A Vision: Mathematics for the Emerging Technologies A ...c.ymcdn.com/sites/ Mathematics for Tomorrow: Recommendations and Exemplary Programs Mary Ann Hovis Robert L. Kimball John C

A Vision: Mathematicsfor the Emerging Technologies

A Report from the ProjectTechnical Mathematics for Tomorrow:

Recommendations and Exemplary Programs

Mary Ann HovisRobert L. KimballJohn C. Peterson

NSF Award #: DUE 0003065

This material is based on work supported by the National Science Foundation under DUE grantno. 0003065. Any opinions, findings, and conclusions or recommendations expressed in this paperare those do not necessarily reflect the views of the National Science Foundation.

PERMISSION TO PHOTOCOPY AND QUOTEGeneral permission is granted to educators to photocopy limited material from A Vision: Math-ematics for the Emerging Technologies for noncommercial instructional or scholarly use. Per-mission must be sought from AMATYC in order to charge for photocopies, to quote materialin advertising, or to reprint substantial portions of the document in other publications. Creditshould always be given to the source of photocopies or quotes by citing a complete reference.

ISBN 0-9643890-1-0

Copyright c©2003 by the American Mathematical Association of Two-Year Colleges

Cover design and internal art by Frances Kimball.

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Individual copies of this document may be obtained by contacting AMATYC. A limited numberare available. When the supply is exhausted, copies will be made available at a moderate charge.

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Contents

Preface iii

A Vision: Mathematics for the Emerging Technologies 1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Pedagogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Keeping Current . . . . . . . . . . . . . . . . . . . . . . . . . 17Maintaining a Successful Program . . . . . . . . . . . . . . . 19Other Participant Recommendations . . . . . . . . . . . . . . 22Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Bloom’s Taxonomy 27

National Conference Participants 31Facilatators and Recorders . . . . . . . . . . . . . . . . . . . . 34

CRAFTY Workshop Participants 35Biotechnology and Environmental Technology . . . . . . . . . 35Electronics, Telecommunications, and Semiconductors . . . . 36Information Technology . . . . . . . . . . . . . . . . . . . . . 36Mechanical and Manufacturing Technology . . . . . . . . . . 37Mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . 38Project Evaluator . . . . . . . . . . . . . . . . . . . . . . . . . 38

Advisory Committee 39

i

ii CONTENTS

Preface

Over 80 people participated May 12–15, 2002 in a national confer-ence to create a vision for the mathematics for the emerging technolo-gies. The meeting was held at the San Remo Casino and Resort, LasVegas, Nevada under the auspices of a National Science Foundationfunded grant “Technical Mathematics for Tomorrow: Recommenda-tions and Exemplary Programs” (DUE #0003065) that was awardedto the American Mathematics Association of Two-Year Colleges (AM-ATYC).

The tone for the meeting was established during the opening sessionwhen Project Director Mary Ann Hovis said,

The purpose of this conference is to identify and promoteexemplary practices and use them as a basis for developing avision and making recommendations for mathematics pro-grams which serve students in highly technical curricula,such as biotechnology, computerized manufacturing, elec-tronics, information technology, semiconductors, and telecom-munications.

Participants in the conference included mathematics educators, tech-nical personnel from business and industry, and technical faculty fromtwo-year colleges. A complete list of participants is on pages 31–34.

Preliminary work on the vision was done during two CRAFTY1 CRAFTYworkshops in October 2000. The CRAFTY workshops were part ofAMATYC’s NSF-funded grant. They were patterned after, and donein collaboration with, a series of workshops conducted by the CRAFTYsubcommittee of the Committee on the Undergraduate Programs inMathematics (CUPM) of the Mathematical Association of America(MAA). Participants in the CRAFTY workshops are listed on pages 35–38.

The CRAFTY workshops were held in discipline areas served bymathematics, and the CRAFTY reports attempt to describe to math-ematics faculty what skills should be developed by the mathematicscurriculum during the first two years of college.

1Originally, CRAFTY was an acronym for Calculus Reform And the First TwoYears. At its January 2002 meeting, CRAFTY elected to rename itself as theSubcommittee on Curriculum Renewal Across the First Two Years.

iii

iv Preface

At AMATYC’s CRAFTY workshops, technicians used a set of ques-tions developed by the MAA to address the mathematics needs of em-ployees and/or students in the emerging technology areas of biotech-nology; electronic engineering technology, semiconductors, and telecom-munications; information technology; and mechanical and manufactur-ing technology. Copies of these reports can be downloaded from theAMATYC web page or the project web page.Project and CRAFTY web

pages The CRAFTY web page contains information about CRAFTY andlinks to all the workshop reports. Additional information as well aslinks to CRAFTY articles that have appeared in MAA’s newsletterFocus can be found at Focus web page.

The national conference began with a keynote presentation on “Com-munity Colleges, Technology, and Economic Opportunity: What’s Nextin the New Economy” by Robert Templin, Senior Fellow at the MorinoInstitute in Reston, Virginia.

The Project had an Advisory Committee of seven members. Thenames and brief biographies of Advisory Committee members are onAdvisory Committeepages 39–40. The Advisory Committee first met in January 2001 andestablished criteria for selecting exemplary programs. In May 2001,the Project’s National Advisory Committee selected ten programs itdeemed exemplary in one or more aspects.

Each of the exemplary programs was honored at the conference.Exemplary programsRepresentatives from seven of the programs were able to attend, andeach gave a short focused presentation on one aspect of his or herschool’s program. These presentations and the abstracts that werein the conference materials provided participants with examples theycould use in developing the vision.

The preliminary part of the conference concluded with presentationsby representatives from each of the emerging technology areas that wereNational conferencethe basis of the CRAFTY meetings in October 2000. These talks pro-vided a synopsis of the thinking by the technology groups. Participantswere also given the latest versions of the CRAFTY reports.

For the majority of the conference, participants outlined a visionfor the mathematics needed in emerging technologies. Each participantwas placed in one of six groups, and each group had a representativefrom at least three of the CRAFTY areas, a member of the AdvisoryCommittee, and at least one representative from an exemplary program.The rest of the people in the groups were mathematics educators.

The groups were led by Facilitators and Recorders, listed on page34, who were chosen by the project investigators. The Facilitators andRecorders met prior to the conference to establish goals for their groups,examine the proposed deliverables, and discuss general procedures.

One of the deliverables was a set of six matrices. Each group wasgiven a set of skeleton matrices to use as discussion guides. Informa-tion from all the matrices was consolidated for this report and helpedto define the vision for the mathematical needs of students in emerging

http://www.amatyc.org

http://www.waketech.edu/~rlkimbal/CRAFTY/

http://www.mathsci.appstate.edu/~wmcb/CFF/

http://www.maa.org/features/currfound.html

v

technologies. In addition, participants generated a set of recommenda-tions for additional activities.

This Vision is being distributed to all two-year colleges, all AM-ATYC members, and all participants in the CRAFTY workshops orthe national conference.

Many people worked to produce this report. Representatives frombusiness and industry and faculty from the technologies gave of theirtime to contribute to the Vision. The Advisory Committee was al-ways prepared for our meetings and had many excellent suggestionsthat made the project more successful. George Vaughn, the Projectevaluator, gave us many helpful suggestions on how to improve thequestionnaires.

The AMATYC leadership and office staff, especially President Su-san S. Wood, who initiated the proposal for this project, Treasurer IlgaRoss, and Accounting Director Christy Hunsucker were instrumental inthe success of the project. We are also grateful for the support of thecurrent AMATYC President, Philip Mahler, the AMATYC officers, andother members of AMATYC’s office staff who gave us their support.

The project’s investigators would also like to thank all the partici-pants in the CRAFTY workshops and the national conference for theirhard work and patience. The Facilitators and Recorders at the nationalconference were asked to do project work before, during, and after theconference. We are especially grateful for their efforts.

This project was interrupted by the events of September 11, 2001.The conference was scheduled for the weekend after those events, and, ofcourse, had to be postponed. Most participants stayed with the projectand attended the 2002 conference. The schedule change did, however,prevent three of the exemplary programs from sending representativesto the conference.

It is also appropriate to thank the hotel staff of the San Remo forallowing AMATYC to cancel the scheduled room block and reschedulethe events without penalty.

Note: All supporting documentation for this vision statement—thematrices, abstracts of the exemplary programs, CRAFTY reports—canbe downloaded from the AMATYC web page or the project web page.

Project Directors:

Mary Ann HovisJames A. Rhodes State College, Lima, Ohio

Robert L. KimballWake Technical Community College, Raleigh, North Carolina

John C. PetersonChattanooga State Technical Community College, Chattanooga,Tennessee

http://www.amatyc.org

http://www.waketech.edu/~rlkimbal/CRAFTY/

vi Preface

A Vision: Mathematics forthe Emerging Technologies

A national conference to establish a vision for the mathematical needs ofstudents in emerging technologies was held as part of an NSF-fundedgrant “Technical Mathematics for Tomorrow: Recommendations andExemplary Programs” awarded to the American Mathematics Associ-ation of Two-Year Colleges (AMATYC). Participants included mathe-matics educators, technical personnel from business and industry, andtechnical faculty from two-year colleges. This Vision document is theresult of that conference.

Introduction

Preliminary work on the vision was done during two CRAFTY2 work-shops in October 2000. The CRAFTY workshops were part of AM-ATYC’s NSF-funded grant and were patterned after, and done in col-laboration with, a series of workshops conducted by the CRAFTY sub-committee of the Committee on the Undergraduate Programs in Math-ematics (CUPM) of the Mathematical Association of America (MAA).

