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Learning and teaching early math: The learning trajectories approach
Abstract
Incl. bibl., index
... Evaluation of students must be related to the thought processes and methods used to solve the problem, not only to the final products (Ministry of Education, 2006). Clements and Sarama found positive longitudinal effects on student learning of curriculum developed around learning trajectories (Clements & Sarama, 2014). In the present study, we explored the design principles of DFA activities in geometry that align with the Israeli national curriculum learning trajectories. ...
... We designed an assessment unit based on curriculum learning trajectories to improve the assessment process. This unit was intended to assist teachers in interpreting the evidence collected and influence their decisions in the classroom about the teaching of the topic, to enhance the efficiency of learning and teaching (Clements & Sarama, 2014;Confrey et al., 2017;Ebby & Petit, 2018). In the design principles, we proposed activities appropriate for the topics that appear in the curriculum (with respect to their timing and the order of topics). ...
... The implementation of the assessments in the classroom is designed as an ongoing process, carried out after teaching each aspect (subtopic), and providing a comprehensive analysis of student work after the completion of the entire topic. This design principle is aligned with the notion of a learning trajectory based on a sequence of activities that are suitable and developmentally effective (Clements & Sarama, 2014). In the design principles of our assessment units, we emphasized the importance of alignment between teaching and assessing student work, so that the assessment is authentic and represents the knowledge of the mathematical topic and not the students' testing skills (Stacey & William, 2013). ...
This study presents a method for constructing guiding principles for designing digital formative assessments (DFAs). It is part of a larger research that explores the efficiency of a digital formative assessment platform for elementary school geometry. Our goal in this study was to describe the design principles of effective and authentic DFA tasks informing teachers on various aspects of geometry learning aligned with the Israeli national curriculum for 5th grade. Using a design-based research approach, we formulated our essential components for DFA activities according to the research literature. Next, after carrying out the activities, we redesigned them and reformulated the design principles. We describe a three-cycle study following an "altitudes of a triangle" activity.
... 공간 방향은 공간 감각의 요소이며 (Clements & Sarama, 2014), 여러 학자들은 공간 방향에 대해 조금씩 다른 정의를 내리고 있다. Chong (2017) 에서는 Clements (1999), Del Grande (1990), McGee (1979) Table 3. Elements for spatial orientation (Chong, 2017, p. 666) Sense of direction Sense of distance Sense of coordinates definition ...
... 같다(Clements & Sarama, 2014). 우리나라의 경우, 점의 이동이 3~4학년군에 처음 도입되므로, 학생들의 공간 방향에 대한 학습 경로는 지도 계획 및 방법을 수립하는데 도움이 될 것이다. ...
... El subnivel elemental como período crítico para el desarrollo matemático El subnivel elemental de educación básica, comprendido generalmente entre segundo y cuarto grado, representa un período crítico en el desarrollo del pensamiento lógicomatemático. Durante esta etapa se establecen las bases conceptuales y procedimentales que determinarán en gran medida la relación futura del estudiante con esta disciplina (Clements & Sarama, 2018). ...
El presente estudio analiza la efectividad del juego como herramienta pedagógica para el aprendizaje significativo de las matemáticas en estudiantes del subnivel elemental de educación básica. La investigación se desarrolló con 125 estudiantes de segundo a cuarto grado, a través de un diseño cuasiexperimental con grupo control y experimental, aplicando estrategias lúdicas sistemáticamente durante un período académico. Los resultados evidencian un incremento significativo en la comprensión de conceptos matemáticos fundamentales, la motivación, la participación activa y el desarrollo de habilidades como el razonamiento lógico y la resolución de problemas en el grupo experimental. Se concluye que la implementación metódica de actividades lúdicas, contextualmente significativas y adecuadamente estructuradas, potencia considerablemente el proceso de enseñanza-aprendizaje de las matemáticas en este nivel educativo, reduciendo la ansiedad matemática y mejorando la percepción de autoeficacia de los estudiantes.
... In the absence of larger samples and studies showing a functional relation, the correlation we found can also be interpreted such that a high level of math skills is required to perform well on the relational tasks. On the other hand, prior research on early mathematical abilities and their developmental trajectories in young children seem to indicate at least some foundations in nonarbitrary relations of sameness, difference, comparison and hierarchy (Clements & Sarama, 2007; see also Clements & Sarama, 2014, for an overview of math skills from birth to grade 3). Each of these frames, for instance, may support children's early understanding of quantities, similarities and differences in patterns, and part/whole relations. ...
