instructional supports and transitions in students’ WoT extend current research on quadratic function. A visual metaphor is leveraged to discuss the role of learning trajectories research in unifying research on teaching and learning.
https://docs.lib.purdue.edu/jpeer/vol10/iss1/4/
Engineering education has increasingly become an area of interest at the P-12 level, yet attempts to align engineering knowledge, skills, and habits to existing elementary and secondary educational programming have been parochial in nature (e.g., for a specific context, grade, or initiative). Consequently, a need exists to establish a coherent P-12 content framework for engineering teaching and learning, which would serve as both an epistemological foundation for the subject and a guide for the design of developmentally appropriate educational standards, performance expectations, learning progressions, and assessments. A comprehensive framework for P-12 engineering education would include a compelling rationale and vision for the inclusion of engineering as a compulsory subject, content organization for the dimensions of engineering literacy, and a plan for the realization of this vision. The absence of such a framework could yield inconsistency in authentically educating students in engineering. In response, this study was conducted to establish a taxonomy of concepts related to both engineering knowledge and practices to support the development of a P-12 curricular framework. A modified Delphi method and a series of focus groups—which included teachers, professors, industry professionals, and other relevant stakeholders—were used to reach a consensus on engineering concepts deemed appropriate for secondary study. As a result, a content taxonomy for knowledge and practices appropriate for P-12 engineering emerged through multiple rounds of refinement. This article details the efforts to develop this taxonomy, and discusses how it can be used for standards creation, curriculum development, assessment of learning, and teacher preparation.
This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth-grade algebra students, and link to results of semi-structured interviews with students before and after the experiment. Results of analyzing the five-week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool-based representations.
In this chapter, we discuss the algebraframework that guides our work and how this framework was enacted in the design of a curricular approach for systematically developing elementary-aged students’ algebraic thinking. Weprovide evidence that, using this approach, students in elementary grades can engage in sophisticated practices of algebraic thinking based on generalizing, representing, justifying, and reasoning with mathematical structure and relationships. Moreover, they can engage in these practices across a broad set of content areas involving generalized arithmetic; concepts associated with equivalence, expressions, equations, and inequalities; and functional thinking.
In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.
This study characterizes s of quadratic function from a variation perspective in the context of a quantitatively rich instructional setting. We studied middle grade (ages 12-13) ceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic function as a growth situation with a constant difference in rate of change. We report five clusters in student thinking: (a) variation without coordination, (b) qualitative and/or implicit coordination of quantities, (c) explicit coordination of change in quantities, (d) attending to how change in quantities depend on other quantities, and (e) dependency relations and their symbolizations. This work contributes to an understanding of what students' rich conceptions of functions can be.
Recent research suggests that children in elementary grades have some facility with variable and variable notation in ways that warrant closer attention. We report here on an empirically developed progression in first-grade children’s thinking about these concepts in functional relationships. Using learning trajectories research as a framework for the study, we developed and implemented an instructional sequence designed to foster children’s understanding of functional relationships. Findings suggest that young children can learn to think in sophisticated ways about variable quantities and variable notation. This challenges assumptions that young children are not “ready” for a study of such concepts and raises the question of whether difficulties adolescents exhibit might be ameliorated by an earlier introduction to these ideas.
Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM)(CCSSI 2010) have taken up this call by reiterating the place of early algebra in children’s mathematics education, beginning in kindergarten. Some might argue that early algebra instruction represents a significant shift away from arithmetic-focused content that has typically been taught in the elementary grades. To that extent, it is fair to ask, “Does early algebra matter?” That is, will teaching children to think algebraically in the elementary grades have an impact on their algebra understanding in ways that will potentially make them more mathematically successful in middle school and beyond?
Third- through fifth-grade students participating in a classroom teaching experiment investigating the impact of an Early Algebra Learning Progression completed pre- and post- assessments documenting their abilities to represent or describe unknown quantities. We found that after a sustained early algebra intervention, students grew in their abilities to represent related unknown quantities using letters as variables.
Third- through fifth-grade students participating in a classroom teaching experiment investigating the impact of an Early Algebra Learning Progression completed pre- and post-assessment items addressing their abilities to engage in functional thinking. We found that after a sustained early algebra intervention, students grew in their abilities to shift from recursive to covariational thinking about linear functions and to represent correspondence rules in both words and variables.
