Content uploaded by Douglas H. Clements
Author content
All content in this area was uploaded by Douglas H. Clements on Mar 26, 2016
Content may be subject to copyright.
Early Childhood Mathematics Education Research: Learning Trajectories for Young Children
Abstract
This important new book synthesizes relevant research on the learning of mathematics from birth into the primary grades from the full range of these complementary perspectives. At the core of early math experts Julie Sarama and Douglas Clements's theoretical and empirical frameworks are learning trajectories-detailed descriptions of children's thinking as they learn to achieve specific goals in a mathematical domain, alongside a related set of instructional tasks designed to engender those mental processes and move children through a developmental progression of levels of thinking. Rooted in basic issues of thinking, learning, and teaching, this groundbreaking body of research illuminates foundational topics on the learning of mathematics with practical and theoretical implications for all ages. Those implications are especially important in addressing equity concerns, as understanding the level of thinking of the class and the individuals within it, is key in serving the needs of all children.
... Spatial reasoning is a set of cognitive skills that allows humans to interact with the physical world and supports our making sense of the world mathematically (Freudenthal, 1973;Clements and Battista, 1992;National Research Council, 2009). It includes skills that clearly relate to mathematics (e.g., composing and decomposing, transforming; and skills that seem more practical in nature and less directly aligned with mathematics education (e.g., mapping, perspective-taking; Muir and Cheek, 1986;Liben and Downs, 1989;National Council of Teachers of Mathematics, 2006;Sarama and Clements, 2009). Nonetheless, skills across this nebulous construct support learning and achievement in other mathematics domains (Hawes and Ansari, 2020;Gilligan-Lee et al., 2022). ...
... First, the spatial reasoning framework used represents only one way to define spatial reasoning and its skills. Aligning the standards with another model, like one more focused on psychological constructs (e.g., the 2×2 classification of spatial skills presented by Uttal et al., 2013 and refined for early education by Đokić and Vorkapić, 2024) or the developmental progression of skill acquisition (e.g., Sarama and Clements, 2009) may have resulted in different findings. Additional research is needed to ascertain if this model best supports EEEs and other educators teaching these skills. ...
Spatial reasoning is critical to early mathematics learning, but it is unclear how early elementary educators learn to teach and are supported in teaching its comprising skills. One view of the available supports can be found by examining the alignment of spatial reasoning skills and mathematics education standards, as standards provide the content of the intended curriculum children are expected to learn at each grade level. This study used content analysis methods to investigate how spatial reasoning might be taught through broadly adopted early elementary education standards in the United States, the Kindergarten through Grade 2 Common Core State Standards for Mathematics. The paper describes the frequency and degree of explicitness with which 38 spatial reasoning skills are therein represented. Findings indicate that most standards implicitly relate to some form of spatial reasoning through a pedagogical reach of teaching expertise, but few standards contain explicit spatial linkages. The implications and limitations of this analysis are discussed in relation to teaching spatial reasoning in early elementary grades and students’ opportunities to learn these critical skills.
... It cannot be conveyed through material action, but only through linguistic description or by reference to representatives from the environment. Therefore, directly or indirectly comparing measures of time is challenging, unlike length measurement (Sarama & Clements, 2009). Time has both cyclical and linear aspects. ...
... Geometrinin etkili bir şekilde kullanılması, özellikle soyut kavramları somut hale getirerek öğrenmeyi kolaylaştırır (Presmeg, 2006). Geometri sayesinde öğrenciler şekillerin, desenlerin ve uzayın yapısını anlamak için geometrik kavramları kullanarak görsel ve pratik deneyimler elde eder (Sarama & Clements, 2009). Ancak ilkokulun ilk yıllarından itibaren bütün sınıf düzeylerinde geometri kazanımlarına yer verilmesine rağmen öğrencilerin geometri konularında zorlandıkları bilinmektedir (Köseleci-Blanchy & Şaşmaz, 2011;Millî Eğitim Bakanlığı [MEB], 2018;. ...