Relational Frame Theory (RFT) sees the operant acquisition of various patterns of relational responding as being key to language and cognition. Interventions that apply RFT have been shown to be effective, are increasingly more prevalent, and seem to have wide-ranging benefits including increases in intelligence scores and academic performance. Nonarbitrary relations (i.e., relations based on formal properties) are said to be a key precursor that facilitates arbitrary relations in young children. Despite the importance of this repertoire, very few studies have investigated its development. Similarly, although relational responding is said to be an overarching higher-order operant that can be applied to any stimulus content, the specific relationship (if any) between nonarbitrary relational responding and early mathematical skills remains unknown. In the present study, we measured nonarbitrary relational responding in a sample of 43 neurotypical and neurodiverse children between the ages of 2 – 6 years across six relational frames and four response topographies and assessed their mathematics problem solving. We found that nonarbitrary relational responding improved with age, was strongly correlated with mathematical ability, and there was a significant difference in scores between the neurotypical and neurodivergent groups. We also found some differences between response topographies and specific contextual cues.
... In addition to its focus on real-world contexts, RME also highlights the significance of applying structured learning trajectory in its instructional design. Learning trajectory refers to the depiction of cognitive processes and learning progression in a specific mathematical subject, facilitated by a sequence of educational tasks intended to generate anticipated cognitive processes or actions (Clements & Sarama, 2009). It is useful for bridging the work of researchers and practitioners. ...
This research aims to determine whether there is an effect of learning trajectory with the traditional congklak game context on mathematical reasoning ability in terms of self-efficacy. This study adopts a quasi-experimental approach using a 2 × 3 factorial design. The target population included all eighth-grade students at SMP Negeri 2 Wonogiri. The research employed cluster random sampling, selecting learners from two classes. The data collection techniques used were documentation, self-efficacy questionnaire, and mathematical reasoning abilities test. This study employed a data analysis approach that involved a two-way analysis of variance. The results of the hypothesis testing concluded that (1) learners who were taught using an RME-based learning trajectory with the traditional congklak game context showed better mathematical reasoning abilities compared to those taught through direct learning in statistics; (2) learners with high self-efficacy have better mathematical reasoning abilities than those with moderate or low self-efficacy, while learners with moderate self-efficacy showed better mathematical reasoning abilities than those with low self-efficacy; (3) there was no significant interaction observed between the treatment and self-efficacy regarding learners' mathematical reasoning abilities.
... This sequence was designed to help students imagine, classify and describe attributes of polygons. According to Clements and Sarama (2014), a polygon is defined as "a closed plane figure bounded by three or more straight sides" (p.144). The sequence had four lesson suggestions, including additional consolidating lessons the teachers could use if needed. ...
This study was designed to report on the questioning strategies and pedagogical moves a Year 1 teacher used during a lesson, specifically focusing on using student work samples to support mathematical reasoning. The single-case research approach considered questions selected by the teacher, their distribution throughout the lesson, and the implementation of anticipated questions. The results show how the teacher leveraged student work samples as they responded to an open-ended problem (a challenging task) and how she employed pedagogical moves to support students’ mathematical reasoning. A qualitative analysis of classroom discussions provided insights and recommendations for pedagogical moves for facilitating, eliciting, responding to and extending student reasoning. We also offer a practical Questioning Move model teachers can refer to when helping students interpret tasks; supporting those who may struggle; clarifying misconceptions or partial conceptions and supporting understanding; exploring multiple strategies, identifying all possible solutions and extending responses, ensuring all learners have opportunities to develop their reasoning skills.
... Early childhood introduces children to mathematics, a branch of science that studies counting, logical thinking, and reasoning (Clements & Sarama, 2020). Cellucci (2022) concurs, asserting that mathematics is a daily necessity for everyone. ...
This study's objective is to determine the effect of the use of Electronic Student Worksheets (E-SW) based on Quizizz on student learning achievement. The type of research used in this study is quantitative, and it uses an experimental method with a pretest-posttest control group design. Where in this research design, students are given a pre-test first to determine students' initial knowledge, and the last is provided a post-test. The study's population was 87 students from grade X at Public Vocational School 1 Bulango Selatan. The research sample was chosen at random from two grades: grade X Askep B was used as an experimental grade to test E-SW based on Quizizz, and grade X Askep A was used as a control grade to compare how E-SW and textbooks are used. The findings showed that in the inferential analysis, pretest-posttest, both sets of data are normally distributed and homogeneous. This led to the covariance test (ANCOVA), which showed that H0 was rejected and H1 was accepted. So the results of this study show that student learning achievement are higher after being taught using E-SW based on Quizizz compared to using regular student worksheets. This means that using electronic student worksheets based on Quizizz has a positive influence on students' mathematics learning achievement.
