Profiles of Software Utilization by University Mathematics Faculty

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PROFILES OF SOFTWARE UTILIZATION BY UNIVERSITY MATHEMATICS FACULTY

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

James Edward Quinlan, B.S., M.S., M.S.

*****

The Ohio State University

2007

Dissertation Committee: Approved by

Douglas T. Owens, Adviser Patricia Brosnan Adviser Bostwick Wyman College of Education c Copyright by

James Edward Quinlan

2007 ABSTRACT

Tools and methods used to maximize mathematics learning are a primary concern for mathematics education. According to the literature, software applications are becoming increasing more prevalent in mathematics. While there exists a consensus establishing technology as a valuable tool in mathematics, and consequently ample recognition that it is important to mathematics education, arbitrary application of technology is both ineﬃcient and ineﬀective. The purpose of this study is to describe software utilization by academic mathematicians.

Survey research was conducted using an electronic questionnaire distributed to members of 47 AMS Group I university mathematics departments (n=2857). Data were used to describe, diﬀerentiate, and explain practices regarding the nature and extent of utilizing mathematical software to facilitate mathematics. Further, beliefs about mathematics education were explored. In addition to increasing awareness in the education community, proﬁles based on participants’ responses (n=422) were created to aid in identifying core practices of mathematicians with respect to technology.

The results indicate that software is increasingly important to mathematics, especially to communicating mathematics. However, participants expressed less favorable views when relating these technologies to learning and understanding mathematics.

MATLAB, Mathematica, and Maple were the three primary software applications utilized. Selection of these software depended on the branch of mathematics but

ii not necessarily on mathematical activities. In addition, results show that computer programming is considered an important aspect of mathematizing.

In conclusion, there are appropriate choices and uses of mathematical software for mathematics and thus for mathematics education. LaTeX should be used for communication, C++ or Fortran for programming, and MATLAB for applied mathematics. Mathematics education researchers should play an important role by moni- toring mathematical activities and software use in order to develop new strategies for teaching and learning.

iii To my mom

iv ACKNOWLEDGMENTS

I would like to acknowledge several people who assisted and supported me in the process of completing this dissertation. First, I would like to thank the members of my committee: Douglas Owens, Patricia Brosnan, and Bostwick Wyman. I would especially like to thank my advisor Douglas Owens; not only for his academic council but also for the interest he and his wife Faye took in my daughter Hannah.

Secondly, I would like to thank those that provided assistance with some of the more technical details. Michael Williams and Jeﬀ McCune from the Ohio State

University Mathematics Department assisted with the survey distribution by writing an e-mail script to send out individual invitations to nearly 3000 potential participants and tracked all undeliverable invitations. Additionally, Michael used the same script to electronically deliver results to all participants. I would also like to thank Michael

Quinlan for helping in the collection of the names and e-mail addresses from the survey population and John Draper for statistical advice.

I would like to thank the Ohio State University Department of Mathematics for their support over the years in the form of an assistantship. In particular, I would like to thank Cindy Bernlohr for working around my schedule and supplying me with an opportunity to teach various courses within the department.

I would like to thank my family, especially my mom and dad, for instilling the importance of education, the assurance that hard work pays oﬀ, and supporting me

v through the entire process. I love and respect them with all my heart. I would also like to than my brother David Quinlan for technical assistance with the back-end network of the survey instrument, for providing an ear so that I could internalize my arguments, and moral support to ﬁnish. I would like to acknowledge my daughter

Hannah for her patience and understanding. She has been my inspiration and a source of motivation.

Finally, I would like to acknowledge my mother, for which this dissertation is dedicated, for being a champion of education. She has taught me that the value of an education is priceless.

vi VITA

1969 ...... Born - East Liverpool, Ohio

1992 ...... B.S. Mathematical Science, Ohio State University 1995 - 1997 ...... Graduate Teaching Associate, Youngstown State University 1997 ...... M.S. Mathematics, Youngstown State University 2004 ...... M.S. Mathematics, Ohio State University 1998 - 2007 ...... Graduate Teaching Associate, Ohio State University 2006 - 2007 ...... Research Consultant, Ohio State University.

PUBLICATIONS

Instructional Publications

J. Quinlan “Delveloping graphical user interfaces for interactive applications in MAT- LAB using GUIDE”. Masters Thesis, 2004.

J. Quinlan “Delveloping graphical user interfaces for interactive applications in MAT- LAB using GUIDE”. School Science and Mathematics Association, Conference Pre- sentation, Atlanta, 2005.

FIELDS OF STUDY

Major Field: Mathematics and Mathematics Education

vii TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables ...... xi

List of Figures ...... xiii

Chapters:

1. Introduction ...... 1

1.1 Background and Motivation ...... 2 1.2 Statement of the Problem ...... 7 1.3 Purpose of the Study ...... 9 1.3.1 Research Questions ...... 10 1.4 Rationale for the Study ...... 11 1.5 Signiﬁcance of the study ...... 14 1.6 Limitations of the study ...... 15

2. Literature Review ...... 20

2.1 Technology Policies and Problems ...... 23 2.2 Technology in Mathematics Education ...... 28 2.3 Mathematics Paradigm Shift ...... 33 2.3.1 Experimental Mathematics ...... 37 2.3.2 Discovery Learning and the New Paradigm ...... 39

viii 2.4 On Mathematical Software ...... 44 2.4.1 System Software ...... 47 2.4.2 General Application Software ...... 47 2.4.3 Speciﬁc Application Software ...... 49 2.4.4 Programming Languages ...... 51 2.5 Summary of Literature Review ...... 53

3. Methods ...... 54

3.1 Participants ...... 57 3.1.1 Conﬁdentiality and Privacy ...... 58 3.1.2 Informed Consent ...... 59 3.2 Instrument ...... 60 3.2.1 Survey Questionnaire ...... 61 3.2.2 Front-end Interface ...... 63 3.2.3 Back-end and Network Topology ...... 66 3.2.4 Testing and Validation ...... 67 3.3 Procedures ...... 70 3.3.1 Data Collection ...... 70 3.3.2 Data Analysis ...... 73

4. Analysis and Results ...... 79

4.1 Demographics ...... 82 4.1.1 Computer Expertise and Experience ...... 83 4.1.2 AMS Mathematical Subject Indexes ...... 87 4.2 Software Selection ...... 90 4.2.1 HTTP-USER AGENT and System Software ...... 90 4.3 Software Activities ...... 102 4.3.1 Software for Proof ...... 111 4.4 Attitudes, Beliefs, and Preferences ...... 114 4.4.1 Interface Preferences ...... 115 4.4.2 Beliefs about the Importance of Technology ...... 116 4.4.3 Beliefs about Collegiate Software Use ...... 123 4.4.4 Beliefs about the Role of Technology in Education . . . . . 124 4.4.5 Beliefs About the Future of Technology in Mathematics . . 125 4.4.6 Additional Comments ...... 129

5. Conclusions and Discussion ...... 134

5.1 Overview of Conclusions ...... 135 5.1.1 Conclusions ...... 138

ix 5.2 Discussion ...... 139 5.2.1 Software in Mathematics ...... 139 5.2.2 Software Selection ...... 141 5.2.3 Communication ...... 142 5.2.4 Programming Languages ...... 143 5.2.5 Mathematical Activities with Software ...... 145 5.2.6 Attitudes Toward Technology ...... 146 5.3 Implications ...... 148 5.3.1 Implications for Research ...... 149 5.3.2 Implications for Practice ...... 154 5.4 Summary ...... 160

List of References ...... 162

Appendices:

A. Group I Institutions ...... 177

B. AMS Mathematical Subject Classiﬁcations ...... 178

C. Institutional Review Board ...... 180

D. Survey Questionnaire ...... 187

E. Responses to Open Response Items ...... 200

x LIST OF TABLES

Table Page

3.1 Overview of the survey...... 64

4.1 Response rate of Public Universities ...... 81

4.2 Response rate of Private Universities ...... 82

4.3 Participant Demographics ...... 84

4.4 Frequency of the number of AMS codes selected ...... 89

4.5 Participants distribution based on Rusin’s Broad Subject Classiﬁca- tions...... 89

4.6 HTTP User Agent Variable showing Operating System and Browser 91

4.7 Users Number of Operating Systems and Combinations ...... 92

4.8 Frequency response of Software Distribution Types ...... 94

4.9 Frequency Distribution of General Application Software ...... 95

4.10 Frequency Distribution of Programming Language Use ...... 96

4.11 Mathematical Software Usage ...... 97

4.12 Standardized Residuals of Chi-square test−Software vs. AMS MSC . 99

4.13 Technologies used in Mathematical Discourse ...... 100

4.14 Technological Support Material ...... 101

xi 4.15 Software usage for higher-order thinking skills ...... 102

4.16 Frequency of Software Utilization for Mathematical Activities . . . . 104

4.17 Cluster Deﬁnitions ...... 107

4.18 Standardized Chi-square residuals between Activities and Subject Ar- eas...... 108

4.19 Factor Loadings of Mathematical Activities ...... 110

4.20 Groupings Based on Factor Analysis ...... 111

4.21 Standardized Residuals of Chi-square test−MATLAB, Mathematics, & Maple and Activity Clusters ...... 112

4.22 Beliefs about the importance of technology to Mathematics and Math- ematics Education ...... 118

4.23 Mathematical course which could beneﬁt from technology ...... 122

4.24 Opinion Themes about Technology in Mathematics Educations . . . 126

xii LIST OF FIGURES

Figure Page

3.1 Entity Relation Database Model of Back-end Survey System . . . . . 67

3.2 Grossman’s Taxonomy of Mathematical Subject Areas ...... 78

4.1 Distribution by Faculty Rank ...... 85

4.2 Distribution of participants’ computer expertise levels on a scale from 1to10...... 86

4.3 Distribution of participants’ by AMS MSC codes ...... 88

4.4 Operating Systems: Response vs. HTTP User Agent ...... 93

4.5 Venn diagram of frequent users of MATLAB, Mathematica, & Maple 98

4.6 Dendrogram of the results of clustering similar activities...... 106

4.7 Dendrogram of the results of re-clustering Cluster 3 activities. . . . . 108

4.8 Raw residuals: Clusters vs. Subject Area ...... 109

4.9 Ways computers have aided mathematicians with proofs ...... 113

4.10 Interface and Input Preferences ...... 117

4.11 Year technology should play a role in K-12 mathematics education . . 119

4.12 Year technology should play a role in collegiate mathematics education 120

xiii CHAPTER 1

INTRODUCTION

. . . we consider it anomalous that an important component of the process of mathematical creation is hidden from public discussion. It is our loss that most of the mathematical community is almost always unaware of how new results have been discovered. - Epstein, Levy, & de la Llave, 1992; p.1

Mathematics education is primarily concerned with what one does while math- ematizing1 and what experiences ensure success (Thompson, 1994). Mathematics depends on tools and methods for its development, but as Epstein, Levy, and de la

Llave pointed out, some of these components are not transparent. A tool2−both a physical and psychological one−that has become central to mathematics, and thus a primary concern for educators, is the digital computer and software technology. In support of that notion, Roger Howe, Professor of Mathematics at Yale University,

1Mathematize - To reason mathematically; to perform mathematical calculations; to adopt a mathematical approach; to consider or treat mathematically; to reduce to mathematical terms (i.e., Doing mathematics) (Oxford English Dictionary, 2006). 2The word “tool” will be used as a metaphor to describe anything that facilitates methods or processes.

1 identiﬁes computer technology as third in a list of ten major issues in mathematics education (Howe, 1998). But seldom is technology not an issue in education; the problem is that an informed consensus on its selection and understanding of its mathematical purpose and utility does not exist in education in general and mathematics education in particular. Moreover, technology used in education is not isomorphic to that used in mathematics in selection or use. Thompson’s (1994) hypothesis should prompt educators to be concerned for the speciﬁc tools that facilitate mathematics and mathematical thinking, because “[i]f the true nature of mathematics is understood, then the use of technology in the learning of mathematics will be seen as natural enhancements and extensions.” (Waits & Demana, 1999, p.7).

1.1 BACKGROUND AND MOTIVATION

In terms of research and education, this is an important period in mathematics.

Utilization of computers as an active tool for mathematical research has become increasingly more prevalent, and new results have been discovered partly or entirely using software (Bailey & Borwein, 2005). According to the National Research Coun- cil (NRC), computers have had a major impact on modern mathematical research.

“Computers have profoundly inﬂuenced the mathematical sciences themselves, not only in facilitating mathematical research, but also in unearthing challenging new

2 mathematical questions.” (NRC, 1991, p. 16). In fact, in some circles3, the development of the digital computer was selected as the top contribution to mathematics of the 20th century (Friedman, 2000).

A common misconception is that computers apply only to applied mathematical

ﬁelds (e.g., ﬂuid dynamics, statistics, physics, computer science) when in fact computers recently have had a major impact on theoretical mathematics (Borwein &

Bailey, 2004). Although contributions are naturally larger to the applied mathematical ﬁelds where numerical approximations are routine, computers are used as tools to make advances in pure ﬁelds such as number theory, group theory, and graph theory.

The quintessential example illustrating the impact computers have on mathematics occurred in 1976 when a computer was used to check a large, but ﬁnite, number of cases that had not been ruled out by humans which transformed the Four-Color conjecture into the Four-Color theorem.

In addition to computers being a central tool in mathematical application and research, computer technology has naturally provided a tool for both teaching and learning mathematics. When used properly, technology fosters mathematical understanding (National Council of Teachers of Mathematics, 2000). Goldenberg, Lewis,

3Michigan Chapter of the Mathematical Association of America (MAA).

3 and O’Keefe (1992) hypothesize that through technology highly abstract, sophisticated, and fundamental mathematical ideas become accessible to unsophisticated students. Computer software can provide visual representations of mathematical objects. Simply put, learning is more diﬃcult when “concrete” representations do not exist (Dubinsky, 1991a).

Computer technology impacts mathematical thinking and understanding, content and curriculum, and teaching and learning, (Connell & Abramovich, 1999; Heid,

1997). Several researchers suggest that the use of technology will enhance conceptual learning and that successful education projects integrate technology into the curriculum (e.g., Harel & Papert, 1990). The use of technology typically underlies mathematics education reform movements. Hill (1993) claims mathematics reform has no chance without technology and Heid (1997) asserts, “the single most important catalyst for today’s mathematics education reform movement is the continuing exponential growth in personal access to powerful computing technology” (p. 5).

According to the National Council of Teachers of Mathematics’ (NCTM) Prin- ciples and Standards for School Mathematics (2000), technology enhances learning mathematics, supports effective teaching of mathematics, and influences what mathematics is taught. “Students can learn more mathematics more deeply with the appropriate use of technology” (p.24). Specifically, and succinctly, the Technology

4 Principle states: “[t]echnology is essential in teaching and learning mathematics”

(NCTM, 2000, p. 24).

Moreover, federal legislation mandates the study of conditions and practices that

enhance learning environments and that increase the ability of teachers to integrate

technology eﬀectively into the curriculum. Speciﬁcally, the No Child Left Behind

Act (NCLB) provides $15 million over 5 years for scientiﬁc research on the eﬀect of

technology on student achievement (U.S. Department of Education, 2002).

Policies regarding technology in education are not limited to the national level, as

48 of the 50 states included technology standards for students in 2004-2005 to ensure

at least a minimal amount of exposure to technology (U.S. Department of Education,

2005). This trend has extended to the local level as well.

Although these directives focus on primary and secondary school mathematics and

not on university mathematics, the Mathematical Association of Americas (MAA)

Call for Change (1991) states that technology is essential in college mathematics.

It recommends that collegiate mathematical experience should have technology used naturally and routinely in teaching, learning, and doing mathematics. It “calls” for technology to be used to enhance the understanding of mathematical ideas and should be included in the entire undergraduate major program.

5 However, in practice, many eﬀorts regress to the established way of learning−especially

advanced mathematical topics−through textbooks and lectures. Undergraduates and

graduates rarely encounter computer technologies incorporated in the curriculum. Of-

ten the end result is exclusion of technology altogether (Papert, 1993). With respect

to the diversity of the mathematical sciences, only a few areas (e.g., calculus and linear

algebra) have seen any concentrated eﬀorts to incorporate technology in coursework

(Artigue, 1999), unlike other related ﬁelds (e.g., statistics) that aggressively integrate

technology into curriculum and instruction. Both the reality of this commitment

to traditional philosophy and the importance of computers to mathematics are con-

tained in one paragraph reported by the National Research Council (1991) concerning

university mathematics.

Nothing in recent times has had as great an impact on mathematics as computers, yet in most college courses mathematics is still taught just as it was 30 years ago as a cerebral, paper-and-pencil discipline for which computers either are irrelevant or can be ignored. Computers serve mathematics these days as indispensable aids in research and application. Yet only in isolated experimental courses has the impact of computing on the practice of mathematics penetrated the undergraduate curriculum (p. 17).

Papert purports that the problem begins when new technologies are used but the epistemology remains constant. Papert’s explanation of the eventual non-use of technology due to the hegemony of tradition provides a highly likely scenario; another possibility might be simply a case of ignorance. Wu (1999) states that teachers are teaching what they do not know. In the case of technology, an analogy can be

6 made to the sentiment presented by Silver, Kilpatrick, & Schlesinger (1990) regarding problem-based learning: “applications of mathematics are often neglected by teachers because they are not familiar with how the mathematics they are teaching might be applied” (p. 9). The analogy in our case is that often teachers neglect technology in mathematics because they are not familiar with what and how the technology is actually used to support mathematics. Moreover, current reports (e.g., U.S. Department of Education, 2005) indicate that students are more technologically sophisticated than the teacher.

1.2 STATEMENT OF THE PROBLEM

The problem this study confronts is framed by the continuing concerns of educators for the tools and methods used to facilitate mathematics. While there exists a consensus establishing technology as a valuable tool in mathematics, and consequently recognizing it as an important issue in mathematics education, arbitrary application of technology is both ineﬃcient and ineﬀective. Local, state, and federal policies and organizations encourage educators to use and explore the variety of possible hardware and software (e.g., “Select and use various . . . ,” “Apply and adapt a variety of

[appropriate] strategies . . . ,” “Apply appropriate techniques and tools . . . ,” “Select

7 and use appropriate . . . .” NCTM, 2000; p. 324), but “beguiled by ever fancier calculators and computers,” teachers are less and less eﬀective (Andrews, 1996 p. 868).

This immediately begs the questions: what are the appropriate tools that should be selected and used and what are the appropriate strategies and techniques for these tools?

It is an exciting time in mathematics education as technology has not only changed how to teach mathematics, but also what mathematics can be taught (NCTM, 2000), and “what is important to teach” (Howe, 1998, p.244), it has changed mathematics itself. Thus, what mathematics needs to be taught has changed, as well. Henry

Pollak said, “with technology, some mathematics becomes more important, some mathematics becomes less important, some mathematics becomes possible” (Cohen,

1995, p. 37). However, a deﬁcient understanding about this component of mathematics, resulting from the current insuﬃcient examination of mathematicians’ uses of technology, may result in widening the gap between mathematics and mathematics education.

Moreover, for better or worse, with the Internet, podcasting, and powerful software, university faculty are no longer students’ main source of knowledge and information (Parker, 1997). If, for mathematicians, half the battle is to know where to start and which tools to use (Borwein & Bailey, 2004), then how much truer is it

8 for students, who most often work independently (N. Flowers, personal communication, 1996)? Thus, an evident concern is the deﬁcient understanding of the proper selection and use of these technological tools as a result of insuﬃcient examination of mathematicians’ uses of technology.

1.3 PURPOSE OF THE STUDY

The purpose of this study is to gain better understanding of the tools and activities used in modern mathematics to provide the best information for any attempt at educational reform or improvements. By investigating mathematicians’ utilization of computer software to facilitate mathematics and mathematical thinking, the objectives are to: (a) describe the appropriate selection and use of technology in mathematics; (b) describe speciﬁc purpose and events, including higher-order cognitive processes, for which computer technology is used for mathematical pursuits; (c) increase the knowledge base and awareness of the mathematics education community to facilitate optimizing the use of technology; and (d) describe faculty’s attitudes and beliefs about technology as “there is precious little evidence about how the university mathematics community as a whole feels about this issue” (Ralston, 2004, p. 406).

9 1.3.1 Research Questions

The following research questions will be used to focus on the general question concerning how mathematicians utilize technology to facilitate mathematical methods and processes:

1. What software applications and distribution types do mathematicians use to facilitate mathematics?

2. What mathematical areas (e.g., combinatorics, number theory) use software most frequently?

3. What are mathematicians’ level of use and expertise with computers in general and software in particular?

4. How do mathematicians utilize software, for what purpose(s), and to what extent (e.g., daily, weekly, etc.)?

5. What are mathematicians’ attitudes toward and beliefs about technology in mathematics and mathematics education?

(a) What role and significance do mathematicians believe software will have on their field in the future? (b) To what extent is technology essential or indispensable to a particular field? (c) What technologies will be relevant to mathematics in the future? (d) Which interface do mathematicians prefer, 1-Dimensional or 2-Dimensional input? (e) Which software should undergraduate and graduate students learn? (f) What are mathematicians’ attitudes toward technology used in education?

10 1.4 RATIONALE FOR THE STUDY

Dubinsky and Noss (1996) emphasize the urgency to examine the computer’s role in mathematics and mathematics education. Computers and software are substantially important in mathematics education because of their impact and inﬂuence on mathematical research, mathematical thinking, and mathematics teaching and learning. It has changed what mathematics is, what methods are used, and what is important to teach (Howe, 1998). Its utility goes beyond mere ampliﬁcation of human ability to compute, but also profoundly transforms mathematical tasks (Pea, 1992).

Computers have led people to reorganize their intellect by providing “objects to think with” (Papert, 1980, p.11), even when separated physically from the machine itself.

As the impetus of a paradigm shift in the methodology of mathematics, technology has permanently changed and redeﬁned the ﬁeld of mathematics. This new paradigm now makes up-to-date research available to novice mathematicians (Bor- wein, Bailey, & Girgensohn, 2004). Michael Atiyah (1966 Fields Medalist) warns that younger mathematicians cannot ignore computers, because not using technology will be increasingly rare for the next generation (Horgan, 2003). Furthermore, technology is becoming both a primary tool for learning and the main source of information for students (Parker, 1997).

11 It is already becoming more prevalent for mathematical research to be done by undergraduates (Suzuki, n.d.) and Cuoco (2001) believes these modern research methods are widely accessible even to high school students. Cuoco (2001) further claims that the best high school teachers are those with experience in mathematical research.

Therefore, supporting educational activities that bring students to the front line of mathematical research sooner in their career should be a focus of modern mathematical education. To meet this goal, the National Science Foundation (NSF) awarded several grants to implement the Vertical Integration of Research and Education (VI-

GRE) program to prepare undergraduate and graduate students for a wide range of opportunities in mathematical sciences enriched with signiﬁcant experience in aspects of mathematical research for which computers are invaluable.

Unfortunately, although federal, state, and local governments and organizations

(e.g., NCLB Act, U.S. Department of Education, 2002; NCTM, 2000; MAA, 1991) recognize the importance of technology in mathematics education and mandate policies specifying its study and use on all levels, its usage is far from universal (Ralston,

2004), and even further from optimal. Wu (1999) recommends “students should be taught the proper use of technology” (p.541). The ﬁrst step is to conduct research

12 on the availability and use of diﬀerent types of software used professionally (Young,

2002).

According to Moore’s Law, technology is advancing exponentially (Moore, 1965).

It is easy to see that “for mathematics education to remain viable in the future, it must include a major role for the computer now” (Shane & Tabler 1981, p. 107).

Thus far, educational eﬀorts have produced extensive research focused on the ef- fects of computer technology on K-12 student achievement (Connors, 1995), attitude

(Melin-Conejeros, 1993), engagement (Apple Computer Company, 2002; Mevarech &

Kramarski, 1992), conceptual development, (Hollar & Norwood, 1999; OCallaghan,

1998; Artigue, 1999), and understanding (Knuth, 2000). Technology-assisted professional mathematical activity, however, is not aﬀorded the same emphasis (Artigue,

1999). Professional mathematicians’ thinking processes are not well understood, and as a result our understanding of the nature of mathematics is impaired (Tall, 1981).

Examining mathematicians’ uses of technology should serve to increase the knowledge base and aid in unifying the ﬁeld, at least with respect to technology, in the following ways: (a) by increasing the awareness of the tools and methods used in mathematics,

(b) by specifying professional software applications, (c) by understanding the mathematical needs of technology, and (d) by working toward the proper and optimal use

13 of technology in mathematics education.

1.5 SIGNIFICANCE OF THE STUDY

This study contributes to mathematics education research in several ways. First, it will provide speciﬁcs about software used by professional mathematicians. Second, it will deliver signiﬁcant information regarding how technology facilitates mathematics and mathematical thinking processes. Third, this study will provide evidence concerning the extent of the methodological shift to “experimental” mathematics.

Fourth, it will contribute to and expand educational research literature. Finally, the research will oﬀer reference and guidance to educators for the incorporation of computer software technologies in mathematics education and to students learning outside the classroom.

This study identiﬁes which mathematical software packages are commonly used, how they are used, for what purposes, and to what extent are they used. Knowledge of the available software and its use will be helpful to mathematicians, mathematics educators, students−mainly graduate students and future graduate students−and anyone interested in pursuing mathematics. Future graduate students faced with a career decision may ﬁnd the results especially enlightening in establishing the direction of mathematical research, the software used for conducting it, areas of specialty

14 where software is indispensable, and the universities actively pursuing such models of research.

This study provides descriptions of particular uses of computational devices that can be implemented at the undergraduate and graduate level mathematics instruction and possibly “trickle down” to school mathematics. Furthermore, the results of this study have theoretical signiﬁcance in cognitive and constructivist learning theories by exploiting the natural isomorphism between discovery learning and experimental mathematics.

Finally, the ﬁndings will support educational software development. Examination of the use of computer technology by mathematicians, it is hoped, will maintain Vy- gotsky’s spirited position that education leads development (Kozulin, 1998). In this analogy, education should lead developments in software technology concerning education, not the other way around. In addition, the ﬁndings might help us understand how cognitive technologies can be designed, starting from the theory of discovery learning and experimental mathematics.

1.6 LIMITATIONS OF THE STUDY

In this research study, access to a web-based survey will be distributed to faculty members at 47 mathematics departments through electronic mail (E-mail). Factors

15 contributing to the limitations of this study are: (a) the nature of the research,

(b) the research instrument, (c) the instrument distribution method and timing, (d) population collection, and (e) selection method and bias.

A major concern in survey research is the typically low response rates that limit generalizations of the ﬁndings. Furthermore, this research study relies on self-reported data, which raises concerns about validity. Cook & Campbell (1979) pointed out that subjects tend to respond according to what they believe are the researcher’s expectations and respond in a way that will reﬂect positively on their own abilities, knowledge, beliefs, or opinions. In addition, cognitive psychologists highlight the fallibility of the human mind (Schacter, 2001), thus questioning responses based on memories or perceptions (i.e., answers to questions based on human memories and perceptions are suspect).

With respect to the data collection instrument, the questionnaire is researcher- designed and was not rigorously tested, although it was subjected to a pilot pre-test prior to distribution. Furthermore, lack of rigorous testing includes concerns for content validity: are the items measuring eﬀectively the intended quantities. Additional concerns for the appearance and behavior of the survey on diﬀerent web browsers

(e.g., Internet Explorer, Firefox, and Netscape Navigator) could contribute to higher drop out rates (non-responses), drop-oﬀ rates, and incomplete responses.

16 The scale−daily, weekly, monthly, and yearly − is a cause for concern. This scale, adopted from a similar study done by MapleSoft for engineering software, was used to quantify the level of use of various software. However, utilization may not be homogenous. In particular, mathematicians may use software daily for extended periods of time and, after it has served its purpose, may not use it again for equally lengthly periods of time. Furthermore, the scale is ordinal and not interval, and thus making some statistical analysis impossible because ordinal data does not qualify under the assumptions of the test.

The survey items were non-adaptive; therefore, only associations, and not direct implications, could be made as to the use of particular software. For instance, although a participant might indicate daily use of MATLAB and daily use of software for data analysis, it can not be concluded the participant uses MATLAB for data analysis.

With respect to the distribution, some e-mail servers may ﬁlter the survey as

“spam” mail. There also exist possibilities for some anomalies in the response rate due to the timing of the survey distribution. There is no common university calendaring system. Some universities follow the quarter system (9-11 weeks) while others use the semester system (15-18 weeks). Even schools on the same system often have diﬀerent breaks and holidays.

17 Furthermore with regard to timing, the delivery of the survey may come at in- convenient times such as sabbatical, medical leave, or out-of-town conferences. Thus, some faculty may not receive the survey and there is no way to determine how many faculty members actually do receive the survey.

Due to expected low response rates, there is concern for selection bias. For example, mathematicians involved in some branches of mathematics (e.g., dynamical systems and numerical analysis), which have a pre-disposition to using computational devices, may be more inclined to participate. These areas of mathematics may be over represented. Therefore the study might lack signiﬁcant representation of the general population. An additional limitation is how faculty members are listed on departments’ published websites. For example, it is possible that some faculty members’ e-mail addresses are not reported on the website or there is no distinction between faculty rank (i.e., emeritus and post-doctorate faculty could be listed under faculty with no distinguishing ranking).

The research study is bound by: (a) the magnitude of available software, (b) the participant’s position on the continuum between theoretical and experimental mathematics, and (c) the selection of the population under study. With respect to the amount of available software, there are thousands of software products and packages available for facilitating mathematics. There are numerous types of software:

18 freeware, shareware, open-source projects, online modules, and commercial software to identify and survey. Unfortunately it is not possible in this study to include in the survey questions all of the possible types of software.

Second, responses will be aﬀected by the participant’s computer expertise and mathematical philosophy. Mathematicians with a predisposition toward computer assisted mathematics will skew the results. In addition, mathematics is in the midst of a paradigm shift. Participants that accept the “authority” of the computer for proofs, may provide more favorable responses.

Finally, with respect to the population, the research study restricts the population to faculty at 47 mathematics departments grouped by the American Mathematical

Society (AMS), which will limit the generalizations of the ﬁndings concerning mathematicians’ use of technology. Group I, as reported by American Mathematical Society

(n.d.), is composed of 25 public and 23 private university mathematics departments that have a rank between 3.01 and 5.0, where 3.01 to 4 points marks a “strong” department and 4.01 to 5 points marks the department as “distinguished,” measured on several factors (e.g., highest degree oﬀered, diversity of the program, number of

Ph.D.s awarded, and quality of research). A measure of scholarly quality of program faculty is a major factor in this ranking system (Goldberger, Maher, & Flattau, 1995).

The next chapter reviews related literature.

19 CHAPTER 2

LITERATURE REVIEW

If the true nature of mathematics is understood, then the use of technology in the learning of mathematics will be seen as natural.

- Waits and Demana

In preparation for this study, literature covering a variety of topics (e.g., mathematical software and methodology, roles of technology in mathematics and mathematics education) was reviewed from multiple sources: books, journal articles, maga- zines, and Web sites. Documents from organizations and government calling for use of technology in mathematics education were examined to determine polices, recommendations, and guidelines of implementing technology in mathematics education and to serve as an indicator of the level of understanding educators have about mathematical technologies. Previous studies using technology as the treatment of educational research were reviewed for their general conclusions as well as to determine the speciﬁc selection of hardware/software used for the treatments. Expositions on the philosophy, tools, and methods of mathematics (and their morphology) were examined to

20 provide a basis for the current state and direction of the ﬁeld with respect to com-

puter software. Case studies using particular examples of computer-based mathemat-

ical discoveries were considered to examine the processes involved and how software

was integral in these processes. Finally, a review of current mathematical software

was compiled to provide an extensive examination of the raw resources available to

mathematicians and to determine the practical uses with respect to mathematics.

From this review, several relevant observations were made. First, educational poli-

cies and research regarding technology focus primarily on K-12 school mathematics;

there is less consideration on all technological issues for undergraduate and graduate

mathematics education (Schoenfeld, 2000; Achacoso, 2003). The focus becomes even

narrower with respect to investigating professional mathematical activity (Artigue,

1999) even though it is a primary concern of math education (Thompson, 1994).

Extending these same policies and implications of results to university mathematics

by the bottom-up4 approach is tenuous. Therefore, this research study will take the top-down approach by examining how mathematicians use computer technology to facilitate mathematics and mathematical activities.

4The “bottom-up” approach is to apply theory, policies, and practices based on research on K-12 to collegiate mathematics, whereas “top-down” considers applying the practices and processes of professional mathematicians to education.

21 Second, using computers for mathematics is increasingly more prevalent and many new mathematical results are discovered, in some part, with the aid of software (Bai- ley & Borwein, 2005; NRC, 1991). Several mathematicians use software such as

Maple as part of their day-to-day work (Borwein & Bailey, 2004). Moreover, the

ﬁeld is experiencing a philosophical paradigm shift (Borwein & Bailey, 2004). This metamorphosis is on the very nature of mathematics itself. The gradient of this methodological shift is the notion of proof, which is the current and traditional foundation of mathematics and the axiomatic system. Keith Devlin, who writes a column on computers for Notices of the American Mathematical Society, posits that in the near future the importance of proof will diminish, saying “you will see many more people doing mathematics without necessarily doing proofs” (Horgan, 2003, p. 652).

Third, the abundance of software is overwhelming. A complete and current list with description of all software is a futile task. In addition to the compilation, such a list would need considerable maintenance due to new software being added to and old software being deleted from the list, as well as current listings updated with the latest versions and patches.

Fortunately, two software repositories maintain an indexed list of available software and Borwein & Bailey (2004) provide a thorough list of Internet-based mathematical resources. The ﬁrst of the repositories, Guide to Available Mathematical

22 Software (GAMS), is a project of the National Institute of Standards and Technol- ogy (NIST). GAMS classiﬁes and displays software as a tree-structured taxonomy by mathematical problems. Oak Ridge National Laboratory and the University of

Tennessee at Knoxville maintain the second repository, netLib. Unfortunately, even these lists are not completely current.

