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CAS in the Classroom:Yesterday, Today, and Tomorrow
TIME 2012, Tartu, Estonia
William C. Bauldry
Mathematical SciencesAppalachian State University
Boone, NC 28608USA
July, 2012
TIME 2012: Page 1 of 20
Outline
CAS in the Classroom:Yesterday, Today, and Tomorrow
1 Introduction• Setting the Stage
2 Yesterday• A Little Bit of History
• First Steps
• Engagement
• CAS and the“Calculus Wars”
3 Today• Current Foci• Commercial versus
Open Source• Online Tools• From Worksheet to
Microworld• The Beginning of CAS
Assessment Tools
4 Tomorrow• Mobility & Personal Power
TIME 2012: Page 2 of 20
Introduction
AbstractWe will consider the development of classroom use of computeralgebra systems (CAS) from different perspectives. A small bit of CAShistory sets the stage, followed by what and how we have arrived attoday’s usage. We close with thoughts on future technology and whereCAS can lead us to go.
Prediction is always difficult. Especially about the future.— Mark Twain (or Yogi Berra or Niels Bohr or Albert Einstein)
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Computer Algebra Timeline
1950 — •
• 1953 Master Theses: Kahrimanian, Nolan
•1955 — •
•1960 — •
• 1963 ALPAK
• 1966 SYMBAL1966 SYMSAM ’66 •• 1965 Buchberger’s algorithm1965 — •
• 1968 ALTRAN, CAMAL, FORMULA, FORMAC, MATHLAB1968 Risch Algorithm •• 1967 Berlekamp’s algorithm
• 1969 MACSYMA• 1970 REDUCE, SchoonShip1970 — •
• 1974 SCRATCHPAD
• 1979 muMath
1975 — •
• 1980 VAXIMA1980 — •• 1981 SMP
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1980 — • • 1980 VAXIMA• 1981 SMP
• 1983 Maple, Symbolics MACSYMA• 1984 DOE-Macsyma, FORM• 1985 MathCAD, Powermath, J of Symbolic Computation1985 — •• 1986 REDUCE/PC• 1987 Milo, HP-28C• 1988 Derive, Macintosh Maple, Mathematica, PARI, Theorist1988 ISSAC •• 1989 MACSYMA/PC
• 1991 Axiom, ALJABR, Mathcad, ParaMacs, SymbMath• 1992 SIMATH
1990 — •
• 1995 MuPAD, TI-921995 — •
• 1997 CASIO-Maple
• 1999 Yacas• 2000 Giac/Xcas, Maxima (DOE-Macsyma)2000 — •
• 2005 Sage2005 — •
• 2009 TI-Nspire, PocketCAS• 2010 iCAS (REDUCE)2010 — •
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Yesterday
The Early Days: Automatic Differentiation1953 H Kahrimanian, Analytical differentiation by a digital
computer, Master’s thesis, Temple Univ1953 J Nolan, Analytical differentiation on a digital computer,
Master’s thesis, MIT
Beginnings• Handling large computations of physics• Emphasis on writing special purpose programs• Beginnings of collections of libraries/routines
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YesterdayThe ’60s: Exponential Growth
1961 SAINT; Slagle, MIT (Symbolic Automatic INTegration)1962 FORMAC; Sammet, Tobey, IBM1963 Schoonschip; Veltman, SLAC1964 ALPAK; Brown, Bell Labs1965 MATHLAB; Engleman, MIT — First interactive system1966 SIN; Moses, MIT: (Symbolic INTegrator)1968 REDUCE; Hearn, Stanford — Still available
MACSYMA; Engelman, Martin, Moses — Still available
1966: First CAS Dedicated ConferencesACM SYMSAC ’66: Symposium on Symbolic & Algebraic
Manip., Washington, D.C.SAME Symbol Manip. Languages & Techniques, Pisa
These conferences merged to become the ISSAC series.