Introdu

ction

The CRAFTY workshops were held in discipline areas served by math-ematics and the CRAFTY reports attempt to describe to mathematicsfaculty what skills should be developed by the mathematics curriculumduring the first two years of college.

At AMATYC’s CRAFTY workshops, technicians used a set of ques-tions developed by MAA to address the mathematics needs of employ-ees and/or students in the emerging technology areas of biotechnology;electronic engineering technology, semiconductors, and telecommuni-cations; information technology; and mechanical and manufacturingtechnology.

The NSF Project had an Advisory Committee of seven members.The Advisory Committee first met in January 2001 and establishedcriteria for selecting exemplary programs. In May 2001, the AdvisoryCommittee selected ten programs that it deemed exemplary in one or

2Originally, CRAFTY was an acronym for Calculus Reform And the First TwoYears. At its January 2002 meeting, CRAFTY elected to rename itself as theSubcommittee on Curriculum Renewal Across the First Two Years.

2 A Vision: Mathematics for the Emerging Technologies

more aspects. Input from the exemplary programs and the CRAFTYreports formed a basis for the dialogue on issues surrounding the math-ematics needs of students in emerging technologies.

Issues discussed included

• the integration of mathematics content with science and technol-ogy;

• the balance between abstract and applied mathematics;

• pedagogy;

• the use of technology;

• prerequisite skills;

• the nature of ongoing professional development for faculty teach-ing technical mathematics;

• connections among mathematics faculty, technical program fac-ulty, business and industry; and

• transferability of the mathematics courses for students in emerg-ing technologies.

In order for students, especially those preparing to become techni-cians, to be prepared for the workplace, they must acquire a set of “softStudents must acquire

a set of “soft skills”that prepare them forthe workplace.

skills” that are often referred to as SCANS skills.3 These skills mustbe developed by a coordinated effort among all faculty.

As mentioned in the Preface, participants were formed into sixgroups. Each group had at least one person from each three of theCRAFTY areas and one or more representative from an exemplary pro-gram. Except for the members of the Project’s Advisory Committee,the rest of the people in each group were mathematics educators.

Each group was given a skeleton matrix to use as a guide for thediscussion during the breakout sessions. The six breakout sessions werein the areas of (a) Content, (b) Pedagogy and Classroom Environment,(c) Delivery, (d) Currency, (e) Successful Programs, and (f) OtherRecommendations.

During the conference, participants decided that the Delivery ma-trix was not needed and that more time should be devoted to resources,including the classroom environment. Several of the groups also cre-ated lists of recommendations of further efforts that AMATYC, NSF,and/or the project staff should consider pursuing.

3SCANS is the acronym for the Secretary’s Commission on Achieving NecessarySkills. Ten years ago, this commission of 31 senior Human Resource executives andeducators and union officials issued two reports, entitled: What Work Requires ofSchools and Learning a Living. The so-called SCANS skills include the ability touse basic and advanced skills, such as mathematics and problem-solving to solveproblems in five domains. The two math-intensive SCANS problem domains areplanning and systems.

Introduction 3

The matrices each group completed were merged into the six com-posite matrices that were used to develop this vision of the mathematicsneeds of students in emerging technologies.


Content

The discussion of the Content recommendations begins with three con-cepts that span the entire mathematics curriculum: critical thinking,problem solving, and communicating mathematically. The discussionof these three areas is followed by recommendations in the more specific

Content

areas of arithmetic, algebra, geometry, trigonometry, statistics, calcu-lus, etc.

Critical Thinking, Problem Solving, and CommunicatingMathematically

Perhaps the most important aspect of the Content recommendationswas the general ideas of critical thinking, problem solving, and commu-Critical thinking,

problem solving, andcommunicatingmathematically shouldbe taught through allmathematics.

nicating mathematically. In fact, as one group said, “Critical thinking,problem solving, and communicating mathematically, while containingcontent objectives of their own, can and should be taught through allof the other topics listed.”

Because of the need to communicate with co-workers, techniciansneed to give both written and oral presentations and should be ableto explain (and interpret) mathematical ideas using the “Rule of 4”—that is, they need to be able to understand and use mathematics withformulas, graphs, and tables, and they need to communicate this un-Technicians need to

give both written andoral presentations.

derstanding. Even though the Rule of 4 is not a new concept (it wasthe foundation of much of the “reform” mathematics in the 1990s), thegroups deemed it important enough to warrant special considerationand mention.

Students in ATE (the National Science Foundation’s Advanced Tech-nological Education) programs should be able to select an appropriatemethod to solve a problem. In the participants’ view, a problem is notjust a traditional “story” problem.

There are different levels of problems. Some problems require stu-dents to do some research, develop their own process, collect data, oruse technology to organize data. Bloom’s taxonomy (see pages 27–29) was mentioned several times during the meetings. While Bloom’sStudents should be able

to handle problemsfrom all six levels ofBloom’s taxonomy.

taxonomy is some 40 years old, participants felt that it was still appro-priate. In particular, in mathematics courses for students in emergingtechnologies, students should be given, and be able to handle, problemsfrom all six levels of Bloom’s taxonomy.

Problem solving also requires that students know some logic. Stu-dents should begin their program of study with the ability to use in-ductive and deductive reasoning. When placed in the context of themathematics that students are studying, deductive reasoning is not thetype of formal proof often encountered in mathematics courses, butrather the ability to form arguments or solve problems based on laws,rules, or other widely accepted mathematical principles.

Content 5

Problem solving means that students must learn how to extractthe relevant information from a problem, identify what, if any, addi-tional information is needed, develop a plan, solve the problem, checkthe reasonableness and accuracy of the solution, present their findingsorally using appropriate visuals (spreadsheets, presentation software,animation, etc.), and submit a written report.

Word problems should often involve “real-life” situations in a stu-dent’s field of study. Using “real-life” situations is often difficult becausemany schools do not have enough students in any one program to allowfor a mathematics course devoted to its students.

It is essential that students learn to use estimation skills to assess It is essential thatstudents learn to use

estimation skills.the reasonableness of answers. While estimation skills have always beenimportant in mathematics, many students today rely on an answerproduced by a calculator or computer. Estimation skills are needed sostudents can determine if they have obtained a reasonable answer.

Dimensional analysis is a universal concept that was stressed byparticipants. Technical faculty stressed the importance of using exam-ples which require units and conversion between units. Dimensional Estimation skills and

dimensional analysis areuniversal concepts.

analysis should get increased emphasis.As mentioned above, students need to be able to communicate

mathematically. Because technicians need to explain a problem, pro-cess, or solution to their colleagues, clients, or supervisors, increasedemphasis must be given to strengthening student skills in communicat-ing mathematically.

Mathematical communication involves more than explanations. Italso means that technicians need to be able to read and understandmathematics expressed through tables, graphs, formulas, and writtenstatements. They also need to be able to translate among these formats.

Arithmetic

Most people do not normally consider arithmetic to be a college-leveltopic, which may explain why it did not get much attention from partic-ipants. The arithmetic topics that need the most attention are measure-ment conversions (metric-English, English-metric, Celsius-Fahrenheit),ratio, proportion, and scientific and engineering notation. Arithmetic topics that

need the mostattention aremeasurement

conversions, ratio,proportion, and

scientific andengineering notation.

Many students have probably not seen engineering notation beforethey started technician training. While they should have been exposedto scientific notation when they were in secondary school, it probablydid not get much emphasis unless they took physics or chemistry.

Students should know how to compute with percents. This topicwas undoubtedly taught several times during their middle school years,but it is an area where most students do not perform well.

Some areas, such as information technology (IT), need a topic thatwas popular during the “modern math” era: number base conversions.While information technology students will be taught, and use, thebinary, hexadecimal, octal, and decimal systems in their IT courses,


it is an arithmetic topic that might be included in some mathematicscourses for students in emerging technologies.

Algebra

When they enter college, students should already have the ability toapply the basic operations of signed numbers, algebraic operations, al-gebraic properties, and basic graphing. They should be able to solvelinear equations in one variable and 2 × 2 systems of linear equations.At present, many students entering college do not have these abilitiesThe ability to apply

mathematics topics isvital.

or cannot apply these skills in a context outside the mathematics class.The ability to apply mathematics topics outside of mathematics,

or in a new setting, is vital. Mathematics educators must constructcurricula that require students to make connections among mathemat-ics topics and with applications from several technical areas. StudentsBasic algebra concepts

should be secondnature.

need to internalize basic algebra concepts so that when they see theseconcepts in an application, the procedures are second nature.

Modeling with functions is the primary algebra topic that partici-pants thought students need to learn. This area has received increasedattention in the past few years. Modeling and problem solving could al-most be considered as natural partners and modeling fits quite logicallywith the goal to use realistic data.Students need to learn

to model withfunctions.

Students have long been given a formula (model) and told, “here isa formula that you should use in this type of situation.” They were thengiven many opportunities to exhibit that they could use that formulain those situations.

In modeling, one of three scenarios generally occurs: (a) studentsare given some data from an actual (real-life) occurrence, (b) they aregiven a problem and told to gather the necessary data to solve theproblem, or (c) they conduct an experiment and generate some data.The data is examined, usually with a scatter plot, and a regressionprocedure is used to generate a formula that fits the data. This for-mula/model is then used for interpolation and extrapolation. Muchof this formula generation would not be possible without the use ofinexpensive scientific/graphing calculators and/or computer software.

In order to use this type of modeling process, students must becomemuch better at analyzing a graph and determining the type of functionthat best fits the data. They must also be able to assess whethera given model makes sense in the context of the problem situation.Understanding a problem’s natural domain and other limitations is partof such assessment.

Modeling situations create much of the other reasons for studentsModeling creates manyof the needs foralgebra.

to take algebra: formula interpretation, familiarity with mathematicsnotation, graphing of all elementary functions, applications of bothlinear and nonlinear models including piecewise-defined functions, andestimation.

