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27
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 36 (13 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 16 (2 self)
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this paper is the first to make the connection with the literature on variablebinding environments.
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.
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.
Construction of Mathematical Knowledge through the use of Guided Collaborative Critiques in Problem Solving. Paper presented at the Redesigning Pedagogy: Culture, Knowledge and Understanding Conference
, 2007
"... This paper presents the use of Guided Collaborative Critiques as an effective aid to develop the construction of mathematical knowledge in a classroom environment. The students were tasked to detect errors/mistakes in problems with "solution " using Guided Collaborative Critique techniques ..."
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Cited by 3 (2 self)
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This paper presents the use of Guided Collaborative Critiques as an effective aid to develop the construction of mathematical knowledge in a classroom environment. The students were tasked to detect errors/mistakes in problems with "solution " using Guided Collaborative Critique techniques. The concept of Guided Collaborative Critique was adapted from Polya's problem solving fourstage model (Polya, 1957). The students worked in pairs to identify the location of errors/mistakes, used strategies to rectify the errors/mistakes and reflected on their problem solving process in a logsheet. The logsheets and interviews with students were analyzed to explain the phenomena of the Guided Collaborative Critique technique. More importantly, this work presents an alternative approach to the learning and the teaching of mathematics, helping students to become more effective in mathematical problem solving.
A Probelm With Problem Solving: Teaching Thinking Without Teaching Knowledge.The
 Mathematics Educator
, 2007
"... ..."
Assessing employability skills: The case of problemsolving. Paper presented at the Sixth Australian VET Research Association Conference: The Changing Face of VET, Australian Technology
 Studies in Continuing Education
, 2003
"... Over the past two years, renewed interest in employability skills has emerged. The lack of effective and efficient assessment and reporting approaches have been identified as impediments to the successful implementation of past employability skills schemes. In order to address these deficiencies, a ..."
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Over the past two years, renewed interest in employability skills has emerged. The lack of effective and efficient assessment and reporting approaches have been identified as impediments to the successful implementation of past employability skills schemes. In order to address these deficiencies, a new approach to the assessment of problemsolving, involving both a new assessment tool and a novel method of administration, has been developed. The approach is argued to be authentic, being based on tasks that participants undertake routinely rather than being developed specifically for assessment. The method is performancebased and, consistent with assessment practices in the VET sector, it is criterionreferenced. The paper describes the development of the ProblemSolving Assessment tool, including its theoretical basis, and the method of its administration. It also presents the results of a study undertaken to validate the tool. The results of the analysis indicate that several bands of performance can be recognised and suggest that the performance bands may provide a basis for profiling individuals ’ employability skills. These profiles may be matched to the needs of industry and to the requirements of occupation types. Implications for the assessment of other employability skills using similar tools and methods are outlined.
1 MATHEMATICAL INVESTIGATION: TASK, PROCESS AND ACTIVITY
"... Many writers believe that mathematical investigation is open and it involves both problem posing and problem solving. However, some teachers feel that there is a sense of doing some sort of investigation when solving problems with a closed goal and answer but they are unable to identify the characte ..."
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Many writers believe that mathematical investigation is open and it involves both problem posing and problem solving. However, some teachers feel that there is a sense of doing some sort of investigation when solving problems with a closed goal and answer but they are unable to identify the characteristics of this type of investigation. Such confusion will affect how teachers teach their students and how researchers conduct their research on investigation. Therefore, this article seeks to clarify the relationship between investigation and problem solving by providing an alternative characterisation of mathematical investigation as a process involving specialisation, conjecturing, justification and generalisation. It also distinguishes between mathematical investigation as a process and as an activity: investigation, as a process, can occur when solving problems with a closed goal and answer, while investigation, as an activity involving open investigative tasks, includes both problem posing and problem solving. Implicit support for this alternative characterisation of mathematical investigation is gathered from some existing literature as these writers did not state this perspective explicitly. The article concludes with some implications of this alternative view on teaching and research.
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.