Past and Future of Computer Algebra in Mathematics Education

Bernhard Kutzler (Austria)

Past and Future of Computer Algebra in Mathematics Education

A personal (& nostalgic) perspective

Bernhard Kutzler (Austria) The PAST

10, 294 days ago… 10, 294 days ago… … on Mayy, 3rd, 1982:

Z Ronald Reagan is US President Z Margaret Thatcher is British Prime Minister Z Helmut Kohl becomes Chancellor of Germany Z Leonid Brezhnev dies, Y. Andropov follows Z Falklands war Z first CDs released to public in Germany Z Computer is Time Magazine‘s Man of Year 10, 294 days ago… … on Mayy, 3rd, 1982: 10, 294 days ago… … on Mayy, 3rd, 1982:

10, 294 days ago… … on Mayy, 3rd, 1982: 10, 294 days ago… … on Mayy, 3rd, 1982:

6, 647 days ago… 6, 647 days ago… … on App,ril 27th, 1992:

6, 647 days ago… … on App,ril 27th, 1992:

International Spring School on the Didactics of Computer Algebra

27-30 April 1992 KAtiKrems, Austria (organized by ACDCA) Bärbel Barzel Josef Böhm

Bärbel Barzel

Helmut Heugl Josef Böhm

Bärbel Barzel Helmut Heugl Josef Böhm

Johann Wiesenbauer

Bärbel Barzel

Helmut Heugl Josef Böhm

Johann Wiesenbauer

Bärbel Barzel

Bernhard Kutzler Helmut Heugl Josef Böhm David Stoutemyer

Johann Wiesenbauer

Bärbel Barzel Eduard Szirucsek

Bernhard Kutzler Klaus Aspetsberger

We worked extremely hard during the sessions … … and used the breaks between sessions for some (mathematical) fun.

David Stoutemyer‘s Dream

Z Using Reduce with his UH students … David Stoutemyer‘s Dream

Z Using Reduce with his UH students …

Z Teaming up with Albert Rich …

David Stoutemyer‘s Dream

Z … to form Soft Warehouse … David Stoutemyer‘s Dream

Z … with its products muMATH and muLISP.

Z muMATH-79 ran on 8080 and Z80 cocomputersmputers aandnd nneededeeded as lilittlettle as 4488 KB memory under CP/M and TRS-DOS. Z muMATH-80 ran also on 6502 based AppApplele II cocomputersmputers.

Z muMATH-83 ran a lso on 8088 computers and needed 300 KB.

Z David wrote less ambitious CASs in Basic, oonene ooff tthemhemwwasas tthehe ssharewarehareware pprogramrogram PicoMath (1980).

Z All four programs begin with 1010 GOGOTOTO 4040 20 A = expression 30 RETURN 40 …

Z The manual tells users to modify the right side of the assignment statement on line number 20 to an expression that they want simplified to the implemented class. Z Derive was released in 1988 as the successor ooffmmuMATHuMATH.. ItIt wwasas menumenu- oriented, offered 2D and 3D plotting, ran under MS-DOS and needed 512 KB .

Z Derive 2 was released in 1990. Z Derive 3 (still DOS) was released in 1993. Z There was a ROM version of Derive 3 for tthehe HP 9595LX.LX.

Z Derive 4 (Derive for Windows) was releasedreleased in 11996.996. SWH impressions

Karen & David Stoutemyer Joan & Albert Rich

SWH impressions About SWH

Z The BG Story ...

Z Bruno Buchberger, U of Linz, & SWHE

Z The Pentium Bug Story …

On Oct 19, 1994 Dr. Thomas Nicely, pprofessorrofessor ooffmmathematicsathematics at LLynchburgynchburg College, discovered what became known as the Pentium Bug . He found that 824,633,702,441 divided by iitselftself …

… gave 0. 999999996274709702 instead of 1.

Later it was found that this was true for any number between 824,633,702,418 and 824, 633, 702, 449.

The unknown part of the story …

Z In 1999 Soft Warehouse, Inc. was acquired by TTexasexas InstrumentsInstruments Inc.Inc. Z In 1999 Soft Warehouse, Inc. was acquired by TTexasexas InstrumentsInstruments Inc.Inc. Z Derive 5 was released in 2000. Z Derive 6 was released in 2003.