Last, while some evidence supports parallels between the use of software in mathematics and mathematics education, the untold part of the story is the non-isomorphic uses. While mathematics education uses the beneﬁts of computer visualization and graphic representations, it is used less for computation, simulations, programming, and proof, especially in university mathematics education.

2.1 TECHNOLOGY POLICIES AND PROBLEMS

Unlike most countries, the United States does not have a Ministry of Education.

The federal government provides some funding and exercises controls and regula- tions for funded programs. However, education is overseen at the state and local levels. Each school district and state is responsible for the curriculum and standards.

However, a number of national commissions, agencies (e.g., NSF, NRC, National

Commission on Excellence in Education, U.S. Department of Education) and organizations (e.g., NCTM, MAA) have signiﬁcant inﬂuence on the states and local school

23 districts. Among all of these entities there exists a consensus that success in education in general, and mathematics education in particular, to a large extent, involves technology.

For instance, the National Council of Teachers of Mathematics (NCTM) published

Principles and Standards for School Mathematics (2000) claiming that “technology is essential in teaching and learning mathematics” (NCTM, 2000, p. 24), and providing standards concerning technology usage in K-12 mathematics classrooms. Now at least

48 of the 50 states have adopted technology principles and standards. To help ensure success, the United States Congress passed legislation that provides $15 million over

5 years for scientiﬁc research on the eﬀect of technology on student achievement (U.S.

Department of Education, 2005).

Calls for reform over the years (e.g., Conference Board of the Mathematical Sci- ences and National Advisory Committee on Mathematics Education, 1975; National

Commission on Excellence in Education, 1983; Mathematical Association of Amer- ica, 1991) have put technology in a central role in both teaching and learning. In

2004, the U. S. Department of Education devised the National Education Technology

Plan that included input from educators, oﬃcials from educational organizations, technology experts, and students, based on the realization that modern education

24 has technology as a central component in education. The plan includes six major recommendations (U.S. Department of Education, 2005, p.39):

1. Strengthen leadership by investing in tech-savvy leaders or provide training for current leaders 2. Create an innovative budget to invest in technology 3. Provide teacher training and improve preparation of new teachers. 4. Support virtual learning environments. 5. Encourage broadband access 6. Move toward digital content shrinking school budgets make it diﬃ- cult to provide textbooks for students

The third recommendation on the list confronts one of the more common educational problems5: the lack of adequate professional development of teachers (pre- service and in-service). Today, many students feel that they are more technologically sophisticated than their teachers. Results of a survey partially funded by the U.S. De- partment of Education was released in 2004. This survey which enlisted over 200,000

K-12 students from all 50 states, Puerto Rico, the District of Columbia, and U.S. military bases worldwide reported students’ feelings and perceptions toward their teachers knowledge and use of technology. These perceptions are made clear by a few of these students comments (U.S. Department of Education, 2005):

• I think that teachers should be required to go to technology training. • I think that the school should provide technology classes to students and teachers because our teachers are falling behind the students, as they are not good with computer programs and software.

5This is the only recommendation in which universities can have a direct impact

25 • I think the teachers could use technology better by learning more about it. I think if they learn more about it they could help the students better.

Other frequently cited problems of incorporating technology into mathematics classrooms and curriculum are the lack of knowledge, awareness and understanding of the software and the role software can have in facilitating mathematics learning. In particular, Wetzel (1993) identiﬁed lack of software knowledge as the main obstacle to classroom integration.

Gilmore (1998) states that ﬁrst it is necessary to increase faculty awareness of the software available to them and their students. Young (2002) also suggests a thorough investigation of the availability and use of diﬀerent types of software. However, educators must realize that integrating technology requires a large time investment

(Cardenas, 1998) and therefore adds to the urgency of this type of investigation (Du- binsky & Noss, 1996).

These concerns are not limited to primary and secondary mathematics. Surveys of university faculty have shown several factors that hinder their use of technology.

Parker’s (1997) results reported faculty indicated the following obstacles to their use of technology: lack of time (25%), lack of software (52%), lack of hardware (58%), lack of knowledge (29%), technology changes too fast (13%), and technology will not enhance the subject material (16%). Results from the University of California’s

26 (1995) faculty survey shows 42% of faculty cite lack of training from preventing them from using technology for teaching but also 39% claim a lack of time for training.

The same survey presents desired outcomes for students as a result of employing technology in teaching. Faculty from the University of California (all branch cam- puses) reported that they wish for students to know the technology critical to their discipline (70%), that students can use technology to think critically (72%), that students can use technology to communicate with experts of the ﬁeld (64%), and that students can use technology to increase their employment prospects (64%).

These research studies surveyed university faculty in general, but do not target speciﬁcally university mathematics faculty. From these reports, and others like them, there is no distinction between mathematics faculty and the faculty as a whole, making it impossible to elicit information indicating responses from mathematics faculty on particular items.

Although there exists research of software selection and use among ﬁelds related to mathematics, similar studies cannot be found in the literature for mathematics.

Previous studies have surveyed software used in engineering and statistics. Maplesoft, the makers of Maple software conducted a survey over three-month period, December

2005 - Febuary 2006 (Maplesoft, 2006). The goal was to shed light on the current state of technology and collaboration tools in engineering ﬁelds. The participants (n=2092)

27 from United States, Canada, Germany, United Kingdom, and Japan completed an extensive online survey. The survey revealed that the community largely considers its

ﬁeld to be “fully modern” and “taking full advantage of modern tools and technology.”

Ironically, the results showed that paper and pencil was the most frequently used tool

(52% on a daily basis). Spreadsheets are the most common software tool used, with

39% using them daily and 31% using them weekly (Maplesoft, 2006).

Jolliffe and Rangecroft (1997) conducted a comparison study of statistical software used by academic and non-academic consultants. They found a significant difference between the two groups on the selection of statistical software. The academic consultants overwhelmingly used Minitab software whereas non-academic consultants used

SAS. A natural question then is why is there such a diﬀerence between these groups?

2.2 TECHNOLOGY IN MATHEMATICS EDUCATION

While mathematics dates back thousands of years, the ﬁeld of mathematics education can be traced back to the 1960s with the ﬁrst issue of Educational Studies in Mathematics or 1970s when the Journal for Research in Mathematics Education

ﬁrst went to press (Schoenfeld, 2000). From the beginning, technology has been investigated and has served as the treatment in several studies (e.g., Papert, 1980;

Dreyfus and Eisenberg, 1983; Mayer, 1994; Vinner, 1992; Eisenberg & Dreyfus, 1992;

28 Porzio, 1994; O’Callaghan, 1998; Hollar & Norwood, 1999; Knuth, 2000; Mevarech &

Kramarski, 1992, etc.).

During this time period, mathematics education has produced extensive research focused on the eﬀects of computer technology on student achievement (Palmiter,

1991), attitude (Melin-Conejeros, 1993), engagement and creativity (Apple Com- puter Company, 2006; Mevarech & Kramarski, 1992), conceptual development, (Hol- lar & Norwood, 1999; O’Callaghan, 1998; Dubinsky, 1991a), multiple representations

(Knuth, 2000; Eisenberg & Dreyfus, 1992), general understanding (Knuth, 2000), and gender based achievement (Connors, 1995). However, conclusions from evaluations are often dated by the time they are completed (Heinecke et al., 1999)

Using meta-analysis to aggregate results of hundreds (n=500) of individual research studies of technology-rich learning environments, Kulik (1994) found that (1) students scored at the 64th percentile on achievement tests compared to 50th percentile for students in control groups, (2) students learn more in less time, and (3) students have more positive attitudes. Among these research studies, approximately

225 consisted of higher education, college, or adult education. Sivin-Kachala (1998) conducted a longitudinal review of 219 research studies from 1990 to 1997 across all learning domains and ages. From this analysis the conclusions match those of Kulik’s on achievement and attitudes at all ages and major subject areas including higher

29 education. These studies show positive results of technology on general constructs such as achievement and attitude; but attempts to extend results to more speciﬁc aspects of higher-order thinking and conceptual understanding have taken over mainstream research. Keller & Russell (1997) found that technology has cognitive and meta-cognitive beneﬁts for mathematics students: They reported that an experimental group using technology had more time to spend on higher-order problem solving processes and less on low-level calculations.

“The feature of computers that has recently caused the most excitement amongst mathematics educators is the ease” the computer has in displaying multiple representations (numerical, graphic, and symbolic) “as the user searches for conceptual understanding and problem solutions” (Fey, 1989, p. 255).

Research suggests that students often use visual representations for mathematical analysis. Studies have shown that graphical representations are beneﬁcial and in some cases necessary for students. However, other researchers (e.g., Knuth, 2000) identiﬁed a tendency for students to rely on algebraic representations even when guided toward the graphic approach.

Keller & Hirsch (1994) found that diﬀerent students have diﬀerent preference for particular representations among tables, graphs, and equations. While often visual learning is connected with weak mathematical ability, Eisenberg & Dreyfus (1991),

30 found that mathematically talented students showed “reluctance to visualize in mathematics (p. 25). Others found that students avoid visualization or graphical representations because they feel that graphical representations are not as valid as algebraic representations and play only a supporting role (Presmeg, 1986; Vinner, 1989).

Knuth (2000) reported that algebraic representation was mostly used (75%) among

ﬁrst year algebra through calculus students (n = 178). He suggests that graphical representation is a function of mathematical sophistication. Others argue that students’ uses of the various representations might be a product of instruction instead.

For example, in a study similar to Dreyfus & Eisenberg (1983), Porzio (1994) studied representations of mathematical functions by students in three diﬀerent treatments:

(a) a calculus course using Mathematica (Math 151 C), (b) calculus course using graphing calculators (Math 151G), and (c) a traditional calculus course (Math 151).

An 18-item test gave students an option to choose which representation (from the three generally accepted representation of functions: symbolic expression, graphical, and tabular) would serve them best for solving the given problem. In addition, inter- views were conducted to further elicit opinions and give insights to student preferences in choosing a representation. In the traditional calculus (Math 151) course, over 90% of the representations presented during instruction were symbolic and about the same

31 percentage in the homework. Math 151G instruction used approximately 70% symbolic and 30% graphical, but 60% of the homework problems used some graphical representations. Math 151C was approximately 50% symbolic and 50% graphical.

The results showed that student preferences were symbolic representation, graphical representation, and equally mixed representations for the Math 151, Math 151G, and

Math 151C, respectively.

An initiative to reformulate mathematical teaching and learning was launched by the National Science Foundation (NSF) funding of the University of Illinois’ Calcu- lus & Mathematica program. The Calculus & Mathematica program goals were to promote: (a) experimentation, (b) conceptual development, (c) problem-solving, and

(d) graphic visualization through the use of Mathematica, a state-of-the-art computer program, (Brown, Porta, & Uhl, 1991). These goals are not only emphasizing conceptual sense-making through experimentation and visualization, but also better parallel the current mathematical methodology by replacing axioms with insight and intuition (Gray, 1998). Nevertheless, these goals are not aggressive enough. Miss- ing from this set of goals are other aspects of mathematical thinking such as critical thinking used for mathematical proof and classiﬁcation, creative thinking for novel ideas and solutions, and decision making when conﬁrming analytically derived results.

32 2.3 MATHEMATICS PARADIGM SHIFT

The Oxford English Dictionary deﬁnes Mathematics as “the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations” (Oxford English Dictionary, 2006, p. 975). Curry (1951) stated that mathematics is the science of formal systems and rules of syntax. However, a distinction is made between mathematics and science.

Broad philosophical diﬀerences separate them, such as the notion of truth and existence of “objects,” but primarily the diﬀerence is their epistemology, the way to the truth. In particular, proofs are a legitimatization process to reach truth (Cussins,

1987). Furthermore, they are the foundation of mathematical research.

Although there are a few schools of thought in mathematical philosophy (realism, formalism, and deductivism), traditionally, mathematics uses deductive reasoning where proof of a mathematical hypothesis is based on logical deductions derived from a set of axioms and/or other proven theorems. This deductive methodology is referred to as rationalism.

Russell (1937) stated that all pure mathematics concepts are deﬁned in terms of logic, and all properties of these deﬁned concepts are deduced from logical principles.

The notion of abstract objects based on an axiomatic system from which other truths

33 are deduced with no concern for the utility of these results helps deem mathematics

as a “pure” science. Furthermore, mathematicians strive to produce the simplest and

most elegant proofs, making mathematics an art as well as a science.

Science, on the other hand, uses inductive reasoning by experiencing a physical

phenomena, quantifying the phenomena, collecting data, analyzing the data, and

making conclusions based on that analysis. This is empiricism. Scientiﬁc proof con-

sists of the agreement (or lack of disagreement) of repeated independent observations.

However, with some disappointment among members of the mathematics commu-

nity, mathematics is experiencing an epistemological paradigm shift, from rationalism

to empiricism. David Berlinski identiﬁes the computer is the impetus of change (Bor-

wein & Bailey, 2004, p. 3): The computer has in turn changed the very nature of mathematical experience, suggesting for the ﬁrst time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen.

Steen (1988, p. 611) also implies computers have drastically changed the nature of mathematics through interaction of computers and applied mathematics. In recent years, computers have ampliﬁed the impact of applications; together, computers and applications have swept like a cyclone across the terrain of mathematics. Forces unleashed by the interaction of these intellectual storms have changed forever and for the better the morphology of mathematics.

However, the idea that computers are only used for applied mathematics is a misconception. In fact computers are being used in theoretical (pure) mathematics.

34 Hersh (1997) states two trends in computer proofs: (1) computers are used as aids in proving theorems; and (2) computers are changing the nature of proof in mathematics.

Hersh (1997, p.55) provides ﬁve reasons why use of computers will continue to increase in pure mathematics:

1. Access to more powerful computers is increasing. 2. Computers are embedded in culture and newer generation mathematicians schooled with computers will replace the elder generation. 3. Computers have had success in mathematics (e.g., Haken-Appel & Landford - Four Color Theorem). 4. Science and engineering are computerized; without computer technology mathematicians could not collaborate or communicate with scientists and engineers. 5. The four preceding reasons (1, 2, 3, & 4) stimulate areas of mathematics that use computation.

First, the computer has produced an ontological shift by providing visual representations of abstract mathematical objects (e.g., cosets) (Dubinsky, 1991b); thus computer software is a means for externalizing an abstract object. Second, even more profound, there has been a major shift in methodology from deductive reasoning to inductive reasoning. Critics claim this shift in methodology fundamentally changes the nature and meaning of proof. G¨odelreassures us that “if mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics” (Borwein & Bailey, 2004, p. 1). Frege described proof as a way to make a proposition true beyond all doubt

35 and also to provide insight into the relationship among various truths (Borwein &

Bailey, 2004).

Mathematical objects are abstract entities in the Platonic universe and exist independent of human activity, but proof related to even Platonic entities involves ordinary reasoning and thus is subjective. Nagel abandoned Platonism and recommended empiricism in mathematics, because arguments are inductive generalizations from perceptual experiences or direct observations (Lee, 2002).

This methodological paradigm shift is labeled as “experimental mathematics” and it is an approach in which systematic computation plays a signiﬁcant role. Critics claim this methodology is the product of “semi-rigorous” mathematics, and they believe the consequence is totally non-rigorous mathematics. But Hadamard assures us: “the object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it” (Borwein & Bailey, 2004, p.

10). Another common negative sentiment aimed at computational-experimentalist is the notion that real mathematicians do not do computations (Borwein, Bailey, &

Girgensohn, 2004). However, there is no disagreement that Gauss used experimental methods, or that he was a real mathematician (Peitgen, 1999).

36 2.3.1 Experimental Mathematics

Borwein & Bailey (2004, p. 2-3) describe experimental mathematics as a methodology of doing mathematics that includes computation for: (1) gaining insight and intuition, (2) discovering new patterns and relationships, (3) using graphical displays to suggest underlying mathematical principles, (4) testing, and falsifying conjectures,

(5) exploring a possible result to determine if it is worth formal proof, (6) suggesting approaches for a formal proof, (7) replacing lengthy hand derivations with computer- based derivations, and (8) conﬁrming analytically derived results. Epstein, Levy,

& de la Llave (1992) emphasize that experimental mathematics should not be confused with, or be associated with, applied mathematics; while in fact the focus is on theoretical work and development of mathematical concepts. Computational- experimentation has been used in the study of a broad scope of mathematical topics such as partial fractions, convexity, log-concavity of Poisson moments, Fourier series, and zeta functions.

There are many advocates of experimental mathematics including numerous Fields

Medalists: Milnor 1962, Atiyah and Smale 1966, Mumford 1974, and Thurston 1982

(Horgan, 2003). A major factor in the transition to experimental mathematics was the development of software, speciﬁcally symbolic mathematics software (Bailey &

Borwein, 2005).

37 One of the primary motivations for using an experimental approach to mathematics is to gain insight and develop intuition, as “the purpose of computing is insight, not numbers” (Hamming, 1985, p. 7). Intuition is a holistic view that is gained through visualization; it is the opposite of a detailed or analytic view; it is high plau- sibility in the absence of proof (Hersh, 1997). Visual exploration and reasoning is an essential analytic tool to mathematicians (Eisenberg & Dreyfus, 1991). Mathemati- cians spend numerous hours analyzing situations and examples (Borwein & Bailey,

2004). It is important to understand how mathematicians combine visual and analytic approaches in solving mathematical problems and conceptual understanding.

For example, consider Mandelbrots description as quoted by Peigen (1992, p. 9):

Needless to say, this view inﬂuenced my way of handling the heavy mathematics homework we were given. After I had become acquainted with the basic outline of a new problem, I did not rush to worry about the questions being asked, but instead hastened to draw some kind of picture. When the problem was stated geometrically, this task was straightfor- ward. When the problem was stated algebraically or analytically, this was the hardest step. Once a picture was available, it received my undi- vided attention; I played with it and introduced all kinds of changes. In particular, I modiﬁed, trying to make it (somehow) richer, more attractive and more symmetric. At some point, geometric intuition rewarded me with a sudden shower of observations. Only then did I look up the questions we had been asked, and as a rule found that all the answers were intuitively obvious. Invariably, again, the formal proofs of these guesses were the quickest and easiest steps in the process. I do not recall having been stumped once

38 2.3.2 Discovery Learning and the New Paradigm

There exists an isomorphic methodology between experimental mathematics and discovery learning. Discovery learning is a type of learning where the learner conducts experiments on a domain to test a hypothesis or to form hypotheses based on the results of the experiment (Joolingen, 1999). Although discovery learning is based on

Plato’s premise that truths are discovered and not invented, aspects of the theory of constructivism are injected, in particular von Glasersfeld’s (1980) first principle: the learner constructs knowledge through active engagement and exploration. However, a difference is that constructivism contends a fallible ontology and is tolerant of different epistemology, whereas the discovery-learning approach assumes an objective reality.

In the basic model of discovery learning there exist dual search spaces: the hypotheses space and the experiment space (Klahr & Dunbar, 1988). These two search spaces are related in such a way that hypotheses can guide the search in an experiment and experiments can generate ideas for searching for new hypotheses.

Therefore, Klahr and Dunbar (1988) identify two diﬀerent search strategies, the theorist and the experimenter. The theorist strategy is to begin with a certain hypothesis and then use the experimental space to ﬁnd evidence that either supports or rejects the hypothesis. The opposite, or dual-strategy, is to collect data before

39 generating a hypothesis. Klahr and Dunbar (1988) point out that the main selection criterion is based on the learners previous knowledge.

Van Joolingen and De Jong (1997) extended the theory put forth by Klahr and

Dunbar by introducing new concepts in order to decompose the original hypothesis space to include the learner’s prior knowledge. The universal hypothesis space contains the learner’s hypothesis space and the learner’s search space.

An essential component in discovery learning theory, especially in the experiment space, is cognitive tools. A cognitive tool can be deﬁned simply as anything that helps in cognition or eases the cognitive load. van Joolingen (1999) deﬁnes a cognitive tool as an instrument that supports or performs cognitive processes. Cognition, or cognitive processes, includes all mental processes, such as thinking and reasoning (e.g., problem solving, critical thinking), representation, and memory functions

(recognition, recall), although for all practical purposes, cognition and thinking can be used interchangeably.

The word “tool” is a metaphor used to describe anything that facilitates cognitive processes. Tools change ways of thinking (Vygotsky & Luria, 1930). According to Vy- gotsky, material tools are not interesting by themselves, but the symbolic aspects of a tool-mediated activity gives rise to a new class of mediators designated as “psychological tools.” Psychological tools are cultural artifacts such as signs, symbols, text,

40 formulae, and graphing devices that help humans master the psychological functions of perception, memory, and attention. There is a natural extension of the class of tools to include higher-order thinking skills. Psychological tools are the cornerstones of Vygotskys theory. One of Vygotskys primary theoretical constructs is the Zone of

Proximal Development (ZPD). The ZPD is deﬁned to be the diﬀerence between what a learner can do by himself or herself and what the learner can accomplish with a mediator (usually a teacher or peer, but in this case an “intelligent” machine).

Vygotsky (1978) contended that cultural tools act as mediators and provide humans with the ability to regulate cognitive processes, ﬁrst externally, and then inter- nally. Examples of “tools include: paper-and-pencil, ruler-and-compass, and computers- and-calculators. Psychological examples of cognitive tools include a mental image of a circle and visual animation of objects. Moreover, visual instruments provide a means for externalizing an argument or object. Computer software provides visual representations of abstract mathematical objects (Dubinsky, 1991), and visual exploration and reasoning is an essential analytic tool to mathematicians (Eisenberg & Drey- fus, 1991). Furthermore, Jonassen, Campbell, & Davidson (1994) claim the visual cues received from cognitive tools reduce the cognitive load and improve functions of higher-order thinking.

41 The primary application of cognitive tools is to extend human intellect. Pea (1992)

writes: “I take it as axiomatic that intelligence is not a quality of the mind alone,

but a product of the relation between mental structures and the tools provided by

the culture. Let us call these tools cognitive technologies” (p. 91). Pea elaborated on

Bruner’s idea of cognitive technologies as amplifiers of the mind. Amplification refers to using technology to allow users to perform a task more quickly and accurately, to profoundly extend human capabilities. Furthermore, Pea (1992) identified technology as a tool that supports mathematical thinking and activities (e.g., developing mathematical concepts, exploration, and multiple representations). For the purposes of this report, computers used to support or represent cognitive processes will be referred to as cognitive technologies. Cognitive technologies are a significant enabler of the discovery learning approach by supporting the experimental space. More specifically, these cognitive technologies become the laboratory instruments in the experiment space.

Several cognitive psychologists have provided taxonomies for general thinking skills. Bloom’s (1956) categorization consisted of knowledge, comprehension, application, analysis, synthesis, and evaluation, whereas Guildford’s (1956) taxonomy consists of units, classes, relations, systems, transformations, and implications. Costa

(1985) adapted a synthesis of Bloom’s (1956) and Guildford’s (1956) taxonomies to

42 create a model of basic cognitive components. This model contains the following

basic thinking skills: (a) causation−used to make predictions, inferences, evalua-

tions, and judgments; (b) transformations−mapping known characteristics to un-

known characteristics for creating meanings by analogies, metaphors, and logical

induction; (c) relationships−detecting regular operations for logical deductions; (d)

classiﬁcation−determining similarities and diﬀerences, grouping and sorting; (e) qual-

iﬁcations−ﬁnding novel meanings (Costa, 1985).

These basic thinking skills, or “micro-skills” are components to the higher-order thinking skills, sometimes referred to as “macro-skills.” The macro-skills are: problem solving, decision-making, critical-thinking, and creative thinking.

Problem solving is deﬁned to be a higher-order thinking skill that uses basic thinking processes to resolve a conﬂict by inference of a solution. Most commonly, doing mathematics is associated with problem solving. According to Beckmann (2005), solving problems is both a means and end of mathematics. Many educational learning theories stress the importance of problem solving; and Thompson (1994) suggests problem-solving skills are perhaps the ultimate measure of intelligence.

Decision making uses the basic thinking skills, provides an optimal solution among several alternatives, and compares advantages and disadvantages of alternative processes of attaining solutions. Critical thinking uses basic skills to analyze arguments

43 by logical reasoning and to develop insight into context and meanings. Finally, creative thinking is a macro-skill to generate novel ideas and products. Presseisen (1985) synthesizes a relationship between the micro-skills and macro-skills. For this relationship, Presseisen identifies micro-skills composing each macro-skill. According to this model, problem solving contains the micro-skills: causation and transformations; decision making includes making relationships and classification; critical thinking implies causation, transformations, and relationships; creative thinking is composed of the micro-skills: transformations, relationships, and qualifications. Each macro-skill has associated a general out come. For problem solving, the outcome is a solution, while for decision making, critical thinking, and creative thinking, the outcomes are response, proof, and new meanings, respectively.

2.4 ON MATHEMATICAL SOFTWARE

“The use of computers has expanded enormously since their commercial introduction in the early 1950s, and new applications continue to emerge” (Rubinstein, 1975, p 127). Computers can be used in several ways with many purposes. In particular, they can be used for ﬁle and information retrieval, data manipulation, calculations, simulations, control, pattern recognition, and problem solving. Data can be sorted, classiﬁed, displayed, and printed in a variety of forms such as tables and graphs.

44 Computers can perform calculations from elementary to sophisticated functions and algorithms. They can operate on symbols and logic. They can be used to generate models of experience by simulating the behavior of various systems, physical and behavioral, deterministic and probabilistic.

The computer can be used to recognize any symbol or combination of symbols

(e.g., letters, numbers, and words) and can be used to discover proofs in mathematics, to generate new combinations, and can employ means-ends analysis in general problem solving.

A natural place to start investigating mathematicians’ selection and speciﬁc use of software is to overview software and distribution types. A module is a software unit and a collection of modules is called a software package. Size and complexity of software can range from a simple task-speciﬁc utility program (e.g., LaTexIT, Trigger) to robust applications (e.g., Mathematica, MATLAB). There are four basic types of software (Craig, 1987):

1. Operating system−manages the computer and peripheral devices (Coburn et. al., 1982). The operating system is also called system software and often referred to as the platform. Currently there are three primary operating systems: Microsoft Windows XP, Mac OS 10.4, and Linux.

2. Computer Languages−speciﬁc code syntax acting as an interface in which the user provides the computer instructions. Languages are either interpreted or compiled. Examples of computer languages include: FORTRAN, Java, C/C++, and BASIC.

3. General Application Software−includes word-processors, spreadsheets, and database management software. Microsoft Oﬃce is an example of a bundle

45 of general application software. MATLAB and Maple are two examples of mathematical general application software.

4. Speciﬁc Application Software−designed to perform a speciﬁc task (i.e., module or package, e.g., MATLABs Image Processing toolkit, Combintorica, including Web-based applications).

There are multiple methods for distributing software. [Mathematical] software is distributed as commercial, open-source software (OSS), shareware, freeware, and

Web-based. Freeware, not to be confused with free software (to be explained later), is usually speciﬁc application software that can be downloaded or obtained for no

ﬁnancial value. Free software (or open-source projects) is not bound by agendas or

ﬁnancial budgets and the software evolves as many programmers contribute to source code. For example, Octave is an open-source clone of MATLAB. Although there are many useful freeware applications and utilities for mathematics, commercial software

(e.g., Mathematica) and open-source projects (e.g., GAP, SAGE) are the two principal sources of mathematics software. Open-source projects are an important consideration when it comes to mathematical software. Open-source software is available using the GNU general public license agreement and is free to use, modify, and redistribute,

(see http://www.gnu.org/licenses/gpl.txthttp://www.gnu.org/licenses/gpl.txt).

46 2.4.1 System Software

Operating systems, although not our primary concern, are an important consideration with respect to mathematical software. There are several platforms: Windows,

Mac, Linux, and BSD. Speciﬁcally, the list of system software includes: Windows

XP, Windows 2000, Windows 98, Windows 95, Windows NT 4.0, Mac OS X, Mac

OS 8/9, Solaris, Linux, Free BSD, Open BSD, Net BSD, Be OS, Irix, and Tru64.

While most software is developed for all major operating systems (Windows, Mac- intosh, and Linux), there are still some applications that are platform speciﬁc. For example, Minitab software for statistical and data analysis is not available on the

Mac OS. Manufacturers and their Web sites advertising these products will specify system requirements.

2.4.2 General Application Software

Mathematica and Maple are two of the most robust general application software available to mathematicians (Borwein & Bailey, 2004). Mathematica and Maple are both categorized as Computer Algebra Systems (CAS). Other CASs include Derive,

Theorist, and Mathcad.

MATLAB (MATrix LABoratory) is both a computer language and a general application environment developed in the 1970s for applications involving matrices (Palm,

47 1999). MATLAB contains a number of tools for visual display of information (Hansel- man & Littleﬁeld, 2001). MATLAB continues to be one of the best applications available for providing both the computational capabilities of generating data and displaying it in a variety of graphical representations (Marchand & Holland, 2003).

In addition to MATLAB’s general application environment, there are approximately

25 speciﬁc application packages, or toolboxes.

Although general application software, by its classiﬁcation, has the ability to perform broad tasks, software typically is used for speciﬁc mathematical activities.

Simulink and Stella are examples of general application software for building dynamic models, SAS, SPSS, MATLAB are for data-analysis, and MAPLE and Mathematica are for analytic derivation. Speciﬁc application software narrows the focus.

Magma, and its predecessor Cayley, is a high-performance, non-commercial software for Unix-like operating systems that cover several areas of mathematics including number theory (and algebraic number theory), group theory, Lie theory, algebraic and arithmetic geometry, coding theory, and commutative algebra. A complete list of features can be found at the Web site: http://magma.maths.usyd.edu.au/magma/Features/Features.html.

Magma Version 2.13 was released on July 14, 2006 and can be downloaded from http://magma.maths.usyd.edu.au/magma. Magma has been used for applications in

48 number theory, group theory, and coding theory (e.g., Reed-Muller codes). More examples of application can be found at the Magma Web site (Some Applications of

Magma): http://magma.maths.usyd.edu.au/magma/AboutMagma/node198.html

2.4.3 Speciﬁc Application Software

GAP (Group, Algorithms, & Programming) is open-source software used in research of algebraic ﬁelds such as groups (e.g., Computational Group Theory), rings, vector spaces, and combinatorics. Currently, there are more than 300 research journal articles that cite GAP software. In addition GAP contains several more speciﬁc software packages: ACE (Advanced Coset Enumerator), ACLIB (Almost Crystallo- graphic Groups), ALNUTH (Algebraic number theory and interface to KANT), AutP-

Grp (Computing Automorphism Group of a p-Group), CARAT (Crystallographic group package), FGA (Free Group Algorithms), and GRAPE (Graph Algorithms using Permutation groups).

The Mathworks (TMW), makers of MATLAB, offer the following specific application software known as toolboxes: signal processing, image processing, communications, system identification, wavelet, filter design, control systems, fuzzy logic, robust control, mu-analysis and synthesis, LMI control, model predictive control, curve fit- ting, optimization, statistics, mapping, neural network, partial differential equation,

49 spline, data acquisition, instrument control, database, Excel link, financial, and financial derivatives. The above list provides insight into the types of tasks and tools needed for mathematics and associated fields or subfields.

Several software are created to provide the user the ability to construct mathematical objects (e.g., DISCRETA allows for the construction of t-designs), construction and visualization, such as subgroup homomorphism (e.g., FGB, Finite Group Be- havior ), proving theorems stated in ﬁrst-order logic (e.g., Otter), manipulation of objects, in particular geometries (e.g., Cabri, EUKLID), and languages for working with sets (ISETLW, Interactive SET language).

Open-source projects that combine and integrate software packages seem to be a recent trend and goal of mathematical software. For instance, SAGE (Software for Al- gebra and Geometry Experimentation) is a “new” open source software project built on Python/Pyrex (www.python.org) that interacts and integrates with many diﬀerent programs. The key components and features of SAGE are a unique Graphical User In- terface – an AJAX application and “wiki” that uses Python’s built-in HTTP Server, interactive shell and interpreted language – using Python, graphics – Matplotlib,

GP/Pari (Number Theory), GAP (Group Theory), Maxima (algebraic computations, similar to Mathematica/Maple), Singular (Commutative Algebra), and you can add

50 programs like Macaulay2 (Algebraic Geometry), GNUPlot (graphics), Octave (Nu- merical Analysis – similar to MATLAB) and KASH (Algebraic Number Theory). It also interacts with commercial software, such as Mathematica and Maple. For more details visit the Website: http://modular.math.washington.edu/sage/features.html.

The project is evolving fast with new stable releases.

Moreover, SAGE integrates well with LaTeX: for example, you can write SAGE code inside a LaTeX ﬁle and it prints the result to a Digital Video Interface (DVI)

ﬁle. It also can show its results in a nice DVI window or give you the result in the

LaTeX syntax (so you can paste it into a LaTeX ﬁle). SAGE has the same sort of syntax and all the powerful features as a programming language. SAGE scripts are easy to write for those that have knowledge of Python. SAGE is an excellent choice for those (e.g., graduate students) who need to learn some sort of algebra software

(Finotti, 2006).

2.4.4 Programming Languages

Programming languages provide a means for a user to supply the computer with a set of instructions on how to execute an algorithm. Because of the nature of mathematical calculations on large data sets, the primary consideration for selection of a programming language in mathematics is execution time of the program. In

51 general compiled languages are faster than interpreted languages at the machine level.

However, development time is another consideration that has led to shift away from procedural language to object-oriented languages. Object-oriented programming is a programming paradigm that has made development more productive by shifting the focus from “actions” to “objects.” Classes are built in which an unlimited number of objects from that class can be used and interact with other objects from that class or other classes.

