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An investigation of teachers' beliefs of students' algebra development
 Cognition and Instruction
, 2000
"... Elementary, middle, and high school mathematics teachers (N = 105) ranked a set of mathematics problems based on expectations of their relative problemsolving difficulty. Teachers also rated their levels of agreement to a variety of reformbased statements on teaching and learning mathematics. Anal ..."
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Cited by 19 (10 self)
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Elementary, middle, and high school mathematics teachers (N = 105) ranked a set of mathematics problems based on expectations of their relative problemsolving difficulty. Teachers also rated their levels of agreement to a variety of reformbased statements on teaching and learning mathematics. Analyses suggest that teachers hold a symbolprecedence view of student mathematical development, wherein arithmetic reasoning strictly precedes algebraic reasoning, and symbolic problemsolving develops prior to verbal reasoning. High school teachers were most likely to hold the symbolprecedence view and made the poorest predictions of students ’ performances, whereas middle school teachers ’ predictions were most accurate. The discord between teachers ’ reformbased beliefs and their instructional decisions appears to be influenced by textbook organization, which institutionalizes the symbolprecedence view. Because of their extensive content training, high school teachers may be particularly susceptible to an expert blindspot, whereby they overestimate the accessibility of symbolbased representations and procedures for students ’ learning introductory algebra. The study of people engaged in cognitively demanding tasks must consider the relation between people’s judgments and actions and the beliefs they hold. Several aspects of people’s decision making are well established. People do not strictly follow the laws of logic and probability when weighing information or following im
Shallow Binding Makes Functional Arrays Fast
 ACM SIGPLAN notices
, 1991
"... this paper is the first to make the connection with the literature on variablebinding environments. ..."
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Cited by 15 (2 self)
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this paper is the first to make the connection with the literature on variablebinding environments.
Problems in Comprehending Recursion and Suggested Solutions
 In Proceedings of the 6th annual conference on innovation and technology in computer science education
, 2001
"... Recursion is a very powerful and useful problem solving strategy. But, along with pointers and dynamic data structures, many beginning programmers consider recursion to be a difficult concept to master. This paper reports on a study of upperdivision undergraduate students on their difficulty in com ..."
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Cited by 5 (1 self)
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Recursion is a very powerful and useful problem solving strategy. But, along with pointers and dynamic data structures, many beginning programmers consider recursion to be a difficult concept to master. This paper reports on a study of upperdivision undergraduate students on their difficulty in comprehending the ideas behind recursion. Three issues emerged as the points of difficulty for the students: (1) insufficient exposure to declarative thinking in a programming context (2) inadequate appreciation of the concept of functional abstraction (3) lack of a proper methodology to express a recursive solution. The paper concludes with a discussion of our approach to teaching recursion, which addresses these issues. Classroom experience indicates this approach effectively aids students' comprehension of recursion.
Epistemic Privacy
"... We present a novel definition of privacy in the framework of offline (retroactive) database query auditing. Given information about the database, a description of sensitive data, and assumptions about users ’ prior knowledge, our goal is to determine if answering a past user’s query could have led t ..."
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Cited by 5 (0 self)
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We present a novel definition of privacy in the framework of offline (retroactive) database query auditing. Given information about the database, a description of sensitive data, and assumptions about users ’ prior knowledge, our goal is to determine if answering a past user’s query could have led to a privacy breach. According to our definition, an audited property A is private, given the disclosure of property B, if no user can gain confidence in A by learning B, subject to prior knowledge constraints. Privacy is not violated if the disclosure of B causes a loss of confidence in A. The new notion of privacy is formalized using the wellknown semantics for reasoning about knowledge, where logical properties correspond to sets of possible worlds (databases) that satisfy these properties. Database users are modelled as either possibilistic agents whose knowledge is a set of possible worlds, or as probabilistic agents whose knowledge is a probability distribution on possible worlds. We analyze the new privacy notion, show its relationship with the conventional approach, and derive criteria that allow the auditor to test privacy efficiently in some important cases. In particular, we prove characterization theorems for the possibilistic case, and study in depth the probabilistic case under the assumption that all database records are considered apriori independent by the user, as well as under more relaxed (or absent) priorknowledge assumptions. In the probabilistic case we show that for certain families of distributions there is no efficient algorithm to test whether an audited property A is private given the disclosure of a property B, assuming P � = NP. Nevertheless, for many interesting families, such as the family of product distributions, we obtain algorithms that are efficient both in theory and in practice.
Mathematical Knowledge Construction through the use of Guided Collaborative Critique in a Quasi Synchronous Chat Environment
"... The study of the process of meaning making is central to Computer Support Collaborative Learning (CSCL). Jurong Junior College and the National Institute of Education (Singapore) Learning Sciences Laboratory have recently conducted several collaborative maths problem solving sessions using the Virtu ..."
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The study of the process of meaning making is central to Computer Support Collaborative Learning (CSCL). Jurong Junior College and the National Institute of Education (Singapore) Learning Sciences Laboratory have recently conducted several collaborative maths problem solving sessions using the Virtual Math Team Chat, a quasisynchronous chat medium used to facilitate the process of mathematical knowledge construction. The maths problems were designed based on the Guided Collaborative Critique (GCC) framework. The GCC framework requires students to analyse maths problems with &quot;solutions &quot; that contained conceptual or oversight errors, critique these problems with mathematical arguments and amend the &quot;solutions &quot; collaboratively. Our past research focused on the analysis of face to face (FTF) social interaction of groups working on maths problems using the GCC framework in traditional classrooms setting. In this study, the GCC framework is extended to the VMT chat medium which consists of a shared whiteboard, chat message box and tools for students to construct mathematical representations. The analyses of these sessions were based on the Collaboration Interaction Model (CIM), a model designed to study the knowledge construction process of complex chat transcripts. This paper will mainly discuss how participants mediate shared understanding of mathematical representations and form mathematical arguments to construct new knowledge in the chat medium, using the CIM as the key instrument of analysis.