Araştırmanın amacı, 2003-2024 yılları arasında Türkiye’de gerçekleştirilen lisansüstü tezlerin geometrik düşünme düzeyleri üzerindeki eğilimlerini çeşitli değişkenlere göre incelemektir. Doküman analizi yöntemiyle gerçekleştirilen bu çalışmanın verileri, toplamda 92 lisansüstü tezden elde edilmiştir. Elde edilen veriler içerik analizine tabi tutulmuştur. Tezler, türüne, yayın yılına, hazırlandığı üniversite ve anabilim dalına, konu alanına, kullanılan yönteme, araştırma desenine, çalışma grubuna, veri toplama aracına ve veri analizine göre on başlık altında sınıflandırılmıştır. Yapılan analiz sonuçlarına göre, yüksek lisans tezlerinin doktora tezlerine göre daha fazla olduğu ve en yoğun araştırmanın 2023 yılında gerçekleştirildiği gözlemlenmiştir. Lisansüstü tezlerin belirli üniversitelerde ve matematik ile ilköğretim anabilim dallarında yapıldığı belirlenmiştir. Başarı, tutum, uzamsal yetenek, kalıcılık gibi konu başlıklarının en fazla tercih edildiği görülmüştür. Nitel ve nicel araştırma yöntemleri arasında benzer bir çalışma sayısının olduğu ve bununla birlikte durum çalışması ve tarama araştırma desenlerinin en sık tercih edilenler olduğu tespit edilmiştir. En fazla çalışmanın ortaokul düzeyinde olduğu ve en az çalışmanın ilkokul düzeyinde gerçekleştirildiği bulunmuştur. Veri toplama araçları olarak test, ölçek, görüşme formu ve veri analiz tekniği olarak t testi, ANOVA, içerik analizi ve betimsel analiz tekniğinin sıklıkla kullanıldığı belirlenmiştir.
... The current study interprets this trajectory within the interdisciplinary context of visual arts and mathematics education by adapting the works of art to the context of this study. Considering the grade level (sixth grade, twelve years old) and the nature of the artworks utilized in this current study, the hypothetical trajectory levels are more complex than those of Sarama and Clements (2009). Several changes have been made to the levels, including the addition of the reversible 'figure disembedder' due to the nature of artworks perceived as both 2D and 3D. ...
This cross-sectional study investigates the role of maths self-concept in spatial skills development among primary school children. The study, conducted on a sample of students (mean age M = 8.48, SD = 1.11) consisting of 70 girls and 74 boys, explores the relationship between maths self-concept and mental
rotation skills, with a focus on gender and the stage of childhood development. Students completed a computerized mental rotation task measuring accuracy and response time and a maths self-concept questionnaire. Results reveal girls and tweens (pre-adolescents) demonstrate lower maths self-concept compared to boys and younger children. Moreover, maths self-concept and the stage of childhood development significantly influence mental rotation performance, with those with higher self-concept and tweens scoring better on the task. These findings emphasize the importance of addressing low maths self-concept, particularly in girls and students transitioning to secondary education, to foster spatial skills development. Targeted support during this critical educational phase is crucial.
... These items also cover the more advanced levels of the LT, including Length Unit Relator and Repeater, Length Measurer, and Integrated Conceptual Path Measurer. Consistent with the theoretical framework underlying the LT and the assessment (Sarama & Clements 2009), development along the progression proceeds from sensory-concrete, implicit levels, at which manipulable concrete objects are necessary or helpful to integrated-concrete understandings relying on internalized mental representations. Such internal cognition, of course, is less likely to be detected than physical movement of objects. ...
