The work of equitable mathematics teaching: Leading a discussion of student solutions

International audienceMathematics teaching produces and reproduces social injustice. It also has the potential to disrupt patterns of inequity and advance just communities of practice. Drawing from literature on equitable mathematics teaching, we analyze the work of leading a discussion of student solutions in ways that nurture healthy identities, relationships and societies. From a conceptual analysis of a Norwegian mathematics lesson, we first identify dynamics of race and gender at play, then identify three key aspects of mathematics teaching that can serve to disrupt these dynamics while creating opportunities for alternative identities, relationships and futures: (i) having regard for property and its use; (ii) taking up student thinking as participatory citizenship; and (iii) orchestrating collective mathematical work. We discuss nuances of this work and implications for research on teaching

Conceptions of teaching and justice as pivotal to mathematics teacher educators’ thinking about mathematical knowledge for teaching

Recent scholarship has explored mathematical demands faced by mathematics teacher educators and ways to support their development, but little attention has been given to the basic question of how mathematics teacher educators think about content knowledge for teaching. Knowing what they think could inform efforts to support them. Our analysis reveals that some think about mathematical knowledge for teaching as an independent, abstracted resource to be taught and learned in relative isolation from teaching, while others think about it as dynamic, situated work. We argue that this key difference matters for how they work with teachers. Further, our analysis reveals that their thinking about both teaching and justice interacts with their thinking about mathematical knowledge for teaching and that their thinking in these other two domains can be a resource for supporting their mathematical development.acceptedVersio

Distributional Latent Variable Models with an Application in Active Cognitive Testing

Cognitive modeling commonly relies on asking participants to complete a battery of varied tests in order to estimate attention, working memory, and other latent variables. In many cases, these tests result in highly variable observation models. A near-ubiquitous approach is to repeat many observations for each test, resulting in a distribution over the outcomes from each test given to each subject. In this paper, we explore the usage of latent variable modeling to enable learning across many correlated variables simultaneously. We extend latent variable models (LVMs) to the setting where observed data for each subject are a series of observations from many different distributions, rather than simple vectors to be reconstructed. By embedding test battery results for individuals in a latent space that is trained jointly across a population, we are able to leverage correlations both between tests for a single participant and between multiple participants. We then propose an active learning framework that leverages this model to conduct more efficient cognitive test batteries. We validate our approach by demonstrating with real-time data acquisition that it performs comparably to conventional methods in making item-level predictions with fewer test items.Comment: 9 pages, 6 figure

Identifying, Measuring, and Defining Equitable Mathematics Instruction.

Many scholars have studied the problem of persistent inequitable educational opportunities and outcomes in the U.S. They have presented analyses of the causes of these inequities and proposed solutions ranging from increasing school funding to studying participation structures in classrooms. This dissertation takes the perspective that inequities are produced inside of classrooms as well as through the complex interplay of social and economic factors and argues that instructional practice is an important site for study and intervention. Therefore, although there exist numerous definitions of and strategies for working toward equity for underrepresented minority students, serious attention to instruction is crucial. This is specifically accomplished by studying the mathematical knowledge and skills, along with cultural awareness and sensitivities that would produce equitable, high quality teaching. In this dissertation, equitable teaching is defined as focused on quality mathematics and distributed intentionally to ensure that all students learn. This study probes the interplay in instruction of attention to equity and to the quality of the mathematical content, with a focus on what constitutes equitable mathematics instruction for students in elementary classrooms. Specific instructional practices are evaluated to determine whether and how particular teaching practices provide leverage and create access to the mathematics content for different groups of learners. This study has two central features. The first details the construction of the set of Mathematical Quality and Equity codes, analytic video codes focused on issues of equity. The second section comprises analyses of three paradigmatic examples of instruction. One is of a teacher with high MKT (Karen); a second a teacher who has clear commitments to students and to equitable access (Rebecca), and a third a teacher who has both high levels of MKT and of commitment to students and to equity (Lauren). The analyses illustrate the central hypothesis in my dissertation, that teaching mathematics in equitable ways requires both attention to the quality of the mathematics combined with sensitivities to issues of equity and diversity for students. This dissertation contributes to the empirical examination of instruction and its contributions to equity.Ph.D.EducationUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77677/1/imasters_1.pd

Accelerating Executive Function Assessments With Group Sequential Designs

Inferences about executive functions (EFs) are commonly drawn via lengthy serial administration of simple independent assessments. Classical methods for EF estimation often require excessive measurements and provide little or no flexibility to dynamically adjust test length for each individual. In order to decrease test duration and mitigate respondent burden, active testing modalities that incorporate more efficient data collection strategies are indispensable. To this end, we propose sequential analysis to improve upon traditional testing methods in behavioral science. In this paper, we show that sequential testing can be used to rapidly screen for a difference in the EF of a given individual with respect to a baseline level. In cognitive tests consisting of repeated identical tasks, a sequential framework can be utilized to actively detect significant differences in cognitive performance with high confidence more rapidly than conventional non-sequential approaches. Ultimately, sequential analysis could be applied to a variety of problems in cognitive and perceptual domains to improve efficiency gains and achieve substantial test length reduction

Scalable Probabilistic Modeling of Working Memory Performance

A standard approach for evaluating a cognitive variable involves designing a test procedure targeting that variable and then validating test results in a sample population. To extend this functionality to other variables, additional tests are designed and validated in the same way. Test batteries are constructed by concatenating individual tests. This approach is convenient for the designer because it is modular. However, it is not scalable because total testing time grows proportionally with test count, limiting the practical size of a test battery. Cross-test models can inform the relationships between explicit or implicit cognitive variables but do not shorten test time and cannot readily accommodate subpopulations who exhibit different relationships than average. An alternate modeling framework using probabilistic machine learning can rectify these shortcomings, resulting in item-level prediction from individualized models while requiring fewer data points than current methods. To validate this approach, a Gaussian process probabilistic classifier was used to model young adult and simulated spatial working memory task performance as a psychometric function. This novel test instrument was evaluated for accuracy, reliability and efficiency relative to a conventional method recording the maximum spatial sequence length recalled. The novel method exhibited extremely low bias, as well as test-retest reliability 30% higher than the conventional method under standard testing conditions. Efficiency was consistent with other adaptive psychometric threshold estimation strategies, with 30–50 samples needed for consistently reliable estimates. While these results demonstrate that similar spatial working memory tasks can be effectively modeled as psychometric functions by any method, the advantage of the novel method is that it is scalable to accommodate much more complex models, such as those including additional executive functions. Further, it was designed with tremendous flexibility to incorporate informative theory, ancillary data, previous cohort performance, previous individual performance, and/or current individual performance for improved predictions. The result is a promising method for behavioral modeling that can be readily extended to capture complex individual task performance