Computer Algebra Systems
(CAS): From Where Did They
Come, and Where Might They Go?
National Council of Teachers of Mathematics
Annual Meeting – San Diego
22 April 2010
Ed Dickey
University of South Carolina
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
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
My First CAS Experiences…
• NSF Conference Report,
October, 1982
• Graphing utilities,
VisiCalc, TK!Solver, and
muMATH in school
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.”
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
To Professor Pollak for the 2010
Mathematics Education Trust
Lifetime Achievement Award
for Distinguished Service to
Mathematics Education
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
My First CAS Experiences…
• 1986 NCTM
Charleston Regional
• Why and How to Use
Symbol Manipulation
• muMATH-80 for the
Apple II
My First CAS Experiences…
• Phone call to Albert Rich
and Dave Stoutemyer
• 1987 USC became
“world distributor” of
• Teacher’s Guide
• $40 ($5 Soft Warehouse
Smithsonian Exhibit
• Slates, Slide
Rules, and
Teaching Math
in America
• muMATH,
Derive, TI-92
History of CAS
• LISP in 1962
• Physicist Martin
Schoonschip (“clean
ship” in Dutch) in 1963
• U.S. Carl Engelman’s
MATHLAB in 1964 led
to Macsyma
CAS in Schools
“Computer Symbolic
Math & Education: A
Radical Proposal”
Bulletin 13, 3
(August 1979), pp.
CAS in Schools
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
NCTM 1989 Standards
• The use of computer utilities to develop conceptual
• The use of technology in instruction should further
alter both the teaching and the learning of
• 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.
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.
CAS in Schools: Science Fair
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!
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
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
Textbook 2009
Al Cuoco and others with EDC
• “platform for experimenting”
• “reducing computational overhead”
• “build computational models”
Dynamic Geometry
• Geometric
Schwartz and
Dynamic Geometry
• Cabri, 1994, Jean-Marie
• Geometer’s Sketchpad,
1995, Nick Jackiw
• “Lifting the Curtain: The
Evolution of The Geometer’s
Sketchpad” by Daniel Scher
Graphing Utilities
• Widespread use
• 1985 Casio 7000G
• Handheld CAS
Ubiquitous access…
• Wolfram
Alpha, free on
• iPhone
Calculator $.99
• PocketCAS
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
• 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
Lynn Steen
Tom Dick
Jeanette Palmiter
Bert Waits
• John Mahoney
• Lin McMullin
• Natalie Jakucyn
Paul Goldenberg
Carolyn Kieran
Paul Drijvers
Luis Saldanha
Mathematics Educators
Jim Fey
Karen Hollebrands
Thomas Edwards
Bob Ronau
Johnny Lott
• Al Cuoco
• Bernhard Kutzler
• Albert Rich
The Questions
1. 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:
a) the development and maturation of the technology;
b) the impact on school or college mathematics
c) teachers’ decisions to make use of the software for
teaching mathematics; or
d) students’ learning of mathematics in relation to the
use of the software?
The Questions
2. How has your concept of symbol
manipulation or computer algebra
software changed since you first learned
about it?
The Questions
3. What development(s) do you foresee
related to CAS software technically or as
a tool for teaching and learning?
The Questions
4. 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
“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
It’s impact on school and college curricula
continues to be a patchwork, which does not
surprise me.”
“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).”
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.
Development & Maturation of CAS
• Waits described the technology as maturing “nicely
including hand-held CAS. However, attention to the userinterface 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..
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.”
Impact on School or College Math
Mahoney summaries reason for lack of impact:
1. “Most teachers are uncomfortable using CAS
2. Most teachers believe that students need to do
problems the old way – without CAS.
3. 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.”
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.”
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.”
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.
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.”
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.”
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.
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
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.”
CAS Relation to Student Learning
Dick discusses the black-box metaphor: “… the main
advantage offered by CAS as seen by many
proponents and opponents is CAS as servant to whom
we can delegate computational tasks
(proponents might think of tasks that are tedious or out
of reach for paper-and-pencil; opponents might think
of tasks that they feel SHOULD be done by paperand-pencil).”
Waits adds “the answer box mentality is seen all too
frequently and inappropriate use is a real problem…”
CAS Relation to Student Learning
Drijvers mentions that “the math education research
community and the math educators community have
not succeeded in convincing authorities and ‘ordinary’
teachers of the benefits of integrating computer
So far, we have not really tackled the issue of the
complex relationship between by-hand skills, insight
and tool use, and this makes it easy for the rest of the
world to neglect computer algebra.”
• Kieran and Drijvers offer insights to deal with these
issues based on their research.
CAS Relation to Student Learning
Kieran: “I have observed many teachers… have a
tendency to focus on the technology and not on
the mathematics. Those who succeed with
enriching students’ learning of mathematics, in
conjunction with this [CAS] technology, are
generally those who have a vision of the way in
which the technology can be used to enhance
students’ conceptual understandings at the same
time that the students are learning mathematical
CAS Relation to Student Learning
“ Our research has designed tasks according to the
Task-Technique-Theory theoretical framework-tasks where teachers have contributed to the design,
before they have used them in class.
