Recent technological developments have given rise to blended learning classrooms. An inverted (or flipped) classroom is a specific type of blended learning design that uses technology to move lectures outside the classroom and uses learning activities to move practice with concepts inside the classroom. This article compares the learning environments of an inverted introductory statistics class with a traditional introductory statistics class at the same university. This mixed-methods research study used the College and University Classroom Environment Inventory (CUCEI), field notes, interviews and focus groups to investigate the learning environments of these two classrooms. Students in the inverted classroom were less satisfied with how the classroom structure oriented them to the learning tasks in the course, but they became more open to cooperative learning and innovative teaching methods. These findings are discussed in terms of how they contribute to the stability and connectedness of classroom learning communities.

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How Learning in an Inverted Classroom Influences Cooperation, Innovation and Task Orientation.

Some usability and interpretability issues for single-strategy cognitive assessment models are considered. These models posit a stochastic conjunctive relationship between a set of cognitive attributes to be assessed and performance on particular items/tasks in the assessment. The models considered make few assumptions about the relationship between latent attributes and task performance beyond a simple conjunctive structure. An example shows that these models can be sensitive to cognitive attributes, even in data designed to well fit the Rasch model. Several stochastic ordering and monotonicity properties are considered that enhance the interpretability of the models. Simple data summaries are identified that inform about the presence or absence of cognitive attributes when the full computational power needed to estimate the models is not available.

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Cognitive Assessment Models with Few Assumptions, and Connections with Nonparametric Item Response Theory:

In educational assessment, we observe what students say, do, or make in a few particular circumstances and attempt to infer what they know, can do, or have accomplished more generally. A web of inference connects the two. Some connections depend on theories and experience concerning the targeted knowledge in the domain, how it is acquired, and the circumstances under which people bring their knowledge to bear. Other connections may depend on statistical models and probability-based reasoning. Still others concern the elements and processes involved in test construction, administration, scoring, and reporting. This article describes a framework for assessment that makes explicit the interrelations among substantive arguments, assessment designs, and operational processes. The work was motivated by the need to develop assessments that incorporate purposes, technologies, and psychological perspectives that are not well served by familiar forms of assessments. However, the framework is equally applicable to a...

On the Structure of Educational Assessments

In One Line and Close. Permutations as Linear Orders. Descents Alternating Runs Alternating Subsequences In One Line and Anywhere. Permutations as Linear Orders. Inversions. Inversions Inversion in Permutations of Multisets In Many Circles. Permutations as Products of Cycles. Decomposing a Permutation into Cycles Type and Stirling Numbers Cycle Decomposition versus Linear Order Permutations with Restricted Cycle Structure In Any Way but This. Pattern Avoidance. The Basics. The Notion of Pattern Avoidance Patterns of Length Three Monotone Patterns Patterns of Length Four The Proof of the Stanley-Wilf Conjecture In This Way but Nicely. Pattern Avoidance. Follow-Up. Polynomial Recurrences Containing a Pattern Many Times Containing a Pattern a Given Number of Times Mean and Insensitive. Random Permutations. The Probabilistic Viewpoint Expectation Variance and Standard Deviation An Application: Longest Increasing Subsequences Permutations versus Everything Else. Algebraic Combinatorics of Permutations. The Robinson-Schensted-Knuth Correspondence Posets of Permutations Simplicial Complexes of Permutations Get Them All. Algorithms and Permutations. Generating Permutations Stack Sorting Permutations Variations of Stack Sorting How Did We Get Here? Permutations as Genome Rearrangements. Introduction Block Transpositions Block Interchanges Block Transpositions Revisited Solutions to Odd-Numbered Exercises References List of Frequently Used Notation Index Exercises, Problems, and Problem Solutions appear at the end of each chapter.

