إيتيت زا إليكتينو a بلين زداج سي فم فاي تي زا إليكتينو a بلين بوديزيل فيسوك، إيل نيشس سي فم دو سروفنفن نابديك أوزيروجوج فس أود زمني دودافاتيل نيكونين بابروفن مم إين: بوروفنم ديستريبون سازبي زا إليكتينو i بلين a فيبيريم تو برو فس نيجفودنج. A بوكود بوديت تشت، زادم زا فس i زمنو دودافاتيل. برمرن إتم 2320 كورون رون نا إليكتين a 2560 كورون نا بلينو. فت، كدي ناجديت نيجليفنج بينزين في سفم أوكول ماي أنو. أبود، ليبانون: الرسوم المتحركة، أداة لفهم الإحداثيات القطبية الطلاب في الصفوف الجامعية لديهم الكثير من الصعوبة في رسم الرسوم البيانية من الوظائف المعطاة في التنسيقات القطبية. في العمل السابق القيام به، تم استخدام الرسوم المتحركة كأداة لفهم كيف يؤثر التحول الخطي على الرسم البياني لوظيفة، وهنا أنا توسيع هذا العمل لتمكين الطلاب من فهم أفضل الإحداثيات القطبية والعلاقة إلى الإحداثيات الديكارتية. في تجربتنا، واستخدام نظام الجبر الكمبيوتر مثل ماثماتيكا لا يضمن في حد ذاته أن الطلاب سوف تحسين مهارات التصور أو فهمهم للمفاهيم الرياضية. ومن الضروري تصميم المشاريع حيث يتم تشجيع الطلاب على مراقبة الظواهر، وجعل التخمينات ثم اختبار ما إذا كانت هذه التخمينات صحيحة حقا. نقدم في هذه الورقة العمل الذي قمنا به والتي يمكن استخدامها مع طلاب حساب التفاضل والتكامل. جورج أدي، السويد: التطبيقات العملية لل كاس باستخدام وظائف جيبية وهناك الكثير من الفيزياء تشارك مع دراسة الاختلاف الجيبية. في هذا الحديث سوف نعرض كيف التكنولوجيا المحمولة مع كاس تغير نهجنا مما يجعل الفيزياء أكثر سهولة في متناول الطلاب والسماح للدورة الفيزياء لتصبح أعمق وأكثر وضوحا. وسوف نناقش أيضا التغييرات المناسبة في دورة ستودينتساكوتماثس. جورج أدي، السويد: المعادلات التفاضلية في الرياضيات والفيزياء بدلا من الأساليب التحليلية التكنولوجيا المحمولة مع كاس يجعل من الأسهل لدراسة الظواهر العلمية المشتركة على مستوى البكالوريوس مباشرة باستخدام المعادلات التفاضلية والتقنيات العددية بدلا من الأساليب التحليلية التقليدية. سوف نسلط الضوء على مجالات الدراسة حيث الاحتياجات تتغير في الفيزياء باستخدام المعادلات التفاضلية الخطية وغير الخطية في واحد أو أكثر من الأبعاد. وهذا يؤدي إلى تغيير المتطلبات في الرياضيات. وستتم مناقشة هذه التغييرات. بينغت أهلاندر، السويد كيفية إجراء اختبارات للطلاب باستخدام أدوات كاس (تي-89) في مدرستي، أوسترابوجيمناسيت مدرسة ثانوية في السويد، أعمل مع فئة الرياضيات حيث يستخدم كل طالب تي-89. سن الطلاب 17 سنة. سيتم شرح أفكاري حول كيفية فحص الطلاب باستخدام هذه الأداة القوية ولا تزال اختبار فهم الرياضيات. أسئلة مثل ما هي جذور المعادلة x2-6x 5 0 لا تختبر الفهم إذا كنت تستخدم تي-89. ولكن إذا كنت تعطي الطلاب الجواب (جذور المعادلة التربيعية هي x 5 و س 1)، يمكنك أن تطلب من الطلاب لإعطاء أمثلة من المعادلات التي من شأنها أن تعطي هذه الإجابات. هذا هو نوع من خطر في الرياضيات واختبارات حقا إذا كان لديهم فهم وراء حلول المعادلات التربيعية. يمكننا أيضا تقديم أسئلة مع بعض الحلول ونطلب من الطلاب للسيطرة وشرح الخطوات في الحل. وهذا اختبار أيضا إذا كان يمكن للطلاب شرح بطريقة سليمة التفكير الرياضي. سأقدم بعض الأمثلة في العرض التقديمي من تجربتي في الفصل الدراسي. مارا ألاجيك، الولايات المتحدة الأمريكية: رسم الخرائط للتعلم: التمييز تعليم الرياضيات للتعلم شخصية في سياق ماذا - كيف - منظمة الصحة العالمية. إذا كان ما هو المنهج القائم على معايير الرياضيات أندور ومنظمة الصحة العالمية هي المتعلمين، ويمكن كيف يفسر طريقتنا في التفكير، لدينا تيشنغلينارنينغريفلكتينغ الفلسفة، أندور عمليات صنع الإحساس أين هو مكان التكنولوجيا في هذه العمليات تحاول هذه الورقة إلى وإعطاء بعض الأمثلة الأجوبة وطرح المزيد من الأسئلة حول قوة التكنولوجيا في تعلم الرياضيات: كيف يمكن للتكنولوجيا أن تحدث فرقا في الطريقة التي نميز التعليم للتعليم شخصية في الرياضيات الصف G. ألبانو، C. دابيس، M. ديسيديريو، إيطاليا: تحويل لابلاس والدوائر الكهربائية: أداة تعليمية متعددة التخصصات هذا العمل موجه إلى طلبة المدارس الثانوية ذات التوجه العلمي ويهدف إلى دعم التلاميذ في تعلم مادتين: حل المعادلات التفاضلية الخطية الثانية ودراسة الدوائر الكهربائية. ويرتبط الموضوعان لأن أحد الأساليب المعروضة لحل المعادلات التفاضلية يستخدم تحويل لابلاس، وهذه هي أفضل طريقة لحل المعادلات التكاملية التفاضلية التي يتم تلبيتها في دراسة الدوائر الكهربائية. يتم إنشاء حزمة باستخدام كاس كما ماثماتيكا. حزمة يوفر إطارا نظريا والعديد من التمارين حيث يتم الرصاص الطلاب خطوة بخطوة لحل المعادلات التفاضلية. باستخدام هذه المعادلات حزمة تصف الدوائر الكهربائية يمكن حلها، وبالتالي الكميات الفيزيائية تطور (كثافة الحالية والجهد) يمكن الحصول عليها. بوركهارد ألبرس، ألمانيا: تطبيق رياضي مشاريع للمهندسين الميكانيكيين - مفهوم، مبادئ توجيهية وأمثلة في هذا المقال، نقدم مفهوم مشاريع التطبيقات الرياضية كوسيلة لتعزيز قدرات طلاب الهندسة لاستخدام الرياضيات لحل المشاكل في المشاريع الكبيرة وكذلك على التواصل وتقديم المحتوى الرياضي. وعلى عكس العديد من دراسات الحالة، نركز على تحديد المعايير وفئات المشروع التي يمكن للمدربين بناء حالات (أي مشاريع محددة). والهدف الرئيسي من هذه الورقة هو تسهيل تعريف مشاريع جيدة جديدة في وضع المناهج الدراسية معينة. هاليل أردهان، تركيا: قضايا حول دمج كاس في تعليم الرياضيات: منهج وظيفي وبرمجي لبعض الأسئلة في السنوات الأخيرة حاولنا دراسة القضايا الرئيسية ومسائل البحث المختلفة حول دمج وتنفيذ الأدوات المعرفية مثل بيئات أنظمة الجبر الكمبيوتر (كاس) ، ولا سيما تي-92 آلة حاسبة في كل من تدريس وتعلم الرياضيات في تركيا. في هذا العرض، بعد استعراض القضايا الرئيسية والعقبات في هذا الموضوع لفترة وجيزة جدا، ونحن بناء وظيفة جديدة، اسمه وظيفة الغيار أرقام (دسف)، ونهج وظيفي إلى رقمين رئيسيين أرقام ونهج البرمجة للعثور على القاسم المشترك الأكبر (غد) من الأعداد الصحيحة. وأخيرا، نقدم بعض المواد التعليمية، والتي تم تصميمها وتطويرها من وجهة نظر نظريات التعلم الجديدة والنماذج، وهي التعلم البناء واكتشاف. دين أرغانبرايت، الولايات المتحدة الأمريكية: رسومات جداول البيانات الإبداعية في الرياضيات التدريس والنمذجة جدول البيانات هو أداة ممتازة ومتاحة بسهولة لتعليم وتعلم الرياضيات. ويمكن تنفيذ النماذج الرياضية والخوارزميات، وتقنيات التصور في جداول البيانات في شكل تفاعلي بطريقة أن عملية الإنشاء نفسها ينقل الرياضيات الأساسية. وتظهر الأمثلة كيفية تعزيز النمذجة الرياضية والتعليم من خلال رسومات جداول البيانات المبتكرة والرسوم المتحركة. وتشمل الرسوم التوضيحية الرياضية التحقيق في الوظائف والهياكل الهندسية والخوارزميات الحسابية، والتصور الرياضي. وتأتي الأمثلة من الهندسة، وحساب التفاضل والتكامل، والطرق العددية، والجبر الخطي، وأبحاث العمليات، فضلا عن المجالات التطبيقية مثل النمذجة السكانية، والتدفق الحراري، والأوبئة، وعلم الوراثة، والأعمال التجارية، والرسومات الثقافية والكمبيوتر. بريجيتا أمب كلاوس أسبتسبيرجر، النمسا: تدريس المناهج الدراسية والتجريب في الرياضيات دورات أمب العلوم باستخدام التكنولوجيات الجديدة عبر تدريس المناهج والتعلم من خلال التجارب هي أهداف مهمة لدورات الرياضيات أمب الرياضيات في المستقبل. يتم تدريب مختلف المهارات العملية وكذلك الرياضية للطلاب من خلال إجراء التجارب وتحليل النتائج وأخيرا باستخدام وظائف لتركيب نقاط البيانات التي تم الحصول عليها من قبل التجارب. يجب على الطلاب الجمع بين المعرفة حول أنواع مختلفة من الوظائف مع المعرفة حول النظريات الكيميائية والفيزيائية. كجانب إضافي، يجب على الطلاب أيضا أن تأخذ الرعاية من دقة في تجربة للحصول على نتائج جيدة. ويدعم جمع قوائم كبيرة من البيانات التجريبية من قبل نظام تي-كبل. ويدعم التجريب الرياضي، والحسابات المعقدة والتصور من قبل الجرافيك حاسبة جيب تي-92. نحن نبلغ عن التجارب التي أجريت مع عدة مجموعات من الطلاب في سن 17-18 وحوالي مجموعة واحدة من الطلاب ذوي القدرة العالية في سن 14. مهارات وقدرات الطلاب لإجراء التجارب عدنان باكي، تركيا: التحقيق تصورات المعلمين حول إعدادهم لاستخدام تكنولوجيا المعلومات في تدريس الفصول الدراسية قامت الباحثة بتدريس مقرر مدته سنتان ضمن برنامج تعليم معلمي الرياضيات لتدريب المعلمين الطلابية والتحقيق في التصورات حول إعدادهم لاستخدام الحواسيب في تعليمهم الخاص. وتصف هذه الورقة القضايا الناشئة عن تحليل الدورة. تم جمع البيانات من خلال الاستبيانات. الطلاب الذين يشعرون استعدادا جعلت العلاقة بين الأنشطة القائمة على الحاسوب والرياضيات المدرسية، وكان أكثر خبرة في البرنامج التعليمي خلال الدورة من غيرها. وتناقش الآثار المترتبة على هذه النتائج لتصميم وتنفيذ المقررات الجامعية القائمة على الحاسوب وإجراء مزيد من البحوث في هذا المجال. ماريا باكو فرنسا: برنامج رياضي في العملية التعليمية للمعلمين الفرنسيين والمجريين تنفق أنظمة التعليم الفرنسية والمجرية الكثير من الطاقة لمواكبة التطورات الجديدة في مجال التكنولوجيا. يتم تدريس المعلوماتية من خلال الخروج من المدرسة الثانوية في جميع أنحاء ولكن أجهزة الكمبيوتر ليس لها دور كاف حتى الآن في عملية التدريس من مختلف المواضيع. هدف الاقتراع، الذي عرض في المقال، هو إظهار مدى ومدى أساتذة الجامعات وطلابهم يعرف ويستخدم البرامج الرياضية. مواضيع هذا الاستطلاع هي الأساتذة والطلاب في كلية الرياضيات من جامعة بول ساباتييه من تولوز وجامعة ديبريسين من المجر. وأدت الدراسة الموازية لهذين البلدين المختلفين ثقافيا واقتصاديا إلى توجيه انتباهنا إلى بعض المشاكل الخاصة والعامة المثيرة للاهتمام التي ترد بالتفصيل في هذه الورقة. هذا والأفكار على الاستبيانات يمكن أن تساعد على وضع أهداف جديدة في تطبيق أجهزة الكمبيوتر في عملية تدريس الرياضيات. يوريكو بالدين، البرازيل: دراسة من المخروطية مع القيقب الخامس و كابري-جيكوتوميغرافيتر إي ويستند العرض المعتاد من المخروطية في التعليم الابتدائي على هندسة الطائرة، بدءا من الخصائص البؤرية ومن ثم ربط الهندسة إلى الجبر عن طريق التعبيرات التربيعية. مع 3 يتم عرض المخروطية كما أقسام الطائرة من مخروط متماثل وخصائص التنسيق الأساسية وعادة ما يكون من الصعب أن يفهم من قبل الطلاب. ومع ذلك، فإن التطبيقات الأكثر جمالا ومحفزة من الكونيات لمشاكل العالم الحقيقي تتطلب المخروطية التي سيتم وضعها في إعدادات 3 الأبعاد. في هذه الورقة، نقدم دراسة مع الجمع بين استخدام كاس (مابل الخامس) و دغس (كابري-جيكوتوميغرافيتر) الذي يدمج كل من النهج في الفصول الدراسية، مؤكدا قدرات كل برنامج يناسب حالات محددة. نحن تشمل تمارين مفيدة على الانشاءات داندلين مع القيقب الخامس وكابري-جيكوتوميغرافيتر، والتي من شأنها أن تساعد المعلمين على بناء مواد التدريس ملموسة حول هذا الموضوع. رافائيل بارباستيفانو، البرازيل: تابولاي ومانجابا: الهندسة الديناميكية مع تويست المسافة نحن تقرير عن التطوير المستمر ل دغس مكملين، للهندسة الطائرة والفضاء. تم تصميم ملخصات تصميم كل من البرمجيات مع الأخذ بعين الاعتبار احتياجات التدريس عن بعد والاتصالات على شبكة الإنترنت. ويرد وصف للتنفيذ الحالي بشيء من التفصيل، ونناقش أيضا بعض القضايا التي أدت إلى اتخاذ قرار بالانخراط في المشروع، فضلا عن الآثار المترتبة على برنامج تدريب المعلمين المدعوم بالتكنولوجيا الذي وفر الحافز الأولي لذلك المشروع. إليزابيث بيلفورت، رافائيل بارباستيفانو، لويز كارلوس غيمارايس، البرازيل: استخدام الحاسوب في برامج تدريب المعلمين في الرياضيات: تأمل في التجربة كجزء من متطلبات دورة الدراسات العليا في جامعتنا، يحضر معلمي المرحلة الثانوية تخصصا فيما يتعلق باستخدام أجهزة الكمبيوتر لتدريس الرياضيات. ومن بين الأنشطة الأخرى، يطلب منهم إنتاج موادهم التعليمية الخاصة، التي ينبغي أن تدعمها إحدى حزم الحواسيب التعليمية المتاحة لهم أثناء الدورة. وقد صاغ المؤلفون هذا الانضباط، وكانوا يزاولونه أيضا على مدى السنوات الثلاث الماضية. وفي الوقت نفسه، كنا نحقق في عواقبه على آراء المعلمين وممارساتهم. في هذه المقالة، نقوم بتحليل نوعي المواد التعليمية التي تم إنتاجها خلال فترة الانضباط من قبل هؤلاء المعلمين، فضلا عن بعض النتائج على المدى المتوسط من هذه الأنشطة لعملهم الفصول الدراسية اللاحقة. ليودميلا بيلوسوفا، أوكرانيا: استخدام جداول البيانات لتطوير المهارات الرياضية ويخصص المقال لأسئلة استخدام جداول البيانات بهدف تشكيل المناهج التعليمية. والغرض الرئيسي هو تطوير مهارات الرياضيات والعادة. تم البحث في نتائج البحث أثناء تدريس عدة فصول من مقرر الرياضيات. وترد في مجموعة من المهام التي تبين أهداف ونتائج العمل مع الطلاب في هذه المادة. في نهاية البحث تم كشف اتصال متعدد التخصصات. ستيفان بيرتكتولد، النمسا: تطوير المدارس - منظور الأنظمة خلال العقد الماضي، ازدادت الحاجة إلى القيام بشيء حيال المدارس بشكل ملحوظ. الأسباب لا تعد ولا تحصى. في هذا العرض سوف يقدم المؤلف ملخصا قصيرا لكيفية وصول المدارس إلى وضعها الحالي. وهذا يبني الأساس لتحليل المدرسة من منظور النظم. أسئلة مثل مدرسة أو مؤسسة يمكن أن ينظر إليها على أنها نظام اجتماعي تسليط الضوء على بعض نقاط الضعف الرئيسية. بناء على 5 سنوات العمل في تطوير المدرسة سوف يقدم مقدم أيضا دراسة حالة قصيرة باستخدام أداة النظم، ما يسمى الرسوم البيانية السببية حلقة، لإظهار كيف يمكن تطبيق أدوات النظم في التنمية التنظيمية. ستيفان بيركتولد، إرنست جيبيتسروثر، ستيفان غيلدنبرغ حلقة السببية الرسم التخطيطي - دورة تحطم عملي في هذا العرض سوف ثلاثة خبراء النمذجة النظم من نظام ديناميكية المجموعة النمسا مع خلفيات مهنية مختلفة (التعليمية والإدارية والعلمية) تقدم دورة تحطم عملي لكل من يريد لتعلم أساسيات السببية حلقة الرسم التخطيطي، أداة بسيطة لكنها مرنة لرسم تخطيطي الحالات النظامية. في المصطلحات الرياضية الرسوم البيانية حلقة السببية هي الرسوم البيانية الموجهة (من العقد والحواف)، مع و - علامات تعلق على حواف. لذلك يمكن اعتبار الرسم البياني حلقة السببية كنوع من نظرية الرسم البياني تطبيقها، والتي أصبحت بارزة في العديد من المجالات من المؤسسات النظامية. نطاق هذه الورشة سيكون: الآثار السببية الإيجابية والسلبية تصاعدية الآثار السببية تصاعد واستقرار حلقات من الآثار السببية النماذج الأساسية من الهياكل حلقة سببية في أنظمة ديتليف برنتزن، ألمانيا: أفلام من تي-بلوس لقطات من تي-92PLUS يمكن أن يكون مرتبة لأفلام صغيرة (سعة تخزين أقل من 30 كيلوبايت) باستخدام أداة جيف البناء. التفاصيل التقنية هي سهلة الاستخدام، وبالتالي من الفائدة لأنشطة التلاميذ في دروس الرياضيات. وسوف تستخدم المحاضرة لإظهار تكنيك وكذلك لمناقشة استخدام في تعليم الرياضيات. الجلسة العامة: جون بيري، المملكة المتحدة: استخدام التكنولوجيا في تطوير مهارات النمذجة الرياضية جزء مهم من تدريس وتعلم الرياضيات على جميع مستويات التعليم هو تطوير المهارات اللازمة لحل المشاكل الحقيقية. وتسمى عملية حل المشاكل في العالم الحقيقي في الرياضيات النمذجة الرياضية. ويمكن تلخيصها من خلال الرسم البياني التالي (انظر صفحة من حبلا، هوتكي للجلسة العامة). وللتكنولوجيا دور هام تؤديه في هذه العملية. استخدام البرمجيات والآلات الحاسبة الطبيعية في مرحلة الحل. ومن الثابت الآن أن مرحلة صياغة النمذجة الرياضية تمثل مرحلة اختناق عملية النمذجة. مساعدة الطلاب على تطوير مهارات حل المشاكل الجيدة غالبا ما ينطوي على الكثير من الوقت والجهد في هذه المرحلة. معدات تسجيل البيانات هي وسيلة قوية لجمع وتحليل البيانات كجزء من مرحلة تفسير العملية. والهدف من هذه المحاضرة العامة هو التفكير في الطرق التي يمكننا أن نقدم التكنولوجيا للتعليم والتعلم وتقييم المهارات الرياضية. جون بيري، أندي سميث، المملكة المتحدة: