Principles for a new mathematics curriculum buildup

Ivan Cnop, 
Brussels, 
icnop@vub.ac.be

Abstract

Regularly we see alarming reports about the declining mathematical knowledge in secondary and early tertiary education [Sefi ]. Accreditation rules [Abet ] for many curricula include mathematical knowledge, proficiency in mathematics and ability to apply (advanced) mathematics, and sometimes specify the broad mathematical subjects which should be covered. They never state precise contents, or how this should be achieved and avoid discussion of the meaning of mathematical thinking [Gardner ] and mathematical proof.  
Everybody agrees that use of technology is here to stay and enhances teaching at all levels [ICMI ]. Published guidelines state that institutes should provide processes to prepare students for achieving the education goals, but they ar never specific about the means to be chosen. Moreover implementation has to be updated regularly following the rapid changes in the technology, and this is expensive in printed formats. 
Besides doing all technical computing and modeling, symbolic platforms allow better insight in concepts and proofs.
A vertical approach in the curriculum by project work is made possible by efficient use of technology. This will be illustrated by a case study around one topic (the spirograph) which from elementary geometry reaches into advanced analysis and its applications. 
The principles outlined below apply to undergraduate tertiary education but can be applied for the final years of a secondary curriculum as well.

Choice of a symbolic system

Acceptable by a large number of teachers

Most symbolic CAS cater to teacher's needs and have commercial offers targeting this group. 
Mathematica offers several options towards them:
	special licensing conditions for teachers (up to community colleges) and their students;
	the Teacher's Edition with extra palettes for a test generating engine;
	Calculation Center with a slimmed-down programming environment for non-computer science teachers.

Lasting a full studying career

The usability of a symbolic system should last for a full studying career and beyond. It is very expensive to retrain students in different softwares at successive levels. Much better is to stick to one symbolic system for the full education cycle. One has to start with a limited set of simple commands and increase gradually the number of features being used as they arise in the course. The use of "black boxes" is avoided and programming of the concepts should remain transparent. Optimising these for speed should come in the later stages of the training.

Syntax should be close to mathematical thinking

It is important that the underlying ideas and buildup of symbolic programming should be similar to the mathematical ideas behind these concepts. It is not efficient if one has to learn one language for teaching the theoretical background of mathematics and a separate language for implementing mathematics in technology. This used to be the case for handheld technology, and a lot of effort addresses the problem of writing out command lines in the right sequence of keyboard  strokes, or to address the machine-dependent anomalies encountered under implementation.

Advanced objects can be programmed in Mathematica  [Wolfram] using a few lines, translating the underlying mathematical construction.

Example: the following implement Julia sets in the plane for different constants and initial values.

coords(z_) := {Re(z), Im(z)}
new(z_) := {(Sqrt[z - c]), -(Sqrt[z - c])}
pts(begin_) := 
      Flatten[NestList[new, begin, 12]]
c =  .3 +  .4 I;
ListPlot[coords /@ pts( .3 +  .5 I), 
      AspectRatio -> Automatic, 
      PlotStyle -> PointSize[ .001]]

This is a typical example of Mathematica  programming in three lines: 
	one line to define a function, 
	a second one to apply this function a number of times, 
	and a final line to plot the result.

Cheap 1-feature system limits the scope of problems

Other limitations are the delivery of CAS on small screens such as those on hand-held devices.

No delivery of dead text or a mere testing package

Some packages only offer static text. 
Other softwares merely offer a database for testing students. This is even more unfair if checking answers is done in a symbolic system, while use of the same system is made impossible or prohibited for the students .

Visual introduction of content

Show what software can do in the classroom

Whenever I show Mathematica notebooks in the classroom, I refer to the Help for syntax of commands and these are also evaluated on different examples or in slightly different assumptions. 
Sometimes I even take time in class to write new commands or programs in a separate notebook.

Hands-on trial

It is unfortunate that the students do not have their own portable computers in class to try out the same material immediately. 
The best I can do is to make available all notebooks developed in classroom on a server for downloads. 
This way they can experiment further and modify this material immediately after class sessions. 
This is very much appreciated by the students, especially since they can work at their own pace.