Content 7

Another key concept that all students should acquire is a betterinterpretation of slope, especially with regard to the units for slope All students should

acquire a betterinterpretation of slope.

in non-traditional or in realistic problems. Many of the modeling sit-uations provide an excellent opportunity to use slope to provide anintuitive introduction to calculus.

No doubts were expressed about the use of technology as an impor-tant tool in the teaching and learning of mathematics. Using technologywill be discussed further in the Pedagogy section (see page 9).

Geometry

Geometric topics important for technicians include the ability to useright triangles and similar triangles to model situations. Students needto be able to solve problems using the Cartesian coordinate system,especially the distance between two points. Participants were also in-terested in three-dimensional geometry and three-dimensional graphs.

Students were introduced to perimeter, area, and volume when theywere in middle school, but these formulas and concepts need to bereinforced using practical problems that require the student to do morethan substitute into a formula that is given to them. They must beable to use geometric concepts to draw pictures of problem scenarios.Geometry also provides an opportunity to show how to approximatethe perimeter, area, and volume of irregular shapes and can provideanother intuitive introduction to calculus. Geometry should be

integrated throughoutall courses.

Geometry should be integrated throughout courses with applica-tions as appropriate. This integration will require students to recallformulas or conduct research to find the needed information.

Trigonometry

The trigonometry needed by technicians primarily falls into three cate-gories: right triangle trigonometry, sine graphs, and vectors. They needto be able to use these concepts to solve problems or model phenomena.

In order to use right triangle trigonometry, students have to knowhow to work with degrees-minutes-seconds. Because of their use ingraphing (and calculus), students should encounter radians and be ableto convert between radians and degrees-minutes-seconds. They will alsoneed to use inverse trigonometric functions.

Statistics

All students need to know some basic concepts from statistics. In par-ticular, they need to be able to use measures of central tendency andvariance to explain and/or interpret data. While they may have beenintroduced to correlation when they were applying regression modelsin algebra, these concepts will be strengthened as they are applied instatistics. Many statistical calculations will be accomplished with the


use of technology. Therefore, the emphasis must be on the ability toapply these concepts to analyze and interpret real data.All students need basic

concepts fromstatistics.

Technicians use technology to organize and display data. Sincemany technicians will use a spreadsheet on the job, mathematics fac-ulty should use this technology as a tool when possible. Spreadsheetsallow students to simultaneously see the data and a graph of the data.A spreadsheet gives students the opportunity to test conjectures easilyby simply changing the values of the parameters in a calculation orformula. Technicians may need an introduction to hypothesis testing,perhaps with a t-test, and confidence intervals should probably be in-Mathematics faculty

should use spreadsheetsas a tool.

troduced. More advanced hypothesis testing might involve ANOVA orχ2.

Among the additional statistical topics needed by some techniciansis SPC (Statistical Process Control). The use of SPC will seldom involveactual calculations because SPC will be done by computers. However,technicians will need to know how to interpret and use SPC information.

Calculus

Most technicians will not need to use calculus. However, they willneed an understanding of some of the fundamental calculus concepts.Those who do need calculus will, of course, need differentiation andintegration. The conceptual introduction to both differentiation andintegration can be done when students study algebra and geometry.Knowing how to differentiate and integrate functions is worthless unlessstudents know how these operations are used and can interpret theresults in the context of a real-life situation.

Pedagogy 9

Pedagogy

As the groups discussed pedagogy, they listed several different aspectsof teaching: lecture/discussion, discovery or Socratic, small group,

Pedagog

y

projects, teams, and various aspects of using technology. Because tech-nology is so fundamental to technology programs, the discussion aboutpedagogy will include the different aspects of technology after a discus-sion about the types of students.

The Learners

The discussion in one group revolved around the nature of the studentswho are enrolled in mathematics courses. For the most part, two-year college students are adults rather than children—and adults havedifferent learning styles than do children. Two-year college

students are adults, notchildren. Adults have

different learning stylesthan do children.

When do children become adult learners? Adult learners are definedsociologically, not chronologically. Adult learners learn differently, andthey need to be ready to learn in order to learn.

Andragogy was developed by Malcolm Knowles (1913–1997) and hasbeen defined as the art and science of helping adults learn. Knowles’instructional model is based on assumptions about how adults learnand can also be viewed as an alternative to pedagogy, a more learner-centered approach for students of all ages.

Knowles has six assumptions4 about adult learners.

1. Adults need to know why they need to learn something beforeundertaking to learn it.

2. Adults have a self-concept of being responsible for their own lives.3. Adults come into an educational activity with both a greater vol-

ume and a different quality of experience from youths.4. Adults become ready to learn those things they need to know.5. Adults are life-centered (or task-centered or problem-centered) in

their orientation to learning.6. While adults are responsive to some extrinsic motivators. . . the

more potent motivators are intrinsic motivators.

These assumptions about how adults learn need to be considered whenone is designing the curriculum and preparing lessons.

Technology (Other than the Internet)

Regardless of the technology that is being used, two considerations needto be kept in mind:

• Technology should be a tool, not the driving force behind thecurriculum. Technology by itself is not the objective.

4Knowles M. S. (1990) The Adult Learner: A Neglected Species (4th edition)Houston: Gulf Publishing.


• Professional development in the appropriate use of technologyas a tool to teaching and as a tool for learning should be madeavailable to all faculty including adjunct faculty. (Professionaldevelopment will be discussed in the Currency section starting onpage 17.)

The use of technology in a mathematics class usually involves eithergraphing calculators or computers. Discussion did not revolve aroundDiscussion was not

about whethertechnology should beused but rather how itcould be used.

whether technology should be used but rather how it could be used.Graphing calculators began appearing in mathematics classrooms

in the late 1980s, but it was not until the introduction of the TI-81 cal-culator in 1990 that graphing calculators began to be widely integratedinto mathematics education.

Graphing calculators do more than just graph functions. Most allowtables to be generated either by the user entering specific data pointsor by the user entering a function which the calculator evaluates at aspecified set of values.

A user-entered set of data points can be graphed on a calculator, andvarious functions can be generated using regression features built intothe calculator. The types of functions include 2nd, 3rd, and 4th degreepolynomials, exponential, logarithmic, sinusoidal, and, with some effortby the user, piecewise.

Spreadsheets are alternatives to graphing calculators. An obviousreason to use spreadsheets is the fact that they are the technology ofchoice for business and industry. An obvious drawback is that personalSpreadsheets are the

technology of choice forbusiness and industry.

computers are not always available in classrooms. Other advantages ofthe spreadsheet over the calculator include its ability to simultaneouslydisplay data and graphics and the ability of the spreadsheets to beinteractive with the users. A spreadsheet can be more easily integratedinto a written (word processed) document.

Computer algebra systems (CAS), such as DeriveTM, Maple©R,MathCad©R, and Mathematica©R, have many of the same advantagesas spreadsheets. However, MathCad©R and Mathematica©R probably arebetter at integrating data, graphs, and mathematical expressions withtext.

All of these types of technology, graphing calculator, spreadsheet,and CAS, allow students to examine, organize, and analyze real data.Because they are pursuing careers in areas that require technology, theability to use technology to examine, organize, and analyze real datashould be a natural extension of the student’s technical program.

The major advantages of using these technologies is that studentsStudents have toorganize their thinkingand select frommultiple solutionstrategies.

can use realistic data, they have to organize their thinking, and theyoften have to select from multiple solution strategies. Justification ofthe method selected affords excellent learning opportunities.

Use of multiple technologies gives students the chance to understandhow to use each technology and its limitations. Technology also pro-vides situations for students to emphasize the need to estimate solutions

Pedagogy 11

and check their work.

The Internet

The Internet can be an excellent source of realistic data. It can also bea source of misinformation. On occasion, examination of different websites will result in different data for the same situation.

Many research activities can be facilitated by using the Internet.Java applets can be used as teaching or computation tools. Teachingstudents to use the Internet will probably do more to promote lifelonglearning—one of the major goals of higher education. The Internet facilitates

many researchactivities.

The Internet can help teach students how to use reference materials(including the textbook). It can also use the problem-solving processas a tool and set up situations where students analyze what is relevantto a particular problem and determine what simplifying assumptionsto make in order to solve the problem.

Teaching-Learning Methods

The lecture-discussion method has been around for centuries, and de-spite all the calls to replace it with other methods, it will undoubtedlycontinue to be an appropriate way to provide information in some cir-cumstances. However, this group calls for lectures to be supplemented Lectures should be

supplemented with avariety of

student-centeredmethods.

with a variety of student-centered methods, such as computer simula-tions, web-assisted activities, collaborative learning activities, and in-structional television. The inclusion of technology in the lecture pro-vides opportunities to enliven the teaching and to get students moreactively involved in the learning process.

Technology also allows a lecture to be interspersed with multiplechoice questions which students anonymously answer. The responsesprovide instructors with instantaneous feedback as to whether studentshave understood a certain idea. Responses can also be used to measurestudent attitudes. Instructors should use

collaborativetechniques.

Instructors should also use collaborative techniques as alternativesto lecture. These methods may do a better job of providing studentswith situations similar to ones they might encounter at work. Whilesome uses of technology mean that students can work alone, othersrequire that they have to collaborate and work in teams.

Collaborative learning opportunities include working in teams. Oneadvantage of collaborative learning and small group work is that theyrequire students to communicate effectively both orally and in writing,not only with each other but with the instructor.

Outside Contributors

Recent graduates, industry retirees, and other practitioners bring ex-periences that students can find valuable.