مراقبة أساليب عمل الطلاب باستخدام الآلات الحاسبة الجرافيكية عندما يعمل الطلاب مع التكنولوجيا المحمولة باليد، مثل آلة حاسبة رسومية، فإننا عادة ما نرى فقط نتائج أنشطتهم في شكل مساهمة في حل مكتوب من مشكلة رياضية. ومن الأصعب التقاط عملية تفكيرهم أو أفعالهم حيث يستخدمون التكنولوجيا لحل المشكلة. في هذه الورقة وصفنا التحقيق التجريبية من أساليب عمل الطلاب مع آلة حاسبة الرسم باستخدام البرمجيات التي تلتقط ضربات المفاتيح التي يتم استخدامها. وبهذه الطريقة تمكن الطلاب من العمل بشكل طبيعي دون الشعور بالاحتفال. بعد جلسة حل المشاكل الطلابية كنا قادرين على تشغيل تسلسل ضربات المفاتيح لاستكشاف كيفية استخدام الطلاب فعلا التكنولوجيا، سواء كانوا يستخدمون التجربة ووضع الخطأ وكيف عملهم المتعلقة بالتدريب التي تلقوها. بيوتر بيالاس، الولايات المتحدة الأمريكية: أنوفا مع حاسبة الرسوم البيانية تي-83 هذا العرض سوف تثبت كيف يمكن استخدام كتالوج وقطع ولصق المرافق من تي-83 الرسوم البيانية حاسبة لإكمال جدول أنوفا عامل واحد. وسيتم توزيع ورقة العمل بما في ذلك مثالين. يمكن أن يشمل التمديد أنوفا اثنين من العوامل مثل 2X2، 2X3، أو 3X3 التصاميم عاملية. (جلسة العمل مع المشاركين باستخدام تي-83 حاسبة الرسوم البيانية المقترحة). سوف يشارك مقدم العرض على آلة حاسبة الرسم تي-89 إذا لزم الأمر. بيوتر بيالاس، الولايات المتحدة الأمريكية: ربط الحواسيب البيانية للإنترنت (لغسي) لغسي يزيد من الوصول إلى ملفات البيانات العددية، لا حاجة لكتابة البيانات في آلة حاسبة الرسوم البيانية، ويجعل من الممكن أن ملفات البيانات المحددة يمكن استخدامها ل إكسيل، مينيتاب ، سبس، والعديد من المنتجات البرمجية الإحصائية الأخرى. سيتم عرض مثال تي-83 لنقل البيانات. وسيتم تزويد المشاركين بالمواد المكتوبة حول نقل البيانات إلى تي-83TI-89 الرسوم البيانية حاسبة. بيوتر بيالاس، الولايات المتحدة الأمريكية: يستخدم جدول البيانات في دورة الإحصاءات الأولية العديد من المفاهيم الإحصائية الهامة التي تبدو غامضة جدا للطالب بداية يمكن فهمها بسهولة من خلال التصور والقدرة على أداء الحسابات المعقدة بسرعة. متاح تجاريا مختلف ملفات جداول البيانات برامج تطبيق الكمبيوتر تسمح إنستروكتورستودنت لإجراء العمليات الحسابية المعقدة، ورسم الرسوم البيانية وتحريك هذه في الوقت الحقيقي. سوف يشارك مقدم العرض نتائج تحقيقه حول آثار جدول البيانات على الإنجاز في مواضيع إحصائية مختارة من طلاب المرحلة الجامعية في دورة الإحصاءات الأولية. جوزيف بوملهم، النمسا من القطب إلى القطب، رحلة رقمية مع وجهة تحليلية و تي-8992 البيانات - محرر هو أداة ممتازة لاتباع نهج عددي لأساسيات حساب التفاضل والتكامل. وتبين لنا كيفية الجمع بين الوسائل العددية والرسومية لإدخال الانقطاعات، والتمايز والانحناء. نجد ليس فقط الحلول العددية، ولكن أيضا التحليلية دون استخدام أي حساب التفاضل والتكامل. نقطة البداية لدينا هي قطب من وظيفة عقلانية ووجهة لدينا هو قطب من إيفولوت. ويمكن بسهولة أن تقدم هذه الوحدة التدريس مع أي كاس الأخرى. فرانسيسكو بوتانا فيريرو، إسبانيا: إعادة النظر في الروابط الثلاثية والرابعة: الرسوم البيانية والمعادلات تستعرض هذه الورقة سلوك النظم الهندسية الديناميكية الحالية (جيوميترز سكيتشباد، كابري جيكوتوميغرافيتر، سندريلا، هندسي إكسيرت و لوكوس) عند التعامل مع اثنين من الروابط البسيطة: ثلاثة وأربعة وصلات شريط. وتناقش النهج المختلفة للتوليد العددي للمكان، وتسليط الضوء على نجاحها والقيود. توليد الربط الديناميكي يمكن استخدامها في التعليم الهندسي والتصميم الحقيقي، والتغلب على احتياجات الكتب للمصممين. دينيس بوهينو، جان-فرانتشيديلواز نيكود، كزافييه بافارد، إيمانويل ساندر، فرنسا: A ميكروورلد لمساعدة الطلاب على تعلم الجبر تصف هذه الورقة مبادئ التصميم للعالم المجهري المكرس لتلاعب التعبيرات الجبرية. هذا المجهر يحتوي على محرر متقدم مع الإجراءات الكلاسيكية والتلاعب المباشر. معظم الإجراءات المتاحة في اثنين أو ثلاثة طرق وسائط العمل الثلاثة هي: وضع النص الذي يتلاعب الأحرف، ووضع هيكل يعتني هيكل جبري من التعبيرات، ووضع التكافؤ الذي يأخذ في الاعتبار التكافؤ بين التعبيرات. كما يسمح العالم الصغير لتمثيل الأشجار المنطق. يتم تقييم معادلة التعبيرات التي بنيها الطالب ويتم إبلاغ الطالب بالنتيجة. وتصف الورقة أيضا الحالة الراهنة لتنفيذ العالم الصغير. وقد تم تحقيق نموذج أولي في بداية فبراير 2001. بير برومان، السويد: النمذجة الرياضية مع كابري الأول سوف تظهر بعض الأمثلة كيف كابري يمكن استخدامها من أجل تشكيل وظائف من الهياكل الهندسية. على سبيل المثال: خصائص واستخدام خطوط ديريكتريكس والدوائر من مخروطي مختلفة. ماذا لو قمنا بإدراج مستطيل في مثلث الزاوية الحاد كيف يمكننا استخدام كابري و ديريف في تركيبة أريد أيضا أن أقول بضع كلمات عن تيم، شبكة الشمال وسلسلة مؤتمرات حول التكنولوجيا في تعليم الرياضيات. دوغلاس بتلر، المملكة المتحدة: إضافة التألق إلى التدريس في الفصول الدراسية - استخدام ورد و إكسيل والإنترنت عرض مباشر على الشاشة الكبيرة للاستخدام الإبداعي لأدوات البرامج العامة سواء في الفصول الدراسية أو في إنشاء أوراق العمل. يمكن إنشاء تعبيرات رياضية سطر واحد معقدة بشكل مدهش في ورد كنص باستخدام مجموعة خطوط ونيكود ومفاتيح ألت المعرفة من قبل المستخدم (في تفضيل لمحرر المعادلة، على الرغم من أن ذلك لا يزال مطلوبا للتعبيرات متعددة الطبقات). يمكن لصق هذه التعبيرات في بيئات الخط الواحد مثل خلية إكسيل أو بريد إلكتروني. أيضا شريط الأدوات الرسم يمكن استخدامها لإنشاء مجموعة واسعة من الرسوم البيانية، على الرغم من أن هناك قيود مخيبة للآمال. وستتم مناقشة استنتاج وتصنيف موارد الويب المفيدة، كما سيتم أيضا تغطية لصق النصوص والرسومات والبيانات (التي غالبا ما تواجه صعوبات للتغلب عليها) من الإنترنت، بما في ذلك السحب من خلال موارد الويب المدهشة المرتبطة من مدرسة أوندل ( المملكة المتحدة) الموقع سيكون هناك أيضا نظرة على بعض من المزالق عند استخدام إكسيل، ومقدمة لمفهوم استخدام الكائنات المرتبطة حيوي لتصور الرياضيات. دوغلاس بتلر، أوك: أوتوغراف: ديناميكية تنسيق الهندسة والإحصاء هذا العرض سوف تظهر كيف يمكن أن تستخدم الكائنات الديناميكية والمعتمدة لتعزيز التفاهم في تدريس الرياضيات على مستوى المدارس والكليات، وكيف أنها تعطي المعلم ذخيرة جديدة ومثيرة من التحرك الصور. تاتيانا بيليافتسيفا، أوكرانيا: دعم الكمبيوتر باور بوينت خلال دروس الرياضيات في المدرسة الثانوية وتشمل هذه المادة تحليلات دعم الكمبيوتر خلال دروس الرياضيات في المدرسة الثانوية. ويتمثل أحد الأهداف الرئيسية في ذلك في تحليل الدروس المخصصة لتطوير المفاهيم الهندسية الأساسية. كما يظهر تأثير تكنولوجيات الحاسوب الجديدة على عملية تحفيز البحث العلمي بين تلاميذ المدارس الثانوية. خايمي كارفالهو e سيلفا، جوزاكيوت كارلوس بالسا، ماريا جوزاكوت راموس، البرتغال: الإنترنت كأداة في إعداد معلمي الرياضيات في المستقبل وصفنا المشروع الذي تم تطويره مع مجموعتين من سبعة معلمين الرياضيات في المستقبل (الصفوف 7-12th) التي عملت في المدارس المختلفة (على بعد 30 كم). أرسلوا رسائل مع تقارير أسبوعية عن أنشطتهم وتعليقاتهم وملفاتهم إلى قائمة بريدية. واعتبرت هذه المشاركة مثمرة جدا، وأصبح هؤلاء معلمي الرياضيات في المستقبل أكثر وعيا من الأنشطة خارج روتينهم اليومي، وتطوير في الوقت نفسه مهارات الاتصال الخاصة بهم وتبادلوا أكثر من 90 رسالة و 50 ملفات (أساسا مع الأنشطة والامتحانات). واعتبر كل هذا المشروع جزءا هاما جدا من إعدادهم كمعلمين للرياضيات، وتبين كيف يمكنهم الحصول على أفكار جديدة ومحاربة عزلتهم باستخدام الإنترنت. وأظهر هذا المشروع أن الإنترنت أداة قوية جدا لإعداد المعلمين وينبغي أن تستخدم على نحو أكثر تواترا. نيل تشاليس، المملكة المتحدة: دور التكنولوجيا في التشخيص الرياضي في المملكة المتحدة وأماكن أخرى، فإن الوصول إلى التعليم العالي آخذ في الاتساع. الطلاب الذين يصلون في نفس الدورة يمكن أن يكون لها خلفيات رياضية مختلفة على نطاق واسع. وتنشأ هذه المسألة من تحديد الاحتياجات الرياضية الفردية للطلاب، ومتابعة بشكل مناسب، فضلا عن جعل الدورات المناسبة لهؤلاء الطلاب. نحن نبلغ عن مشروع في جامعة شيفيلد هالام لمعالجة هذه المسألة، وخاصة دراسة دور التكنولوجيا، سواء للتعلم والقيام الرياضيات، ويمكن ولا يمكن أن تقوم به. الجلسة العامة: أليسون كلارك-جيفونس روزالين هايد، المملكة المتحدة: وضع مخطط غني من الناحية التكنولوجية لعمل الأطفال الذين تتراوح أعمارهم بين 11 و 12 سنة في مجال الرياضيات من أجل التوصيل الإلكتروني هناك تغيير كبير يحدث في نظام التعليم الإنجليزي والويلزي فيما يتعلق باستخدام التكنولوجيا. عموما، في السنوات القليلة الماضية، انتقلت المدارس من استخدام مجموعة متنوعة من منصات الكمبيوتر، بما في ذلك، أرخميدس عادة إلى استخدام شبكات متوافقة مع عب من أجهزة الكمبيوتر الشخصية. وقد حدثت زيادة هائلة خلال هذه الفترة في عدد المدارس المتصلة بالإنترنت، رغم أن مستوى الوصول إلى المدارس يختلف. ومن الشائع الآن أن تجد الفصول الدراسية الرياضيات مجهزة واحد أو اثنين من أجهزة الكمبيوتر، وهناك مخططات لمساعدة المعلمين شراء أجهزة الكمبيوتر المحمولة للاستخدام الشخصي. كما بدأت المدارس تبني أشكال أخرى من التكنولوجيا. بعض المدارس لديها الآن بعض الوصول إلى ألواح الكتابة الإلكترونية وأجهزة العرض البيانات. وتساعد الحكومة على تأجيج هذه التطورات في استخدام تكنولوجيا المعلومات والاتصالات من خلال إدارتها للتعليم والتوظيف التي تنفذ مبادرات مختلفة، يرد وصف أحدها هنا. كان استخدام الآلات الحاسبة، وأربع وظائف والعلمية والرسومات، على جميع مستويات المناهج الدراسية مسألة للمناقشة كبيرة في انكلترا وويلز لبعض الوقت. كما أن المسائل المرتبطة باختيار البرمجيات وتدريب المعلمين على استخدام هذه التكنولوجيا هي أيضا أمور ينبغي النظر فيها. وفي نهاية أيلول / سبتمبر 2000، نشرت الاستراتيجية الوطنية للعدويات مشروع إطار لتعليم الرياضيات للمرحلة الرئيسية 3 (11 - 14 سنة من العمر). وينبغي أن يكون لهذا تأثير كبير على استخدام التكنولوجيا في تدريس الرياضيات كما أنه يحتوي على مثال على استخدام أجهزة الكمبيوتر (أساسا جداول البيانات والهندسة الديناميكية) وآلات حاسبة الرسومات. من أجل الاستجابة بشكل استباقي لهذا المناخ من التكنولوجيا المتغيرة، وقد كلفت وزارة التعليم والعمالة آلات البحوث بلك لتطوير السنة 7 (التلاميذ الذين تتراوح أعمارهم بين 11-12) مخطط عمل للرياضيات التي تستخدم على نطاق واسع من هذه التقنيات. ويتم تسليم المواد التي تشكل مخطط العمل إلى المدارس التجريبية ال 20 إلكترونيا. وقد تم تجهيز كل من هذه المدارس التجريبية مع 3 أجهزة الكمبيوتر الفصول الدراسية، وكمبيوتر محمول للمعلم، وسبورة إلكترونية، وجهاز عرض البيانات، و 15 الآلات الحاسبة الرسومات. من حيث البرمجيات، والمدارس لديها مايكروسوفت أوفيس، و جيوميتس لوحة الرسم، مسو لوغو، إيسيتيتش لاستخدام السبورة الإلكترونية، وبعض البرامج المكتوبة حسب الطلب. وقد تم تطوير المشروع لتحفيز الطلاب وإشراكهم، ويهدف إلى تقييم مساهمة تكنولوجيا المعلومات والاتصالات في رفع المعايير في تدريس الرياضيات. ويعد تطوير مواد لاستخدام هذا المستوى من التكنولوجيا في الفصول الدراسية تحديا حقيقيا، وهو إقليم غير محدد، وبالتأكيد لمشروع بهذا الحجم ومع هذا المستوى من التأثير على الصعيد الوطني. وتتاح الفرصة لتطوير أساليب التدريس من أجل الاستخدام الملائم للتكنولوجيا وتؤدي إلى تأثير حقيقي على تعليم وتعلم الرياضيات. وستدرس الورقة الخلفية لهذا العمل وتربط البحوث التي أجريت مؤخرا بشأن آثار أنواع مختلفة من الوصول إلى تكنولوجيا المعلومات والاتصالات في عملية التعلم. وسيضع الأساس المنطقي لتطوير هذه المواد ويدرس الآثار المترتبة على هذا التطور وآثاره. وستعرض المحاضرة العامة هذا العمل، فضلا عن عرض المواد التي أعدها المشروع وعرض بعض النتائج الأولية. أليسون كلارك-جيفونس، المملكة المتحدة: لماذا دغس هي أداة فعالة في تعليم الرياضيات العديد من المناهج الدراسية تدافع عن استخدام البرمجيات جيمويتري ديناميكية. هذا العرض سوف يوضح لماذا دغس هو مثل أداة فعالة في الفصول الدراسية الرياضيات، المتعلقة وجهات النظر الحالية حول كيفية تعلمنا في بيئة تكنولوجيا المعلومات والاتصالات. سيقترح مقدم العرض طرقا عامة يمكن من خلالها استخدام البرنامج لتعزيز التعلم من أجل الفهم. بيتر كوبر، الولايات المتحدة الأمريكية: تصميم المحتوى الأنظمة التعليمية المستقلة في تصميم وتنفيذ الأنظمة التعليمية للاستخدام عن بعد، تسمح بيئة التطوير الأكثر تطورا باستخدام محتوى الوسائط المتعددة في بيئة معبأة. في هذه الأنظمة، يكون كونتينينتيرفاس ملزما بالمحتوى في وقت الترجمة وقبل التوزيع. وكجزء من مشروع مشترك مع مختبر أبحاث فيلق المهندسين التابع للولايات المتحدة، قام الباحثون بالتحقيق في أساليب فصل الواجهة وحاوية البيانات عن المحتوى بطرق تدعم نهجا أكثر ديناميكية للحفاظ على عملة المحتوى وتوزيع المواد التعليمية الموزعة. وسوف تشمل جلسة العرض مظاهر التطبيق التدريب، وتطبيقات إدخال البيانات ونظرة على التدريب القائم وضعت من خلال النظام. جون كوسغراف، ايرلندا فيرماتس نظرية صغيرة للاحتفال بالذكرى السنوية ال 400 (في 17 أغسطس 2001) من ولادة بيير دي فيرمات سأقدم ورقة المسح - باستخدام القيقب - على نظيره الشهير قليلا. وسأعالج النظرية نفسها، وأقدم الأفكار المتعلقة بتطبيقاتها على فترات التوسعات العشرية، وحلول التطابق، واختبار بريماليتي، بولاردز ف -1 طريقة التخصيم، والتشفير المفتاح العمومي. وسأعتبر أيضا بعض الأسئلة المفتوحة المتعلقة فيرماتس نظرية قليلا. وسأطرح حديثي في جمهور عام غير متخصص. جان جاك داهان، فرنسا: كابري جافا: اتصال جديد وأداة تربوية 1. عرض البرنامج كابريويب: سأعرض كيفية إنشاء كابريجافا الصغير (ملف الإنترنت) بدءا من ملف كابري وما يمكن القيام به مع هذا أبليت (الرسوم المتحركة التلقائي والتلاعب في الرقم على شبكة الإنترنت). 2. أمثلة من المشاكل تحت كابريجافا: مثل المشاكل التي عثر عليها التي يمكن أن تكون مشتركة مع أشخاص مختلفين في بلدان مختلفة (صناديق سوداء هي مشاكل معزولة خاصة ولكن سوف تقدم الآخرين من مستويات مختلفة). 