Computer Algebra Systems (CAS): From Where Did They Come, and Where Might They Go?

1. Computer Algebra Systems (CAS):From Where Did They Come, and Where Might They Go? Ed Dickey University of South Carolina National Council of Teachers of Mathematics Annual Meeting – San Diego 22 April 2010

2. Jacques Barzun, Teacher in America “ I have more than an impression- it amounts to a certainty- that algebra is made repellent by the unwillingness or inability of teachers to explain why… There is no sense of history behind the teaching, so the feeling is given that the whole system dropped down ready-made from the skies, to be used by born jugglers.”

3. Plan for this Session • A Look Back… • General and Personal History of CAS • Contrast with dynamic geometry • Input from key researchers and educators • A Look Forward… • What needs to happen for CAS to have an impact on school mathematics • Views from key mathematicians, mathematics educators and researchers, developers, and teachers

4. My First CAS Experiences… • NSF Conference Report, October, 1982 • Graphing utilities, VisiCalc, TK!Solver, and muMATH in school algebra

5. From 1982… • “A new role for manipulative skills.” As calculators impact arithmetic, “computers seem to offer the same promise in algebra; diminishing the importance of developing student skill in algebraic manipulations.” • “The preceding proposals of topics to be deleted or given reduced attention in high school algebra will certainly provoke vigorous dissent from all corners of the mathematical community.”

6. My First CAS Experiences… “The biggest need at the moment, in my opinion, is to have a good, thorough look at the total elementary and secondary curriculum… in mathematics, to see how the priorities of topics, and pedagogical possibilities, and the interaction among the topics change in light of current technological possibilities.. How do they change what we ought to do?” Henry Pollak (interviewed in Steen and Albers, 1981 and quoted p.3 of Computing and Mathematics)

7. Congratulations! Lifetime Achievement Award for Distinguished Service to Mathematics Education To Professor Pollak for the 2010 Mathematics Education Trust

8. My First CAS Experiences… • 1984 from NCTM Year Book • “Imperatives and Possibilities for New Curricula in Secondary School Mathematics” by Fey and Heid • Page 23

9. My First CAS Experiences… 1986 NCTM Charleston Regional Why and How to Use Symbol Manipulation Software muMATH-80 for the Apple II

10. My First CAS Experiences… Phone call to Albert Rich and Dave Stoutemyer 1987 USC became “world distributor” of muMATH-80 Teacher’s Guide $40 ($5 Soft Warehouse royalty)

11. Smithsonian Exhibit • Slates, Slide Rules, and Software: Teaching Math in America • muMATH, Derive, TI-92

12. History of CAS • LISP in 1962 • Physicist Martin Veltman’s Schoonschip (“clean ship” in Dutch)in 1963 • U.S. Carl Engelman’s MATHLAB in 1964 led to Macsyma

13. CAS in Schools “Computer Symbolic Math & Education: A Radical Proposal” ACM SIGSAM Bulletin 13, 3 (August 1979), pp. 8-24.

14. CAS in Schools

15. NCTM 1989 Standards The Information Society. … The use of this technology has dramatically changed the nature of the physical, life, and social sciences; business; industry; and government. …. The new technology not only has made calculations and graphing easier, it has changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them.

16. NCTM 1989 Standards • The use of computer utilities to develop conceptual understanding • The use of technology in instruction should further alter both the teaching and the learning of mathematics. • Computer software can be used effectively for class demonstrations and independently by students to explore additional examples, perform independent investigations, generate and summarize data as part of a project, or complete assignments.

17. South Carolina Standards Elementary Algebra A-1.7 [2007] • Understand how to represent algebraic relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs) • Also mentioned in Intermediate Algebra and Precalculus.

18. CAS in Schools: Science Fair

19. Even among mathematicians… • Even this year, a former editor of a SIAM journal and current math department chair did not realize a CAS could compute 1000! in full precision. • Have you seen it? All 2,658 digits? • Thank you, Derive 5!

20. Mathematics Teacher, Sept 1989 “How Symbolic Mathematical Systems Could and Should Affect Precollege Mathematics.” M. Kathleen Heid • muMATH, Derive, IBM Math Exploration Toolkit, HP28S • “… school mathematics may be able to progress toward better understanding of concepts, mathematical modeling, superprocedures, numerical and symbolic patterns and the equivalence of mathematical representations.”