When researchers code behavior that is undetectable or falls outside of the validated ordinal scale, the resultant outcomes often suffer from informative missingness. Incorrect analysis of such data can lead to biased arguments around efficacy and effectiveness in the context of experimental and intervention research. Here, we detail a new Bayesian mixture approach that analyzes ordinal responses with undetectable/uncodable behaviors in two stages: (1) estimate a likelihood of response detection and (2) estimate an Explanatory Item Response Model for the ordinal variable conditional on detection. We present an independent random effects and correlated random effects variant of the new model and demonstrate evidence of model functionality using two simulation studies. To illustrate the utility of our proposed approach, we describe an extended application to data collected during a length measurement teaching experiment (N = 186, 56% girls, 5-6 years at preassessment). Results indicate that students assigned to a learning trajectories instructional condition were more likely to use detectable, mathematically relevant problem-solving strategies than their peers in two comparison conditions and that their problem-solving strategies were also more sophisticated.
... En intervention kan även generera en kunskapsprodukt i form av lärarhandledningar eller utbildningsprogram med mer eller mindre detaljerade beskrivningar av undervisningsaktiviteter (Mulligan m.fl., 2010;Sarama & Clements, 2009). Ett exempel på hur en longitudinell undervisningsutvecklande intervention har resulterat i en kunskapsprodukt, är det franska programmet: Programmation et Progression ACE, som utvecklats av Sensevy med kollegor (2015). ...
Syftet med artikeln är att bidra med kunskap om hur en intervention, med fokus på undervisning om tal och talrelationer i förskoleklass, kan utveckla matematikundervisning. I interventionen har en undervisningsutvecklande modell tagits som utgångspunkt, där planering, genomförande och reflektion över undervisning har skett i en cyklisk process tillsammans med kollegor och forskare. Då interventionen sträckte sig över fyra år, gavs möjligheter att identifiera och förändra komponenterna i relation till syfte och kontext. En nyckelkomponent är de reflektionsunderlag som utvecklades och hur de bidrog till reflektion kring undervisning och lärande. De vetenskapligt förankrade aktiviteterna som förfinades är en annan nyckelkomponent. Kunskapsprodukten som har genererats innehåller en undervisningsguide med beprövade och designade undervisningsaktiviteter, med tillhörande reflektioner kring lärares matematikdidaktiska handlingar samt den undervisningsutvecklande modellen. Developing the teaching of numbers and number relations in preschool class The purpose of this article is to contribute knowledge about how an intervention, focused on teaching numbers and number relations in preschool class, can develop teaching. In the intervention, we employed a practice-oriented model based on planning, teaching, evaluating, and reflecting on teaching in an iterative process, in collaboration with colleagues and researchers. We carried out this intervention over four years, during which we identified key components of the model that were essential for the development of mathematics teaching. We also changed these components based on the purpose and local context. A key component is the reflection documents that were developed and how they contributed to reflection on teaching and learning. Another key component is the research-based activities that we refined. The knowledge product includes a teaching guide with documented and designed teaching activities with reflections on the teacher's enactment of the mathematical activities as well as the teaching model.
This study was conducted to optimize the designs of learning guides embedded in a computer-based simulation environment. The research was based on the Cognitive Theory of Multimedia Learning and Cognitive Load Theory. We investigated computer simulations under four conditions that combined representation and imagination learning strategies. This study recruited 244 fifth-grade Taiwanese students from various backgrounds (indigenous vs. nonindigenous, 98 vs. 146; boys vs. girls, 131 vs. 113) and examined the interactions and main effects of the four experimental conditions on the students’ learning of geometric area concepts. The results revealed that the computer-based simulation platform can serve as a cognitive tool to help students to explore graphs and identify formulas. Moreover, the dynamic representations and trace changing imagination strategy embedded in the learning platform as guiding tools were highly beneficial for learners; this tool moderately reduced the learners’ extraneous cognitive load and improved their learning performance at the application level. The static representation and concept imagination strategy embedded in the platform as a learning guide was also noted to increase the learners’ internal cognitive load, which impaired their learning performance. Notably, boys and girls adopted different learning strategies, and the overall platform appeared to be more beneficial for nonindigenous students and students with middle to high levels of cultural stimulation than for indigenous students and students with low levels of cultural stimulation.
