Visions For Mathematics Instruction, Instructional Practices, And Common Core: Individuality In Large-Scale Reform

The Common Core content standards and standards for mathematical practice (SMP) have introduced an unprecedented opportunity in U.S. history to consider the implications of a national scale standards reform effort that provides teachers the individualistic opportunity to select instructional practices aimed at achieving the standards. The content standards and SMP specifically lay out what students should be able to do by the end of each grade but do not describe how teachers should support students through instructional practices in order to achieve these goals (CCSSI, “Myths About Implementation,” para. 1). Seemingly in contradiction to the Common Core State Standards Initiative’s proclamation that instructional practices should be left to individuals and schools, Cobb and Jackson (2011) asserted that large-scale mathematics reform efforts are likely to be successful if “a detailed vision of high-quality mathematics instruction specifies concrete instructional practices that have the potential to lead to the attainment of learning goals” (p. 13). In the absence of a national-scale vision of high-quality mathematics instruction to accompany the Common Core Standards for Mathematics (CCSSM), the purpose of this study was to describe how teachers reasoned about specifying instructional practices aimed at addressing the goals of the CCSSM. Four case study teachers were selected from an initial interview as to categorically represent the visions for mathematics instruction (V4MI) described by 10 secondary mathematics teachers from one mathematics department in one high school located in the Midwestern region of the United States within an urban city. The four case study teachers participated in a pre-lesson and post-lesson interview surrounding a lesson of their choice. In the initial, pre-lesson, and post-lesson interview, teachers reasoned about ideal, planned, and instructional practices. Using the method of constant comparative analysis, I found emerging themes in teachers’ reasoning and described them as categories of reasoning about instructional practices. Themes emerged among the four case study teachers in the following forms of reasoning: prior knowledge, building conceptual knowledge, external sources of influence, establishing classroom culture, cultivating general learner qualities, and cultivating mathematics learner practices. During which interviews these categories of reasoning emerged (i.e., initial, pre-lesson, post-lesson interviews), the type of language case study teachers used when discussing each category of reasoning (i.e., teacher-centered or student-centered), the alignment or misalignment of the category of reasoning with teachers’ V4MIs, and teachers explicit reasoning about V4MIs and CCSSM are explored. Findings suggest that teachers and those leading large-scale Common Core mathematics reform (e.g., administration, state-level lawmakers) should explore the external sources of influence that teachers adhere to when specifying instructional practices. Furthermore, teachers should be provided opportunities to build explicit connections between their V4MIs, instructional practices, and the goals of the CCSSM

A Formal Architecture of Shared Mental Models for Computational Improvisational Agents

This paper proposes a formal approach of constructing shared mental models between computational improvisational agents (improv agents) and human interactors based on our socio-cognitive studies of human improvisers. Creating shared mental models helps improv agents co-create stories with each other and interactors in real-time interactive narrative experiences. The approach described here allows flexible modeling of non-Boolean (i.e. fuzzy) knowledge about scene and background concepts through the use of fuzzy rules and confidence factors in order to allow reasoning under uncertainty. It also allows improv agents to infer new knowledge about a scene from existing knowledge, recognize when new knowledge may be divergent from the other actor’s mental model, and attempt to resolve this divergence to reach cognitive consensus despite the absence of explicit goals in the story environment

Reasoning about order errors in interaction

Reliability of an interactive system depends on users as well as the device implementation. User errors can result in catastrophic system failure. However, work from the field of cognitive science shows that systems can be designed so as to completely eliminate whole classes of user errors. This means that user errors should also fall within the remit of verification methods. In this paper we demonstrate how the HOL theorem prover [7] can be used to detect and prove the absence of the family of errors known as order errors. This is done by taking account of the goals and knowledge of users. We provide an explicit generic user model which embodies theory from the cognitive sciences about the way people are known to act. The user model describes action based on user communication goals. These are goals that a user adopts based on their knowledge of the task they must perform to achieve their goals. We use a simple example of a vending machine to demonstrate the approach. We prove that a user does achieve their goal for a particular design of machine. In doing so we demonstrate that communication goal based errors cannot occur

