Effectiveness of Schema-Based Instruction for Improving Seventh-Grade Students’ Proportional Reasoning: A Randomized Experiment

This study examined the effect of schema-based instruction (SBI) on seventh-grade students’ mathematical problem solving performance. SBI is an instructional intervention that emphasizes the role of mathematical structure in word problems and also provides students with a heuristic to self-monitor and aid problem solving. Using a pretest-intervention-posttest-retention test design, the study compared the learning outcomes for 1,163 students in 42 classrooms who were randomly assigned to treatment (SBI) or control condition. After 6 weeks of instruction, results of multilevel modeling indicated significant differences favoring the SBI condition in proportion problem solving involving ratios/rates and percents on an immediate posttest (g = 1.24) and on a six-week retention test (g = 1.27). No significant difference between conditions was found for a test of transfer. These results demonstrate that SBI was more effective than students’ regular mathematics instruction

Development of intuitive rules: Evaluating the application of the dual-system framework to understanding children's intuitive reasoning

This is an author-created version of this article. The original source of publication is Psychon Bull Rev. 2006 Dec;13(6):935-53 The final publication is available at www.springerlink.com Published version: http://dx.doi.org/10.3758/BF0321390

Improving Students' Proportional Thinking Using Schema-Based Instruction

This study investigated the effectiveness of an instructional program (schema-based instruction, SBI) designed to teach 7th graders how to comprehend and solve proportion problems involving ratios/rates, scale drawings, and percents. The SBI program emphasized the underlying mathematical structure of problems via schematic diagrams, focused on a 4-step procedure to support and monitor problem solving, and addressed the flexible use of alternative solution strategies based on the problem situation. Blocking by teacher at three middle schools, the authors randomly assigned the 21 classrooms to one of two conditions: SBI and control. Classroom teachers provided the instruction. Results of multilevel modeling used to test for treatment effects after accounting for pretests and other characteristics (gender, ethnicity) revealed the direct effects of SBI on mathematical problem solving at posttest. However, the improved problem solving skills were not maintained a month later when SBI was no longer in effect nor did the skills transfer to solving problems in new domain-level content

Composable Probabilistic Inference with Blaise

Probabilistic inference provides a unified, systematic framework for specifying and solving these problems. Recent work has demonstrated the great value of probabilistic models defined over complex, structured domains. However, our ability to imagine probabilistic models has far outstripped our ability to programmatically manipulate them and to effectively implement inference, limiting the complexity of the problems that we can solve in practice.This thesis presents Blaise, a novel framework for composable probabilistic modeling and inference, designed to address these limitations. Blaise has three components: * The Blaise State-Density-Kernel (SDK) graphical modeling language that generalizes factor graphs by: (1) explicitly representing inference algorithms (and their locality) using a new type of graph node, (2) representing hierarchical composition and repeated substructures in the state space, the interest distribution, and the inference procedure, and (3) permitting the structure of the model to change during algorithm execution. * A suite of SDK graph transformations that may be used to extend a model (e.g. to construct a mixture model from a model of a mixture component), or to make inference more effective (e.g. by automatically constructing a parallel tempered version of an algorithm or by exploiting conjugacy in a model). * The Blaise Virtual Machine, a runtime environment that can efficiently execute the stochastic automata represented by Blaise SDK graphs. Blaise encourages the construction of sophisticated models by composing simpler models, allowing the designer to implement and verify small portions of the model and inference method, and to reuse model components from one task to another. Blaise decouples the implementation of the inference algorithm from the specification of the interest distribution, even in cases (such as Gibbs sampling) where the shape of the interest distribution guides the inference. This gives modelers the freedom to explore alternate models without slow, error-prone reimplementation. The compositional nature of Blaise enables novel reinterpretations of advanced Monte Carlo inference techniques (such as parallel tempering) as simple transformations of Blaise SDK graphs.In this thesis, I describe each of the components of the Blaise modeling framework, as well as validating the Blaise framework by highlighting a variety of contemporary sophisticated models that have been developed by the Blaise user community. I also present several surprising findings stemming from the Blaise modeling framework, including that an Infinite Relational Model can be built using exactly the same inference methods as a simple mixture model, that constructing a parallel tempered inference algorithm should be a point-and-click/one-line-of-code operation, and that Markov chain Monte Carlo for probabilistic models with complicated long-distance dependencies, such as a stochastic version of Scheme, can be managed using standard Blaise mechanisms