3. How to write an article using this tool in order to get a dynamic communication between us. Hans Dirnboeck, Austria The Evolvente-Curve of a Circle, Used for Gear-Wheels. You Need It Everyday Gear-wheels are an important chapter of Kinematic Geometry. The terms to construct or to plot the evolvent curve of a circle are given. The fundamental law of gearing is explained. On two wheels we fix two evolvent curves we proof that they can work as profiles of two gear-wheels. Special case: An evolvent curve fixed on a wheel and a straight line fixed on a rack are working as profiles. This gear-wheel mechanism You are using everyday in Your car, in the railway, the aircraft in Your coffee-mill etc. You need it and You need Geometry. DERIVE, drawings, models to visualize it. Alfred Dominik, Austria: Taylor Series and Finding Zeros with DERIVE and MATHEMATICA The meaningful use of the two Computer - Algebra - Sytems MATHEMATICA and DERIVE in the Calculus Curriculum for 16 to 18 year old students in Austrian Grammar Schools will be demonstrated with the help of Taylor - Polynomials, Bisection - and Newtons method. Specially prepared functions help the students to get better insights into basic ideas of Calculus such as approximation and limit. Additionally the influence of inital values to iteration - processes will be discussed. Plenary: Tommy Dreyfus, Israel: The construction of meaning for abstract algebraic concepts The teaching and learning of algebra, whether elementary, linear or modern algebra, seems to virtually cry out for computer support, for several reasons: A large variety of multi-representational tools are available, the heavier calculations can easily be taken over by the computer, and most importantly, appropriate software can be used to bridge the existing gap between the concrete and the abstract (see, e. g. Schwarz amp Dreyfus, 1995). Indeed, there are examples of success in using technology for students construction of meaning for abstract algebraic concepts but there are also examples of failure. In the lecture, I will examine a number of possible reasons for failure, including inadequate task design (Sierpinska, Dreyfus amp Hillel, 1999) and the ambiguity of representatives for mathematical objects (Dreyfus amp Hillel, 1998 Schwarz amp Hershkowitz, in press). I will conclude that there is no simple explanation. I will then make the point that in order to deepen our understanding of the relevant learning processes, a re-conceptualisation of abstraction is in order, as well as a research program that allows describing processes of abstraction. Such a re-conceptualisation will be proposed and a research program will be outlined (Hershkowitz, Schwarz amp Dreyfus, in press). Timo Ehmke, Germany: Geometria: A Tool for the Production of Interactive Worksheets on the Web With this contribution I will introduce the Java-Applet Geometria, a tool for interactive worksheets to be presented on web-pages. Worksheets generally contain a dynamic figure together with some kind of geometric learning content. This content is described by means of a script-language (GeoScript) which provides the possibility to construct a euclidian figure and also supports the analytical definition of points, vectors and curves. A special feature is the feedback given to the student, while heshe is interacting with the figure. A tutoring component enables Geometria to evaluating and commenting on the students answer. Hans-Juumlrgen Elschenbroich, Germany: Teaching and Learning Geometry: dynamic and visual A generation has grown up that may be far more visual than verbal. The state of mind of young mathematicians is not what it was fifty or hundred years ago. (Davis) Dynamic Geometry Software like Cabri II, Cinderella or Euklid-Dynageo offers new chances by using dragmode and loci to learn and to teach geometry in a visual and dynamic way. Classical ideas can be brought to life. DGS is not seen as a substitute, but as a complement to and an extension of the classic tools compass and ruler. Electronic worksheets will give a safe basis, which avoids lengthy phases full of mistakes and will support experimental and heuristic activities of the students. After some basic reflections about visual learning and teaching, well-tried examples of electronic worksheets and pre-formal, visual-dynamic proofs will be presented. Joachim Engel, Marcus Otto, Germany: Simulation and Modelling with Lisp-Stat: A Flexible Software for Teaching Statistics We illustrate how a simulation based use of computers supports conceptual learning in statistics. We focus on three areas of application: 1. simulation via bootstrap 2. modelling functional relationships between two variables that are corrupted by noise 3. demonstration of the central limit theorem. The basis is the programming environment Lisp-Stat. Bjoumlrn Felsager, Denmark: Through the Looking Glass: A glimpse of the Minkowski Geometry The Minkowski geometry offers the possibility of seeing well-known concepts from high school mathematics in a new perspective. The investigation of Minkowski Geometry requires the use of a Hyberbolic compass. This is introduced using Cabrii, which supports conic sections as a primitive geometric object. Thus the use of modern technology makes it feasible to investigate Minkowski Geometry in almost the same elementary way as Euclidean Geometry. Roger Fentem, UK: The impact of training for students on learning mathematics Training for teachers in the use of graphing calculator technology is widely accepted. To what extent are the training needs of the users of the technology addressed i. e. the students This paper introduces a research project designed to investigate the issues of technology training for both teacher and student in studying mathematics post 16. Attitude, relative achievement and practice are studied, recorded and analysed. Roger Fentem, UK: Investigation into Student Attitudes to using Calculators in Learning Mathematics In many countries curriculum designers, educators and examiners receive mixed messages about the role that should be played by information technology: imposition of severe restrictions to active encouragement of CAS. We present an international study exploring student attitudes to the use of technology, their training needs, and their ability in mathematics when learning in a CAS intensive but assessment hostile environment. Isabel Fevereiro Carmo Belchior, Portugal: Changing the Classroom Practices. The use of Technology in Mathematics Teaching Since 199798 the Department of Secondary Education, Ministry of Education of Portugal, has created a training teachers NET constituted by 80 mathematics teachers to improve meetings and promote training sessions and workshops with mathematics teachers in all secondary schools in Portugal. The aim of this project is to change the classroom practices according to the curriculum guidelines, which focus on experimental teachinglearning process, centred in the students themselves, in knowing how to do, and with a strong emphasis in the use of technology. Since then, after three years, the use of the graphic calculators in the classroom is generalised (the use of the graphic calculators is compulsory in the final national 12ordf grade exams). We will take a look at the final exams. Since our truing teachers are working with many different activities in the classroom we will take a look in some of these activities. Ruhal Floris, Switzerland: Evolution of mathematical tasks in a CAS-classroom We will propose a study on how technology modify the kind of work made by students and teacher in the lesson and the kind of mathematical objects discussed. We will analyse some new possible tasks such as studying equations as objects, or solving function interpolation problems we will also study the consequences of the constant use of CAS and graphic technology, tending to modify some didactical situations, with the production of complex outputs and discuss the ways and conditions to convert these outputs in learning objects. Plenary: Jean Flower, UK: Interactive web-based resources and a new perspective on algebra and geometry This paper will reflect upon the use of DaC (dynamic geometry and computer algebra software) in two contexts - two undergraduate Linear Algebra courses taught at different UK universities. The main questions of this strand will be considered in the light of this experience. It is hard to compare the two linear algebra modules and claim that one was more successful than the other. One covered more pure algebra topics, whereas the other included more applications of Linear Algebra. Both used DaC. One used Maple and JavaSketchpad, and the other used TI92s algebra and Sketchpad on the PC. The students on one module were mainly training to become teachers, whereas the students on the other were studying for a mix of maths degrees, heading for business. Is it necessary to achieve widespread use of DaC throughout a course for best benefits The students who had a wider exposure to Sketchpad in a range of modules over many semesters made better use of the Linear Algebra images than the students who were unfamiliar with DGS. How do the costs (time as well and money) of introducing DaC in a single module compare with the benefits Is it necessary to integrate DaC into assessment at the same time as its introduction to the teaching The students whose assessment included a Maple test learned to use Maple mainly for the purposes of completing the test, whereas the students with TI92s used them more widely to shortcut rote algebra. Use of the handheld technology was not required for successful completion of the course, but the TI-92s were used more widely. How can we tie in a DaC approach to a subject whose key texts take a more traditional approach There is a mismatch between the students experience of Linear Algebra in the classroom (and in the website) and the students experience of Linear Algebra from books. Does this contribute to confusion Can we make use of this contrast to deepen understanding of the different facets of a subject The use of DaC allows for revitalisation of some tough topics which were getting taught later on in a degree. Tasks which required intensive numerical calculation can now be completed quickly, allowing more space for understanding the results of the calculation. The use of technology itself can provide relevant applications for study (eg. computer graphics). Different approaches to proof and argument contrasts axiomatics (a traditional way in to Linear Algebra) with investigation (assisted by DaC). What is the relationship between working on the computer and working with paper and pencil This question is critical when introducing DaC into courses which maintain traditional assessment strategies like exams, where students may not have access to DaC. p Looking at the changing nature of algebra and geometry is like trying to gaze into a crystal ball. But we can have some fun looking there. Mathematics Teacher Development That Works Several characteristics make for an effective professional development program for secondary mathematics teachers. The program needs a clearly focused purpose that is relevant to the participants. It must have expert and stimulating presenters with participant involvement and reflection. Participants should create and implement an action plan, with ongoing support. Good facilities and organisation are important. U. S. examples will illustrate these key features. Ruth Forrester, UK: Data Collection and Manipulation using Graphic Calculators with 10-14 year olds A teacher researcher group at the Edinburgh Centre for Mathematical Education is currently investigating the use of graphic calculators in Mathematics classes for pupils aged 10 -14 years. One focus has been on the development of data handling skills. Activities have been devised where pupils use graphic calculators in the collection of data and its subsequent analysis. Classroom implementation has produced positive results. Evidence has been found of gains in understanding of statistical concepts attributable to the use of this technology. Positive motivational effects were also seen. The graphic calculators enabled the use of pupils own data and allowed the teacher to pace and vary the learning experience appropriately. Wolfgang Fraunholz, Frank Postel, Germany: A Computer Learning Environment in Linear Algebra using CAS MuPAD The Computer Learning Environment in Linear Algebra offers an introduction to Linear Algebra (vector space, basis, dimension, matrices, determinants, systems of linear equations, linear operators, dot product, vector product). Representing the development and the examples, the solution of exercises step by step and the controlling of solutions is done by the Computer Algebra System MuPAD. Important is also a three-dimensional graphic tool, which visualises vectors, vector algebra, linear equations, mappings in three dimensions. The talk will give aspects of math education (Wolfgang Fraunholz) as well as those of programming and software (Frank Postel). Nils Fruensgaard, Denmark: Danish experiences with technology in mathematics teaching in upper secondary school A co-operation between the Association of Mathematics Teachers and the Ministry of Education has resulted in building a new experimental curriculum, based on enhanced use of ICT, which teachers are encouraged to use in their classes in grade 10-12 in the Gymnasium. The old and new curricula are presented, with emphasise on the national written assignments and the intended use of technology in the experimenting classes. The general curriculum problems that ICT has created in teaching mathematics are discussed. Karl Josef Fuchs, Christian Kraler, Austria: Programming in the Age of CAS and the Algorithm as Fundamental idea in mathematics education The authors will concentrate on the basic question of the Special Group by taking How much Programming (knowledge skills) must a Mathematics - teacher have in the Age of CAS as their theme. Reasons for the motivation and necessity of this question for the process of teaching mathematics with new technology will be given. Different accents in defining the term of Programming will show that fundamental ideas of mathematics such as algorithm, function or modelling are essential parts of these terms. Further discussions will mainly focus on the idea of the algorithm and its importance as a connecting piece between mathematics and computerscience. The Role of the Graphic Calculator in Early Algebra Lessons This is a study of first algebra lessons at secondary school using the lettered stores of a graphic calculator to form a model of a variable. The calculator provides a tool for thinking and for building up concepts. In this paper is a discussion of what happened in the classroom, and how the calculator