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Incorporating a Digital Game Into the Formal Instruction of Algebra
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| Pengarang | : | Shulamit Kapon, Angela Halloun, and Michal Tabach |
| Nama Majalah/Jurnal | : | Journal for Research in Mathematics Education |
| Volume / Edisi | : | 50 (No. 5) |
| Halaman | : | 555-591 |
| Abstrak | : | We compared students' learning gains in authentic seventh-grade classrooms (N = 144) in 4 different interventions that incorporated a computer game that aims to teach players to solve linear equations. Significantly higher learning gains were measured in the implementations that were specifically designed to mediate the attribution of algebraic meaning to objects, actions, and rules in the game by engaging students in analogical mapping between these constructs and their algebraic counterparts and an exploration of the boundaries of this isomorphism. These findings suggest that learning disciplinary content and skills from a digital game requires learners to attribute disciplinary meaning to objects, actions, and rules in the game. Moreover, this process does not necessarily occur spontaneously and benefits from instructional mediation. |
Who Gets in Examining Inequality in Eighth-Grade Algebra
| Pengarang | : | Karisma Morton and Catherine Riegle-Crumb |
| Nama Majalah/Jurnal | : | Journal for Research in Mathematics Education |
| Volume / Edisi | : | 50 (No. 5) |
| Halaman | : | 529-544 |
| Abstrak | : | Using data from a large urban district, this study investigated whether racial inequality in access to eighth-grade algebra is a reproduction of differences in prior opportunities to learn (as evidenced by grades, test scores, and level of prior mathematics course) or whether patterns reflect an increase in inequality such that racial differences in access remain when controlling for academic background. We considered how this varies by the racial composition of the school; further, we examined differences in access between both Black and Hispanic students and their White peers as well as differences between Black and Hispanic students. The results point to patterns of reproduction of inequality in racially integrated schools, with some evidence of increasing inequality in predominantly Hispanic schools |
An A Priori Measure of Visual Difficulty of 2-D Sketches Depicting 3-D Objects
| Pengarang | : | Mirela Widder, Avi Berman, and Boris Koichu |
| Nama Majalah/Jurnal | : | Journal for Research in Mathematics Education |
| Volume / Edisi | : | 50 (No. 5) |
| Halaman | : | 489-528 |
| Abstrak | : | Aiming to enhance understanding of visual obstacles inherent in two-dimensional (2-D) sketches used in high school spatial geometry instruction, we propose a measure of visual difficulty based on two attributes of the sketches: potentially misleading geometrical information (PMI) and potentially helpful geometrical information (PHI). The difficulty of 12 normatively oriented cube-related sketches was theoretically ranked according to their ratios, #PHI/#PMI. The ranking was compared to the actual visual difficulty as measured by the percentage of correct or desired comprehension, individual spatial ability, and study-time allocation. This procedure was repeated for unnormatively oriented sketches, obtained by vertically flipping the original sketches. In both cases, the findings substantiate #PHI/#PMI as an a priori measure of visual difficulty. Practical, theoretical, and methodological implications are inspected and discussed. |
Factor Copula Models for Replicated Spatial Data
| Pengarang | : | Pavel Krupskii |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 113 (No. 521) |
| Halaman | : | 467-479 |
| Abstrak | : | We propose a new copula model that can be used with replicated spatial data. Unlike the multivariate normal copula, the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process. Moreover, the proposed copula can model tail dependence and tail asymmetry. The model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature, such as the Matérn model. For some choice of common factors, the joint copula density is given in closed form and therefore likelihood estimation is very fast. In the general case, one-dimensional numerical integration is needed to calculate the likelihood, but estimation is still reasonably fast even with large datasets. We use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors. We apply the proposed model to spatial temperature data and compare its performance with some popular geostatistics models. Supplementary materials for this article are available online. |
Mathematics Majors Diagram Usage When Writing Proofs in Calculus
| Pengarang | : | Juan Pablo Mejiía-Ramos and Keith Weber |
| Nama Majalah/Jurnal | : | Journal for Research in Mathematics Education |
| Volume / Edisi | : | 50 (No. 5) |
| Halaman | : | 478-488 |
| Abstrak | : | We report on a study in which we observed 73 mathematics majors completing 7 proof construction tasks in calculus. We use these data to explore the frequency and effectiveness with which mathematics majors use diagrams when constructing proofs. The key findings from this study are (a) nearly all participants introduced diagrams on multiple tasks, (b) few participants displayed either a strong propensity or a strong reluctance to use diagrams, and (c) little correlation existed between participants' propensity to use diagrams and their mathematical achievement (either on the proof construction tasks or in their advanced mathematics courses). At the end of the report, we discuss implications for pedagogy and future research. |
Multi-Armed Bandit for Species Discovery: A Bayesian Nonparametric Approach
| Pengarang | : | Marco Battiston |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 113 (No. 521) |
| Halaman | : | 455-466 |