Erken çocukluk döneminden itibaren matematik becerilerinin kazandırılması, matematiği öğrenme sürecinden olumlu bir tutum geliştirmeye ve akademik başarıya kadar çeşitli alanlarda destek sağlar. Bu nedenle, çocukların matematiği öğrenmelerini ve sevmelerini sağlayacak ortamların oluşturulması önemlidir. Bu bağlamda, 54 ila 66 ay arasında değişen toplam 25 çocuktan oluşan bir çalışma grubuyla gerçekleştirilen tek gruplu deneysel desende, erken çocukluk döneminde geliştirilen açık alandaki matematik programının çocukların matematiği sevme durumları üzerindeki etkisi incelenmiştir. Verilerin toplanmasında Dağlı ve Dağlıoğlu (2018) tarafından geliştirilen “Çocuklar İçin Matematiği Sevme Ölçeği (ÇMSÖ)” kullanılmıştır. Açık alanda uygulanan matematik programı 15 etkinlikten oluşmaktadır. Her bir etkinlik süresi yaklaşık 25 dakika olarak planlanmıştır. Haftada 3 gün, 5 hafta süren açık alanda matematik uygulamalarında rakamlar, geometri, parça bütün, gruplama, eşleştirme ve ölçme konuları yer almaktadır. Araştırmada açık alanda uygulanan matematik programının çocukların matematik sevme durumları üzerinde etkili olduğu, sayma, geometrik şekiller, parça bütün, gruplama ve ölçme konularında son test lehine anlamlı farklılıklar elde edilmiştir.
Teori Piaget telah banyak berpengaruh terhadap desain pembelajaran. Pembelajaran yang berorientasi pada guru (teacher centere) berubah menjadi berorientasi pada siswa (student centere). Hal ini berarti bahwa faktor siswa menjadi hal yang utama dan harus diperhatikan dalam membuat suatu desain pembelajaran. Perumusan Hypothetical Learning Trajectory sebagai pedoman pelaksanaan pembelajaran sekaligus sebagai suatu tindakan antisipatif terhadap kemungkinan masalah yang dihadapi siswa dalam proses pembelajaran. Artikel ini menyajikan contoh perumusan Hypothetical Learning Trajectory untuk pembelajaran nilai tempat di kelas 1 sekolah dasar. Piaget's theory has had a lot of influence on learning design. Teacher-oriented learning (teacher center) changed to student-oriented (student center). This means that the student factor is the main thing and must be considered in making a learning design. Formulation of Hypothetical Learning Trajectory as a guideline for implementing learning as well as an anticipatory action against possible problems faced by students in the learning process. This article presents an example of the formulation of a Hypothetical Learning Trajectory for place-value learning in grade 1 elementary school.
The framework of Tharp and Gallimore (1988) was adapted to form a ZPD (Zone of Proximal Development) Model of Mathematical Proficiency that identifies two interacting kinds of learning activities: instructional conversations that assist understanding and practice that develops fluency. A Class Learning Path was conceptualized as a classroom path that includes a small number of different learning paths followed by students, and it permits a teacher to provide assistance to students at their own levels. A case study illustrates this model by describing how one teacher in a Japanese Grade 1 classroom assisted student learning of addition with teen totals by valuing students' informal knowledge and individual approaches, bridging the distance between their existing knowledge and the new culturally valued method, and giving carefully structured practice. The teacher decreased assistance over time but increased it for transitions to new problem types and for students who needed it. Students interacted, influenced/supported one another, and moved forward along their own learning paths within the Class Learning Path.
The first half of the twentieth century has seen many studies made by educators to determine the most desirable time to start a planned sequential instructional program in arithmetic for the young child. These studies have presented findings that vary from initial instruction starting in the first grade to a recommendation that beginning formalized instruction be delayed until the seventh grade. However, these studies indicate that, in general, the young child benefits when systematic instruction starts during his first-grade experiences.