FORTRAN, C, C++, and Java are four common high-level programming languages with broad acceptance. C++ and Java are object-oriented programming languages and all four are compiled languages. FORTRAN (standing for Formula Trans- lator), programming language was developed by IBM Corporation and is widely used for scientific computing, however it is not object-oriented making development time longer. Another advantage of the C/C++ programming languages over FORTRAN is the use of “pointers.” Pointers are variables that contain memory addresses to other variables. Pointers enable efficient use and management of memory that could affect the performance of the program during execution. Improper memory allocation can significantly affect the execution time of a program. For this reason, C is often selected over FORTRAN.

52 Modern mathematical software named in previous section (e.g., MATLAB) are also programming languages. Many mathematicians program with MATLAB, Math- ematica, or Maple.

2.5 SUMMARY OF LITERATURE REVIEW

The literature review has provided an overview of current research, policies, and practices regarding technology, speciﬁcally software technology, in mathematics and mathematics education. We have seen it has had a major impact on mathematics and has changed the nature of mathematics. Moreover, this trend is expected to continue and will have future impacts on mathematics and thus mathematics education.

To reiterate Henry Pollak’s sediments, technology makes some mathematics more important, some less important, and some mathematics possible. Therefore it is necessary that mathematics education focus signiﬁcant attention on this issue. As a beginning to that end this research study is devoted.

53 CHAPTER 3

METHODS

Software technologies are becoming more central to mathematics education, but there is a lack of understanding of exactly how these technologies are central to mathematics itself. In particular, attempts to integrate technology into mathematics curriculum (e.g., Calculus & Mathematica) are numerous. Equally numerous are policies

(e.g., NCTM Principles and Standards for School Mathematics , 2000) as to what, how, and why these technologies should be incorporated. However, the rationale for these actions is ﬂawed because the literature does not contain nor attempt to understand how professional mathematicians have utilized technology for mathematics.

Moreover, mathematics education researchers mainly focus their eﬀorts on primary and secondary school mathematics with little to no attention paid to undergraduate and graduate mathematics students who would possibly beneﬁt the most from technology. The purpose of this study is to gain understanding of the uses professional

54 university mathematicians make of technology in their research and teaching. Ex- amining and describing the selection and use of these technologies among university mathematicians will be a step toward accomplishing this goal. Proﬁles of university mathematicians’ professional mathematical software activity will be created.

The following basic research questions are used to study this problem.

1. What software applications and distribution types do mathematicians use to facilitate mathematics?

2. What mathematical areas (e.g., combinatorics, number theory) use software most frequently?

3. What are mathematicians’ level of use and expertise with computers in general and software in particular?

4. How do mathematicians utilize software, for what purpose(s), and to what extent (e.g., daily, weekly, etc.)?

5. What are mathematicians’ attitudes toward and beliefs about technology in mathematics and mathematics education?

Survey research was conducted using the Internet (also known as the World Wide

Web (WWW), or “Web” for short) and the data collected were used to describe, diﬀerentiate, and explain practices and attitudes regarding the nature and extent

55 of utilizing mathematical software among a speciﬁc population of university math-

ematics faculty. An Internet survey was used to exploit advantages in time, cost

(Cobanoglu, Warae, & Morec, 2001), and ease in analysis by having all data auto-

matically returned in electronic format. Furthermore, for some populations that use

the Internet regularly, the Web has been found to be a useful means of conducting

research (Couper, Traugott, and Lamias, 2001; Sills & Song, 2002). In particular, all

department faculty members have free access to the Internet and an also e-mail ac-

count and are expected to use e-mail to interact and communicate with students and

administrators (Kaplowitz, Hadlock, & Levine, 2004). Results include both quan-

titative descriptors and qualitative accounts. More speciﬁcally, in describing the

population’s use of software technology, diﬀerences between groups (e.g., mathemat-

ical subject area and rank) in the population, associations among software selection

and use, and issues that arise while exploring other less well known aspects of the

population’s use of software were reported.

This chapter contains and deﬁnes: (a) the participants, (b) the research instru-

ment including, (i) the questionnaire, (ii) the front-end6 Web-interface, (iii) the back- end7 service applications, and (iv) the back-end hardware, and (c) the procedures, (i)

6A front-end application refers to a computer interface that the user interacts with directly (i.e., the electronic questionnaire). 7A back-end service, or application, is a program that supports the front-end service.

56 data collection methods, (ii) procedures for validating the instrument, and (iii) the

data analysis techniques. A copy of the questionnaire is provided in Appendix D.

3.1 PARTICIPANTS

The population for the study consists of university mathematics faculty from

Group I mathematics departments (see Appendix A). A participant, or respondent is

deﬁned as a responding member of the survey population.

Group I, as reported by American Mathematical Society (n.d.), is composed of 25

public and 23 private university mathematics departments that have a rank between

3.01 and 5.0, where 3.01 to 4 points marks a “strong” department and 4.01 to 5 points

marks the department as “distinguished,” measured on several factors (e.g., highest

degree oﬀered, diversity of the program, number of Ph.D.s awarded, and quality of

research). A measure of scholarly quality of program faculty is a major factor in this

ranking system (Goldberger, Maher, & Flattau, 1995).

The choice of Group I mathematics department faculty for this research study was

based on several factors: (a) Goldberger, et al. (1995) rankings, (b) Group I institu-

tions account for the largest percentage (12%) of the mathematical community among

the top three groups, (c) these universities also have the largest percent of postdoctoral

positions, and (d) the researcher’s perception of the institutions’ academic reputation

57 and inﬂuence on the ﬁeld of mathematics. From these 48 departments, Ohio State

University (N=120) was removed because of potential bias due to the researcher’s af-

ﬁliation. Population data were collected in preparation for this research study during the summer of 2006. The population consisted of 2857 mathematicians (1836 public,

1021 private) and according to the American Mathematical Society’s annual survey of the mathematical sciences in the United States (2005), there are 2,811 total full- time faculty at Group I institutions (1835 public, 976 private), a relative error of 1.6%.

3.1.1 Conﬁdentiality and Privacy

Research reveals Internet security and receipt of unsolicited electronic mail (a.k.a.

“spam”) are among major concerns of potential participants (Sills & Song, 2002).

Therefore, all participants’ responses are confidential and warehoused on a secure private network (see Section 3.2.3). Although no guarantee of Internet security can be given, as it is possible for transmissions to be intercepted and for IP addresses to be identified, with this survey a breach of confidentiality is extremely unlikely to occur. Survey response data came in directly from the web site file without any personal identifying information. The only way that confidentiality could be breached would be if an outside party intercepted the data coming from the participant’s computer during the time they were completing the survey. However, since there

58 was no identifying information included on the online survey, even if the data were intercepted, conﬁdentiality remains assured. In addition, after data were collected and before any analysis and results were produced, all responses were de-identiﬁed from the raw data. Therefore, it was impossible to identify, directly or indirectly, participants’ responses.

Conﬁdentiality was chosen over anonymity in order to diﬀerentiate between respondents and nonrespondents (pre-analysis), to verify respondents’ trustworthiness, and to minimize the number of survey items by importing from the population database (a separate encrypted database) known information about each participant

(e.g., university aﬃliation) without actually increasing the number of questions.

3.1.2 Informed Consent

The population, thus each participant, is 18 years of age or older. The population received an invitation via electronic mail that explains the nature and purpose of the research, along with any associated risks of Internet security. In order to participate, the recipient must select the included hyperlink to the URL or manually type the

URL in their Internet browser to initiate the survey. Furthermore, the survey application contained a login screen, upon which the nature and purpose of the research, including the risk associated with Internet transmission, was reiterated. They chose

59 whether or not they wanted to respond. The act of participation served as informed

consent. The Institution Review Board (IRB) has declared this research exempt and

has granted a waiver of signed informed consent (See Appendix C).

3.2 INSTRUMENT

A researcher-designed Web-based survey system was developed and served as

the instrument. The instrument consisted of three components: (a) the question-

naire: question and response items, (b) the front-end interface, and (c) the back-end

database and network used to deploy the questionnaire and warehouse the response

data. This probability-based survey (Couper, 2000) was a self-reporting electronic

questionnaire. Table 3.1 provides an overview of the survey instrument details.

The design of a survey can aﬀect response rate, attrition rate, and the responses

themselves (Couper, et al., 2001). HTML8, CSS9 , ASP10, Javascript, and Microsoft

SQL Server were selected for the technical design to allow the researcher to: (a) sup-

port multiple platforms and browsers (Yun and Trumbo, 2000), (b) prevent multiple

submissions while allowing multiple logins (Smith, 1997), (c) collect both open-form

8HyperText Mark-up Language 9Cascading Style Sheet−A ﬁle linked to a Webpage to add style (e.g., fonts, colors, spacing) to the Web document 10Microsoft’s Active Server Page technology embeds scripts in HTML and is executed on the server prior to being sent to the client’s browser

60 and closed-form response types (Yun and Trumbo, 2000), and (d) provide immediate feedback to respondents by indicating how many questions were completed and how many questions remained.

3.2.1 Survey Questionnaire

A primarily closed-response questionnaire was designed to provide an eﬃcient and reliable format for completion (Fink, 1995). Thirty items: twenty closed-response, four open-response, and six hybrid-response type (closed-response with an option for an additional free-response) composed the questionnaire (see Appendix D). Several items were arranged in table format, thus actually increasing the number of items without the appearance of such an increase. There were two types of closed-response types: inclusive and exclusive. Inclusive choice form allowed participants to select more than one response (e.g., “Which operating system(s) do you use? Select all that apply”). Exclusive response items solicited exactly one choice (e.g., Gender). Nominal data were code with “1” if the item was selected. For ordinal response data, the survey uses a 5-point rank scale (i.e., 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, and 1 =Never). This 5-point rank scale was adopted from MapleSoft (2006) and attempts to quantify software utilization. A phrase completion continuum 10-point

61 scale was used to gauge participants’ computer expertise to increase diﬀerentiability and variability among the responses.

The questionnaire consists of four sections. The ﬁrst section collects demographic information: (a) gender, (b) age group, (c) level of service (full, associate, assistant, postdoctoral professor), (d) mathematical subject area(s) of interest, (e) preference of system software, and (f) overall computer expertise.

Section two contains 6 items that probe for specific information concerning the selection and extent of use of mathematical software. Software were listed from general to specific as questions progressed. Participants had the option to provide details of specific software if not listed in the item choices by selecting an “Other” option.

The third section investigates speciﬁc tasks and activities mathematicians use technology to facilitate, such as computational analysis, visual reasoning, and simulation. In addition to determining what activities beneﬁt from computer technology, a measurement of the extent of utilization was taken using the same scale as the questions from Section 2. This section also considers the connection between higher- order cognitive tasks (e.g., critical thinking) and computer software. Participants were asked to select their level of software utilization to perform these higher-order mental functions.

62 Section 4 records participants’ attitudes, beliefs, and preferences with respect to technology in mathematics and mathematics education. Questions specific to mathematical courses and the appropriate grade level to introduce technology were included in this section. In addition, Questions 5 and 6 inquired about their preferences toward the computer interface and input methods. There were four open-response items for which participants provided opinions about: (a) which software applications undergraduate and graduate mathematics students learn, (b) what mathematics education should do to better prepare mathematics students for use of technology, (c) what future technological advancement might be significant to the field of mathematics, and (d) any other issues regarding technology in mathematics and/or mathematics education. This section also provided participants with opportunities to indicate issues that may limit their use of technology in mathematics.

3.2.2 Front-end Interface

The questionnaire was presented as a dynamic web site linked to a back-end SQL database where the data were automatically warehoused. A custom ASP application was deployed on the Web server to act as the interface to the survey questionnaire.

The electronic questionnaire was designed with a simple layout and navigation strategy to add credibility, to reduce the dropout rate, and to make it easy to complete

63 Elements Description Purpose To gain understanding of the tools and methods used in modern mathematics by identifying faculty selection and use of technologies

Respondents Mathematics faculty at Group I institutions

Responses 87% closed; also contains open response and hybrid response types

Surveyor Self-reporting online survey

Number of sections 4

Number of questions 30

Number of web pages 7

Timing 15-20 minutes

Resources Access to the Internet and E-mail

Privacy Conﬁdential

Table 3.1: Overview of the survey.

(Couper, Traugott, and Lamias, 2001; Dillman, Totora, Conradt, and Bowker, 1998;

Preece, Rogers, and Sharp, 2002) by using an oﬃcial university logo as the header of each page and by using basic principles of human-computer interaction: (1) keep the design simple, (2) maintain consistency throughout the interface, (3) provide feedback to the user, (4) allow recovery from mistakes and reversible action such as an “Undo” or “Back” button, (5) allow user control, and (6) apply graphic design principles such

64 as aligning objects, using plenty of white space, and use high contrast between objects

(Apple Computer, Inc., 1992). Javascript was used to provide “client-side” features such as highlighting multiple-choice rows and transitioning between sections.

Zanutto (2001) identiﬁed the diﬀerences in appearances from Web browser to

Web browser (e.g., Internet Explorer, Firefox, Safari) as a concern for online surveys as poor appearance and navigation aﬀect the response rate and attrition rate. To circumvent this issue, the ASP application used a global CSS to provide a consistent format style and presentation of the document interface. Furthermore, the CSS was validated for compliance to the W3C (World Wide Web Consortium) design standards by the W3C quality assurance markup validation service v.0.7.2 not only to ensure accessibility and consistency among diﬀerent browsers, but also to ensure accessibility for the disabled.

The front-end Web interface contained standard graphical interface objects: option buttons for exclusive closed choice responses; and check boxes for inclusive closed choice responses. Textboxes were used to accept open-response items, and a coupling of either checkboxes or radiobuttons with textboxes−depending on whether the item was inclusive or exclusive−allowed respondents to include “other” responses not covered by the presented choice responses.

65 3.2.3 Back-end and Network Topology

The back-end and network are crucial components of the instrument. These components are responsible for delivering the questionnaire and storing data, as well as maintaining respondent conﬁdentiality. Furthermore, a well-designed database model ensured easy querying and analysis of data.

Two separate computer servers on the network were utilized for the data collection phase (herein after referred to as the Web Server and the Data Server). Both employed the Microsoft Windows Server 2003 operating system and are updated automatically with the latest system and security patches. Figure 3.2.3 shows the entity relation database model of the survey.

All software components that compose the instrument reside on a state-of-the-art network divided into zones. Network address translation (NAT) mapped traffic to the appropriate zone (i.e., web server or data server) and zone access rules permit only specific services. All ports not specifically authorized are actively blocked and appear stealth.

The network is located behind an enterprise-class network security appliance. This appliance has a comprehensive set of security applications (automatically updated daily at 3:00 A.M. CT), including Gateway anti-virus, Gateway anti-spyware, and intrusion detection system to further protect conﬁdentiality of the participants and

66 their responses. Fiber-optic lines provided the network bandwidth. The network is a private network and is not owned, maintained, or administered by a third party.

Figure 3.1: Entity Relation Database Model of Back-end Survey System

3.2.4 Testing and Validation

The instrument was tested for internal validity and consistency, understandability, and ease of completion through multiple stages as suggested by Dillman (2000).

First, the questionnaire was distributed to knowledgeable colleagues and members of the Ohio State University mathematics department for question relevance, appropri- ateness, ambiguity, and sensibility. Finally, reﬂection on the stages and responses to

67 the stages provided an opportunity to “conceptualize and re-conceptualize the key elements of the study and to prepare for the ﬁeldwork and analysis” (Oppenheim,

1992, p. 64).

The second stage involved testing the instrument using a pilot study to emulate the procedures of the main study including the e-mail invitation, the Web application, the database, and the network (see Section 3.3). Members of Arizona State University mathematics department (n = 54) and University of Pittsburgh (n = 35) mathematics department served as the population for the pilot study. These members were selected to maximize the effectiveness of the pilot study. First, both have relatively large size departments. Second, the members of the department are actively involved in research in computational mathematics, nonlinear dynamics, differential equations, mathematical biology, discrete mathematics, and number theory. Faculty working in these fields would be predisposed toward technology use, and therefore nonresponse would be minimized. The timeline of the pilot study was during early fall of 2006.

The pilot followed the same procedures as the main study and served the primary purpose of validating the instrument. It ensured the clarity of the questions and ease of completion, debugged the Web application, and estimated response rate by distributing to a large number of a characteristically similar population. During the pilot study, one individual participated using “think aloud” protocols and thereby helped

68 the researcher evaluate understanding, logical sequencing, and motivational qualities.

This individual made contact with the researcher by email and agreed to participate

in the survey over the phone. This participant read the questions aloud, vocalized

his thoughts, and made comments as to the understandability and sensibility of each

question and response. The researcher listened over the phone and took notes. How-

ever, due to notoriously low response rate and high attrition rate, examining only the

response data did not provide the desired rigor of validating the instrument. There-

fore, in addition to analyzing the pilot response data, a thorough examination of the

pilot’s participants’ log ﬁles provided further insight and helped identify problematic

survey items by analysis of the seven response/non-response scenarios: (1) complete

responders, (2) unit non-responders11, (3) item non-responders, (4) dropouts, (5) item non-responding dropouts, (6) lurkers,12 and (7) lurking drop-outs (Bornjak & Tuten,

2001). Moreover, this study assumed a most favorable distribution between optimiz- ers13 and satisﬁcers14 (Krosnick, 1991) because of relevance of the content to each participant. Of the 89 invited to participate, there were 13 complete responders, 2

11Unit nonresponder-Participant that does not respond to an item (or an item nonresponder). 12Lurkers-Nonparticipant that read and reviewed the survey but do not answer any of the questions. 13Optimizers are participants that thoughtfully answered all the questions, usually because they care about the subject which the survey is based. 14Participants that only go through the motions of taking the survey. They do not have any great connection to the subject understudy and only participate because they feel obligated.

69 unit non-responders, 3 to 10 item non-responders, 2 dropouts, and 12 lurkers. From the analysis, several questions were rephrased and additional Javascript was added to highlight (in blue) the row of the response item in order to aid making selection easier for participants, and attempting to circumvent item non-respones.

3.3 PROCEDURES

From each department (including ASU and UP for pilot study), a census of faculty members was obtained. The census information included: name and e-mail address as reported by their institutions published department homepage. This information was stored in an encrypted database ﬁle on a personal computer with no network access. A computer generated random number was associated with each record in the database. The last name, e-mail address, and random number were extracted to form a new table, called the Population Table.

3.3.1 Data Collection

Data were collected using the Internet survey instrument outlined in Section 3.1.

The survey was available from 12:00 AM Central Time October 1, 2006, through 12:00

AM Central Time October 30, 2006. There are two primary reasons for this timeline.

First, by this time universities on both quarters and semesters were in session, and

70 second, survey availability for 30 days provided ample time to bolster response rate by making follow-up contact electronically.

Each faculty member in the Population Table was sent an invitation to participate via the e-mail server at The Ohio State University Mathematics Department. The invitation (see Appendix C) contained a brief background and purpose of the study along with the Universal Resource Locater (URL) hyperlink to the survey login and the computer-generated password. Recipients followed an e-mailed link to a login page and entered a 6-digit password. Requiring a password login and enabling cookies15 controlled multiple submissions and identiﬁed non-responders so that follow-up e-mail reminders can be sent to help bolster the response rate. An increase in e-mail contact resulted in increase rate for responses (Smith, 1997).

In particular, to increase response rates, multiple contacts were made with potential participants. First, e-mail was sent to establish initial contact. Second, one week after the survey was delivered, a follow-up e-mail reminder was sent to members of the population that had not yet participated.

Similar to the e-mail invitation, the survey begins with an introductory page with the purpose to (1) establish the authority and credibility of the researcher, (2) explain the survey purpose, (3) explain the signiﬁcance of the research, (4) ensure respondent

15A Web “cookie” is a small ﬁle on the client’s computer containing information about the user, such as preferences and/or session management variables.

71 conﬁdentiality and privacy, (5) provide contact information to the researcher, (6) explain the sampling method, (7) provide the Institutional Review Board approval,

(8) establish authenticity of the research (Cho and LaRose, 1999), and (9) explain the risk associated with Internet transmission. Participation is voluntary and informed consent was assumed after the respondent logged into the server (see Appendix C).

Participants followed an intuitive interface to complete the questionnaire. All data obtained from the survey questions−in real time−was transferred to the Data

Server. Each response was stored as a vector: participant ID, section number, question number, choice number, and weight. Microsoft SQL Server 2005 warehoused the data. The data collection phase utilized both a Web Server and a Data Server on the network. Both employ the Microsoft Windows Server 2003 operating system and are updated automatically, and regularly, with the latest system and security patches.

In addition to collecting items contained on the questionnaire, server technology provides the means to collect some relevant information, unbeknown to the user, without making the questionnaire longer. The data collected includes the users Internet browser and system software.

72 3.3.2 Data Analysis

All responses were de-identiﬁed using a one-way hash function from the raw data before any analysis and results were produced, and therefore it is impossible to identify, directly or indirectly, individual participant’s responses. The database ﬁle storing the participants’ responses was then exported from Microsoft’s SQL Server to a

Microsoft Access database to query information. When appropriate, convenient, or necessary, data were exported to other software. In particular, data were exported to

MINITAB for cluster analysis, factor analysis, and chi-square analysis.

Coding of the responses were automated by the design of survey system. Each response item included a weight. Nominal variables were assigned a weight of “1” whereas ordinal data and interval data were assigned “1”-“5” or “1”-“10” respectively.

Descriptive statistics (frequency, mean, median, mode, and standard deviation) were reported for each response item.

Cross-tabulated queries were used to examine relationships between response variables and a Chi-squared test was performed to determine independence. In particular, we considered: software versus mathematical activities, branch of mathematics versus mathematical activities, branch of mathematics versus software, and computer expertise versus faculty rank.

73 Hierarchical cluster analysis using Wards linkage method was used to reduce the software activities list and to determine associations among the activities. Chi- squared tests for independence were conducted on activities versus grouped mathematical subject area classiﬁcations, software versus mathematical activities, and professor rank versus computer expertise. Factor analysis, using principal components as the method of extraction and equimax rotation for highest loading on one factor, were conducted to supplement and verify the groupings of the cluster analysis.

Mathematical Subject Classification (MSC) codes, as defined by AMS (see Ap- pendix B), were used as a grouping variable. There are several ways to partition or group Mathematical areas, (e.g., pure/applied, continuous/discrete, or algebra/geometry/analysis, etc.). Furthermore, there exists some overlap in mathematical areas from partition to partition. For example, a natural partition exists for algebraic areas and geometric areas; however some fields could be placed into two groups (e.g., algebraic geometry (AMS 14)).

A standard division in mathematics is the “pure” or “applied” dichotomy of Math- ematical subjects. One partition presented by Grossman (2005) of the Mathematical subject areas based on this dichotomy is to ﬁrst separate pure and applied mathematics, and then subdivide pure mathematics into “discrete” and “continuous” (see

Figure 3.2).

74 For our analysis we adopt, but further synthesize, the following partition into

distinct groups from Dave Rusin (2000) that broadly divides the AMS Mathemat-

ical Subject Classiﬁcations (MSC) subﬁelds into general allied areas: foundations

(i.e., logic and set theory), algebraic areas, geometric areas, analytical areas, prob-

ability and statistics, computer sciences, scientiﬁc application areas, and other. See

Appendix B for AMS MSC.

• Logic and Set theory - 03

• Algebraic Areas - 05, 06, 08, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22

• Geometric Areas - 51, 52, 53, 54, 55, 57

• Analytical Areas: Calculus and Real Analysis - 26, 28, 33, 39, 40 Complex Variables - 30, 31, 32 Diﬀerential and integral equations - 34, 35, 37, 45, 49, 58 Functional Analysis - 42, 43, 44, 46, 47 Numerical analysis and optimization - 41, 65, 90, 91

• Probability and Statistics - 60, 62

• Computer science and Information theory - 68, 94

• Applications to science - 70, 74, 76, 78, 80, 81, 82, 83, 85, 86, 92, 93

• Other - 00, 01, 97

Three different methods were used to pigeonhole participants into Rusin’s subject classifications. The first method involved placing each participant into every relevant category. For instance, a participant selecting AMS codes 14 and 35 were placed into both algebra and analysis.

75 The second method placed participants in Rusin’s categories based on “predom- inance.” If a participant selected AMS codes 11, 14, and 35, they were placed in algebra because 2 of their 3 selections are classiﬁed as algebraic ﬁelds. The main disadvantage of this method was that participants with an even split between subject

ﬁelds were removed from the analysis.

The third method used a selection pass ﬁlter set at one-third. In particular, participants selecting multiple AMS codes were placed in a Rusin general group if they had selected at least one-third of the general area. For example, a participant selecting 05, 11, 15, and 35 would be placed in algebra whereas a participant selecting

11, 14, and 35 would be placed in both, algebra and analysis.

Fortunately, these methods only applied to a few participants. As it turned out, all three methods produced the same chi-square result.

Open-response items were grouped and reported verbatim (see Appendix E) unless response seemed oﬀensive, inappropriate, or irrelevant (as judged by the researcher).

Frequencies of common responses were reported in tabular format. Frequency counts of the removed response items were not reported and no further consideration for these responses was given.

The next chapter presents the results using tables and ﬁgures that reﬂect participant responses. Tables report responses in terms of percents for each ordinal or

76 categorical measurement along with total responses per item and the weighted average for each item. Data presented are used to describe and reﬂect current situations and relationships regarding software usage by mathematicians.

77 Figure 3.2: Grossman’s Taxonomy of Mathematical Subject Areas

78 CHAPTER 4

ANALYSIS AND RESULTS

The purpose of computing is insight, not numbers. - Richard Hamming

The following results are based on information gathered by an electronic questionnaire distributed to all faculty members at 47 university departments of mathematical sciences in the United States (n=2857). The data reported in this chapter provide a snapshot of technology usage in mathematics and mathematics education by academic mathematicians. Analysis of the data provides proﬁles of mathematicians’ software use for the general guidelines set forth by mathematics education organizations and governmental education policies.

Of the 2857 invitations, 432 signed into the survey system, however only 422 participated by answering at least one question. Ten potential participants logged in, possibly even read all questions, but responded to none. Therefore the overall response rate was approximately 14.8%, however due to a researcher error in the collection of the e-mail addresses from one university, speciﬁcally in the e-mail domain name

79 (e.g., “university.edu” instead of “math.university.edu”), 66 e-mails were returned

as undeliverable (a.k.a. “bounced”). In addition to those 66, there were 38 other

bounced e-mails, returned for one of the following reasons: (1) user mailbox full,

(2) user out of town and not accepting e-mails, (3) e-mail redirection to unknown

address, or (4) user unknown. User unknown errors are typically errors in the e-mail

address itself, such as incorrect domain name or misspelling of user’s name and e-mail

address. These latter errors were most likely the researcher’s error in collecting the

e-mail address. A total of 104 e-mail invitations were returned (bounced). Upon

excluding the bounced e-mails, the population assumed to receive an invitation was

actually 2749, thus bolstering the response rate to 15.12%. Although 422 participants

logged in to the survey system and responded to at least one item, responses were

consistently taken for 385, thus making a drop oﬀ rate around 9%.

There was at least one participant from 44 of the 47 universities. Response rates

varied per university with a maximum of 38% and an average of 14.6%. Table 4.1

lists the public universities included in the survey−partitioned into public and pri-

vate institutions− their population, their percent of the population, the number of responses from each university, and each university’s response rate. Table 4.2 lists

80 the same data for private institutions. The highest responses rate of a single university was 38.64%. Response rates were higher among private universities (17%) than public universities (14%).

Group I Mathematics Departments Population Pop. % Responses Rate % Public 1836 64.26 258 14.05 City University of New York 65 2.28 3 4.62 Georgia Institute of Technology 77 2.7 4 5.19 Indiana University 53 1.86 0 0 Michigan State University 85 2.98 12 14.12 Pennsylvania State University 74 2.59 2 2.7 Purdue University 95 3.33 17 17.89 Rutgers University 68 2.38 14 20.59 State University of New York 68 2.38 12 17.65 University of California, Berkeley 126 4.41 9 7.14 University of California, Los Angles 122 4.27 25 20.49 University of California, San Diego 44 1.54 0 0 University of California, Santa Barbara 83 2.91 9 10.84 University of Illinois, Urbana-Champaign 82 2.87 18 21.95 University of Illinois, Chicago 67 2.35 7 10.45 University of Maryland 71 2.49 6 8.45 University of Michigan 133 4.66 49 36.84 University of Minnesota 74 2.59 0 0 University of North Carolina, Chapel Hill 43 1.51 2 4.65 University of Oregon 37 1.3 5 13.51 University of Texas, Austin 87 3.05 3 3.45 University of Utah 82 2.87 15 18.29 University of Virginia 39 1.37 8 20.51 University of Washington 90 3.15 23 25.56 University of Wisconsin 71 2.49 15 21.13

Table 4.1: Response rate of Public Universities

81 Group I Mathematics Departments Population Pop. % Responses Rate % Private 1021 35.74 174 17.04 Boston University 37 1.3 3 8.11 Brandeis University 15 0.53 3 20 Brown University 25 0.88 1 4 California Institute of Technology 14 0.49 2 14.29 Carnegie Mellon University 37 1.3 10 27.03 Columbia University 57 2 9 15.79 Cornell University 72 2.52 21 29.17 Duke University 44 1.54 17 38.64 Harvard University 57 2 7 12.28 John Hopkins University 27 0.95 8 29.63 Massachusetts Institute of Technology 109 3.82 20 18.35 New York University, Courant Institute 57 2 6 10.53 Northwestern University 44 1.54 8 18.18 Princeton University 49 1.72 6 12.24 Rensselaer Polytechnic Institute 29 1.02 7 24.14 Rice University 24 0.84 2 8.33 Stanford University 58 2.03 5 8.62 University of Chicago 37 1.3 1 2.7 University of Notre Dame 59 2.07 15 25.42 University of Pennsylvania 40 1.4 5 12.5 University of Southern California 68 2.38 10 14.71 Washington University, St. Louis 27 0.95 6 22.22 Yale University 35 1.23 2 5.71

Table 4.2: Response rate of Private Universities

4.1 DEMOGRAPHICS

The participants consisted of 362 (86%) males and 57 (14 %) females, with 3 per- sons not indicating their gender. Respondents with the title/rank of Professor (either

Full, Associate, or Assistant) represented 67% with visiting assistant professor and professor emeritus composing nearly 10%. Post-Doctorates made up approximately

22% of the participants. There were 35 respondents (8%) categorizing themselves as

82 “Other” rank and speciﬁed their title as either: Visiting Professor, Visiting Assistant

Professor, Senior Research Scientist, Research Scientist, Recalled Emeritus, Senior

Lecturer, or Instructor.

Overall, the mode interval age was 31-40, the mode rank was professor, and the mode interval for the number of years as a professor was 0−2 years. Table 4.3 shows demographic information sorted by respondents university rank (Professor, Associate,

Assistant, Post Doctorate, Emeritus, Other) and Figure 4.1 displays the distribution of participants by faculty rank. The ﬁrst numeric column speciﬁes the total number of respondents in each row category. For example, there were 362 male respondents, or 86.4% male. Relative percentages can be calculated for each demographic within and between groups (e.g., 41.2% of the male respondents were professors and 94% of the professors are male).

4.1.1 Computer Expertise and Experience

Results show that considerable amount of time is spent working with computers as more than one-quarter (25.6%) of the participants reported interacting with a computer for more than 30 hours per week and almost 60% spend more than 20 hours per week using a computer in some capacity. The mode interval for the number of hours per week using a computer was “30 or more hours.”

83 N Professor Assoc. Assist. Post Doc Emeritus Other Total 422 158 48 77 90 14 35 Gender Male 362 149 40 62 72 14 25 Female 57 9 5 15 18 0 10 Age 18 - 25 2 0 0 1 1 0 0 26 - 30 79 1 1 24 48 0 5 31 - 40 128 5 24 46 40 0 13 41 - 50 66 39 15 5 2 0 5 51 - 60 61 50 3 1 0 0 7 > 60 84 63 2 0 0 14 5 Years as Professor 0 - 2 108 3 7 47 45 0 6 3 - 5 55 6 13 27 4 0 5 6 - 10 30 8 17 3 0 0 2 11 - 20 43 40 3 0 0 0 0 21 - 30 46 41 3 0 0 1 1 > 30 74 59 2 0 0 12 1 Does not apply 64 1 0 0 43 1 19

Table 4.3: Participant Demographics

The results also indicate an overall high level of technical expertise. Approximately

40% of the participants selected 8 or above on a 10-point scale indicating level of expertise with computers. Only 20% rated themselves below a 5 on the 10-point scale. Furthermore, approximately 10% consider themselves as an expert computer user (i.e., 10), or a computer “guru” (See Figure 4.2). The overall mode of this distribution was 8 and the median was 7.5. Nearly identical distributions were found across university rank groups suggesting “newer” generation mathematicians are not more technologically savvy than the older generation mathematicians. Chi-square test

84 Figure 4.1: Distribution by Faculty Rank

veriﬁed that expertise level and faculty rank are independent (χ2 = 6.297, df = 6, p = 0.391). There was a moderate positive correlation between time spent per week using computers and level of computer expertise (ρ = 0.44) indicating a relationship between hours of use and expertise level.

The skew in the distribution could indicate either a biased sample or mathematicians are generally computer savvy. If a bias exists, it could be that mostly mathematicians with the propensity to use computer/software responded to the survey

85 Figure 4.2: Distribution of participants’ computer expertise levels on a scale from 1 to 10

whereas mathematicians that do not use software opted out of the survey. However even with this bias present, this report will not be severely compromised, but possibly improved since the main purpose is to consider how mathematicians that use software, use the software, and not to merely gather data indicating how many use software or not. Section 4 would suﬀer the greatest from the presence of this bias due to questions regarding attitudes toward software and its use(s), in particular in teaching and learning across all grade levels.

