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Conceptualizations and Issues related to Learning Progressions, Learning Trajectories, and Levels of Sophistication
... The results demonstrated significant differences among clusters within each FT mode (the F values ranging from 149.02 to 520.36, with all p values below 0.01), confirming the validity of the clustering results. Finally, the sophistication of each cluster's performance on FT was analyzed to characterize the five clusters as different levels within the LP of FT (e.g., Battista, 2004Battista, , 2011Duschl et al., 2011). ...
... Unlike previous studies (e.g., Blanton et al., 2015a;Stephens et al., 2017a), where the levels within LPs of FT form a weak levels-hierarchy with no necessary class inclusion relationship between the levels, the levels identified in this study form a strong levels-hierarchy, requiring class inclusion relationships between the levels (Battista, 2011). This implies that students' reasoning at one level is assumed to have progressed through the preceding levels. ...
... This study has several limitations. Firstly, students' performance and development in FT are influenced by local curricula and instructional methods (Battista, 2011). Therefore, local traditions, textbook series, and teacher practices should be considered when interpreting an LP. ...
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 the context of general mathematics education, assessments built on learning trajectories (LT) gained considerable attention. LT, sometimes called learning progressions (Battista, 2011), are cognitive models capturing how learning of specific contents typically develops. The models are used as the "underlying scale" (Harris et al., 2022), and the assessments are designed to tap into certain misconceptions or inform about students' proficiency levels regarding specific curricular content (Harris et al., 2022;Stacey et al., 2018). ...
... Researchers must delineate components of the cognitive model and coherently conceptualize them, for example, as different knowledge components, which may be further characterized based on levels of understanding (Battista, 2011;Confrey et al., 2017). A component may play different roles in the learning process, for example, as a learning outcome or prerequisite. ...
... A component may play different roles in the learning process, for example, as a learning outcome or prerequisite. Roles and hypothesized linkages between different components (e.g., prediction of specific outcomes by specific prerequisites) should be described according to the underlying cognitive model (Battista, 2011;Harris et al., 2022). For sophisticated cognitive models like LT, research may focus on the central components of the model first. ...
Regularly assessing students’ learning using evidence-based cognitive models can enhance learning by supporting teachers’ monitoring and scaffolding decisions. In the case of mathematics, this is considered especially essential for topics renowned as being complex for students, like fractions. Although previous findings from paper-based studies yielded sophisticated cognitive models, they were not consistently related so far to evidence-based assessment approaches. Also, as paper-based testing appears resource-consuming, its practical use is hindered. Addressing this gap, we developed instruments for digital evidence-based assessment. This paper focuses on the psychometric quality of two tests assessing prior knowledge of fractions: symbolic multiplication and division skills (SMDS) and proportional reasoning skills (PRS). Using item-response-theory-methods to analyze data from N = 147 fifth graders, the digital SMDS-test showed good reliability and validity. The findings suggest a specific item subset suited as an efficient short test. Regarding PRS, further adaptation of the digital format is advised.
... On the contrary, different individuals handle the various obstacles in different ways that lead to a variety of personal developments, some of which allow the individual to progress through increasing sophistication in a meaningful way while others lead to alternative conceptions, or even failure" (Tall, 2004, p. 286). Battista (2011) ...
... • The second level comes after the introduction of "big ideas" or "core ideas" (Battista, 2011). During this phase, the teacher can use DGGA problems posed for investigation and proof in a DGS environment. ...
... This is why students are not able to solve static geometry problems, when they belong at the lower van Hiele levels. According to Battista (2011) "Selecting/creating instructional tasks, adapting instruction to students' needs, […] require detailed, cognition-based knowledge of how students construct meanings for the specific mathematical topics targeted by instruction" (p.527). ...
In the current study, I will be presenting a literature review regarding the importance of students building a problem's representation and the role modeling a real-world problem plays in students' progressive mathematization. I shall introduce five types of geometrical problems applying the meaning of Linking Visual Active Representations (LVARs). Concrete examples will be presented in the next sections (i.e., Euclid's proof of the Pythagorean Theorem, Vecten's theorem, Gamow's problem). I shall also introduce the meanings of hybrid object and diagram, as well as the meaning of dynamic section in a dynamic geometry environment, through examples. To summarize, I created an empirical classification model of sequential instructional problems in geometry. Its contribution to our knowledge in the area of the didactics of mathematics lies in the fact that this sequence of problems is regarded as a process whereby students develop a sequentially deeper understanding and increasingly more coherent reasoning that raises their van Hiele level.
... Por otra parte, la progresión en el aprendizaje del concepto de recta tangente se entiende como un camino a lo largo del cual los estudiantes alcanzan el objetivo de aprendizaje y puede ser descrita mediante una descomposición genética del concepto, vista como una hipótesis del investigador para justificar las decisiones de enseñanza (Orts, Llinares & Boigues, 2016). En esta investigación nos hemos apoyado en la relación entre la descomposición genética de un concepto y la idea de trayectoria de aprendizaje (Battista, 2011;Clements & Sarama, 2004;Simon, 2014), ya que entendemos que una descomposición genética puede verse como una forma de describir la progresión en el aprendizaje que forma parte de una trayectoria hipotética de aprendizaje. En particular, la descomposición genética usada en esta investigación considera la linealidad local de la función como un elemento que debe relacionarse con la tendencia de la sucesión de las rectas secantes para llegar a conocer el concepto de recta tangente como objeto. ...
... Para ello se deben considerar diferentes tipos de tareas, como presentar gráficas de funciones cuya recta tangente corte en varios puntos y ejemplos de rectas tangentes en puntos de inflexión, como han sido consideradas en el experimento de enseñanza diseñado y en otras investigaciones(Biza et al., 2008;Tall, 1985;Vivier, 2010). En este sentido, actividades como la número 5 de la sesión 4 han mostrado ser un buen indicador de cómo los estudiantes han tematizado el esquema de recta tangente al usar la concepción leibniziana no solo como definición del concepto, sino también en un entorno diferente.En segundo lugar, esta investigación aporta información sobre el debate en educación matemática acerca de los constructos de trayectoria de aprendizaje (hipotética o real) y niveles de progresión, en el sentido de integrar la perspectiva cognitiva que considera las fases en el aprendizaje conceptual en el desarrollo de las trayectorias de aprendizaje(Battista, 2011;Clements & Sarama, 2004;Simon, 2014;Weber, Walkington & McGalliard, 2015). Los elementos usados para describir las progresiones en el aprendizaje de la recta tangente permiten estar en disposición de fundamentar el diseño de futuros experimentos de enseñanzaClements & Sarama, 2004). ...
... La descripción de la descomposición genética del concepto de recta tangente mostrada en este trabajo es una manera de caracterizar la progresión hipotética en el aprendizaje y, por lo tanto, puede ser vista como una parte de una trayectoria hipotética de aprendizaje(Simon, 1995), al definir un objetivo de aprendizaje (la tematización del concepto de recta tangente con alumnos de Bachillerato). Así, los resultados empíricos vinculados a una secuencia didáctica determinada, y por lo tanto a un contexto curricular, permiten considerar en qué medida las características de las trayectorias de aprendizaje identificadas (la progresión en el aprendizaje) podrían o no modificarse al introducir cambios en la secuencia instruccional(Battista, 2011). ...
El objetivo de esta investigación es caracterizar trayectorias de aprendizaje del concepto de recta tangente en estudiantes de Bachillerato en un experimento de enseñanza. Se considera un modelo de progresión del aprendizaje del concepto de recta tangente que usa la idea de linealidad local (concepción leibniziana) para apoyar la transición desde la concepción euclídea hasta la concepción cartesiana. Identificamos tres trayectorias de aprendizaje caracterizadas por dos aspectos: i) la relación entre los registros gráfico y analítico que permite progresar desde la concepción euclídea a la cartesiana vía la concepción leibniziana, y ii) la aproximación al valor de una función en el entorno del punto de tangencia mediante la recta tangente. Los resultados obtenidos sugieren que la interiorización de la concepción leibniziana es necesaria para superar el obstáculo epistemológico que supone la concepción euclídea para el aprendizaje del concepto de recta tangente.
... That means that the level hierarchy implies a relation of dependence and inclusion between the levels: First, children need the knowledge of lower levels to develop successive levels (dependence). Second, children who have developed a certain level are supposed to have developed the prior levels, too (inclusion) (Battista, 2011). The levels of the model are not distinct classes of place value understanding that suddenly change. ...
... The model construction followed a four-step circle as suggested by Battista (2011). The first step was the literature review and content analysis described above. ...