TIME 2012: Page 7 of 20
YesterdayThe ’80’s: Desktop Systems & Graphical User Interfaces
1979 muMATH; Stoutemyer, Rich — The first on desktops! 1
1980 Maple; Gonnet, Geddes — A Macintosh version in ’85
1988 Derive; Stoutemyer, RichMathematica; WolframMilo; AvitzurTheorist; Bonadio
CAS Arrives in Education“At the Fifth Int’l Congress on Mathematical Education (8/84) held inAdelaide, Australia, ... four meetings on symbolic mathematical systems andtheir effects on the curriculum. The report ... computer algebra will affecteducation: (i) courses about computer algebra, (ii) courses on how to usecomputer algebra and (iii) pedagogical uses of computer algebra.” —
from “Computer Algebra: Past And Future,” Caviness, EUROCAL ’85
1For IBM-PC & Jr, MS-DOS, CP/M-80, Apple II, & TRS-80. Also PICOMATH-80 (16k).
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YesterdayEarly Classroom UsageReports• B Buchberger, Special Section on Symbolic Mathematical
Systems and Their Effect on the Curriculum, SIGSAM Bull.,18/19 (4), 1984
• F Demana, J Harvey, B Waits, Proc. of the First ICTCM, 1988
PhD Dissertations• K Heid, An exploratory study to examine the effects of
resequencing skills and concepts in an applied calculuscurriculum through the use of the microcomputer, U of Maryland,1984
• J Palmiter, The Impact of a Computer Algebra System onCollege Calculus, Ohio State U, 1986
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Yesterday
1984: The CallIn his SIGSAM presentation “A Radical Proposal For ComputerAlgebra In Education” David Stoutemyer 2 calls for “computer algebra[to be] used in math education” for:• Computer-Aided Instruction
• Automatic Drill, Testing & Record Keeping
• Enrichment and Motivation
• Preparation & Checking of Teaching Materials
• Computer Algebra Courses
2David R. Stoutemyer, “A Radical Proposal For Computer Algebra In Education,”SIGSAM Bull., 18/19 (4), 1984, pp 40-53.
TIME 2012: Page 10 of 20
Yesterday
The ’90’s: CAS Enters Mainstream Classrooms
- Initial Classroom Uses• Algebraic and graphical demonstrations• Arbitrary precision calculator• Powerful symbolic calculator• Extended and “real-world” exercises• “Laboratory” inquiry sessions
- Many Initial Projects Were Grant Funded
- Early 90’s: CAS Texts Began to Appear (1990, 2nd ed 1996)
- Late 90’s: CAS Becomes Widespread
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Yesterday
The “Calculus Wars”• Not the 17th century Newton v Leibniz controversy
• Ubiquitous inexpensive CAS:• Added a powerful mathematics expert to the classroom!
• Changed the approach to topics!
• Changed the topics to approach!
• Changed the answers to didactic questions!
• Changed the questions needing didactic answers!
• Large scale and emotional turmoil in mathematics:• What is important in mathematics instruction?
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Today
Current Focus• Powerful Symbolic Manipulators• Elegant Graphics• Huge Mathematics Knowledge Base with ‘Math-Search’ & Texts• Mathematical Microworld Platforms• ‘Friendly’ GUI for Input and Output & doc structure: iBooks, CDF
Current Popular Systems
Commercial Open Source
Desktop Maple, Mathematica, MuPAD Giac/Xcas, SageHandheld TI-Nspire, Casio ClassPad PocketCAS, iCASa
Online Wolfram Alpha Sage
aCommercial front-ends with Xcas and Reduce, respectively.
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TodayCurrent Pedagogical StandardsIn the US, the “Common Core State Standards3 for Mathematics”includes 7 references to CAS in the standards areas:
• Use appropriate toolsstrategically
• Numbers & NumberSystems
• Algebra: expressions
• Rewrite rational expressions
• Functions: graphing
• Geometry
• Mathematical Modeling
Today’s Research & Retrospectives• The Didactical Challenge Of Symbolic Calculators• Research on Tech and the Teaching and Learning of Math• The Int’l J for Tech in Math Education (formerly Int’l J of CA in Math Ed)
And of course: Austrian Center for Didactics of Computer Algebra
3Adopted by 45 of the 50 states, Washington D.C., and 2 US territories.