... Notwithstanding these findings, some parents and educators criticize direct math instruction and do not believe it is developmentally appropriate. However, with a high-quality learning environment, mathematical concepts typically grasped by children age seven or older can be learned by younger children (Clements & Sarama, 2014). Students' everyday mathematics as well as any additional math skills students acquire before the onset of kindergarten are critical indicators of their future success in both reading and math Engel et al., 2013). ...
Given the importance of early math instruction on later reading and math ability (Claessens and Engel in Teach Coll Rec 115:1–29, 2013), we sought to determine whether preschoolers’ preference for math could be changed. A counterbalanced pre- and post-intervention ABAB/BABA design and a delayed multiple probe across dyads design were used to test the effect of an individualized reinforcement intervention on the indirect and direct reinforcement value of math. Indirect measures referred to the participants’ rate of responding to a performance task during a 1-min timing when Play-Doh® was delivered as a consequence compared to when math problems were delivered as a consequence. Direct measures referred to the number of 5-s intervals (out of 60) each participant completed math when given both math worksheets and Play-Doh®. Preference for math was defined as selecting math for 80% of intervals. The individualized reinforcement intervention consisted of a sequence of procedures. First learn units were delivered, then stimulus–stimulus pairing, and then observational conditioning-by-denial, until a defined successful outcome resulted. Learn units alone were successful in establishing preference for math for two participants, while the stimulus pairing procedure was required for the remaining participants. The results suggest that increasing the reinforcement value of math is indeed possible for individuals as young as four years old and that different procedures are necessary for different students. These results strengthen the need for further research on the possible impact on rate of learning math and literacy.
Despite more than a decade of Science, Technology, Engineering, and Mathematics ( STEM ) reform initiatives toward integrated STEM (iSTEM) education, there remains a paucity of mathematics-focused iSTEM curricular materials in which mathematics is the anchor discipline. To explore the nature of mathematical thinking within the context of an iSTEM instructional approach, the authors investigated four preservice mathematics teachers’ ( PMT s’) spatial thinking as they engaged in a task-based digital activity involving the concept of radian angle measure in the context of light reflection. Findings indicate that PMT s utilised spatial thinking as well as mathematical terminology, concepts, and processes to mathematically describe the science phenomenon of light reflection. Results from this study support the argument that mathematics-focused iSTEM approaches provide an opportunity for a productive understanding of mathematics and of how mathematics relates to other STEM disciplines.
Penelitian tinjauan literatur ini bertujuan untuk mengidentifikasi dan menganalisis kesulitan-kesulitan umum yang dihadapi siswa Sekolah Menengah Pertama (SMP) dalam memahami materi matematika. Matematika, sebagai mata pelajaran fundamental, seringkali menjadi sumber tantangan bagi siswa di jenjang SMP. Tinjauan ini mensintesis temuan-temuan dari berbagai penelitian yang meneliti area-area spesifik dalam matematika yang secara konsisten menimbulkan kesulitan bagi siswa SMP, termasuk pemahaman konsep aljabar, geometri, aritmetika, dan pemecahan masalah. Penelitian ini juga mengeksplorasi faktor-faktor yang berkontribusi terhadap kesulitan-kesulitan tersebut, seperti kemampuan kognitif, latar belakang pendidikan, metode pengajaran, dan sikap siswa terhadap matematika. Hasil sintesis literatur diharapkan dapat memberikan wawasan yang mendalam bagi para pendidik, pengembang kurikulum, dan peneliti mengenai akar permasalahan dalam pembelajaran matematika di SMP, sehingga dapat dirumuskan strategi intervensi dan pendekatan pengajaran yang lebih efektif untuk mengatasi kesulitan-kesulitan umum ini. Implikasi dari tinjauan ini menyoroti pentingnya pengembangan kurikulum yang lebih adaptif dan penggunaan metode pengajaran yang inovatif untuk meningkatkan pemahaman matematika siswa SMP.
The performance of 210 2nd graders in different areas of mathematical cognition was examined. Children were divided into 4 achievement groups: children with difficulties in mathematics but not in reading (MD-only), children with difficulties in both mathematics and reading (MD/RD), children with difficulties in reading but not in mathematics, and children with normal achievement. Although both MD groups performed worse than normally achieving groups in most areas of mathematical cognition, the MD-only group showed an advantage over the MD/RD group in exact calculation of arithmetic combinations and in problem solving. The 2 groups did not differ in approximate arithmetic and understanding of place value and written computation. Children with MD-only seem to be superior to children with MD/RD in areas that may be mediated by language but not in ones that rely on numerical magnitudes, visuospatial processing, and automaticity. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The main objective of this study is to investigate the role of four different modes of representation in mathematical problem solving (MPS), and more specifically to develop a model, which provides information about the effects of these representations in the solution procedures of one-step problems of additive structures. Data were collected from 1447 pupils in Grades 1, 2 and 3 of elementary school in Cyprus. Confirmatory factor analyses (CFA) affirmed the existence of four first-order representation-specific factors indicating the differential effects of each particular type of representation and a second-order factor representing the general ability to solve mathematical problems. Results provided support for the invariance of this structure across the three groups of pupils.