helped in the remediation of a specific misconception without any need for teacher intervention. There is also discussion of what ideas the students bring to the work, and how these ideas change during the lessons. Giuliano Gargiulo, C. DApice, R. Manzo, Italy: Mathematica and symbolic-numerical methods for solving first order ODEs The use of information technology in addition to traditional lectures affords a means to develop student intuition and curiosity, reaching in the same time a deep knowledge of the subject of study. The aim of this work is to show the didactic use of a Computer Algebra System, as Mathematica 4.04.1, to illustrate and compare different symbolic-numerical methods for solving first order ordinary differential equations (ODEs). In particular, we apply, relate and compare the built-in functions of Mathematica, the method of integration by series, the Picard process and the linearization method in solving some first order ODEs. Moreover, numerical solutions are compared with symbolical ones at the various stages of computation. This includes use of numerical methods (internally adaptive) to look for and analyse singular points for maximal solutions. Ernst Gebetsroither, Austria: Modelling Carbon Dioxide Pollution in Austria One of the major ecological issues world-wide is the increase on carbon dioxide in the atmosphere. At the international Climate Conference in Kyoto 1991 Austria has committed itself to reduce until 2018 the Carbon Dioxide emissions by 13, compared to the emissions of 1990. To fulfil this goal it is necessary to understand the national carbon cycle of Austria, an interdisciplinary team lead by Austrian Research Centre Seibersdorf (ARCS) has developed a system dynamics model of the Austrian Carbon Balance Cycle. The findings of this modelling are being used by political decision-makers in Austria. Ernst Gebetsroither, the co-ordinator of this project, will report about the experiences of this project. Luiz Carlos Guimaratildees, Brazil: Tabulae and Mangaba: Dynamical Geometry with a Distance Twist We report on the ongoing development of two complementary DGS, for plane and space geometry. The design briefs of both softwares were tailored bearing in mind the needs of distance teaching and Web communication. The current implementation is described in some detail, and we also discuss some of the issues that brought about the decision to engage in the project, as well as the implications for the technology driven teacher training program that provided the initial motivation for it. Stefan Gueldenberg, Werner H. Hoffmann, Austria: Leadership, Management and Management Control - a System Dynamics Approach The purpose of leadership is to create a viable organisation capable of development that is both internally guided and externally oriented. Normally leadership is understood as the capability of a person - the charismatic leader. In this manner leadership is given someone by birth and not teachable. In contrast to this personal and determined view we understand leadership as a capacity of an organisation, a human community, to create its own future and can be built by its members. Utilising a system dynamics interpretation of the term leadership, we aim to identify in our work the current challenges to companies from their environments, and toexplain the consequences of these challenges for company design and control. For a company to achieve sustained development, there must be a healthy proportion of growth and balance. The conclusion of our work is that system dynamics is a prerequisite for educating successful organisational leaders to help them to understand complex organisation and design viable structures. Samer Habre, Lebanon: The ODE Curriculum: Traditional vs. Non-Traditional - The Case of One Student A Traditional course in ordinary differential equations consists of tricks to find formulas for solutions with very little emphasis on the geometry of the solutions or on an analysis of the outcomes. Since differential equations are important in many fields, educators have come to believe that this approach is obsolete. With the advancement of computer graphics, it is now possible to offer a course on differential equations using a qualitative approach. This paper examines the two approaches as offered by the same instructor at the Lebanese American University in Lebanon. In particular, the point of view of one student who took the course twice using a different approach each time is presented. Results show that the qualitative approach is more appreciated, and that technology plays an essential role in the understanding of the material. Mary. S. Hall, USA Creating and Teaching Online Mathematics Courses As distance learning has expanded, so also has the use of the Internet. More and more we are seeing the expansion of course material to the Internet. What are the issues for teaching course material on the Internet What students will benefit from such opportunities These are some of the issues addressed in creating an online developmental mathematics course and other mathematics courses. This presentation will provide both resources and methods for teaching a course on the Internet as well as an emphasis on the new technologies becoming available. Several online mathematics courses will be used to demonstrate some basic forms of communication and evaluation that are necessary for a course to be successful. Andre Heck, Netherlands: Modelling Human Growth Many a pupil at secondary school asks oneself questions like Am I too thick or too thin, Am I short or tall in comparison with persons of my age, and What adult length may I expect to reach. To answer such questions one needs real data. We have used the recent Dutch growth study to create learning material for pupils in upper general secondary education (age 15-16 yrs.) to carry out practical investigation tasks. A mathematical highlight is the ICP-model that models length for age within millimetres. It is used in the medical literature and yet consists of growth models that are studied at school, viz. exponential growth, quadratic growth, and logistic growth. We shall present the learning material and discuss the classroom experiences. Andre Heck, Netherlands: A Practical Investigation Task with the Computer at Secondary School: Bridges and Hanging Chains Almost everywhere you can come across hanging chains and cables. Examples are necklaces, high-voltage cables, and cables that support a bridge surface. Do these cables all hang in the same mathematical shape The first thought of many a pupil will be: this is a parabola, isnt it In the computer learning environment Coach you can easily measure this on digital images. It will turn out that the parabolic shape quite often occurs with bridges, but that an ordinary chain does not hang as a parabola. Can you understand this We shall show that a key idea for solving the problem can be discovered by measuring digital images and that it can be theoretically explained with basic physics afterwards. It also leads to a simple computer model of hanging chains. We shall discuss our learning material and classroom experiences, and in this way present an example of how ICT and context situations can contribute to the realisation of challenging mathematical investigation tasks. Judith Hector, USA: Teaching Probability and Statistics via the Internet The author has taught a one-semester Probability and Statistics course via the Internet four times. The course is offered for university transfer credit at an American community college. The course is conducted totally online for students at a distance, but local students may meet for an orientation, midterm exam and final exam. From her experiences and research, the author discusses basic principles of teaching and learning mathematics on the Internet. Judith Hector, USA: Programming Principles for MathematicsEngineering Students The author has taught computer programming since 1970. She has developed an introductory programming course for mathematicsengineering students. Students develop structured programs on a computer using FORTRAN and the same programs for a TI-92 calculator. Students learn to program certainnumerical techniques such as Newtons method of root finding and Eulersmethods of solving a differential equation. Such techniques are available preprogrammed as black boxes in CAS. Guido Herweyers, Belgium Elimination of Parameters and Substitution with Computer Algebra Elimination of parameters and substitution with computeralgebra. Starting with the geometrical concept of parametric equations of lines and planes, we illustrate the method of elimination to obtain a cartesian equation. This elimination can be done in a direct and simple way by using the procedures solve and substitute (the basic algebraic manipulations of formulas) of a CAS. Without a CAS this method is difficult to realize by hand (e. g. solution of a system of two equations in a context with different letters). Therefore it was necessary to introduce in advance more elegant (but also more sophisticated) algebraic techniques like determinants. The result was that, for a lot of pupils, the meaning of the elimination process disappeared behind these algebraic manipulations. Later on in the educational process, we have the opportunity to show the equivalence and strength of the new algebraic techniques. These ideas will be illustrated in a few (geometric) examples. Iavor Hristov, Bulgaria: Model of deformations of fluid particles due to electric field A mathematical model of finite deformations of compound drop containing another drop due to electric field are obtained. The fluids are homogenous, incompressible and Newtonian. The cases of concentric and eccentric particles are investigated together. Creating a Professional Development Network Recent developments in T3 (Teachers Teaching with Technology) in England will be used as a case study to explore the setting up of formal and informal networks for professional development in the context of an increased emphasis from government on continuing professional development for teachers. The intention is to explore the creation of networks that are enabling and empowering for teachers and that provide teachers with the support and resources they need to take responsibility for their own professional development. Nicholas Jackiw, USA: Functions as First-Class Dynamic Geometry Objects The Geometers Sketchpad version 4.0, arriving Summer 2001, includes support for functions as first-class objects in Dynamic Geometry, allowing users to define, combine, and differentiate functions symbolically, evaluate them numerically, and plot them through a variety of coordinate projections. While in isolation, these capabilities have been long present in other mathematics technologies (e. g. graphing calculators and CAS packages), their meaning is altered by the rich possibilities of interaction and manipulation afforded by the dynamic geometry environment. In this talk, Sketchpads designer will summarize the research leading to these new developments, demonstrate some models of their classroom use observed in software field tests, and outline possibilities for how representations of functions as first-class dynamic geometry objects engage various strands of a secondary-level mathematics curriculum. Youngcook Jun, Austria Theorema-based TI-92 Simulator for exploratory learning One of the Theorema systems capabilities provides computing environment which can simulate the existing graphing calculator such as TI-92. Moreover, the deductive reasoning facility of Theorema allows the simulator to deal with propositional and predicate logic for pedagogical purposes. We present how to apply the use of such a simulator to help students explore mathematical ideas in terms of black boxwhite box principle. This experimental approach is demonstrated with our prototype by explicitly generating the sequences of calculator keystokes. Exploratory learning as a part of cretivity cycle is realized with algorithmic and logical empowerments built in the Theorema system. Henryk Kakol, Poland: Integrated Teaching Mathematics with Elements of Computer Science At present nearly all Polish schools have computer rooms well equipped while teaching mathematics generally is traditional. During school lessons chalk and blackboard are still teaching instruments frequently applied. What are the reasons for such a situation There are many of them. It will be list some of them. Special Programme of Teaching Mathematics with Elements of Computer Science in Gymnasium eliminates many of the above mentioned problems. It offers teaching mathematics and elements of computer science in the form of one thematic block. Jan Kaspar, Czech Republic: Programming as a tool for the precision Using the TI-83 graphing calculator I would like to demonstrate how programming requires precision in step-by-step description of mathematics tasks. Karl-Heinz Keunecke, Germany Curvature of Functions as a Limit A road sign Dangerous Curve can introduce to the problem. A car driving through a curve must not cut but osculate the road. For a short while, when the steering wheel is in a certain position the car moves on a arc of a circle. From this discussion all the expressions are available to define the curvature of a function by means of the radius r of the osculating circle as k 1r. We will realize the teaching unit using DERIVE 5acutes new