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Cell[CellGroupData[{
Cell["Principles for a new mathematics curriculum buildup", "Subtitle"],

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

Cell[CellGroupData[{

Cell["Abstract", "Subsection"],

Cell[TextData[{
  "Regularly we see alarming reports about the declining mathematical \
knowledge in secondary and early tertiary education [Sefi ]. Accreditation \
rules [Abet ] for many curricula include ",
  StyleBox["mathematical knowledge, proficiency in mathematics and ability to \
apply (advanced) mathematics",
    FontColor->RGBColor[0, 0, 1]],
  ", 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.  \nEverybody 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. \n\
Besides doing all technical computing and modeling, symbolic platforms allow \
better insight in concepts and proofs.\nA 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. \n\
The principles outlined below apply to undergraduate tertiary education but \
can be applied for the final years of a secondary curriculum as well."
}], "Text",
  FontWeight->"Bold"]
}, Closed]],

Cell[CellGroupData[{

Cell["Choice of a symbolic system", "Section"],

Cell[CellGroupData[{

Cell["Acceptable by a large number of teachers", "Subsection"],

Cell[TextData[{
  "Most symbolic CAS cater to teacher's needs and have commercial offers \
targeting this group. \n",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " offers several options towards them:\n\tspecial licensing conditions for \
teachers (up to community colleges) and their students;\n\tthe Teacher's \
Edition with extra palettes for a test generating engine;\n\tCalculation \
Center with a slimmed-down programming environment for non-computer science \
teachers."
}], "Text",
  FontWeight->"Bold"]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[{
  "Lasting a ",
  StyleBox["full studying career",
    FontColor->RGBColor[1, 0, 0]]
}], "Subsection"],

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

Cell[CellGroupData[{

Cell[TextData[{
  "Syntax should be close to ",
  StyleBox["mathematical thinking",
    FontColor->RGBColor[1, 0, 0]]
}], "Subsection"],

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

Cell[TextData[{
  "Advanced objects can be programmed in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "  [Wolfram] using a few lines, translating the underlying mathematical \
construction. "
}], "Text",
  FontWeight->"Bold"],

Cell["\<\
Example: the following implement Julia sets in the plane for \
different constants and initial values.\
\>", "Text"],

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Cell["Cheap 1-feature system limits the scope of problems", "Subsection"],

Cell["\<\
Other limitations are the delivery of CAS on small screens such as \
those on hand-held devices.\
\>", "Text",
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}, Closed]],

Cell[CellGroupData[{

Cell["No delivery of dead text or a mere testing package", "Subsection"],

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

Cell[TextData[{
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  " what software can do in the classroom"
}], "Subsection"],

Cell[TextData[{
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    FontSlant->"Italic"],
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and these are also evaluated on different examples or in slightly different \
assumptions. \nSometimes I even take time in class to write new commands or \
programs in a separate notebook."
}], "Text",
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Cell[TextData[{
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Cell["\<\
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. \
\>", "Text",
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Cell[TextData[{
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These documents are revisited often and expanded, each time a new \
viewpoint is introduced.\
\>", "Text",
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Cell["Experimenting and problem-oriented approach", "Section"],

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Cell[TextData[{
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intrest has been in the prediction of behavior of complex systems, from \
Babylonian astronomy to post-bubble finances.\n\nThanks to the introduction \
of technology this functionality is again possible [see Uhl, Kawski] \
following the paradigm\n\n\t",
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    FontSize->16],
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to the software \nand focus on the qualitative analysis of the result and \
understanding. \nSimulations indicate tendencies when no exact results can be \
obtained for a particular problem. \nGradually increasing the complexity of \
the model achieves the goals expected by their future professional \
environment."
}], "Text",
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Cell["\<\
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].\
\>", "Text",
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Cell["\<\
Simulation example: the following implement some Random \
distribution on random length intervals, and the graph it obtains raises \
interesting problems about random distributions.\
\>", "Text",
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    \(\(xesRight = Take[xes, {2, howmany + 1}];\)\), "\n", 
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    \(\(slots = 
        Map[slotFunction, Transpose[{xesLeft, xesRight}]];\)\), "\n", 
    \(\(values = Table[Random[], {howmany}];\)\), "\n", 
    \(\(l = Flatten[Transpose[{slots, values}]];\)\), "\n", 
    \(Clear[f]; f[x_] := l /. Head[l] \[Rule] Which\), "\n", 
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Cell[CellGroupData[{

