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Notebook[{
Cell["\<\
ATCM 2003
Hsinchu, Taiwan (ROC)\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell[TextData[{
  "The influence of CAS on curricula\nCase Study: ",
  StyleBox["e",
    FontSize->36],
  "  in the curriculum"
}], "Subtitle"],

Cell["\<\
Ivan Cnop
icnop@vub.ac.be\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Abstract",
  FontWeight->"Bold"]], "Subsubsection"],

Cell["\<\
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.\
\>", "SmallText"]
}, Closed]],

Cell[CellGroupData[{

Cell["Guidelines for curriculum development under technology", "Section"],

Cell[CellGroupData[{

Cell["\<\
Final EU Report FP6 from the Working Party on Education and \
Training\
\>", "Subsection"],

Cell["\<\
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 
\tteacher training [which is an outstanding issue everywhere] 
\tthe huge lack of useful and appropriate content for e-learning, 
\tand the need for continued learning. 
They recommend (quoting from the text):\
\>", "Text"],

Cell["Content development and the e-learning as such.", "Text",
  FontWeight->"Bold"],

Cell["\<\
Research should be driven by learning pull rather than technology \
push.\
\>", "Text",
  FontWeight->"Bold"],

Cell["Modular learning objects and their management are required.", "Text",
  FontWeight->"Bold"],

Cell["\<\
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.\
\>", "Text",
  FontWeight->"Bold"],

Cell["\<\
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.\
\>", "Text",
  FontWeight->"Bold"]
}, Closed]],

Cell[CellGroupData[{

Cell["Symbolic system to support the mathematics curriculum", "Subsection"],

Cell["acceptable by a large number of teachers", "Subsubsection"],

Cell["\<\
a symbolic system should last for a full studying career and beyond\
\
\>", "Subsubsection"],

Cell["that avoids \"black boxes\" ", "Subsubsection"],

Cell[CellGroupData[{

Cell["stays close to mathematical thinking", "Subsubsection"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\(\[IndentingNewLine]\)\)\)], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Visual introduction", "Subsection"],

Cell[CellGroupData[{

Cell["shows how technology can enhance mathematical content", "Subsubsection"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\)\)], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Experimenting", "Subsection"],

Cell["\<\
the research buildup paradigm:       experiment  \[Rule]   \
conjecture  \[Rule]  proof\
\>", "Subsubsection"],

Cell["qualitative analysis of the result ", "Subsubsection"],

Cell[CellGroupData[{

Cell["\<\
simulations indicate tendencies when no exact results can be \
obtained \
\>", "Subsubsection"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\(\[IndentingNewLine]\)\)\)], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Control", "Subsection"],

Cell["graphical output is checked faster ", "Subsubsection"],

Cell["qualitative considerations are  important", "Subsubsection"],

Cell[CellGroupData[{

Cell["programming answers  to deepen  understanding ", "Subsubsection"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\(\[IndentingNewLine]\)\)\)], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Assessment", "Subsection"],

Cell["need for a new exam system including project work", "Subsubsection"],

Cell[CellGroupData[{

Cell["exam questions depend on a random entry ", "Subsubsection"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\)\)], "Input"]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Definitions of \[ExponentialE]", "Section"],

Cell["\<\
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 \[ExponentialE] stands for. 
A small survey points out that\
\>", "Text"],

Cell["\<\
\t60 % of my students have learned \[ExponentialE] as a limit
\t15 %of my students have learned \[ExponentialE] as approximated by a \
sequence
\t25 %of my students have learned \[ExponentialE] by a property of \
derivatives.\
\>", "Text"],

Cell[CellGroupData[{

Cell["The  number  \[ExponentialE]  numerically", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["As a limit", "Subsection"],

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Cell[BoxData[
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Which is still way off any approximate value for  \[ExponentialE] .\
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It is the value at    x = 1.   in the expressions of the form\
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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\
\>", "Text"],

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its\ own\ \(\(derivative\)\(.\)\(\ \)\)\)], "Text",
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Cell[BoxData[
    \(There\ is\ no\ circular\ reasoning\ here, \ 
    since\ slopes\ can\ be\ defined\ for\ graphs\ of\ functions\ defined\ \
only\ on\ rationals\ and\ \ \ \[ExponentialE]\ \ \ can\ be\ obtained\ as\ a\ \
rational\ approximation\ by\ binary\ search\ or\ \(\(guessing\)\(.\)\)\)], \
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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.\
\>", "Text"],