features to enable the students producing their own notebooks combining text, graphs and calculations. Mark Klespis, USA: An on-going program of professional development program in hand held technology for instructors of prospective teachers The Mathematics Teacher Educator (MTE) program is an on-going professional development program of Teachers Teaching with Technology (T3) and is designed to assist US college faculty integrate technology into their mathematics content courses for prospective elementary teachers. The paper focuses on a collaboration of the MTE program with a similar NSF-funded program directed by the author. This collaboration began in 1998 and has provided a forum for 39 faculty interested in restructuring their mathematics courses for prospective elementary teachers. Data collected at the workshops indicate participant improvements in teaching and using technology. In May 2001, members of the original cohort and new faculty will participate in an updated workshop. Longitudinal data will be collected and included in the paper. Heiko Knechtel, Germany Mathematic with Graphic and Symbolic Calculators - Teacher Training in Lower Saxony, Germany History - organisation - contents of teachertraining in Lower Saxony: In Lower Saxony a new concept of teacher-training was developed from the mathematics advisers: Every math-teacher at highschool have to take part in 4 math workshops within 3 years. They should learn, how to integrate the new technology of the handheld calculators and dynamic geometry in their own math lessons. Interested teachers were trained within 2 years for math-multipliers. The math-multiplier-groups were divided in teams of two persons. Each team is responsible for six schools in their region. Each team focussing on special interests for each school and go ahead for four times with the groups. They will visit the colleagues in their own school and give several workshops there. Items of the workshos are handling with graphic and symbolic calculators and dynamic geometry developing units with the new technology basing on their traditional math lessons by their own. After testing their own units during half a year the last 2 workshops give them a view on new possibilities in math lessons, specially in advanced or real-world mathematics. Supplementary every year in each region there are Regional Tsup3-Conferences with a main lecture and up to 15 workshops all over the day. Mykola M. Kolodnytsky a. o. Ukraine Teaching Elementary Number Theory with a Software System In this paper we show how to teach and to solve some computational problems of elementary number theory including modular arithmetic using the software tool DSR Open Lab 1.0 designed and developed by the authors. We consider such computational problems as follows: to run the prime number test, to determine all prime numbers in some range (the sieve of Eratosthenes), to factorise a number into primes, to compute the GCD for a pair (or more) of numbers, to solve the systems of linear or polynomial congruences, i. e. polynomials modula m, to compute residue classes, i. e. modulo m, as well as the Euler phi-function, quadratic and power residues, reciprocal number modulo m, primitive roots modulo m, modular exponential, indexes, discrete logarithm, etc. We also give the comparison of the user interface implemetation of our software with the following: Maple V release 5, Mathematica 4 and DERIVE. The shown examples convince that the process of elementary number theory problem solving and teaching became easier now due to the visual interface of the presented software. Michael Kourkoulos, Marianne A. Keyling: Self-correction in algebraic algorithms with the use of educational software: an experimental work Our work points out that self-correction is a complex but fruitful activity concerning the learning of elementary algebraic algorithms. Pupils who have worked with an adequate software (laquoArithmraquo), both in Greece and in France, present a significant improvement of their strategies of localisation of errors, which are an essential element of the self-correction procedures. Furthermore, the work done led these pupils to a significant amelioration concerning the treatment of the examined algorithms. The software allowed teachers to be alone in their class (or in a half-class in the case of weak pupils) but nevertheless to offer adequate individual support to the pupils in their self-correction work, which is very difficult to realise in usual teaching conditions. Konrad Krainer, Austria: Innovations in Mathematics, Science and Technology Teaching (IMST 2 ) - First outcomes of a nation-wide initiative for upper secondary schools in Austria The bad results of Austrian high school students with regard to the TIMSS achievement test led to a research project where the results were analysed and additional investigations into the situation of mathematics and science teaching were started. As a consequence, a pilot project called IMSTsup2 - Innovations in Mathematics, Science and Technology Teaching - was launched in the school year 2000-01. The project aims at supporting mathematics and science teachers efforts for raising quality in learning and teaching. 126 Austrian schools participated in this project, about one quarter collaborated more intensively with the IMSTsup2-team and documented one or more innovations at their school. The concept, experiences and findings of IMSTsup2 will be presented and discussed. Krivsky Stefanie, Germany: Didactic innovations of teaching by internet While in the beginning the internet was designed by scientists for the purpose of exchanging information, it is nowadays more and more adopted by entertainment and commercial use. The internet project matheprisma (math prism) tries to combine these two objectives with the aim to simplify learning of complex mathematics using multimedia and entertainment aspects of internet. Matheprisma is a collection of modules addressing several mathematical questions on different educational levels. Technical and didactic possibilities of internet pages are presented by means of some examples of matheprisma-modules. Ewa Lakoma, Poland : On the impact of hand-held technology on mathematics learning - from the epistemological point of view Recently in the most of countries, mathematics became to be treated as one of the most important components of general education and general culture. Thus it is extremely important to enable students to develop their own mathematics as a language for communication. Thus, it is necessary to consider a process of mathematics learning from the epistemological perspective and to recognise students ways of mathematical thinking, especially when students use information technology. In this presentation I would like to show the main results of my educational research, concerning exploring and analysing a process of mathematics learning from epistemological point of view - at secondary and tertiary level - in which graphing calculators TI-83 and TI-92 are used as supporting tools. Duncan Lawson, J. Reed, and S. Tyrrell, UK: Extending a Mathematics Support Centre via the Web The Mathematics Support Centre at Coventry University offers support to any student in the University who wants help with any area of mathematics, statistics or quantitative methods. The support offered by the Centre is in addition to that routinely received in lectures, tutorials, seminars, problems classes, etc. The primary mechanism of support is one-to-one contact with students offered on a drop-in basis. This support is staff intensive and in order to optimise the use of staff time alternative methods of supporting students are continually under review. A recent development has been the introduction of a web-site for the Centre. This paper describes the background to the Mathematics Support Centre, the development to-date of the web-site and an evaluation of its use. Duncan Lawson, UK: A Discrete Introduction to Modelling In applications focused mathematics degree courses there is an understandable desire to introduce students to the ideas and practice of mathematical modelling at an early stage. However, many mathematical models depend on a level of mathematical sophistication, such as differential equations, which most undergraduates do not have on entry to university. Furthermore, it is often the case with such models that specialist mathematical software is required for the solution of the model equations. This combination of sophisticated mathematics and unknown software can be a considerable deterrent to new undergraduates. This paper describes a way of introducing a range of key ideas within modelling, initially without using any new mathematical concepts, and relying on software which is both familiar and not specifically mathematical, namely the spreadsheet. A short description is given of a number of models which are easily explored with spreadsheets. Josef Lechner, Austria Standardisierung der Normalverteilung - ein Anachronismus Waumlhrend der numerische Taschenrechner alle anderen Funktionstabellen aus dem Schulunterricht vertrieben hat, hat bis zum heutigen Tag die Tabelle fuumlr PHI(z) mit den Parametern 0 und 1 fuumlr den Erwartungswert bzw. die Standardabweichung bei der Normalverteilung in den Lehrbuumlchern uumlberlebt. Welche Ursachen hat dieser Anachronismus (traditionelle, technische oder andere) Was wuumlrde es bedeuten, auf die mehr oder weniger aufwendige Skalentransformation im Unterricht zu verzichten Carl Leinbach, USA Using a CAS to Teach Algebra - Going Beyond the Manipulations In this paper I will examine two of the basic theorems from a first year algebra class, the Division Algorithm and its corollary, the Remainder Theorem for polynomials. These two theorems are the basis of much of the teaching and learning in a first course in algebra. Unfortunately, most of the students efforts are devoted to factoring polynomials and finding their roots with little gained in terms of insight as to why they are performing these tasks. In this paper we will show how we can use these theorems to write expansions of polynomials about x a for a not equal to 0. Once this is done, students can learn about the idea of local linearity and tangent lines to the graphs of polynomials. I intend to develop two applications of these ideas. One is an application to pure mathematics, the other is to more real world settings. Pavel Leischner, Czech Republic: The collection of interactive solids figures and spatial situations in the Cabri - geometry The article gives information on the collection of interactive solid figures and spatial situations in the program Cabri-geometry. These aids would facilitate the teaching of stereometry at high and elementary schools. It is intended for the spatial imagery evolving. It should make students pass from experimental manipulations with the spatial situation to mental ones. Key Words: High school stereometry, spatial imagery, teaching with software, Cabri-geometry. Gisegravele Lemoyne, Canada: Cognitive and didactic ideas in ICT environments for the learning and teaching of mathematics Over the past few years, we have designed computer environments for the teaching of arithmetic, pre-algebra and algebra. We describe some of these to demonstrate how cognitive and didactic ideas are put into practice and how these environments engage both learners and teachers in non trivial problem-solving activities. The first environment is devoted to additive and multiplicative problems. Three different tasks were planned: construct an iconic representation of a problem, using the tools in the environment write a mathematical sentence that corresponds with an iconic representation of a problem write a problem that corresponds with a mathematical sentence. In the second environment, teachers have access to a calculator and can create problems by specifying numbers and operations and then choosing on the key pad of the calculator which keys will be non functional. Each subgroup of students receives specific calculations. The third environment consists of a task of abstraction of properties and characteristics of numbers and operations Auxencia Limjap, Philippines: Current Educational Theories amp New Tech: Development of a Training Programme for Math Teachers in the Philippines Reform movements on mathematics education in different parts of the world point out to the need to adopt a cognitivist view of instruction that focuses on the nature and process of mathematics learning. Proponents advocate constructive learning and gear teaching towards the development of meaningful quantitative thinking. They adhere to the social origins of cognition and situate learning in realistic settings. They harness technology as a learning resource that provides both context and support for meaningful problem solving activities. Consequently, learner centred educational theories proliferated with the advances in educational technologies. These developments in pedagogy and didactics pose a big challenge to school mathematics teachers especially those who have neither experienced the constructive process of learning mathematics in the classroom, nor employed the current educational technologies. Wolfgang Lindner, Germany: The Digraph-CAS-Environment and Misconceptions around Matrixoperations A longtime research at the University Duisburg, Germany, studies the impact of CAS on the belief structur of high school