Collaboration with other departments and staff help faculty developinterdisciplinary skills that are expected of the students (oral, written,and other communications). Working with other faculty members helpsmathematics instructors incorporate familiar content from other disci-plines.

Pedagogy Affected by Material

Activities

The use of activities to engage students in the learning process shouldbe a normal occurrence in the mathematics classroom. Activities mayActivities should be a

normal occurrence inmath class.

require students to do a short experiment to collect data or show aphenomenon. An activity may be a problem posed to groups of studentsor it may be some type of assessment of students or instruction. Anactivity may involve technology (animations or graphics). Activitiesmay require the students to discuss, plan, or write and should providea chance to engage students with various learning styles.

Projects

Projects require a student or group of students to design, plan, or-ganize, research, calculate, model, and/or present. They differ fromactivities in that they require more than a class period to complete andoften require students to meet or communicate outside of the classroom.Projects can engage

faculty from otherdisciplines.

Projects provide opportunities to engage faculty from other disciplines.The context of a project may come from one of the technologies. Tech-nology faculty may make a guest appearance to set up the project.English faculty may assist students in the written or oral part of theproject.

Scenarios (Problem-Based Learning)

Scenarios or case studies provide another way to engage students inmeaningful activities. The problems are often vaguely defined, andthere may be several acceptable solutions. Scenarios or case studies aremore often used within an integrated curriculum (or within a learningcommunity). The problem may be very technical in nature but requirethe student to use skills in mathematics, science, and English.

Pedagogy Affected by Industry/ Workplace Environment

All students must learn to be responsible. They must learn to at-tend class (on time), be involved with the discussion, be aware of allAll students must learn

to be responsible. requirements, and meet deadlines. While mathematics instructors can-not achieve these goals alone, they must contribute to the end result.Mathematics faculty must set high standards for students and require

Pedagogy 13

work to meet those standards without excuses. Mathematics instruc-tors must demonstrate these attributes by being prepared, using bestpractices, and meeting deadlines.


Resources

Resources are the tools and other material that support mathemat-ics instruction. There are three major categories: physical resources,

Resourc

es

human resources, and fiscal resources.

Physical Resources

The physical resources have several subcategories such as the classroomenvironment, equipment needs, classroom management systems, andtextbooks.

Classroom Environment and Equipment

Most students in the technical mathematics classroom today use graph-ing calculators. In the near future we would expect that all studentsin these classes would have a laptop computer or a personal digital as-sistant (PDA). All classrooms should have network connectivity withmultimedia and projection capabilities built in.

Students and faculty should be able to exchange files using wire-less connections and should be able to integrate the use of computersoftware, calculators, PDAs, and the world wide web.

Mathematics faculty must encourage the use of technology and fos-ter computer literacy. The web allows students to communicate withMathematics faculty

must encourage the useof technology.

each other and their instructors as well as allowing faculty to commu-nicate with each other.

Equipment needs might include smart boards, and smart classrooms—projection unit, DVD, computer connections, document camera, digitalcamera—interactive classrooms with video conferencing capabilities tooff-campus students and to industry, climate control and energy sen-sitive lights and equipment, “holographic” instructors, and electroniclinking to labs.

The furniture in the room must be different from the desk-chairsthat have been de rigueur for over 50 years. At present, students cannothave an open textbook, laptop computer, and graphing calculator onthe desk at the same time. In fact, in some rooms, it is difficult tosimultaneously use a notebook and a graphing calculator.

Rooms need to be equipped with furniture that fosters collaborationRooms should havefurniture that is largeenough to use both alaptop computer andthe text.

and is large enough for students to conveniently use both a laptopcomputer and the text.

Classroom Management Systems

Management systems, such as WebCT and Blackboard, enable thequick dissemination of syllabi, assignments, and updates. These web-based management systems provide instructors with new ways of pro-moting daily homework using quick on-line quizzes and better ways of

Resources 15

helping students review and learn material (animations, access to othertexts, and access to previous exams).

Many of these systems also have chat rooms and whiteboard capa-bilities. The chat room allows students to discuss, either with peers orthe instructor, areas where they are having problems. However, it isoften difficult to type mathematics in these systems. Students need to be

comfortable withtechnology that uses

both symbolicmathematics and

arithmetic.

Students need to be comfortable with technology that uses bothsymbolic mathematics and arithmetic. It may be that students don’tneed a specific software package, but they do need the skills and theexperience with these types of technologies that they can readily learnand adapt when confronted with a specific software package.

Textbooks

Participants would like to see a reform movement in mathematics books.Textbooks should include writing assignments, projects, technology-based activities, a sufficient amount of skill-and-drill, useful web mate-rials, and information that is relevant to the technologies representedin their mathematics courses.

To get texts to change, faculty must tell publisher’s representativeswhat they want and what they need. The need for textbook reformmust be said to representatives who visit faculty and when faculty visit Publishers must be told

that textbook reform isneeded.

a publisher’s booth at a conference.Good, sound, basic textbooks are needed that form the basis for

web and other ancillary materials. When a textbook offers the newdirection, the ancillary materials will follow.

Books need to include problems that more realistically reflect realworld applications. For example, some problems need to provide toomuch information and other problems should omit some information.Students need to use the appropriate technology to model actual situ-ations.

Human Resources

In addition to full-time faculty, the human resources section has twoprimary subcategories: learning communities and adjuncts.

Learning Communities

Learning communities5 may be thought of in two ways:

• Learning communities can be topic experts, industry personnel,local professional organizations, and the student’s classmates whocan serve as instructional aides.

5The information technology area refers to a learning community as a “communityof interest.”


• Learning communities may also be a group of instructors whoplan, coordinate, and offer a set of courses with a common themeor shared projects.

A learning community can provide a supportive environment forstudents. Mathematics departments should help form learning commu-nities either entirely within mathematics or in conjunction with one orMathematics

departments shouldhelp form learningcommunities.

more technical areas.The formation of various learning communities, either by the stu-

dents themselves or by faculty assistance, can help many students.Since, in some cases, it may be especially difficult for commuting stu-dents or part-time employed students to form such groups, on-line bul-letin board or chat rooms can be used to enhance student communica-tion.

Adjuncts

With “reformed” curricula and classrooms, in-service training (possiblypaid) for adjuncts is essential. Adjuncts need offices and access tocomputers for e-mail and instruction. They also need class training,technical support, orientation, recognition, and monetary support forprofessional development activities.Adjuncts and full-time

faculty must remaincurrent.

Both adjuncts and full-time faculty must remain current. Whilethese concerns were discussed by some groups in this section, the rec-ommendations will be deferred to the next Section, “Keeping Current.”

Fiscal Resources

Faculty, including adjuncts, need to be compensated as professionals.This issue is more critical in some states than others. The salary ofFaculty need to be

compensated asprofessionals.

two-year college mathematics faculty should be high enough to attractand retain the type of people who will provide positive role models forour future citizens and workforce.

The support for faculty must also be shown in the teaching loadrequired of faculty members. Faculty who teach 15–20 hours or morefind it difficult to remain current or find the time to do research and totry new ideas.

Keeping Current 17

Keeping Current

Mathematics faculty must remain current if they are to provide an up-to-date curriculum, hone their pedagogical skills, and use the latesttechnology and other resources. These are problems for both adjunctsand full-time faculty. The following were among the specific suggestions Keep

ing Curr

ent

for professional development.

• The institution should have an institutional coordinator of andan institutional commitment to professional development.

• Both adjuncts and full-time faculty should have opportunities forprofessional development.

• Instructor evaluation should include one’s professional develop-ment.

• Technical support people should have opportunities for profes-sional development.

• Faculty should receive support for release time to participate inprofessional development.

Two of the best ways to keep current are to read the professionalliterature and participate in professional meetings. Not only should Read the professional

literature andparticipate in

professional meetings

instructors read mathematics journals such as The AMATYC Review,The College Mathematics Journal, NCTM’s Mathematics Teacher, andThe Journal of Online Mathematics and its Applications but they shouldalso read non-math journals such as Prism and the Journal of Engi-neering Technology.

Two-year college mathematics faculty should attend AMATYC na-tional conferences and meetings of affiliate organizations. Conferencesprovide an opportunity to share ideas and discuss challenges with col-leagues. The conferences also provide a venue where faculty can learnabout innovative programs and pedagogy. Mathematics faculty, espe-cially technical mathematics faculty, should participate in meetings oftechnical associations like ASEE (the American Society for Engineer-ing Education) and IEEE (the Institute of Electrical and ElectronicsEngineers).

Since not all faculty are able to attend national conferences, fac-ulty who do should share findings and experiences from professional Faculty should share

findings andexperiences from

professional meetings.

meetings with others in the department by giving presentations duringdepartment meetings, with internal written reports, and/or by writingarticles in newsletters.

Many professional organizations offer workshops that help facultykeep current. One example is AMATYC’s “Traveling Workshops.”Some of these can be offered on, or near, the local campus and areusually less than a day long. Others, such as AMATYC’s summer in-stitutes, can last a week or more and require traveling a great distance.

Participants felt that technical mathematics faculty should gainsome experience in industry, which could involve tours of industry,


talking to graduates, and job shadowing. As an alternative, indus-try representatives could speak at faculty and advisory meetings.

Maintaining a Successful Program 19

Maintaining a Successful Program

In order to maintain a successful program, departments need to identifymethods of attracting students and to generate graduates who are ableto find employment in their field of study.

Maintain

ing a

Success

ful Prog

ram

One group felt that in order to have a successful program we shouldeliminate the perception of “Mathematics = Barrier.” This perceptionwill be mentioned first. The other major categories of recommendationsare partnerships, articulation, and community involvement.

