ATCM 2003
Hsinchu, Taiwan (ROC)

The influence of CAS on curricula
Case Study: e  in the curriculum

Ivan Cnop
icnop@vub.ac.be

Abstract

Accreditation rules for many curricula [see Abet] include mathematical knowledge, proficiency in mathematics and ability to apply (advanced) mathematics. They specify which broad mathematical subjects should be covered, but never state precise contents, or how this should be achieved, and they avoid discussion on the relevance of mathematical thinking and mathematical proof. An overview of curricula in European countries confirms this. 
Everybody agrees that use of technology enhances teaching at all levels. Published guidelines state that institutes should provide processes to prepare students for achieving education goals without specific recommendations. Moreover content implementation has to be updated regularly following the rapid changes in the technology, and this is expensive in printed formats. Symbolic platforms are doing all technical computing and modeling. They also allow better insight in concepts and proofs. It is necessary to study how these capabilities can be transferred to the learners. The presentation will concentrate on what learner attitudes should be cultivated for maximal profit. The question of organization of exams in a technology environment will also be addressed. 
Finally, 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) that reaches from elementary geometry into advanced analysis and its applications. 
These guidelines for curriculum achievement will be illustrated by a case study on the introduction of the number  e  and the exponential function.

Guidelines for curriculum development under technology

Final EU Report FP6 from the Working Party on Education and Training

states in its paragraph on "Policies and initiatives" that, while many countries involved in the round FP5 projects have developed activities, some issues should be adressed in the future, especially 
	teacher training [which is an outstanding issue everywhere] 
	the huge lack of useful and appropriate content for e-learning, 
	and the need for continued learning. 
They recommend (quoting from the text):

Content development and the e-learning as such.

Research should be driven by learning pull rather than technology push.

Modular learning objects and their management are required.

The support to the teachers is seen as an element in future advanced "Community Knowledge Networks" where the school, the library the local government and local industry is working together with centers of excellence supporting and involving pupils, the teachers and parents, the learning community.

Support to teachers/educators in using ICT in educational contexts: on the school level more and better support to the teachers concerning the use of ICT in the learning process is needed.

Symbolic system to support the mathematics curriculum

acceptable by a large number of teachers

a symbolic system should last for a full studying career and beyond

that avoids "black boxes"

stays close to mathematical thinking

Visual introduction

shows how technology can enhance mathematical content

Experimenting

the research buildup paradigm:       experiment  ->   conjecture  ->  proof

qualitative analysis of the result

simulations indicate tendencies when no exact results can be obtained

Control

graphical output is checked faster

qualitative considerations are  important

programming answers  to deepen  understanding

Assessment

need for a new exam system including project work

exam questions depend on a random entry

Definitions of e

In our country, the secondary curriculum covers part of the one variable calculus, and all of my students know before starting their first semester what the number e stands for. 
A small survey points out that

60 % of my students have learned e as a limit
	15 %of my students have learned e as approximated by a sequence
	25 %of my students have learned e by a property of derivatives.

The  number  e  numerically

E == 2.71

E == 2.71828

N[E, 12]

N[E, 20]

% == E

N[E, 100]

% == E

As a limit

Limit[(((1 + 1/n))^n), n -> Infinity]

Table[(((1 + 1/n))^n), {n, 30}];

N[%] // TableForm

Table[N[(((1 + 1/n))^n), 30], {n, 300}] // TableForm

Which is still way off any approximate value for  e .

As a series

Table[(∑_(k = 0)^n) (1. /(k!)), {n, 20}] // 
      TableForm

It is the value at    x = 1.   in the expressions of the form

Series[(E^x), {x, 0, 12}]

(∑_(k = 0)^∞) (1. /(k!))

And the exact value is recognised by the software:

(∑_(k = 0)^∞) (1/(k!))

By a derivative

Plot some exponential functions

f(x_) := (a^x)

a = 2.5;Plot[{f(x),f'(x)},{x, -1, 3},PlotStyle -> {Hue[0], Hue[ .6]}]

a = 2.8;Plot[{f(x),f'(x)},{x, -1, 3},PlotStyle -> {Hue[0], Hue[ .6]}]

Checking the value at one point is sufficient. Interpolating values in a binary way will approximate the value of  e .

a = 2.718;f(10)≤f'(10)

a = 2.719;f(10)≤f'(10)

a = 2.7185;f(10)≤f'(10)

a = 2.71825;f(10)≤f'(10)

a = 2.718375;f(10)≤f'(10)

The next guess coud be around  a=2.718325 , and so on. This iteration is faster than the limit definition. 
Finally it turns out that

((∂(E^x))/(∂x))

\(This\ \ e\^x\ \ is\ the\ only\ power\ function\ that\ is\ its\ own\ \(\(derivative\)\(.\)\(\ \)\)\)

\(There\ is\ no\ circular\ reasoning\ here, \ 
    since\ slopes\ can\ be\ defined\ for\ graphs\ of\ functions\ defined\ only\ on\ rationals\ and\ \ \ e\ \ \ can\ be\ obtained\ as\ a\ rational\ approximation\ by\ binary\ search\ or\ \(\(guessing\)\(.\)\)\)

One other advantage of this approach is  the holistic consideration of graphs rather than (huge sets ) of numbers, or excel-like value tables (fishbone diagrams) that tell little about the qualitative behavior of the functions.