This research study was specifically concerned with the development, testing, and revision to an instructional theory for studying the mathematical concepts of equivalence and equation solving with multiple representations and multiple tools. Following a design research approach, a collaborative teaching experiment was conducted with a ninth-grade algebra teacher in which instruction was guided by a specifically designed sequence of tasks, techniques using paper-and-pencil and computer algebra systems (CAS), and theory on a hypothesized progression of learning. Retrospective analyses of data informed revisions to a resulting progression of learning and activity sequence that are being tested with pre-service secondary mathematics teachers.
This chapter provides an overview of research about algebraic reasoning among relatively young students (6-12 years), It focuses on mathematics learning and, to a lesser extent, teaching. Issues related to educational policy, epistemology, and curriculum design provide a backdrop for the discussion.
We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional sequence of tasks based on objectives concerning content and algebraic thinking practices; assessments; and posited levels of sophistication in children's reasoning about algebraic concepts within big ideas of early algebra. This research balances the goals of longitudinal research on supporting students' preparedness for algebra while attending to the practical goals of establishing connections among curriculum, instruction, and student learning.
learning trajectories
IOM/NRC
Knowledge of the equal sign as an indicator of mathematical equality is foundational to children's mathematical development and serves as a key link between arithmetic and algebra. The current findings reaffirmed a past finding that diverse items can be integrated onto a single scale, revealed the wide variability in children's knowledge of the equal sign assessed by different types of items, and provided empirical evidence for a link between equal-sign knowledge and success on some basic algebra items.
Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction aims to provide: A useful introduction to current work and thinking about learning trajectories for mathematics education An explanation for why we should care about these questions A strategy for how to think about what is being attempted in the field, casting some light on the varying, and perhaps confusing, ways in which the terms trajectory, progression, learning, teaching, and so on, are being used by the education community.
Specifically, the report builds on arguments published elsewhere to offer a working definition of the concept of learning trajectories in mathematics and to reflect on the intellectual status of the concept and its usefulness for policy and practice. It considers the potential of trajectories and progressions for informing the development of more useful assessments and supporting more effective formative assessment practices, for informing the on-going redesign of mathematics content and performance standards, and for supporting teachers’ understanding of students’ learning in ways that can strengthen their capability for providing adaptive instruction. The authors conclude with a set of recommended next steps for research and development, and for policy.
In this commentary, we first outline several frameworks for analyzing the articles in this issue. Next, we discuss Clements and Sarama's overview and the issue hypothetical learning trajectories (HLTs) in general. We then analyze each of the other contributions. We conclude our commentary by offering a vision of HLTs that includes a key role for "big ideas."
Coherent curricula are needed to help students develop deep understanding of important ideas in science. Too often students experience curriculum that is piecemeal and lacks coordination and consistency across time, topics, and disciplines. Investigating and Questioning our World through Science and Technology (IQWST) is a middle school science curriculum project that attempts to address these problems. IQWST units are built on 5 key aspects of coherence: (1) learning goal coherence; (2) intraunit coherence between content learning goals, scientific practices, and curricular activities; (3) interunit coherence supporting multidisciplinary connections and dependencies; (4) coherence between professional development and curriculum materials to support classroom enactment; and (5) coherence between science literacy expectations and general literacy skills. Dealing with these aspects of coherence involves trade-offs and challenges. This article illustrates some of the challenges related to the first 3 aspects of coherence and the way we have chosen to deal with them. Preliminary results regarding the effectiveness of IQWST's approach to these challenges are presented.
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.
In this article, the authors first indicate the range of purposes and the variety of settings in which design experiments have been conducted and then delineate five crosscutting features that collectively differentiate design experiments from other methodologies. Design experiments have both a pragmatic bent—“engineering” particular forms of learning—and a theoretical orientation—developing domain-specific theories by systematically studying those forms of learning and the means of supporting them. The authors clarify what is involved in preparing for and carrying out a design experiment, and in conducting a retrospective analysis of the extensive, longitudinal data sets generated during an experiment. Logistical issues, issues of measure, the importance of working through the data systematically, and the need to be explicit about the criteria for making inferences are discussed.