Recent studies have shown that children’s proficiency in writing numbers as part of the so-called transcoding correlates with math skills. Typically, children learn to write numbers up to 10,000 between Grade 1 and 4. Transcoding errors can be categorized in lexical and syntactical errors. Number writing is thus considered a central aspect of place value understanding. Children’s place value understanding can be structured by a hierarchical model that distinguishes five levels. The current study investigates to what extent a profound understanding of the place value system can explain individual differences in number writing. N = 266 s and third graders (126 girls) participated in the study. The children wrote down 28 verbal given numbers up to 10,000 and completed a place value test based on a hierarchical model to assess number writing skills and place value understanding. Second graders made more number writing errors than third graders and transcoding errors were mostly syntactical errors. In both grades, transcoding performance and place value understanding correlated substantially. In particular complex numbers were more often solved correctly by children with a more elaborated place value understanding. The effect of place value understanding on error rate was smaller regarding lexical errors than syntactical errors. This effect was also comparably small regarding inversion-related errors. The results underpin that writing numbers is an integral part of early place value understanding. Writing numbers can be assumed to be mostly based on the identification of the place values. However, variance in transcoding skills cannot totally be explained by place value understanding, because children with an elaborated place value understanding differed in transcoding performance, too. The differences between the grades indicate that children’s development of writing numbers is also driven by instruction in school. Thus, writing numbers and place value understanding overlap but exceed each other. We discuss how an understanding of the place value relations can be integrated in existing frameworks of place value processing. Since writing numbers is a basic skill in place value understanding, it might serve as an efficient screening method for children, who struggle severely with understanding the decimal place value system.
... After constructing a theory-based model, this model was operationalized and tested. The results of the pilot studies were discussed within the context of the model, which led to a critical revision of the model (Battista, 2011). This process was completed when the operationalization did not contradict the theoretical assumptions anymore. ...
... The presented validation of the level sequence draws on quantitative methods exclusively and especially on item-responsetheory (IRT). While this method is often used to evaluate models of children's developmental trajectories (Balt et al., 2020;Clements et al., 2008;Fritz et al., 2018;Schulz et al., 2020), some researchers also raise criticism (Battista, 2011). As children can only respond to the given stimuli in the chosen paradigm, the test can only assess the extent to which the a-priori described levels are reached. ...
Research has shown that a profound place value understanding is crucial for success in learning mathematics. At the same time, a substantial number of students struggles with developing a sustainable place value understanding. In this regard, two aspects of the place value system appear especially relevant: First, the knowledge of the decimal structure of numbers, and second, the relation between the bundling units. To support teaching place value understanding, a developmental model of place value understanding focusing on both aspects has been constructed and validated in Germany. The model comprises five levels of place value understanding that build upon each other hierarchically. This study aims at validating the level hierarchy in Turkey to prepare the usage of the developmental model as learning trajectory for Turkish primary schools. N=437 Turkish students from Grades 2 through 4 completed a translated version of the item collection of the German validation study as well as a Turkish place value test that is not based on a developmental model. In a Rasch analysis, most items of the translated item collection showed item difficulties as predicted by the model. In a regression analysis, item difficulties were well predicted by their allocation within the level hierarchy, while other item characteristics did not inform about the item difficulties. Substantial correlations between the translated item collection and the Turkish test underpin the claim that the model is appropriate to assess Turkish students' place value understanding as well as structuring place value instruction in Turkish primary schools.
... Exemples de définitions du spatial dans ces recherches Battista (2008Battista ( , 2011, développant plutôt une approche cognitive, intègre le spatial dans le géométrique. Il distingue en effet trois « niveaux de sophistication » (level of sophistication 18 ) dans le raisonnement géométrique. ...
... Son modèle s'appuie sur les trois premiers niveaux de Van Hiele (rappelés dans Battista, 2011), un modèle universellement cité dans les travaux relatifs à la pensée géométrique 19 . ...
... Typically, a learning progression starts with knowledge that students are expected to bring to a learning experience and ends with a societal norm of what students should know. Between these endpoints are intermediate stages representing progress toward the overall goal [5]. A less linear model, called Pieces of Knowledge [24], expresses independent kernels of knowledge that one could learn in any order, but that all must be learned to gain full understanding of a concept. ...
... An LT has three parts: (1) an overall learning goal, (2) a predicted pathway to the learning goal (with periodic waypoints), and (3) instructional activities that help students move along the path [8]. A key difference between a learning progression and an LT is the latter's attention to activities [5]. This paper focuses on the first two components of our Debugging LT, goals and pathways, because full discussion of activities is beyond the scope of this paper. ...
Curriculum development is dependent on the following question: What are the learning goals for a specific topic, and what are reasonable ways to organize and order those goals? Learning trajectories (LTs) for computational thinking (CT) topics will help to guide emerging curriculum development efforts for computer science in elementary school. This study describes the development of an LT for Debugging. We conducted a rigorous analysis of scholarly research on K-8 computer science education to extract what concepts in debugging students should and are capable of learning. The concepts were organized into the LT presented within. In this paper, we describe the three dimensions of debugging that emerged during the creation of the trajectory: (1) strategies for finding and fixing errors, (2) types of errors, and (3) the role of errors in problem solving. In doing so, we go beyond identification of specific debugging strategies to further articulate knowledge that would help students understand when to use those techniques and why they are successful. Finally, we illustrate how the Debugging LT has guided our efforts to develop an integrated mathematics and CT curriculum for grades 3-5.
... Bu doğrultuda öğretim programı tasarlayanların ve yazanların, program materyallerinin kalitesini artırmaları için öncelikle öğrencilerin konu ile ilgili ne bildiklerini, öğrenmede karşılaştıkları zorlukları ve o konuda yaygın olarak sahip oldukları kavram yanılgılarını temele almaları önem kazanmaktadır (Maskiewicz & Lineback, 2013). Bazı yazarlar da öğrencilerin (yerçekimi, buharlaşma, kimyasal değişim gibi)bazı kazanmaktadır (Alonzo & Steedle, 2008;Battista, 2011;Duschl, Maeng & Sezen, 2011). ...
... Öğrenme yörüngeleri, öğrenme progresyonlarının tasarlanması ve geçerliliğinin saptanmasına yönelik yöntemlere katkıda bulunan önemli kaynaklardan biri olmasına rağmen aralarında nüanslar olduğundan ayrıştırılması gerekmektedir. Çünkü öğrenme progresyonları daha büyük materyal yığınlarını ifade ederek öğretim programı gibi daha geniş bağlamlarda ele alınırken öğrenme yörüngeleri düşüncelerin, akıl yürütme yollarının ve stratejilerin gelişimsel olarak sıralanmasıdır (Battista, 2011;Confrey & Maloney, 2015 ...
Öğrenme progresyonları öğretim programı ile ilgili hipotezlerin dikkatli bir şekilde tasarlanmasına ve test edilmesine dayanan kanıt temelli modellerden biridir ve son yıllarda öğrencilerin merkezi bilimsel kavramları anlamasını destekleme ve o alandaki okuryazarlıklarını artıracak öğretim materyallerini tasarlama anlamında popülerlik kazanmaktadır. Fen eğitiminde öğrenme progresyonları ile ilgili olan bu inceleme çalışması konuyla ilgili araştırma, geliştirme ve inceleme çalışmaları hızla ilerlerken Türkçe alanyazındaki erişilebilir kaynakları artırmak ve daha fazla araştırmacı ve uygulayıcının dikkatini çekmek gereksinimlerinden yola çıkılarak gerçekleştirilmiştir. Bu bağlamda çalışmada konuyla ilgili yapılmış çalışmalar incelenerek (a) öğrenme progresyonlarının tanımı, genel özellikleri, potansiyel kullanım alanları, (b) öğrenme progresyonlarının geliştirilmesi, geçerliliği ve revizyonu ve (c) öğrenme progresyonlarının değerlendirmeler ile ilişkisine yer verilmektedir. Çalışmanın hem profesyonel programların tasarlanmasında hem de öğretim programlarına yeni bir bakış açısı ile bakılarak öğretim materyallerinin ve değerlendirmelerin programla daha uyumlu hale getirilmesinde öğrenme progresyonlarını incelemeye ve kullanmaya teşvik etmesi umulmaktadır.
... Her öğrencinin tek bir düzeyde akıl yürütme sergilemediğini, bazılarının görevin zorluk düzeyine bağlı olarak aynı anda birkaç düzeyde tepkiler ortaya koyduğunu gözlemlemişlerdir. Battista (2011), bu durumun tam olarak aynı olmasa da Siegler'in (2005) örtüşen dalgalar metaforuna benzediğini söylemiş, öğrencinin farklı konularda farklı düzeylerde olabileceğini belirtmiştir. Araştırmacılar bu durumun düzeylerin hiyerarşik yapısının reddedildiği anlamına gelmediğini ancak bir öğrencinin farklı geometri içeriklerinin hepsinde tek bir düzeye atanmasının da doğru olmadığını belirtmişlerdir. ...