21. Textbook 1990s • Fey, Heid, Good, Sheets, Blume, Zbiek, Janson Publications,1995 • Table of Contents • Variables and Functions • Functions and Computing Technology • Linear Functions • Quadratic Functions • Exponential Functions • Rational Functions • Systems of Functions and Equations • Symbolic Reasoning : Equivalent Expressions • Symbolic Reasoning : Equations and Inequalities

22. Textbook 2009 Al Cuoco and others with EDC • “platform for experimenting” • “reducing computational overhead” • “build computational models”

23. Dynamic Geometry • Geometric Supposer, 1985,Judah Schwartz and Michal Yerushalmy

24. Dynamic Geometry • Cabri, 1994, Jean-Marie Laborde • Geometer’s Sketchpad, 1995, Nick Jackiw • “Lifting the Curtain: The Evolution of The Geometer’s Sketchpad” by Daniel Scher

25. Graphing Utilities • Widespread use • 1985 Casio 7000G • Handheld CAS

26. Ubiquitous access… • Wolfram Alpha, free on web • iPhone Symbolic Calculator $.99 • PocketCAS $4.99

27. Some 40 years have passed… • 25 years since my own first exposure to CAS • … and SO WHAT? • What can teachers, mathematicians, mathematics educators, researchers, and developers tell us about… • Where we are today… • And where we might go tomorrow

28. Method • Wrote to 35 of the “best and brightest” involved in CAS work for at least the past 15 years. • Four Questions about CAS • 19 generous responses

29. Mathematicians • Lynn Steen • Tom Dick • Jeanette Palmiter • Bert Waits

30. Teachers • John Mahoney • Lin McMullin • Natalie Jakucyn

31. Researchers • Paul Goldenberg • Carolyn Kieran • Paul Drijvers • Luis Saldanha

32. Mathematics Educators • Jim Fey • Karen Hollebrands • Thomas Edwards • Bob Ronau • Johnny Lott

33. Educators/Developers • Al Cuoco • Bernhard Kutzler • Albert Rich

34. The Questions • To what degree has the promise you envisioned when you first became aware of symbol manipulation or computer algebra software been realized in terms of: • the development and maturation of the technology; • the impact on school or college mathematics curriculum; • teachers’ decisions to make use of the software for teaching mathematics; or • students’ learning of mathematics in relation to the use of the software?

35. The Questions • How has your concept of symbol manipulation or computer algebra software changed since you first learned about it?

36. The Questions • What development(s) do you foresee related to CAS software technically or as a tool for teaching and learning?

37. The Questions • If you feel the full potential of computer algebra software has NOT been realized, what impediments must be removed or what issues must be addressed for CAS to have a more significant impact on the teaching and learning of school mathematics?

38. “Promise Realized?” Steen responded by stating that his initial vision was “a threat to the status quo in mathematics education from roughly grades 9 through 14. This threat held the potential for stimulating positive change, but also for inhibiting deep understanding…. It’s impact on school and college curricula continues to be a patchwork, which does not surprise me.”

39. “Promise Realized?” Palmiter sees “very little change from the early 1990s when many colleges were first embracing CAS… If anything, the move has been to withdraw from using CAS …Teaching with CAS is/was viewed (with good reason) as labor intensive, not lending itself well to large lecture halls or even adjunct/TA taught sections. I see schools/colleges creating different tracks, one in which CAS is not part of the course (intended for weaker students where they can just spend their time on algebraic manipulations) and another track where CAS is an integral part of the course (intended for stronger students, in which we don’t mind putting a bit more effort).”

40. Development & Maturation of CAS • 12 of the 13 responding cited increased capabilities (“power”) and the improved “ease of use.” • Lott and others cited the “leap forward” coming with the TI-92. • McMullin saw the mid 90s as a “peak” in CAS use at the high school level. • Ronau described the maturation as moving from “cumbersome and limited” to “cumbersome and powerful” to “accessible and powerful” where CAS is now available on many platforms with a more natural user interface.