“Mathematics = Barrier” Perception

Many people perceive mathematics as a barrier that needs to be hurdledor maybe just climbed over in order to gain entrance to their selectedcareers. Perhaps the biggest key to maintaining a successful programis the ability to reduce this perception. The relevance, utility

and criticality ofmathematics to a

career must bedemonstrated.

To help show that “Mathematics 6= Barrier,” the relevance, util-ity and criticality of mathematics to a career must be demonstratednot only to students but to their parents, advisors, faculty in otherdisciplines, and other stakeholders such as boards of trustees.

Recognizing and addressing issues of student diversity (diversityincludes race, gender, age, economic conditions, ADA, ESL and otheraccess issues) are other ways to reduce the “Mathematics = Barrier”perception. Because so many students come to two-year colleges with Recognizing and

addressing issues ofstudent diversity will

reduce the“Mathematics =

Barrier” perception.

poor mathematics skills, we need to develop ways to remediate thatare more in-depth and applications-based. Increased diversity offaculty within technical mathematics and technical programs will alsohelp eliminate the “Mathematics = Barrier” perception.

Additional suggestions for individual mathematics faculty and lead-ership include: getting involved in, and attending, division meetings oftechnical programs; volunteering to be on advisory boards; establishingpartnerships with business, high schools, industry, civic groups, andtrade unions; giving presentations about technical mathematics withapplications; teaching selected topics in high schools; and holding orhosting monthly mathematics seminars for all faculty.

Given the large number of possible activities and actions, it wasnoted that no one person could do or implement each and every recom-mendation. A faculty member should start by doing one or two things Start by doing one or

two things differently.differently, or choose to change something, not everything. Change isa gradual process, not a sudden revolution. Change can be either con-scious or unconscious. We think it should be conscious and directed,even if a little at a time.

Partnerships

Three types of partnerships were suggested: with technology faculty,with business and industry, and with high schools.


Partnerships with Technology Faculty

Mathematics programs should establish partnerships with technologyfaculty by including technology faculty on mathematics hiring commit-Mathematics programs

should establishpartnerships withtechnology faculty.

tees. Technology areas should reciprocate by including mathematicsfaculty on technology hiring committees.

The mathematics faculty should solicit applications from the tech-nology faculty. They should visit and observe technology classes to seewhat mathematics is used and how it is used. Individual faculty mem-bers should invite technology faculty to explain and/or demonstrateapplications that use certain mathematical topics. Similarly, technol-ogy faculty could invite mathematicians to occasionally teach certainmathematics concepts.Math faculty should

regularly meet withfaculty in technicalareas.

Mathematics faculty should regularly meet with faculty in all tech-nical areas to discuss content, to learn about new technology and newuses of old technology, and to discuss any modifications that need tobe made in the curriculum.

Partnerships with Business and Industry

Partnerships should also be established with business and industry.Mathematics departments should establish mathematics advisory com-mittees that include both technology faculty and business and indus-try representatives. These advisory committees would be an excellentmeans for keeping programs up-to-date, for providing useful informa-tion for planning and assessment, and for recruiting guest speakers andadjunct faculty.Mathematics faculty

should participate inlocal professionalorganizations.

Business and industry partnerships would also provide opportuni-ties for offering specialized courses and programs for workforce training.Mathematics faculty should participate in local professional organiza-tions, and industry representatives should be invited to share with themathematics department ways in which they use math.

Partnerships with High Schools

College mathematics faculty should participate in Tech Prep partner-ships and career fairs. They can also provide in-service professionaldevelopment for high school faculty.

High school math teachers often are not fully aware of the mathe-matics expectations of collegiate faculty and the extent that technologyis used in college mathematics classes. College faculty often complainabout the poor preparation of entering students. Improving commu-nication between the area high schools and the college mathematicsprograms can help reduce these misunderstandings.

College mathematics departments should invite high school coun-Invite high schoolpersonnel to activitiesdemonstrating technicalprograms.

selors, administrators, and teachers to hands-on activities demonstrat-ing technical programs. College faculty members could take students

Maintaining a Successful Program 21

in the technical programs into high schools for talks, demonstrations,and “college” career days.

Articulation

A big problem encountered by schools where a large percentage of stu-dents transfer to four-year programs is the fact that two-year courses donot transfer to four-year colleges and universities. It is interesting thatmany colleges have lower-level mathematics courses that do transfer,but that technical mathematics courses which often include the samecontent as precalculus courses, but with an applied viewpoint, do nottransfer. Many two-year colleges have changed their technical mathe-matics courses to one in college algebra and trigonometry, often a mixedblessing. While college algebra and trigonometry courses will transfer,they often do not contain the applications that make the mathematicsmeaningful.

We need to explore ways of promoting transferability of mathemat-ics courses for students in emerging technologies to baccalaureate pro- We need to promote

the transferability ofmathematics courses to

baccalaureateprograms.

grams. More dialogue between two- and four-year colleges is necessaryon this issue. Students choosing technical curricula need workplace rele-vance in their mathematics courses and should not be unduly penalizedfor their choice if they opt to continue their education.

Community Involvement

In addition to the above suggestions, mathematics departments shouldestablish and/or strengthen connections with civic organizations. Notonly should faculty speak at meetings of these organizations, but thedepartment should also invite members of organizations such as theLions and Rotary Clubs to serve on advisory boards and provide guestspeakers.

The college should encourage civic organizations to establish stu-dent scholarships at the college and encourage the organizations toparticipate in the evaluation and judging of general education presen-tations and papers.


Other Participant Recommendations

The following are items that either are not contained in the matricesor were felt to be worthy of special emphasis.

• Help mathematics departments in the nation’s community col-leges begin to examine how, as a department, faculty can worktogether to improve the teaching and learning of mathematics.

Other P

articip

ant

Recommend

ations

Identify departments that have successfully implemented and in-stitutionalized innovation in their mathematics programs. An-alyze the group dynamics and team building skills that lead tothe transformation of mathematics teaching throughout an entiredepartment, and share this knowledge with other mathematicsdepartments.

• Establish a national clearinghouse for (a) sharing informationabout reform of technical mathematics and (b) sharing authenticapplication problems. The clearinghouse should have a dedicatedstaff, probably with initial government funding, but ideally tobe eventually supported on a permanent basis through industryfunding. The clearinghouse should contain

– Projects– Case studies– Web sites– Assessments– Problem-based learning activities– Software

• Establish Internet-based capability for faculty to share appliedproblems they develop.

• Establish a Technical Mathematics journal emphasizing authenticindustrial applications. The journal should include but not berestricted to applications for courses below calculus.

• Explore ways of promoting transferability of technical mathemat-ics courses to baccalaureate curricula. Students choosing tech-nical curricula need workplace relevance in their mathematicscourses, but they should not be unduly penalized for their choiceif they opt to continue their education (as many eventually will).

• Collect assessment data on outcomes of a reformed technical math-ematics course sequence, including affective as well as cognitiveoutcomes. Subsequent performance in mathematics-intensive tech-nical courses may be equally as important as performance inhigher level math courses. Enlist the support of technical fac-ulty to help convert resistant mathematics faculty.

• Adopt the “reform” recommendations proposed by this confer-ence in higher level mathematics courses taken by technical stu-dents.

Other Participant Recommendations 23

• Provide ways to continue discussions similar to those of this con-ference and the CRAFTY workshops.

• Put time and money into tracking what reform progress is hap-pening, following students, and assessing the effect. Should wetry to make connections to ABET? Should we have a joint effortwith ABET?

• Increase our visibility. Connections to industry organizations willbe a next step. The Society for Human Resource Management isa suggestion. Look at skill standards generated by industry.

• Use a system that includes several different types of evaluation.Good assessment involves a variety of techniques.

• Provide professional development to K–12 teachers to help themmore realistically provide meaningful examples from the technolo-gies. The professional development should also provide examplesand experience in using appropriate technology for teaching andlearning.

• Increase our political power, particularly on the state level, whichis where curriculum and course content are dictated.


Summary

In many ways, the following statements apply to the entire mathematicscurriculum in the first two years of college, not to just the mathematicscourses for students in emerging technologies.Sum

mary

• Critical thinking, problem solving, and communicating mathe-matically transcend all mathematics.

The general ideas of critical thinking, problem solving, and com-municating mathematically can and should be taught in all levelsof mathematics. Expected student outcomes should be based lesson specific content skills and more on problem solving, criticalthinking, and communicating mathematically.

• Traditional content is NOT appropriate today.

All students need statistics, and many will not need the heavydose of algebra that is traditionally required. Additional topicsin discrete mathematics and modeling should be added to thecurriculum, not necessarily as separate courses, but through in-tegration with existing material. Content should be addressedin ways that demonstrate connections between mathematics andother areas, as well as among mathematics topics.

• Memorizing isn’t all there is to learning!

Students must acquire a deeper understanding of mathematicaltopics. They must apply the topics to non-traditional problemsand be able to apply the appropriate tools to unfamiliar situ-ations, not just those found at the end of a chapter. Having“covered” a section in a course isn’t a measure of success—it’swhat the student learned and thus can apply that is important.

• Applications should have local depth and global breadth.

Students must be able to transfer the mathematics knowledge orskills to applications within their disciplines. Focusing on localbusinesses gives immediate relevancy to applications. Yet, exam-ples must be transferable outside the local community.

• Faculty must use technology as a tool for teaching and, at thesame time, develop the ability of students to use that technologyto solve problems.

Students will use technology in the workplace. The classroomshould model the workplace if we are to equip students to succeedin a variety of settings. Instructors must use technology to enablestudents with different learning styles to grow mathematically.Technology enables us to solve more realistic problems that aren’talready designed and trivial.

• The classroom environment must use all the resources possible tofacilitate learning.