An e-mail from one potential participant, it was suggested that there might exist a systematic bias in the survey with those highly concerned about computer security excluded due to the use of JavaScript (used to navigate between sections). The

86 suggested bias was hypothesized, as the ones most concerned with computer security are also the most knowledgeable about computers. However, this concern is only a conjecture. It is also possible that less knowledgeable users have the same level, if not higher, concern for security, and heightened concern for security does not imply higher knowledge. In fact the results show the opposite to be true. According to the self-reported data, overall participants’ computer expertise ranks high, with a median

7 and mode 8.

4.1.2 AMS Mathematical Subject Indexes

Responses included at least one participant with interests from each AMS Math- ematical Subject Classification (MSC). The distribution of the participants among mathematical subject areas was not uniform (see Figure 4.2). Although most participants selected only one area of interest, it is common to have more than one area of interest. The field of partial differential equations (PDE), AMS Code 35, was well represented (n=62) in the survey; it was the mode of the distribution. Algebraic geometry had the second highest number of participants (n=59). Table 4.4 shows the frequency of the number of different AMS codes selected. The most frequent number of AMS codes selected is 1 (n=174), median is 1.5, and the maximum number selected was 16. The mean was slightly larger than 2, (2.38) with standard deviation of 1.87.

87 Figure 4.3: Distribution of participants’ by AMS MSC codes

88 Number of Subject Areas Selected 1 2 3 4 5 6 7 8 10 13 16 Frequency 174 98 73 31 17 11 5 3 2 1 1

Table 4.4: Frequency of the number of AMS codes selected

Table 4.5 shows the frequency distribution and percent distribution of participants

AMS MSC codes based on Rusins (2000) groupings. The results show more than 35% of the participants were from analytical areas. In particular, slightly more than 16% of the participants’ interest is diﬀerential equations.

Mathematical Area Frequency Percent Foundations 19 1.90 Algebraic areas 257 25.73 Geometric areas 131 13.11 Analytic areas 353 35.34 Real analysis 24 2.40 Complex analysis 30 3.00 Diﬀerential equations 163 16.32 Functional analysis 68 6.81 Numerical analysis 68 6.81 Probability and Statistics 45 4.50 Computer Sciences 33 3.30 Applied Sciences 121 12.11 Other 40 4.00 Note: Rusin’s Broad Subject Classiﬁcations

Table 4.5: Participants distribution based on Rusin’s Broad Subject Classiﬁcations

89 4.2 SOFTWARE SELECTION

One of the primary research questions asked which software do mathematicians use to facilitate mathematics including system software and application software.

Data for items in Section 2 (see Appendix D) were reported based on a 5-point scale representing extent of use: 5=Daily, 4=Weekly, 3=Monthly, 2=Yearly, and 1=Never.

Each question contained multiple independent items, however the items were not mutually exclusive (i.e., participants may report using more than one software type).

4.2.1 HTTP-USER AGENT and System Software

For each request made by a user to the server (e.g., visiting a home page or logging into a survey system), several key pieces of information were gathered (Smith, 2000).

In this study, the HTTP-USER AGENT was recorded which includes the participants operating system, browser type, and in many instances their micro processing hardware chip (e.g., Mac OS X Intel Chip vs. Mac OS X Power PC).

Table 4.6 reports the cumulative results of the contents of this variable for all participants. Table entries are listed as raw numbers so that totals of column and row entries are cumulated. Browser versions (e.g., Firefox 1.5.0.7) are not diﬀerentiated nor reported. Mac OS X (combining PPC and Intel) was the mode operating system

90 used (n=157) while participating in the survey. Firefox was the most popular browser, with 188 participants.

MS IE Firefox Safari Netscape SeaMonkey Others Total Mac OS X PPC − 25 99 3 − 5 132 Mac OS X Intel − 5 20 − − − 25 Mac OS 8/9 2 − − 2 − − 4 Windows XP 59 65 − 6 − 3 133 Windows ME 1 − − − − − 1 Windows 9x 1 − − 2 − − 3 Linux i686 − 78 − 1 14 11 104 Linux x86 64 − 4 − − 1 1 6 Linux PPC − 1 − − − − 1 Linux sparc64 − − − − − 1 1 Sun OS − 10 − − − 11 21 Sun OS i86pc − − − 1 − − 1 Total 63 188 119 15 15 32 432

Table 4.6: HTTP User Agent Variable showing Operating System and Browser

Besides the HTTP User Agent, participants were asked to identify their use of various operating systems by selecting from a non-mutually exclusive list. Of the operating systems in use, Windows XP, Mac OS X, and Linux were approximately equally distributed among participants’ responses, (n=209, 203, and 223 respectively). Sun’s

Solaris operating system was the fourth most frequently reported (n=56). Although the mode number of operating systems per participant was one, it is important to note the use of multiple operating systems by the participants. In fact, 57% (n=241) of the participants reported using multiple platforms (See Table 4.7). Figure 4.4

91 shows the various operating systems reported by the participants and recorded by the HTTP User Agent.

Frequency Percent 1 - Single Platform Users 179 42.62 Mac OS X 78 18.57 Mac OS 8/9 1 0.24 Linux 44 10.48 Solaris 7 1.67 Unix 3 0.71 Windows XP 44 10.48 Windows NT 1 0.24 Windows 2000 1 0.24 2 - Double Platform Users / Combinations 141 33.57 Linux / Windows XP 57 13.57 Linux / Mac OS X 38 9.05 Mac OS X / Windows XP 16 3.81 Solaris / Windows XP 9 2.14 3 - Triple Platform Users / Combinations 77 18.33 Linux/Mac OS X/Windows XP 30 7.14 Linux / Solaris / Windows XP 10 2.38 Solaris / Mac OS X / Windows XP 4 0.95 Note: Four or more operating systems not shown

Table 4.7: Users Number of Operating Systems and Combinations

More than 33% of the participants (n=141) use two operating systems, 18%

(n=77) reported using three, and 5% (n=23) indicated using four or more diﬀerent operating systems. The median number of operating systems used by participants was 2. Of the single platform users, Mac OS X had the largest percentage with

18.57% (n=78), while Windows XP and Linux were tied, each with 10.48% (n=44).

92 Figure 4.4: Operating Systems: Response vs. HTTP User Agent

However, of the double operating system participants, Linux−Windows XP was the most frequent pairing with 13.57% (n=57).

On the item regarding software distribution types−commercial, open-source, and freeware−overall open source software was used both most (93%) and most frequent

(65% daily use), with a weighted average of 4.28 (interpreted as, “at least weekly”; see Table 4.8). Freeware and/or shareware programs were reported as the second most frequently used, while commercial software was ranked the lowest daily use

93 (25.9%). Table 4.8 summarizes the frequency distribution, the mode of each item and an approximate median can be quickly obtained.

Percent N 5 4 3 2 1 M SD Commercial 409 25.9 26.9 11.49 21.5 14.18 3.28 1.41 Open Source 402 65.4 18.7 2.49 6.2 7.21 4.28 1.22 Freeware / Shareware 380 38.2 19.7 7.9 18.7 15.53 3.46 1.52 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.8: Frequency response of Software Distribution Types

Table 4.9 summarizes participants’ level of use on broad software categories. Ac- cording to the results, specialized mathematics software are used by 90% of the participants, approximately 18% daily, 26% weekly, 19% monthly, and 27% yearly. Pro- gramming languages are used frequently by approximately 20% of the participants and used, although less frequently, by another 44%. Daily use of spreadsheets were reported by 7.8%.

Of the various programming languages, C/C++ was reported as the language used most with approximately 54% using it throughout the year (See Table 4.10). The results also show C/C++ is used most frequently among the alternatives with nearly

6% participants report using C/C++ on a daily basis and approximately 17% use it monthly, if not weekly. FORTRAN, the traditional programming languages of choice

94 Percent N 5 4 3 2 1 M SD Programming Languages 406 8.13 11.6 12.07 32.3 35.96 2.23 1.27 Specialized math software 410 17.8 26.6 18.54 27.6 9.51 3.15 1.27 Statistical tools 400 1.25 1.00 4.00 15.3 78.5 1.31 0.71 Spreadsheets 410 7.8 33.2 19.51 22.7 16.83 2.92 1.24 Database / Database Mgn. 402 0.25 2.49 3.23 13.2 80.85 1.28 0.66 Hand-Held Calculators 399 1.25 1.75 1.5 13.5 82 1.26 0.69 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.9: Frequency Distribution of General Application Software

among mathematicians and scientist, was selected by 26% of the 369 participants with approximately 6% using it at least weekly. Other programming languages that were reported as being used frequently are: C#, Mathematica Programming, MATLAB programming, php, RealBasic, S, Scheme, awk (handles text and data ﬁles), shell scripting, R (free version of S statistical software), Mma (to study reaction-diﬀusion), and gp (Genetic Programming).

Using a cross-tab query, results show programming language use is not isolated to scientific fields. Participants using C / C++ were mostly from analytic fields such as differential equations, numerical analysis, and Fourier analysis, and few scientific

fields, namely fluid dynamics, biology, and natural sciences. However, results show some algebraic fields also use C/C++ (e.g., number theory, commutative algebra, and algebraic geometry). Fortran use is more sparse and isolated mostly to numerical analysis and differential equations.

95 Percent N 5 4 3 2 1 M SD Ada 357 0.00 0.00 0.00 1.4 98.6 1.01 0.12 BASIC 363 0.00 0.28 0.55 16.8 82.4 1.19 0.43 C / C++ 387 5.9 5.7 11.4 30.2 46.8 1.94 1.15 Fortran 369 2.16 2.98 4.61 16.26 74.01 1.43 0.88 Java 360 0.55 1.39 3.89 17.22 76.9 1.31 0.67 O’Calm 354 0.00 0.85 0.00 1.13 98 1.04 0.29 Perl 365 1.1 2.74 5.21 13.42 77.5 1.36 0.76 Python 359 0.56 1.67 1.95 9.47 86.4 1.21 0.61 Others 193 6.73 5.7 4.66 5.7 77.2 1.59 1.22 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.10: Frequency Distribution of Programming Language Use

Several specialized mathematical software choices were presented as items on the survey, among the list are three commercial software, MATLAB, Mathematica, Maple

(see Table 4.11). MATLAB had the highest frequency of use, 8% daily and 13% weekly. However 45% of the participants also indicated never using MATLAB (also the largest non-use among the top three). Although MATLAB has a symbolic toolbox add-on, (i.e., Maple Kernel), Maple and Mathematica are more comparable as they are both Computer Algebra Systems (CAS) and do symbolic manipulation. Among these top three, Maple had the highest reported percentage of use overall with a weighted average of 2.26. Twenty-one percent (21%) of the participants reported using MATLAB at least weekly, Mathematica 13% at least weekly, and Maple 20% at least weekly. Other software used by at least 10% of the participants−at some

96 point of the year−include: GAP (13%), Macaulay (13%), Magma (13%), and PARI-

GP (11.2%).

Percent N 5 4 3 2 1 M SD MATLAB 396 8.08 13.4 8.56 25 44.95 2.15 1.33 Mathematica 404 5.45 8.42 12.6 37.1 36.37 2.09 1.15 Maple 398 7.04 13.3 12.31 33.68 33.68 2.26 1.24 Cabri Geometry 381 0 0.79 0.79 2.89 95.54 1.07 0.35 GAP 382 0.26 2.88 2.09 7.59 87.17 1.21 0.64 Macaulay 382 1.05 0.52 2.62 9.16 86.65 1.20 0.61 Magma 381 0.79 2.62 2.36 7.09 87.14 1.23 0.69 Octave 380 0.53 1.05 2.36 4.74 91.32 1.14 0.54 PARI-GP 384 1.56 1.82 1.56 6.25 88.8 1.21 0.70 SAGE 380 0.53 1.05 1.58 2.89 93.95 1.11 0.51 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.11: Mathematical Software Usage

Software not included in the list but speciﬁed by participants in the Others category include: SnapPea, Singular, Axiom, Cocoa, Geometers Sketchpad, gnuplot, R,

S, Eigenmath, Mupad, Graphviz, Xﬁg, LIE, tecplot, MacMath, GeoGebra, JSCL,

Isabelle, Graphing Calculator, and Excel.

The selections are not mutually exclusive, for example, some participants use both Mathematica and Maple (daily, etc.). In particular, there were 241 participants reporting at least monthly use (3, 4, or 5 on the 5-point scale) of MATLAB, Mathe- matica, and Maple. Of these participants, the data show that there are 40 participants

97 that use both Maple and Mathematica, 54 use both and Maple and MATLAB, 42 use both MATLAB and Mathematica, and 21 use all three, Maple, MATLAB, and

Mathematica. The Venn diagram (see Figure 4.5) shows Maple with largest frequency of use, nearly one-third more exclusive users than Mathematica.

Figure 4.5: Venn diagram of frequent users of MATLAB, Mathematica, & Maple

Cross-referencing MATLAB, Mathematica, and Maple users with their ﬁeld(s) of study reveals that Maple is used by more participants in combinatorics, commutative

98 algebra, and algebraic geometry; Mathematica is used by more participants from partial diﬀerential equations and dynamical systems, MATLAB use is more prevalent in numerical analysis and computer science, probability theory, statistics, and biology and natural science. A Chi-square test on a subset of mathematical subject areas with greater than 20 responses distributed among the three software (MATLAB,

Mathematica, and Maple) did show a dependence relation (χ2 = 52.82, df = 34,

p = 0.02). Table 4.12 shows standardized residual values (i.e., Expected√ −Observed ). Expected

MATLAB Mathematica Maple 05 - Combinatorics -1.52 -0.07 1.60 11 - Number Theory -2.42 2.33 0.40 14 - Algebraic Geometry -1.89 1.29 0.78 34 - Ordinary Differential Equations 0.59 -0.62 -0.07 35 - Partial Differential Equations 0.63 -0.41 -0.29 37 - Dynamical Systems 0.07 -0.62 0.46 53 - Differential Geometry -1.05 0.79 0.36 55 - Algebraic Topology -0.95 0.53 0.49 57 - Manifolds/Cell Complexes -1.30 1.85 -0.31 58 - Global Analysis -0.84 0.65 0.28 60 - Probability 1.40 -0.70 -0.80 62 - Statistics 1.35 -0.23 -1.17 65 - Numerical Analysis 1.72 -0.99 -0.87 68 - Computer Science 0.47 -0.07 -0.41 76 - Fluid Mechanics 0.99 -1.43 0.25 92 - Natural Sciences 1.43 -1.18 -0.42 97 - Mathematics Education 0.15 0.30 -0.41

Table 4.12: Standardized Residuals of Chi-square test−Software vs. AMS MSC

99 Speciﬁc software such as LaTeX or general software types (e.g., HTML/Internet) used for communication or discourse and the percentage of their weighted use are listed in Table 4.13 in the same order as it appeared on the survey. LaTeX is used most, and most often, with a weighted average of 4.50 and standard deviation of 0.97.

Percent N 5 4 3 2 1 M SD TeX / LaTeX 408 70.10 20.10 2.70 3.68 3.43 4.50 0.97 Word processors 403 18.86 19.4 10.7 18.9 32.3 2.73 1.54 Scientiﬁc Workplace 389 1.03 2.06 4.03 4.63 91.3 1.16 0.64 MathType 385 2.34 4.16 2.08 3.9 87.5 1.29 0.89 HTML / Flash / Web 398 13.07 33.7 14.8 12.56 25.9 2.95 1.42 Internet Newsgroups 386 5.96 10.1 8.29 16.06 59.6 1.87 1.27 Word2TeX 388 0.26 0.26 0.26 4.9 94.3 1.07 0.34 Matlab Report Generator 383 0 0 0.26 3.39 96.3 1.04 0.21 Maple HTML / MapleML 388 0 0.52 1.29 9.02 89.2 1.13 0.41 Other(s) 140 10 3.57 2.86 2.14 81.4 1.58 1.31 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.13: Technologies used in Mathematical Discourse

Some participants selecting the “other” speciﬁed they use Wikis, Google Chat,

Scheme to generate HTML, WinEdt, Keynote, LaTeXIT, StarOﬃce, TeXPoint, Macro-

Media Freehand, OpenOﬃce, and Sketchpad to JavaSketchpad. In summary, LaTeX, typesetting software for mathematical formula (similar to HTML in the sense of a mark-up language) is the most frequently used. Over 70% of the participants use it daily and 20% at least weekly. Word processors and HTML/Web-technologies are

100 also frequently use technologies by the participants, approximately 38% and 45%, respectively. Internet message boards, newsgroups (e.g., sci.math), or other electronic mathematical forums were reported being used at least weekly by 16%.

Materials mathematicians use to support and facilitate their work are described in

Table 4.14. Printed references, such as books, were reported as used frequently by 90% of the participants. However, electronic reference (e.g., pdfs) are used as frequently by only slightly smaller number of participants (88%). Web references were also reported as frequently used, approximately 85% at least weekly. However, similar to the MapleSoft report (2006), paper and pencil is still the most often used tool to support mathematics. Mathematicians found hand-held calculators signiﬁcantly less useful with only 16% indicating weekly use.

Percent N 5 4 3 2 1 M SD Printed references 406 64.29 26.85 4.93 2.22 1.72 4.50 0.83 Electronic reference 404 63.86 24.5 3.47 5.9 2.23 4.42 0.97 Web reference 407 57.25 28.7 7.13 3.69 3.19 4.33 0.96 Software “Help” ﬁles 392 12.5 25.8 20.9 23.72 17.1 2.93 1.29 Hand-held calculators 392 5.1 11.5 5.1 29.34 49 1.94 1.21 Paper and pencil 403 91.56 4.47 1.49 0.99 1.49 4.84 0.64 Other(s) 21 61.9 9.52 9.52 9.52 9.52 4.05 1.43 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.14: Technological Support Material

101 4.3 SOFTWARE ACTIVITIES

Participants indicated their level of use of software for higher-order thinking skills outlined by Presseisen (1985). Problem solving was selected over decision-making, critical thinking, or creative thinking as the higher-order thinking skill for which software was most useful. Thirty-ﬁve percent (35%) reported using software for problem solving either daily or weekly. Table 4.15 reports the number of participants answering each item, the percent selected for each category (daily, weekly, etc.), the weighted average, and the standard deviation of each item.

Percent N 5 4 3 2 1 M SD Problem solving 382 16.00 19.00 14.00 21.00 30.00 2.70 1.47 Decision-making 371 6.50 8.60 9.40 26.00 49.06 1.97 1.23 Critical thinking 372 8.10 8.90 11.00 21.00 51.08 2.02 1.31 Creative thinking 367 11.00 9.80 10.00 26.00 43.32 2.19 1.37 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never

Table 4.15: Software usage for higher-order thinking skills

Software for communicating mathematics was reported by 375 of the 394 participants (95%), by far the largest percentage for any single task. More than half of the responses (54.3%) use software daily for communication purposes and more than three- quarters (79.9%) at least weekly. Teaching and presenting results, sub-categories

102 of communicating, also yielded large percentages for usage. Other tasks with high

percentage of daily responses include: gaining insight, computational reasoning and

calculation, visual representation and reasoning. Tasks with lowest percentage of use,

daily or otherwise, were logical induction and deduction, proof automation, creat-

ing new meanings, ﬁnding similarities or diﬀerences, verifying analytical results, and

visual reasoning.

It can be seen from Table 4.16, by subtracting Column 7 (1 =never) from 100

to calculate percent of users, that a majority of participating mathematicians do use

software−at least occasionally−for all tasks listed except: (a) proof checking and automation, (b) detecting diﬀerences/similarities, (c) creating new representations,

(d) logical induction or deduction, (e) making predictions, (f) verifying analytical results, and (g) visual reasoning. Alternatively, items with mean less than 2.0 are approximate cut-off for “majority of users” classification. For example, software for data analysis (mean 2.15) is used to some extent by approximately 54% (100% - 46%) of the 380 respondents while using software for finding difference and similarities

(mean 1.94) is only used by approximately 44% of the 370 respondents.

There was no signiﬁcant relationship found between mathematical software selection and mathematical activities (χ2 = 11.26, df = 8, p = 0.18). In fact, the results

indicate mathematical activities are independent of the software used. However, this

103 result is likely because of the design of the questionnaire did not link software selection to particular activities.

Percent N 5 4 3 2 1 M SD 1. Checking proofs 385 4.16 7.79 8.83 16.1 63.13 1.74 1.16 2. Communicating mathematics 394 54.3 25.6 10.7 4.8 4.57 4.2 1.10 3. Computational reasoning 381 13.4 17.8 15.5 17.59 35.7 2.56 1.46 4. Visual representations 388 11.6 21.4 30.4 17.27 19.3 2.89 1.27 5. Data analysis 380 7.11 13.2 13.2 20.26 46.3 2.14 1.32 6. Find diﬀerence/similarities 370 7.3 9.19 10.3 16.49 56.8 1.94 1.30 7. Discover new patterns 383 7.31 12.5 17.5 23.76 38.9 2.26 1.29 8. Falsify conjectures 376 3.99 9.57 16.5 23.4 46.5 2.01 1.17 9. New meaning/representation 365 3.84 7.95 7.67 20.55 60.00 1.75 1.13 10. Gain insight/intuition 381 12.3 17.6 20.7 22.83 26.5 2.66 1.36 11. Logical deduction 366 2.19 3.83 4.37 12.57 77.00 1.42 0.91 12. Logical induction 365 2.47 5.48 4.38 12.05 75.6 1.47 0.98 13. Making predictions 367 5.99 11.2 13.1 19.07 50.7 2.03 1.28 14. Perform experiments 373 8.58 18.5 13.9 19.3 39.7 2.37 1.38 15. Perform calculations 383 16.4 22.2 19.3 20.37 21.7 2.91 1.40 16. Perform simulations 371 9.43 10.5 10.2 21.56 48.21 2.11 1.36 17. Presenting results 375 16.8 25.1 27.5 12.27 18.4 3.10 1.33 18. Proof automation 363 0.55 0.83 1.65 8.26 88.72 1.16 0.53 19. Searching for solutions 373 7.51 10.5 13.1 22.52 46.40 2.10 1.30 20. Symbolic manipulation 371 8.63 13.7 17.5 25.34 34.81 2.36 1.31 21. Teaching & Instruction 378 18.5 29.1 20.4 21.16 10.8 3.23 1.28 22. Verify analytical results 368 5.71 9.24 10.6 19.84 54.64 1.92 1.24 23. Visual reasoning 368 7.07 7.07 10.9 18.48 56.52 1.91 1.26 24. Other(s) 75 4 0 0 0 96 1.16 0.79 *Note: 5 =Daily, 4 =Weekly, 3 =Monthly, 2 =Yearly, 1 =Never **Item number was inserted for reference in the Dendrograms

Table 4.16: Frequency of Software Utilization for Mathematical Activities

104 Cluster analysis was used to combine items from Section 3 Question 1 of the questionnaire (see Appendix D) to determine which activities are similar and to simplify the analysis to create proﬁles (see Figure 4.6). Clusters were formed based on similarity in the responses using Ward’s linkage method. The result of this analysis grouped the 23 software activity items into 5 clusters with a similarity greater than

70%. Participants’ data were categorized based on these derived clusters. Cluster

1 consisted of response items 1 and 18 (proof checking and automation). Cluster 2

(communication cluster) included items 2, 4, 17, and 21. Cluster 3 was composed of response items 3, 6, 7, 10, 13, 14, 19, 20, 22, and 23. Cluster 4 contained response items 5, 15, and 16, while Cluster 5 had response items 8, 9, 11, and 12. Cluster 3 was the largest cluster with 10 activities (See Table 4.17 and Figure 4.7).

Cluster 3 contains 10 activities/purposes. Further analysis identiﬁes clusters within this cluster to be (similarity > 80%): items (3, 19, & 20), (6 & 7), (10,

13, & 14) and (22 & 23). Figure 4.7 shows the dendrogram with Ward’s linkage with the clusters from Cluster 3.

The clusters were renamed to reﬂect a general composite task. Clusters 1-5 are renamed to: Proof, Communication, Experimental Mathematics, Data Analysis, and

Logic, respectively.

105 Figure 4.6: Dendrogram of the results of clustering similar activities.

Table 4.18 shows the standardized residuals of the chi-square (χ2 = 51.39 df=16, p < 0.001 ) test for independence between general ﬁeld of study16 and the categories

resulting from the cluster analysis. These residuals show a signiﬁcant diﬀerence be-

tween ﬁeld of study and activities for which software is utilized. The results show,

with respect to software, that branch of mathematics and mathematical activities

are not independent. Diﬀerent areas of mathematics use software for diﬀerent ratio-

nale. For instance, there were 19 more responses than expected in Cluster 3 among

16General areas deﬁned by Rusin (2000). See Section 3.3.2

106 Item Number Response Item Group Mean Cluster 1 1.45 1 Checking Proofs 18 Proof Automation Cluster 2 3.36 2 Communicating Mathematics 4 Visual Representation 17 Presenting Results 21 Teaching & Instruction Cluster 3 2.21 3 Computational Reasoning 6 Finding Diﬀerences & Similarities 7 Discover new patterns 10 Gain insight 13 Making Predictions 14 Performing experiments 19 Searching for solutions 20 Symbolic Manipulation 22 Verify analytical results 23 Visual reasoning Cluster 4 2.39 5 Data Analysis 15 Perform Numerical Calculations 16 Perform Simulations Cluster 5 1.66 8 Falsify conjectures 9 Create Novel Meanings 11 Logical deduction 12 Logical induction

Table 4.17: Cluster Deﬁnitions

algebraists, but 9 less than expected in Cluster 1 among the same group. Geometric

fields use software (for communication, visual representation, presentation, and/or teaching) more than other fields. Significant residuals, (absolute value greater than

2) are shown in bold face.

107 Figure 4.7: Dendrogram of the results of re-clustering Cluster 3 activities.

Algebra Analysis Geometry Applications Other Proof 0.65 -0.03 -1.16 1.18 -0.77 Communication -0.51 -2.23 4.09 -0.67 0.98 Experimental Mathematics 1.07 0.45 -1.22 0.14 -1.09 Data Analysis -2.94 2.74 -2.47 0.52 1.31 Logic 2.37 0.29 -2.41 -0.24 -1.06

Table 4.18: Standardized Chi-square residuals between Activities and Subject Areas

Factor analysis was performed to support the cluster analysis. Table 4.19 shows the loading coeﬃcients for each item using equimax rotation to force highest loadings on a single factor. The largest absolute value for each activity item is bold faced.

108 Figure 4.8: Raw residuals: Clusters vs. Subject Area

Table 4.20 shows the grouping of mathematical activities based on the results of the factor analysis. Although the groupings are more equally distributed, this result complements the Clusters derived above. Cluster 1 (Proof) corresponds to Factor 4 and Cluster 2 (Communication) corresponds to Factor 5. Some items from Cluster 3

(Experimental Mathematics) were split among Factors 1 and 3. Cluster 5 (Logic) is most similar to Factor 2.

Table 4.21 shows MATLAB has more than expected users associated with activities contained in Data Analysis and Mathematica and Maple have less than an

109 Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Checking Proofs 0.301 0.174 0.011 -0.740 0.033 Communication 0.027 0.049 -0.245 0.012 0.826 Computational Reasoning 0.647 0.110 0.279 -0.284 0.268 Visual Representation 0.283 -0.014 0.336 -0.227 0.592 Data Analysis 0.250 0.239 0.632 -0.186 0.202 Find Diﬀerences/Similiaries 0.365 0.496 0.473 -0.195 0.117 Descover New Patterns 0.638 0.470 0.237 -0.147 0.110 Falsify Conjuctures 0.707 0.366 -0.039 -0.262 0.039 New representations 0.331 0.641 0.175 -0.313 0.077 Gain insight 0.677 0.216 0.432 -0.202 0.242 Logical deduction 0.085 0.767 0.065 -0.392 0.174 Logical induction 0.149 0.822 0.116 -0.285 0.154 Making predictions 0.444 0.525 0.479 -0.102 0.116 Performing experiements 0.514 0.324 0.555 -0.108 0.186 Perform numerical calculations 0.460 0.118 0.624 -0.071 0.230 Perform simulations 0.131 0.100 0.722 -0.320 0.205 Presenting results 0.108 0.102 0.255 -0.091 0.686 Proof automation -0.073 0.162 0.073 -0.763 0.090 Searching for solutions 0.505 0.445 0.343 -0.211 0.179 Symbolic manipulation 0.538 0.205 0.257 -0.386 0.220 Teaching and Instruction -0.235 0.196 0.438 -0.053 0.455 Verify analytic results 0.269 0.380 0.379 -0.455 0.169 Visual Reasoning 0.105 0.390 0.449 -0.405 0.228

Table 4.19: Factor Loadings of Mathematical Activities

expected number of users associated with activities from the same cluster. However, based on a Chi-square test, software and clusters are independent (χ2 = 11.26, df = 8, p = 0.187).

110 Item Number Response Item Factor 1 3 Computational Reasoning 7 Discover New Patterns 8 Falsify Conjectures 10 Gain insight 19 Searching for solutions 20 Symbolic manipulation Factor 2 6 Find Diﬀerences/Similiaries 9 New representations 11 Logical deduction 12 Logical induction 13 Making predictions Factor 3 5 Data Analysis 14 Performing experiements 15 Perform numerical calculations 16 Perform simulations 23 Visual Reasoning Factor 4 1 Checking Proofs 18 Proof automation 22 Verify analytic results Factor 5 2 Communication 4 Visual Representation 17 Presenting results 21 Teaching and Instruction

Table 4.20: Groupings Based on Factor Analysis

4.3.1 Software for Proof

Literature has shown that software has aided mathematicians in proofs (Borwein

& Bailey, 2004). Figure 4.9 presents a histogram of reported responses identifying the

111 MATLAB Mathematica Maple Proof -0.70 0.88 0.00 Communication -0.08 -0.49 0.52 Experimental Mathematics -0.71 0.97 0.06 Data Analysis 2.09 -1.49 -0.92 Logic -0.58 0.44 0.38

Table 4.21: Standardized Residuals of Chi-square test−MATLAB, Mathematics, & Maple and Activity Clusters

ways software had aided participants in mathematical proofs. Thirty percent have reported computers have checked large sets of data in aiding their proofs.

Of the 556 selected responses on this item, there were 25 (4.5% of the total) selecting some “other” way software has aided in proof. The details of the 4.5%

“Other(s)” category are listed verbatim below. One of the 25 was removed, as it was

Never, and therefore combined in the “Has NOT” category.

• By allowing me to express the proof in written form

• By automating symbolic computations used in proofs

• By checking formulas

• By providing approximate solutions

• By solving small cases of the problem

• By suggesting the extremal solution, which I then verify by hand

• By suggesting what is true

• By weeding out incorrect conjectures, by showing me an expected pattern

• Carrying out otherwise intractable calculations

• Check computation

112 Figure 4.9: Ways computers have aided mathematicians with proofs

• Check one or two examples

• Checking/aiding intuition, and in calculating an approximation

• Computing examples

• Double checking

• Helping to calculate / simplify expressions quickly

• Intuition building

• Numerical experiments

• Numerical proofs via interval arithmetic

• Providing experimental evidence for conjectures and suggesting proofs

• Providing numerical evidence for truth or falsehood of conjectures

• Simplifying a computation using Mathematica

113 • Suggesting conjectures

• Suggesting counterexamples

Although approximately 7% reported that there was nothing that limited their use of software, twenty-one percent (21%) said it was because the lack of necessity

(15%) indicated a lack of expertise limited their use, and (18%) cited a lack of time to learn the software. Thirteen percent (or approximately 13%) point to a lack in performance capabilities in either hardware (6%) or software (7%) and 3.5% referenced a lack of availability. The lack of ﬁnancial resources to obtain software accounted for about 4% of these rationales. Two individuals acknowledged simply a lack of interest, although they too cited lack of necessity.

4.4 ATTITUDES, BELIEFS, AND PREFERENCES

Section 4 of the survey gathered data about mathematicians’ opinions about technology in mathematics and mathematics education. In particular, these data indicate how important participants believe software/technology is to various aspects, levels, and courses in mathematics. Furthermore, two questions explored participants’ computer interface preferences. Concluding this section were four open response questions allowing participants to elaborate on items relating to technology in mathematics and

114 mathematics education.

4.4.1 Interface Preferences

Overall, forty-three percent (43%) preferred Graphical User Interfaces (GUI) to

27% Command Line Interfaces (CLI), and 29% were unsure of their preference. Of the participants that were sure (n=262), 61% compared to 39% preferred GUIs. The relative high percentage of participants preferring CLIs might be explained by the responses to their preference to one-dimensional input instead of two-dimensional input or WYSIWYG - What You See Is What You Get, (pronounced - “Wiz E Wig”).

Nearly half (48%) indicated their preference to one-dimensional input compared to about 22% preference of two-dimensional input. In particular, the diﬀerence is in the way the formula or equations are entered into the software. For instance, the volume of a sphere in one-dimensional input format may be entered as: (1) in MATLAB syntax:

V = (4/3)*pi*r^3 or (2) in LaTeX syntax:

V=\ frac{4 \ pi}{3} r^3}

The same formula entered using a WYSIWYG two-dimensional template, such as

Equation Editor or recent releases of Maple, would be entered using a template where

115 for the right hand side of the equation, the user would select a “fraction” template, select the numerator and place the symbols “4” and “π,” select the denominator and place the symbol “3,” then type “r” and select a superscript tool option to place the

“cube.” The output form is identical to the input form (i.e., WYSIWYG).

4π V = r3 3

Figure 4.10 shows participants’ preferences for computer interface (i.e., command line interface, CLI or graphical interface, GUI) and mathematical formula input method (i.e., one dimensional input or two dimensional input). The results indicate preference toward GUI and 1-dimensional input methods.

4.4.2 Beliefs about the Importance of Technology

A majority of participating mathematicians indicated that technology was significantly important to the ﬁeld of mathematics (78%), by either selecting a 4 or 5 on the Likert-type scale. Additionally, participants indicated a high level of importance to their speciﬁc area(s) of research (62%). Only 7.5% responded that technology has little to no importance to mathematics, but 22% indicated technology has very little or no importance to their area(s) of research.