Combinatorics of Permutations

The authors propose and test an exemplar-based random walk model for predicting response times in tasks of speeded, multidimensional perceptual classification. The model combines elements of R. M. Nosofsky's (1986) generalized context model of categorization and G. D. Logan's (1988) instance-based model of automaticity. In the model, exemplars race among one another to be retrieved from memory, with rates determined by their similarity to test items. The retrieved exemplars provide incremental information that enters into a random walk process for making classification decisions. The model predicts correctly effects of within- and between-categories similarity, individual-object familiarity, and extended practice on classification response times. It also builds bridges between the domains of categorization and automaticity.

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An Exemplar-Based Random Walk Model of Speeded Classification

From the Publisher:
Knowledge Spaces offers a rigorous mathematical foundation for various practical systems of knowledge assessment, applied to real and simulated data. The systematic presentation extends research results to new situations, as well as describes how to build the knowledge structure in practice.

Knowledge spaces

The information regarding a particular field of knowledge is conceptualized as a large, specified set of questions (or problems). The knowledge state of an individual with respect to that domain is formalized as the subset of all the questions that this individual is capable of solving. A particularly appealing postulate on the family of all possible knowledge states is that it is closed under arbitrary unions. A family of sets satisfying this condition is called a knowledge space. Generalizing a theorem of Birkhoff on partial orders, we show that knowledge spaces are in a one-to-one correspondence with AND/OR graphs of a particular kind. Two types of economical representations of knowledge spaces are analysed: bases, and Hasse systems, a concept generalizing that of a Hasse diagram of a partial order. The structures analysed here provide the foundation for later work on algorithmic procedures for the assessment of knowledge.

Spaces for the assessment of knowledge

This article gives a comprehensive description of a theory for the efficient assessment of knowledge. The essential concept is that the knowledge state of a subject with regard to a specified field of information can be represented by a particular subset of questions or problems that the subject is capable of solving. The family of all knowledge states forms the knowledge space. It is assumed that if 2 subsets K and K′ of questions are assumed to be states in a knowledge space , then KU K′ is also assumed to be a state in K. Such a theory is consistent with the idea that at least some of the notions in the field may be acquired from different sets of prerequisites. Various aspects of the theory are discussed. In particular, the problem of constructing a knowledge space in practice is analyzed in detail. A first sketch of the knowledge space can be obtained by consulting expert teachers in the field. The mathematical theory necessary to render this consultation efficient is given. This preliminary construction can then be tested and refined on the basis of empirical data. To this end, a probabilistic version of the theory is developed, which is similar in spirit to some psychometric models, but it is grounded on the concept of a knowledge space rather than on that of skill or ability. An exemplary application of this probabilistic theory to a high school mathematics test is described, based on a sample of several hundred students. By standard likelihood ratio methods, it is shown how the preliminary knowledge space can be gradually refined, and the number of possible knowledge states substantially reduced. Two classes of Markovian knowledge assessment algorithms are outlined. Most of the results presented summarize previous articles published in various technical journals. The application of the probabilistic theory to the high school mathematics test is original to this article.

Introduction to knowledge spaces: How to build, test, and search them

Adaptations from a book and many scholarly articles are presented. It reviews the main ideas of a novel theory for the assessment of a student's knowledge in a topic and gives details on a practical implementation in the form of a software system available on the Internet. The basic concept of the theory is the 'knowledge state,' which is the complete set of problems that an individual is capable of solving in a particular topic, such as elementary algebra. The task of the assessor consists in uncovering the particular state of the student being assessed. Even though the number of knowledge states for a topic may exceed several hundred thousand, these large numbers are well within the capacity of current home or school computers. The result of an assessment consists in two short lists of problems, which may be labelled: 'what the student can do' and 'what the student is ready to learn'. In the most important applications of the theory, these two lists specify the exact knowledge state of the individual being assessed.

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Jean-Paul Doignon

Papers

Knowledge spaces

Spaces for the assessment of knowledge

Introduction to knowledge spaces: How to build, test, and search them

The assessment of knowledge, in theory and in practice

On realizable biorders and the biorder dimension of a relation