students and on the development of conceptions and skills of Elementary Linear Algebra with special consideration of animated visualisations and algorithmic semiautomations. The design of a Digraph-CAS-Environment (realized in MuPAD) is shown, which represents e. g. airline connections in an informal-visual way. The usual matrixoperations on the quadratic adjacency matrices are introduced and programmed to enhance understanding. Afterwards the extracted concepts and intuitions are transfered to rectangular matrices and the effect of this singular local perturbation of the individual knowlege net is studied. We compare the handling of misconceptions by the students with and without CAS. Alex Lobregt, Netherlands Introducing Fourier Series with DERIVE In Electrical Engineering Courses functions such as the square wave Sq(t) and the sawtooth Saw(t) are frequently used. These periodic functions may well be approximated by a so-called Forier Series. In a workshop we will present some examples leading to an application, which can be shown by means of DERIVE as a first step in the filtering theory. Marie-Theacuteregravese Loeman Belgium: To learn from and make history of maths with the help of ICT Results from the EEP Comenius Action 1. The history of some aspects of mathematics like: history of mathematical persons, symbols, algorithms. Looking through different aspects of history of maths, in co-operation with people from other nationalities and cultures, convinced our students that maths, having its special common language and symbolic notations, has no boundaries. Digging in history of maths and working cross-subject ( English, religion, philosophy, chemistry, geography, physics. ) revealed to them that as it comes to solve a problem, not only the solution is to be appreciated but certainly getting to know a nice, perhaps different and original way of reasoning can be a source of inspiration for the scientist being superior to the machine. In addition they were encouraged to learn from the stronger elements in each partner country. Victor Lysytsya, Ukraine: University level Geometry Course and DG Computer experiments within the course of Analytical Geometry are suggested. This course is taught at the Department of Mechanics and Mathematics of Kharkov National University. The most interesting are the tasks devoted to the geometrical sets of points on the plane. The experiments are constructed with the help of geometrical packet DG, which has been worked out at Kharkov State Pedagogical University. Supervision of students projects This paper concerns supervising students projects in information technology. Maple and a unified approach This paper will discuss the use of Maple in teaching Linear Algebra and Calculus as a unified approach. Technology and History of Mathematics This paper will discuss some aspects of using technology in teaching history of mathematics. Eoghan MacAogain, Ireland: A CAS-index applied to engineering mathematics Ps A CAS-index is applied to a set of first year university engineering mathematics examination papers the results are analysed. The CAS-index is an index of suitability its purpose is to try to answer the following question: given a mathematics examination paper which was written for a CAS-free environment how suitable is that examination paper for use in a CAS-supported environment Tom G Macintyre, UK: A CAS project carried out in Scotland with 16-17 year olds using TI-92s This study explored the impact of using hand-held technology throughout a course of study in a year 12 mathematics course - leading towards the Scottish Higher Grade. Students in the study sample had dedicated access to Texas Instruments TI-92 calculators, utilising the built in Computer Algebra System (CAS) as they developed their knowledge of the various components of mathematics studied. Both quantitative and qualitative data was gathered from the study sample students and teachers, who were based in three secondary comprehensive schools. Additionally, data was gathered from the three paired-control groups, providing evidence of algebraic ability at the start and end of the period of intervention. Performance in algebraic skills was of particular interest in this study, ascertaining whether extended use of technology had a positive or negative impact on students abilities. The quantitative findings, taken from the two assessments administered at the start and end of the one-year course, demonstrate a significantly better performance in the study sample compared with the control group. This affected performance in items that were common to both assessments, resulting in a 7 increase in the study sample compared to the control (p0.004). A similar trend was noted in new items that assessed mathematics studied during the course of the year taking the base level of performance into consideration there was a 5 increase in the study sample compared with the control (p0.046). Some underlying reasons for these differences in algebraic ability are explored. The discussion includes consideration of: the teaching approaches promoted by the staff the impact of mathematical rigour and syntax demanded by the technologies the emphasis on equivalence when interpreting screen displays and the general motivational effect that dedicated access to the technology has had on the students in the study. A number of questions remain, for current debate and future research into the use of a CAS in mathematics education. Katherine Mackrell, UK: The role of dynamic geometry packages in visualisation and animation This session will comprise a report of discussions held at the CabriWorld conference in Montreal in June 2001 regarding the use of Cabri-Geometre to create interactive teaching materials using visual imagery and animation to introduce mathematics from a wide range of areas. Giora Mann, Nurit Zehavi, Israel: Virtual Experiments and Probability A good model in probability must agree with observations. It is not practical to perform the real experiment many times. In a CAS environment we can perform a virtual experiment many times with relative ease. This changes modelling in probability to be twofold - programming a virtual experiment which controls the traditional modelling. Robert Mayes, USA: Cinderella: Software Tool for Euclidean and Non-Euclidean Geometry Although axiomatics account for a small part of the current boom in geometric research, the study of the axiomatic approach dominates the geometry taught in high school and college. The result is a curriculum where the geometry of plane figures is developed from a very narrow point of view. Students view geometry as an intellectual game of proof that has little or no relation to the real world. In addition, many students do not see a connection between geometry and other areas of mathematics. If teachers present solely an axiomatic approach, they will propagate this approach among their students. The outcome is an isolated and outdated geometry course that serves to turn students off, rather than demonstrating the beauty and utility of geometry in our world. Breaking away form the current narrow curriculum provides for a variety of societal and mathematically desirable goals. Modern Geometry should aspire to attain some of the goals recommended by the NCTM in the Curriculum and Evaluation Standards for School Mathematics and the NCTM 1987 Yearbook: Learning and Teaching Geometry, and by COMAP in Geometrys Future. Michael McCabe, Ann Heal, Alison White, UK Computer Assisted Assessment of Mathematical Proof Proof of Computer Assisted Assessment. An Integrated Approach to Higher Level Learning using Group Response Systems and On-Line Assessment In the School of Computer Science and Mathematics at the University of Portsmouth, computer assisted assessment (CAA) has been used successfully in support of maths teaching for almost 10 years. CAA is most commonly used for first year university modules, where the numbers of students are greatest and the topics covered are basic. This leads to the common conception that CAA is only appropriate for low-level learning. Mathematical proof is a topic which students find difficult to grasp and involves a higher level of learning. Traditional exam questions on proof are time-consuming to mark, but CAA can provide an efficient and effective alternative. The speed and accuracy of marking objective questions and the ability to give immediate feedback are among its obvious benefits. It remains to demonstrate that CAA can generate results equivalent to those of a written, hand-marked examination. We will explain how this has been achieved: by carefully designing test questions and considering learning objectives by exploiting both on-line assessment and group response systems (also referred to as an audience (or class or personal response) system by integrating both public and private practice of CAA into learning by analysing the results of computer marked exams Claus Meyer-Bothling, Germany: Thinking the Unthinkable - Understanding 4 Dimensions The existence of a fourth spatial dimension is confirmed by the Theory of General Relativity. Furthermore some simple properties of 4-dimensional objects, say of a 4-D-cube, can be deduced by analogy. The 3-D-projections of such objects can even be illustrated. Although we can state the properties of a 4-D-cube, we cannot picture the object itself. Our brain is not equipped to do that - following todays accepted wisdom anyway. My claim is that with the aid of modern resources we will probably be able to overcome this obstacle: With tod ays technology of illustration it should be possible to train our perception in such a way that we will be able to imagine 4-D-bodies. Claus Meyer-Bothling, Germany: More is more More is less Does IT really improve the educational process, or does it merely get in the way of communication between teachers and students Why do so many teachers persistently neglect or even refuse the use of IT in their day-to-day teaching practice - doesnt the great number of successful pilot projects prove that IT enhances the educational process Drawing mostly on examples from Schools of Baden-Wrttemberg, and reporting on good as well as poor practice, I shall try to provoke a search for criteria that will be continued in our following sessions. Criteria that are sufficiently complex to be useful, but sufficiently simple to be practical, in order to distinguish between success and failure with IT in teaching. Eva Milkova, Milan Turcani, Czech Rep.: Integration ICT into teaching and learning the subject Discrete Mathematics ICT enables new approach to the education of various subjects, also of mathematics. The education with help of visualisation is interesting and more understandable. Because our faculty disposes with good and modern equipment and there are several students who are able to prepare nice programs, we decided to improve lectures of the subject Discrete mathematics with help of students teaching packages. In our article three programs developed by students as part of their thesis will be briefly introduced. Kent Neuerburg, USA: Introductory statistics with spreadsheets Spreadsheets are ideally suited for use in an introductory statistics course. These programs have the ability to handle large amounts of data and are easy to use. As an added benefit, a working knowledge of spreadsheets is a marketable skill for many students. We will focus on our experience in using spreadsheets to teach an introductory statistics course. In section one, we consider the pedagogical strengths and weaknesses of using spreadsheets in statistics. In section two, we discuss the computational strengths and limitations of spreadsheets. Finally, in section three, we provide some resources for real data and offer suggestions as to how to integrate these data into the course by demonstrating a few applications of spreadsheets to descriptive and inferential statistics. Walther A. Neuper, Austria: What teachers can request from CAS-designers The basic functionality of computer algebra systems (CAS), increasingly introduced to math classes, is considered not yet optimal for education: CAS show up with the final result in one go, and do not show their built in knowledge. concept for re-engineering the interactive features of CAS is presented from the users point of view: An example session illustrates what a teacher (and a student) can request w. r.t. the assistance in modelling and specifying a problem, and w. r.t. the user-guidance in stepwise solving a problem. Brief explanations point out, how the concept presented makes the example session work and tasks for teachers are mentioned. Erich Neuwirth, Austria: The spreadsheet paradigm as a new mathematical notation One of the fundamental properties of spreadsheets is creating formulas by relative an absolute references. These references represent spatial relationships, and therefore mathematical structures are represented visually and geometrically. Some exaples (e. g. from combinatorics and difference equations) will demonstrate how using these representations as conceptual tool can help in not only performing calculations in a very user friendly way, but also in gaining mathematical and structural insights. Erich Neuwirth, Austria: Let the spreadsheet throw the dice - Spreadsheets as Monte Carlo simulation engines Monte Carlo simulation (using computer generated pseudo random numbers) is an extremely