Participants rated technology more important to teaching than to learning with a weighted average of 3.69 to 3.47 respectively. Table 4.22 shows that approximately

116 Figure 4.10: Interface and Input Preferences

59% rated technology “important” (4) or “very important” (5) to teaching, while learning received only 49% rated similarly. Slightly more than 5% indicated that technology was not at all important to learning mathematics.

The results also show the participants believe that technology becomes increasingly important from primary to secondary, and from secondary to tertiary level as indicated in Table 4.22. Furthermore, at the tertiary level the increase continues from undergraduate to graduate level, although there are still slightly more than 5% that reported technology is not at all important even in graduate school.

117 Percent N 5 4 3 2 1 M SD Mathematics 386 43.26 34.71 14.52 6.48 1.04 4.13 0.96 Speciﬁc area(s) of research 385 35.58 26.5 15.8 15.62 6.49 3.69 1.27 Mathematics teaching 385 26.23 33.2 26.51 11.43 2.61 3.69 1.06 Mathematics learning 383 21.15 28.7 31.63 13.31 5.22 3.47 1.12 Primary school mathematics 368 8.42 10.9 29.9 29.9 20.91 2.56 1.17 Secondary school mathematics 368 10.05 20.4 36.71 20.94 12.00 2.96 1.13 Undergraduate mathematics 374 19.25 31.8 34.49 9.89 4.55 3.51 1.05 Graduate mathematics 377 25.46 33.2 24.11 12.22 5.04 3.62 1.13 *Note: 5 =Very, 4 =Moderate, 3 =Somewhat, 2 =Not Very, 1 =Not at All

Table 4.22: Beliefs about the importance of technology to Mathematics and Mathe- matics Education

Although participants indicated rather strongly the importance of technology to mathematics (and their area of research), ratings were considerably lower overall on all items related to mathematics education (i.e., teaching, learning, primary, secondary, and tertiary mathematics education). But participants feel that technology is more important to teaching mathematics (59%) than learning mathematics (49%).

Moreover, less than 20% of the responses rated technology of signiﬁcant importance to primary school mathematics education with a slight increases in importance from primary (19%) to secondary (30%), and from secondary (30%) to tertiary (either 50% or 58%)

118 Figure 4.11 shows the distribution of the responding mathematicians beliefs regarding technology in primary and secondary school mathematics. This figure displays that participants believe technology has increasing significance at higher-grade levels in primary and secondary school mathematics. By Grade 12, about 12% of the responses indicate technology should play a significant role in mathematics curriculum.

Figure 4.11: Year technology should play a role in K-12 mathematics education

119 Figure 4.12: Year technology should play a role in collegiate mathematics education

120 Figure 4.12 is an extension of Figure 4.11 and shows the percent distribution of mathematicians beliefs about which year technology should play a signiﬁcant role in collegiate mathematics. Although this distribution is more uniform than primary and secondary school mathematics distribution, there is still a slight increasing trend.

There is less variance, and thus more uniform agreement regarding technology at the tertiary level.

Approximately 50% reported high importance of technology in undergraduate mathematics education. For “service courses” (i.e., algebra, trigonometry, and calculus), extending to courses that typically compose the core of a mathematics degree, participants rated differential equations as the course for which technology plays the largest role. Approximately 25% reported technology playing the highest level of significance to differential equations, and 43% second highest, combining to 68% (see

Table 4.23 ). Lower level courses, (e.g., college algebra and trigonometry) had approximately 5% of the participants rating technology as being very signiﬁcant.

Number theory, discrete mathematics, linear algebra, and calculus also received large percentages indicating technology as signiﬁcant to the course (approximately

36%, 51%, 53%, and 45% respectively). Table 4.23 shows a summary of the relative frequency in response, the highest and lowest means are in bold. It is interesting to

121 note that the two of the lowest weighted averages are in courses in which mathematics education has pushed the technological agenda (i.e., often TI-83 calculators are required for these courses - at least at one large Midwestern university).

Percent N 5 4 3 2 1 M SD College Algebra 368 5.16 14.91 32.1 32.93 14.9 2.62 1.07 Trigonometry 369 5.15 19 35 26.32 14.53 2.74 1.08 Calculus 372 9.95 34.7 40.01 10.77 4.57 3.35 0.95 Diﬀerential Equations 372 24.73 43.3 21.5 5.65 4.84 3.77 1.03 Linear Algebra 371 15.63 38.02 28.8 11.32 6.23 3.45 1.07 Abstract Algebra 367 2.18 14.21 39.22 33.5 10.89 2.63 0.93 Real Analysis 369 1.63 13.79 34.96 39.61 10.01 2.58 0.91 Complex Analysis 372 3.49 16.41 37.11 32.5 10.49 2.70 0.98 Discrete Mathematics 369 15.72 35.5 19.79 10.8 18.19 3.21 1.33 Number Theory 366 9.02 27.6 23.52 15.32 24.54 2.81 1.32 *Note: 5 =Major, 4 =Moderate, 3 =Minor, 2 =None, 1 =Do Not Know

Table 4.23: Mathematical course which could beneﬁt from technology

Seventy-two percent (71.7%) of the responding mathematicians agree, to varying degrees (e.g., strongly agree, agree, or somewhat agree) that the mathematics profession is taking full advantage of modern tools and technology, (i.e., 28% disagree).

That is, twenty-eight percent of the participants think that mathematics could make better use of technology.

122 4.4.3 Beliefs about Collegiate Software Use

Participants were asked their opinion about which software undergraduates and graduate students should learn in Section 4 Question 8. The question should have included, as it was the intention of the researcher, the word “mathematics,” and should have read “...which software applications should undergraduate and graduate mathematics students learn?” According to the responses, the majority interpreted the question as intended.

At least 80% of the responses included a subset of MATLAB, Maple, Mathemat- ica, LaTeX, and C/C++, where the subsets varied depending level of study (i.e., graduate or undergraduate) and area(s) of interest. Broad recommendations for collegiate mathematics students were basic knowledge and skill of a computer algebra system (CAS) (e.g., Maple or Mathematica), a programming language (e.g., C++), a statistical package, a spreadsheet program, basic HTML for web development, and typesetting software (i.e., LaTeX) for communication. “TeX (or LaTeX is really best these days”) was strongly emphasized and included in nearly all subsets. Moreover, it was the only singleton that was indicated as required, essential, or a “must learn”

(in particular for graduate mathematics students).

123 Learning a high-level programming language was recommended in a large percentage of responses. The most common speciﬁc listings included C/C++, Fortran, or

Java.

Other recommendations for mathematics graduate students (depending on area of study) were Macaulay2, python, SAGE, GAP, MAGMA, and SAS or Minitab for statisticians. For a scientiﬁc ﬁeld, it was recommended learning at least one of the following: MATLAB, MathCAD, and Mathematica or Maple. A smaller percentage of responses reported software usage for undergraduates was not useful and reported undergraduates do not need to learn any mathematics software.

4.4.4 Beliefs about the Role of Technology in Education

The open-ended question read, “In your opinion, with respect to technology, what should the education system do to better prepare mathematics students to meet the challenges of the future?” Many responses highlight mathematicians’ passion opposing, and contempt for technology used in K-12 education. Although varying in degree, a common sentiment or theme emerged: use less technology in mathematics education

(n=36, 16%). The range in this theme covered “do not overuse” to “do not use.” Fur- thermore, the responses indicate that mathematicians believe mathematics education is concerned only with K-12, as the majority seems to imply not to use technology,

124 even though the responses to the prior question were more favorable toward using technology, in some cases “essential,” at higher levels.

Besides this theme, the second notable theme that emerged was for students to learn a programming language (see also Section 4.4.6). Table 4.24 lists recommendations for, and opinions about mathematics education regarding technology use. The data were categorized in common themes and both the frequency and corresponding percents are reported. Large portions of the responses suggest limiting the use of technology (rows 3 and 4). Programming, or teaching programming, was the second most frequent recurring theme suggested by mathematicians, one that is not as prevalent as it once was in the curriculum (at all levels).

4.4.5 Beliefs About the Future of Technology in Mathematics

Section 4 Question 10 of the questionnaire asked what technological advancements were anticipated that will have a direct impact on respondents mathematical work.

Participants’ beliefs about future advancements in technology having direct impact on their mathematical work include an increase in a computer’s central processing unit (CPU), speed through multi-processors (e.g., dual-, triple-, etc.- “cores”), digital libraries and more online availability of literature (e.g., electronic journals), software

125 Main Sentiment Frequency Percent Integrate / add computer lab 23 12.64 Teach Programming 22 12.09 Do not overuse technology 22 12.09 Do not use 16 8.79 Provide support and training 14 7.69 Teach advantages, possibilities, and pitfalls of technology 12 6.59 Teach appropriate use of software 6 3.29 Train Professors to use technology 4 1.82 Re-think curriculum around technology 2 0.91 Use technology to increase critical thinking 3 1.36 Give students experience with software / earlier 7 3.18 Theory ﬁrst, software second 4 1.82 Justify mathematics using computers 2 0.91 Use technology for experimentation and creative thinking 6 2.73 Supplement lessons with technology 8 3.64

Table 4.24: Opinion Themes about Technology in Mathematics Educations

development for topics such as abstract algebra, increase in user-friendliness and intuitive software, hypothesis checking software and better automated provers/veriﬁers, wider availability of integrated software (e.g., SAGE), and an increase in Internet software (e.g., Google Book to connect areas of knowledge). Responses were labeled and categorized accordingly.

A speciﬁc comment−useful for software makers−that illustrates a limitation on mathematicians without advancement of technology is:

I hope to see a commutative algebra package as powerful as MacCaulay II but with the easy interface of commercial symbolic computation software. I have often avoided computational exploration of commutative algebra problems because the coding eﬀort would be too extensive.

126 There is less variability in the responses about future technologies, with the majority citing either an increasing in “power/speed” of computers/Internet and/or major advancements in communication. The largest anticipation of the future (33%), with respect to technology in mathematics, reported by the participants is dramatic increases computing power. Comments were both general, “faster Central Processing

Units (CPUs),” and speciﬁc, “faster computers will lead to faster simulations/parallel simulations, ability to check more cases, and more extensive experimentation that will yield insight.” Additionally, faster computers will be both the means and the ends,

“[t]he emergence of algebraic structures, which are so complex that they can only be manipulated with help of a computer.”

Beyond the consequences of Moore’s Law17 (1965), most responses center on communicating mathematics. More than one-ﬁfth (22%) of the responses look toward advancements in communication technology over the next 10 years. Better video-conferencing, virtual whiteboards, e-Ink, portable “ePaper,” electronic libraries,

Google Books to connect disparate areas of knowledge, and e-Journals were reoccur- ring suggestions. One participant “wishes” for holographic telephony expressing: “the ability to virtually visit a colleague instantaneously anywhere in the world and have a reasonable approximation to instantaneous face-to-face contact would be the biggest

17“Technology” doubles every 18-24 months.

127 boon to mathematics since e-mail.” A common expression was for the ability to typeset mathematics on the Web (i.e., LATEX to HTML). Other insightful comments were: • Proof veriﬁcation software will have a signiﬁcant impact on the dissemination of Mathematical knowledge.

• More expressive programming languages.

• My own software will enable me to model the systems I study, so that I can check hypothesis.

• Communication software (Arxive, Numdan, GDZ). Electronic Whiteboards.

• More centralized access of publications.

• Email is an indispensable tool; mathematics has become a much more collaborative discipline than years ago. Email has enabled greater communication from a group of introverts.

• Faster speed for abstract algebra computations.

• Integration of symbolic computation and numerical simulation. Basically, adding MATLAB simulation tools to scientiﬁc workplace.

• More sophisticated linear algebra solvers.

• Videoconferences.

• Improvements for abstract algebra computations

• TeX → HTML for e-mail and newsgroups.

• Multicore / multiprocessor systems will be common place. Thousand cores on a chip. Cluster cores.

• Computers for 4 dimensional modeling and visualization

• Systems that will beneﬁt from the interplay between numerical, probabilistic, and purely symbolic methods.

• Translation of symbols and voice recogition.

• Uniﬁed platforms for exploring multifaceted problems (e.g., SAGE)

• Commutative algebra package as powerful as MacCaulay II but with an easy interface. I have avoided computational exploration of commutative algebra problems because the coding eﬀort would be too extensive.

128 4.4.6 Additional Comments

Question 11 of Section 4 was a free response item aﬀording participants an opportunity to provide additional comments about any (technological) issue(s) in mathematics or mathematics education, or concerns that the previous items of the questionnaire might not have covered. There were 115 responses (see Appendix E), many reiterating or re-conﬁrming previous open responses. In particular, statements that technology is “over used,” “math education relies too much on technology.” “students are overly dependent on technology.” “it can be overused,” and “technology is overrated” were common and is obviously an opinion held be many mathematicians (see

Section 4.4.9). For instance one participant wrote:

Computer techniques are, of course, useful for certain areas of mathematics, often for ad hoc reasons, in that they make available for inspection and resulting conjecture examples that are too complicated or tedious to work out “by hand.” But this is an opportunistic technique, and there are large swaths of mathematics where they simply don’t apply.

In basic math education, however, computer skills have been greatly over- emphasized, to the detriment of student’s absorption of basic ideas. For instance, despite my protests, a MAPLE component is included in my multivariable calculus courses. The result, as far as I can see, is that students are distracted from basic and diﬃcult ideas by the technicalities of MAPLE. The use of the latter doesnt even include graphic methods that might elucidate certain ideas, e.g., alternative coordinate systems for volume integrals, or vector ﬁelds and line integrals.

Philosophically, I think it would be much better for students to master the mathematics ﬁrst, and only subsequently take a dedicated course on computer methods in calculus, diﬀerential equations, and linear algebra. The latter might be better taught in a computer science department though the mathematics department could certainly handle it.

129 Other participants were even more critical of using technology and believe that it should never be used in mathematics education, writing: “calculators in schools should be forbidden,” “get calculators out of the hands of kids under 10,” and “it is better to keep calculators away from kids.”

The basis for this opinion, provided by the participants, seems to be that the technology “is not helping students to think critically,” “technology...makes students less likely to think about the material but to try simply to do brute force computations, leading to less understanding,” “educators often regard technology as a magic bullet that can replace thinking,” and/or technology is used as a crutch, replacing thought, and substitutes the ability to perform basic tasks (e.g., adding fractions was often cited). Some participants believe technology is less important in primary and secondary school or beginning collegiate mathematics, but is useful in more advanced courses (e.g., “there is no need for math students to use a computer until they get to graduate school”). One participant responded that “after the students learn to think, then the computer is useful to amplify their thoughts,” and another elaborated:

First you must learn to think. And thinking is like any strenuous exercise, it has to be nurtured and rewarded for it to continue. This is why technology has little role in the formative years (K-12), while in more advanced courses, it begins to integrate better with the thought processes (visu- alizations) and even to show Mathematicalapplications, which motivates students to study further, harder and deeper.

130 Negative opinions about technology used in mathematics education are not shared

by all however, there were several opinions expressed more favorable toward technol-

ogy, and opposing the opinions of the critics, such as:

• Computers and software are not used as much as they should be in most undergraduate courses.

• I use MAPLE during my lectures in calculus and I am ﬁnding that the good graphics help signiﬁcantly in the understanding of the topics.

• Technology has a major role to play in mathematics; I hope to see it more fully utilized by students for producing, say, 3-D images in several variable calculus, or understanding diﬀerential equations.

• Its a great tool for illustrating various examples in a classroom, especially visualization of graphs, PDE behavior, etc.

• Technology is essential to the advancement of Mathematical ideas. More and more areas of science are relying on Mathematical techniques and tools to get answers, and educating students early help to promote a healthy discourse on the usefulness of technology in education.

Others take a more cautious approach stating that technology is essential, but warn that it should not “wag the dog.” Still others express value in technology, yet specify it is time-consuming. Additionally, some participants criticized fellow mathematicians and math departments that opposed technology and their lack of ability to portend technology. Three participants remarks were:

Mathematicians are late to realize the importance of computers. The ﬁrst computer scientists were mathematicians who lost patience with the typical mathematicians disregard of computers, which have completely revolutionized science and technology. There is similar friction between pure mathematics and applied mathematics. Change is beginning to happen faster: funding agencies realize the importance of mathematics to science and engineering and are shifting the mix of funding in ways that favor the parts mathematics that mixes with science and engineering. These parts

131 of mathematics are technology intensive (relying as they do on strong computational hardware and very fast networks).

Most mathematics departments (including the best and most prestigious) are very reactionary in the use of technology. Many professors believe technology is actually harmful to students. Universities don’t provide up to date computer equipment to their faculty.

Kakutani used to say (something like) “Doing mathematics is like climbing a mountain. The more tools you have, the higher you can climb - the higher you can climb, the greater the view.” All tools are valid if the student can use it to climb higher and to connect with students. Mathe- maticians need to be more open to showcasing all the tools out there.

Other major concerns for mathematicians is how the technology is, and/or will be, supported, one participant said: “I have some concerns about long term support of some software programs−particularly free ones like GAP but even ones that charge like MAGMA.” Another participant is concerned about the “correctness” of current

(Web-based) software:

The math department here has examined many, many web-based homework/testing applications offered by textbook publishers in support of their low-level texts over the past 7 years. None of them have been Math- ematically correct−for each program, for some problems, the program either accepts incorrect answers as correct, or rejects correct answers. These are design problems−there is no semantics behind the formula manipulation (or the floating-point calculations, in the case of (WebWorks) in the programs. It is incredible (literally) how easily one starts to accept this type of behavior, telling students they have to type answers in special forms (“sqrt (100)” instead of “10”, e.g., because every other answer is “sqrt (something)”), etc. I’ve talked to one publisher that had a programming team in Moscow, one in the US, and one in at least one other country, which I forget, and their program while visually attractive and with a great database backend to process grades, etc., still suffered

132 from these problems. There needs to be more rigorous attention paid by mathematicians to these problems.

The results presented in this chapter provide insight into mathematicians’ selection and use of software to facilitate mathematics as well as their attitudes about its use in mathematics education. The next chapter states the conclusions and discusses possible implications for future research and practice.

133 CHAPTER 5

CONCLUSIONS AND DISCUSSION

The ultimate goal of mathematics is to eliminate any need for intelligent thought.

- Alfred North Whitehead

The results of this survey presented descriptions of mathematicians’ uses of software and provided data to contribute to the literature in the ﬁeld of mathematics education. Evidence from demographic data shows that the participants are representative of the population. For instance, the sample closely matches the population ratios on gender, age, rank, and mathematical area(s) of interest. If we assume the participants form a random sample that is representative of the population, then Kre- jcie & Morgan (1970) indicate the number of responses in this survey (n=420) from the population (N=2857) is more than suﬃcient to make inferences.

However, it needs to be reiterated, because of the relatively low response rate

(15%) and possible selection bias due to the likelihood that participants of this online

134 survey may be predisposed to computer use, caution must be taken when generalizing results to the population (faculty members at Group I university departments of mathematical sciences) of this study or to the mathematics population as a whole.

Regardless, this study reveals some interesting results that warrant further research and some have immediate implications for practice.

5.1 OVERVIEW OF CONCLUSIONS

The purpose of the study was to determine and describe academic mathematicians’ software use by asking such questions as: What software do mathematicians use?

Which areas of mathematics use software to facilitate activities? For what purposes do mathematicians use software? What are mathematicians’ attitudes toward technology in mathematics and mathematics education? Below is a brief summary of the answers to these questions obtained from the data.

Question 1: What software applications do academic mathematicians use (frequently)? Academic mathematicians at Group I university department of mathematical sciences reported using MATLAB, Mathematica, Maple, LaTeX, and C++ most frequently. There was an approximately equal distribution between the three major

135 operating systems; however, it is noteworthy that many participants reported using multiple operating systems. Software distribution types were also approximately equally distributed among commercial, open-source, and freeware applications.

Question 2: What branches of mathematics most frequently use software? The branches of mathematics with the greatest number of frequent users were: combinatorics, number theory, algebraic geometry, differential equations (both ordinary and partial), dynamical systems, differential geometry, algebraic topology, manifolds and cell complexes, global analysis, probability and statistics, numerical analysis, computer science, fluid mechanics, natural science, and mathematics education.

Question 3: What are mathematicians’ level of use and expertise with computers in general and software in particular? Overall, the participants in this survey ranked themselves high on a scale to indicate computer expertise. On a scale from 1 to 10, where 1 is low, the mode was 8 and the median was 7.5. Only 20% rated themselves below a 5 on the 10-point scale. Approximately 60% of the participants reported spending more than 20 hours a week using a computer. However, the more important

ﬁnding of this study, related to the level of software use, is that a large percentage of participants (45%) indicated frequent use of mathematical software.

Question 4: For what purposes do mathematicians use software? The main activities for software use among participants were: communicating, data analysis and

136 numerical computation, symbolic manipulation, performing experiments and simulations, and gaining insight, although there were frequent users of software on all

“tasks.” Problem solving and creative thinking were the two most frequently reported usages of software for higher-order thinking processes; critical thinking and decision-making ranked third and fourth respectively.

Question 5: What are mathematicians’ attitudes toward and beliefs about technology in mathematics and mathematics education? In particular: How important is technology to the ﬁeld of mathematics and speciﬁc areas of mathematics? What technologies will be relevant to mathematics in the future? Which software should undergraduate and graduate students learn? What are mathematicians’ software interface preferences? And what are mathematicians’ attitudes toward technology used in education? Results show that mathematicians believe technologies are important

(or very important) to mathematics in general and to their speciﬁc area of research

(77% and 62% respectively). However, 30% or less believe technology is important in primary or secondary school mathematics. Approximately 50% felt it is important

(or very important) in undergraduate mathematics education and 59% have similar beliefs with regard to graduate school mathematics. Participants indicated the importance of technology increases with grade level. There were more respondents that believe technology is important (or very important) in teaching mathematics (59%)

137 than learning mathematics (49%). Respondents claimed that the communication technologies will have the biggest impact on mathematics in the future.

With regard to which software undergraduate and graduate mathematics students should learn, 80% of the participants stated that they should learn a subset of

MATLAB, Mathematica, Maple, LaTeX, and a programming language (e.g., C++).

Graphical interfaces were preferred to command line interfaces (43% to 27%) and one-dimensional input was preferred to two-dimensional input (48% to 22%). Finally, participants seemed to have negative attitudes toward students’ use of technology, especially calculators in elementary school.

5.1.1 Conclusions

The main conclusions of this study are:

1. Software is used by a signiﬁcant portion of mathematicians to facilitate mathematical activities. Software will play a signiﬁcant role in the future of mathematics and thus, is a primary concern for mathematics education (Thompson, 1994).

2. Mathematicians use multiple software products for both applications and operating systems for a variety of purposes and depend on several factors. These issues warrant further research and consideration.

3. Software is an essential component in modern mathematical communication.

4. Computer programming is an important element in the relationship between mathematics and software. Programming is required for advanced utilization of mathematical software. Mathematics courses should combine a software component within the current curriculum.

138 5. There are significant differences in software utilization between fields of mathematics and mathematical activities; therefore appropriate choices for software selection and utilization can be identified.

6. Attitudes of mathematicians toward technology used in mathematics education are negatively skewed. Speciﬁcally, mathematicians consider mathematics education to be concerned mainly with primary and secondary school; and they are less favorable toward the use of technology in mathematics education. However, not all remarks were negative. There were several positive opinions expressed and valuable comments provided about using technology for mathematics education.

5.2 DISCUSSION

The conclusions above are based on the data reported in Chapter 4 and are high- lights of the results that most closely connected to the research questions for which this study was framed.

5.2.1 Software in Mathematics

The data reported in this survey provide evidence conﬁrming the literature (e.g.,

Bailey & Borwein, 2005; NRC, 1991) that claims technology is important to mathematics and that some mathematicians use software such as Maple as part of their day-to-day work (Borwein & Bailey, 2004). For example, approximately 60% of respondents stated that technology is important to their area(s) of research, and over

139 77% stated that technology is important to the ﬁeld of mathematics in general. Fur- thermore, nearly all participants (i.e., 90%) indicated some degree of use of discipline- speciﬁc software while some use MATLAB (8%), Mathematica (5%), or Maple (7%) as part of their day-to-day work.18 Nine-percent of the participants reported using some “other”, (not listed), type of software on a daily basis.

The scale (daily, monthly, etc.) was adopted from the Maplesoft (2005) study regarding engineers’ utilization of software. This scale was an attempt to quantify extent of use. However, in practice, usage may be more variable as one participant pointed out. “I am spending a huge amount of time on the project (using Mathe- matica). But once I get insight into what is going on, I won’t need Mathematica any more. . . . so the standard deviation is huge.”

This trend should continue. The literature describes four reasons why the use of computers will continue to increase in mathematics: more powerful computers, a newer generation computer culture, previous success in mathematics, and the applied mathematical fields are computerized (Hersh, 1997). While respondents of this survey expect an increase in the power of computer technology to affect mathematics in the future, the results did not show a significant difference between the older and newer generation of mathematicians. That is, there was no significant difference between

18There was overlap in the responses. Some use a combination of two or all three.

140 department rank and computer use or expertise, as the results indicated younger and older mathematicians had equal expertise. The results cannot support or re- fute claims of previous success in mathematics, as the survey did not address such concerns. However, the results do verify applied mathematical fields or fields with particularly close connection to applied mathematics (e.g., PDE) made more use of technology more frequently than theoretical fields.

5.2.2 Software Selection

Among the three most frequently used mathematical software applications, there were differences in percentage of responses per field of study (see Table 4.12). In general, usage of Maple and Mathematica was orthogonal to MATLAB. That is, areas of mathematics for which Maple or Mathematica were used, saw very little use of MATLAB, and vice-versa. In some areas, one particular software was reported more useful than the others. For example, Maple had more than twice the number of responses for combinatorics than either MATLAB or Mathematica among frequent users19 while the opposite was the case for probability theory where MATLAB more than doubled the number of frequent users. Although these numbers provide a general indication of the magnitude of utilization of specific software applications for

19Daily or weekly users.

141 mathematical activity, the questionnaire design did not link speciﬁc software to spe- ciﬁc activities. Instead, only associations of software and activities were made from participant responses.

5.2.3 Communication

One of the primary ﬁndings of this research study is the prominent role software plays in mathematical communication. Each survey item relating software and communication had a high frequency of frequent users. For instance, of the 394 responses to the survey item speciﬁcally inquiring about level of software use for communication,

75% of the participants reported daily or weekly use. Results also show that software had similar frequency rates on activities in which communication is necessary (e.g., teaching, presenting results) or part of the deﬁnition (e.g., visual representations).

LaTex is the primary tool for mathematical communication. Web-centric technologies and software such as Internet newsgroups, HTML, and Macromedia Flash are also frequently used by the survey respondents.

Moreover, comments made to an open response item asking about future advancements in technologies for mathematics centered around communication technologies.

For example, participants suggested the future would bring about better video conferencing and the ability to embed mathematical formulas in Internet communications

142 (e-mail, newsgroups, and Web sites).

5.2.4 Programming Languages

An old adage goes something like, “you never understand something completely until you teach it.” A new adage is, “in order to understand something really deeply, you should program it” (Zeilberger, 2006, p. 1). Computer programming is an activity that has beneﬁts beyond the result of the program. In addition to providing a desired end result, programming is also a means for learning and understanding.

Programming increases logic, sequencing, and problem solving abilities. Zeilberger

(2006) recommends codifying all mathematical knowledge.

Basic programming skills are fundamental and augment general application software such as Mathematica or Maple. The development of mathematical software packages has encapsulated many processes and eliminated the need to code many algorithms such as Fourier transformations. However, programming is also an essential skill for advanced work with these software applications. The encapsulated processes need to be used as part of the solution to the overall problem. Thus, programming is necessary in order to bundle and incorporate these other processes.

Results show that approximately 21% of the participants used various programming languages frequently (i.e., daily or weekly). The percentage increases to 33%

143 if including participants that use programming languages daily, weekly, or monthly.

Incorporating programming into the curriculum for linear algebra and diﬀerential

equations is a strong recommendation.

Several courses could beneﬁt by incorporating a technological component. Re-

sults show that participants believe software technology is important and should play

a major role in several core mathematics courses, namely: calculus, diﬀerential equa-

tions, linear algebra, discrete mathematics, and number theory. However, in open

response items, participants also indicated the need to support such initiatives by

supplementing the course with a computer lab that uses the appropriate software

instead of taking valuable class time to focus on the particulars of that software.

Additional support for technology may come in the form of training professors

on software applications, creating additional computer laboratory components to the

existing course, and/or adding computer programming as part of the core curriculum.

This is not to say that other courses could not beneﬁt from a technology component.

Visual representations of abstract objects such as cosets could be valuable to many students (Dubinsky, 1991).

144 5.2.5 Mathematical Activities with Software

The results indicate signiﬁcant use of software for a variety of mathematical activi-

ties. Besides communication purposes (see Section 5.2.3), there were several activities

for which software is utilized. Results suggest algebraic areas use software more than

expected to gain insight, for symbolic manipulation, and to falsify conjectures (Clus-

ters 3 and 5). Analysts use software more than expected for data analysis, numerical

calculations, symbolic manipulation, and gaining insight (Clusters 3 and 4). Geomet-

ric ﬁelds loaded extremely high on Cluster 2, communication and visual representation

(χ2 = 51.39, df = 16, p < 0.001).

The results show that a relationship exists between mathematical subject area and

mathematical activity (χ2 = 51.39, df = 16, p < 0.001). Although MATLAB had higher frequencies on activities related to numerical calculation, data manipulation, and numerical analysis, no signiﬁcant relationship was detected using a chi-square test for independence between particular software and mathematical activities (p = 0.18).

The activities reported in the survey can be compared to the activities reported in the literature. For example, Knuth (2000) suggests that graphical representation is a function of mathematical sophistication and that students ﬁrst use symbolic representation and then graphical representations later. The results of this study show that 32% of the most sophisticated mathematicians use software for visual

145 representations frequently. However, the results do not support any conclusions about similar sequencing of symbolic to graphical representations.

Gaining insight and developing intuition was another primary purpose participants reported using software related to learning. Participants (30%) reported using software frequently for developing insight.

The results of the cluster analysis and factor analysis show certain activities are more associated than others (see Table 4.17 and Table 4.20). For example, software use for symbolic manipulation and computational reasoning are more similar than performing numerical calculations and computational reasoning.

5.2.6 Attitudes Toward Technology

In many cases, attitudes toward technology used for education were less than

“positive.” Ironically, participants indicated strong importance of software in mathematics but not in mathematics education. The primary concern was that technology would become a “crutch” that would ultimately interfere with student learning and understanding. Participants believe that calculators produce laziness and, as evidence, oﬀer examples where students instinctively reach for the calculator to perform simple calculations. Overall, their recommendation was to remove calculators from

K-12 mathematics altogether.

146 From the data on the open-item responses, it appears participants are working under a few basic assumptions. The first assumption is that mathematics education is associated exclusively with K-12 school mathematics. When asked about technology in mathematics education, many participants automatically assumed it referred to technology usage by the K-12 segment. Howson (1994) states that such a belief divides the mathematics community and leads to inadequate teaching at the tertiary level. Confounding matters, the field of mathematics education has surrendered all claims to education at the advanced level. One goal of mathematics education should be to actively pursue the unification of these fields.

Second, many assume technology is associated exclusively with calculators or graphing calculators. When asked about technology in mathematics education, many participants commented only about calculators, even though calculators are the least profound technology available to students today. Rarely mentioned were Internet technologies or more advanced visualization software that are easy enough for children to use (Papert, 1980).

The third assumption is that calculators (technology) produce laziness in students.

However, if used properly, calculators can be very thought provoking and produce the opposite eﬀect. Technology can amplify students’ abilities and enable work on more advanced problems (Pea, 1985; NCTM, 2000). Technology can be used for many

147 of the same activities for which participants use it daily: pattern recognition20 or data analysis. If teachers are unable to ﬁnd activities for which the “opposite” eﬀect occurs, then could it be the teacher who is lazy?

It must be pointed-out that some of these negative attitudes are not unfounded.

Many students come to the university class ill-prepared even for basic mathematics.

Too often, students seem to heavily depend on calculators to perform basic operations on simple numbers. However, this is no reason to condemn all technology. A major issue for mathematics education to confront is how to modify these perceptions by having students better prepared.

5.3 IMPLICATIONS

This research produced several implications for further research and current practices. For example, recommendations based on data obtained about mathematical communication can be incorporated immediately into practice while others need further research.

20Approximately twenty percent (20%) of the participants reported using software for pattern recognition daily or weekly.

148 5.3.1 Implications for Research

Continued research of mathematical communication is necessary. Mathematics is an inherently social and cultural activity (Cobb, Jaworski, & Presmeg, 1996); however, society and culture are evolving. New technologies are changing the way mathematics is and can be communicated. Researchers need to continually study how these new technologies impact mathematics education. For instance, the Internet is quickly becoming a central medium for education. The success of the Web drastically changes how information is communicated and the way scholarship is conducted

(Miner, 2005). Mathematics, rich in symbolic notation, is diﬃcult to express using standard text editors or word processors. One potential technology for communicating mathematically that warrants further research is MathML, described below. If such technology is to be used eﬀectively, research is needed to understand potential problems and limitations associated with its use.

MathML, used in electronic learning systems such as MapleTA and WebCT, is an

XML-based encoding standard for mathematical notation on the World Wide Web

(WWW). MathML is composed of two separate markup components: presentation markup, for the visual appearance (e.g., superscripts, etc.), and content markup, which describes mathematical semantics (Miner, 2005). XML21 is “the dominant

21Extensible Markup Language is a mark-up language to interface data sharing between various information systems.

149 data format underlying the information infrastructure of the World Wide Web” and is

“deeply embedded in software systems and work ﬂows that will shape the information landscape for years to come” (Miner, 2005, p. 532). Miner (2005) provides an example illustrating the diﬀerence in syntax between the presentation mark-up and the content markup for the expression (x + 2)3 that can also serve as an example of a potential research problem for communicating mathematically.