41. Development & Maturation of CAS • Waits described the technology as maturing “nicely including hand-held CAS. However, attention to the user-interface lags. More attention needs to be paid to the KISS principle.” • Kutzler offered a different view: that the development and maturation was not nearly “as much as would have been possible” and offered a “plea” for Pedagogical CAS (PeCAS) in which emphasis is placed on supporting “teachers and students in teaching, learning, and doing mathematics” and replaces CAS with PeCAS by implementing system features that support the pedagogy and facilitate access and application of mathematical (algebraic, numeric, graphic) operations..

42. Impact on School or College Math • To the question on impact on school and college curriculum, 11 responded with terms like • Edwards “fails to meet my expectation.” • Lott “far less” impact • Hollebrands not much “uptake” • Mahoney “appears CAS related change won’t occur” • McMullin “far too little” • Fey “modest” impact… • Goldenberg “In my own work, I don’t see CAS anywhere… Few curricula seem to know what to do with it (… obvious exceptions like Fey’s work and Cuoco’s).” • Drijvers “marginal” • Waits “ZIP, NONE, NADA... in the US for school mathematics… some bright spot... in some regions in Asia, Europe, and Canada.”

43. Impact on School or College Math Mahoney summaries reason for lack of impact: • “Most teachers are uncomfortable using CAS • Most teachers believe that students need to do problems the old way – without CAS. • Problems on national tests (PSAT, SAT, AP Calc) have become CAS-proof. In other words, having a CAS unit won’t help a student do the problem. So in some ways, it isn’t particularly useful for a student to have a CAS handheld.”

44. Impact on School or College Math • Others (Cuoco, Jakucyn, Lott, McMullin and Drijvers) agree that traditional expectations and assessment issues have worked against impact of CAS in the mathematics curriculum. • Lott offered that “schools are not using it to the degree that they could be; colleges are not accepting of CAS; and teacher preparation programs have not taught prospective teachers how to use it effectively.”

45. Impact on Math Curriculum • Anonymous: “… the policy-powerful traditionalists in mathematics are forcing us to miss (or at least delay) a once in a lifetime opportunity for a great leap forward in empowering all students to learn and come to use really powerful mathematical ideas. • They won’t even consider experiments to see the potential benefits and possible pitfalls. • They continue to flog a curriculum perhaps appropriate for some in the 1950’s as the common core expectations for all students.”

46. Teachers’ Decisions to Use CAS • 11 responded citing reasons like fear of or lack of familiarity as reasons for not using CAS. • Also mentioned by some were teachers’ beliefs about mathematics learning.

47. Teachers’ Decisions to Use CAS Edwards: most teachers still fear the use of technology to support teaching mathematics. Among those who claim to be open to such use, we are still hearing things like, “… but they have to learn it by hand first.” Lott: “Many teachers at the high school level are as afraid of CAS as elementary teachers have been of calculator usage in grade schools. Many still will not consider it as a valuable learning option for students. Or maybe worse they will not allow ‘slower’ students to use the CAS, reserving it for use only by the more advanced students. This latter decision is likely the ‘most wrong’ of all.”

48. Teachers’ Decisions to Use CAS McMullin: “The state tests drive the curriculum (actually they are the curriculum) and, by and large, teachers will not make use of CAS until and unless the state testing forces them to. Don’t hold your breath.” Ronau: “If the software improves to the point of being seamless … as mathematical ideas move from representation to representation, and that the process is natural and intuitive, then teachers are likely to embrace this technology, provided that they have assurances that it will not cause their students to falter on whatever assessments are driving their lives.”

49. CAS Relation to Student Learning • 12 responded often citing the metaphor of the “black box” vs. “glass box” CAS (pops out an answer with not insight vs. provides insights into the process). • Also cited was a lack of definitive research on student learning with CAS.

50. CAS Relation to Student Learning Lott: “Ironically, I don’t think that we have gotten to the point yet where we know enough about the use of CAS and its affect on student learning.” Fey: “We really have not studied this carefully enough.” Ronau mentions the current “learning curve for CAS being a true multi-representational tool is too steep. “Students are not driving yet. Again, the software is not seamless or intuitive yet.”