This study draws from the National Center for Early Development and Learning's Multi-State Pre-Kindergarten Study to examine the extent to which program, class- room, and teacher attributes of the program ecology predict observed quality and teacher-child interactions in a sample of 238 classrooms representing 6 states' pre-kindergarten programs. Quality was assessed observationally at the global level and for specific teaching practices. Quality was lower in classrooms with more than 60% of the children from homes below the poverty line, when teachers lacked formal training (or a degree) in early childhood education, and held less child-centered beliefs. Program and teacher attributes were statistically significant, albeit quite modest, predictors of observed quality. Location of the program in a school building, child:staff ratio, and length of day had no relation to quality. State-level factors not attributable to the teacher, program, and classroom factors examined accounted for the majority of explained variance in observed quality. Re- sults suggest that the association between distal features of programs and teachers and quality in pre-kindergarten is more similar to elementary school settings than to child care settings and that quality appears most closely related to proximal teacher and child characteristics.
Harry completed his first year of primary school (Kindergarten) during 2004 in New South Wales, Australia. He enjoyed school; made great friends; played lots of sport; continued to read quite successfully; was well-liked by his teachers; participated in many activities; and, on reportedly rare occasions, did some mathematics. In this paper, comparisons are made between the mathematics Harry was capable of doing before he started school and what his parents were told he actually did during his first year of school. The paper was stimulated by Harry's response to his parents when asked, near the end of his first year, what he had learned in mathematics at school: "I know that you don't have to work hard".
The reliability and, to a lesser extent, the validity of the newly created Number Sense Test was evaluated with a group of 40 3- to 5-year-old children attending Head Start. Six number sense skills (e.g., counting, number identification, addition-subtraction) and children's feelings about school were assessed both before and after instruction either in 2 of these skills (i.e., counting and number identification) or in insects (i.e., an attention control condition). The Number Sense Test was found to have high internal consistency and to be a valid means of assessing pre- to posttest improvements in instructed skills as a function of condition. Except for addition-subtraction, training effects did not generalize to noninstructed number sense skills. In neither condition did instruction have a harmful effect on children's largely positive feelings about school.
This study investigated the developmental dynamics of mathematical performance during children's transition from preschool to Grade 2 and the cognitive antecedents of this development. 194 Finnish children were examined 6 times according to their math performance, twice during each year across a 3-year period. Cognitive antecedents, that is, counting ability, visual attention, metacognitive knowledge, and listening comprehension, were tested at the first measurement point. The results indicated that math performance showed high stability and increasing variance over time. Moreover, the growth of math competence was faster among those who entered preschool with an already high level of mathematical skills. The initial level of math performance, as well as its growth, was best predicted by counting ability. Although quite a large body of research has been carried out on the learning of math, most studies have focused on specific math disabilities (Geary & Hoard,. However, little is known about the development of chil-dren's overall math performance (Ai, 2002), particularly during their early years of formal education. The aim of the present study was to examine the developmental dynamics of children's math performance during the transition from preschool to Grade 2 and the roles that a variety of cognitive antecedents play in this development. At the same time, heterogeneity in the developmen-tal trajectories that children show in their math performance was also investigated.
Acknowledgments The National Institute on Early Childhood Development,and Education expresses its sincere appreciation to the early childhood and education leaders who participated in the meeting we convened to discuss school involvement,in early childhood. Their discussion and insightful comments,are the basis for this publication. We especially thank
Chinese students often outperform U.S. students on international tests in mathematics. Chinese students’ mathematics performances are assumed to be related directly to their teachers’ deep mathematics understanding and ability to represent concepts flexibly in their classrooms, which, in turn, are thought to be influenced by Chinese mathematics curriculum and policies. The authors examine this theoretical assumption through a systematic review of relevant literature and attempt to identify the relationship between Chinese students’ mathematics performance and the factors that contribute to their achievement. On the basis of their review, the authors raise questions about the assumption and propose research that can lead to a better understanding of the relationship between the quality of students’ mathematics learning and the contexts in which their learning occurs.