(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ \(\[ExponentialE] == 2.71\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ExponentialE] == 2.71828\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[ExponentialE], 12]\)], "Input"], Cell[BoxData[ \(2.718281828459045`\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[ExponentialE], 20]\)], "Input"], Cell[BoxData[ \(2.718281828459045235360287471352662`20\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(% \[Equal] \[ExponentialE]\)], "Input"], 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Interpolating values \ in a binary way will approximate the value of \[ExponentialE] .\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{\(a = 2.718\), ";", RowBox[{\(f(10)\), "\[LessEqual]", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "10", ")"}]}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{\(a = 2.719\), ";", RowBox[{\(f(10)\), "\[LessEqual]", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "10", ")"}]}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{\(a = 2.7185\), ";", RowBox[{\(f(10)\), "\[LessEqual]", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "10", ")"}]}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{\(a = 2.71825\), ";", RowBox[{\(f(10)\), "\[LessEqual]", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "10", ")"}]}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{\(a = 2.718375\), ";", RowBox[{\(f(10)\), "\[LessEqual]", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "10", ")"}]}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Closed]], Cell["\<\ 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"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ FractionBox[\(\[PartialD]\[ExponentialE]\^x\), \(\[PartialD]x\), MultilineFunction->None], TraditionalForm]], "Input"], Cell[BoxData[ \(\[ExponentialE]\^x\)], "Output"] }, Closed]], Cell[BoxData[ \(This\ \ \[ExponentialE]\^x\ \ is\ the\ only\ power\ function\ that\ is\ \ its\ own\ \(\(derivative\)\(.\)\(\ \)\)\)], "Text", FontFamily->"Times"], 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\ \(\(guessing\)\(.\)\)\)], \ "Text", FontFamily->"Times"], Cell["\<\ 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.\ \>", 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