Beliefs About Algebra Development 2
"... Mathematics teachers and mathematics educational researchers were asked to rank order arithmetic and algebra problems for their predicted problemsolving difficulty for students. ..."
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Mathematics teachers and mathematics educational researchers were asked to rank order arithmetic and algebra problems for their predicted problemsolving difficulty for students.
Learning to Program: Going PairShaped
"... Abstract: Students continue to struggle with learning to program. Not only has there been a significant drop in the number of students enrolling in IT courses, but the attrition rate for these courses continues to be significant. Introductory programming subjects in IT courses seem to be a stumbling ..."
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Abstract: Students continue to struggle with learning to program. Not only has there been a significant drop in the number of students enrolling in IT courses, but the attrition rate for these courses continues to be significant. Introductory programming subjects in IT courses seem to be a stumbling block for many students. How do we best engage students in the learning of a programming language? How can our current teaching and learning methods be improved to provide a better experience for them? Issues that have a detrimental effect on students ’ learning outcomes include more than simply the cognitive. Although programming really is complex and difficult to learn, there are also cultural and social influences on students presenting to introductory computer science courses. This paper highlights the advantages of intensive collaboration between students by exploiting the students ’ own ability and desire to interact with their peers. Peer interaction can lead to very strong learning experiences. This paper reflects on the current approaches to teaching programming by the Queensland University of Technology (QUT), Australia, with a short summary of the current focus of QUT’s first programming subject and the methods used to teach it. An overview is then given of the webbased Environment for Learning to Program (ELP) which provides scaffolding for students while learning to program. The authors propose the introduction of tools to present a collaborative environment for students to actively engage in the course material through interaction with each other.
Do We Teach Them How to Think?*
"... In today's marketplace there is an urgent need for innovative “outofthebox ” thinkers with teaming, communication, and interpersonal skills. Many college courses focus on knowledge acquisition and less on thinking. Some students are losing basic skills for defining, understanding and solving prob ..."
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In today's marketplace there is an urgent need for innovative “outofthebox ” thinkers with teaming, communication, and interpersonal skills. Many college courses focus on knowledge acquisition and less on thinking. Some students are losing basic skills for defining, understanding and solving problems while some others struggle with logical and critical thinking. Teaming and communication skills are being addressed in a relatively small number of college courses. In order to get students who can solve real problems, we must address the need for development and implementation of course modules in innovation and inventiveness in different disciplines, especially engineering and technology. Such modules can and should be designed to enhance teaming, communication and interpersonal skills. This paper discusses some of the problems in teaching innovative problem solving and suggests some possible solutions based on experience in an undergraduate course at Florida Atlantic University titled: “Introduction to Inventive Problem Solving in Engineering”. Its goal is to enhance innovative and inventive thinking abilities of undergraduate students resulting in skills
Teri Rysz Metacognition in Learning Elementary Probability and Statistics METACOGNITION IN LEARNING ELEMENTARY PROBABILITY AND STATISTICS
, 2004
"... This study used qualitative research methods to identify metacognitive thoughts adult students had while learning elementary probability and statistics concepts and while problem solving, alone and with other students. From the 49 students observed in a classroom setting, seven were purposefully sel ..."
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This study used qualitative research methods to identify metacognitive thoughts adult students had while learning elementary probability and statistics concepts and while problem solving, alone and with other students. From the 49 students observed in a classroom setting, seven were purposefully selected to be interviewed outside the classroom three times: a review of the student’s notes taken during a class immediately preceding the interview, the student solving a problem alone, and a group of three or four students solving a problem together. Classroom observation notes were organized according to categories of metacognitive thinking—orientation, organization, execution, and verification—and a fifth category labeled “lack of metacognition. ” Interviews were recorded, transcribed, and coded according to the same categories. During data analysis four themes found in the literature emerged from the data: novice vs. expert problem solving, statistics as a viable subject, selfreporting, and a cognitivemetacognitive framework. The interviewed students could be classified into two groups by similar
Classification and Resolving Urban Problems by Means of Fuzzy Approach
"... Abstract—Urban problems are problems of organized complexity. Thus, many models and scientific methods to resolve urban problems are failed. This study is concerned with proposing of a fuzzy system driven approach for classification and solving urban problems. The proposed study investigated mainly ..."
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Abstract—Urban problems are problems of organized complexity. Thus, many models and scientific methods to resolve urban problems are failed. This study is concerned with proposing of a fuzzy system driven approach for classification and solving urban problems. The proposed study investigated mainly the selection of the inputs and outputs of urban systems for classification of urban problems. In this research, five categories of urban problems, respect to fuzzy system approach had been recognized: control, polytely, optimizing, open and decision making problems. Grounded Theory techniques were then applied to analyze the data and develop new solving method for each category. The findings indicate that the fuzzy system methods are powerful processes and analytic tools for helping planners to resolve urban complex problems. These tools can be successful where as others have failed because both incorporate or address uncertainty and risk; complexity and systems interacting with other systems. Keywords—Classification, complexity, Fuzzy theory, urban problems.