Summary 25

The classroom must be designed and equipped with appropriatefurniture and technology so that a variety of formats can be usedto facilitate learning. Students and instructors will use many toolsin class, so the space must enable, not hinder, the learning process.The environment must also be conducive to collaboration, bothverbally and electronically. Students should be able to share theirideas and findings using a variety of tools.

• Teaching math is not just teaching math.

Mathematics faculty must help provide students with a variety ofhabits (soft skills) that will enable them to succeed in the work-place as well as be good citizens and life-long learners. The skillsthat employers want from their employees rarely include specificcontent. They want instructors to strengthen the students’ abilityto think, to communicate, and to be responsible. Mathematicsclasses must be a part of equipping the student with employabilityskills.

• Teaching can’t be done in isolation.

Mathematics faculty must keep current with the changes andneeds of students who are majoring in other disciplines. Schoolsand departments must find ways to motivate mathematics facultyto communicate regularly with faculty in other departments andwith practitioners in industry.

Faculty must be involved in professional activities that keep themcurrent and provide them with new ideas and with informationabout best practices. Schools and departments need to exam-ine the current structure that promotes the status quo and isola-tion. Education institutions must find ways to promote innova-tion, sharing, and personal growth.

• Decisions should not be made in isolation

Educators who attempt to provide the appropriate experiencesfor students who will soon enter the workplace must be aware ofthe skills required in the workplace.

Mathematics faculty must find ways to learn about the employ-ability skills their students will need to be successful employeesor employers. By performing workplace research, better decisionswill be made regarding the appropriate content, technology, andpedagogy for tomorrow’s classrooms. The old answers to the ques-tion of “Why do we need to know this?” (“Because it is on thenext test.” or “Because you’ll need it in the next course.”) arenot sufficient. We need to be able to provide realistic examplesthat motivate content.

• Institutions must set the example

The technology students use in the classroom must help equipthem to deal with the highly technical workplace of the future.


Schools must find ways to fund up-to-date equipment and soft-ware as well as ways to ensure faculty are using those resourcesappropriately.

Students should be able to communicate with each other and withthe instructor in a wireless environment that makes formative as-sessment and instructive communication easy. The infrastructureshould promote methodology that utilizes the capabilities of thepersonal computer, especially as it relates to ways to aid expla-nations visually through pictures, videos, or animations.

• All mathematics faculty are equal.

Adjunct faculty should have the same benefits as full-time faculty.This especially applies to opportunities and monetary support forprofessional development.

Bloom’s Taxonomy

Competence Skills DemonstratedKnowledge • Observation and recall of information

• Knowledge of dates, events, places• Knowledge of major ideas• Mastery of subject matter• Question Cues: list, define, tell, describe, identify, show, label, collect,examine, tabulate, quote, name, who, when, where, etc.Mathematics example:Given D(m) = 0.00002m3 − 0.0002m2 + 0.0003m, find D(50).

Comprehension • Understand information• Grasp meaning• Translate knowledge into new context• Interpret facts, compare, contrast• Order, group, infer causes• Predict consequences• Question Cues: summarize, describe, interpret, contrast, predict, asso-ciate, distinguish, estimate, differentiate, discuss, extendMathematics example:A cantilever beam is supported on one end, and an object is placed onthe other end, as shown in Figure 1. The distance the beam is bentfrom the horizontal position is called the deflection. The formula D(m) =0.00002m3−0.0002m2 +0.0003m uses the weight of the object to determinethe deflection. Find the deflection for a weight of 2 pounds.

D

Figure 1

Continued on next page

28 Bloom’s Taxonomy

Competence Skills DemonstratedApplication • Use information

• Use methods, concepts, theories in new situations• Solve problems using required skills or knowledge• Questions Cues: apply, demonstrate, calculate, complete, illustrate, show,solve, examine, modify, relate, change, classify, experiment, discoverMathematics example:Attach different weights to the end of the cantilever beam in Figure 1. De-termine quadratic, cubic, and quartic formulas for the amount of deflectionin the beam in Figure 1 as a function of the weight on the end.

Analysis • Recognition of patterns• Organization of parts• Recognition of hidden meanings• Identification of components• Question Cues: analyze, separate, order, explain, connect, classify, ar-range, divide, compare, select, explain, inferMathematics example:As shown in Figure 2, a simply supported beam is placed across an openspace and is not supported at either end. Explain how you would design anexperiment to find the deflection of the beam as a function of the weightplaced on the beam.

Figure 2

Synthesis • Use old ideas to create new ones• Generalize from given facts• Relate knowledge from several areas• Predict, draw conclusions• Question Cues: combine, integrate, modify, rearrange, substitute, plan,create, design, invent, what if?, compose, formulate, prepare, generalize,rewriteMathematics example:A 24-inch long beam is supported on one end (a simply supported beam).An object placed on the end of the beam will cause the beam to bend(deflection). The amount of deflection is dependent on several factors. Youwill be provided plastic “beams” with the three different cross-section shapesshown in Figure 3—all with a cross-sectional area of 0.25 square inches.

Continued on next page

Bloom’s Taxonomy 29

Competence Skills DemonstratedSynthesis (cont.)

0.5 inch

0.5 inch0.282 inch0.25 inch

1.0 inch

Figure 3

Each group of three students will determine a model for the deflection ofeach beam as a function of the mass on the end (using masses between 50and 100 grams). Write a report on the process used to determine the model,describe the model and the extent that the model is reasonable.Use the model to predict the deflection with a 200-gram mass. AFTERmaking the prediction, find the deflection and calculate the percent error inyour estimate.Which design of the beam is “best”? (Discuss)

Evaluation • Compare and discriminate between ideas• Assess value of theories, presentations• Make choices based on reasoned argument• Verify value of evidence• Recognize subjectivity• Question Cues: assess, decide, rank, grade, test, measure, recommend,convince, select, judge, explain, discriminate, support, conclude, compare,summarizeMathematics example:Compare the accuracy of the quadratic, cubic, and quartic regression equa-tions you developed in the Applications Example. Explain which formula isthe most accurate.

30 Bloom’s Taxonomy

National Conference Participants

The following people attended the May 12–15, 2002 national conference to establish a vision forthe mathematical needs of students in emerging technologies.

Darrell Abney, Professor of Mathematics,Maysville Community College, Maysville,Kentucky

Ashok Agrawal. Dept. Chair & Professor, St.Louis Community College, St. Louis,Missouri

Catherine Aust, Head of Department ofMathematics, Clayton College & StateUniversity, Morrow, Georgia

W. David Baker, ProfessorEmeritus,Rochester Institute ofTechnology/Past Chair of TAC of ABET,Kingsport, Tennessee

Bob Bixler, Electronic Technology faculty,San Juan Basin Technical School,Durango, Colorado

Constance Blackwood, Consultant, IdahoFalls, Idaho

Gail Burgess, Math Instructor, SouthCentral Technical College—Mankato,North Mankato, Minnesota

Paul Calter, Professor of MathematicsEmeritus, Vermont Technical College,Randolph Center, Vermont

Robert Campbell, Chief Information Officer& Executive Dean of IT Services, RockValley College, Rockford, Illinois

Cheryl Cleaves, Professor, DevelopmentalMathematics, Southwest TennesseeCommunity College, Memphis, Tennessee

Ray Collings, Mathematics Faculty Member,Georgia Perimeter College, Lawrenceville,Georgia

Dennis Davenport, NSF Program Officer,National Science Foundation, Arlington,Virginia

Andrea Faber, Instructor, Mathematics,James A. Rhodes State College, Lima,Ohio

Michael Farley, Marine Designer, NationalSteel & Shipbuilding Co., Spring ValleyCalifornia

Linnea Fletcher, Bio-Link Regional Director,Austin Community College, Rio GrandeCampus, Austin, Texas

Elisabeth Gambler, Associate Professor ofMathematics, Vermont Technical College,Northfield, Vermont

Judy Giffin, Lecturer, Mathematics, JamesA. Rhodes State College, Lima, Ohio

Lawrence Gilligan, Department Head ofMathematics, Physics & Computing,College of Applied Science, University ofCincinnati, Cincinnati, Ohio

32 National Conference Participants

Carol Gudorf, Mathematics Instructor,Edison State Community College, Piqua,Ohio

Martha Haehl, Administrative Intern, BlueRiver Community College, Blue Springs,MO

Chuckie Hairston, Mathematics Instructor,Wake Technical Community College,Raleigh, North Carolina

Eric Hiob, Mathematics Instructor, BritishColumbia Institute of Technology,Burnaby, British Columbia

Patricia Hirschy, Associate Professor ofMathematics, Asnuntuck CommunityCollege, Enfield, Connecticut

Jan Hoeweler, Mathematics Instructor,Cincinnati State Tech & C.C., Cincinnati,Ohio

Amy Hoffman, Mathematics Instructor,Community College of Baltimore County,Essex Campus, Baltimore, Maryland

Linda Hollstegge, Instructor, Info &Engineering Tech, Cincinnati StateCollege, Cincinnati, Ohio

Wayne Horn, Director Computer SciencePrograms, Pensicola Junior College,Pensicola, Florida

Mary Ann Hovis, Math Chair, James A.Rhodes State College, Lima, Ohio

Robert Hovis, Professor and ProgramDirector (Computer Science), OhioNorthern University, Ada, Ohio

Don Hutchison, Mathematics Chair,Clackamas Community College, OregonCity, Oregon

James Hyder, F11X Training Developer(Thin Films), Intel Corporation,Hillsboro, Oregon

Gerald Janey, Chairman of EngineeringTech, Massasoit Community College,Brockton, Massachusetts