helpful tool for illustrating concepts in probability and statistics. It is surprisingly easy (and surprisingly unknown) that this kind of simulation can easily be done with spreadsheet programs. We will show some simple examples from probability and some moderately advanced examples from inductive statistics (testing and estimation) to demonstrate how simulation can help getting the feeling for randomness convergence of frequencies to probabilities. Hitoshi Nishizawa, Japan: Remedial Education of Quadratic Functions Using a WWW-based On-line Exercise System The method and the effectiveness of remedial education using a WWW-based on-line exercise system are reported. The system displays a graph of a quadratic function and requests the student to express it in a symbolical expression. Six students were selected to attend the remedial course using the system. Although they used only one formula to express the graphs before the exercises, they have extended the variety of their expressions through the exercises. Vladimir Nodelman, Israel: Parametric nature of mathematics objects and computer environment Although the simplest mathematics objects may be considered as based on parameters. Most of parameters are numeric. In computer software it is a regular task to implement numeric input. The problem is in: visually discrete nature of an input box entry opposite to continuity of most mathematics notions parameters, not friendly interface with static changes in correspondence to entered values. We present an approach which let the student DYNAMICALLY enter and change parameters in not pure numeric way, even prepare such input by himself in order to analyse parameters rule and mathematics objects behaviour. Plenary: Walter Oberschelp, Germany: Chances and limits for teaching in the information age - human mind models and society demands Successful IT-based teaching requires motivation, understanding, training and didactic sugar. The main problem is to adapt the problem structure to the intellectual structure of the learner and to his needs. Moreover there must be results which are useful for the society. We experience more and more, that the charm of having huge information resources e. g. via internet is only temporary: The present IT scratches only the surface of the human and social demands. The main need of man is not the consumption of news, but production of and interaction with personal signals on a reliable basis in order to be sure of ones own uniqueness. Surfing for information through open and heterogeneous nets will loose importance against new types of devices, which guarantee, e. g. security of transmission, legal control of transactions and semantic reliability of information. The task to keep the society in good order is incompatible with unrestricted informational liberalism, and the society needs more than a netiquette without obligations. New problems for jurisdiction arise: Information crimes cannot be judged by simply counting bits like peas. Some epistemological problems which are connected with the concept of information are discussed. And the realisation of a global justice will have to be recognised as one fundamental basis for the global society. In particular, we investigate, how math-learning will have to develop: The special problems of math-teaching are the alienation by formalism, the lack of personal appeal and the somewhat metaphysical nature of mathematics, whereas its pragmatic value is often invisible. Since mathematical ideas are often very compact, the abundant information of the internet is hard to combine with mathematical thinking. And yet, mathematical teaching establishes useful tools for the complex existence in the global society. We exemplify problems in private and global economy and in our real physical world and discuss essential and obsolete material. We sketch, how methods for self-guided instruction may be improved. But we emphasise, that, due to the anthropological situation, personal instruction and care will become even more important in the future. The satisfaction of really understanding an argument from the scratch and the experience of responsibly solving problems without the assistance of non-transparent tools will become a source of creativity and a well accepted motive in the education of independent and mature citizens. Regis Ockerman, Belgium: Probability simulation with TI-83 Taking advantage of the possibilities of the TI-83, its easy to do simulations, dealing with problems of probability. In this workshop, we will use programs for those simulations. This will be done in a way, that you can also use these things in class. Tatyana Oleinik, Ukraine: Project on critical thinking development using technology This paper represents the results of special courses given to undergraduate teacher students of laquomathematics-computer scienceraquo speciality. A general problem of its study is understanding the possibilities of technologies for realisation of ideas of Project on Critical Thinking development. The most interesting and significant aspect of this study was modification of views on the essence and kinds of teaching and learning activity. Obviously it is necessary to modify curricula and methodical frameworks which should focus to formation successful learners. So and why CAS like DERIVE and dynamic geometry software like DG are good medium for encouragement of pupils interests and reflection. Besides new standards of the mathematics education require to understand how meaningful classroom dialog can stimulate collaboration of teacher, students and software. Technology as a Vehicle for Updating Middle Grades Content and Pedagogy Technology has certain unique capabilities that support the learning, doing, teaching, and assessing of mathematics. Accepting that these capabilities are ever changing as new tools are developed, the design of innovative and effective professional development programs for motivating and inspiring the current and next generation of mathematics teachers is a continuously evolving and stimulating endeavour. A description of the guiding principles, planning and development phase, and initial implementation and evaluations efforts that support technology training from a slightly different perspective will be presented. Guenther Ossimitz, Austria: System Dynamics modelling: a new perspective for math classes - An introduction for all who are interested in this field In this presentation I will give an introduction to the topics of the working group System Dynamics and Systems Thinking. I will address the following issues: What is systems thinking What are the basic ideas of System Dynamics Modelling Can Systems Thinking Systems Dynamics be a topic for math classes SDST: a section in Austrian Mathematics curriculum Results of empirical studies concerning SD ST in math classes. Guenther Ossimitz, Austria: Practical Examples for Teaching System Dynamics in Mathematics Classes In this presentation I will give an overview about some practical examples of teaching System Dynamics Modelling in Math Classes. Each example will include some application context. I plan to present some of the following examples: A variety of simple growth models: linear, exponential, logistic, overshoot and collapse. A homeostatic feedback model and how simple time delays may cause even an elementary model to run into (deterministic) chaos. Population dynamics: development of the age-structure in Austria, problem of an over-aged society A model of balanced age structures of faculty staff: how to keep a healthy relation between assistant, associate and full professors over a longer period of time The ecological balance between deer and mountain lions in the Kaibab Plateau (USA) and how human protection of the deer induced their doom. Marcus Otto, Joachim Engel, Germany : Design and Use of a Computer Language for Teaching Mathematics - Some Examples from Statistics During the last years, we designed a computer language and used it in mathematics education. Our aim was to establish a tool for learning and doing mathematics. The language can be shaped to meet the needs of a course. Besides using such a language for algorithmic purposes, one can create its own mathematical structures based on their features, relations and operations. Students can use this to investigate the concepts presented in a course. Taking concepts from probability and statistics as examples, we illustrate how to incorporate our language into mathematical teaching. Bronislav Pabich, Poland: Close your eyes and imagine that you are connecting the midpoint of a cube with its vertices by line segments, creating in this way six congruent square pyramids, which will completely fill this cube. Now duplicate each of these pyramids by reflecting each of them on the plane given by its base. You get now 6 square pyramids positioned onto the faces of the cube outside. The cube together with these six pyramids perform a new polyhedron. Draw this polyhedron in that way you can imagine it. Then answer the following questions: How many vertices, faces, edges does have this new polyhedron Which kind of polygonal shapes are its faces of Are its faces congruent Is this polyhedron a regular one Whats its volume (Compare the volume of this polyhedron with the volume of the cube in regard with the method you did create it.). John Pappas Greece: Integrating Mathematics, Physics and Interactive Digital Video Previous research on Digital Interactive Video Technologies (DIVT) is limited to the domain of kinematics and graph interpretation in particular. This pilot study is part of a full-scale research that aims to extend the field of investigation using Digital Video Technologies as a connecting link for the Integration of Mathematics and Science. Five students participated in this study, which consisted of two parts, one without and one with DIVT support. The analysis of data gathered indicate that being able to manipulate the reference frame in the environment of the DIVT software and notice how it affects co-ordinates, graphs and equations improves the students conceptual knowledge on this subject, in two levels: Students realise that there is a dynamic linking of the reference frame position and orientation and the way that graphs and the matrix of co-ordinates look. By bringing the reference frame to particular positions of special interest, such as positioning one of the axes to be parallel to an inclined level, they can deal with their misconceptions and gain a better understanding and insight to the role of a co-ordinate system. Pavel Pech, Czech Rep.: Cubics and quartics on computer In basic courses of geometry at universities are mainly linear and quadratic objects studied. Using computers enables us to include into this courses also objects, which are described by an algebraic equation of the order higher than two. With the co-operation with the students of the Pedagogical Faculty at the University of South Bohemia the software has been developed by means of which cubics and quartics (and conics as well) can be mapped in a high quality. Valentyna Pikalova, Ukraine: Learning Explorations and its DG Support in Geometry Course for Secondary School The article includes the analyses of DG support in geometry course for secondary school. As a result the Dynamic Demonstrative library was developed. It includes sketches for learning explorations in geometry. This library is recommended to use in geometry course by the minister of science and education of Ukraine. The attention is also paid to the methodological questions of implementing learning explorations in secondary school curriculum. Neil Pitcher, UK: Evaluating the Effectiveness of Computer-Based Learning in Mathematics This session will discuss effective ways of integrating computer-based learning environments into university Mathematics courses. The system Mathwise will be used as an exemplar. Mathwise contains materials both for learning and for assessment. Such a system needs to be used carefully if it is to promote good study skills. Different teaching methods will be examined and some evaluation results presented. Rein Prank, Eno Tonisson, Estonia: Computers in School Mathematics - a pilot training program for Estonian Mathematics teachers Most of the software for the national schools computerisation program called Tiigrihuumlpe (Tiger Leap) has been acquired in such a way that the programs are available to allmost of the schools in Estonia. This will also simplify the training of teachers. Each county has a well-equipped pilot school, which shall organise local training and consultation for teachers of different subjects. This report describes the training cycle (9 sessions with 144 hours plus homework in the scope of more than 300 hours) conducted for 40 teachers in 2000. The cycle consisted of thematic modules based on special packages (StudyWorks, dynamic geometry, computer algebra systems, graphing functions, proofs in geometry, probability theory and statistics, spreadsheets, testing software, Internet and distance education tools) and the final integrative module on the use of computers. Pavel Prazak, Czech Rep.: Software Maple and Matlab in teaching of ordinary differential equations Matlab and Maple are the powerful interactive numerical computation programmes. They have efficient built in routines enabling wide variety of computations. They have also easy to use graphical commands to make