Z In 1999 Soft Warehouse, Inc. was acquired by TTexasexas InstrumentsInstruments Inc.Inc. Z Derive 5 was released in 2000. Z Derive 6 was released in 2003.

Z Derive was taken off the market in 2007. The iffiilinofficial SWH museum iAlbin Albert Rich‘ scondo

David Stoutemyer‘s Dream

ZDavid had looked for a partner for a handheld CAS.

Z In 1992 David and TI started working on the TI-92 project Z(and Albert Rich took over the development of Derive). Z The TI-92 was released in 1995. Z The TI-92+ was released in 1998. Z The TI-89 was released in 1998. Z The Voyage 200 was released in 2002.

Z TI-Nspire CAS was released in 2007. History of CAS

Z It began in the mid 60s …

More CAS

Z Casio CFX -9970G, Casio Algebra FX 2. 0, Casio ClassPad Z Z WIRIS Free CAS

Z Reduce Z Z Axiom Z (= Yet another CAS) Z Geogebra (?) Z PocketCAS for (iPhone) Z [Rubi – you need Mathematica 6] http://www.apmaths.uwo.ca/~arich/ David Stoutemyer‘s CAS Code

David Stoutemyer‘s Dream CAS as Teaching Tool

Z Teacher seminar 1982 in Klagenfurt, Austria

Z muMATH for teaching 12 year old students at Wilhering High School in Austria in 1982/83. Klaus Aspetsberger

1984

Klaus Aspetsberger CAS as Teaching Tool

Z Helmut Heugl, Austria, 1986: using HP28C with high school students Z Kathy Heid, USA, 1988: CAS in undergraduate math teaching

Z Bruno Buchberger 1985/90: „Should Students Learn Integration Rules“ – introduced White-Box/Black-Box Principle Z BK: Scaffolding Method Z 1991: Austrian MoE (Eduard Szirucsek) buys Derive for all Austrian high schools

Z Slovenia, United Arab Emirates, Slovakia, five German states follow

Z France, Italy, …

Z 1991/92: ACDCA founded

Z 1992-96: ACDCA 1 – 2000 stud, Derive Z 1997-98: ACDCA 2 – 1700 stud,D, D + TI92 Z 1999-00: ACDCA 3 – 2000 stud, D + TI92 Z 2001-02: ACDCA 4 Z 2003-04: ACDCA 5

Z Many natilional projects … www.acdca.ac.at

Z Herget/Heugl/Kutzler/Lehmann 2000: „Indispensable Manual Calculation Skills in a CAS Environment“ 1992: Krems, Austria

Terence Etchells

1993: Krems, Austria

Philip Yorke

Helmut Heugl‘s Country Buskers 1994: Plymouth, UK

John Berry

1994: Plymouth, UK

SWH & SWHE @@y Plymouth 1995: Honolulu, USA

Wade Ellis

1996: Bonn, Germany

Austrian Students

Walter Klinger 1997: Sarö, Sweden

Singing in David Sjöstrand‘s home

1998: Gettysburg, USA Carl Leinbach Eugenio Roanes

Carl Leinbach

Eugenio Roanes

Michel Beaudin

Carl Leinbach Eugenio Roanes Guido Herweyers

Michel Beaudin

Carl Leinbach

1999: Gösing, Austria 1999: Gösing, Austria

Theresa Shelby

2000: Portoroz, Slovenia

Vlas ta KklKokol-VljVoljc

Deepa Jeswani, India 2000: Liverpool, UK

2002: Vienna, Austria 2002: Vienna, Austria

2002: Vienna, Austria

BBhbBruno Buchberger 2002: Vienna, Austria

2002: Vienna, Austria

Sergey Biryukov, Russia 2004: Montreal, Canada

Gilles Picard Kathlyn Pineau

2004: Montreal, Canada 2004: Montreal, Canada

2006: Dresden, Germany 2008: Buffelspoort, South Africa

2008: Buffelspoort, South Africa

Rainer Heinrich

Steve Joubert Karsten Schmidt 2008: Buffelspoort, South Africa

Gosia Brothers

Jean-Jacques Dahan Koen Stulens

2008: Buffelspoort, South Africa

Dirk Warthmann

Wolfgang Moldenhauer 2010: Malaga, Spain

José Luis Galan

meecas.org CAS in ME Conferences

Z 1992-***: ACDCA/Derive/TIME conf.