Bölümde öncelikle van Hiele geometrik düşünme düzeyleri modelin ortaya çıkışından bahsedilmiştir. Sonrasında geometrik düşünme düzeyleri örneklerle
anlatılmış ve düşünme düzeylerinin kendine has özelliklerine yer verilmiştir. Ardından düşünme düzeyleri arasındaki geçiş aşamaları tanıtılmıştır. Daha sonra
bu teorinin uygulamaya bakan yönünde araştırmacıları ve öğretmenleri en çok
ilgilendiren konulardan biri olan geometrik düşünme düzeylerinin ölçülmesi ile
ilgili açıklamaların olduğu alt bölüme yer verilmiştir. Bu modeli incelemek ve
değerlendirmek üzere yapılan araştırmalar hakkında kısa bir derlemeyi takiben
düşünme düzeylerine uygun etkinlik önerilerine yer verilmiştir. Bölümde ayrıca
yaygın kabul görmesine rağmen teoriyle ilgili akla gelen bazı eleştiriler açıklanmıştır. Son kısımda ise, öğretmenler için teoriyle ilgili uygulama önerileri sunulmuştur
... Learning trajectories, according to Simon (1995), account for unexpected tendencies, recognizing that each student's learning process can be idiosyncratic, though often follows similar paths due to regularities in learning. Battista (2011) further suggested that learning trajectories are simplified representations of the complex learning paths taken by individual students, which often involve substantial back-and-forth movement between cognitive levels. ...
This paper conducts a systematic literature review of research on mathematics learning trajectories within the Korean context and suggests directions for future learning trajectory research. Specifically, the aim of this review is to illuminate the trends of existing and emerging research on learning trajectories in the Korean context, ultimately guiding future research directions in the international research and practice field. This review critically examines 32 articles published in Korea Citation Index journals and reports on national research projects, focusing on the necessities, purposes, and types of research categorized by mathematical content area, as well as the theoretical frameworks used. The results reveal that the types of learning trajectory research conducted in Korea are diverse, encompassing exploration, development, and application of learning trajectories. We suggest that the learning trajectories research warrants further investigation, including discussion on different perspectives in the international contexts. We also posit that these types of research have the potential to create synergy when conducted in a connected series, rather than as disconnected studies.
... LPs theory emphasized that students' understanding of the scientific concept is a gradual development process from one state to another. With the continuous leap of understanding state, students will form a better cognitive structure related to the concept, a clearer way of thinking and a stronger ability to solve situational problems (Alonzo & Steedle, 2009;Battista, 2011). In addition, Linn et al. expressed the students' understanding level of the scientific concept as the degree of knowledge integration, that is, whether and to what extent meaningful links among the concept and other relevant scientific concepts, scientific principles, scientific methods, and scientific thinking have been established in students' cognitive structure (Linn, 2006;Liu et al., 2022). ...
Many research studies have shown that lower-secondary school students’ scientific literacy depends on their understanding of corresponding scientific concepts. Therefore, it is of great significance to model and assess students’ conceptual understanding in learning specific scientific concepts, because it can reflect and diagnose students’ cognitive characteristics, so as to provide evidence for targeted educational interventions. However, on the topic of Moon Phases, existing research studies have deficiencies in the evaluation and analysis of different levels of students’ conceptual understanding, which restrict teachers to take effective educational interventions to help them. Thus, this study explored a new level division and an assessment test for revealing and explaining students’ different conceptual understanding in learning Moon Phases. This study was conducted among 768 Chinese lower-secondary school students. Through a variety of quantitative and qualitative analyses, it was found that students’ conceptual understanding of Moon Phases can be divided into three levels of continuous improvement, including the mechanical memory of different phenomena of moon phases, the formation of a preliminary mental model of Moon Phases, and the construction of a complete mental model of Moon Phases. Accordingly, this study made some precise suggestions for effectively improving students’ conceptual understanding of Moon Phases. Keywords: assessment of conceptual understanding, level division, Moon Phases, quantitative analysis, qualitative analysis
... Most of the experiments are conceptualised as case studies, oriented to support the learning of groups of students in a particular content domain. The theoretical intention is to identify and describe how the different groups of students learn during a period of time (Battista, 2011), adapting the curricular goals of the research and using theoretical models to justify the design of the tasks (Steffe & Thompson, 2000). ...
In this article, we share the design and implementation of the preparatory phase of a didactical experiment with teachers of mathematics, using informatics technologies, in classrooms intended for young teenagers from 12 to 17 years old from the province of Buenos Aires, Argentina. We based this design on the contributions of different theoretical frameworks (MTSK, Mathematical Working Space, Collaborative Research, Teaching experiment methodology) and analysed the data obtained from the administration of an instrument created ad hoc, to generate information that will allow us to build the didactical experiment. We conclude that the design of the experiment requires a joint reflection between the groups of teachers and researchers, to make the teacher's specialised knowledge explicit, guarantee its performance and goal achieving.
... We frame students' evolving enumeration schemes as suggestive of an initial learning trajectory (LT) for understanding permutations. Consistent with both Steffe's (2004) and Battista's (2011) formulations, we take a LT to be a researcher-constructed model of (a) students' initial concepts and mental operations for reasoning about a particular logico-mathematical idea, (b) an account of how their concepts and operations transformed over a fixed period of time, and (c) an account of the instructional tasks and interactions that seem to be linked to these transformations. ...
Permutations are fundamental to combinatorics and other areas of mathematics, and it is important that students develop efficient and conceptually supported ways of mentally constructing, listing, and enumerating them. To date, there is still much to learn about how students reason about enumerating permutations, and how instruction can support students’ conceptual development. We address this gap in the research literature by carefully tracing the evolution of two preservice middle school teachers’ permutation enumeration strategies and conceptualizations, which led to the formulation of levels of sophistication for combinatorial reasoning. These levels are explained by applying and extending a constructivist theory of levels of abstraction. Additionally, we outline an instructional approach that was instrumental in facilitating student learning. Together, the proposed levels and linked instructional approach constitute an initial learning trajectory for permutations that we believe could be useful for understanding and supporting post-secondary non-STEM students’ meaningful conceptualizations and enumerations of permutations.
... Critics of the sequentially ordered model of learning progressions argue that LPs are, at best, constructs imposed by the investigators and not a valid description of learning (Sikorski and Hammer, 2010). Instead, they suggest a complex view of the learning process consisting of hosts of possible pathways that are sensitive to curriculum, instruction, social status of the learners, and other contextual characteristics (Battista, 2011). Moreover, the non-unitary view of LPs expects that the highest-level performances in the progression -its' ''upper anchors'' -should represent rich and sophisticated ways of reasoning about a topic, that resemble those of experts (Sikorski, 2019). ...
Learning progressions (LPs) are novel models for the development of assessments in science education, that often use a scale to categorize students’ levels of reasoning. Pictorial representations are important in chemistry teaching and learning, and also in LPs, but the differences between pictorial and verbal items in chemistry LPs – is unclear. In this study, we examined an Ordered Multiple Choice (OMC) LP assessment of explanations of physical properties and processes in matter, that included equivalent verbal and pictorial items. A cohort of 235 grade 7 students that learned the particle model of matter, responded to these assessments and the data was analyzed in terms of their apparent levels of reasoning. We employed two analyses to examine the role of pictorial items in the level-based model of the LP: a polytomous RASCH analysis of the multiple-choice responses, and a verbal analysis of the students’ explanations of their choices. We found that our data does not fit a fine-grained, four-level model, but that it does fit a coarse-grained three-level model. In addition, when fitting the data to the three-level model, the pictorial items placed more students in the midlevel than their verbal counterparts. The verbal analysis showed that explanations of selections of pictures that represent a partial, midlevel understanding, were significantly less aligned with the details in the picture, than explanations of the correct, upper-level selections. Finally, the proportions of student explanations of both upper-level choices and midlevel choices that were aligned with the information in the pictures were correlated with the overall difficulty of the items. This suggests that complex pictorial representations of processes are less likely to reveal coherent reasoning.
... Introduced in science education in the early 21st century (e.g., Smith et al., 2006), the term LP is now common across disciplines (e.g., Heritage, 2008). It is sometimes used interchangeably with LT in mathematics (e.g., Confrey, 2019), although some note definitional distinctions between these terms (Battista, 2011). Jin et al. (2019) described LPs as "cognitive models" (p. ...
This systematic review examined evidence of the utility of learning progression (LP)–based assessments to inform teaching and student learning in classroom contexts. Fifty-nine studies met inclusion criteria and were analyzed against four research questions. Evidence highlighted their potential for supporting judgments about learning, informing instructional and learning decisions, and improving teacher learning and development. Although 23 studies measured student achievement, reporting positive overall effects, only 6 adopted the experimental designs necessary for causal claims. Using LP-based assessment for formative purposes was well supported. Limited evidence was found regarding summative and accountability uses. Findings show that LP-based assessment design and use requires trade-offs relating to standardization and scale. Teachers need opportunities for negotiation when making judgments and integrating LP-based assessments into existing curriculum and policy contexts. Future research should examine student use of LP assessments and find a balance between standardization and customization to meet the needs of diverse learners and local contexts.
... 214). An LP for a topic (a) starts with the informal, pre-instructional 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) (Battista, 2011(Battista, , 2012. Learning progressions are playing an increasingly important role in mathematics and science education (National Research Council, 2001Smith et al., 2006). ...