Early algebra differs from algebra as commonly encountered in high school and beyond in that it: (1) builds heavily on background contexts of problems; (2) only gradually introduces formal notation and (3) is tightly interwoven with the traditional topics from the early mathematics curriculum. Here we discuss these three distinguishing characteristics, drawing on examples from our longitudinal investigations of four classrooms in an ethnically diverse school in the Greater Boston area. From the second half of Grade 2 to the end of Grade 4, we designed and implemented weekly early algebra activities in the classrooms. The project documented how the students worked with variables, functions, directed numbers, algebraic notation, function tables, graphs, and equations in the classroom and in interviews. To highlight the nature of the progress students can make in early algebra, we compare the same students' reasoning and problem solving at the beginning of Grade 3 and in the middle of Grade 4. We will show how mathematics educators can exploit topics and discussions so as to bring out the algebraic character of elementary mathematics.
Topics that receive broad coverage with little integration provide a fragile foundation for integrated knowledge growth. In order to support the development of integrated understanding in science, coherent instructional materials should be developed to emphasize not only the learning of individual topics, but also the connections between ideas and across topics and disciplines. To build coherent instructional materials, designers can use empirically tested learning progressions as a ready-made artifact. However, well-developed coherent instructional materials should be designed, implemented, and tested as part the process of empirically tested learning progressions as well. Because the process of building such learning progressions is complex and iterative, research-based guidelines require fully articulating the process for designers use in their development of instructional materials. In this paper, we discuss six guidelines needed for learning progressions to inform the development of coherent curricula over the span of K-12 science education, focusing on organizing, identifying and specifying critical concepts within big ideas. We illustrate how these guidelines are applied to develop learning progressions and associated coherent instructional materials using a single principled and systematic design process, Construct- Centered Design. We conclude by stating major challenges for the development of a coherent curriculum based on LPs.
Given its important role in mathematics as well as its role as a gatekeeper to future educational and employment opportunities, algebra has become a focal point of both reform and research efforts in mathematics education. Understanding and using algebra is dependent on understanding a number of fundamental concepts, one of which is the concept of equality. This article focuses on middle school students' understanding of the equal sign and its relation to performance solving algebraic equations. The data indicate that many students lack a sophisticated understanding of the equal sign and that their understanding of the equal sign is associated with performance on equation-solving items. Moreover, the latter finding holds even when controlling for mathematics ability (as measured by standardized achievement test scores). Implications for instruction and curricular design are discussed.
We examined children's development of strategic and conceptual knowledge for linear measurement. We conducted teaching experiments with eight students in grades 2 and 3, based on our hypothetical learning trajectory for length to check its coherence and to strengthen the domain-specific model for learning and teaching. We checked the hierarchical structure of the trajectory by generating formative instructional task loops with each student and examining the consistency between our predictions and students' ways of reasoning. We found that attending to intervals as countable units was not an adequate instructional support for progress into the Consistent Length Measurer level; rather, students must integrate spaces, hash marks, and number labels on rulers all at once. The findings have implications for teaching measure-related topics, delineating a typical developmental transition from inconsistent to consistent counting strategies for length measuring. We present the revised trajectory and recommend steps to extend and validate the trajectory.
Our paper is an analytical review of the design, development and reporting of learning progressions and teaching sequences. Research questions are: (1) what criteria are being used to propose a ‘hypothetical learning progression/trajectory’ and (2) what measurements/evidence are being used to empirically define and refine a ‘hypothetical learning progression/trajectory’? Publications from five topic areas are examined: teaching sequences, teaching experiments, didaktiks, learning trajectories in mathematics education and learning progressions in science education. The reviewed publications are drawn from journal special issues, conference reports and monographs. The review is coordinated around four frameworks of Learning Progressions (LP): conceptual domain, disciplinary practices, assessment/measurement and theoretical/guiding conceptions. Our findings and analyses show there is a distinction between the preferred learning pathways that focus on ‘Evolutionary LP’ models and the less preferred but potentially good LP starting place curriculum coherence focused ‘Validation LP’ models. We report on the respective features and characteristics for each.
