creased by 0.47 SDs under active learning (n = 158 studies), and that the odds ratio for failing was 1.95 under traditional lecturing (n = 67 studies). These results indicate that average examination scores improved by about 6% in active learning sections, and that students in classes with traditional lecturing were 1.5 times more likely to fail than were students in classes with active learning. Heterogeneity analyses indicated that both results hold across the STEM disciplines, that active learning increases scores on concept inventories more than on course examinations, and that active learning appears effective across all class sizes—although the greatest effects are in small (n ≤ 50) classes. Trim and fill analyses and fail-safe n calculations suggest that the results are not due to publication bias. The results also appear robust to variation in the methodological rigor of the included studies, based on the quality of controls over student quality and instructor identity. This is the largest and most comprehensive metaanalysis of undergraduate STEM education published to date. The results raise questions about the continued use of traditional lecturing as a control in research studies, and support active learning as the preferred, empirically validated teaching practice in regular classrooms.

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Active learning increases student performance in science, engineering, and mathematics

Software reuse is the process of creating software systems from existing software rather than building software systems from scratch. This simple yet powerful vision was introduced in 1968. Software reuse has, however, failed to become a standard software engineering practice. In an attempt to understand why, researchers have renewed their interest in software reuse and in the obstacles to implementing it.This paper surveys the different approaches to software reuse found in the research literature. It uses a taxonomy to describe and compare the different approaches and make generalizations about the field of software reuse. The taxonomy characterizes each reuse approach in terms of its reusable artifacts and the way these artifacts are abstracted, selected, specialized, and integrated.Abstraction plays a central role in software reuse. Concise and expressive abstractions are essential if software artifacts are to be effectively reused. The effectiveness of a reuse technique can be evaluated in terms of cognitive distance—an intuitive gauge of the intellectual effort required to use the technique. Cognitive distance is reduced in two ways: (1) Higher level abstractions in a reuse technique reduce the effort required to go from the initial concept of a software system to representations in the reuse technique, and (2) automation reduces the effort required to go from abstractions in a reuse technique to an executable implementation.This survey will help answer the following questions: What is software reuse? Why reuse software? What are the different approaches to reusing software? How effective are the different approaches? What is required to implement a software reuse technology? Why is software reuse difficult? What are the open areas for research in software reuse?

Software reuse

On Educational psychology: A cognitive view.

Introduction Part I. Discourse on Thinking: 1. Puzzling about (mathematical) thinking 2. Objectification 3. Commognition: thinking as communicating 4. Thinking in language Part II. Mathematics as Discourse: 5. Mathematics as a form of communication 6. Objects of mathematical discourse: what mathematizing is all about 7. Routines: how we mathematize 8. Explorations, deeds, and rituals: what we mathematize for 9. Looking back and ahead: solving old quandaries and facing new ones.

/pdf/thinking-as-communicating-human-development-the-growth-of-2vx23bbaa2.pdf

Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing

This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

https://www.math.ksu.edu/~bennett/onlinehw/qcenter/lzs.pdf

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

Our purpose in this chapter is to propose that the concept of reflective abstraction can be a powerful tool in the study of advanced mathematical thinking, that it can provide a theoretical basis that supports and contributes to our understanding of what this thinking is and how we can help students develop the ability to engage in it. To make such a case completely, it would be necessary to do at least several things:

Reflective Abstraction in Advanced Mathematical Thinking

Our goal in this paper is to make two points. First, college students, even those who have taken a fair number of mathematics courses, do not have much of an understanding of the function concept; and second, an epistemological theory we have been developing points to an instructional treatment, using computers, that results in substantial improvements for many students. They seem to develop a process conception of function and are able to use it to do mathematics. After an introductory section we outline, in Section 2, our theoretical epistemology in general and indicate how it applies to the function concept in particular. In Sections 3, 4, and 5 we provide specific details on this study and describe the development of the function concept that appeared to take place in the students that we are considering. In Section 6 we interpret the results and draw some conclusions.

Development of the Process Conception of Function

The Concept of function : aspects of epistemology and pedagogy

The programming language SETL is a relatively new member of the so-called "very-high-level" class of languages, some of whose other well-known mem bers are LISP, APL, SNOBOL, and PROLOG. These languages all aim to reduce the cost of programming, recognized today as a main obstacle to future progress in the computer field, by allowing direct manipulation of large composite objects, considerably more complex than the integers, strings, etc., available in such well-known mainstream languages as PASCAL, PL/I, ALGOL, and Ada. For this purpose, LISP introduces structured lists as data objects, APL introduces vectors and matrices, and SETL introduces the objects characteristic for it, namely general finite sets and maps. The direct availability of these abstract, composite objects, and of powerful mathematical operations upon them, improves programmer speed and pro ductivity significantly, and also enhances program clarity and readability. The classroom consequence is that students, freed of some of the burden of petty programming detail, can advance their knowledge of significant algorithms and of broader strategic issues in program development more rapidly than with more conventional programming languages."

Programming with Sets: An Introduction to SETL

In this paper, we have mentioned six ways in which a theory can contribute to research and we suggest that this list can be used as criteria for evaluating a theory. We have described how one such perspective, APOS Theory, is being used in an organized way by members of RUMEC and others to conduct research and develop curriculum. We have shown how observing students’ success in making or not making mental constructions proposed by the theory and using such observations to analyze data can organize our thinking about learning mathematical concepts, provide explanations of student difficulties and predict success or failure in understanding a mathematical concept. There is a wide range of mathematical concepts to which APOS Theory can and has been applied and this theory is used as a language for communication of ideas about learning. We have also seen how the theory is grounded in data, and has been used as a vehicle for building a community of researchers. Yet its use is not restricted to members of that community. Finally, we point to an annotated bibliography (McDonald, 2000), which presents further details about this theory and its use in research in undergraduate mathematics education.

Ed Dubinsky

Papers

Reflective Abstraction in Advanced Mathematical Thinking

Development of the Process Conception of Function

The Concept of function : aspects of epistemology and pedagogy

Programming with Sets: An Introduction to SETL

APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research