... Science educators usually use the term 'learning progression' and developed these for core science ideas and practices. Battista (2011) makes the distinction that learning trajectories are more detailed than learning progressions because learning trajectories include descriptions of instruction. ...
Australia has a National Numeracy Learning Progression (NNLP) that is strongly aligned with the Australian Curriculum: Mathematics. This article examines how a sub-element within this progression could be impacting students’ learning of Science. This sub-element is firmly based on Mathematics education research as to how students build their understanding of geometric measurement (the structure of length, area and volume). Mathematics educators subsequently researched children’s measurement of mass and included it within the same sub-element of the NNLP. The contexts in which mass and volume are measured in Mathematics are different to those used in teaching Science. This article presents two studies that used variation theory and task-based interviews of children in Years 5 and 6, to explore their thinking about mass and volume in a Science context. The findings suggest that mathematical constructs in geometric measurement could be constraining the development of scientific ideas about matter. This research has implications for furthering the development of the NNLP to encompass scientific aspects of measuring matter.
... In elementary teacher education in relation to mathematics, the making of kinds is annealed at the most fundamental level with processes of learning to make ken -or a focus on individual and collective sense-making (Khan, 2006) as well as on teacher understanding of student understandings most notably through frameworks like Variation Theory (Marton, 2014), Learning Trajectories (Battista, 2011;Clements & Sarama, 2014; and Mathematical Knowledge for Teaching (MKT/M4T) (Ball, 2017;Davis & Renert, 2014). However, for many of the pre-service teachers I have worked with, make ken is a prelude to their (teachers') attempts to 'classify' and 'sort' learners into petrified categories often based on learning styles (kinesthetic, visual), and much less frequently called upon, multiple intelligences, which ignores the shameful ethnocentric origins of the concept (Fallace, 2019). ...
This study was primarily concerned with the concepts that are relevant in determining the possible role of visualization towards students' mathematical abstraction and representation in probability. In particular, we highlighted the value of mathematical abstraction, and the discussions about representation and visualization. The subject of the study were six Grade 11 students enrolled in different academic strands in one of the Senior High Schools in Metro Manila. Results show that students' experiences help them establish rich representations with numerous meaningful visual imageries. In particular, students were able to interpret and reflect rather than just recall and rely solely on established procedures in solving problems involving probability. Through visualization, students were able to reach the level of mathematical abstraction needed to understand a new concept.
... In elementary teacher education in relation to mathematics, the making of kinds is annealed at the most fundamental level with processes of learning to make ken -or a focus on individual and collective sense-making (Khan, 2006) as well as on teacher understanding of student understandings most notably through frameworks like Variation Theory (Marton, 2014), Learning Trajectories (Battista, 2011;Clements & Sarama, 2014; and Mathematical Knowledge for Teaching (MKT/M4T) (Ball, 2017;Davis & Renert, 2014). However, for many of the pre-service teachers I have worked with, make ken is a prelude to their (teachers') attempts to 'classify' and 'sort' learners into petrified categories often based on learning styles (kinesthetic, visual), and much less frequently called upon, multiple intelligences, which ignores the shameful ethnocentric origins of the concept (Fallace, 2019). ...
The conventional way of teaching probability, often using coins, dice, and the standard deck of cards has continually been a challenge for many students. Most of the time, students are unable to relate to these learning materials causing them to hold numerous misconceptions. This study focuses on how visualization can be developed as a way to bring students to the level of mathematical abstraction expected of them when they finish high school by using a specially designed learning material. The conceptualization process of the learning material that taps on two important unique characteristics of Filipino students, namely visuals and humor, is also discussed. The result shows that student's affinity with the material addresses misconceptions and helps students develop a more powerful conceptual understanding of probabilities.
... Burada bir başka husus ise öğrencilerin farklı rotalardan (basamaklardan) geçerek de hedefe ulaşabilecekleridir dolayısıyla bütün öğrenciler için geçerli olan tek bir rota tarif etmek güçtür. Öğretim programlarında öğrencilerin ulaşması gereken hedefler ve bu hedeflere giden etkili yollar tarif edilmiş olsa da öğrencilere bireysel olarak odaklandığımızda öğrencilerin başarılı bir şekilde atabilecekleri bir sonraki adımı spesifik olarak görebilmekteyiz (Battista, 2011). Bu bakımdan öğrencilerin gerekçelendirmelerinin anlaşılması hangi seviyede olduğunun ortaya konması ve yorumlanması önemlidir. ...
In this study, students’ thinking about measuring and comparing length
was investigated. Sample of the study is consists evenly distributed 204
students from 4, 5 and 6th grade. A test, including open-ended question
about length was used as data collection instrument. In some questions,
students asked to find the length of shapes, in others they asked to
compare the length of given two shapes. Data was analyzed qualitatively.
The results revealed that the majority of students use units when they
measure length while they decide with appearance when comparing
length. According to the findings obtained from the study, the
frequencies of the participants seem to be low in the category of answers
based on measurement. This indicates that the participants rather make
visual-holistic assessment and do not make much use of mathematical
iterations and individual comparison or recombination of the parts of the
shapes. The number of responses that coded as incomprehensible or not
justified was far high. Further, It has also been seen that as the class level
increases, the number of answers that use mathematical basis in their
explanations increases.
... Several researchers have pointed out that person classification is not a straightforward process (or may even have potential pitfalls) and that the level of a student's performance is to be expected to fluctuate across performances, depending on both the demands of the task and the student's perception of what is required. For example, Battista (2011) wrote, "Research on learning suggests that quite often, the state of student learning is not neatly characterized as 'being at a specified level,'" (p. 542), and Corcoran et al. (2009) observed that "At any given time, an individual may display thinking/practices characteristic of different points on the path, due to features of both the assessment context and the individual's cognition" (p. ...
A learning progression, or learning trajectory, describes the evolution of student thinking from early conceptions to the target understanding within a particular domain. As a complex theory of development, it requires conceptual and empirical support. In earlier work, we proposed a cycle for the validation of a learning progression with four steps: 1) Theory Development, 2) Examination of Empirical Recovery, 3) Comparison to Competing Models, and 4) Evaluation of Instructional Efficacy. A group of experts met to discuss the application of learning sciences to the design, use, and validation of classroom assessment. Learning progressions, learning trajectories, and how they can support classroom assessment were the main focuses. Revisions to the cycle were suggested. We describe the adapted cycle and illustrate how the first third of it has been applied towards the validation of a learning progression for the concept of function.
... Σημαντική θέση μεταξύ αυτών έχει ασφαλώς το μοντέλο των Pierre και Dina Van Hiele (Fuys, 1985), καθώς έχει χρησιμοποιηθεί ως βάση και πλαίσιο μελέτης της γεωμετρικής σκέψης από πολλούς ερευνητές (π.χ. Battista, 2011. Clements, 2003. ...
Geometry is a structural component of mathematics, with increased spatial and design requirements that cannot be easily met by students with mild disabilities. Systematic investigation of the difficulties encountered by students with mild disabilities in their effort to learn Geometry is a prerequisite for the implementation of effective intervention programs. However, research on this issue is relatively scarce. The aim of the present study was to assess the geometric knowledge of 54 students with mild disabilities (learning disabilities or ADHD) who attended the two last classes of elementary school. Participants were asked to recognize, describe and categorize geometric shapes and solid bodies that were presented in tactile mode and through pictorial representations. Semi-structured clinical interviews were used for gathering the data in the context of Curriculum Based Assessment and the Van Hiele’s model of geometrical thinking. Participants of both categories of mild disabilities presented difficulties in distinguishing shapes and bodies, properly using the terminology, and formatting inductive geometrical reasoning. Participants with learning disabilities had higher achievement when dealing with haptic relative to pictorial representations of geometric shapes and bodies. Sixth graders performed better than fifth graders. Results are discussed in terms of the differences between the two categories of mild disabilities as well as with regard to the implementation of intervention programs.
... 186-187). In Battista's (2011) research, the sequence of tasks and ordered levels of sophistication in students' understanding are central to both what is conveyed in a learning trajectory and how it is communicated. Figure 1b offers a visual depiction of Battista's metaphor of "plateaus" of levels of understanding, which are complemented by narrative descriptions of related task type and instruction in his research. ...
... Essentially an LP is a description of how students typically progress from naïve to sophisticated levels of reasoning about a concept, and most content standards are not designed to reflect the full range of student reasoning. Battista (2011) elaborates other assumptions behind LPs: ...
Approaches to test score use and test purpose lack the well-developed methodological guidelines and established sources of evidence available for intended score interpretation. We argue in this paper that this lack fails to reflect the ultimate purpose of a test score—to help solve an important problem faced by intended test users. We explore the treatment of intended test purpose and test score use under the chain of assumption/inferences perspective identified within an argument-based approach to validity. Next, we revisit the notion of test score use and argue that, at least for classroom assessments based on complex constructs, such as learning progressions in math and science, test score use can be more effectively conceptualized as part of a potential solution to solving a problem, or “job-to-be-done.”. We argue for shifting from the definition of validity to the concept of effectiveness. Finally, we illustrate an argument- based approach to test score effectiveness by contrasting effectiveness arguments for interim assessments based on a conventional test blueprint or a test blueprint augmented with learning progressions.