| Abstrak | : | Let (P1, …, PJ) denote J populations of animals from distinct regions. A priori, it is unknown which species are present in each region and what are their corresponding frequencies. Species are shared among populations and each species can be present in more than one region with its frequency varying across populations. In this article, we consider the problem of sequentially sampling these populations to observe the greatest number of different species. We adopt a Bayesian nonparametric approach and endow (P1, …, PJ) with a hierarchical Pitman–Yor process prior. As a consequence of the hierarchical structure, the J unknown discrete probability measures share the same support, that of their common random base measure. Given this prior choice, we propose a sequential rule that, at every time step, given the information available up to that point, selects the population from which to collect the next observation. Rather than picking the population with the highest posterior estimate of producing a new value, the proposed rule includes a Thompson sampling step to better balance the exploration–exploitation trade-off. We also propose an extension of the algorithm to deal with incidence data, where multiple observations are collected in a time period. The performance of the proposed algorithms is assessed through a simulation study and compared to three other strategies. Finally, we compare these algorithms using a dataset of species of trees, collected from different plots in South America. Supplementary materials for this article are available online. |
So What Justifying Conclusions and Interpretations of Data
| Pengarang | : | Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang, Victoria Robison, Michelle Cirillo, Steven L. Kramer, and James Hiebert |
| Nama Majalah/Jurnal | : | Journal for Research in Mathematics Education |
| Volume / Edisi | : | 50 (No. 5) |
| Halaman | : | 470-477 |
| Abstrak | : | Although often asked tactfully, a frequent question posed to authors by JRME reviewers is “So what?” Through this simple and well-known question, reviewers are asking: What difference do your findings make? How do your results advance the field? “So what?” is the most basic of questions, often perceived by novice researchers as the most difficult question to answer. Indeed, addressing the “so what” question continues to challenge even experienced researchers. All researchers wrestle with articulating a convincing argument about the importance of their own work. When we try to shape this argument, it can be easy to fall into the trap of making claims about the implications of our findings that reach beyond the data. |
Group Regularized Estimation Under Structural Hierarchy
| Pengarang | : | Yiyuan She |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 113 (No. 521) |
| Halaman | : | 445-454 |
| Abstrak | : | Variable selection for models including interactions between explanatory variables often needs to obey certain hierarchical constraints. Weak or strong structural hierarchy requires that the existence of an interaction term implies at least one or both associated main effects to be present in the model. Lately, this problem has attracted a lot of attention, but existing computational algorithms converge slow even with a moderate number of predictors. Moreover, in contrast to the rich literature on ordinary variable selection, there is a lack of statistical theory to show reasonably low error rates of hierarchical variable selection. This work investigates a new class of estimators that make use of multiple group penalties to capture structural parsimony. We show that the proposed estimators enjoy sharp rate oracle inequalities, and give the minimax lower bounds in strong and weak hierarchical variable selection. A general-purpose algorithm is developed with guaranteed convergence and global optimality. Simulations and real data experiments demonstrate the efficiency and efficacy of the proposed approach. Supplementary materials for this article are available online. |
The Spike-and-Slab LASSO
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 113 (No. 521) |
| Halaman | : | 431-444 |
| Abstrak | : | Despite the wide adoption of spike-and-slab methodology for Bayesian variable selection, its potential for penalized likelihood estimation has largely been overlooked. In this article, we bridge this gap by cross-fertilizing these two paradigms with the Spike-and-Slab LASSO procedure for variable selection and parameter estimation in linear regression. We introduce a new class of self-adaptive penalty functions that arise from a fully Bayes spike-and-slab formulation, ultimately moving beyond the separable penalty framework. A virtue of these nonseparable penalties is their ability to borrow strength across coordinates, adapt to ensemble sparsity information and exert multiplicity adjustment. The Spike-and-Slab LASSO procedure harvests efficient coordinate-wise implementations with a path-following scheme for dynamic posterior exploration. We show on simulated data that the fully Bayes penalty mimics oracle performance, providing a viable alternative to cross-validation. We develop theory for the separable and nonseparable variants of the penalty, showing rate-optimality of the global mode as well as optimal posterior concentration when p > n. Supplementary materials for this article are available online. |
Correlated Random Measures
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 113 (No. 521) |
| Halaman | : | 417-430 |
| Abstrak | : | We develop correlated random measures, random measures where the atom weights can exhibit a flexible pattern of dependence, and use them to develop powerful hierarchical Bayesian nonparametric models. Hierarchical Bayesian nonparametric models are usually built from completely random measures, a Poisson-process-based construction in which the atom weights are independent. Completely random measures imply strong independence assumptions in the corresponding hierarchical model, and these assumptions are often misplaced in real-world settings. Correlated random measures address this limitation. They model correlation within the measure by using a Gaussian process in concert with the Poisson process. With correlated random measures, for example, we can develop a latent feature model for which we can infer both the properties of the latent features and their dependency pattern. We develop several other examples as well. We study a correlated random measure model of pairwise count data. We derive an efficient variational inference algorithm and show improved predictive performance on large datasets of documents, web clicks, and electronic health records. Supplementary materials for this article are available online. |