Elaine Johnson, Bio-Link Director ATECenter, City College of San Francisco,San Francisco, California

Wendie Johnston, Director, NetBiotechnology Center, Pasadena CityCollege, Pasadena, California

Vern Kays, Assistant Professor Mathematics,Richland Community College, Decatur,Illinois

Robert L. Kimball, Department Chair ofMath/Physics, Wake TechnicalCommunity College, Raleigh, NorthCarolina

Susan La Fosse, Assistant Professor inMathematics, Dutchess CommunityCollege, Poughkeepsie, New York

Margaret Ann Leonard, AssistantProfessor, Mathematics, Hudson ValleyCommunity College, Troy, New York

Philip Lootens, Dean for Sciences,Engineering and Communication Arts,Edison State Community College, Piqua,Ohio

Lynn Mack, Director of InstructionalDevelopment, Piedmont TechnicalCollege, Greenwood, South Carolina

Philip Mahler, President AMATYC,Mathematics professor, MiddlesexCommunity College, Bedford,Massachusetts

Robert Malena, Professor of Mathematics,Community College of Allegheny County,West Mifflin, Pennsylvania

Cyrus McCarter, Instructor of Mathematicsand Physics, Wake Technical CommunityCollege, Raleigh, North Carolina

National Conference Participants 33

Lyle McCurdy, Professor Electrical &Computer Engineering Technology, Dept.of Engineering Technology, CaliforniaState Polytechnic University, Pomona,California

Carol McVey, Technical MathematicsInstructor, Florence-Darlington TechnicalCollege, Florence, South Carolina

Michael Miller, Mathematics Instructor,Wake Technical Community College,Raleigh, North Carolina

Jeanette Mowery, Biotechnology Instructor,Madison Area Technical College,Madison, Wisconsin

Jennifer Nichols, Assistant Professor,Mathematics, Hudson Valley CommunityCollege, Troy, New York

Rodney Null, Associate Professor ofMathematics, James A. Rhodes StateCollege, Lima, Ohio

Carolyn Pabian, Instructor/Mathematics,Delaware Technical and CommunityCollege, Wilmington, Delaware

Edward Pabian, Coordinator, IndustrialTraining Division, Delaware Technical andCommunity College, Newark, Delaware

Arnold Packer, Senior Fellow Institute forPolicy Studies, Johns Hopkins University,Baltimore, Maryland

John Palmer, Professor of ManufacturingEngineering/Electronics TechnologyProgram, Industrial/Electronics, CentralArizona College, Signal Peak Campus,Coolidge, Arizona

Jesse Parete, Chair, Department ofMathematics, Edison State CommunityCollege, Piqua, Ohio

Mary Parker, Professor of Mathematics,Austin Community College, Austin, Texas

James Payne, Supervisor Tech Support &Mgmt Systems, Oak Ridge NationalLaboratory, Oak Ridge, Tennessee

Kathleen Peak, Mathematics Instructor,Rochester Community & TechnicalCollege, Rochester, Minnesota

John C. Peterson, Professor of Mathematics,Chattanooga State Technical CommunityCollege, Chattanooga, Tennessee

Ellena Reda, Mathematics Instructor,Dutchess Community College,Poughkeepsie, New York

John Reed, President, The LearningBusiness, Baton Rouge, Louisiana

Stephen Rodi, Professor of Mathematics,Austin Community College, Austin, Texas

Branisalv Rosul, Professor of Mecomtronics,College of DuPage, Glen Ellyn, Illinois

Nancy Sattler, Associate Dean forCurriculum, Terra Community College,Fremont, Ohio

Albert Schwabenbauer, AerospaceManufacturing Consultant, Wallingford,Connecticut

Dan Schwartz, Manager, Supply &Transportation Systems DevelopmentGroup within IT, Marathon AshlandPetroleum, Findlay, Ohio.

Lisa Seidman, Biotechnology Instructor,Madison Area Technical College,Madison, Wisconsin

Witold Sieradzan, Dean, ComputerInformation Systems, Wake TechnicalCommunity College, Raleigh, NorthCarolina

34 National Conference Participants

Gary Simundza, Professor of Mathematics,Wentworth Institute of Technology,Brookline, Massachusetts

Donald Small, Professor of Mathematics,United States Military Academy, WestPoint, New York

Deanne M. Sodergren, Interim DepartmentChair, Mathematics Department, HudsonValley Community College, Troy, NewYork

Joseph Stearns, Jr., MathematicsInstructor, ATE Faculty, Tri-CountyTechnical College, Pendleton, SouthCarolina

Ronald Talley, EET Department Head,Tri-County Technical College, Pendleton,South Carolina

Robert Templin, Senior Fellow, MorinoInstitute, Reston, Virginia

Gwen Turbeville, Instructor of Mathematics,J. Sargent Reynolds Community College,Richmond, Virginia

Emin Turker, Dean of EngineeringTechnologies, SUNew York Canton,Canton, New York

George Vaughan, Professor of HigherEducation, North Carolina StateUniversity, Raleigh, North Carolina

Jesse Williford, Instructor of Mathematicsand Physics, Wake Technical CommunityCollege, Raleigh, North Carolina

Darlene Winnington, MathematicsInstructor, Delaware Technical &Community College, Newark, Delaware

Jim Wood, Division Chair, Ind’l & Eng Tech,Tri-County Technical College, Pendelton,South Carolina

William Woodruff, Department HeadBiotechnology, Alamance CommunityCollege, Graham, North Carolina

Deborah Woods, Associate Professor ofMathematics, University of Cincinnati,Raymond Walters College, Cincinnati,Ohio

Howard Woods, Adjunct Instructor inInformation Technology, College ofApplied Science, University of Cincinnati,Cincinnati, Ohio

Michelle Younker, Associate Professor andLead Teacher of Mathematics, TerraCommunity College , Fremont, Ohio

Facilatators and Recorders

The following people functioned as either a Facilatator or Recorder at the national conference.

Cheryl Cleaves

Ray Collings

Lawrence Gilligan

Chuckie Hairston

Jan Hoeweler

Vern Kays

Lynn Mack

Rodney Null

Nancy Sattler

Gary Simundza

Deborah Woods

Michelle Younker

CRAFTY Workshop Participants

The following people attended the October 5–8, 2000, CRAFTY workshop at Los Angeles PierceCollege, Los Angeles, California, or the workshop at J. Sargeant Reynolds Community College,Richmond, Virginia, October 12–15, 2000.

Biotechnology and Environmental Technology

Brenda Breeding, Professor of Biology,Oklahoma City CC, Oklahoma City,Oklahoma.

Daniel R. Brown, Lead Instructor ofMicrobiology and Biotechnology, Santa FeCC, Gainesville, Florida.

Sybil Chandler, Lead faculty for theEnvironmental Health and SafetyTechnology Associate Degree program ofMetropolitan CC’s (MCC) Maple WoodsCC, Kansas City, Missouri.

Lois Dinterman, Director of Bioprocessingfor Biolex, Inc., Pittsboro, NorthCarolina.

Linnea Fletcher, Austin CC, Austin, Texas,and South Central Regional Director forBio-Link

Leland Grooms, Adjunct faculty forEnvironmental Sciences, Maple WoodsCC, Kansas City, Missouri.

Elaine Johnson, City College of SanFrancisco, San Francisco, California, andNational Director for Bio-Link, anAdvanced Technological Education (ATE)Center of Excellence of the NationalScience Foundation

Wendie Johnston, Pasadena City College,Pasadena, California: Program Directorand Ed>Net Director

JoAnne Marzowski, Bristol Myers Squibb,Oncogene, Seattle, Washington.

Donald McAfee, Chair and CEO ofDiscovery Therapeutics, Inc., Richmond,Virginia.

Jim McGillem, Biomedical, FacilityDirector, St. Mary’s Warrick Health CareFacility, Boonville, Indiana

Lisa Seidman, Biotechnology Program,Madison Area Technical College,Madison, Wisconsin; North CentralRegional Director for Bio-Link

William H. Woodruff, Department Head,Biotechnology Department, AlamanceCC, Graham, North Carolina, andSoutheast Regional Director for theBio-Link.

Steve Yelton, Program Chair, Electronics,Biomedical Electronics & ComputerNetwork Engineering Technology,Cincinnati State Technical and CC,Cincinnati, Ohio.

36 CRAFTY Workshop Participants

Electronics, Telecommunications, and Semiconductors

David Baker, Professor Emeritus, RochesterInstitute of Technology/Past Chair ofTAC of ABET, Kingsport, Tennessee

Elizabeth Basinet, President, BarrettResource Group, San Diego, California

Bob L. Bixler, Workforce Consulting, AustinCC, Austin, Texas (Now at San JuanBasin Technical School, Cortez,Colorado.)

Steve Cooperman, Science Teacher Lecturer,Westridge School (high school), Pasadena,California

Christopher Dennis, Consultant, SteamingKettle Consulting, Portland Community,Portland, Oregon College

Linda Hollstegge, Instructor, Information &Engineering Technology, Cincinnati StateTechnical & CC, Cincinnati, Ohio

James Hyder, F11X Training Developer(Thin Films), Intel Corporation,Hillsboro, Oregon

Gerald Janey, Chairman of EngineeringTechnology, Massasoit CC, Brockton,Massachusetts

Rino Mazzucco, Instructor, Department ofTechnology: Electronics, Mesa CC, Mesa,Arizona

Lyle B. McCurdy, Professor Electrical &Computer Engineering Technology, Dept.of Engineering Technology, CaliforniaState Polytechnic University, Pomona,California

John Palmer, Professor of ManufacturingEngineering/Electronics TechnologyProgram, Industrial/Electronics, CentralArizona College, Signal Peak Campus,Coolidge, Arizona

James Payne, Supervisor Tech Support &Management Systems, Oak RidgeNational Laboratory, Oak Ridge,Tennessee

Branisalv Rosul, Professor of Mecomtronics,College of DuPage, Glen Ellyn, Illinois

Tim Woo, Instructor,Electronics/Engineering, DurhamTechnical CC, Durham, North Carolina

Jim Wood, Division Chair, Industrial &Engineering Technology, TricountyTechnical College, Pendleton, SouthCarolina

Information Technology

Connie Blackwood, Director of Educationand Research Initiatives, IdahoEngineering Laboratory, Idaho Falls,Idaho.