Composable probabilistic inference with BLAISE

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 185-190).If we are to understand human-level cognition, we must understand how the mind finds the patterns that underlie the incomplete, noisy, and ambiguous data from our senses and that allow us to generalize our experiences to new situations. A wide variety of commercial applications face similar issues: industries from health services to business intelligence to oil field exploration critically depend on their ability to find patterns in vast amounts of data and use those patterns to make accurate predictions. Probabilistic inference provides a unified, systematic framework for specifying and solving these problems. Recent work has demonstrated the great value of probabilistic models defined over complex, structured domains. However, our ability to imagine probabilistic models has far outstripped our ability to programmatically manipulate them and to effectively implement inference, limiting the complexity of the problems that we can solve in practice. This thesis presents BLAISE, a novel framework for composable probabilistic modeling and inference, designed to address these limitations. BLAISE has three components: * The BLAISE State-Density-Kernel (SDK) graphical modeling language that generalizes factor graphs by: (1) explicitly representing inference algorithms (and their locality) using a new type of graph node, (2) representing hierarchical composition and repeated substructures in the state space, the interest distribution, and the inference procedure, and (3) permitting the structure of the model to change during algorithm execution. * A suite of SDK graph transformations that may be used to extend a model (e.g. to construct a mixture model from a model of a mixture component), or to make inference more effective (e.g. by automatically constructing a parallel tempered version of an algorithm or by exploiting conjugacy in a model).(cont.) * The BLAISE Virtual Machine, a runtime environment that can efficiently execute the stochastic automata represented by BLAISE SDK graphs. BLAISE encourages the construction of sophisticated models by composing simpler models, allowing the designer to implement and verify small portions of the model and inference method, and to reuse mode components from one task to another. BLAISE decouples the implementation of the inference algorithm from the specification of the interest distribution, even in cases (such as Gibbs sampling) where the shape of the interest distribution guides the inference. This gives modelers the freedom to explore alternate models without slow, error-prone reimplementation. The compositional nature of BLAISE enables novel reinterpretations of advanced Monte Carlo inference techniques (such as parallel tempering) as simple transformations of BLAISE SDK graphs. In this thesis, I describe each of the components of the BLAISE modeling framework, as well as validating the BLAISE framework by highlighting a variety of contemporary sophisticated models that have been developed by the BLAISE user community. I also present several surprising findings stemming from the BLAISE modeling framework, including that an Infinite Relational Model can be built using exactly the same inference methods as a simple mixture model, that constructing a parallel tempered inference algorithm should be a point-and-click/one-line-of-code operation, and that Markov chain Monte Carlo for probabilistic models with complicated long-distance dependencies, such as a stochastic version of Scheme, can be managed using standard BLAISE mechanisms.by Keith Allen Bonawitz.Ph.D

Formative Assessment for Middle School Mathematics Instruction: An Evidence-based Approach to Evaluating Teacher Posing, Pausing, and Probing Moves

This study involved empirical investigation of a moves-based conceptualization of teacher practices of planning, enacting, and reflecting on formative assessment (FA) in mathematics classrooms in a high-needs school district in California. A qualitative case study of six middle school mathematics teachers’ practices of posing questions, pausing to foster equity of participation and quality of response, and probing student thinking, the study provides empirical evidence of qualitatively distinct levels of teacher posing, pausing, and probing moves. The study utilized a National Research Council-based educational assessment design framework that employed construct maps, multi-faceted items design, and scoring guides to examine teacher practice and to provide feedback protocols for teachers engaged in FA. Guided by the 2014 Standards for Educational and Psychological Testing, the study provides evidence for content validity and tools for future rater reliability investigations. The study found levels of teacher questioning practice, operationalized as posing, pausing, and proving moves, could be represented along generalized continua in the context of middle school mathematics instruction. The study’s work toward the development of a teacher learning progression framework in the formative assessment domain has implications for establishing an empirically-based, common grammar of practice in mathematics instruction and preparation

Advances in the sociology of trust and cooperation: theory, experiments, and field studies

The problem of cooperation and social order is one of the core issues in the social sciences. The key question is how humans, groups, institutions, and countries can avoid or overcome the collective good dilemmas that could lead to a Hobbesian war of all against all. Using the general set of social dilemmas as a paradigmatic example, rigorous formal analysis can stimulate scientific progress in several ways. The book, consisting of original articles, provides state of the art examples of research along these lines: theoretical, experimental, and field studies on trust and cooperation. The theoretical work covers articles on trust and control, reputation formation, and paradigmatic articles on the benefits and caveats of abstracting reality into models. The experimental articles treat lab based tests of models of trust and reputation, and the effects of the social and institutional embeddedness on behavior in cooperative interactions and possibly emerging inequalities. The field studies test these models in applied settings such as cooperation between organizations, informal care, and different kinds of collaboration networks. The book will be exemplary for rigorous sociology and social sciences more in general in a variety of ways: There is a focus on effects of social conditions, in particular different forms of social and institutional embeddedness, on social outcomes. Theorizing about and testing of effects of social contexts on individual and group outcomes is one of the main aims of sociological research. Modelling efforts include formal explications of micro-macro links that are typically easily overlooked when argumentation is intuitive and impressionistic Extensive attention is paid to unintended effects of intentional behavior, another feature that is a direct consequence of formal theoretical modelling and in-depth data-analyses of the social processe

An Instructional Analogy Between Unitizing and Fraction Division: Seventh-Graders’ Conceptual Understandings of Division and Interpretations of Fractional Remainders

Fraction division is not a simple concept. In fact, both students and teachers struggle to understand and explain this operation. This study examined whether using instructional analogies between unitizing and division were effective when teaching fraction division to seventh-grade students. The extent to which the analogies were explicitly supported varied by condition. The aim was to investigate whether condition differences existed in participants’ understanding of fraction division, in their interpretations of fractional remainders, and in their understanding of unitizing. Fifty-one participants were randomly assigned to the explicit-links condition (n = 17), the implicit-links condition (n = 17), or the control condition (n = 17). Students in the explicit and implicit-links conditions were first presented with a unitizing problem in which six was regrouped using various units and then they were taught how to solve a fraction division equation. The instructor in the explicit-links condition made connections between both concepts by using gestures, spatial cues, relational language, and other cognitive supports. The link between both concepts was in no way emphasized for the participants in the implicit-links condition. Finally, participants in the control condition saw the fraction division instruction twice and did not receive unitizing instruction. No condition differences were found at posttest in participants’ division explanations nor in their understandings of unitizing. When interpreting a fractional remainder, participants in the implicit-links condition referred to the original unit significantly more than participants in the other two conditions, but no differences were observed in their uses of the referent unit when interpreting the remainder. The present study is relevant to teaching professionals as it provides new information about the ways children think of fraction division and the fractional remainder