A signiﬁcant application of MathML and a primary concern for educators is electronic learning. MathML is a standard by which mathematical formulas can be expressed in distance learning software. Communication via WWW, e-mail, and newsgroups that incorporate such an encoding standard will require understanding the structure of the content semantics, how students learn this “language,” and problems that may arise.

Other issues regarding technical communication of mathematics are evident. The computer interface is an essential component in considering how mathematical communication is achieved. Results indicate that the user interface is an important aspect of mathematical software choice and use. Therefore, computer interface issues need further examination.

More research needs to be conducted on the various input methods. Currently, there are two forms of input: one dimensional (1-D) and two-dimensional (2-D).

150 In most, but not all cases, two-dimensional input is more time consuming than one- dimensional input. An example from differential equations for which two-dimensional input required fewer keystrokes than one-dimensional input is the first-order nonlinear differential equation. In 1-D input the equation would be entered as:

diff (y(x), x) + y(x) = sin(x)

The same equation in 2-D requires 13 fewer keystrokes:

y0 + y = sin(x)

However, the results of this study indicate that participants have a preference for

1-dimensional input. In fact, many expressed contempt for software applications that have recently changed their interface from 1-dimensional input to 2-dimensional input.

Mathematicians’ preference for 1-dimensional input should be explored further.

Several other aspects of computer interfaces require research. For example, graphical user interface (GUI) elements provide the user with control over the processes of variation. Students construct meanings based on these kinesthetic controls (Kynigos,

Koutlis, & Hadzilacos, 1997). New releases of mathematical software (e.g., Maple 10 and Mathematica 6.0) provide these controls as part of their interface. Little research has been conducted on how these controls aﬀect students’ understanding and no research has been conducted on the incorporation of interface controls on professional mathematical activity. One participant’s response to Question 11 of Section 4 stated

151 software companies have added “lots of ﬂuﬀ” to their latest software and claimed that it detracts from their usefulness in attempt to appease mathematics education causes. It was suggested that there should be distinction between research tools and teaching tools.

Another implication for research based on the results of this study is to understand the rationale of why mathematicians use multiple operating systems. The results show that two-thirds of the participants use more than one operating system. In the past, most software was platform speciﬁc. For example, Minitab statistical software is not available for the Macintosh operating systems. However, today this is almost an exception and not the rule. All three major mathematical application software,

MATLAB, Mathematica, and Maple, are developed and supported for all three major operating systems (i.e., Mac OS, MS Windows, Linux).

One possible explanation might simply be preference. For example, although

MATLAB runs on both Windows and Macintosh, it is possible there are subtleties in the interface in which the use of the underlying operating system for this particular software is deemed necessary. Another possible explanation is performance. In any case, the question remains as to why multiple platforms are so commonly used.

The current study needs to be replicated to eliminate limitations. Some of the limitations were: a potential selection bias, restriction of the population to Group I

152 university departments of mathematics, and no linkage between speciﬁc software and speciﬁc activities.

Replicating this study with Group II and Group III mathematics departments would examine practices and beliefs for a more general population. Furthermore, a replication could be used to compare Group I software uses to Groups II and III software uses.

A similar study needs to be conducted for which specific activities are related to specific software. For instance, this study does not make inferences about which activities are performed by a specific software, but only makes associations. The results of the cluster analysis and factor analysis show certain activities are more associated than others. However, these activities were not linked to particular software. For this reason, it is believed the results of this study show that there is no significant relationship between software and activities; however, intuition and experience places this belief in doubt.

Research needs to be conducted that further decomposes standard mathematical activities and examines more closely software’s contribution to those activities. In- structional programs that promote technologies for gaining insight, such as Calculus

& Mathematica (Brown, Porta, & Uhl, 1991), need further research. A new study

153 could rephrase the question so as to solicit not only the software title but also the intended activity. This can be easily done modifying the current survey system.

Lastly, continued research needs to be conducted to track changes in mathematics faculty’s utilization of software and attitudes associated with use in mathematics education.

5.3.2 Implications for Practice

Results of this research provide several practical consequences for mathematics education: (a) use software for mathematical communication; (b) incorporate computer programming in the curriculum and oﬀer/require students to take at least one programming course; (c) incorporate appropriate software into the core curriculum; (d) carefully consider system software choices; (e) consider open-source alternatives, use appropriate software and hardware; and (f) provide technology training and support to students and faculty.

One of the most important ﬁndings of this research study was the extent for which software is used for mathematical communication. Therefore, it is recommended communication software and technologies be incorporated and supported.

154 The National Council of Teachers of Mathematics in the Principles and Stan- dards for School Mathematics (NCTM, 2000) emphasizes the importance of developing communication in mathematics. However, this communication standard focuses on handwritten (paper and pencil) and spoken communication and not on electronic communication. Moreover, the context for this communication is in the classroom, even though it is known that most learning occurs outside the classroom (Parker,

1997). In particular, Internet newsgroups and message boards provide platforms for discussions. Participation in Internet newsgroups and messages boards which already exist should be encouraged, if not required. These forums provide “anytime” opportunities for open mathematical discussions.

Web development is another recommendation for practice that can augment mathematical communication. Web development can be used as a mechanism for increasing electronic communication and presenting mathematics. Learning to code HTML would serve as a precursor for future electronic communication by enabling students to understand basic markup languages that are essential for more advanced mathematical communication software such as LaTeX and MathML. Dreamweaver software

(or Nvu as an open-source alternative) makes web development easy by providing a

WYSIWYG editor. In addition, this type of software has a ‘split’ window in which to

155 view the raw HTML code and the ﬁnal product simultaneously which can facilitate the learning process.

For secondary school mathematics, it is recommended that students be required to develop a Web page that includes mathematical topics. This would have multiple beneﬁts. In addition to communicating mathematics, it involves learning Web development and HTML. Furthermore, it would cause problems and issues of presenting mathematical notation electronically to surface.

Finally, with respect to communication, results show that LaTeX is the principal software for mathematical communication and therefore can not be ignored. Under- graduate, or at least graduate mathematics students need to know LaTeX. However, little or no training is provided. Too often students have to teach themselves. Adding an additional course for training on LaTeX (and other relevant software) to the core curriculum for university mathematics students is recommended. For example, during a 10 week quarter, this course could devote two weeks each to LaTeX, MATLAB,

Mathematica or Maple, a programming language such as C++, and Web development software. Afterwards, it is recommended that assignments be typeset for collection.

One ﬁnding that was not expected was the degree to which participants utilize computer programming. Computer programming constitutes a considerable percent of mathematicians’ day-to-day work. Results show that 21% of participating academic

156 mathematicians use at least one computer programming language at least weekly and nearly 65% of all participants partake in programming sometime throughout the year.

Furthermore, open response items indicate participants have more favorable views toward students learning to program and using programming languages over many other technologies.

For students, because computer programming strengthens problem solving and teaches logical and sequential reasoning, programming classes need to be added to the general curriculum of secondary school and as a basic education requirement in college. In some cases, programming may be appropriate for even elementary school students (see Papert, 1980). In addition, programming is essential to advanced applications of mathematical software such as MATLAB and Mathematica.

The results support the need for an integration of computer programming and mathematics courses as programming is an essential skill among many professional mathematicians. Several participants in this research study also support such a notion by their comments regarding what mathematics students should know or learn (e.g.,

“require students to master basic programming,” and “everybody should know at least two programming languages”).

Incorporating appropriate software into the core curriculum is recommended. Re- sults show, as expected, that software use depends on subject area. We can take

157 advantage of courses for which software use is both more accepted and expected.

For instance, responses were more favorable toward software use in calculus, diﬀer- ential equations, linear algebra, discrete mathematics, and number theory. These courses provide a starting point for incorporating software. However, experimenting with software (e.g., Groups & Graphs 3.1) in such courses as abstract algebra is also recommended and needed (Artigue, 1999).

Another recommendation based on the results is that there needs to be ﬂexibility in the selection and use of system software, and thereby, hardware. It is not uncommon for mathematicians to make use of more than one operating system. In fact, two- thirds of the respondents use more than one operating system. Education must provide alternatives to the mainstream operating systems that place rigid constraints on some mathematical activities or make those activities more diﬃcult to perform.

System software has a major impact on the selection of application software.

For years, a decision to purchase one piece of hardware over the other was made based upon the software available to the hardware. However, this is less of a concern today. The greatest ﬂexibility of system software is provided by the new Intel-based

Apple Macintosh. These machines can run all three major operating systems. In particular, the hardware can be partitioned to natively run Microsoft Windows operating system in addition to the Mac OS X operating system. Furthermore, since the

158 Mac OS X is built on a solid Unix system, any “Unix-like” operating system may also be installed. Schools and universities now enjoy a choice when deciding which new hardware to purchase, spurring competition between the Intel-based Apple computers or native Windows-based machines.

Another implication for practice is the use of open-source software. Not only is open-source software free, in the ﬁnancial sense, it is free to be modiﬁed and distributed. It provides an alternative to expensive software. There is an open-source equivalent to nearly all major mathematical software (e.g., Octave for MATLAB, R for

S). Currently, there are three primary software products covering a broad spectrum of mathematical ﬁelds and activities. There are alternatives however. The results imply that the open-source software movement is increasing in popularity and relevance.

Integrated software packages such as SAGE are increasing in utility and demand.

One participant stated that it is disgraceful that open-source is not more widely known and available and that we are doing a disservice by recommending packages like Mathematica which violate all principles of academia.

Lastly, there needs to be training and a support system for students and faculty alike. Nearly 25% of the participants report that they do not use software because of a lack of training or time to learn the software. Additionally, 14% speciﬁed a lack of expertise to use software eﬀectively. Additionally, literature shows in many cases

159 that students are more advanced than their teachers.

5.4 SUMMARY

The concern of this study was what software applications academic mathematicians at Group I university mathematics departments utilize while mathematizing, how it facilitates their work, and what attitudes they have toward technology in mathematics and mathematics education. The results show: (a) Matlab, Mathemat- ica, and Maple−depending on the branch of mathematics−were their three primary

“tools”; (b) software, in particular LaTeX, is central to mathematical communication, both currently and expected to have a larger role in the future; (c) programming is an essential skill for both basic and advanced work with software and should be incorporated in students’ curriculum; and (d) mathematicians’ main concern with technology use into education is that it detracts from students’ thinking skills.

In summary, technology is redeﬁning mathematics. Technology has changed what mathematics is, what mathematics is important, and what mathematical activities are performed. In particular, results of this study verify that various software applications are used as tools by mathematicians while mathematizing, but participants still

160 expressed negative opinions about technology use in mathematics education. Mathe- matics educators must remain current in their investigation of software and for which mathematical activities beneﬁt from software use.

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176 APPENDIX A

GROUP I INSTITUTIONS

Boston University Stanford University Brandeis University State University of New York, Stony Brook Brown University University of California, Berkeley California Institute of Technology University of California, Los Angles Carnegie Mellon University University of California, San Diego City University of New York University of California, Santa Barbara Columbia University University of Chicago Cornell University University of Illinois, Urbana-Champaign Duke University University of Illinois, Chicago Georgia Institute of Technology University of Maryland Harvard University University of Michigan Indiana University University of Minnesota John Hopkins University University of North Carolina, Chapel Hill Massachusetts Institute of Technology University of Notre Dame Michigan State University University of Oregon New York University, Courant Institute University of Southern California Northwestern University University of Texas, Austin Ohio State University22 University of Utah Pennsylvania State University University of Virginia Princeton University University of Washington Purdue University University of Wisconsin Rensselaer Polytechnic Institute Washington University, St. Louis Rice University Yale University Rutgers University

177 APPENDIX B

AMS MATHEMATICAL SUBJECT CLASSIFICATIONS

00 General 01 History and biography 03 Mathematical logic and foundations 05 Combinatorics 06 Order, lattices, ordered algebraic structures 08 General algebraic systems 11 Number theory 12 Field theory and polynomials 13 Commutative rings and algebras 14 Algebraic geometry 15 Linear and multilinear algebra; matrix theory 16 Associative rings and algebras 17 Nonassociative rings and algebras 18 Category theory; homological algebra 19 K-theory 20 Group theory and generalizations 22 Topological groups, Lie groups 26 Real functions 28 Measure and integration 30 Functions of a complex variable 31 Potential theory 32 Several complex variables & analytic spaces 33 Special functions 34 Ordinary diﬀerential equations 35 Partial diﬀerential equations

178 37 Dynamical systems and ergodic theory 39 Diﬀerence and functional equations 40 Sequences, series, summability 41 Approximations and expansions 42 Fourier analysis 43 Abstract harmonic analysis 44 Integral transforms, operational calculus 46 Functional analysis 47 Operator theory 49 Calculus of variations, optimal control; optimize 51 Geometry 52 Convex and discrete geometry 53 Diﬀerential geometry 54 General topology 55 Algebraic topology 57 Manifolds and cell complexes 58 Global analysis, analysis on manifolds 60 Probability 62 Statistics 65 Numerical analysis 68 Computer science 70 Mechanics of particles and systems 74 Mechanics of deformable solids 76 Fluid mechanics 78 Optics, electromagnetic theory 80 Classical thermodynamics, heat transfer 81 Quantum theory 82 Statistical mechanics, structure of matter 83 Relativity and gravitational theory 85 Astronomy and astrophysics 86 Geophysics 90 Operations research, mathematical programming 91 Game theory, economics, social-behavior sciences 92 Biology and other natural sciences 93 Systems theory; control 94 Information and communication, circuits 97 Mathematics education

179 APPENDIX C

INSTITUTIONAL REVIEW BOARD

180

APPLICATION FOR EXEMPTION PROTOCOL NUMBER: REVIEW BY THE INSTITUTIONAL REVIEW BOARD 2006E0630 The Ohio State University, Columbus OH 43210

Principal Investigator Name: Douglas T. Owens Phone: 292-8021

University Title: Department or College: Education E-mail: Professor [email protected]

Campus Address: 333 Arps Hall 1945 N. High St. Columbus, Ohio 43210

Signature: Fax: 292-7695

Date:

Co-Investigator Name: James Quinlan Phone: 262-6331

University Status: Campus Address: E-mail: Graduate Student 256 Math Tower [email protected] 231 W. 18th Ave Columbus, Ohio 43210

Signature: Fax: 292-7695

Date:

Source of Funding Self For office Use Only Research has been determined to be exempt under these categories: ___2___. Approved Research may begin as of the date of determination listed below. The proposed research does not fall within the categories of exemption. Disapproved Submit an application to the appropriate Institutional Review Board for review.

Date of determination: 09/19/2006 Signature: _Tani Colvin______Office of Research Risks Protection

181 EXEMPT CATEGORY: 1 2 3 4 5

Does any part of the research require that subjects be deceived? Yes No

Will research expose human subjects to discomfort or harassment beyond levels encountered in daily Yes No life?

Could disclosure of the subjects’ responses outside the research reasonably place the subjects at risk of Yes No criminal or civil liability or be damaging to the subjects’ financial standing, employability, or reputation?

Will fetuses, pregnant women, human in vitro fertilization, or individuals involuntarily confined or Yes No detained in penal institutions be subjects of the study?

For research proposed under category 2, will research involve surveys, interview procedures, or Yes No observation of public behavior with individuals under the age of 18?

For research proposed under category 4, will any of the data, documents, records, pathological Yes No specimens, or diagnostic specimens be collected or come into existence after the date you apply for exemption?

For research proposed under category 4, will any of the information obtained from data, documents, Yes No records, pathological specimens, or diagnostic specimens that come from private sources be recorded by the investigator in such a manner that subjects can be identified directly or through identifiers linked to the subjects?

182

CONFIDENTIALITY

7. Provide a brief description of the measures you will take to protect confidentiality. Please describe how you will protect the identity of the subjects, their responses, and any data that you obtain from private records or capture on audiotape or videotape. Describe the disposition of the data and/or the tapes once the study has been completed.

Potential participants information will be housed in a separate (encrypted) database than the response database located on the network described below. Participants will log into a server using a computer generated random password. All responses will be de-identified from the raw data before any analysis and results are produced, therefore it will be impossible to identify, directly or indirectly, participant responses.

Network Topology State-of-the-art fiber-optic lines provide the network bandwidth. The network resides behind an enterprise-class network security appliance. This appliance has a comprehensive set of security applications, including gateway anti- virus, gateway anti-spyware, and intrusion detection system. Each application is automatically updated daily.

The network is divided into zones. Network address translation (NAT) maps traffic to the appropriate zone/computer. Zone access rules permit only specific services. All ports not specifically authorized are actively blocked and appear stealth.

Computer & Software Environment Both the Web Server and the Data Server employ the Microsoft Windows Server 2003(r) operating system and are updated automatically with the latest system and security patches. Both servers are protected by anti-spyware and anti-virus systems (in additional to the gateway security services). A custom application will be deployed on the Web Server to act as the survey instrument. All data obtained from the survey questions - in real time - will be transferred to the Data Server. Microsoft SQL Server 2005(r) will warehouse the data.

183

INFORMED CONSENT

8. What information do you plan to give to your subjects before you ask for their consent? Use a style of language that simply and clearly explains the research to your subjects. Respond in the space provided here, or attach a copy of the information you plan to provide to your subjects and/or their parents or guardians. (Note: if you use more than one method of recruitment, you may check more than one box)

Statement attached. Participants will receive, electronically, the attached letter (Exhibit A) and will also be directed to the (attached) html document upon login (Exhibit B).

9. How do you plan to document informed consent?

The subjects are 18 years of age or older. I am distributing a survey to the subjects. They can choose whether or not they want to respond. Additionally, participants will receive an invitation via electronic mail that explains the nature and purpose of the research. In order to participate, the recipient must select the included hyperlink to the URL or manually type the URL in their internet browser to initiate the survey. Furthermore, the survey application contains a login screen, upon which the nature and purpose of the research, including the risk associated with internet transmission will be reiterated. I am requesting a waiver of written consent.

184 Exhibit A EMAIL Invitation document

Dear Colleague,

This is a request to participant in a survey research project. The project is being undertaken as a doctorial dissertation by James Quinlan, a Ph.D. candidate in Mathematics Education at The Ohio State University in the College of Education. As the academic advisor, Dr. Douglas T. Owens is the supervisor of the project, while I serve on his dissertation committee. Completion of the 30-question survey is estimated to be less than 20 minutes, but you will be able to return to the survey page as many times as necessary to complete your entries before the survey closes on September 30, 2006. A hyperlink to the survey login and your participation code (password) is provided below. http://www.math-ed.org/surveyLogin.asp Password: 123456

The purpose of this survey is to explore the professional use of computer/software for facilitating mathematics with the objective to improve the use of technology in mathematics education at all levels. The survey is being sent to faculty members at 50 mathematics departments across the United States. Your email address was obtained from your department published website and stored in an encrypted database. There was no use of automation to obtain your address; James Quinlan recorded all emails “by hand”. Furthermore, be assured, your information will not be revealed to any third parties or sold. This project is self-funded and there is no corporate influence.

PARTICIPATION IS VOLUNTARY; you may refuse to participate simply by not selecting the hyperlink above. If you do choose to participate, you are free to refuse to answer any questions that you do not wish to answer. Furthermore, you may withdraw at anytime with no stipulations, penalties, or repercussions.

ALL RESPONSES WILL BE CONFIDENTIAL, although no guarantee of Internet security can be given; it is possible for transmissions to be intercepted and for IP addresses to be identified. With this survey, a breach of confidentiality is extremely unlikely to occur. Survey response data will come in directly from the website file without any personal identifying information. The only way that confidentiality could be breached would be if an outside party intercepted the data coming from your personal computer during the time while you complete the survey. However, since there will be NO identifying information included on the online survey, even if the data is intercepted, confidentiality remains assured.

Although there is no compensation for completion, as an incentive for your participation, you will be mailed an electronic copy of the survey results. If you need any assistance filling out the survey or have questions or concerns regarding the survey, please do not hesitate to contact James Quinlan through any or all of the following means:

231 W. 18 Ave Math Tower 256 Columbus, Ohio 43210 (614) – 292-7401 [email protected]

Sincerely Bostwick Wyman

185 Exhibit B HTML document

Dear faculty member,

Thank you for taking the time to participate in this survey. There are 30 questions. Completing the survey should take approximately 15-20 minutes. Please complete the survey by September 30, 2006. You will be able to return to the survey page as many times as necessary to change your entries and complete before the survey closes.

BACKGROUND:

Utilization of technology has become increasingly more prevalent in facilitating mathematics. As such, its professional use is a primary concern for mathematics education. Federal, state, and local governments and organizations have mandated its use in modern mathematics curriculum; however, a consensus on its utility still does not exist; arbitrary application of technology is both inefficient and ineffective. Math educators are usually unaware of the selection and use of discipline specific software as a result of insufficient examination of the mathematician’s uses of these technologies.

The purpose of this survey is to explore the selection and use of computer/software to facilitate mathematics, mathematical thinking, and mathematical discourse. The results will be valuable in providing a description of the current uses of technology in mathematics, thus expanding the field of mathematics education. The project is being undertaken as a doctorial dissertation by James Quinlan, a Ph.D. candidate in Mathematics Education at The Ohio State University in the College of Education.

DIRECTIONS:

Clicking the CONTINUE button at the bottom of this page will redirect you to the beginning of the survey. The answer choices are selected using various interface objects such as “checkboxes” and “radiobuttons”. Checkboxes represent inclusive choices while radiobuttons are for an exclusive choice. Navigate the survey by using the “left” and “right” arrow buttons located at the bottom of each page. The answers you have selected for each section/page will be saved every time you use the arrow buttons to move to a new section and/or when you click on the SAVE button located at the bottom of each section/page. Questions containing a ten- point scale have the typical meaning, 1 lowest and 10 highest.

Be sure to click the SAVE button before you leave the survey if you wish to record your answers.

WARNING:

1. If you use the "Forward" or "Back" buttons of your web browser (instead of the arrow buttons provided in the survey) answers that have not been saved will be lost.

2. If you do not make any entries for 15 minutes, the web site will time out and answers that have not been saved will be lost.

If you need any assistance filling out the survey or have questions or concern regarding the survey, please do not hesitate to contact us via email at: [email protected].

186 APPENDIX D

SURVEY QUESTIONNAIRE

1. Select your gender. Male Female

2. As of your most recent birthday, which interval best describes your age? 18 - 25 26 - 30 31 - 40 41 - 50 51 - 60 > 60

3. What is your level of service to your department? Professor Associate Professor Assistant Professor Post Doctorate Emeritus Other

187 4. How many TOTAL years have you held the rank of assistant, associate, and/or full professor? 0 - 2 3 - 5 6 - 10 11 - 20 21 - 30 > 30 Does not apply

5. Which interval best describes the number of hours per week you spend interacting with a computer, in any capacity? 0 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 > 30

6. On a scale from 1 to 10, complete the following sentence. I would consider myself as a computer:

Novice Guru 1 2 3 4 5 6 7 8 9 10

188 7. Which operating system(s) do you use to run your application software? Select all that apply.

Windows XP Macintosh OS X Solaris Linux Windows 2000 Windows 98 Windows NT Macintosh OS 8/9 Open BSD Free BSD Net BSD Be OS Irix Tru64 Other(s) (please specify)

8. Which AMS subject classiﬁcation best describes your primary area(s) of interest? Select all that apply.

00 01 03 05 06 08 11 12 13 14 15 16 17 18 19 20 22 26 28 30 31 32 33 34 35 37 39 40 41 42 43 44 46 47 49 51 52 53 54 55 57 58 60 62 65 68 70 74 76 78 80 81 82 83 85 86 90 91 92 93 94 97

189 SECTION 2 - Software selection and use.

1. Indicate your extent of use on each of these software distribution types for research and/or presentation.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Commercial Open Source Freeware / Shareware

2. Indicate your extent of use on each of these general categories of software.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Programming Languages Specialized math software Statistical tools Spreadsheets Database/Database management Software on hand-held calculators

190 3. How often do you use the following software

Daily Weekly Monthly Yearly Never 5 4 3 2 1 MATLAB Mathematica Maple Cabri Geometry Cinderella EUKLID GAP ISETLW Macaulay Magma Mathcad Maxima Nauty Octave Otter PARI-GP SAGE Shur Simulink XPPAUT Other(s) - please specify

191 4. Specify your extent of utilization of the following programming languages.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Ada BASIC C/C++ Fortran Java O’Calm Perl Python Other(s) - please specify

5. Specify your extent of utilization on each of the following software for mathematical discourse.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 TeX/LaTeX Word processors Scientiﬁc Workplace MathType HTML/Flash/Web-technologies Internet message boards/Forums Word2TeX Matlab Report Generator Toolbox Maple HTML (MapleML) Other(s) - please specify

192 6. Specify your extent of utilization on each of the following software that supports your mathematical work.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Print reference Electronic reference (e.g., pdf) Web reference Software Help Files Hand-held calculators Paper and pencil Other(s) - please specify

193 SECTION 3 - Software for Mathematical Activities

1. I use software for:

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Checking proofs Communicating mathematics Computational reasoning Creating visual representations Data analysis Determining diﬀerences and similarities Discovering new patterns Falsifying conjectures Creating new representations Gaining insight Logical deduction Logical induction Making predictions Performing experiments Performing numerical calculations Performing simulations Presenting results Proof automation Searching for solutions Symbolic manipulation Teaching & Instruction Verifying analytical derivations Visual reasoning Other(s) - please specify

2. Indicate your level of utilization of software for the following processes.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Problem solving Decision making Critical thinking Creative thinking

194 3. Jeﬀ Suzuki (MAA Online, 2006) claims “the vast majority of mathematical research falls into one of ﬁve categories.” Indicate your level of utilization of software for these categories outlined by Suzuki.

Daily Weekly Monthly Yearly Never 5 4 3 2 1 Proof Extensions Applications Classiﬁcation Existence

4. How has the computer aided you in mathematical proof and justiﬁcation? Se- lect all that apply.

By suggesting a formal proof By providing visual justification By checking a large, but finite number of cases By automation It has not aided me in proof or justification Other(s), please specify.

5. What limits or restricts your use of software for mathematics? Select all that apply.

Lack of necessity Lack of availability Lack of of ﬁnancial resources to obtain software Lack of hardware performance Lack of software capabilities Lack of expertise Lack of time to learn program Lack of time for set-up and installation Lack of training Nothing limits my use Other(s), please specify.

195 SECTION 4 - Attitudes and Beliefs

1. How important would you say technology is to:

Very Moderately Somewhat Not Very Not 5 4 3 2 1 Mathematics Your area(s) of research Mathematics teaching Mathematics learning Primary school mathematics Secondary school mathematics Undergraduate mathematics Graduate mathematics

2. In what year of college should technology play a signiﬁcant role in mathematics students’ development? Select all that apply.

Freshmen Sophomore Junior Senior Graduate None of the Above

196 3. How large of a role should technology play in the following courses:

Major Moderate Minor None Do not know 5 4 3 2 1 College Algebra Trigonometry Calculus Diﬀerential Equations Linear Algebra Abstract Algebra Real Analysis Complex Analysis Discrete Mathematics Number Theory

4. In what year of school should technology play a signiﬁcant role in mathematics students’ development? Select all that apply.

1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade 10th Grade 11th Grade 12th Grade None of the Above

5. Do you prefer a command line interface (CLI) or graphical user interface (GUI)?

CLI GUI Not sure

197 6. Do you prefer 1 or 2-dimensional input (WYSIWYG, “What you see is what you get”)? One-dimensional input parses a string such as:

2*x^2+1

Two-dimensional input uses templates to produce a more natural representation by using superscripts and subscripts for example, 2x2 + 1.

1-D 2-D Not sure

7. How strongly do you agree with the statement, “I believe that the mathematics profession is fully modern and is taking full advantage of modern tools and technology.”?

SA A N D SD 5 4 3 2 1 Mathematics is fully modern

8. In your opinion, which software applications should undergraduate and graduate mathematics students learn?

9. In your opinion, what should the education system do to better prepare mathematics students to meet the challenges of the future?

198 10. In your opinion, in the next 10 years, what do you anticipate will be the greatest technology advancement that will have a direct impact on your mathematical work?

11. Please feel free to provide additional comments regarding technology in mathematics and/or mathematics education.

199 APPENDIX E

RESPONSES TO OPEN RESPONSE ITEMS

4.8 In your opinion, which software applications should undergraduate and graduate students learn?

1. For mathematics majors, some computer algebra system with graphing capability, such as Maple or Mathematica, and a programming language. For engineers, a signal processing program such as Matlab in addition to the above. All undergraduates should be able to use spreadsheets to solve basic computational problems.

2. Matlab and maybe Maple

3. C++, Mathematica (free version octave), pari-gp

4. Undergraduates need be asked to learn little, but they will learn applications anyway. My graduate students need a computer algebra package.

5. Any (not necessarily all) of Mathematica, Maple, and Matlab.

6. Matlab & others depending on interest

7. C/C++ for numerical exploration

8. R for statistics

9. Grad students need TeX

10. Excel, LaTex

11. Graphical techniques for illustrating dynamical systems phenomena–discrete, ODE, and PDE.

12. Undergrads should have some familiarity with one of Matlab/Maple/Mathematics, and with scientiﬁc word processing. Grad students should learn LaTeX

200 13. Communication tools

14. They should be able to write programs, say in c.

15. I have opinions about types of software, not speciﬁc applications, so I have listed several possible applications for each purpose. I assume that you mean students majoring in mathematics or closely related ﬁelds.

16. Undergraduate – a symbolic computation environment (Mathematica, Maple, MATLab, MAGMA...)

17. software for producing mathematical documents: Word w/ Equation Editor, MathML, LaTeX, Mathematicas presentation capabilities....

18. Graduate – LaTeX and an editing environment for LaTeX (emacs, vi, TeXShop, MikTeX...)

19. Figure drawing software (xﬁg, Freehand, some CAD software....)

20. A programming language: either learn to use the programming capacities of a symbolic computation package such as Mathematica, Maple ... or learn a standard language such as Python, Lisp, Java, ...

21. Minimal HTML, just enough to maintain a basic webpage for a course or seminar

22. Database skills: learn how to search mathscinet and arXiv eﬀectively.

23. Matlab or equivalent math prototyping software that can be up scaled for applications. I am not a big fan of mathematica, as I dont see it as a suﬃciently expressive language.

24. Maple and/or Mathematica, Matlab

25. Do you mean undergrad math majors or do you include art and drama?

26. For math majors: they should learn a programming language and gain some facility with numerical solutions to linear algebra and diﬀ eq. For grad students, they should also learn some ﬂavor of TeX.

27. A programming language (such as C), Matlab and/or Mathematica.

28. MATLAB

29. The three Ms, Maple, Mathematica, Matlab

30. Of course, it depends on the material they are learning.

31. Basic/Intermediate general mathematical software

201 32. Matlab, Mathematica and Maple

33. Some basic knowledge of Mathematica, Maple, or similar. Graduate students should have excellent skills in using Latex.

34. Undergraduates - Speaking for Pure Mathematics, none. Graduate students - LaTeX. Other software only as speciﬁc needs arise in the course of research work

35. Those that can be used to better understand the courses that they take (ex. Matlab for Linear Algebra course.)

36. Matlab, Maple, Mathematics

37. Matlab & Mathematica (or Maple) but only in upper division classes!

38. Mathematica or Maple

39. Graduate students should learn latex

40. Mathematica or maple, if they are involved in diﬀerential equations or numeric analysis. Gap if they are studying group theory to support understanding.

41. Graduate students should learn TeX.

42. Maple seems useful for both graduate and undergraduate students, but should not be over-emphasized.

43. Matlab, something like Maple or Mathematica, something like SPSS

44. Maple or Mathlab, Magma or GAP...but only if they are going into math. For everyone, probably Word, Excel, and a web browser.

45. Any programming language, and LaTeX.

46. Maple and Mathematica

47. Like many of the questions, this is too broad. It depends strongly on the goal of the students, and their area of specialization.

48. It depends on the context. Graduate students in math “must” learn LaTeX or the equivalent. Most undergraduates and graduates should learn a system such as maple or mathematica at a signiﬁcant level of expertise. Other specialized software is important, depending on the area.

49. Mathematica

50. Maple or Mathematica, Matlab, C with some number of + signs

202 51. Undergraduate: Maple and Mathematica. Graduate: above plus Latex/Tex

52. Should: none. Useful: mathematica or maple

53. C

54. Mathematica for symbolic manipulation, MATLAB for computation & modeling, DFIELD & PPLANE for diﬀerential equations (see Rice Univ web site). Graduate students should learn basic Tex (or at least Sci Workplace)

55. One of the computer algebra packages such as maple, mathematica, ...

56. A programming language

57. SAGE since its open source, built on a mainstream programming language, and is polyglot.

58. Latex, maple, mathematica

59. Matlab (or equivalent), Latex, maybe Mathematica

60. Nothing that they *should* learn - they learn what they need when they need it

61. e.g. Maple, Mathematica

62. Symbolic computation, scientiﬁc computation

63. Maple or mathematica

64. LaTeX

65. Programming, LaTeX, Maple or Mathematica

66. TeX

67. TeX

68. Any software to latex

69. Maple

70. Maple, Mathematica, LaTex

71. Undergraduate: Maple, Mathematica or Matlab (pick one)

72. Graduate: LaTeX

73. Mathematica or Maple

203 74. Matlab, Mathematica, LaTex

75. TeX (and its variants) is the widely accepted format for typesetting papers and presenting mathematical notes. Graduate students should learn to use it if they intend to become professional mathematicians.

76. I imagine that certain sub disciplines of mathematics have other vital software applications, and some minimal knowledge of programming syntax is always useful regardless of ones intended profession.