Robert D. Campbell, Chief InformationOfficer & Executive Dean of InformationTechnology Services, Rock Valley College,Rockford, Illinois.

C. Fay Cover, Director, Division of LearningTechnologies, Pikes Peak CC, ColoradoSprings, Colorado (Currently with SunMicrosystems).

Robert Hillery, Chairman, InformationSystems Technology, NHCTC PeaseInternational Tradeport Campus,Portsmouth, New Hampshire.

CRAFTY Workshop Participants 37

Denecia Merritt-Damron, InformationTechnology Division Director andInstructor of Desktop ApplicationsSoftware, Marshall Community andTechnical College, Huntington, WestVirginia.

Keith Morneau, Assistant Professor,Information Systems Technology,Northern Virginia CC, Annandale,Virginia.

Scott Nicholas, MCSE and Instructor ofComputer Networking, MarshallCommunity and Technical College,

Huntington, West Virginia.

Randy Robertson, Network Instructor,Wake Technical CC, Raleigh, NorthCarolina.

Dan Schwartz, Manager, Supply &Transportation Systems DevelopmentGroup within IT, Marathon AshlandPetroleum, Findlay, Ohio.

Dick Thomas, Independent DatabaseConsultant, Littleton, Colorado.

Nancy Thomas, Independent DatabaseConsultant, Littleton, Colorado.

Mechanical and Manufacturing Technology

Ashok Agrawal, Department Chair andProfessor, Plastics Technology, St. LouisCC, St. Louis, Missouri

Gregg Anderson, Materials Engineer, AirForce Research Lab (AFRL/MLPJ)USAF, Wright Patterson Air Force Base,Dayton, Ohio

Larry Earnhart, Technical Leader,Performance, Polymers and Chemicals,Honeywell International, Inc., ElSegundo, California.

Mike Farley, Marine Designer, National Steeland Shipbuilding (NASSCO), San Diego,California.

Reece Gibson, CNC Instructor, MetropolitanCCs, Kansas City, Missouri.

Nick Johnson, Technical Instructor, BoschAutomotive, Anderson, South Carolina.

Tracy Koss, Engineering Manager,Marathon/ Ashland Corporation, Findlay,Ohio.

Jack Mclellan, Quality AssuranceCoordinator, Mott CC, Flint, Michigan.

Gregory Meserve, Professor ofManufacturing Technology, NewHampshire Technical CC, Berlin, NewHampshire.

Bob Monter, Technology Specialist, WrightTechnology Network, Dayton, Ohio.

John Reed, Training and DevelopmentSpecialist, ExxonMobil, Baton Rouge,Louisiana.

Frank Rubino, Professor and Chair,Department of MechanicalCivil/Construction EngineeringTechnology, Middlesex County College,Edison, New Jersey.

Al Schwabenbauer, Vice President, SikorskyAircraft, United TechnologiesCorporation.

Jim Shimel, Machine Tool Instructor,Metropolitan CCs, Kansas City, Missouri.

Emin Turker, Dean of EngineeringTechnologies, Lakeland CC, Mentor, Ohio(Currently Dean of EngineeringTechnologies, SUNY Canton, Canton,New York).

38 CRAFTY Workshop Participants

Mathematicians

Gayle Childers, J. Sargeant ReynoldsCommunity College, Richmond, Virginia

Deborah L. Cohen, John Tyler CommunityCollege, Chester, Virginia

Susan Ganter, Clemson University, Clemson,South Carolina

Bill Haver, Virginia CommonwealthUniversity, Richmond, Virginia

Mary Ann Hovis, James A. Rhodes StateCollege, Lima, Ohio

Harvey Keynes, University of Minnesota,Minneapolis, Minnesota

Robert L. Kimball, Wake TechnicalCommunity College, Raleigh, NorthCarolina

Roland Moore, J. Sargeant ReynoldsCommunity College, Richmond, Virginia

John C. Peterson, Chattanooga StateTechnical Community College,Chattanooga, Tennessee

Elizabeth Teles, National ScienceFoundation, Washington, D.C.

Gwen Turbenville, J. Sargeant ReynoldsCommunity College, Richmond, Virginia

Bruce Yoshiwara, Los Angeles PierceCollege, Los Angeles California

Kathy Yoshiwara, Los Angeles PierceCollege, Los Angeles California

Susan Wood, J. Sargeant ReynoldsCommunity College, Richmond, Virginia

Julia Woodbury, J. Sargeant ReynoldsCommunity College, Richmond, Virginia

Project Evaluator

George Vaughan, Professor of Higher Education, North Carolina State University, Raleigh,North Carolina

Advisory Committee

Seven people were members of the AdvisoryCommittee. A short biography of each Commit-tee member is below.

Catherine C. Aust, Head of the Departmentof Mathematics, Clayton College & StateUniversity, Morrow, Georgia. She is therepresentative to the Advisory Commit-tee from the Mathematical Association ofAmerica (MAA).

David Baker, Professor Emeritus, RochesterInstitute of Technology. Past Chair of TACof ABET (Technology Accreditation Com-mission of the Accreditation Board for En-gineering and Technology, Inc.)

Baker began his teaching career at theSUNY College of Technology in Alfred,New York, after receiving his undergrad-uate degree from Manmouth College ofWest Long Branch, NJ. While he pursuedhis graduate degree at the Rochester Insti-tute of Technology (RIT), Baker continuedteaching at SUNY. He earned his Master ofScience degree in 1972.

In 1981, Baker left SUNY to become theDirector of the School of Engineering Tech-nology and a Professor of Electrical Engi-neering Technology at Rochester Instituteof Technology, where he was responsiblefor overseeing 1300 students and 47 facultyand staff.

As a tireless supporter of engineering tech-nology education, Baker has spent 25 years

striving to develop a niche for this emerg-ing specialization. He has championed nu-merous initiatives to bring greater qual-ity, diversity, and innovation to engineeringtechnology programs and has also served asa leader in a number of professional soci-eties, including the Institute of Electricaland Electronics Engineers (IEEE) and theAmerican Society for Engineering Educa-tion (ASEE), where he chaired both theEngineering Technology Council and theEngineering Technology Leadership Insti-tute.

Elaine A. Johnson, City College of San Fran-cisco, Director of Bio-Link. BS Phys-ical Science/Math, University of Min-nesota, 1960; MA Biology/Microbiology,San Francisco State University, 1976; MSBiology/Clinical Nutrition, 1982; PhD,Educational Administration, University ofTexas at Austin, 1997.

Johnson is the PI for the Bio-Link Na-tional Advanced Technological EducationCenter for Biotechnology and has partici-pated in national efforts to improve mathe-matics education for students preparing forcareers in the emerging technologies, par-ticularly for biotechnology technicians.

Arnold Packer, Senior Fellow, Institute forPolicy Studies, the Johns Hopkins Univer-sity, Baltimore, Maryland.

40 Advisory Committee

Packer was executive director of the U.S.Labor Department’s Secretary’s Commis-sion on Achieving Necessary Skills orSCANS. He is co-author of Workforce2000, which, along with the SCANS re-ports, has altered the national conversa-tion about human resource education. Hehas had a distinguished career in the fed-eral government having worked for the U.S.Office of Management and Budget, theSenate Budget Committee, and as assis-tant secretary for policy, evaluation, andresearch at the U.S. Department of La-bor. Dr. Packer holds a Ph.D. in economicsfrom the University of North Carolina atChapel Hill and a master’s in business ad-ministration.

John Reed, President of The Learning Busi-ness, specializing in linking learning withbusiness. Former Training and Develop-ment Specialist for ExxonMobil, BatonRouge, Louisiana.

Albert Schwabenbauer, Aerospace Manufac-turing Consultant. Former Vice-President,Planning at Sikorsky Aircraft, UnitedTechnologies Corporation. He has broadmanufacturing and customer service ex-ecutive background, having served asVice-President, Manufacturing, and Vice-

President, Customer Service, over a 33-year career in aerospace.

Schwabenbauer holds a bachelor’s andmaster’s in Electrical Engineering fromRensselaer Polytechnic Institute, and amaster’s in Administrative Sciences fromYale University.

Ronald Talley, Department Head, ElectronicEngineering Technology, Tri-County Tech-nical College, Pendleton, South Carolina,and leader in the South Carolina ATE (Ad-vanced Technological Education) Center.

Talley has three years experience in sec-ondary education and over twenty-six yearsin postsecondary. He is presently headof the Electronic Engineering TechnologyDepartment at one of sixteen campusesin the South Carolina technical educationsystem. He has been a member of SouthCarolina’s Exemplary Faculty of AdvancedTechnological Education for the past threeyears, which is funded by the NationalScience Foundation. He was recognizedas a distinguished educator by the Gover-nor’s Office in the State of South Carolina.Talley was named one-of-ten outstandingtechnical educators by the American Tech-nical Education Association.

Documents

A Vision: Mathematics for the Emerging Technologies A ...c.ymcdn.com/sites/ Mathematics for Tomorrow: Recommendations and Exemplary Programs Mary Ann Hovis Robert L. Kimball John C