Simon's (1995) development of the construct of hypothetical learning trajectory (HLT) offered a description of key aspects of planning mathematics lessons. An HLT consists of the goal for the students' learning, the mathematical tasks that will be used to promote student learning, and hypotheses about the process of the students' learning. However, the construct of HLT provided no framework for thinking about the learning process, the selection of mathematical task, or the role of the mathematical tasks in the learning process. Such a framework could contribute significantly to the generation of useful HLTs. In this article we demonstrate how an elaboration of reflective abstraction (i.e., reflection on activity-effect relationships), postulated by Simon, Tzur, Heinz, and Kinzel (in press), can provide such a framework and thus a theoretical elaboration of the HLT construct.
We discuss an emerging program of research on a particular aspect of mathematics learning, students’ learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which students pass in the context of particular instructional approaches or learning environments. Generally missing from the literature is research that examines the process by which students progress from one of these conceptual steps to a subsequent one. We provide a conceptualization of a program of research designed to elucidate students’ learning processes and describe an emerging methodology for this work. We present data and analysis from an initial teaching experiment that illustrates the methodology and demonstrates the learning that can be fostered using the approach, the data that can be generated, and the analyses that can be done. The approach involves the use of a carefully designed sequence of mathematical tasks intended to promote particular activity that is expected to result in a new concept. Through analysis of students’ activity in the context of the task sequence, accounts of students’ learning processes are developed. Ultimately a large set of such accounts would allow for a cross-account analysis aimed at articulating mechanisms of learning.
Modeling is a core practice in science and a central part of scientific literacy. We present theoretical and empirical motivation for a learning progression for scientific modeling that aims to make the practice accessible and meaningful for learners. We define scientific modeling as including the elements of the practice (constructing, using, evaluating, and revising scientific models) and the metaknowledge that guides and motivates the practice (e.g., understanding the nature and purpose of models). Our learning progression for scientific modeling includes two dimensions that combine metaknowledge and elements of practice—scientific models as tools for predicting and explaining, and models change as understanding improves. We describe levels of progress along these two dimensions of our progression and illustrate them with classroom examples from 5th and 6th graders engaged in modeling. Our illustrations indicate that both groups of learners productively engaged in constructing and revising increasingly accurate models that included powerful explanatory mechanisms, and applied these models to make predictions for closely related phenomena. Furthermore, we show how students engaged in modeling practices move along levels of this progression. In particular, students moved from illustrative to explanatory models, and developed increasingly sophisticated views of the explanatory nature of models, shifting from models as correct or incorrect to models as encompassing explanations for multiple aspects of a target phenomenon. They also developed more nuanced reasons to revise models. Finally, we present challenges for learners in modeling practices—such as understanding how constructing a model can aid their own sensemaking, and seeing model building as a way to generate new knowledge rather than represent what they have already learned. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 46: 632–654, 2009
This article presents an Exponential Growth Learning Trajectory (EGLT), a trajectory identifying and characterizing middle grade students’ initial and developing understanding of exponential growth as a result of an instructional emphasis on covariation. The EGLT explicates students’ thinking and learning over time in relation to a set of tasks and activities developed to engender a view of exponential growth as a relation between two continuously covarying quantities. Developed out of two teaching experiments with early adolescents, the EGLT identifies three major stages of students’ conceptual development: prefunctional reasoning, the covariation view, and the correspondence view. The learning trajectory is presented along with three individual students’ progressions through the trajectory as a way to illustrate the variation present in how the participants made sense of ideas about exponential growth.
The study of functions has traditionally received the most attention at the secondary level, both in curricula and in standards documents—for example, the Common Core State Standards for Mathematics (CCSSI 2010) and Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM] 2000). However, the growing acceptance of algebra as a K–grade 12 strand of thinking by math education researchers and in standards documents, along with the view that the study of functions is an important route into learning algebra (Carraher and Schliemann 2007), raises the importance of developing children’s understanding of functions in the elementary grades.
The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.
This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district's regularly planned mathematics instruction. We share and discuss students' responses to a written pre-and post-assessment that addressed their understanding of several big ideas in the area of early algebra, including mathematical equivalence and equations, generalizing arithmetic, and functional thinking. We found that the intervention group significantly outperformed the nonintervention group and was more apt by posttest to use algebraic strategies to solve problems. Given the multitude of studies among adolescents documenting students' difficulties with algebra and the serious consequences of these difficulties, an important contribution of this research is the finding that-provided the appropriate instruction-children are capable of engaging successfully with a broad and diverse set of big algebraic ideas.