Cell["Cultivate control", "Section"],

Cell[TextData[{
  "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 ",
  StyleBox["graphical ",
    FontColor->RGBColor[1, 0, 0]],
  "output is checked faster than formulas.\nQualitative considerations are in \
this respect more important than computational techniques.\nStudents 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 ",
  StyleBox["basic ideas",
    FontColor->RGBColor[1, 0, 0]],
  " from mathematics and computer science overlap. \nIt is unfortunate that \
programming courses do yet not use (part of the syntax of) Mathematica in \
their lab sessions. "
}], "Text",
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Cell[CellGroupData[{

Cell["Qualitative considerations", "Section"],

Cell["\<\
Versus computational techniques; leave computing to the \
computer\
\>", "Subsubsection",
  CellDingbat->None],

Cell[CellGroupData[{

Cell["Link to computer science", "Subsection"],

Cell["\<\
Programming your answers is a good exercise to deepen the \
understanding and to avoid false belief pitfalls.\
\>", "Subsubsection",
  CellDingbat->None],

Cell["\<\
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.\
\>", "Subsubsection",
  CellDingbat->None]
}, Closed]],

Cell[CellGroupData[{

Cell["Encourage implementation of symbolic proofs", "Subsection"],

Cell[TextData[{
  "Students are quite positive about learning programming principles in their \
mathematics course. \nMathematica gives them the opportunity to practice \
facts from their computer science courses and they acknowledge that many \
ideas overlap. \nIt is unfortunate that programming courses do not yet use \
(part of the syntax of) ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
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Cell["WYTIWYG: test symbolic literacy and appetite for problems", "Section"],

Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["Open",
    FontColor->RGBColor[1, 0, 0]],
  " problems depending on random entry."
}], "Subsection"],

Cell["\<\
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.\
\>", "Text",
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Cell[CellGroupData[{

Cell["The Sefi curriculum revisited", "Section"],

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

Cell[CellGroupData[{

Cell["Core zero", "Subsection"],

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

Cell["what importance should be placed on number operations", "Text"],

Cell["\<\
ambiguity between allocations, equations and replacements \
(rule)\
\>", "Text"],

Cell["\<\
recognising type of an equation is more important than solving it\
\
\>", "Text"],

Cell["factoring is nontrivial", "Text"],

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Cell["Inverse functions from parametric plots", "Text"],

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Cell["\<\
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Cell["\<\
Underestimate discrete mathematics and relationships discrete - \
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Here explicit reference is made to using software. However, much \
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\>", "Text"],

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hyperbolic functions have no interest as such; they are \
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\>", "Text"],

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rational functions with linear denominators are too \
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\>", "Text"],

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

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\>", "Text"],

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\>", "Text"],

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Most topics at this level can be handled by students who have \
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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, \
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Cell["References", "Section"],

Cell[TextData[{
  "Abet:  www.abet.org\nI. Cnop: \"Computer Screens and Toys\". Proc. of the \
6th ATCM, Melbourne, 2001.\nICMI study: \"The future of the teaching and \
learning of Algebra\", to appear.\nM. Kawski: \"Vector Field Analyser\", \
interactive website at asu.edu.\nSefi Core curriculum: Sefi-Mathematics \
Working group. \n\thttp://learn.lboro.ac.uk/mwg/core.html or the sefi-mwg \
website\nSefi 2002 Gotheborg report.  See Sefi website.\nJ. Uhl: \"Software \
that works\", Mathematics Everywhere Inc.\nS. Wolfram: \"",
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