This leads to

Graphical introduction

Plotting the limit

Plot[(((x + 1))^(1/x)), {x, 0, 4}]

Plot[(((x + 1))^(1/x)), {x, 0, 0.001}]

Because no CAS can compute millionth powers of reals accurately [and if they can in the future, the same problem will arise by the hundred millionth or many billionth powers]

Plot[(((x + 1))^(1/x)), {x, 0, 
        0.000001}]

Plotting in logaritmic scale

The following does not yield much information:

Plot[(E^x), {x, -100, 100}]

To visualise the behavior of the exponential (and similar functions) over a large interval one needs logarithmic scales on the coordinate axes. 
Early introduction of such rescaling is very useful in many other situations as well.

Plotting the derivative

Is done in many education packages. It should be stressed that the crucial point is that differentiable functions allow linear approximations locally, i.e. zooming in will eventually yield a straight line. 
Better still is the holistic consideration of the derivative of a function at all points, and the investigation (at least on bounded intervals) of uniform convergence of the differential quotients involved in the definition of derivative.

\(Taylor\ polynomials\ and\ series\ are\ derived\ using\ the\ derivative\ of\ \ e\^x\)

Series[g[x], {x, 0, 12}] // TraditionalForm

Series[(E^x), {x, 0, 12}] // TraditionalForm

Here the numerators are factorials

Experimenting with slow convergence

Slow convergence of the sequence

Table[(((1.00000000000000000 + (1/n)))^n), {n, 20}] // 
      TableForm

Even computing 2000 terms does not give any precision:

Table[(((1.00000000000000000 + (1/n)))^n), {n, 100, 2000, 100}] // 
      TableForm

and recall that

E // N

Instability in the plot

Since the plot above is highly unstable and convergence is slow, we will no longer use the limit defintion for  e  .

Experimenting with rapid convergence

Rapid convergence of the series

Let us specify we want accuracy of 38 decimal digits and compute a few partial sums:

Table[(∑_(k = 0)^n) (1.00000000000000000000000000000000000000/(k!)), {n, 40}] // 
      TableForm

Accuracy[%]

An application in probability

The following converges rapidly to a ratio slightly over 63 %:

Table[1 - (1. /((∑_(k = 0)^n) (1.0000000000/(k!)))), {n, 12}] // 
      TableForm

This means that we can make money betting the following odds:

Gambling

and the number e appears in many other problems of probability as well.

Control

The Stirling formula

Stirling

Recovering the instability of the plot by the series

Limit

Extending the definition

Trigonometry and vectors in the complex plane

Many facts from trigonometry can be obtained and simplified using complex exponentials.

Directions

The series

Knowing one power series is a good starting point for developing the theory of (Mc Laurin and) Taylor series, even in the complex case.
Exponential generating functions can be introduced as straightforward generalisations of the exponential and these are useful in many problems in discrete mathematics, where order is important. 
For instance: what is the number of 12 letter words that can be written with four characters A B C D, if the first  character A has to appear an even number of times and the last character D has to appear an uneven number of times. The answer is the coefficient of  ((x^12)/(12!))  in the expression of

(1/4) (E^x) (E^x) ((E^x) + (E^(-x))) ((E^x) - (E^(-x)))

Series[%, {x, 0, 15}];

12! Coefficient[%, (x^12)]

So with these 12 letterwords we could roughly make all words in every language.

The derivative

All kinds of derivatives involving powers can be computed from the derivative of the exponential by use of the chain rule.

((∂(E^(g(x))))/(∂x))

and knowing that (in fact the only other useful formula):

a =. ; 
    (a^x) == (E^(x log(a)))

For students used to working on a computer screen, extending variables on the (real) line to the plane should not be too difficult.
The definition of the exponential extends immediately to complex variables:

((∂(E^z))/(∂z))

((∂(E^(I y)))/(∂y))

and this can be visualised using the limit of the same differential quotient (vector) in the complex plane.

Take the limit of the vectors

(((E^(I y)) - 1)/y)

for y tending to zero. The derivative is again the basis for setting up the (complex) Taylor approximation.

Applications

Where do we go from here?

Trigonometry revisited

We know how to specify directions in the plane. Recall how to color directions:

<<Graphics`ArgColors`

a = 200

un(
        k_) := (E^(((2 π I k)/a)))

Show[{Plot[0,{x,-1,1}],
	Graphics[
		Table[
   			{ArgColor[un[j]],
   				Disk[{Re[un[j]], Im[un[j]]}, 0.03]},
  		 {j, a}]
  		 ]}, AspectRatio -> Automatic
  	]

Having avoided all trigonometry so far, it is possible to recover any expression involving sin and cos (and all other functions as well) using the definitions (different from those in the software) for the sum and the difference of conjugate directions

cos[
          x_] = (((E^(I x)) + (E^(-I x)))/2);

sin[
          x_] = (((E^(I x)) - (E^(-I x)))/(2 I));

where the last division by  i  is nothing else than a clockwise rotqation over a right angle in the plane. 
All properties are obtained using the derivative of the exponential.

((∂cos(x))/(∂x))

Simplify[% == - sin(x)]

Differential equations and modeling

Especially those of first order and second order, including the study of periodicity (the complex exponential case).
It is important that students learn to deal with complex roots early in their curriculum. Handling these is not any more difficult than

Complex mappings

Hydrodynamic flows and behavior of complex maps around singularities.

The discrete Fourier Transform

DFT

Including the smoothing of phenomena, such as a noisy sound:

DFT2

Geometrical shapes defined by rational images of the unit circle

A crash course in design using (complex) trigonometric polynomials

Shapes

Inquiries:

			icnop@vub.ac.be

ATCM website

http://we.vub.ac.be/exploot/