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Cell[CellGroupData[{

Cell["Graphical introduction", "Section"],

Cell[CellGroupData[{

Cell["Plotting the limit", "Subsection"],

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Cell["\<\
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]\
\>", "Text"],

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Cell["\<\
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.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Plotting the derivative", "Subsection"],

Cell[TextData[{
  "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. \nBetter still is the \
",
  StyleBox["holistic",
    FontColor->RGBColor[1, 0, 0]],
  " 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."
}], "Text"],

Cell[BoxData[
    \(Taylor\ polynomials\ and\ series\ are\ derived\ using\ the\ derivative\ \
of\ \ \[ExponentialE]\^x\)], "Text",
  FontFamily->"Times"],

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          Times[ 
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          Times[ 
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}, Closed]],

Cell["Here the numerators are factorials", "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Experimenting with slow convergence", "Section"],

Cell[CellGroupData[{

Cell["Slow convergence of the sequence", "Subsection"],

Cell[CellGroupData[{

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}, Closed]],

Cell["Even computing 2000 terms does not give any precision: ", "Text"],

Cell[CellGroupData[{

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          {"2.7148917443812869193244966896`14.7001"},
          {"2.7155685206517259295998487022`14.6029"},
          {"2.716020048880588862682784`14.5236"},
          {"2.716342737729735040949849`14.4566"},
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Cell["and recall that", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["Instability in the plot", "Subsection"],

Cell["\<\
Since the plot above is highly unstable and convergence is slow, we \
will no longer use the limit defintion for  \[ExponentialE]  .\
\>", "Text"]
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Cell["Experimenting with rapid convergence", "Section"],

Cell[CellGroupData[{

Cell["Rapid convergence of the series", "Subsection"],

Cell["\<\
Let us specify we want accuracy of 38 decimal digits and compute a \
few partial sums:\
\>", "Text"],

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Cell[BoxData[
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Cell[CellGroupData[{

Cell["An application in probability ", "Subsection"],

Cell["\<\
The following converges rapidly to a ratio slightly over 63 \
%:\
\>", "Text"],

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          {"0.6321167883211679`"},
          {"0.632120144889189`"},
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Cell["\<\
This means that we can make money betting the following odds:\
\>", \
"Text"],

Cell[BoxData[
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Cell["\<\
and the number \[ExponentialE] appears in many other problems of \
probability as well.\
\>", "Text"]
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Cell[CellGroupData[{

Cell["Control ", "Section"],

Cell[CellGroupData[{

Cell["The Stirling formula", "Subsection"],

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Cell["Recovering the instability of the plot by the series", "Subsection"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["Extending the definition", "Section"],

Cell[CellGroupData[{

Cell["Trigonometry and vectors in the complex plane", "Subsection"],

Cell["\<\
Many facts from trigonometry can be obtained and simplified using \
complex exponentials.\
\>", "Text"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["The series", "Subsection"],

Cell[TextData[{
  "Knowing one power series is a good starting point for developing the \
theory of (Mc Laurin and) Taylor series, even in the complex case.\n\
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. \nFor 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  ",
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  "  in the expression of   "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
So with these 12 letterwords we could roughly make all words in \
every language.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["The derivative", "Subsection"],

Cell["\<\
All kinds of derivatives involving powers can be computed from the \
derivative of the exponential by use of the chain rule.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell["and knowing that (in fact the only other useful formula):", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
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:  \
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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and this can be visualised using the limit of the same differential \
quotient (vector) in the complex plane.\
\>", "Text"],

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Cell[BoxData[
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Cell["\<\
for y tending to zero. The derivative is again the basis for \
setting up the (complex) Taylor approximation.\
\>", "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Applications", "Section"],

Cell["Where do we go from here?", "Text"],

Cell[CellGroupData[{

Cell["Trigonometry revisited", "Subsection"],

Cell["\<\
We know how to specify directions in the plane. Recall how to color \
directions:\
\>", "Text"],

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Geometrical shapes defined by rational images of the unit \
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A crash course in design using (complex) trigonometric \
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Inquiries:

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ATCM website

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*)



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End of Mathematica Notebook file.
*******************************************************************)