Children's Fractional Knowledge elegantly tracks the construction of knowledge, both by children learning new methods of reasoning and by the researchers studying their methods. The book challenges the widely held belief that children's whole number knowledge is a distraction from their learning of fractions by positing that their fractional learning involves reorganizing not simply using or building upon their whole number knowledge. This hypothesis is explained in detail using examples of actual grade-schoolers approaching problems in fractions including the schemes they construct to relate parts to a whole, to produce a fraction as a multiple of a unit part, to transform a fraction into a commensurate fraction, or to combine two fractions multiplicatively or additively. These case studies provide a singular journey into children's mathematics experience, which often varies greatly from that of adults. Moreover, the authors' descriptive terms reflect children's quantitative operations, as opposed to adult mathematical phrases rooted in concepts that do not reflect-and which in the classroom may even suppress-youngsters' learning experiences. Highlights of the coverage: Toward a formulation of a mathematics of living instead of beingOperations that produce numerical counting schemes Case studies: children's part-whole, partitive, iterative, and other fraction schemes Using the generalized number sequence to produce fraction schemes Redefining school mathematics This fresh perspective is of immediate importance to researchers in mathematics education. With the up-close lens onto mathematical development found in Children's Fractional Knowledge, readers can work toward creating more effective methods for improving young learners' quantitative reasoning skills. © Springer Science+Business Media, LLC 2010 All rights reserved.
This paper addresses the role of learning progressions in informing many international standards documents, discussing the affordances and limitations of building standards adn curricula from a learning progression model. An alternate model, the hypothetical learning trajectory, is introduced and contrasted with learning progressions. Using the example of exponential functions, learning progressions are compared to learning trajectories in terms of their theoretical origins adn practical implications. Recommendations for further work building learning trajectories in secondary mathematics are discussed
This paper reports results from a written assessment given to 290 third-, fourth-, and fifth-grade students prior to any instructional intervention. We share and discuss students’ responses to items addressing their understanding of equation structure and the meaning of the equal sign. We found that many students held an operational conception of the equal sign and had difficulty recognizing underlying structure in arithmetic equations. Some students, however, were able to recognize underlying structure on particular tasks. Our findings can inform early algebra efforts by highlighting the prevalence of the operational view and by identifying tasks that have the potential to help students begin to think about equations in a structural way at the very beginning of their early algebra experiences.
As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).
The equal sign is perhaps the most prevalent symbol in school mathematics, and developing an understanding of it has typically been considered mathematically straightforward. In fact, after its initial introduction during students' early elementary school education, little, if any instructional time is explicitly spent on the concept in the later grades. Yet research suggests that many students at all grade levels have not developed adequate understandings of the meaning of the equal sign. Such findings are troubling with respect to students' preparation for algebra, especially given Carpenter, Franke, and Levi's (2003) contention that a "limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra. Virtually all manipulations on equations require understanding that the equal sign represents a relation." This article describes middle school students' understanding of the equal sign and the relationship between their understanding and performance when solving algebraic equations. (Contains 6 figures and 1 table.)
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.)
The first in a three-volume set, this book focuses on young children between the ages of four and eight as they construct a deeper understanding of numbers and the operations of addition and subtraction. Rather than offer unrelated activities, this book provides a concerted, unified description of development with a focus on big ideas, progressive strategies, and emerging models. Drawing from the work of the Dutch mathematician Hans Freudenthal, it defines mathematics as "mathematizing"--the activity of structuring, modeling, and interpreting one's "lived world" mathematically. It also describes teachers who use rich problematic situations to promote inquiry, problem solving, and construction, and children who raise and pursue their own mathematical ideas. Chapters include: (1)"'Mathematics' or 'Mathematizing?'"; (2) "The Landscape of Learning"; (3) "Number Sense on the Horizon"; (4) "Place Value on the Horizon"; (5) "Developing Mathematical Models"; (6) "Addition and Subtraction Facts on the Horizon"; (7) "Algorithms versus Number Sense"; (8) "Developing Efficient Computation with Minilessons"; (9) "Assessment"; and (10) "Teachers as Mathematicians." (Contains 73 references.) (ASK)
The purpose of this article is to suggest ways of using research on children's reason-ing and learning to elaborate on existing national standards and to improve large-scale and classroom assessments. The authors suggest that learning progres-sions—descriptions of successively more sophisticated ways of reasoning within a content domain based on research syntheses and conceptual analyses—can be useful tools for using research on children's learning to improve assessments. Such learning MEASUREMENT, 14(1&2), 1–98 progressions should be organized around central concepts and principles of a disci-pline (i.e., its big ideas) and show how those big ideas are elaborated, interrelated, and transformed with instruction. They should also specify how those big ideas are enacted in specific practices that allow students to use them in meaningful ways, en-actments the authors describe as learning performances. Learning progressions thus can provide a basis for ongoing dialogue between science learning researchers and measurement specialists, leading to the development of assessments that use both standards documents and science learning research as resources and that will give teachers, curriculum developers, and policymakers more insight into students' scien-tific reasoning. The authors illustrate their argument by developing a learning pro-gression for an important scientific topic—matter and atomic-molecular theory— and using it to generate sample learning performances and assessment items.