... In elementary teacher education in relation to mathematics, the making of kinds is annealed at the most fundamental level with processes of learning to make ken -or a focus on individual and collective sense-making (Khan, 2006) as well as on teacher understanding of student understandings most notably through frameworks like Variation Theory (Marton, 2014), Learning Trajectories (Battista, 2011;Clements & Sarama, 2014; and Mathematical Knowledge for Teaching (MKT/M4T) (Ball, 2017;Davis & Renert, 2014). However, for many of the pre-service teachers I have worked with, make ken is a prelude to their (teachers') attempts to 'classify' and 'sort' learners into petrified categories often based on learning styles (kinesthetic, visual), and much less frequently called upon, multiple intelligences, which ignores the shameful ethnocentric origins of the concept (Fallace, 2019). ...
This paper aims to discuss how play in pre-kindergarten and kindergarten schools can develop in young children mathematical thinking for future complex mathematics by engaging them in doing
fun mathematical activities. By using two examples from the literature, we will explain the important role of teachers as facilitators to use play for developing children’s mathematical thinking. Play might help teachers to engage students in interesting activities and allow teachers to interact with their students. Learning throughout play has different stages. Those stages consist of introduction to play and its objects, children’s free exploration, asking young children questions to provoke their learning, teaching them concepts, letting them explore more and apply their learning in similar activities, and supporting their learning for the parts that are not clear for them.
... La trayectoria de aprendizaje sobre fracciones (información teórica proporcionada a los estudiantes para maestro en el módulo de fracciones) ha sido diseñada teniendo en cuenta el currículum español y los estudios empíricos sobre el desarrollo del pensamiento de los estudiantes de educación primaria sobre fracciones (Battista, 2011;Steffe, 2004;Steffe y Olive, 2010). El objetivo de aprendizaje de nuestra trayectoria se deriva del currículum de educación primaria y consiste en dar sentido a la idea de fracción y su interpretación como parte-todo para dotar de sentido a los algoritmos con fracciones. ...
La competencia mirar profesionalmente se ha identificado como una competencia necesaria que todos
los maestros deben adquirir como medio para identificar situaciones relevantes de aprendizaje en el
aula. La conceptualización de esta competencia se ha abordado desde diferentes perspectivas, aunque
desde todas ellas se subraya la importancia de identificar los elementos matemáticos relevantes en
las respuestas de los estudiantes, interpretar la comprensión de los estudiantes, y fundamentar las
decisiones de acción que se tomarán en función de la comprensión previamente identificada. Con el
objetivo de desarrollar esta competencia en los futuros maestros de educación primaria, en el ámbito
particular del pensamiento fraccionario, hemos desarrollado un experimento de enseñanza que contempla
un ciclo de investigación en tres fases: diseño de la instrucción y planificación (módulo de
enseñanza), implementación y análisis. En este trabajo presentamos como resultado de este ciclo de
investigación, el diseño de una de las tareas profesionales del módulo de enseñanza.
... Dicho experimento tiene como objetivo caracterizar trayectorias de aprendizaje del concepto de polígono en estudiantes de educación primaria. Las trayectorias de aprendizaje están compuestas por descriptores de la manera de pensar de los estudiantes cada vez más sofisticadas, condicionadas por la enseñanza y los tipos de tareas propuestas (Battista, 2011;Clements & Sarama, 2004;Seah & Horne 2019;Simon, 1995). El análisis de las producciones de los estudiantes permite comprobar las hipótesis que justifican el diseño de las tareas. ...
El experimento de enseñanza es una aproximación metodológica útil en la investigación en educación matemática para estudiar cómo los estudiantes aprenden. El objetivo de este artículo es mostrar las características de esta aproximación metodológica, a través del desarrollo de un experimento de enseñanza en tercer curso de educación primaria con estudiantes de 9 años, que está dirigido a apoyar el aprendizaje del concepto de polígono. Ejemplificamos las características de esta aproximación a la investigación en educación matemática subrayando (i) el uso de teorías sobre el aprendizaje en el diseño de las actividades, y (ii) la diversidad necesaria de fuentes de información que favorece la triangulación de las inferencias que se puedan generar. Finalmente, se apuntan posibles usos de los registros de la práctica derivados de los experimentos de enseñanza para la formación de profesores.
... A few studies looked at student reasoning about variability through the SOLO taxonomy, finding a general increase in reasoning with the age of the student (Watson, 2001), however, researchers have not focused on reasoning at the individual student level apart from a learning trajectory (Ben-Zvi, 2004). Other studies have shown that students' geometric reasoning and degree of acquisition of knowledge can be at multiple levels simultaneously (Battista, 2011;Clements & Battista, 1992), though these studies used van Hiele levels instead of the SOLO taxonomy to classify reasoning levels. The multi-level reasoning of individual students at any moment shows the complexity of learning and has important implications to teaching. ...
... In Mathematics and STEM education field, a variety of researches have been accomplished, investigating successful and effective methods of instruction, teaching and learning. As results of such studies, several Hypothetical Learning Trajectories (HLTs) (Clements & Sarama, 2009;Battista, 2011) have been structured based on different subjects and/or different target groups of students. The aim of this study is to build an HLT at primary education level of a STEM project based on electricity, giving emphasis on the differentiation of learning goals and outcomes, considering the different levels of pupils' readiness. ...
... Agreement among teachers with respect to classification of student responses into levels of an LP has been examined by van Rijn et al. (2018). It has been argued that placing a student into a particular level of an LP is not necessarily an appropriate task, because his or her performance may show aspects of multiple levels (Battista, 2011;Corcoran, Mosher, & Rogat, 2009;Daro et al., 2011). For example, in a discussion of the hierarchic interactionalism framework of Clements and Sarama, Daro et al. (2011) wrote the following with respect to placing students into levels of an LP: ...
Learning progressions (LPs) describe the development of domain‐specific knowledge, skills, and understanding. Each level of an LP characterizes a phase of student thinking en route to a target performance. The rationale behind LP development is to provide road maps that can be used to guide student thinking from one level to the next. The validity of an LP cannot be taken for granted, however. LPs evolve from a synthesis of multiple research studies, subject‐matter expertise, and standards documents. They are working models of student development that may require revision in light of critique and empirical evidence. The formulation of an LP is an iterative process in which expert feedback is elicited, data are collected, and the LP is revised accordingly. We developed an LP for the concept of function both because the concept of function is challenging to attain and because it is central to the study of algebra and higher mathematics. We report early findings with respect to the validity of the concept of function LP, based on small‐scale cognitive interviews and expert reviews.
... Actual learning trajectory is obtained from implementing the hypothetical learning trajectory in the teaching experiment and analyzing students' learning path. Battista [12] focuses the idea of learning trajectory on the framework of a cognition-based assessment. This notion emphasizes the level of the model for a topic that not only describes a student's cognitive process, but also what things students can or cannot do, students' reasoning and conceptualization, a cognitive obstacle, and mental processes for progressing to higher levels. ...
Learning trajectory becomes the main issues in mathematics education research. However, there have been limited studies of students' learning trajectory for triangle construction. Therefore, this design research was conducted to investigate students' learning trajectory for the topic of the triangle. The study involved 22 students from 7th grade in Malang, Indonesia. Data were collected through a videotaped, a student's worksheet, and a classroom observation. The results showed that students discovered the requirement of forming a triangle given three side lengths. In this condition, the starting point of students' learning trajectory was the drawing of a line segment from the given three side lengths. Students examined two side lengths whether these side lengths can be joined to the line segment as a triangle or not. Students used rulers for determining those three side lengths that could form a triangle. They made a statement that the sum of any two sides of a triangle must be greater than the third side. Furthermore, teachers should consider students' learning trajectory for achieving successfully the learning goal.
This document primarily investigates the structure of the Pythagorean Tree, integrating my previous research and writings with earlier insights and contemporary ideas related to its fractal nature, which is produced through instrumental schemata. To illustrate the concept of instrumental schemata, I will present the example of Generator tools that can be created within dynamic geometry software environments. I will briefly discuss the van Hiele theory, which is crucial for classifying students based on their understanding and interpretation of geometric figures. In one section, I will investigate a fresh perspective on the Pythagorean tree, concentrating on the figures that arise when one carefully observes the empty spaces formed between the branches of the tree. Finally, I will delineate the essential elements of an innovative Fractal-based Dynamic Program (FDP) that I designed, developed and implemented. I will propose its effective implementation, as it has the potential to serve as an informal curriculum centered on the principles of transformation geometry and fractals for educational projects. It is anticipated that educators will regard the incorporation of fractal geometry into the standard curriculum as a beneficial didactic and pedagogic framework for fostering students' curiosity and demonstrating the dynamic character of the field.
