Walter M. Stroup

Understanding Qualitative Calculus: A Structural Synthesis of Learning Research

More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features of Piagets analyses of the development of operational thought. In particular, the intensification of rate and two kinds of reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative calculus.

Scaleable Integration of Educational Software: Exploring The Promise of Component Architectures

Technology-rich learning environments can accelerate and enhance core curriculum reform in science and mathematics by enabling more diverse students to learn more complex concepts with deeper understanding at a younger age. Unfortunately, todays technology research and development efforts result not in an richly integrated environment, but rather with a fragmentary collection of incompatible software application islands. In this article we ask: how can the best innovations in technology-rich learning integrate and scale up to the level of major curricular reforms? A potential solution is component software architecture, which provides open standards that enable plug and play composition of software tools produced by many different projects and vendors. We describe an exploratory effort in which four research groups produced software components for the mathematics of motion. The resulting prototypes support (a) integration of the separately produced tools into the same windows, files, and interfaces, (b) dynamic linking across multiple representations and (c) drag and drop activity authoring without programming. We also summarize an extended Internet discussion which raised critical issues regarding the future of component software architecture in education, and speculate on the future need for components for devices other than the desktop computer and for virtual communities that coordinate design teams. Reviewers: David Redmiles (U.California Irvine), Royston Sellman (Hewlett Packard Labs.)

On the Embedded Complementarity of Agent-Based and Aggregate Reasoning in Students’ Developing Understanding of Dynamic Systems

Placed in the larger context of broadening the engagement with systems dynamics and complexity theory in school-aged learning and teaching, this paper is intended to introduce, situate, and illustrate—with results from the use of network supported participatory simulations in classrooms—a stance we call ‘embedded complementarity’ as an account of the relations between two major forms of systems-related learning and reasoning. The two forms of systems reasoning discussed are called ‘aggregate’ and ‘agent-based.’ These forms of reasoning are presented as distinct yet we also outline how there are forms of complementarity, between and within these approaches, useful in analyzing complex dynamic systems. We then explore specific ways in which the embedded complementarity stance can be used to analyze how learner understandings progress in science, technology, engineering, and mathematics-related participatory simulations supported by the HubNet (Wilensky and Stroup 1999c) learning environment developed with support from the National Science Foundation. We found that the learners used and built on the interdependence of agent and aggregate forms of reasoning in ways consistent with the discussion of embedded complementarity outlined in the early parts of the paper.

An analysis of horizontal and vertical device design for school-related teaching and learning

This paper reviews and analyzes 12 horizontal and vertical design attributes of devices for school-related use. Horizontal design is the familiar all-things-to-all-people, ‘just-in-case’ design associated with desk-top, laptop, and mainframe computing. Less familiar is the ‘vertical’ or ‘just-enough’ dimension of computational design, where device functionality is tightly coupled with specific needs in identifiable speciality markets. We believe that equitable access to key forms of learning functionality for all students and issues of total cost of ownership will provide the impetus for K–12 schooling to integrate horizontal and vertical technologies. We use two physically similar but functionally distinct portable handheld devices to illustrate the 12 design attributes. Representing horizontal design are Palm™ operating system-based handhelds and representing vertical design are graphing calculators.

Implementing schema-based assessment in engineering statistics courses

In classes like statistics, a staple for many beginning engineering students, deep interdisciplinary understanding as well as critical thinking is required to successfully learn the material. In this sense, traditional assessment, which may consist of multiple choice questions and free response problems, can be drastically inadequate to fully assess student comprehension of complex concepts. We propose a different method of assessment: schema-based items. These items are multiple choice items; however, there may be one or more correct answers based on the proposed question (hence, the items are non-dichotomous). The answer choices are nuanced and designed to measure not simply the students understanding of a single concept but their progress on an entire learning progression, which may include a single concept as well as several adjacent ones. After some prior promising implementations of these types of assessment items, we are eager to transition these types of items to the post-secondary engineering classroom, where students are required to understand and assimilate content knowledge that is not focused on a single subject area and has a vast interconnection between subjects. Due to the interdisciplinary nature of statistics, these topics can be quite difficult to assess accurately and thoroughly.

Computing the Average Square: An Agent-Based Introduction to Aspects of Current Psychometric Practice.

This paper summarizes an approach to helping future educators to engage with key issues related to the application of measurement-related statistics to learning and teaching, especially in the contexts of science, mathematics, technology and engineering (STEM) education. The approach we outline has two major elements. First, students are asked to compute an “average square.” Second, students work with an agent-based simulation that helps them to understand how aspects of the central limit theorem might be integrated into a much larger conversation about the appropriateness, or validity, of current psychometric practices. We are particularly interested in how such practices and interpretive frameworks inform the construction of high-stakes tests. In nearly all current high-stakes test development, tests are thought of as being built-up from individual items, each of which has known statistical properties. The activity sequence outlined in this paper helps future educators to understand the implications of this practice, and the sometimes problematic assumptions it entails. This instructional sequence has been used extensively as part of a core course in a university-based certification program in the United States (UTeach) recognized for its innovative approaches to developing a new generation of secondary STEM educators.