Z 2003-***: USACAS ( by meecas)

Z 2007: Regional NCTM Conference with CAS strand

Tech in ME Conferences

Z ICTCM (Int Conf on Technology in Collegiate Mathematics) Z ICTMT ( Int Conf on Technology and Mathematics Teaching) Z ATCM (Asian Technology Conference in Mathematics) Z T^3 Conferences IJTME

eJMT NCTM on technology

ClltCalculators an dththliltld other technological tools, such as computer algebra systems, interactive geometry software, applets, spp,readsheets, and interactive p resentation devices, are vital components of a high- quality mathematics education. ClltCalculators an dththliltld other technological tools, such as computer algebra systems, interactive geometry software, applets, spp,readsheets, and interactive p resentation devices, are vital components of a high- quality mathematics education. With guidance from effective mathematics teachers, stu dents at different lev els can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts …

… andhd enhance compu ttilfltational fluency. … andhd enhance compu ttilfltational fluency. In a well-articulated mathematics program, students can use these tools for compp,utation, construction , and representation as they explore problems.

… andhd enhance compu ttilfltational fluency. In a well-articulated mathematics program, students can use these tools for compp,utation, construction , and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making. BK, 2008

The FUTURE Computers are exceptionally wonderful tools for mankind . In the times of DOS, computer use was restridicted to techno-phile people. Advanced operating systems made computers accessible to techno-phobe people also .

Sim ilar ly, CASs are excep tiona lly won der fu l tools for math education. But today’s CAS products are still only used by techno-phile teachers. We do need interfaces on top of CAS engihines that ma khike this great tec hlhnology accessible also to the techno-phobe teachers … … – and that’ s by far the biggest crowd . IdI don ’t see a s ing le too l on the mar ke ttt to day that makes CAS accessible to techno- phobe teachers. As long as we need teacher training for CAS (or other math) products, we are not producing for the mass market. How manyyy cell phones would the industry sell, if users would need cell phone training before being able to make good use of them?

After wor king in this area an d bus iness for 28 years, I wonder about one thing: Why does the industry test their products with techno-phile teachers? More than ten years ago Helmut Heugl from thAhe Austr ian MEMoE said: > If something does not work for my expert teachers, I know it does not work for any of my other teachers. > If it does work for my expert teachers, I know nothing about my other teachers. The average ma th t each er i s weakik in mathematics and techno-phobe.

ZWhy this answer? ZWhy answer different from solution book? ZWhy no answer? b Provide context sensitive background info on math (fitting syn tax an d his tory o f expression) … like spell-checker in MSWord

The average ma th t each er i s wea k in mathematics and techno-phobe.

Z The user interface is non-intuitive. Z Math via keyboard (and mouse) is cumb&lbersome & clumsy. b Provide math-intuitive, pen-based user interface. INPUT:

Z Algebra Z Graphs Z Geometry Z Text

Alge bra: as i+is + • Handwritinggg recognition • Intuitive way of applying operations: Alge bra: as i+is + • … Intuitive wayyppygp of applying operations:

GhGraphs: as i+is + •Draw pp(p)oints (with pen) and choose from available models

• OR choose a model, then draw the necessary number of points GtGeometry: as i+is + •Oppggtion to change a configuration such as in Geometry Expression

GtGeometry: as i+is + • Handsketch with the option to turn into precise object TtText: as i+is + • Handwriting + hand drawn sketches

OUTPUT:

Z Mirror Window Mirror Win dow:

• Eg graphic mirror of algebra window for equation solving

Graphic mirror

2x – 5 = - x + 1 Graphic mirror

2x – 5 = - x + 1 | +x

Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

3x = 6 Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

3x = 6

Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

3x = 6 Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

3x = 6 | /3

Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

3x = 6 | /3 x = 2 Graphic mirror

2x – 5 = - x + 1 | +x

3x – 5 = 1 | +5

3x = 6 | /3 x = 2

„No knowledge seems harder to acquire than the perception of when to stop.“

(Jonath an S wift) The essence of what I learned in these 10,000 days:

Teaching = accompany personal growth