77. Matlab, Mathematica

78. Maple, matlab

79. Depends on their needs and major

80. Scientiﬁc workplace

81. LaTeX-like, Maple-like

82. Grad students: latex

83. Mathematica, maybe Maple, Excel

84. Mathematica

85. Maple, Mathematica, Matlab, Latex

86. TeX, LaTeX, AMS-TeX

87. They should learn basic programming skills that they can apply to all applications.

88. Matlab and Maple or Mathematica (and Excel and Word) - Undergrad

89. TeX - grad and specialized software for ﬁeld such as Macauley

90. Matlab, maple, TeX, and/or sage (depending on major)

91. As needed for projects they may do. All needs no one program.

92. Matlab, Maple and R

93. Descriptive geometry

94. Whatever they find convenient. Its hard to predict what theyll need later, its not hard to learn how to use new software, and anyway lots of tasks can be done using many different systems, so it doesnt make a big difference where they start out.

204 95. Matlab because it takes a vectorial approach, and Mathematica/Maple because of their symbolic capabilities.

96. For science students (Maple or Mathematica) and MATLAB.

97. LaTeX, Maple or Mathematica or a similar program

98. CAS like Mathematica/Matlab/Maple, and TeX

99. I assume you mean math majors or minors. Then I’d say Tex, some math package like Maple or Matlab, spreadsheets for everyone, and for some ﬁelds, much more (but I don’t know what because I’m not in such ﬁeld, e.g. numerical analysis). For other majors, I don’t know.

100. In question above about what year technology should play a role depends strongly on major and subspecialty. K-12, should be minor role not “significant”.

101. Matlab, maple

102. Once their foundation for understanding is in place, students should learn how to deal with computer simulations, learn how to program, etc. The applications depend on their interest.

103. Maple/Mathematica

104. Some version of TeX; some computer algebra package (Maple or Matlab or Mathematica)

105. Maple, Matlab, and some programming language

106. Depends on what they want to study

107. Depends on their ﬁeld. Every grad student should know basic LaTeX

108. For math purposes: maple or equivalent; matlab; C/C++

109. Mathematica, excel, word

110. Maple, MATLAB

111. GAP (or magma)

112. OTTER (or Prover9)

113. LaTeX

114. Mathematica or Maple or MatLab

205 115. Mathematica or Maple or MatLab

116. Matlab, mathematica, maple, spreadsheet

117. TEX, maple

118. Spreadsheet, word processing, mathematica or similar

119. Computer algebra system, at least one general purpose programming language

120. Undergrad and Grad: Maple or Mathematica, Excel, MatLab, C or Java, and PowerPoint. In addition Grad students should learn LaTeX,

121. Matlab

122. Matlab and Mathematica (or Maple). One for real programming, the other for algebraic computation.

123. They should learn not to get too caught up with any one application – it doesn’t matter a whole lot which one they use

124. None is required, Mathematica may be useful.

125. TeX, and a little bit Mathematica/Maple/Matlab

126. Some familiarity with a package like Matlab that has both symbolic and numeric capabilities. Beyond that, I don’t think theres any one package for all. All serious statistics students should learn some SAS.

127. Matlab OR Mathematica OR Maple, TeX, LaTex

128. Undergraduates: Maple, Mathematica

129. Graduate students: tex

130. Matlab, Maple

131. TeX and its ﬂavors. Symbolic language (Macaulay, Maple, Matlab, Mathemat- ica), Matlab

132. TeX, Perl, Maple or Mathematica

133. In liberal arts/ business, a spreadsheet, in math, Maple, Mathematica, matlab in engineering, CAD programs use should be sparing the level of junior/senior level classes in math 1st year and sophomore students are in my experience more frustrated with Maple and Mathematica than enthrall

206 134. They should learn mathematica or the equivalent (maple) everything about tex and typesetting and some specialized computational software such as Lie or Macaulay

135. Should have some basic knowledge of how to work with computers, but not more. What they will need later in life for their job they will pick up easily on the go.

136. Maple, tex

137. LaTeX

138. Sage

139. Maple or Mathematica

140. Mathematica or Maple

141. LaTeX

142. LaTeX for writing mathematics

143. There is nothing that all undergrads or all grads need.

144. Matlab, maple, XPP

145. Depends on the major, question does not make sense in this form. If this is asking about math majors, it also depends on their inclination. Question is to general to yield a meaningful answer.

146. Linear algebra and numerical analysis system like MatLab.

147. Symbolic algebra system like Maple.

148. Statistical analysis system like SAS.

149. Mathematics writing system like LaTeX.

150. Microsoft oﬃce; matlab and Minitab

151. Maple and Matlab

152. Maple/Matlab depending on the ﬁeld of concentration

153. LaTeX

154. A CAS like Maple, and some version of TeX

155. Programming language for students in applied math

207 156. (I wonder why you failed to include applied math in No. 3)

157. LaTeX

158. LaTeX

159. Maple

160. TeX and at least one of Mathematica, Maple or MATLAB. All mathematics students should learn a high-level programming language such as C++.

161. Graduate students should learn some form of TeX.

162. Mathematica

163. Graphing calculators, mathematica-maple, more specialized as appropriate. I have taught number theory and logic with serious computer support in form of illustrating concepts and motivating conjectures. But there is a delicate balance of demanding understanding not just easing computation.

164. Mathematica (or Maple), TeX, Matlab, access to the WWW, statistical software.

165. Students should be taught a wide array of applications. The goal should be to teach the students how to learn new applications (as opposed to simply teaching them some favorite program).

166. Maple, Matlab

167. Can’t say without reference to a speciﬁc major.

168. There is no speciﬁc software I think that all ”students of Mathematics” should learn. Mathematics is, for the most part, a service discipline: students learn it so as to be able to use in the study of something else. For each such other discipline there is relevant software that the students should learn.

169. That depends on what classes we are talking about. Differential equations or linear algebra students would probably benefit most form Maple (I am biased as that was the one that I used, no saying that something else wouldnt fulfill the purpose). On the other hand some engineers I know use Matlab a lot, or SigmaPlot.

170. Both should learn: Matlab and Mathematica or Maple, graduates should know these very well. Graduates should also learn Scientiﬁc Workplace or other Latex software

171. Maple, matlab

208 172. Latex and mathematica

173. Any programming language (Java, C, Fortran, etc.)

174. Mathematica, Maple, MathCAD, C/C++

175. Undergraduate majors in math should learn very early one of Mathematica, Matlab or Maple. I also believe they should be exposed early on to LaTeX etc, especially at the graduate level. Word processing/spreadsheet apps are an obvious thing that I would hope they have signiﬁcant experience with prior to entering college.

176. Matlab, Mathematica

177. Tex, Maple

178. Matlab/Octave, Maple

179. Symbolic calculators like Maple or Mathematica, which are good for learning basic programming.

180. MatLab, Maple, mathematica

181. LaTex, matlab, mathematica

182. Maple, mathematica, Matlab

183. Diﬀerential equations with Maple, Matlab, or Mathematica

184. Linear algebra and numerical analysis with Matlab or Maple

185. Number theory and cryptography with Maple or Mathematica

186. Complex analysis (conformal mapping) with Maple (or Matlab or Mathematica)

187. Probability and stochastic simulation with a specialized simulation language (such as SIMULINK or GPSS)

188. Diﬀerential geometry with specialized software

189. Group theory and abstract algebra with specialized software such as GAP

190. Graphing Calculator

191. Latex, matlab

192. MATLAB

193. Depends on the course

209 194. Not sure. I dont know ﬁner diﬀerences between the various software.

195. It would be good if undergraduate students learnt general-purpose software, such as mathematica.

196. Some form of CAS, spreadsheets, some technical word processor

197. TeX, MATLAB

198. Matlab, maple

199. MATLAB, Maple or Mathematica

200. Those relevant to their area of interest

201. For mathematics, some version of Tex is very useful.

202. For math students: Tex, Matlab, Mathematica

203. Matlab for computation and Maple for symbolic manipulations

204. Tex, SAS, and Matlab

205. Maple (or similar), LaTeX, C

206. R, Latex, Word, Html

207. All: spreadsheets, computer algebra

208. graduate students: TeX

209. I think, working with TeX should not be neglected

210. TeX and some computational package (Mathematica, Maple); and DO NOT OVERUSE THE SECOND−i.e., learn how to integrate the classical way rather than stick the question in Mathematica ﬁrst thing, it helps the imagination.

211. Maple, Mathematica

212. TeX, Matlab, R, Excel, programming (C+, Java), PowerPoint, Mathematica,

213. Learn basic numerical analysis, and implement the analysis in any programming language

214. Maple/Mathematica, Matlab

215. Latex (for papers and presentations), emacs or another editor, web and document search. Also, a low-level language like C, a prototyping language like matlab, and a scripting language like PERL or a UNIX shell

210 216. Focusing on learning applications is generally the wrong way to go, because they are constantly changing. That said, graduate students should learn at least LaTeX.

217. It would be useful for them to learn at least one package such as Mathematica or Maple.

218. Programming language such as C/C++, word processors (Latex, Word, or other)

219. Matrix calculations

220. Software fashions change, so it’s not so important which.

221. I do not have the knowledge to give a meaningful answer. Possibly anything that produces graphical examples.

222. Matlab, Mathematica, and in addition, Latex for grads.

223. LaTeX, maple or mathematica, ﬁrefox, xﬁg

224. Tex/Latex

225. Mathematica for symbolic manipulation, R for simulation and statistical analysis, latex for document preparation

226. It depends on the subject they are studying. For diﬀerential equations class I would recommend MATLAB.

227. Graphing calculator, Excel, Minitab

228. Mathematica

229. Software like maple, mathematica, which allows plotting graphs and solving simple equations.

230. Matlab and LaTex

231. MATLAB; a proper programming language, e.g. java, c++; a proper operating system, e.g. LINUX

232. Maple, matlab, perhaps mathematica

233. Matlab, maple or mathematica

234. Maple, Matlab or Mathematica, Octave or equivalent, JavaScript or Java, TeX, Excel or equivalent spreadsheet, Photoshop or Gimp

211 235. A student should be comfortable with some package such as MATLAB (our choice because it ﬁts with educating engineers). Our approach is to integrate it into the sophomore courses (DE, MVCalc, LA), use in num. anal. Courses, and let other courses demand what is appropriate. E.g. SAS in certain Stat courses.

236. Spreadsheets, graphing, matrix manipulation are absolutely basic; graphic differential equation solvers are essential for post-calculus courses.

237. None

238. Matlab/octave, maple/mathematica, gnu plot

239. Maple

240. TeX

241. At least one programming language (e.g. basic, java, c,) at least one general package (e.g. matlab, maple,

242. TeX and www and Mathematica/Maple/Matlab

243. Maple, matlab

244. Probably a basic knowledge of Mathematica, Matlab is useful, but not essential.

245. Maple and/or Mathematica

246. Undergrads: None required, though TeX/LaTex and computational packages Maple/Mathematica/Magma will not hurt. Grad students: TeX/LaTeX required. Computational packages would not hurt

247. Matlab and maple

248. No recommendation

249. At least one general package such as maple or mathematica.

250. Mathematica (or equivalent), C, SAS

251. Programming

252. They should learn how to program in both a high-level language and a low-level language.

253. TeX, Maple or Mathematica, some webpage creation software

254. Tex

212 255. e-mail

256. Depends upon the students and their majors. Duh! Grad students in math: tex, others depending upon ﬁeld

257. Matlab, TeX

258. Programming (e.g. C or Fortran), Matlab, Maple or Mathematica, Latex, HTML

259. A computer algebra system like Maple or MathCAD. In my opinion, the best computer algebra system is Mathematica

260. Depends on the student, though grad student should learn Tex. MY students might beneﬁt from Macaulay (some have).

261. Tex, Mathematica, some computer algebra package

262. Maple and/or Mathematica along with some Matlab. An introduction to LaTeX should be emphasized for advanced math majors.

263. C, matlab, maple, perl/python

264. I have no opinion on undergrad. All grad students *must* learn LaTeX and something like Maple.

265. At least one programming language. Tex/LaTeX

266. Some exposure to Matlab or equivalent

267. This depends on their ﬁeld. In mathematics, exposure to Maple, Mathematica and/or Matlab, or equivalents, is important.

268. Maple or Mathematica

269. For those in a scientific field: at least one of MATLAB, MathCAD, Mathemat- ica, or Maple. For graduate students: they should learn at least one application in their field of study such as Macaulay2, GAP, MAGMA, for mathematicians or SAS or Minitab for statisticians.

270. A minimum should include MATLAB, Maple, and Mathematica. A programming language, in addition to these, would be even better.

271. Some CAS, depending on ﬁeld, MATLAB, how to program.

272. Matlab or Octave, Maple or Mathematica (or Maxima)

273. Not sure (certainly LaTeX)

213 274. Undergrads need not learn much; maybe maple or mathematica for fun. Grad students should learn LaTeX.

275. Tex

276. No opinion

277. C/C++ or Fortran programming should be mandatory as should be LaTeX (probably via LyX). Matlab/Mathematica should be standard for science students. Graduate students should be familiar with Gap/Pari/Magma.

278. C++, Maple

279. Maple and/or Mathematica, Latex of course. Programming languages can be useful for many (although not all) graduate students.

280. Maple or mathematica, basic programming (C, C++).

281. It is useful for most students to know a little of Matlab, Maple, or Mathematica. For much of the core curriculum, these wont be very useful, but there are some rare occasions where they are. In certain areas involving numerical simulations, these sorts of programs become very useful. Graduate students should also all learn some form of TeX (LaTeX is really best these days).

282. Tex, at least one of MATLAB, Maple, and Mathematica, possible one statistical package also

283. Undergrad: none. Graduate students: Latex

284. Matlab, mathemtica, maple c, c++,

285. How to mix ideas and representations: symbolic, numerical, graphical−and an environment like Maple or Mathematica is useful. Then how to USE these representations in learning and communicating mathematics.

286. Matlab, mathematica

287. Not sure. It should be learned only as applications come up.

288. Mathematica, matlab, emacs, C, TeX

289. I prefer Mathematica and the soon to be released v.6.0 will be a big advance.

290. Mathematica or maple

291. TEX, matlab, java or C

292. Undergraduate: maple, matlab. Graduate: maple, matlab, tex

214 293. Tex, Matlab/Octave, Python

294. Matlab, maple

295. Latex, matlab, maple

296. I think they should learn MATLAB. However, this is a stupid question. The skills they develop in learning any software applications should be good in ac- quiring other software applications. So it doesn’t matter that much.

297. Students should not be “taught software” Rather they should learn that computers can be a good computational aid. Matlab is a good choice for implementation.

298. Whatever is useful...maple, mathematica, Macaulay,

299. Maple/matlab; also a programming language.

300. Maple; LaTeX

301. Matlab

302. Some programming language

303. SKETCHPAD, A SYMBOL MATHPULATION PACKAGE, FATHOM, SPREAD- SHEET

304. Maple and Matlab.

305. Maple and/or Mathematica

306. C and/or C++ and/or Fortran

307. Python and/or Perl

308. Java

309. Latex

310. Any software that gives experience with use of a command line.

215 4.9. In your opinion, with respect to technology, what should the education system do to better prepare mathematics students to meet the challenges of the future?

1. Give students experience using software

2. Should be comfortable with computers and know something about programming.

3. Students should be using computers at an earlier age than they are now.

4. Not overuse technology in the classroom

5. Teaching above software packages

6. Mathematics students should understand the theory behind the mathematics ﬁrst, and should then be able to use software to perform operations.

7. Keep calculators out of the hands of K-10 students and use them minimally for computations of exponentials, logs and trig functions after that.

8. Students should be aware of the possibilities and what software/technology is out there and what it can do.

9. Allow students to learn more computer language like FORTRAN and C/C++ and encourage them to justify their mathematical ideas using computers as much as possible.

10. Encourage its use in some classes.

11. Teach BOTH advantages and pitfalls of technology

12. Require students to master basic programming (e.g. in C++)

13. Better integration of computer use with lectures and research opportunities.

14. If we knew, wed be doing it

15. Teach more basic math

16. Focus less on technology and more on understanding the math

17. Use computers to experiment and think creatively

18. Everybody should know at least two programming languages.

19. Introduce them to programming. Introduce them to software applications

216 20. Teach creativity

21. Think critically; use software as an extension of their thoughts.

22. Don’t rely too heavily on it

23. To learn math in school and college one needs to use the head and not some technology...the way one learns is by thinking about it, not by letting something else think about it

24. Train them to use mathematical software of any kind.

25. More rigor, more proofs!

26. We should not over-rely on technology. It does not replace good teaching.

27. It is useful for mathematics students to take a course or two that involves sig- niﬁcant programming, but this need not be from the mathematics department.

28. We should present material in a way so that the students gain an intuition for the objects that they are studying. Whether this process requires computers, of course, depends on the situation. Often launching into a computer computation gives NO intuition that would be gotten from pencil and paper.

29. Give a sense of the unity-of-mathematics that pervades research at the cutting edge rather than pigeon holing them into “analysts”, “combinatorialists” etc. (the meaninglessness of pigeonholing can be seen in the broad array of research contributions of the most recent Fields medalists)

30. Teach the underlying mathematics.

31. Despite the way I replied to question 4, I think there can be appropriate use of technology in elementary school. But just giving students calculators without preparing teachers to use the machines for other than avoiding learning algorithms, is counter-productive.

32. Teach them to understand what types of problems are appropriate for technological use, and which arent. When technology is appropriate, how to use appropriate technology.

33. Incite them to program & try out examples, explore

34. Emphasize that technology is a means, not an end. Let students know that the real game is still played with the human brain.

35. It seems to be doing ok as is

217 36. More computer practice with examples.

37. Depends on what profession the student chooses.

38. For pure math. Pencil and paper is what counts.

39. It should make students ﬂuent in computer use at all levels. Rather than trying to pursue a willo the wisp state of the art it should attempt to make them competent evaluators of their needs. i.e. the stress should be on basic skills.

40. Enhance CRITICAL and deductive thinking.

41. Use these programs to SUPPLEMENT education, not replace it. Students should be able to do basic calculations without resorting to computers.

42. Wean them from their calculators.

43. Make students familiar with, but less dependent on, technology.

44. Learn to use the technology when needed and learn to judge when it gives wrong answers.

45. Familiarize them with computers in general.

46. Use the technology in the classroom

47. to teach the basic mathematics before students learn how to use symbolic math. Software like Maple or Mathematica. The use of calculators should be very limited at schools as otherwise students do not learn how to think

48. No suggestions

49. Provide them with earlier and more appropriate access and meaningful application

50. Emphasize basic concepts and clear thinking, not technology.

51. Nothing, they should rather increase the use of proofs in schools and undergraduate education (the current trend is “decrease” instead of “increase”)

52. Teach them to think. They can waste time with a computer later.

53. At senior/graduate level, should learn numerical analysis, linear algebra, PDEs, and other applied subjects with a system like Matlab

54. Give at least one computer assignment in each course.

218 55. I dont think any particular technology should be taught to student. A better general understanding and better general reasoning skills should help students be more ﬂexible when they need to use technology.

56. Make the math we teach more directly relevant to the needs of technology.

57. Recognize that mathematics is not used exclusively by physicists

58. Stop using calculators in grades 1-12.

59. Teach them to think critically about results that technology gives them

60. Teach students how to proactively use the various systems (not only the basic commands)

61. Start integrating technology into the classroom as early as possible.

62. Make sure that calculators are BANNED in K-12 education.

63. Teach basic programming concepts.

64. Revise teaching to use technology to play with he ideas before formalizing the,

65. Understand the big picture of mathematics and applications.

66. Students should be encouraged to learn Matlab, Mathematica and/or Maple to initially help check their work. Profs should give example code - e.g. for solutions - to help students learn by good code examples.

67. Help them learn how to use programs such as mathematica

68. Rethink curriculum in light of existence of various technologies

69. Avoid the kind of technology that degrades learning and critical thinking, such as calculators (at lower levels of education). Technology is best used for communication when it comes to mathematics. Students would beneﬁt by being able to learn from people who are far away from them.

70. Strong analytical skills applied to understanding what can be usefully computed via machine, and what are the limits of machine simulation or computation.

71. Provide a solid theoretical basis for students and then giving them computer tools.

72. It seems to me students see plenty of technology and their mathematical training is a more serious problem.

73. Go back to the basics

219 74. I don’t think technology is the major limitation. My ﬁrst guess is that resources could be put to better use elsewhere

75. Nothing. Using technology is the one thing they do anyway.

76. I don’t really know the US system

77. Teach them how to use technology and show them how it may be helpful. We talk about it, but there are not many classes (say geometry classes) where they are required to use any software.

78. Hold classes that involve computer labs that illustrate points made in (math) lecture.

79. Learn to use technology to check their reasoning; use reasoning to check and verify their use of technology.

80. No idea. Use of dynamic geometry software is a good start, but thats something already largely in place (I think) rather than a job for the future.

81. Teach them the advantages and drawbacks of mathematical software.

82. Undergraduates should be provided with laptops capable of handling symbolic manipulator programs

83. Make them ﬂuent in one or more software applications (1). Students need to be taught the motivation and applications of the theory. (2). The current problems and questions in the scientiﬁc world, (3) Asked to use computational tools routinely for homework and presentations.

84. The message should be very strong that cognition at the level of analytical thinking is paramount, and that any technology should *only* be used to support analysis and *never* used to replace it.

85. It should use it minimally when concepts and basic calculational ﬂuency is being learned. To tell a second grader that arithmetic is done by a calculator is an aberration. Most students need very little time to learn computer skills, so they should wait to the end to do so.

86. De-emphasize computer techniques and skills in favor of a greater emphasis on conceptual development of various subjects. This is especially true in basic areas like calc., diﬀ. eq., and linear algebra.

87. Take the time to teach these technologies - too much of it has to be self-taught

220 88. Help the “professors” understand that technology is another instructional tool, and can be used to support, not supplant, more traditional instruction. Some- times technology can be used to expand greatly the types of examples and, even, the types of investigations that can be done.

89. More emphasis on handling data and numerics – more work with statistical software, more work with software like Matlab

90. Spreadsheets

91. Teach students substantive ideas. Illustrate those ideas with technological support.

92. Avoid calculator dependence in the early grades.

93. Students must thoroughly master the arithmetic of rational numbers (be able to do fractions by hand) in order to have a hope of mastering basic algebra without basic algebra mastery, students are hosed no matter how fancy the software they have

94. Train teachers better.

95. Use it but not rely on it. Mathematics is not just computation.

96. Introduction to simple parallel computing

97. For many areas, including mathematics, typing is of course necessary. For many mathematicians, the computer knowledge required to use LaTeX, email, mathscinet, etc. is suﬃcient. But those who use numerical simulations, computational algebra, etc. should be very comfortable with various mathematical platforms and often some programming languages. It really depends on what the student is studying.

98. The education system should prepare mathematics students to think. As such technology should only be used to automate skills the students already have rather than to mask their lack of skills and understanding. Students who know how to solve problems could easily use new technology (or create it when needed); students who learn to use technology rarely learn to understand whats going on.

99. Reliance on software should be minimized. If done in a way that does not encourage dependence, facility with software is deﬁnitely a beneﬁt.

100. Integrate much more computation and numerical analysis into courses to illustrate test implementations of theory

221 101. Stop telling people that technology makes you understand math.

102. Don’t waste time on it.

103. I dont think that (pure) mathematics necessarily has anything to do with computers.

104. I don’t know, but I think that graphic calculators, calculators and computers have actually damaged students. They learned much more when they had to do any calculations by hand. Ask them now to do a basic operation and....

105. Provide computers as a visual aid to lectures, so that students can develop their visualization skills. My freshman calculus students seem to think that no diagram can have useful mathematical meaning.

106. For the majority if math major entering the nonacademic workplace, a familiarity with the software used in the private sector is important.

107. Teach them to understand the results coming out of the computer.

108. Focus on thought! We need people that know what the computer is doing.

109. To rely less on technology, and more on sharpening mental abilities to visualize, categorize, analyze. Mathematicians see the objects they work on with their inner eye. Paper and pencil are used for calculations and writing up results, but rarely for the creative process. No technology can signiﬁcantly enhance a sharp and ﬂexible mind; the thinking process itself must do that.

110. Students should be able to understand what the technology is doing for them.

111. Teach them to use it thoughtfully, not blindly

112. Avoid blind overuse.

113. Emphasize that the use of technology is not a substitute for thinking.

114. Students should be able to use a computer algebra system

115. Require HW solution by computer.

116. Uncertain

117. I don’t know

118. Not sure

119. More training in using software

222 120. Teach Matlab, particularly for linear algebra where its hard to do examples by hand. All students interested in applied math should learn to program in a high level language such as C/C++ or Java or FORTRAN. 121. Exploit the power of visualization to learn intuitively−interactive illustration that lets you see changes immediately: geometers sketchpad, graphic DE solvers, graphing calculator, etc. with point-click interface to change parameters. 122. Always incorporate cutting edge technology into a class whenever feasible 123. The education system should learn to use technology appropriately, which in most cases means using it somewhat less than is currently done, and in a different way from what is currently done. 124. Allow time in courses, or in additional courses, for students to solve significant problems using software. But you cant rush it - they needs time to understand the theory first, or else the computer becomes a crutch rather than a tool. 125. Introduce them in sophomore level students; after they have mastered the concepts of the material and computations by hand. 126. Use technology when appropriate in instruction and assignments 127. Teach programming. Get students very comfortable 128. Teach students about the basics of HTML, Unix, and possibly C/C++. 129. Use more applications 130. Show them how to use CASs, and when 131. Encourage more interaction between applications and theory with the aid of technology 132. I feel that the use of calculators can be fruitful in a low level undergraduate course. However, I feel that the current methodology relegates the calculator to nothing more than a crutch the student relies on whenever any sort of arithmetic arises. The idea behind using a calculator is that the student will need to know how to program a computational device to solve protracted problems. In theory this is sound but in practice most problems of this type are solved using Computer Algebra Systems (Mathematica, Maple, ...) not a calculator. The input methods are strikingly different between a handheld calculator and a full- blown computer so the main concept for using calculators in class is lost. From here it degenerates to using the calculator anytime the student needs to add two numbers. In this instance I feel removing calculators from the instruction and teaching students how to estimate arithmetic expressions without actually calculating the value precisely is a much more useful skill.

223 133. Depends heavily on field of mathematics. But all math programs should encourage familiarity with use in teaching, and communicating mathematical ideas to larger audience, be they students or public in general. For some fields, such as applied math or statistics, and even some pure fields such as group theory, the level of expertise required is much, much higher. In other areas, such as abstract geometry and topology, technology is only used as a communication tool. It might make sense for all students to be required to take some sort of mathematical modeling course, as part of their general cultural literacy - and because such courses might be a lot of fun.

134. Provide programming courses.

135. Only use technology where it actually advances the learning of the student.

136. Students in elementary school through undergraduate education need to do more problems. There should be a lower emphasis on the use of technological tools.

137. Be sure students can use it as an aid, and that students do NOT get in the habit of using it prematurely or where gains from doing so are minimal. Its very sad, for example, to see a calculus student who cant graph y = sin (3x+2) without her calculator.

138. Make sure students have the foundations in algebra, geometry, and trigonometry rather than relying on calculator.

139. I’d be happy if I thought that use of technology could help to engage students and force them to think carefully. I dont see much evidence of this, however.

140. Cut way back on the use of calculators.

141. Include software use in courses, with supporting tutorials.

142. Teach more programming courses

143. Provide workshops or short courses for basic programming techniques for, say, Mathematica or Matlab.

144. More math classes should have a lab so the students can explore the consequences of ﬁddling with parameters, etc, and develop a more concrete feeling for the material.

145. Should be competent to use some basic packages and should be able to write eﬃcient codes.

224 146. Give more emphasis to understanding computing, as opposed to uncritical acceptance of computer output

147. Programming

148. To equip students ability to program as far as they need

149. I don’t know

150. To use less technology

151. How to use computers in conjecture building.

152. Provide core courses that show how to use software packages like mathematica, maple or GAP to model and solve highly computational problems, or express approximations, so that the student can see examples of theory in action.

153. Be sure that students know what is computable, what is not, and what it means for something to be computable or not in the ﬁrst place (e.g., approximately, symbolically, or intractable). That is, learn what questions should reasonably be asked to a computer.

154. Mainly we should focus on teaching math. Isolated exposure should be enough to prepare students for future. It doest make sense to try to anticipate speciﬁc software to train them on, etc.

155. Spend more time on theory than just the applications

156. Create more classrooms that are technology ready. Have instructors who know how to use the technology properly. Teach traditional problem solving techniques along with computer-based techniques. Make every mathematics student learn a programming language.

157. Make it clear that technology must be used carefully, with attention to round oﬀ errors and other sources of confusion, and that even when there is a computational approach to a problem, thought may be better.

158. Some programming experience.

159. I think the education system needs to (continue to) evaluate how to productively use the available technology to enhance student learning.

160. Develop their brainpower; give them a good understanding of the diﬀerent parts of mathematics, a great source for inspiration. The use of computers became trivial and can be done by technicians. The mathematicians should be prepared to come with the ideas to solve the problems of the future.

225 161. Focus on the larger picture and take advantage of modern software to have more classes where students work in teams to solve substantial problems. You can use a telephone without knowing how to build one! Research mathematicians use many tools (e.g., the typical algebraic geometers use desingularization) without ever having gone through anything like the full details behind the tools. Yet, the typical curriculum makes students learn details to an extent that a good undergraduate engineering student sees and uses more advanced mathematics than a good undergrad math major. 162. Care and precision of learning and emphasis on concepts is much more important than learning to use technology. Technology may be useful to give students a better feeling for some concepts, but is often used without caution about its results. 163. Being able to compute, being able to learn math without computing are both important. But who has the time for it all? 164. Support it 165. Make more textbooks available on line, in order to reduce costs to students. 166. Make sure students can do their work with and without technology 167. Learn to use technology, but know its limitations 168. Include some computer science classes (languages, algorithms) and some math classes with computer elements. 169. Better integration of software with problem sets 170. Have lectures or discussions in a computer lab or with computer presentation 171. In an ideal world, many classes should have access to the computer during the class. 172. Use less of it 173. Eliminate calculators in K-5 and graphing calculators in K-10. Stop teaching algebra as a push-buttons-on-calculator procedure. Create hundreds of 1-5 minute videos that teachers can use to launch topics (available free, created with major input from mathematicians, and with clear ordering, list of prerequisites and support material). 174. Technology is an excellent tool for conﬁrming intuition, but it should not be seen as a replacement for computational skills. Reliance on calculators is killing students abilities to conduct the back-of-the-envelope calculations that are vital to practical usage.

226 175. Use and educate more on free software and programming: without access to the tools underlying computers, software will remain in its current bug ridden state.

176. Teach more computer usage

177. Certainly students should have nontrivial exposure to software tools. But the emphasis in mathematics education, except in areas of specialization like scien- tiﬁc computing, has to remain on the development of students conceptual and logical capabilities. Except for a few percent at the top, these remain shaky for students at every level up through graduate school.

178. Nothing

179. Understand that technology is a tool, not a replacement for a grasp of concepts

180. Stop stressing fashionable technology; return to teaching mathematics & use. Technology only as a (secondary) tool to attain that goal; modern students are already much more comfortable with technology than with the fundamental ideas.

181. Students should be introduced to various applications software at early stage of their educational experience at colleges and Univ. Mathematical Science Depts. Should be equipped with state of the art computers and application software.

182. Use and teach how to us the computer for transmitting information, retrieving information and visualization.

183. Have software that does the basics well, rather than adding bells and whistles that are of limited usefulness

184. They are already doing it

185. Use it and invest in the supporting infrastructure.

186. Maybe some introductory courses.

187. Programming training, numerical analysis, training in mathematical modeling. This should complement theoretical training in analytic critical thinking skills and methods of proof.

188. Present bigger, more realistic examples by using computers. Illustrate how computation can be used to check conjectures.

189. More courses and/or tutorials should emphasize the computational aspects of mathematics, and the use of computational methods in mathematical research.

227 190. Better integration of software in the textbooks

191. Stop using calculators

192. No opinion.

193. I don’t have any thoughts on this.

194. Critical thinking about what results you get using technology

195. Avoid making students into software operators who forsake understand of the mathematics they are working on.

196. Stop teaching self-explanatory trivialities (such as “technology” for example)

197. Depend less on calculators, so the students actually know how to think and calculate without them ﬁrst.

198. Include computation in courses.

199. Teach them some mathematics

200. Should make clear that technology is useful, but does not replace reasoning.

201. Teach them how to teach themselves software using help, logic, etc.

202. Don’t use technology until the student reaches a certain level of maturity.

203. Introduce them to some technology so that they have the option of using it.

204. Make sure that Maple and/or Mathematica is available in middle school

205. Make powerful programs more available. At the same time do not let students replace understanding with technology.

206. Learn appropriate mathematical software but not too early

207. Use technology to massage data, to perform case studies motivating analytic approaches.

208. Ensure that students have a good understanding of things from arithmetic to calculus by hand before introducing technology that is all too often used as a crutch by students.

209. Reduce over reliance on calculators

210. Get rid oﬀ ignorant school teachers (which means 99% of them)

211. Revise current course syllabus, a lot...

228 212. Emphasize the usefulness of CAS such as Maple for solving complicated equations coming from Physics/Engineering etc.

213. Teach programming, not just how to use software

214. Encourage the use of the applications in 8

215. Omit the technology.

216. Teach how to perform a computation (at least in principle) before pushing student to use some software. Computers must be used only because they allow you to save time, not because they do what you dont know how to do.

217. Teach them there are resources out there, and how to use them. Teach them the importance of being able to use software.

218. Make it clearer to faculty, teachers, parents, that one works WITH the computer rather than having it substitute for thought.

219. Make clear the proper role of technology (calculators)−an experimenting tool, rather than a substitute for critical thinking

220. Teach mathematics

221. Teach them logical thinking.

229 4.10. In your opinion, in the next 10 years, what do you anticipate will be the greatest technology advancement that will have a direct impact on your mathematical work?

1. Portable low power ePaper

2. Faster computing power.

3. Expansion in memory and speed capability will allow scripting languages to perform at levels suﬃcient to deal with applications.