Repeat often, each time a new viewpoint is introduced

These documents are revisited often and expanded, each time a new viewpoint is introduced.

Experimenting and problem-oriented approach

Problem-driven experimentation

Mathematics has always been an experimental science and its foremost intrest has been in the prediction of behavior of complex systems, from Babylonian astronomy to post-bubble finances.

Thanks to the introduction of technology this functionality is again possible [see Uhl, Kawski] following the paradigm

	experiment -> conjecture -> proof.
	
Students can rephrase (simple) real-world problems, leave computing to the software 
and focus on the qualitative analysis of the result and understanding. 
Simulations indicate tendencies when no exact results can be obtained for a particular problem. 
Gradually increasing the complexity of the model achieves the goals expected by their future professional environment.

Example: what is the most efficient movement of a hinged robot arm? 
Try the ellipse and analyse the movement along its circumference. 
This example can be linked to the understanding of Fourier analysis [see Cnop: SpiroFourier].

Simulation example: the following implement some Random distribution on random length intervals, and the graph it obtains raises interesting problems about random distributions.

howmany = 50; left = 0; right = 10;
xes = 
        Flatten[{left, 
            Sort[Table[left + (right - left) Random[], {howmany - 1}]], 
            right}];
xesLeft = Take[xes, howmany];
xesRight = Take[xes, {2, howmany + 1}];
slotFunction[{a_, b_}] := a < x < b
slots = 
        Map[slotFunction, Transpose[{xesLeft, xesRight}]];
values = Table[Random[], {howmany}];
l = Flatten[Transpose[{slots, values}]];
Clear[f]; f[x_] := l /. Head[l] -> Which
Plot[f[x], {x, left, right}, AxesOrigin -> {left, 0}]

Cultivate control

It is important to cultivate an attitude of control in the learner. This is done by motivating results, checking results in particular cases, obtaining the same result by a different method. The format of the output can be an indication for correctness and graphical output is checked faster than formulas.
Qualitative considerations are in this respect more important than computational techniques.
Students are quite positive about learning programming principles in their mathematics course. Mathematica gives them the opportunity to practice facts from their computer science courses and they acknowledge that basic ideas from mathematics and computer science overlap. 
It is unfortunate that programming courses do yet not use (part of the syntax of) Mathematica in their lab sessions.

Qualitative considerations

Versus computational techniques; leave computing to the computer

Link to computer science

Programming your answers is a good exercise to deepen the understanding and to avoid false belief pitfalls.

Try simulations whenever no exact result can be expected. Either simulations give a sufficient answer to the problem, or they can be used to refine the statement of the problem or isolate interesting cases that can then be solved by symbolic methods.

Encourage implementation of symbolic proofs

Students are quite positive about learning programming principles in their mathematics course. 
Mathematica gives them the opportunity to practice facts from their computer science courses and they acknowledge that many ideas overlap. 
It is unfortunate that programming courses do not yet use (part of the syntax of) Mathematica  in their lab sessions.

WYTIWYG: test symbolic literacy and appetite for problems

Open problems depending on random entry.

What you test is what you get is one of the basic priniples of education. The technology environment creates a need for a new exam system, using the software during the exam. Meaningful exams should not consist of drill and practice question and therefore certainly NO multiple choice questions. The exam should offer open-ended problems in line with the teaching method. Making exam questions depend on a random entry allows testing of a normal class size in parallel while avoiding cheating or outside help.
The random entry can depend on some numbers such as date of birth, student registration number, or last 5 digis in telephone or bank account number. These numbers may have to be verified beforehand. The question then depends slightly on the random entry and students are unaware of the impact this number may have for their peers on the formulation of the question.
At the end of the exam session, students mail their documents (preferrably with outputs deleted to reduce the size of the document) to the teaching assistant and myself. Grading is fast since running the software regenerates the outputs immediately and a straightforward check tells if the answers make sense.

The Sefi curriculum revisited

The Sefi organisation [see Sefi ] has collected central topics compulsory in engineering education in three categories. This was done from a very traditional point of view. The result is a valuable guideline for qualitative education in a traditional context, and could even be used for mathematics majors. But it misses a lot of opportunities created by symbolic software and does not address the natural curiosity for experiments encountered in the behavior of today's young generations. 

