The
Missing Link: a Technology Curriculum in Mathematics
Ivan Cnop
icnop@vub.ac.be
Mathematics
Vrije Universiteit Brussel
Belgium
Abstract
While
technology advances at a rapid pace, curricula have changed little
over the last decades. To fill in the gap a new curriculum has to
be developed that uses the capabilities of state of the art technology,
that exploits the eagerness of experimenting in learners, and that
achieves faster the learning objectives required by modern society.
The logical linking of modules under development provides an answer.
Introduction.
We
see increasing gaps between teaching methods and technology use
(gaming, surfing, cell phone use, ...), and between mathematics
curriculum content and technology capabilities. In
MacSyma [1] , a software that at that time was running on very expensive
Symbolics workstations, the symbolic power was limited to little
more than standard calculus content: working with (finite) series,
immediate answering of drill questions, rendering formulas in fixed
width fonts and some graphing capabilities. Its performance was
comparable to that of currently available handheld calculators.
In twentyfive years, software capabilities have increased nearly
a thousand fold and prices have been cut by a factor one hundred,
both for hardware and software. We also see a widening gap between
the volume of mathematics taught in schools, and the available knowledge.
Researchers, teachers and students can obtain information over the
internet from MathWorld [2], dedicated internet sites (e.g. [3]),
functions database [4] and other electronic publications. Unlike
in other disciplines, a major part of this electronic information
in mathematics is freely accessible. There is a huge gap between
the simplicity of models and the complexity of many realworld phenomena.
While a classic paper and pencil approach is limited to simple applications,
technology makes complex phenomena tractable. We also have to get
rid of the belief that everything is deterministic since reality
is more stochastic. Technology makes this uncertainty tractable
through simulations. Our
students do not seem to realize to what extent their learning environment
and goals have changed. Part of this inertia can be attributed to
the teachers: it takes an entire generation to implement changes.
Efficient
teaching.
To
fill the gap it is not sufficient to teach a technology manual.
A specific technology manual will be obsolete in a couple of years
as new versions appear, and when new technologies take over. But
adapting the old curriculum does not help either. Many curricula
have been around and little changed since the seventies, after the
dispute over introducing "new" mathematics was more or less settled.
I still recognize in most curricula the same material I studied
long before technology was first introduced in the late sixties
with the Compucorp "portable" calculator or the Canola desktop calculator
with LED and one programming JMP key. The
first (very expensive) mathematical symbolic software was running
on "Symbolics" workstations some twentyfive years ago. Symbolic
computer software (CAS) and graphics calculators only became popular
in the late eighties and nineties, and their introduction in education
has been uneven and slow. Innovation has since occurred at an increasing
speed. In spite of the rapid evolution of technology, curricula
have little changed over the last twentyfive years. As a consequence,
society is questioning the relevance of mathematics education today,
and the popularity of mathematics is diminishing. Technology use
should not be limited to provide testing in drill questions. This
way of using technology falls short of attainable objectives both
in mathematics and in technology. All parties concerned, teachers,
students and the society, are calling for a new curriculum that
is better adapted to the needs of the new millennium.
Gradual
buildup.
Introduction
of technology should start early after a playful introduction to
counting, reasoning and (plane) geometry in primary schools. This
is the content of mathematics for everyone, even people that may
never encounter any technology. There are indeed some activities
involving a little computing (exact, approximate or guessing) where
a calculator or computer is not welcome. Plane geometry is well
adapted for a first introduction of technology since it fits a computer
screen and the basic needs for structuring the percieved reality
of objects. Excellent interactive material is available [5, 6].
It prepares learners for more advanced commands in current powerful
symbolic softwares such as MatLab, Maple or Mathematica [7].
Technology
driven curriculum.
A more
advanced mathematics curriculum has to concentrate on features that
are imbedded in every symbolic technology: e.g. a plane representation
(on a computer screen) in stead of numeric tables, fast graphing
and audiovisual capabilities (including web delivery), 3D viewing
capabilities , the usefulness of constructive proofs, fast simulations
that mimic the "for all" clause in mathematics, the switching between
continuous and discrete phenomena, list or vector handling, ...
For
instance, a calculus curriculum can be organized using the following
principles that use technology capabilities:
 the
ability to make computations with arbitrary precision, e.g. exploit
continuity at a point to locate zeroes or attain limits;
 the growth of functions (including O and o notations of Landau):
no growth as seen with horizontal tangents (as in Rolle's theorem),
the order of contact between graphs, estimating growth by derivatives;
 fitting data by wellchosen functions, depending on the nature
of the data: polynomial approximation near a point, Fourier approximations
for nearly periodic data, bell shapes or wavelets for humplike
data;
 the speed of approximations and fastness of algorithms in general;
 modeling of real world problems and the quality of the model;
 trying to do analysis on data for which the prescribed function
is not yet known;
 the everpresent interplay between continuous and discrete mathematics;
 uniform continuity on closed intervals and uniform limits on closed
intervals [8], and even finer convergence criteria whenever needed.
Set
high achievement goals.
