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**11 - 19**of**19**### What’s experimental about experimental mathematics?

, 2008

"... From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, dur ..."

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From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computer-based tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging sub-discipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the so-called PSLQ algorithm.

### The Use of Experimental Mathematics in the Classroom

"... The use of computers is gaining importance in education today. Experimental Mathematics is particularly suitable for teaching (and learning) mathematics in a computer supported learning environment. In this manuscript, we show how Experimental Mathematics works in the classroom. We demonstrate that ..."

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The use of computers is gaining importance in education today. Experimental Mathematics is particularly suitable for teaching (and learning) mathematics in a computer supported learning environment. In this manuscript, we show how Experimental Mathematics works in the classroom. We demonstrate that computers can be used in each phase of the whole learning process (formulating definitions, problems and proofs, detecting finite patterns, conjecturing, falsifying, and applying math knowledge). 1.

### ABSTRACT SREMACK, JOSEPH C. Formalizing Computer Forensic Analysis: A Proof-Based Methodology.

"... Computer forensics is an important subject in the field of computer security. Impenetrably secure systems are not a reality- hundreds of thousands of security breaches are reported annually. When a security breach does occur, certain steps must be taken to understand what happened and how to recover ..."

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Computer forensics is an important subject in the field of computer security. Impenetrably secure systems are not a reality- hundreds of thousands of security breaches are reported annually. When a security breach does occur, certain steps must be taken to understand what happened and how to recover from the incident, including data collection, analysis, and recovery. These responses to an incident comprise one part of computer forensics. A successful forensic investigation of any security breach requires a sound approach. Forensics literature provides a general model for conducting an investigation that can acts as a template for forensic investigations. The current literature, however, has primarily focused on two extremes of forensics: technical details and high-level procedural guidelines. By focusing on the extremes, many of the intermediate steps and logical conclusions that a forensic investigator must make are omitted. This omission leaves the burden of forming the logical structure of an investigation to the investigator. Such ad hoc approaches can lead to inefficient investigations with extraneous investigatory steps, and possibly less accurate results. This thesis explores the formalization of existing computer forensic analysis techniques

### Is Mathematics A Science?

, 1994

"... . Mathematics is not a science, but there are grey areas at the fringes. Mathematics is certainly a science in the broad sense of "systematic and formulated knowledge", but most people use "science" to refer only to the natural sciences. Since mathematics provides the language ..."

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. Mathematics is not a science, but there are grey areas at the fringes. Mathematics is certainly a science in the broad sense of "systematic and formulated knowledge", but most people use "science" to refer only to the natural sciences. Since mathematics provides the language in which the natural sciences aspire to describe and analyse the universe, there is a natural link between mathematics and the natural sciences. Indeed schools, universities, and government agencies usually lump them together. 1 On the other hand, most mathematicians do not consider themselves to be scientists and vice versa. So is mathematics a natural science? 2 The natural sciences investigate the physical universe but mathematics does not, so mathematics is not really a natural science. This leaves open the subtler question of whether mathematics is essentially similar in method to the natural sciences in spite of the difference in subject matter. I do not think it is. A disclaimer is in order. This es...

### Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm

"... From a philosophical viewpoint, mathematics has traditionally been dis-tinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the ..."

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From a philosophical viewpoint, mathematics has traditionally been dis-tinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the

### Accessed: 26/07/2011 09:59

, 1993

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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.

### TEACHERS ’ CONCEPTIONS OF PROOF IN THE CONTEXT OF

"... ABSTRACT. Current reform efforts in the United States are calling for substantial changes in the nature and role of proof in secondary school mathematics – changes designed to provide all students with rich opportunities and experiences with proof throughout the entire secondary school mathematics c ..."

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ABSTRACT. Current reform efforts in the United States are calling for substantial changes in the nature and role of proof in secondary school mathematics – changes designed to provide all students with rich opportunities and experiences with proof throughout the entire secondary school mathematics curriculum. This study examined 17 experienced secondary school mathematics teachers ’ conceptions of proof from their perspectives as teachers of school mathematics. The results suggest that implementing “proof for all” may be difficult for teachers; teachers viewed proof as appropriate for the mathematics education of a minority of students. The results further suggest that teachers tended to view proof in a pedagogically limited way, namely, as a topic of study rather than as a tool for communicating and studying mathematics. Implications for mathematics teacher education are discussed in light of these findings. KEY WORDS: proof, reform, secondary mathematics, teacher conceptions

### A New Paradigm for Exploration in . . .

, 1999

"... This dissertation examines how the computer can aid the creative human endeavour which is data visualization. That computers now critically aid many fields is apparent, as is evidenced by the breadth of contemporary research on this topic. Indeed, computers have contributed widely to the whole area ..."

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This dissertation examines how the computer can aid the creative human endeavour which is data visualization. That computers now critically aid many fields is apparent, as is evidenced by the breadth of contemporary research on this topic. Indeed, computers have contributed widely to the whole area of data comprehension, both in performing extensive computations and in producing visual representations of the results. Computers originally aided mathematicians who could both write the instructions necessary to direct the computer and interpret the resulting numbers. Even though modern computers include advanced graphical capabilities, many issues of access still remain: the users of data visualization software systems may not be experts in any computer-related field, yet they want to see visual representations of their data which allow them insight into their problems. For example, today's mathematicians who are generally expert in exploiting computational opportunities for experimentation may lack similar experience in opportunities for visual exploration. Of particular concern is how a computer-aided visualization tool can be designed to support

### Education: A Critical Look

"... Abstract: Over the past two-plus decades there have been numerous publications asserting the virtues of information technology in mathematics teaching and learning. Despite these claims a broad look at technology supported mathematics classroom practice suggests that implementation is not always smo ..."

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Abstract: Over the past two-plus decades there have been numerous publications asserting the virtues of information technology in mathematics teaching and learning. Despite these claims a broad look at technology supported mathematics classroom practice suggests that implementation is not always smooth and results may not match intentions. This article, using the context of Canadian mathematics classrooms, explores the consequences of mandated ICT use, unanticipated outcomes when calculators and mathematics software are employed, opportunities for expanding student experience, shifting images of mathematics, and the university’s role as a model for ICT use in mathematics research and education and in the preparation of future teachers.