These tasks for Grade 10 algebra engage students in
theoretical reflection at the same time that they are
learning new paper-and-pencil and CAS techniques.
What we have found is that the process of learning
technique in a CAS environment, where theoretical
ideas are being co-stimulated, has led to deeper
mathematical insights than are the usual for students
at this level.”
CAS Relation to Student Learning
Kieran’s research (with Damboise) has
illuminated CAS can help “weak” students in
both technique and theory. “The CAS provided
a certain security to these weak students that
led them to examine the CAS outputs for
structure and patterns. They were provided
with tasks, of course, that encouraged such
Has Your Concept of CAS
• 12 commented.
• Some (Mahoney, Hollebrands, Palmiter)
indicated no significant change. Fey
mentions the basic functionality has
remained constant but the interface is more
• Many (Edwards, Goldenberg, McMullin,
Waits, Palmiter) cited the move away from
“paper-and-pencil” work, the “drudge part,”
freeing up time for the “thinking part.”
Has Your Concept of CAS
• Goldenberg explained: “What I do believe … is that the
purpose of such skills as long division and algebraic
manipulation has changed, and that the change should be
acknowledged in curricula.
• It used to be that we needed these skills for the sake of getting
answers. That’s no longer true. If all we want is the answer, use
the calculator, especially if the computation is too long or
complex without technology.
• Kids no longer need to be super-fast and super-skilled at
[make your own list]. But some of these skills are still needed
just in order to think about what calculations we’d like the
calculator to make.
• If I see no pattern in a sequence of numbers, even roughly, I
don’t know what experiment I’d like to perform with the
“Value-added” of CAS
Cuoco sees three overlapping use of CAS
to help students develop algebraic habits of
– Algebraic laboratory: “… making patterns
– Algebraic calculator: “… reduces computational
– Modeling tool for algebraic structures: “…
medium of expressing abstract algebraic
Has Your Concept of CAS
• Drijvers described a gradual change in his
conception “from a fascinating expert
system in symbolic mathematics into a
functionality that is, or should be, part of a
whole range of integrated technological
tools for teaching and learning
Has Your Concept of CAS
Jakucyn has “… come to appreciate CAS
more for its aid as a pedagogical tool and less
as a checker or to simply manipulate the
mathematical symbols…”
and finds that “…students who are of average
or low mathematics ability can benefit
tremendously from using a CAS to develop
mathematical concepts and skills.”
Has Your Concept of CAS
Waits confessed to being “very naïve in the late 1980s”
thinking CAS use in schools would follow the growth of
graphing technology.
• “Computer graphing was an enhancement of the
mathematics and could be used to promote better
understanding. ...
• However, computer symbolic algebra was viewed as a
replacement of “paper and pencil” manipulative
mathematics… which posed a huge threat to teachers”
particularly those “who knew nothing about mathematics
other than ‘manipulation’” of symbols.
• “At the college level, math prof troglodytes came out of
the woodwork against the use of CAS... for many of the
same reasons.”
New Developments in CAS?
• 12 responded.
• Several (Drijvers, Mahoney, McMullin,
Palmiter, Fey, Hollebrands) mentioned
the increase in portability (iPhone apps
tablets) and …
• improvements in user-interfaces
particularly related to the entry of
mathematical expressions.
New Developments in CAS?
• Kutzler expressed skepticism that “what the market would
need” will not be met by “the industry’s willingness to
respond to this.”
• Steen: “At the college level, faculty have sufficient
knowledge and authority to make use of CAS when and if
they feel it would be helpful. Even within the same course,
the degree of CAS use will differ greatly, but so do many
other things. I see this variety as a strength, not as a
problem. In the K-12 sector, however, the increasing political
pressure for standardization of curricular and testing will
create impediments for constructive development of CAS in
teaching and learning. CAS needs flexibility to develop
effectively in educational contexts, but don’t see how that is
compatible with incessant high stakes testing. “
New Developments in CAS?
Drijvers more optimistically expects “CAS to come out
of its isolated position, and get more integrated,
sometimes visible and sometimes invisible, into other
technological environments. …
From the pedagogical perspective, I guess CAS will
act rather on the background as an engine behind
educational tools for math, than an independent and
front-end tool for users. …
The challenge for designers, educators and
researchers is to find ways to exploit the potentials of
computer algebra.”
New Developments in CAS?
Rich described his work on Rubi: an online, rule-based
repository of mathematical knowledge.
“Unlike the current proprietary CAS, such a repository
would be an open-source knowledge base accessible
to everyone. The knowledge required to automate
mathematics will be stored in the form of high-level
reduction rules.
These rules are expressed as mathematical formulas
with precise application conditions when they are to be
applied. Therefore mathematicians can contribute
knowledge to the repository in their area of expertise
without having to program in a low-level computer
Related to CAS
• 15 responses
• While there was overlap two items
–teachers’ beliefs about learning and
–the need for professional
Related to CAS
• Related to beliefs, Anonymous mentioned “the
most common and strongly held [beliefs] that
you have to first master the basics before using
technology. This belief constrains imagination
about CAS impact.”