Teaching
should offer the fastest approach to learn new techniques, bypassing
old habits or redundant material. For each topic we have to ask
is it still relevant? What is the fastest way to introduce a topic
from material the targeted student group already knows? A new curriculum
will be overloaded if it follows the linear structure of the old
one, delaying the introduction of new concepts until all more general
concepts (or even all simpler concepts) have to be handled first.
Mathematical curricula should look like directed graphs, mapping
the prerequisite  sequel order, in which the fastest way to attain
a goal can be selected. This requires from the teacher a deep understanding
of mathematics and the logical interdependency links between topics.
Example: for a learner who is familiar with elementary geometry
in the plane (e.g. by using a geometry package) it would take a
short period (i.e. less than ten hours over a couple of weeks) of
guided experiments to grasp the concept of discrete Fourier transform
and its applications in probability and signal filtering [9]. Once
this is understood, the learner has access to many state of the
art applications.
Experiments.
Modern
teaching has to exploit the eagerness for experimenting in students
[10]. Old curricula and study material are static. Even modern delivery
over web pages does not change this fact. Interactivity is limited
for security or copyright reasons. We should allow students to have
an impact on the material, to develop new simulations, to convince
themselves and their peers. It is surprising that most students
learn advanced technology commands faster than their teachers or
parents. But experimenting should not be limited to predefined commands.
We can offer templates to encourage structured use of commands.
To allow further experimenting we should not try to hide content
programming from them. Students should have access to the full functionality
of technology. This is no longer a problem since most software manufacturers
have licenses available at student prices.
Historical
developments.
The
teacher should concentrate on historical developments that are considered
milestones in mathematics. The relevance of a subject in mathematics
can be graded by its stability over a very long period. It turns
out that essential content is linked to great names in the history
of mathematics. Historically, approximations were always done by
polynomials. Technology does handle polynomial expressions in the
same way as it handles linear expressions. Functions that do not
allow polynomial approximations tend to exhibit pathologies that
cannot be handled by calculus methods. Naming of concepts is less
important, and the curriculum should refrain from trying to introduce
new names for each slight difference between concepts.
Programming.
A modern
curriculum should link mathematical ideas to algorithms and programming.
This is now less difficult than used to be the case with early procedural
programming languages on typed data. Over the past decades we have
seen a convergence of programming paradigms with the underlying
mathematics in symbolic mathematical technology. Similarities can
be used to streamline both mathematics and basic computer science
curricula. Mathematics is a rulebased system in which computing
function values (lambdacalculus) is possible if the variable matches
the required pattern. For an individual familiar with symbolic mathematics
technology this reasoning is a basis for a programming course [11].
Control.
A modern
curriculum should teach the student a control attitude and warn
him for simplifications or shortcomings in technology. This is not
different from learning control over one's own results in classical
mathematics teaching, as is shown in [12]. This does not mean that
a curriculum should include a list of warnings against "errors"
in the computations. Too many texts focus on warnings of the kind
"if you push this button after that one for such value, you may
get a wrong answer". Since performance is better in advanced symbolic
systems, it takes less effort to warn users. Nevertheless, some
of the technology sideeffects are imbedded in a hard way because
of the finite nature of all computing engines and users should permanently
be aware of this. It is too late if it is cultivated in a numerical
methods course at the end of the curriculum.
Life
long learning.
A good
mathematics curriculum prepares for a life long schooling and applicability
of mathematics. No learner is ever going to use mathematics without
technology, and therefore it is an illusion to continue teaching
mathematics without advanced technology. Since we cannot predict
what kind of technology will be available in twentyfive years from
now, we can only develop in our students an attitude of acceptance
and eagerness for new technology improvements.
References.
[1]
Macsyma history on http://maxima.sourceforge.net/ , MIT, 1982.
[2] MathWorld: http://mathworld.wolfram.com/
[3] StAndrews University: http://wwwhistory.mcs.standrews.ac.uk/
[4] The Wolfram functions site: http://functions.wolfram.com/
[5] Cabri II: http://www.cabri.com/en/
[6] Geometer’s Sketchpad: http://www.keypress.com/sketchpad/
[7] Mathematica: http://www.wolfram.com/
[8] I. Cnop: “A uniform computerassisted approach to analysis:
process, concept, and proofs” Proceedings of the International
Conference on the Teaching of Mathematics, Samos, Greece, pp 65
68, 1998.
[9] I. Cnop: “Computer screens and toys” Proceedings
of the ATCM 2001 Conference, Melbourne, pp 17 – 33, 2001.
[10] .Exploot project guidelines: http://we.vub.ac.be/exploot/summary.html
[11] R. Maeder: “Programming in Mathematica”, AddisonWesley,
3rd ed. 1996, 4th ed. 2001.
[12] I. Cnop & F. Grandsard : “More efficient teaching
and learning of Mathematics: Problem solving and technology”
, in “Trends and Challenges in Mathematics Education”,
ed. Jianpan Wang and Binyan Xu , East China Normal University Press,
pp 223234, 2004.
