The Missing Link: a Technology Curriculum in Mathematics

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 hand-held calculators. In twenty-five 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 real-world 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 twenty-five 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 twenty-five 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 well-chosen functions, depending on the nature of the data: polynomial approximation near a point, Fourier approximations for nearly periodic data, bell shapes or wavelets for hump-like 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 ever-present 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 rule-based system in which computing function values (lambda-calculus) 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 side-effects 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 twenty-five 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://www-history.mcs.st-andrews.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 computer-assisted 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”, Addison-Wesley, 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 223-234, 2004.