This study examined backward transfer , which we define as how students’ ways of reasoning about previously encountered concepts are modified when learning about new concepts. We examined the backward transfer produced when students learned about quadratic functions. We were specifically interested in how backward transfer may vary for students whose incoming conceptions about linear functions were at different levels of development. Our study comprised a two-week quadratic functions instructional unit emphasizing covariational reasoning bracketed by pre- and postassessments and interviews. Our analysis focused on four students with incoming linear functions conceptions at different levels of development. Findings revealed that students experienced different kinds of backward transfer. This study generated new insights into backward transfer in the context of mathematics education.
In prior research, we proposed an initial learning trajectory resulting from analyses of two undergraduate students' schemes for enumerating permutations. We explained the levels of this learning trajectory using an elaborated theory of levels of abstraction for both operations on combinatorial composites and, in more advanced levels, symbolic representations of computational reasoning about combinatorial composites. In this preliminary report, we elaborate on this initial learning trajectory by incorporating two additional permutation enumeration schemes, identified from subsequent data that were collected in a recent teaching-experiment research study of students' developing combinatorial reasoning.
Lockwood has argued that taking a set-oriented perspective is critical for successful combinatorial enumeration. To date, however, the research literature has not yet captured the cognitive processes involved in taking such a perspective. In this theoretical paper, we elaborate the constructs of spatial structuring and spatial-numerical linked structuring from within our constructivist theoretical perspective to provide a cognitive account of how students conceptualize and generate combinatorial composites. Our elaborations emerged from analyzing and reflecting on data from two constructivist teaching experiments examining undergraduate students’ combinatorial concepts, mental actions, and strategies. We define multiple constructs in our theoretical approach, including spatial-temporal-enactive (or S*-) structuring, two specific forms of S*-structuring (intra-composite and inter-composite structuring), numerical structuring, and S*-numerical linked structuring. We then illustrate how our framework can be used to analyze students’ physical/mental actions and operations when engaging in combinatorial inquiry. Critically, our findings suggest that successful and meaningful combinatorial enumeration depends on constructing intra-composite S*-structuring for single combinatorial composites, then abstracting, reflecting on, generalizing, and linking this intra-composite structuring to an inter-composite structuring to generate the set of combinatorial composites. We conclude with a discussion of how our paper contributes to existing research on combinatorial reasoning.
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.
Learning path and learning progression have received extensive attention from broad disciplines. The existing research In the field of learning path is rarely applied in curriculum learning and teaching. Learning progression is usually constructed through observations, interviews but not quantitative analyses. With 726 Grade 8 students’ mathematical knowledge in TIMSS-2015 as the research object, this research adopted a newly generated assessment theory – cognitive diagnosis assessment as the research tool and exploited methods such as K-means clustering analysis to construct learning path by combing the relationships among the attributes. We obtained the students’ ability θs for each classified group through the 3PL model in the Item Response Theory (IRT) and constructed the learning progressions based on the θs and the attribute relationships. From a data-driven approach, this method has provided a new perspective as well as the data support for the construction of the learning paths and the learning progressions.
This study investigated the process of instructional change required to translate data on student progress along learning trajectories (LTs) into relevant instructional modifications. Researchers conducted a professional development session on ratio LTs, which included analyzing 3 years of district-level data from Math-Mapper 6–8, a digital LT-based diagnostic assessment application, with fifteen 6th and 7th grade teachers. Teachers subsequently conducted a lesson study to enact what they had learned, allowing researchers to study how teachers used data on student progress along ratio equivalence LTs to design, implement, and evaluate the lesson study. Researchers applied a framework for LT-based data-driven decision making to analyze video data of the lesson study activities. Teachers successfully scanned data reports to pinpoint the LT levels at which to target modified instruction. In one instance, they focused too narrowly on a single item resulting in excessive lesson time on tasks on graph literacy external to the LT. In the other, their data interpretation was overly general and resulted in the design and implementation of a sequence of tasks that reversed the order implied in the LT and relied on the use of more sophisticated strategies from subsequent LTs. Results suggest a need for more data interpretation skills, a deep understanding of the learning theory underpinning LTs, and more precision in teacher discourse around LTs.
Background and Context: We explored how learning trajectories (LTs) might be used to design variables instruction.
Objective: We aimed to develop an LT for variables and use it to guide curriculum development for fourth graders working in Scratch in an integrated mathematics+CS curriculum.
Method: We synthesized learning goals (LGs) and levels of thinking about variables from existing research, then designed instruction to support progress along the levels of thinking while addressing a subset of the LGs.
Findings: Our literature synthesis articulated four levels of thinking and eight LGs about variables. Curriculum development allowed us to support movement through the levels of thinking while addressing five LGs in an integrated math+CS course for fourth grade using Scratch.
Implications: Our model of curriculum development around levels of thinking can support the creation of customized instructional sequences.
In this article, we describe how students structured 2-dimensional space with square and nonsquare units. We employed a cross-sectional design, interviewing 5 students from each of 4 different grade groups: Grades 1, 3, 5, and 7 (ages 7, 9, 11, and 13) in structured, task-based interviews. Our findings about students’ ways of measuring area fit a hypothetical route of learning, validating 4 specific levels of thinking about area units across tasks, regardless of unit shape and dimension indicators. Implications for instruction and research are discussed.
In this study, we investigated the potential of a hypothesized geometrical measurement learning progression (LP) to examine students' thinking and understanding in this domain. We interviewed 30 third to fifth graders using 3 LP‐based cognitive tasks that asked the students to find the length, perimeter, area, surface area, and volume measurement of a given object. We analyzed the students' responses to the tasks to examine variation in levels of the students' geometrical measurement understanding and found evidence of understanding at 5 successive levels of a geometrical measurement LP in 1, 2, and 3 dimensions. From these findings, we concluded that an LP can be a practical tool for understanding students' existing thinking and understanding in a targeted domain and has the potential to support students' further learning in the domain.
We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflection on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970). The mechanism is an elaboration of Piaget's (2001) reflective abstraction and is potentially useful for addressing some of the more intractable problems in teaching mathematics. Implications of the mechanism for lesson design are discussed and exemplified.
We define spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. It is an essential mental process underlying students' quantitative dealings with spatial situations. In this article, we examine in detail students' structuring and enumeration of 2-dimensional (2D) rectangular arrays of squares. Our research indicates that many students do not "see" the row-by-column structure we assume in such arrays. We describe the various levels of sophistication in students' structuring of these arrays and elaborate the nature of the mental process of structuring.
The present study extends previous research in this area by providing a more elaborate and theoretical description of students' solution strategies and errors in dealing with 3-D cube arrays. It describes several cognitive constructions and operations that seem to be required for students to conceptualize and enumerate the cubes in such arrays, exploring in depth general cognitive operations such as coordination, integration, and ''structuring'' as they are manifested in a spatial context. It describes how, in dealing with 3-D rectangular arrays, students' spatial thinking is related to their enumeration strategies. The findings suggest that students' initial conception of a 3-D rectangular array of cubes is as an uncoordinated set of faces. Eventually, as students become capable of coordinating views, they see the array as space filling and strive to restructure it as such. Those who complete a global restructuring of the array use layering strategies. Those in transition use strategies that indicate that their restructuring is local rather than global. Finally, the data suggest that many students are unable to enumerate the cubes in a 3-D array because they cannot coordinate the separate views of the array and integrate them to construct one coherent mental model of the array.
School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathematized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum. Usiskin (Z. Usiskin, 1987), for instance, has described four dimensions of geometry: (a) visualization, drawing, and construction of figures; (b) study of the spatial aspects of the physical world; (c) use as a vehicle for representing nonvisual mathematical concepts and relationships; and (d) representation as a formal mathematical system. The first three of these dimensions require the use of spatial reasoning.
Learning trajectories are presented of 2 fifth-grade children, Jason and Laura, who participated in the teaching experiment, Children's Construction of the Rational Numbers of Arithmetic. 5 teaching episodes were held with the 2 children, October 15 and November 1, 8, 15, and 22. During the fourth grade, the 2 children demonstrated distinctly different partitioning schemes-the equi-partitioning scheme (Jason) and the simultaneous partitioning scheme (Laura). At the outset of the children's fifth grade, it was hypothesized that the differences in the 2 schemes would be manifest in the children's production of fractions commensurate with a given fraction. During the October 15 teaching episode, Jason independently produced how much 3/4 of 1/4 of a stick was of the whole stick as a novelty, and it was inferred that he engaged in recursive partitioning operations. An analogous inference could not be made for Laura. The primary difference in the 2 children during the teaching episodes was Laura's dependency on Jason's independent explanations or actions to engage in the actions that were needed for her to be successful in explaining why a fraction such as 1/3 was commensurate to, say, 4/12.