The Received Method for Ruling Out Brain Areas from Being NCC Undermines Itself

Research into the neural correlates of consciousness (NCC) aims to identify not just those brain areas that are NCC, but also those that are not. In the received method for ruling out a brain area from being an NCC, this is accomplished by showing a brain area’s content to be consistently absent from subjects’ reports about what they are experiencing. This paper points out how this same absence can be used to infer that the brain area’s content is cognitively inaccessible, in which case we would expect its content to be absent from subjects’ reports whether its content is (phenomenally) conscious or not. If so, such reports cannot count as evidence against that brain area being an NCC, and the received method fails. An alternative method (one suggested in Block, 2007) is considered

The Search for Invariance: Repeated Positive Testing Serves the Goals of Causal Learning

Positive testing is characteristic of exploratory behavior, yet it seems to be at odds with the aim of information seeking. After all, repeated demonstrations of one’s current hypothesis often produce the same evidence and fail to distinguish it from potential alternatives. Research on the development of scientific reasoning and adult rule learning have both documented and attempted to explain this behavior. The current chapter reviews this prior work and introduces a novel theoretical account—the Search for Invariance (SI) hypothesis—which suggests that producing multiple positive examples serves the goals of causal learning. This hypothesis draws on the interventionist framework of causal reasoning, which suggests that causal learners are concerned with the invariance of candidate hypotheses. In a probabilistic and interdependent causal world, our primary goal is to determine whether, and in what contexts, our causal hypotheses provide accurate foundations for inference and intervention—not to disconfirm their alternatives. By recognizing the central role of invariance in causal learning, the phenomenon of positive testing may be reinterpreted as a rational information-seeking strategy

Towards an understanding of the impact of resources on the design process

Considerable effort has been devoted within the design research community to understanding the structure of design processes and their development for different design problems. Whilst much work has examined the impact of design goals upon the structure of a design process, less attention has been paid to the role that design resources can play. This paper describes an experiment directed towards gaining an understanding of the impact that both active resources (which perform design tasks) and passive resources (which are used by active resources) can have upon design process structure. Main outcomes from the experiment were the conclusive identification that resources can significantly impact design process structure and a number of examples of how these impacts manifest themselves. The main conclusion of the paper is that given the sizeable impact resources can have upon process structure, there is a considerable need to obtain a greater understanding of these impacts to facilitate the development of techniques that can support design process definition based upon an understanding of the design resources being used to solve a design problem

Character and theory of mind: an integrative approach

Traditionally, theories of mindreading have focused on the representation of beliefs and desires. However, decades of social psychology and social neuroscience have shown that, in addition to reasoning about beliefs and desires, human beings also use representations of character traits to predict and interpret behavior. While a few recent accounts have attempted to accommodate these findings, they have not succeeded in explaining the relation between trait attribution and belief-desire reasoning. On my account, character-trait attribution is part of a hierarchical system for action prediction, and serves to inform hypotheses about agents’ beliefs and desires, which are in turn used to predict and interpret behavior

Reconstructing the Past: The Case of the Spadina Expressway

In order to build resilient systems that can be operational for a long time, it is important that analysts are able to model the evolution of the requirements of that system. The Evolving Intentions framework models how stakeholders’ goals change over time. In this work, our aim is to validate applicability and effectiveness of this technique on a substantial case. In the absence of ground truth about future evolutions, we used historical data and rational reconstruction to understand how a project evolved in the past. Seeking a well-documented project with varying stakeholder intentions over a substantial period of time, we selected requirements of the Toronto Spadina Expressway. In this paper, we report on the experience and the results of modeling this project over different time periods, which enabled us to assess the modeling and reasoning capabilities of the approach, its support for asking and answering ‘what if’ questions, and the maturity of the underlying tool support. We also demonstrate a novel process for creating time-based models through the construction and merging of scenarios

Moral nihilism and its implications

Philosophers have identified a number of principles that characterize morality and underlie moral judgments. However, philosophy has failed to establish any widely agreed-upon justification for these judgments, and an “error theory” that views moral judgments as without justification has not been successfully refuted. Evolutionary psychologists have had success in explaining the likely origins and mechanisms of morality but have also not established any justification for adopting particular values. As a result, we are left with moral nihilism -- the absence of any unarguable values or behaviors we must or should adopt. The philosophical and psychological implications of this nihilism suggest accepting shared, non-absolute values as “good enough”; a revised, humbler view of moral and other value judgments; and the possible acceptance of the hard truth of a value nihilism