Data Management Plan for Preparing Secondary Mathematics Teachers with Video Cases of Students’ Functional Reasoning

Data management plan for 2016 NSF Education and Human Resources (EHR) Improving Undergraduate STEM Education: Education and Human Resources (IUSE: EHR) Program Proposal

Impact of the Order of EAC and SOS During Instruction on Ratios

The purpose of this study was twofold. First, a set of instructional materials surrounding proportional reasoning with ratios (particularly the understanding of the multiplicative relationship between the quantities within the ratio, referred to as functional reasoning throughout this thesis) were created using the free online tool, Desmos, with a goal of determining the impact of the lesson materials on student understanding. The second goal was to explore the impact of the order in which two instructional strategies, Explicit Attention to Concepts (EAC) and Students’ Opportunity to Struggle (SOS), had on student understanding. The lesson materials consisted of 5 lessons. These 5 lessons had two forms: EAC then SOS or SOS then EAC. In each of these instructional groups, all EAC and SOS sections were identical in each of the five lessons, the difference between materials in each of these groups was the order in which the EAC and SOS sections occurred. Students’ understanding was assessed anonymously, and answers were scored dichotomously (i.e. correct or incorrect). There was a total of 22 items on the full assessment with 8 items addressing functional reasoning specifically. The major findings of this study include that the lesson materials led to an increase in understanding for both overall understanding and the sub-area of functional reasoning, and the EAC then SOS instructional group’s understanding of functional reasoning was higher than that of the SOS then EAC instructional group.

A Mathematical Functional Decomposition Approach Through Granularity Partition Process in Quotient Space

Over the past decades, several methodologies have coalesced around the functional decomposition and partial solution manipulation techniques. These methodologies take designers through steps that help decompose a design problem and build conceptual solutions based on the intended, product functionality. However, this kind of subjective decomposition restricts solutions of conceptual design within designers’ intended the local, rather the whole, solution space. In such cases, the ability for AI-based functional reasoning systems to obtain creative conceptual design solutions is weakened. In this paper, a functional decomposition model based on the domain decomposition theory in quotient space is proposed for carrying out functional decomposition without needing functional reasoning knowledge to support. In this model, the functional decomposition is treated as a granularity partition process in quotient space composed of three variables: the domain granularities, the attribute properties, and the topological structures. The closeness degrees and the attribute properties in fuzzy mathematics are utilized to describe the fuzzy equivalence relations between the granularities in the up-layer and in the lower-layer of the functional hierarchies. According to the order characteristics in the partially sequential quotient space, based on the homomorphism principle, the attribute properties and the topological structures corresponding to the lower-layer of the functional hierarchies are constructed then. Here, the attribute properties are expressed with membership functions pointed to the lower-layer from the up-layer of the functional hierarchies, and the topological structures are expressed with matrixes and the directed function network represent the topological connections among the subfunctions in the lower-layer of the functional hierarchies. Through refining the functional decomposition process step by step, and traversing all tree branches and leaf nodes in the functional decomposition tree, the functional hierarchies are obtained. Since the functional decomposition process not need the user to indicate or manage desired functionality, the model presented in this paper can reduce designers’ prejudices or preconceptions on the functional hierarchies, as well as extend the solution space of conceptual design.

Problem Pressure and Social Policy Innovation: Lessons from Nineteenth-Century Germany

In studying how to best understand social program introduction, political scientists have built up a laundry list of contributory factors. We suggest, however, that “objective” problem pressure has been incorrectly neglected by many scholars in recent decades—and the well-known case of Germany’s nineteenth-century introduction of social insurance legislation provides a clear illustration of this point. In explaining the origins and design of German social insurance, the interplay of three factors is key: first, exceptionally high problem pressure, connected to both labor market- and state-building processes; second, a fragile institutional context dominated by Prussia; and third, the party political constellation. In making this argument, we draw on “open functional reasoning” and extract implications from the case study to further refine the underlying theory. Specifically, we find that goal-oriented action may both be more common and more prone to compromise than the theory suggests. As such, we not only present an argument for considering the potential impact of problem pressure, but also lay out and refine an approach to doing so. In contrasting this approach to the problematic functionalism that initially inclined many scholars to neglect of problem pressure, we hope to help rehabilitate the concept—and in the process strengthen the explanatory power of research in sociology and political science.

Innovation of Matching Structures Through Clustering and Reconstructing Basic Operation Actions in the Form Layer

Due to a lack of essential knowledge to support functional reasoning from the design requirements of the kinematic compound mechanisms to their constituent mechanisms, the creative conceptual design of kinematic compound mechanisms based on functional synthesis approach is still a challenging task. Through introducing the dynamic partition-matching process between the function layer and the form layer to substitute for the direct function-structure matching in the FBS model, the function-structure matching problem corresponding to deficient functional reasoning knowledge for kinematic compound mechanisms is solved by the authors. The following challenge is how to cluster the divided subset of basic operation actions generated in the form layer during the partition-matching process into a well-organized and complete kinematic behavior that can be matched by the sub-function in the function layer and implemented by a structure in the database. The adopted strategies in this paper are: through defining the correlation indexes between basic operation actions, the basic operation action and its realized function behavior, and its embodied structure, as well as its dynamic behavior characteristics, the clustering possibility for a group of basic operation actions is determined. With the aid of the compatibility conditions between basic operation actions in the form layer and the consistency of the order relations between basic operation actions in the function layer and the form layer respectively, the consistency of the order relations among basic operation actions between the sub-functions in the function layer and the sub-behaviors in the form layer are guaranteed. Then, the optimal matching structures corresponding to the sub-functions in the function layer are determined based on the maximum matching coefficients of basic operation actions. In this way, the subsets of basic operation actions in the form layer are clustered into a number of complete behaviors that can be realized by mechanisms in the structure database and matched by the sub-functions in the function layer. Since multiple functional behaviors of each constituent basic mechanism take part in matching, some novel schemes of compound mechanisms with fewer and simpler constituent mechanisms to implement the overall function may be dug out.

Challenges in Using a Delphi Method to Formalize Conceptual Understanding in Functional Reasoning

A project to develop a concept inventory for functional reasoning in engineering design using a Delphi method encountered some unanticipated difficulties at the outset. The anonymous elicitation of a list of candidate concepts from an expert panel was expected to be a straightforward first step in a process in which the main work would be the creation and validation of multiple choice questions to probe understanding of these concepts. Low response rates and inconsistent concept descriptions led to a reconsideration of the nature of the concept inventory project. The early results from the project are discussed and compared with a similar stage of a different concept inventory project in thermodynamics. Implications of this comparison for the field of functional reasoning and the development of concept inventories for relatively immature fields are discussed.