In this article we describe the development, analysis, and interpretation of a novel item format we call Ordered Multiple-Choice (OMC). A unique feature of OMC items is that they are linked to a model of student cognitive development for the construct being measured. Each of the possible answer choices in an OMC item is linked to developmental levels of student understanding, facilitating the diagnostic interpretation of student item responses. OMC items seek to provide greater diagnostic utility than typical multiple-choice items, while retaining their efficiency advantages. On the one hand, sets of OMC items provide information about the developmental understanding of students that is not available with traditional multiple-choice items; on the other hand, this information can be provided to schools, teachers, and students quickly and reliably, unlike traditional open-ended test items.
Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.
This paper focuses on the psychometric modeling of a specific item format known as ordered multiple choice (OMC). The OMC item format was developed to facilitate diagnostic assessment on the basis of levels from an underlying learning progression that is linked to constrained item response options. Though OMC items were developed by following the building blocks of BEAR Assessment System (BAS) developed by Mark Wilson and colleagues, it is shown that these items have features that seem to make them a poor match with the Rasch-based modeling approach typically taken in BAS applications. Some questions are raised about the use of Wright Maps, a key element of the BAS, to facilitate diagnostic classifications. An alternative modeling strategy is proposed, based on a recent extension to Tatsuoka's Rule Space Method known as the Attribute Hierarchy Method (AHM; Leighton, Gierl, & Hunka, 2004). While the AHM has not previously been applied in the context of learning progression assessments with OMC items, it has some promising features that are illustrated at some length.
Ways in which children think of 10 are considered first. Then a study with 14 second graders is reported; students were placed at three levels with respect to their addition and subtraction concepts. Findings are detailed, along with implications for instruction. (MNS)
This article presents a latent trait model for analyzing data generated by theories of cognitive development that postulate discontinuities. The Saltus model is an extension of the Rasch model for the analysis of dichotomously scored responses. A representation of the model parameters is developed to aid the assessment of first- and second-order discontinuities. The description ends with the application of the model to data sets based on the cognitive development theories of Gagné, Siegler, and van Hiele. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The basic theory of Rasch measurement applies to situations where a person has a certain level of a trait being investigated, and this level of ability is what determines (to within a measurement error) how well the person does on each item in a test. This paper responds to frequent suggestions from colleagues that the use of Rasch measurement would be profitable in analysing a set of data on students' understanding of decimal notation. We demonstrate misfit to the Rasch model by showing that item difficulty estimates show important variation by year level, that there is significant deviation from expected score curves, and that success on certain splitter items does not imply a student is more likely to score well on other items. The explanation given is that conceptual learning may not always be able to be measured on a scale, which is an essential feature of the Rasch approach. Instead, students move between categories of interpretations, which do not necessarily provide more correct answers even when they are based on an improved understanding of fundamental principles. In this way, the paper serves to highlight the assumptions built into the Rasch model and to discuss its applicability to describing the progress of learning with various characteristics.
This report provides evidence of the influence of professional development and curriculum on upper elementary students' understandings
of fractions. Three groups of teachers and their students participated. Two groups implemented a fractions unit that emphasized
problem solving and conceptual understanding. The Integrated Mathematics Assessment (IMA) group participated in a program
designed to enhance teachers' understandings of fractions, students' thinking, and students' motivation. The Collegial Support
(SUPP) group met regularly to discuss strategies for implementing the curriculum. Teachers in the third group (TRAD) valued
and used textbooks and received no professional development support. Contrasts of student adjusted posttest scores revealed
group differences on two scales. On the conceptual scale, IMA classrooms achieved greater adjusted posttest scores than the
other two groups, with no differences between SUPP and TRAD groups. On the computation scale, contrasts revealed no differences
between IMA and TRAD, although TRAD achieved greater adjusted scores than SUPP (p < 0.10). Our findings indicate that the benefits of reform curriculum for students may depend upon integrated professional
development, one form exemplified by the IMA program.
This paper presents a new theory of the development of angle concepts. It is proposed that children progressively recognise deeper and deeper similarities between their physical angle experiences and classify them firstly into specific situations, then into more general contexts, and finally into abstract domains. An angle concept is abstracted from each class at each stage of development. We call the most general angle concept the standard angle concept. To investigate the role of the standard abstract angle concept in conceptual development, 192 children from Grades 2 to 8 were tested to find how they used it in modelling 9 physical angle situations and in expressing similarities between them. It was found that the standard angle concept first develops in situations where both arms of the angle are visible. Even at Grade 8, there are still significant proportions of students who do not use standard angles to represent turning and sloping situations. Implications for theory and practice are explored.
The purpose of this study was to rethink the conceptualization of pedagogical content knowledge based on our descriptive research
findings and to show how this new conceptualization helps us to understand teachers as professionals. This study was a multiple
case study grounded in a social constructivist framework. Data were collected from multiple sources and analysed using three
approaches: (a) constant comparative method, (b) enumerative approach, and (c) in-depth analysis of explicit PCK. The results
indicated that (a) PCK was developed through reflection-in-action and reflection-on-action within given instructional contexts,
(b) teacher efficacy emerged as an affective affiliate of PCK, (c) students had an important impact on PCK development, (d)
students’ misconceptions played a significant role in shaping PCK, and (e) PCK was idiosyncratic in some aspects of its enactment.
Discussion centres on how these five aspects are related to teacher professionalism.
Children's thinking is highly variable at every level of analysis, from neural and associative levels to the level of strategies, theories, and other aspects of high-level cognition. This variability exists within people as well as between them; individual children often rely on different strategies or representations on closely related problems presented close in time. Recognizing such variability can help us both describe development more accurately and better explain how cognitive change occurs.
There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items. Our results suggest partial success in measuring this domain among practicing teachers but also identify key areas around which the field must achieve conceptual and empirical clarity. Although this is ongoing work, we believe that the lessons learned from our efforts shed light on teachers' knowledge in this domain and can inform future attempts to develop measures.
This study examined changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period in which the teachers participated in a CGI (Cognitively Guided Instruction) teacher development program that focused on helping the teachers understand the development of children's mathematical thinking by interacting with a specific research-based model. Over the 4 years, there were fundamental changes in the beliefs and instruction of 18 teachers such that the teachers' role evolved from demonstrating procedures to helping children build on their mathematical thinking by engaging them in a variety of problem-solving situations and encouraging them to talk about their mathematical thinking. Changes in the instruction of individual teachers were directly related to changes in their students' achievement. For every teacher, class achievement in concepts and problem solving was higher at the end of the study than at the beginning. In spite of the shift in emphasis from skills to concepts and problem solving, there was no overall change in computational performance. The findings suggest that developing an understanding of children's mathematical thinking can be a productive basis for helping teachers to make the fundamental changes called for in current reform recommendations.
The fraction concept was analyzed into a set of 16 selected subconcepts, which were arranged into a hypothesized hierarchy. A Fraction Concept Test of 68 items, constructed to measure students' attainment of the 16 selected subconcepts was administered to 279 students in grades 4, 5, and 6. The data were analyzed by using two ratios to determine sequencing validity. 18 of the 23 dependencies in the hypothesized hierarchy were supported. On the basis of these results, it was concluded that certain stated subconcepts were prerequisite to others.
In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described here most participants knew how to divide fractions but could not explain the procedure. The prospective teachers were unaware of major sources of students' incorrect responses in this domain. One conclusion is that teacher education programs should attempt to familiarize prospective teachers with common, sometimes erroneous, cognitive processes used by students in dividing fractions and the effects of use of such processes.
Recent research in cognitive psychology has drawn attention to the important role that students' personal understandings and representations of subject matter play in the learning process. This chapter briefly reviews some of this research, and contrasts the kind of learning that results in an individual's changed conception or view of a phenomenon with the more passive, additive kind of learning assessed by most traditional achievement tests. To be consistent with a view of learning as an active, constructive process, educational tests are required that focus on key concepts in an area of learning, and that take into account the variety of types and levels of understanding that students have of those concepts. In these tests, scoring responses right and wrong is likely to be less appropriate than using students' answers to infer their levels of understanding. This will require not only imaginative new types of test items, but statistical models that permit inferences about students' understandings once their responses have been observed. Psychometric approaches are sketched to construct measures of achievement from such tests.
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).
A successful practice grounded in the principles that guide the current mathematics education reform effort requires a qualitatively different and significantly richer understanding of mathematics than most teachers currently possess. However, it is not as clear how teachers' mathematical understandings develop and how those understandings affect instruction. This paper explores two avenues for K-6 teachers' mathematical development, (a) engagement in inquiry into mathematics itself, and (b) investigation of children's mathematical thinking, illustrating how the need for these two kinds of investigations arises in classroom situations and how they can be pursued in a professional development setting.