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TodayCurrent Questions of Didactics
CAS Mathematics Assessment
• About CAS
• How to use CAS
• Which topics• How to incorp-
orate CAS
• Via CAS(background)
• Using CAS
The Math Assoc of America’s Curriculum Foundations Project states:• “Mathematics majors ... should, in particular, have experience
with a computer algebra system during the first two years ofundergraduate training.” pg 110
• “Courses for future teachers make use of computer algebrasystems to investigate equivalence of expressions andtransformations of functions.” pg 150
• “Computer algebra: Material chemists expect mathematicians todevelop. Material in mathematics for which chemists haveresponsibility at the advanced course level.” pg 35
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AssessmentThe beginnings of CAS-supported assessment:• CAS interpreted student free-response answers
• Allowing margins of error in student responses
• New types of questions:• Items algorithmically generated by CAS• Adaptive question streams analyzed by CAS• Clickable image questions
• Students can preview mathematical and graphical answersbefore submitting
• Simple, high-quality, low-workload, student diagnosticself-assistance
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Today2012 CAS Conferences
T3 March 2, Chicago, Illinois, USA
ICTCM 24 March 22, Orlando, Florida, USA
PCA 2012 Polynomial CA 2012, April 23, St. Petersburg, Russia
ECCAD East Coast CA Day, May 12, Rochester, Michigan, USA
SCA Symbolic Comp & its Appl’s, May 17, Aachen, Germany
WCA 15th Workshop On CA, May 23, Dubna, Russia
CADGME CA & Dyn Geo Soft in Math Ed, June 22, Novi Sad, Serbia
ACA 2012 June 25, Sofia, Bulgaria
TIME 2012 July 10, Tartu, Estonia
ISSAC 2012 July 22, Grenoble, France
CASC 2012 September 3, Maribor, Slovenia
ASCM 2012 Asian Symp on Comp Math, October 26, Beijing, China
ACTM Asian Tech Conf in Math, December 16, Bangkok, Thailand
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Tomorrow
Coming Soon to Your Classroom!• MOBILE
• On 28 April, 1999, Maple and Casio announced the A-22T palmcomputer with a version Maple V (and Geometer’s Sketchpad).
• On 10 November, 2010, iCAS (based on Reduce) was released tothe iTunes App Store for iPads and iPhones.
• NATURAL LANGUAGE SYSTEMS
• “What is the Buchberger algorithm?”• “Integrate cos(
√x ) cos(x)/
√x for x from 0 to infinity” (Math Mag)
• “Integrate 1/(x3 − x+ 5 + tan(1/10)) for x from −2 to 2”
• CONVERGENCEA single device: Phone + PDA + Email + Web browser
+ Graphing Calculator + CAS!How do we use these in classes?
• Instant, always-on access to powerful CAS servers online.
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Tomorrow
Imagine...A student takes her ‘CAS-pad’ from her backpack,unfolds it, and begins to enter mathematics and textwith a stylus receiving instantaneous responses, for-mulas and graphs from the CAS server in the cloud.Her teammate across campus appears in her CAS-pad’s video message window to add his computationto their solution on the shared portion of her screen.
She annotates their work adding references with hot-links. Sheactivates the voice mode for finishing touches, editing their report andrecording her narrative explaining the problem and their solution’stechniques. She then tells her CAS-pad to email their report to herinstructor and a copy to her partner.
Her CAS-pad generates a Sudoku puzzle for her to enjoy...
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TomorrowImagine...
A student comes to the Assessment Centerto qualify for his mathematics credits. Heunfolds his CAS-pad, authenticates with hissmartcard, and begins the test. The first prob-lem requires some background computations.As he writes on the board, his CAS-pad captures his work. Thestudent encounters an unfamiliar term, but when he attempts to usehis math search engine, the assessment system blocks his access.He adds several voice annotations to his responses, then submits hiswork. The CAS-enhanced system analyzes his answer; the systemdetermines the next question based on his previous response andpresents the student with the new challenge. At the end of hissession, the student receives a ‘Credit Awarded’ certificate in hisemail.
Our student goes to meet his friends to celebrate... Terviseks!
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As CAS becomes faster, more powerful,comprehensive, pervasive, and easier to use,
we need to ask:
What mathematics do ourstudents need to learn ...
Where will CAS help us to goin didactics and pedagogy...
Tomorrow?
T anan Vaga
William C. Bauldry, Ph.D.
ProfessorMathematical Sciences
Appalachian State University
ph: (828) 262-2355email: [email protected]
web: mathsci2.appstate.edu/∼wmcb/