This curriculum relies little on the new capabilities offered by symbolic computing software, and avoids taking into account the surprising outputs computers will deliver for the earliest inputs. Students will be caught off-guard when they see finding roots will automatically involve complex numbers (either under root form or as approximate numbers). The solution is to imbed these number systems earlier in the curriculum, which is not difficult since these are exactly points in the (plane) computer screen [see Cnop]. 
Some topics are overemphasised for historical reasons. Examples of these are matrix calculus and trigonometry in core zero, linear functions or low degree polynomials in core one, z transform and Laplace transform in core two. These are all topics that can be perfectly left to the computing capabilities of the software [see Uhl], or that can be handled in much more general settings.

Most topics in core three level can be handled by students who are learning core two material and who are used to a powerful symbolic system. 
It is not efficient to forcefully delay their encounter to the end of a (bachelor) degree programme. Students enjoy encountering advanced topics such as higher degree differential equations, graph theory, computer geometry,  experimental design, filtering, Markov processes, chaos and modelling. Technology enables the teacher to show these at a much earlier stage.
It is difficult to develop a problem-oriented approach with meaningful applications of mathematics if such topics are delayed to a later stage in tertiary education.

Core zero

The Sefi curriculum does not impose topic content but states goals to be achieved after completing of instruction by  the means  available. This does not interfere with using CAS symbolic packages. It is not clear if goals should be tested with or without the symbolic package.It is probable that the learning objectives will be achieved more rapidly ifthe testing is done by the same methodology as the learning.

what importance should be placed on number operations

ambiguity between allocations, equations and replacements (rule)

recognising type of an equation is more important than solving it

factoring is nontrivial

Factor[26]

Factor[(x^2) - 1]

Factor[(x^24) - 1]

Length[Factor[(x^24) - 1]]

Divisors[24]

Experimenting with lines , quadratics and polynomials: using the screen as a slate. Interactive geometry can be used, but has limited scope

Rescaling coordinates using graphics

Inverse functions from parametric plots

Experimenting with rules and graphs to understand composites

Recursively defined sequences

Earlier introduction of complex numbers is wanted and possible; this is underestimated in classical curricula

Solve[(z^6) == 3, z]

Solve[(z^6) == 3. , z]

Adress role of proof and geometric reasoning.

Underestimate discrete mathematics and relationships discrete - continuous.

Replace sketching by meaningful input

Overestimates role of trigonometry

TrigExpand[A Cos[2 x  +  B]]

TrigReduce[%]

Core level one

Here explicit reference is made to using software. However, much more can be done in software than is marked as such.

hyperbolic functions have no interest as such; they are conbinations of expnentials

rational functions with linear denominators are too restricted.

experiment with function graphing

methods of integration is superficial; whatever there is can be done in software

why is there a separation between sets and logic?

conic sections and geometry look old fashioned

vector products (a 3D feature only) is overemphasised

Core level two

Little use is made of software and no recommendations are formulated.

Overemphasis on z and Laplace transform while no theoretical background for computer implementation is delivered

Some topics in discrete math come too late to be usefully linked to computer implementations

Geometry at this level is not done without computer.

Core level three

Most topics at this level can be handled by students who have mastered core 2 material and are used to a powerful symbolic system.It is wrong to forcefully delay its encounter to the end of a (bachelor) degree programme.Students enjoy encountering advanced topics such as higher degree diff eq., graph theory, computer geometry,  experimental design, filtering, Markov processe, chaos, modelling.

References

Abet:  www.abet.org
I. Cnop: "Computer Screens and Toys". Proc. of the 6th ATCM, Melbourne, 2001.
ICMI study: "The future of the teaching and learning of Algebra", to appear.
M. Kawski: "Vector Field Analyser", interactive website at asu.edu.
Sefi Core curriculum: Sefi-Mathematics Working group. 
	http://learn.lboro.ac.uk/mwg/core.html or the sefi-mwg website
Sefi 2002 Gotheborg report.  See Sefi website.
J. Uhl: "Software that works", Mathematics Everywhere Inc.
S. Wolfram: "Mathematica 5", 2003, http://wolfram.com .