Pathology of science occurs when the normal processes of scientific investigation break down and a hypothesis is accepted as true within the mainstream of a discipline without a serious attempt being made to test it and without any recognition that this is happening. It is argued that this has happened in psychometrics: The hypothesis upon which it is premised, that psychological attributes are quantitative, is accepted within the mainstream, and not only do psychometricians fail to acknowledge this, but they hardly recognize the existence of this hypothesis at all. It is suggested that certain social interests, identifiable within the history of modern psychology, have produced this situation because of the ideological and economic secondary gains derived from presenting psychology as a quantitative science. The question of whether modern item response models are exempt from this critique is considered, and it is concluded that they have not yet faced up to the challenges of seriously testing the relevant hypothesis or even bothered to recognize its existence.
This study compared the pedagogical contentknowledge of mathematics in U.S. and Chinesemiddle schools. The results of thiscomparative study indicated that mathematicsteachers' pedagogical content knowledge in thetwo countries differs markedly, which has adeep impact on teaching practice. The Chineseteachers emphasized developing procedural andconceptual knowledge through reliance ontraditional, more rigid practices, which haveproven their value for teaching mathematicscontent. The United States teachers emphasizeda variety of activities designed to promotecreativity and inquiry in attempting to developstudents' understanding of mathematicalconcepts. Both approaches have benefits andlimitations. The practices of teachers in eachcountry may be partially adapted to helpovercome deficiencies in the other.
The Van Hiele level theory offers an explanation and a remedy for student difficulty with higher-order cognitive processes required for success in secondary school geometry. This document reports results of a study which involved about 2700 students in 13 high schools, selected to provide broad representation of community socio-economics in the United States. The investigation looked at: (1) How are entering geometry students distributed with respect to the levels in the Van Hiele scheme; (2) What changes in Van Hiele levels take place after a year's study of geometry; (3) To what extent are levels related to concurrent geometry achievement; (4) To what extent do levels predict geometry achievement after a year's study; (5) What generalizations can be made concerning the entering Van Hiele level and geometry knowledge of students who are later found to be unsuccessful in their study of geometry; (6) To what extent is geometry being taught to students appropriate to their level; and (7) To what extent do geometry classes in different schools and socio-economic settings differ in content appropriateness to student level. (MP)
In the attempt to improve the quality of geometry instruction in schools, researchers and teachers alike have given considerable attention to the van Hiele theory of geometry learning and development, which proposes a series of cognitive levels through which every geometry student passes. This paper reports a study to determine the extent to which factors of age, gender, grade point average, standardized achievement test scores, and geometry achievement acted as predictors for van Hiele levels in two groups of 8th through 11th grade students (N=328): entering geometry (i.e., algebra students) and exiting geometry students. Individual variables determined to be significantly correlated with van Hiele levels among all subjects tested were class level (geometry or algebra), standardized achievement test scores, and geometry achievement test scores. For the geometry subgroup, significant variables were standardized achievement test scores, geometry achievement test scores, and age (negative correlation). For the algebra subgroup, only the geometry achievement test scores were determined to have a significant relationship with van Hiele levels. Grade point average, grade level, and gender were determined to have no significance within any of the groupings. Appendices include Van Hiele Geometry Test, Entering Geometry Test, Exiting Geometry Test, and answer sheets. Contains 28 references. (Author/MKR)
Learning progressions are ordered descriptions of students' understanding of a given concept. In this paper, we describe the iterative process of developing a force and motion learning progression and associated assessment items. We report on a pair of studies designed to explore the diagnosis of students' learning progression levels. First, we compare the use of ordered multiple-choice (OMC) and open-ended (OE) items for assessing students relative to the learning progression. OMC items appear to provide more precise diagnoses of students' learning progression levels and to be more valid, eliciting students' conceptions more similarly to cognitive interviews. Second, we explore evidence bearing on two challenges concerning reliability and validity of level diagnoses: the consistency with which students respond to items set in different contexts and the ways in which students interpret and use language in responding to items. As predicted, students do not respond consistently to similar problems set in different contexts. Although the language used in OMC items generally seems to reflect student thinking, misinterpretation of the language in items may lead to inaccurate diagnoses for a subset of students. Both issues are less problematic for classroom applications than for use of learning progressions in large-scale testing. © 2008 Wiley Periodicals, Inc. Sci Ed93: 389–421, 2009
Assessments associated with learning progressions are designed to provide diagnostic information about the level and nature of student understanding. Valid interpretations of such diagnoses are only possible when students consistently express the ideas associated with a single learning progression level. Latent class analysis was employed to evaluate whether patterns of expected responses to diagnostic multiple-choice items afforded valid interpretations of learning progression level diagnoses. Results indicated that students with scientifically accurate understanding of the forces acting on an object with constant speed usually reasoned systematically across items, but many other students did not. Consequently, interpretations of learning progression level diagnoses on a proposed learning progression would often be invalid. Analyses of this sort would be useful in developing and validating future learning progressions. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 46: 699–715, 2009
National and international studies have found U.S. elementary students to be weak in their understandings and applications of geometric concepts. The University of Chicago School Mathematics Project's (UCMSP) Everyday Mathematics Program is one of the current reform-based elementary curricula incorporating geometry throughout the K-6 curriculum, with an emphasis on hands-on and problem-solving activities. In this study, the geometric knowledge of fifth and sixth graders using the UCSMP curriculum is compared to the knowledge of students using more traditional curricula. Because UCSMP students had been in the program since kindergarten, this research attempts to measure the longitudinal effects of such an approach. Along with an overall score, a subset of test items was used to assign each student a van Hiele level for geometric thinking, as well as a reasoning score. On all measures, UCSMP students substantially outperformed their counterparts, and nearly all differences were significant. Aspects of the UCSMP curriculum and the van Hiele model for learning geometry are discussed relative to these results.
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.
The saltus model was developed in dichotomous form by Wilson (1989), and expanded to polytomous form by Draney (1996) as a
method for detecting and analyzing discontinuities in performance that are hypothesized to occur as a result of rapidly occurring
person growth (e.g.,Fischer, Pipp, & Bullock, 1984). Such discontinuities are often theorized to occur as the result of progression
through developmental stages or levels. The most influential such theory was developed by Jean Piaget (e.g., Piaget, 1950;
Inhelder & Piaget, 1958). Although Piagetian theory has been somewhat controversial of late (e.g., Lourenço & Machado, 1996),
there is still a strong interest in stagelike development in a number of areas, including moral and ethical reasoning (e.g.,
Dawson, 2002; Kohlberg & Candee, 1984), evaluative reasoning (e.g., Dawson-Tunik, 2002; Armon, 1984), adult development (e.g.,
Commons et al., 1998; Fischer, Hand, & Russel, 1984), and cognitive development (e.g., Bond, 1995b, a; Bond & Bunting, 1995;
Demetriou & Efklides, 1989, 1994; Hiele, 1986).
The construction of schemes of action and operation involving composite units is crucial in the mathematical development of children. In the article, six different composite units are identified along with two general categories of schemes involving composite units—units-coordinating schemes and unit segmenting schemes. The schemes within these two categories produced by the three children who are subjects of study in the article are the results of accommodations the children made in their number sequences. The three children each began a two-year teaching experiment with a specific type of number sequence. One of them had constructed the initial number sequence (INS), one the tacitly nested number sequence (TNS), and one the explicitly nested number sequence (ENS). The accommodations of these number sequences by the involved children yielded schemes within the two categories that were qualitatively distinct. Based upon these distinctions, the roots of multiplication and division in children are explained. The first multiplicative concept occurred as an accommodation of the TNS, which is explained as a recursive counting scheme. The operation of recursion of the TNS was used to produce the first units-coordination that can be called multiplicative. Preconcepts and concepts of multiplication and division are identified in terms of the schemes in the two categories and are contrasted with how multiplication and division are understood in contemporary school mathematics programs.
How do children learn mathematics? Research and reform in mathematics education
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Highlights of research on learning school geometry
- M T Battista
Battista, M. T. (2009). Highlights of research on learning school geometry. In T.
Understanding Geometry for a Changing World
Craine & R. Rubenstein (eds.), 2009 Yearbook, Understanding Geometry for a
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At What Grade Should CCSSM Expect Students to Achieve the Properties Level in the van Hiele Learning Progression?
- M T Battista
Battista, M. T. (manuscript in preparation). At What Grade Should CCSSM Expect
Students to Achieve the Properties Level in the van Hiele Learning Progression?
Cognition Based Assessment for elementary school mathematics: Length and length measurement
- M T Battista
Battista, M. T. (accepted). Cognition Based Assessment for elementary school
mathematics: Length and length measurement. Heinemann.
A framework for growth points as a powerful professional-development tool
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professional-development tool. Annual meeting of the American Educational
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Statistics for social and behavioral sciences: Multivariate and mixture distribution Rasch models
- Carstensen
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mixture distribution Rasch models (pp. 119-130). Springer.
Teachers' knowledge and its impact
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Designing learning environments for developing understanding of geometry and space
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about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning
environments for developing understanding of geometry and space (pp. 137-167). Mahwah,
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design of educational assessment. Committee on the Foundations of Assessment.
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synthesis of two models. In R. Lehrer & D. Chazan (Eds.), Designing learning
environments for developing understanding of geometry and space (pp. 109-133).