4. I hope to see a commutative algebra package as powerful as MacCaulay II but with the easy interface of commercial symbolic computation software. I have often avoided computational exploration of commutative algebra problems because the coding eﬀort would be too extensive.

5. I do not expect direct impact of technology on my math.

6. Proof veriﬁcation software, if its promise is realized, will have a signiﬁcant impact on the dissemination of mathematical knowledge.

7. Digitization of library collections.

8. MATLAB

9. Development of more user-friendly, intuitive software applications.

10. Bigger computational experiments, better access to data and other web resources.

11. Development of more uniﬁed platforms for exploring multifaceted problems, e.g. Sage. Also Development of more expressive programming languages and faster computers — the limiting factor is always the time available to write the program in the ﬁrst place.

12. I have no idea.

13. I dont know

14. cpu speed

15. I cant think of any.

16. Easier implementation of mathematical experiments.

17. Preprint archives and electronic journals.

18. More (and cheaper) memory (RAM).

230 19. Unsure

20. My own software will enable me to model the systems I study, so that I can check hypothesis, etc.

21. Increases in computational speed

22. EJournals & pre-print servers will signiﬁcantly change the modern culture of scientiﬁc publications.

23. Universal access to math journals and mathematical communication (e.g. Arx- ive, Numdan, GDZ, etc)

24. Maybe better-developed web “whiteboards”?

25. Faster and more centralized access and processing of publications. The Internet has done wonders for research. Due to the increased access speed of current results, the speed of research has grown exponentially.

26. Email is an indispensable tool; mathematics has become a much more collaborative discipline that years ago. Many mathematicians are somewhat introvert in direct contact with other people, but via email, they communicate a lot.

27. Many of my research works are related to the physical problems, which lead to large linear systems. These systems should be solved fast and accurately. Therefore,

28. fast solvers for such large scientiﬁc problems will be a signiﬁcant impact on my work.

29. Universal digital library made available online

30. Probably none. Really useful innovations seem more than 10 years oﬀ.

31. Faster desktop machines with multiple processors

32. Integration of symbolic computation and numerical simulation. Basically, adding matlab simulation tools to scientiﬁc workplace.

33. Vast improvement in speed and memory for abstract algebra computations

34. Faster computers

35. On-line classes

36. Probably computations will simply get faster.

37. Better hardware. More sophisticated linear algebra solvers.

231 38. I dont know, no clue

39. Thats a good question. The technology I use now is primarily for communication rather than research.

40. Some automated proofs for some subjects

41. A better way to do videoconferes

42. There wont be any.

43. More electronic journals and faster communication of ideas directly through the Internet. Better TeX −− > HTML for things like newsgroups and e-mail.

44. Proof veriﬁcation

45. Improved symbolic manipulation software

46. Dont know...probable impact will be limited

47. Internet, etc, will allow for improved course information dissemination. Im- proved classroom facilities will allow for easy computer demonstrations & modeling.

48. Faster computers will enable more extensive experimentation in order to yield insight

49. Google Books in the way we will be able to connect seemingly disparate areas of knowledge

50. I believe that the Internet will revolutionize the way articles are published.

51. I believe that e-Ink will change the everyday handling of math research.

52. Maybe videoconferencing with large screens and some kind of writing tablet that all the people can see. If done right, this could encourage long-distance collaboration in a more natural way. Email is great, but it doesnt allow brainstorming together.

53. Over the next couple of years, multiprocessor (or multicore) systems will move from being mildly exotic to being commonplace.

54. More comfortable chairs.

55. I dont expect there to be any technological advancement that will be terribly relevant to my work.

56. Faster computation

232 57. Greater Internet access across the globe – besides aiding in communication, it will allow for more research to be done outside the oﬃce and home.

58. Improvements in available mathematical software.

59. None.

60. Workstations with CPUs with a thousand cores on a chip. My numerical work uses algorithms that parallelize almost perfectly. Having such chips (technology is already up to quad-core chips) will mean that standard workstations will give Beowulf cluster performance for such algorithms.

61. Higher computational speeds

62. Faster machines with more memory.

63. Faster computers, better training being available

64. Faster machines with more memory. This will probably be more important than software advances.

65. The placement of the past mathematical literature in scannable form onto the web.

66. Application to undergraduate teaching (computers in the lecture room)

67. Probably none on my work speciﬁcally. Faster computation and better graphical interfaces may be helpful, mostly in applied areas of mathematics and perhaps geometry.

68. Speed of computers.

69. 4 dimensional modeling and visualization

70. SAGE and the capacity to experiment that results from such software

71. Less use of computers in education.

72. I expect computational software to become more transparent and user friendly so that we will all check more examples.

73. Better computers.

74. I am not optimistic that advances in technology will be a signiﬁcant help in my research.

75. Faster computers

233 76. Research software in geometry (such as the current SnapPea, Opti). Dont anticipate much change at the teaching level.

77. Eﬃcient proof-checkers

78. Systems that will beneﬁt from the interplay between numerical, probabilistic and purely symbolic methods

79. Easily created ﬁlms illustrating concepts

80. Petascale computing

81. A better way of expressing mathematical symbolism like MathML was supposed to get us to.

82. Quantum computer

83. Much cheaper computers and universal freeware.

84. The emergence of algebraic structures, which are so complex that they can only be manipulated with help of a computer.

85. Faster computers and hence faster simulations.

86. The greatest impact will be advances in communication technologies, which enable me to communicate with students outside the classroom. For instance, I look forward to conducting online oﬃce hours with a virtual whiteboard.

87. To answer this question would be akin to pretending that I could have predicted the advent of DVDs. Who knows what technology will be available in ten years? That being said, the highest item on my wish list (as distinguished from my prediction list, which is empty) is holographic telephony: the ability to virtually visit a colleague instantaneously anywhere in the world and have (a reasonable approximation to) instantaneous face-to-face contact would be the biggest boon to mathematics since email. Note that this would not be a technology that would itself “do mathematics”, in the sense of producing models, solving equations, or verifying examples; instead it would facilitate the human interaction that ultimately is the source of all of the interesting mathematics produced by humanity. (For an opposing view, Google Doron Zeilberger and read his opinions page.)

88. Representation of dynamics of holomorphic functions (e.g. Julia sets)

89. Visualization tools still have a long, long way to go. Seamless manipulation of parameters while looking at visualization would be critical at all levels of mathematical learning.

234 90. Better graphical displays

91. Easy access to information

92. Ability of hardware to give very quick results.

93. Electronic libraries

94. Instant access to large data bases of information (digitization of libraries)

95. I dont think it will change too much in the next 10 years.

96. Probably, computers will get faster and mathematical “experiments” will give new hints about some particular open problems in mathematics.

97. Faster machines

98. Faster computers

99. Faster computers

100. High-speed broadband for video conferencing.

101. Easier to use visualization and computation software

102. Progress in quantum computing

103. Faster hardware for bigger numerical experiments

104. Greater portability of equipment; better materials on the Web

105. I think that new hardware and software will make information and communication much faster, allowing such substantially more time for reﬂection

106. I think it will have less to do with symbolic computation and the like and more with modes of electronic communication. My work is highly collaborative, and I think I will ﬁnd the development of new collaborative environments useful.

107. Improved programming tools, Improved free symbolic manipulation and graphing tools

108. Really useful teleconferencing software will support my work.

109. My work in theoretical computer science will support analysis of data gathered by advancing technology (1) Advances in the theory of mathematical and information approach to access and organize massive amounts of information/data. (2) Faster more powerful computers.

110. Larger hard drives to eventually make Tex available on laptops.

235 111. New system of access to electronic journals and books. If they end up controlled by greedy corporations, this could have a negative impact.

112. High speed internet, Visualization

113. Automated theorem provers

114. Teaching of statistics, linear algebra, diﬀerential equations, etc.

115. Increased speed and memory, increased bandwidth on internet, and algorithmic advances

116. I dont know. It will probably have more to do with whether I take a research tack which brings me into more involvement with the computer.

117. Online publishing will ﬁnally be the norm; access to all mathematical works from prior years (1900 to present); TeX writing via the spoken word (i.e. dictation of technical papers).

118. But if you ask about how technology will impact Math in general, data searches, mathematical modeling, VI, and many other areas will beneﬁt for the rapid rise in computational power and understanding of data processing. It will open up new areas of mathematical explorations, to understand new phenomenon discovered ”in the data”.

119. Improved symbolic logic software so that maple (or equivalent) is more likely to give me information in the form that I need

120. Improved visualization in higher dimensions

121. More parallelization (dual core, triple core, and so on)−more processing power.

122. Good specialized softwares, which are very well coded.

123. Internet search engine for mathematical formulas

124. Higher speed, better graphics, friendlier interfaces, easier access and download from Internet.

125. High speed internet searching and e-literature

126. An AI librarian

127. Converting handwriting INTO Latex

128. Computing power

236 129. Since my area is Computational Commutative Algebra as it is related to solving statistical problems the progression of CPU speeds to ever-higher numbers alone will have a great impact on my work. Since Buchbergers algorithm is doubly exponential the exponential growth of processing power is itself a great step forward.

130. More powerful laptops and universal wireless connections everywhere

131. Easy access to parallel processing

132. A little voice recognition would be nice but not likely.

133. Improvements in Maple and/or TeX

134. Better search for math results!

135. 64-bit processors

136. None. Mathematics doesn’t require technology.

137. I fell that pencil and paper are the most important tools for a mathematician. In the modern age, one has to use a computer to assist in the computations, to check a large number of cases, etc. My opinion is that the use of computer technology in research/education is useful (and I use it daily), but at the same time not essential.

138. Computationally little or none. However, breakthroughs in communications will have an impact on the spread of ideas.

139. Voice transcription software.

140. Computational speed and memory increase.

141. Electronic presentation of journals.

142. Better algorithms

143. Availability of cheaper and faster cluster computers for research

144. Better access to internet (e.g., wireless access at conferences and business loca- tions)

145. Computer projectors in each mathematics classroom. Computer labs so students can become familiar with at least one good symbolic calculator like Maple (and even program a little in it to experiment on their own).

237 146. Since I work almost entirely in an area of pure mathematics removed from computation, the most direct impact would be in communication. It is already possible to scan in equations and send them as graphics, but software recognition may allow these to be translated to symbols. This method would be more eﬃcient than LaTeX (but errors in recognition may make it impractical).

147. Mathematical on-line search - so far tools like Google can only search through words or images, more contextual search is necessary to be able to search for mathematics.

148. More, better: I anticipate there will be a continued improvement in algorithms and implementations. I also anticipate (and this will be more diﬃcult for my personal “adjustment”!) that interfaces will continue to change (what a pain!). I do not anticipate any (sigh!) convergence of standards for syntax etc. (It WOULD be really neat if {} [ ] ( ) turns out to mean the same in various systems.) I do anticipate that CPUs etc. will get faster, but notice that I mentioned hardware last – for me, the other things are more signiﬁcant.

149. Googles scanning of textbooks.

150. Not sure; things change so rapidly that predictions may be useless.

151. Computing speed

152. 10 years is a LONG time. Faster, better graphics available on laptops will have a good inﬂuence.

153. Better PDE solvers

154. Advanced in communications technology (video over IP and virtual whiteboards) will assist remote ecollaboration.

155. Greater computational speed and faster memory

156. Speed! I would like to be able to check some more cases

157. More specialized software; increased processing speed.

158. Improvements of existing technology, including video conferencing. Better parallel environments for simulations.

159. Maybe proof checking

160. Development in the technology that my area of applied math is applied to.

161. Large scale computation

238 162. Increases in computational speed, possibly quantum computing.

163. Smaller, faster, more powerful computers/calculators...

239 4.11. Please feel free to provide additional comments regarding technology in mathematics and/or mathematics education.

1. Computer techniques are, of course, useful for certain areas of mathematics, often for ad hoc reasons, in that they make available for inspection and resulting conjecture examples that are too complicated or tedious to work out ”by hand.” But this is an opportunistic technique, and there are large swaths of mathematics where they simply dont apply.

In basic math ed, however, computer skills have been greatly over-emphasized, to the detriment of students absorption of basic ideas. For instance, despite my protests, a MAPLE component is included in my multivariable calc courses. The result, as far as I can see, is that students are distracted from basic and diﬃcult ideas by the technicalities of MAPLE. The use of the latter doesnt even include graphic methods that might elucidate certain ideas, e.g., alternative coordinate systems for volume integrals, or vector ﬁelds and line integrals.

Philosophically, I think it would be much better for students to master the mathematics ﬁrst, and only subsequently take a dedicated course on computer methods in calc., diﬀ. eq., and linear alg. The latter might be better taught in a comp. Sci. dept., though the math dept. could certainly handle it.

2. I think that one should rarely use computers to investigate a field which one is still learning the basics. They should mainly be used to perform computations in fields which one already understands in order to be able to more effectively study new fields.

3. We teach what we were taught, not what students need.

4. I do think that pencil and paper is the best way to learn to calculate. Doing- Yourself-by-Hand is the only way to develop the right intuition for mathematical concepts. The use of computers in math education is a perfect way to hide the lack of mathematical intuition.

5. Technology in low-level classes is more of a hindrance. In calculus and diﬀer- ential equations, students need to understand concepts and there are enough examples that can be done by hand. The computer is a waste of valuable class time.

6. Computers are a great tool for mathematical research. Calculators are a guarantee of lazy and ignorant students.

7. Mathematical journals freely available on the web

240 8. More parallel computing available. Far too many universities have many computers in clusters for people to read email, surf the web or look up books; these can be converted to a parallel computing cluster (while still allowing students to access these basic functions).

9. I believe that it will be increasingly important to 1. Convince students that the math they are learning is relevant to the real world; 2. Oﬀer math modeling courses; 3. Highlight the connections to other branches of math & to other sciences.

10. Enhanced visual implementation of lectures and presentations of basic undergraduate math courses.

11. Generally, there is too high a barrier to entry for eﬀective use of technology. A GUI interface in the beginning, transitioning to command line for advanced work, would be useful for things like maple and mathematica. This may now exist, as I have not looked lately. Too busy.

12. I think that in terms of education, children and students should use computers to learn a programming language, because this teaches logical and sequential reasoning.

13. Otherwise, I don’t think computers or any other technology will ever play a big role in making a person a better mathematician, apart from what I mentioned above (communication, email, etc). On the contrary, I think that the present education based on fancy demonstrations and colored textbooks (that look like childrens books) has been responsible for the decline in math performance.

14. Mathematics is an art that is performed before the inner eye, and a mind needs years of training to become really powerful in it. No tool can replace of the mind. Trying to simplify the visualization process is detrimental to the development of this mental skill.

15. As a comparison, technology has not played a big role for professional musicians in furthering their abilities to be creative. For the human ability to become so good in an area to pass from the level of being strong to being profoundly creative, technology can at most serve as a tool to simplify some simple tasks, or to improve the quality of the given tools. It can never by itself be very important in leading a person to mastery.

16. However, being strong in using computers to simulate complex systems, and to ﬁnd good conjectures about such systems certainly is extremely important in certain areas of mathematics. However, “experimental mathematics” is an area in itself, and should be distinguished from other ﬁelds in mathematics.

241 17. The technology should never come ﬁrst. It should be used to back up, illustrate, support research and teaching.

18. The math department here has examined many, many web-based homework/testing applications offered by textbook publishers in support of their low-level texts over the past 7 years. None of them have been mathematically correct—for each program, for some problems, the program either accepts incorrect answers as correct, or rejects correct answers. These are design problems—there is no semantics behind the formula manipulation (or the floating-point calculations, in the case of WebWorks) in the programs. It is incredible (literally) how easily one starts to accept this type of behavior, telling students they have to type answers in special forms (“sqrt(100)” instead of “10”, e.g., because every other answer is “sqrt(something)”), etc. Ive talked to one publisher that had a programming team in Moscow, one in the US, and one in at least one other country, which I forget, and their program, while visually attractive and with a great database backend to process grades, etc., still suffered from these problems. There needs to be more rigorous attention paid by mathematicians to these problems.

19. Some of the questions (eg, #4 on this page) I cant answer because you might mean diﬀerent things. For example, “appropriate” technology is appropriate at all levels. However, over-reliance on inappropriate technology (eg, graphing calculators) before understanding can actually hinder learning. Students who use calculators heavily in early grades often fail to learn number sense, and so have no feeling for reasonableness of answers. Similarly, while graphing calculators and CAS systems open a lot of interesting avenues to students in algebra and calculus, many of them fail to learn “function-sense” because of over-reliance on the technology. They

20. They are two unrelated things.

21. Mathematicians are late to realize the importance of computers. The ﬁrst computer scientists were mathematicians who lost patience with the typical mathematicians disregard of computers, which have completely revolutionized science and technology. There is similar friction between pure mathematics and applied mathematics. Change is beginning to happen faster: funding agencies realize the importance of mathematics to science and engineering and are shifting the mix of funding in ways that favor the parts mathematics that mixes with science and engineering. These parts of mathematics are technology intensive (relying as they do on strong computational hardware and very fast networks).

22. Use of technology for grade students or undergraduates, often does not help them understand mathematics.

23. There has to be developed math skills and then technology added

242 24. I think computers are over-used in the lower levels in math education. Theres no need for a math student to use a computer until they get to graduate school. Students grow too dependent on their calculators and dont learn the basic concepts. But when they start to get into the harder stuﬀ in grad school, they need computers to do advanced calculations. Computers should come later, not earlier.

25. Most students eventually learn how to manipulate mathematical symbols, or get a calculator/computer to do it for them.

26. Every problem in the “real world,” be it in engineering or management or personal ﬁnance, naturally appears as a ”word problem.” The ability to translate these into symbolic notation is a separate and equally important skill. Perhaps there is (or soon will be) a technology for doing this.

27. I hope not – at that point it will be possible for a student to “learn” all elementary mathematics without understanding anything.

28. Technology can be a great facilitator - but today it is often misused. Calculators in schools should be forbidden.

29. In my experience students did not have any problems learning how to use math software. They only had trouble trying to translate a concrete math problem to a problem solvable by a machine.

30. Math software is great for showing students things they cant draw on their own. It becomes harmful when they rely on it too heavily, as they become unable to function when denied access to a computer or calculator.

31. My calculus students cannot add rational functions. They have learned to add number fractions on a calculator and do not understand common denomina- tors. Technology is a wonderful tool but introducing it too early can stunt the development of important mathematical skills.

32. Dont overdo technology in classrooms. Blackboard and chalk go a LONG way and are unfortunately underrepresented in education.

33. Do proofs in high schools instead of only pretty pictures.

34. Students in K-12 must develop ”number sense” by learning to do computations without calculators. Calculators should be a supplement only.

35. Sometimes students get excited about using computer tools and do not realize that before meeting the computer they need to have a clear understanding of the problem they are trying to solve.

243 36. Get calculators out of the hands of kids under 10! They should be learning how to think, not how to use a calculator. Everyone should be able to add two digit numbers in their head, and perform “order of magnitude” calculations e.g. 30 times 40 is on the order of 1000. These are essential skills to understanding the world around us. But these skills wont be gained while kids are given the easy way out with a calculator.

37. The use of calculators in elementary schools has been very harmful to the math education of this country. A college freshman tends to reach for his/her calculator automatically when he/she sees 1/2 + 1/3 in a calculus problem.

38. Teachers of mathematics should be careful not to allow their students to replace understanding of concepts with technological facility; both are important. Technology should be used in instruction, but used sparingly.

39. I dont understand the idea of splitting into 5 categories of mathematical en- deavor. One of the categories listed is proof. Proof is used in all the other four categories, showing existence, generalizing, characterizing, and making applications. In applications, one makes generalizations. To categorize, one often generalizes an idea from one case. And so on, the division lines among the ﬁve are artiﬁcial.

40. By the way, this took me 30 minutes, not 15. In my experience, the estimates of time to do surveys always seem to be based on the assumption that one does not need to think about the answers. Is that really what you would like your respondents to do?

41. One ﬁnal question: why does this form remove all my apostrophes? It makes me look illiterate!

42. It is a great tool for illustrating various examples in a classroom, especially visualization of graphs, PDE behavior, etc. (if you can get the technology to function– blackboard never fails...) For my research it is utterly insigniﬁcant.

43. The graphing calculator is not help students to think critically the process and procedure of problem-solving. Students appear to be able to do rather advance mathematics processes like diﬀerentiation and integration without knowing what they mean, nor can they add fractions well.

44. Technology can only be eﬀective if it is used properly in the classroom. Creating a generation of students who are unable to add two fractions together without the help of a calculator would be a failure of technology.

45. I think that elementary math education relies too much on technology. Technol- ogy should only be brought in later, once students understand basic principles.

244 46. Of course, technology should be used to support and improve mathematical understanding, not instead of it.

47. I think several of the questions suggest a relationship between computers and mathematics, while it is not clear that there is such a relationship (depending on ones interests). I didnt want to answer question 6 for this reason.

48. I see too many students in the lower-level math courses who cant add fractions anymore, or even multiply 2 whole numbers, because they have been too dependent on calculators. They need to be taught how to become proﬁcient at these skills ﬁrst & then use calculators only to add to their knowledge

49. It can be overused.

50. The questions above seem to assume that all uses of technology produce like results. The real issue is not the speciﬁc technology used but rather the way it is used,

51. Should not substitute for learning concepts

52. Although future scientists will require considerable training in the basics, it is more common and easier to expect too much of technology. It is naive to believe that it will be the center of the world of science. It is a tool, perhaps as basic as the book, but still a tool.

53. Students should learn programming and computer algebraic systems.

54. I have some concerns about long term support of some software programs – particularly free ones like GAP but even ones that charge like MAGMA.

55. I really dont think technology has much place in math education. We all tried this experiment with freshman calculus throughout the 1990s and we found that its just a distraction and doesnt in any meaningful way actually help students learn math.

56. There is a role for technology, but it is overdone at the expense of more valuable training of young minds.

57. Kakutani used to say (something like) ”Doing mathematics is like climbing a mountain. The more tools you have, the higher you can climb - the higher you can climb, the greater the view.” All tools are valid if the student can use it to climb higher and to connect with students. Mathematicians need to be more open to showcasing all the tools out there.

58. We must do a better job integrating mathematics software into our courses while still emphasizing the theory.

245 59. More students will bring laptops to class?

60. The main challenge of the future (and of the present) for most students (and not only in mathematics) is the critical lack of critical thinking and learning skills due to the absurdly incompetent mathematics education in K-12. To handle the problem, one should start - ﬁnally! - teaching mathematics and other deep and non-trivial subjects (such as programming, physics, chemistry, biology, history, languages, etc.) Teaching ”technology” only takes away valuable time and intellectual resources of students - and teachers.

61. Technology as introduced today makes students less likely to think about the material but to try simply to do ”brute force” computations, leading to less understanding. Unfortunately, in Calculus, it seems impossible to input really interesting (and often discontinuous) functions.

62. Need more development of web materials as aid in teaching. I dont favor all-web instruction except where classroom training is impractical.

63. Computers are terriﬁc to produce results. They are much less useful for understanding things. Therefore, I am very skeptical of the usefulness of computers in mathematical education.

64. My responses about the importance of technology to Mathematics are based on my view of Mathematics as a whole, not necessarily on my day-to-day work in Mathematics. Thus, while I view technology as very important to the development of mathematics, or even to my area of mathematics, this does not mean I use – or more to the point, enjoy using – computers a lot. (I sometimes write algorithms; but I leave implementation to others.) As for the teaching of Math- ematics, I think it is very important for students to learn about the interactions between mathematics and computation. That does not mean, however, that I think lectures are best given in PowerPoint. (Far from it!)

65. Technology has a major role to play in mathematics; I hope to see it more fully utilized by students for producing, say, 3-D images in several variable calculus, or understanding diﬀerential equations

66. Computers and software are not used as much as they should be in most undergraduate courses.

67. Technology should not be a subject. When suitable tools are developed (e.g. maple at the undergraduate level, cognitive tutor at the high school level, they should be integrated into the teaching.

246 68. Mathematics happens in peoples heads, not on screens or hard disks. At present, still the best educational materials are excellent books that lead students through a subject in a systematic and comprehensive way that develops extensive cognitive skills and knowledge base.

69. Calculators are abused in lower school education and the result is that students do not calculate well

70. Many outstanding conjectures in arithmetic geometry (like BSD) were inspired in part by numerical evidence. The most common attitude among workers in this area appears to be that, now that the general pattern seems to be known, the acquisition of further numerical evidence is usually more trouble than its worth. This bias probably arises at least in part from a lack of computer literacy on the part of many older mathematicians (who are often the leading ﬁgures in the ﬁeld).

71. Bringing technology into the classroom is time-consuming for faculty, and can be distracting for the students (I have spent a lot of time doing this over the last twenty-ﬁve years). It is hard to ﬁnd the right balance between the use of technology and traditional mathematical reasoning.

72. Technology should aid mathematics education but not drive it.

73. Of course, all of this begs the question, that first you must learn to think. And thinking is like any strenuous exercise, it has to be nurtured and rewarded for it to continue. This is why technology has little role in the formative years (K-12) IMHO, while in more advanced courses, it begins to integrate better with the thought processes (visualization) and even to show mathematical applications, which motivates students to study further, harder and deeper. And of course, in Applied Math fields, it can define the limits of understanding, such as modeling discrete processes or physical processes. At the upper level undergraduate and graduate levels, and postdoctoral levels, I expect we aint seen nothing yet.

74. Regarding graphing calculators, I think calculus students need to be taught an extensive methodology for graph sketching, which requires only the application of the mind. This needs to come ﬁrst, and use of a graphing calculator should only be taught once the learner is an expert at sketching an astounding variety of functions by hand (and ”by mind”). I think students need this level of sophistication before they can truly appreciate what a graphing calculator *really* needs to be used for, viz., plotting functions which are much too diﬃcult to analyze with pen-and-paper methods. Also one needs this level of experience to understand what the grapher is doing, how its generating the plot, and why the plot is sometimes misleading.

247 75. Technology will play an increasing role, but traditional capabilities and methods must also be retained.

76. Mathematics is losing ground (and respect) in being connected to the science and engineering world. This should not be, and it is harmful for both the math world and the science world. The solution I think is in making sure students are completely conversant in computer technology & science/engineering technology.

77. We all know we should use machines in teaching. e.g., calculus; but it takes so long to prepare ”formal presentations” for this purpose, that we dont. The problem has been solved for writing with LaTeX and formal lecturing with easy-to-use packages (e.g., beamer), but not yet for teaching.

78. Technology can harm mathematical learning by giving the illusion that the computer can understand the problem for you, as when grade-schoolers use calculators instead of learning the fundamental algorithms of arithmetic. It can help when used modestly to work out basic cases, perform experiments, and manage large data sets.

79. Computers are seldom useful in writing proofs, and I dont expect this to change. An exception is the Wilf-Zeilberger method for verifying identities, described in the book ”A=B”.

80. Some students rely on the calculator too much.

81. Matlab can be a useful tool in Calculus II and III.

82. After the students learn to think, then the computer is useful to amplify their thoughts. Introducing software too early distracts from the self-development process.

83. Technology is essential to the advancement of mathematical ideas. More and more areas of science are relying on mathematical techniques and tools to get answers, and educating students early help to promote a healthy discourse on the usefulness of technology in education.

84. I responded earlier that technology is very important to mathematics. There was not there an opportunity for precision. More precisely, I think technology is critical for some areas and unimportant for others.

85. Make all the info available online.

248 86. I think technology should be used more in presenting math to students. The problem with that is that it takes too much time to prepare a good presentation. It is, however, clear to me that one can make many ideas more clear and accessible by nice graphic presentations.

87. For my research I write computer programs to test my conjectures on small examples, which are either impossible or too messy to do by hand. I had good experiences with some specialized softwares. Sometimes I write my own code from scratch. General softwares were too slow for most of problems I was interested in.

88. Take the calculators away from high school students and calculus students! That was the worst idea in math education since the ”new math” of the 50s and 60s.

89. Too much software focuses too much on learning syntax, not mathematics. But ﬁnancial support is too hard to obtain – it is impossible to maintain good software, like MacMath (Springer-Verlag) to keep pace with changing systems. Program updates tend to add features I dont want, and take away features I do –especially egregious is the loss of keyboard shortcuts and necessity to do intricate mousing which is physically exhausting and requires a lot more coordination and control than I often have.

90. Encourage its use

91. Technology is no silver bullet. If you cant compute toy examples by hand, you do not understand the concept. There is no understanding without technique.

92. I think, technology is extremely important (also) in mathematics, but its weight is not the same in diﬀerent ﬁelds. It should be teached and used, but not fetishized.

93. I think we are light years from where we should be with respect to technology. This questionnaire is limited by the scope of todays software.

94. Why didnt you ask speciﬁcally about Google? Thats the biggest advancement in research helped by technology by far!

95. In designing the survey, it would have been better to separate out e-mail, TeX, and the World-Wide Web from computation used directly in research.

96. In short I feel that many of the technological innovations that have come around that deal with the presentation of mathematics to students is fairly decent. The innovations that have come out to let students interact with technology, on the other hand, have been sort of ﬂops. One reason I guess is that with only one

249 person in each class using the technology for presentation and roughly 20-30 people in each class using it for interaction the chance for encountering bugs is much greater in the second scenario. In this sense the ﬁrst type of use can function while being much less robust, the second type of use, however, must be fairly robust to even begin to be useful. (Of course once a tech fails in class its usefulness becomes suspect and this will drive people away from its use.)

97. I think technology has some place, but perhaps a slightly overrated one.

98. Technology is often used as an excuse for students having studied, while nothing can replace working through problems with P&P in my opinion.

99. It doesnt necessarily help (sometimes it is a hindrance, e.g. computer grading systems now replacing human graders).

100. I do not feel comfortable answering questions containing the word ”should.” How can I know all the circumstances that aﬀect someone else?

101. Most mathematics Departments (including the best and most prestigious) are very reactionary in the use of technology. Many professors believe technology is actually harmful to students. Universities dont provide up to date computer equipment to their faculty.

102. I think you will not get a good response to this survey by administering it *using technology*. Your respondents are going to be pre-disposed to using technology more than those who ignore your request.

103. A little technology is good for education, but theres no substitute for pencil- and-paper and interaction between student and instructor

104. Software companies, Wolfram and Maple for instance (but theyre not the only ones), have added lots of ﬂuﬀ to their latest software (graphical or otherwise), which detracts from their usefulness as research tools and sends the wrong message that they are meant for teaching mainly. Perhaps the time is ripe for them to make distinction between their research and their teaching products.

105. Your questionaire talks about daily/monthly/etc. In fact, my use of computer technology is wildly variable. Right now I am working on a problem in pseudo- Riemannian geometry and using Mathematica for the ﬁrst time. I will be spending a huge amount of time on the project. But once I get insight into what is going on, I wont need mathematica any more. I have written FORTRAN code in the past to study problems. When I have a relevant problem, I use the computer. When I dont, I dont. So the standard deviation is huge and the categories daily/monthly/ are very misleading. I dont believe in the sorts of questions you give above and provided no answers.

250 106. I think its very dangerous that educators often regard technology as a magic bullet that can replace thinking (both by the students and by the teacher)

107. I do not understand what you are trying to do. What mathematicians do with technology has no bearing on what should be done with technology in Math Education. These are activities with completely diﬀerent goals done by people with enormously diﬀerent levels of education and expertise.

108. I am very concerned about students dependence on technology as a crutch rather than a tool.

109. I think too few people understand the technology they use. This leads to a dumb ”worship the machine” culture, most visible in students having absolute trust in absurd results of nonsensical computer calculations. A computer can do certain things much faster (or more accurately) than I can. For example, numerically integrate an ODE or typeset my papers. However I can perform all these tasks myself, in principle (only much slower). Whether I write my own Runge-Kutta solver, use one from the ”Recipes” or use the one supplied with Maple is a matter of choice, but all uses are based on me being able to do the same by hand with pencil and paper. I believe students should not be allowed to use (or exposed) to technology they do not fully understand. In most cases, a student who has not obtained full understanding of a skill should not be using a computer to cover for that lack of understanding. For example, students who were exposed to calculators before completely understanding arithmetic never learn it. The completely lose the feel of numbers as concepts and of the algebra underlying the integers. Of course, in practice a calculator is the way to perform arithmetic, but a student who cannot tell how large a product of two numbers will be, or a student who cannot form a strategy for dividing two large numbers [understanding that carrying out the calculation might take a while], are cripples, no less than a healthy student who nevertheless insists on riding in a wheelchair.

110. I do use maple during my lectures in calculus and I am ﬁnding that the good graphics help signiﬁcantly in the understanding of the topics.

111. Technology should be less commercial

112. I feel strongly that 1 dimensional input is far superior to 2 dimensional input (which should be eradicated).

113. I think it essential to use technology, but important not to let it wag the dog.

114. There are lots of ambiguous questions in this questionnaire. For example, the question of whether I prefer a GUI or CLI. Maple just junked a completely

251 workable CLI for a broken GUI. So I prefer a GOOD interface to a BAD one, and I dont care what kind it is.

115. Making us choose, in this survey, between GUI and command line, is unneces- sary. I use a Mac with UNIX underlying it. I use both Gui and command line constantly. In my research if I need to make a computation with maple of C I do it.

116. Most use of technology is email and word processing (latex). I am very involved in K-12 math education and although it might be possible to use calculators wisely, wisdom is a common as common sense, i.e. not very common. It is better to keep calculators away from kids, when they need them it will take them 15 minutes to learn them (ever watch them with a 50 button remote without a manual?). Let them learn some mathematics.

117. I think it is truly disgraceful that the most popular general-purpose symbolic manipulation packages are not free software and are not available with source code; without source code, any results produced remain unreliable. Likewise in education, we are doing students a disservice by recommending packages like Mathematica, which violates all principles of academia.

118. Technology is a double-edged sword. It can be used very eﬀectively to demon- strate examples of mathematical principles, and animation can lead to a better geometric intuition. It can also lead students to be lazy if all of the problems only require a student to ask the computer to compute rather than require that the student understand the mathematics.

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