• McMullin, Jakucyn, Hollebrands, Palmiter,
and Mahoney suggest that new curriculum and
state accountability tests might serve to combat
the beliefs held by some teachers.
Related to CAS
• Goldenberg thinks “… the “game” must be fundamentally changed.
Mathematics tends to be about learning things and it needs to be
about learning thinking. That’s not news—we’ve all said it, and even
hoped that the technology would help force that game change by
doing the “drudge work,” thus removing all need to focus on “things”
and skills, leaving only the thinking to learn. But it’s not my
impression that things have worked out that way. If people still
believe that the goal is to get answers, the problems get hairier, and
the extra “hair” (arithmetic or algebraic) is relegated to the
calculator, but no additional attention is given to the thinking. We’ve
simply replaced mental (or paper) skills with button-pressing skills.
• I think that the major impediment is the public view that
mathematics is something most people can’t learn, except perhaps
at a skill level, and it is all about application, not reasoning. If the
point is to get answers in applications, why not use a tool? And the
public is mixed.”
Related to CAS
• Edwards, Waits, Jakucyn, Kieran and
Saldanha address the importance of
professional development.
• Jakucyn states that “… teachers (and parents)
are not familiar with the capabilities of a CAS
and thus view the use of a CAS as a form of
‘cheating’ …. This may stem from teachers not
being trained to use a CAS or from teachers’
and parents’ views of what it means to ‘do the
Related to CAS
Saldanha calls for “more knowledge about
what is entailed in incorporating a CASoriented perspective” and “… how teachers
might restructure instruction in way that
support such learning.” He mentions that “few
teachers have models of these possibilities,
and they need intensive assistance (in the
form of resources and training) to develop
models and reconceive the mathematics they
teach so as to be able to incorporate CAS use
in their instruction.”
Related to CAS
• Kieran claims that “Most CAS professional
development focuses on the button-pushing of the
technology and not on the issue of how to construct
tasks that will lead to deeper learning of mathematics,
with the aid of this technology. Teachers who have
tended to use mathematics textbooks as their prime
resource for teaching have never really learned how to
generate rich mathematical tasks… Teachers need a
kind of professional development that will yield a
broad vision of the ways in which CAS tasks can be
constructed so as to yield the mathematical benefits
that the technology can foster.”
Related to CAS
Kutzler feels that “computers are an exceptionally wonderful tool for
mankind. In the times of DOS, computer use was restricted to
techno-phile people. As operating systems improved, computers
became more accessible to techno-phobe people.
“Similarly CAS is an exceptionally wonderful tool for math education.
But today's CAS products are still only used by techno-phile
teachers. We need interfaces on top of CAS engines that make
this great technology accessible also to the techno-phobe
teachers - and that's by far the biggest crowd.
“I don't see a single tool on the market that makes CAS accessible to
a techno-phobe teacher. As long as we need teacher training for
CAS products, we are not doing the right thing. How many cell
phones would the industry sell if users would need cell phone
training before being able to make good use of them?”
Related to CAS
• Waits cites the importance of attracting better prepared
students to the teaching profession and better professional
development on using technology appropriately. He addresses
the changing of beliefs by professional development for high
school teachers claiming that teachers “feel comfortable and
are mostly good at teaching paper and pencil… manipulations
(most now obsolete).”
• They feel very threatened if they would have time to teach what
we know is really important about mathematical understanding,
problem solving, thinking, and reasoning… “things CAS give
them time to do.
• “As far as college and universities go: fire-breathing, troglodyteeating dragons need to be let loose in college mathematics
departments!!!  And you can quote me!”
Related to CAS
• Dick, more calmly, sees promise in the “dynamic
linking of objects” that is now available in CAS
environments. “By dynamic linking, I mean linkages
between and within representations (including
symbolic) that are updated instantaneously - changes
made to one object or form are immediately reflected
in changes to a linked object or form.
• When employed across representations this could
allow manipulation on the entry side of numeric data
or graphic representations or some virtual physical
scenario with the linked result in symbolic form
Related to CAS
• Employed within symbolic representations, we'd
have two (or more) dynamically linked symbolic
forms where a change in one immediately
results in a change in the linked form
• To Dick, this opens up power to ask sense-making crossrepresentational questions, and opportunities to actually use CAS as a tool to
gain insight into symbolic processes - transforming
it from the black box task servant to a glass box
Dynamic Linking Across
Nspire CAS Dynamics
• Media4Math Mini-Tutorial on Quadratic
with Sliders
Dynamic Linking Example
Related to CAS
By Peter
By Thomas Edwards & Asli Özgün-Koca
By John Losse
Resources for Follow-Up
Computer Algebra Systems
in Secondary School
Mathematics Education
$12(!) from NCTM
Edited by James T. Fey, Al Cuoco,
Carolyn Kieran, Lin McMullin, and
Rose Mary Zbiek
Resources for Follow-Up
Professional organize to “encourage
active interest in the use of CAS..”
June 26-27, 2010 at
New Trier High School, Illinois
Computer Algebra Systems
(CAS): From Where Did They Come,
and Where Might They Go?
• Questions or Comments.?
• Powerpoint and References
available at
• [email protected]

PRESENTATION NAME - University of South Carolina