We describe efforts toward the development of a hypothetical learning progression (HLP) for the growth of grade 7–14 students' models of the structure, behavior and properties of matter, as it relates to nanoscale science and engineering (NSE). This multi-dimensional HLP, based on empirical research and standards documents, describes how students need to incorporate and connect ideas within and across their models of atomic structure, the electrical forces that govern interactions at the nano-, molecular, and atomic scales, and information in the Periodic Table to explain a broad range of phenomena. We developed a progression from empirical data that characterizes how students currently develop their knowledge as part of the development and refinement of the HLP. We find that most students are currently at low levels in the progression, and do not perceive the connections across strands in the progression that are important for conceptual understanding. We suggest potential instructional strategies that may help students build organized and integrated knowledge structures to consolidate their understanding, ready them for new ideas in science, and help them construct understanding of emerging disciplines such as NSE, as well as traditional science disciplines. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 47:687–715, 2010
This study reports on our steps toward achieving a conceptually coherent and empirically validated learning progression for carbon cycling in socio-ecological systems. It describes an iterative process of designing and analyzing assessment and interview data from students in upper elementary through high school. The product of our development process—the learning progression itself—is a story about how learners from upper elementary grades through high school develop understanding in an important and complex domain: biogeochemical processes that transform carbon in socio-ecological systems at multiple scales. These processes: (a) generate organic carbon (photosynthesis), (b) transform organic carbon (biosynthesis, digestion, food webs, carbon sequestration), and (c) oxidize organic carbon (cellular respiration, combustion). The primary cause of global climate change is the current worldwide imbalance among these processes. We identified Levels of Achievement, which described patterns in the way students made progress toward more sophisticated reasoning about these processes. Younger learners perceived a world where events occurred at a macroscopic scale and carbon sources, such as foods and fuels, were treated as enablers of life processes and combustion rather than sources of matter transformed by those processes. Students at the transitional levels—levels 2 and 3—traced matter in terms of materials changed by hidden mechanisms (level 2) or changed by chemical processes (level 3). More advanced students (level 4) used chemical models to trace matter through hierarchically organized systems that connected organisms and inanimate matter. Although level 4 reasoning is consistent with current national standards, few high school students reasoned this way consistently. We discuss further plans for conceptual and empirical validation of the learning progression. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 46: 675–698, 2009
Scholarship on learning progressions (LPs) in science has emerged over the past 5 years, with the first comprehensive descriptions of LPs, on the nature of matter and evolution, published as commissioned reports (Catley, Lehrer, & Reiser, 2005; Smith, Wiser, Anderson, & Krajcik, 2006). Several recent policy reports have advocated for the use of LPs as a means of aligning standards, curriculum, and assessment (National Research Council [NRC], 2005, 2007). In some ways, LPs are not a new idea; developmental psychologists have long been examining the development of childrens' ideas over time in several scientific domains. However, the emerging research offers renewed interest, a new perspective, and potentially new applications for this construct. For these reasons, this special issue of the Journal for Research in Science Teaching is timely. Our goal in this introduction is to explain the motivation for developing LPs, propose a consensual definition of LPs, describe the ways in which these constructs are being developed and validated, and finally, discuss some of the unresolved questions regarding this emerging scholarship. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 46: 606–609, 2009