visualisation available. In our contribution we will focus on selected possibility of using symbolic calculations, numerical and graphical methods for support and illustration of the subject of ordinary differential equations and outline various possibilities of visualisation of the solutions of these equations and show the samples of application of above mentioned problems Stefan Priselac, Nancy Priselac, USA: Technologically Presented Learning Material: The Communiversity Project in Maryland, USA. The presentation is multi-media in nature and can last from fifteen minutes to one hour depending on the allocation of time. The Communiversity at Garrett Community College provides diverse ways to deliver training, courses, programs and interaction across the globe as we redefine access from set time to anytime and from one place to many places as we create a new future in education. Wolfgang Proumlpper, Germany The TI-8992 as a Tool for Analytic Geometry The CAS calculators by Texas Instruments seem to be primarily suited for algebra and calculus at a first glance. The home screen menus give special emphasis to operations like factor and comDenom or limit and taylor respectively. For problems that typically appear in Analytic Geometry assistance is scarcely found. Solving vecshytorial equations can only be achieved after a large-scale (and by that faulty) rewriting into systems of equations or into matrices. Functions of vector algebra are not available in the home screen but must awkwardly be looked for in a catalog. Texas Instruments however took care for a way out of that dilemma when designing the operating system. The user can easily create customized menus and complete not available functions by proshygrams of his own. In the contribution a menu together with some desirable functions is presented and shown how it can be put into action for solving problems that usually occur in classical Analytic Geometry. Chantal Randour, Belgium: Cabri and anamorphoses Des eacutelegraveves de 17-18 ans ont traiteacute le problegraveme des anamorphoses, tant perspectives que celles utilisant des miroirs. La principale source utiliseacutee est La Perspective Curieuse du Pegravere Niceron (1652). La litteacuterature peu abondante traite uniquement ce sujet sur le plan analytique. Nous avons preacutefeacutereacute utiliser la geacuteomeacutetrie descriptive pour concevoir des constructions simples pouvant ecirctre ensuite communiqueacutees agrave Cabri. Les eacutelegraveves ont ainsi reacutealiseacute des anamorphoses perspectives, coniques, cylindriques et pyramidales. Le travail matheacutematique sest accompagneacute dune recherche artistique en bibliothegraveque, dans les museacutees et sur internet. Un CD-rom (en power-point) montre quelques extraits de cette recherche. Une exposition des travaux a eu lieu dans leacutecole. Je me propose dexpliquer les diffeacuterentes figures Cabri creacutees pour ce travail et de montrer le diaporama (- 20 min.) reacutealiseacute. Quelques modegraveles danamorphoses reacutealiseacutees par les eacutelegraveves seront visibles, ainsi quun pantographe (Scheiner-Parreacute) permettant de reacutealiser un type particulier danamorphoses coniques. T adeusz Ratusinski, Poland: The role of the computer in discovering mathematical theorems Pedagogical University, where I work, prepares mathematicians for being mathematics teachers in the future. The pre-service teachers ought to be educated enough to work in a modern school. In this paper I would like to present my observation I made during my classes with Four Year mathematics students (approx. 22-year-old). The students were supposed to discover, using computer, some properties of the monotonic functions. I would like to show the results the students work and also a few characteristic errors they made formulating mathematical hypothesis. Plenary: Eugenio Roanes-Lozano, Spain: Co-operation Between Dynamic Geometry Systems and Computer Algebra Systems - Investigating, Guessing, Checking and Proving with the computer Computer Algebra Systems (CASs), like Maple, Derive, Mathematica, Axiom, Macsyma, Reduce, MuPad. are specialised in exact and algebraic calculations. They use Exact Arithmetic and can handle non-assigned variables (i. e. variables in the mathematical sense, not in the usual sense in Computer Science). Many extensions like symbolic differentiation and integration, linear and non-linear equation and polynomial systems solving, 2D and 3D plotting. are usually included too. II, Cinderella, Euklid, Dr. Geo, WinGeom. are specialised in rule and compass Geometry. The adjective dynamic comes from the fact that, once a construction is finished, the first objects drawn (points) can be dragged and dropped with the mouse, subsequently changing the whole construction. They usually incorporate animation and tracing too. Unfortunately CASs and DGSs have evolved independently. Some CASs like Maple include specific and powerful packages devoted to Euclidean Geometry, but no CAS has incorporated Dynamic Geometry capabilities. On the other hand, Dynamic Geometry Systems cant handle (at least from the point of view of the user) non-assigned variables. Therefore, what can be saved from a DGS is only live graphic (to be read by the DGS), a geometric algorithm (script or macro, to be interpreted by the DGS) or a dead (fixed) graphic in one of the standard graphic formats. More precisely, what is missing in the DGSs is the possibility to handle and export parametric data about the plot: co-ordinates of points (allowing parameters as co-ordinates), equations of objects (allowing parameters as coefficients), length of objects (depending on parameters). Some DGSs (like Cabri Geometry II or Cinderella) include theorem-checking capabilities. This theorem-checking is based in altering the initial data: they find counterexamples if the result is false and suppose that the result is true if they find no counterexample (i. e. they are not proofs from the mathematical point of view). This lack of co-operation is more surprising in cases like the TI-92, where both technologies are simultaneously available. A straightforward application of this co-operation would be to treat with the computer the whole mathematical process of discovery (or re-discovery): Investigating - Guessing - Checking - Proving. The talk will begin presenting an overview of the main capabilities of CASs and DGSs. A basic introduction to Automatic Theorem Proving in Geometry (Groumlbner bases method and Wus pseudoremainder method) will follow. The missing co-operation between CASs and DGSs will be detailed afterwards. Finally, the (ideal) whole mathematical process of discovery mentioned above will be presented. All steps will be illustrated with adequate examples. Jarmila Robova, Czech Republic: Graphic solutions of equations and their systems The contribution deals with using graphing calculator TI-83 in teaching of algebra in secondary school. Several techniques of graphic solution are presented (geometric representation of problems, boolean function, graphic substitution). Ana Rosendo, Jaime Carvalho e Silva, Portugal: Computers and calculators in the preparation of future mathematics teachers - an experience We will describe how future mathematics teachers are being prepared to use technology in mathematics teaching (at the Mathematics Department of the University of Coimbra) Anna Salvadori, Primo Brandi, Italy: A modern approach of limit process A new approach to limit process is proposed. The aim is to drive students from perception to usual epsilon-delta definition gradually. This path involves the three fundamental aspects: geometric, numeric, algebraic. To supply the graphic support a software ad hoc is implemented. Susanne Saminger, Austria: IMMENSE - a tool for visualization and mathematical experiments Csaba Saacutervaacuteri, Mihaacutely Klincsik, Haacutemori Ildikoacute Perjeacutesineacute, Hungary : How can we combine the CAS with authoring system tools to create a flexible learning environment Using Maple CAS as a powerful mathematical tool and the Toolbook Instructor object oriented authoring system we can create new learning environments. We illustrate with case studies the step by step learning methods within an example and from the easier examples towards the complicated ones. With these new methods the user can be focus, concentrate on the local and the global know-ledges together. Our applications particularly applicable via Internet and local network, too. Ralf Schaper, Germany: Mathematica graphics in the internet An extended version of LiveGraphics3D will be presented. Franz Schloumlglhofer, Austria: Teaching System Dynamics Modelling in Secondary Schools: The Teachers perspective In this presentation the following issues will be addressed: What are the basic ideas of the didactics of System Dynamics What aspects of math teaching are involved when teaching system dynamics What are the main ideas of the section Investigation of interrelated Systems (Untersuchung vernetzter Systeme) of Austrians math curriculum at 11 th grade for a science-oriented subtype of high-school (Realgymnasium) What are the experiences with practical teaching SD in math classes Karsten Schmidt, Germany The Use of CAS in the Thuringian School System: Present and Future Based on a recent survey carried out in all 450 secondary schools in the state of Thuringia, Germany, the following questions will be investishygated: Which level of computer equipment is available for classroom use Which kinds (simple scientific graphical symbolic) of pocket calculators are used in which grad es Does the school possess a license for a CAS In a second part of the survey, the person filling in the questionnaire is asked to give some of hisher personal attitudes, which will also be analysed in the presentation: Which kinds (simple scientific graphical symbolic) of pocket calculators should be used in which grades Which knowledge does heshe have of symbolic calculators and CAS What are the advantages and disadvantages associated with the use of symbolic calculators and CAS in the classroom Alfred Schreiber, Germany: Project ZERO: Developing Online Material for Mathematics Teacher Education This paper reports about a project dealing with the conception and production of supplementary learning material for mathematics teachers. It surveys the various types of courseware-modules presented herein online (e. g. dynamic geometry, computer-based-training-like frames, paper-and-pencil-exercises), and discusses their specific purpose and use. Emphasis is put on the problem of how to embody appropriate functions that provide the opportunity to evaluate user inputs - thus enabling an author to give local feedback to the student. Finally, some questions are raised concerning the form that should be used in the future to represent both data and logical structure of the underlying content. Monika Schwarze, Germany: Self directed learning in maths - szenarios, material from a german case study Information about a german case study of self-directed learning in high schools supported in different ways by new media, e. g. interactive tools or learning environments: there will be an exemplarily presentation of szenarios of learning linear algebra, statistics, analysis or geometry and some results of evaluation of the first projects. Angela Schwenk, Germany: Mathematical Abilities of University Entrants Looking the future of mathematical teaching should also include a view on the situation at the moment: University entrants to engineering courses have poor knowledge in mathematics. The presented results base on investigations in 1995 and 2000: Comparison of the results from 1995 and 2000 Comparison of entrants with 12 (Fachabitur) and 13 (allgemeine Hochschulreife) years of high school education Influence of a mathematical bridging course Peter Sedlmeier, Germany: Improving statistical reasoning: a computer program for high-school students New results in research on judgment under uncertainty show a way of how to improve the teaching of statistical reasoning (Sedlmeier, 1999). The implications of this research are that (i) successful learning needs doing, and (ii) that the format in which information is represented plays a decisive role. Statistical problems are, for instance, solved much better if the relevant pieces of information are presented as frequencies rather than probabilities. It also helps a lot if random processes can be observed rather than only read about. A computer program is presented that incorporates these implications from psychological research (Sedlmeier amp Koumlhlers, 2001). The software accompanies an elementary text book on probability theory to be used in high school. Mazen Shahin, USA: Modelling with Difference equations using DERIVE In this discussion we share the pedagogy and the methodology of modelling real life situations with difference equations using the computer algebra system Derive. This is a part of a reform finite mathematics course in which students explore and discover mathematical ideas on their own as they complete specially designed tasks whose emphasis on applications helps them see the relevance of the abstract concepts. We will emphasise the use of graphical and numerical techniques, rather than theoretical techniques, to investigate and analyse the behaviour of solutions of the difference equations. We will investigate interesting linear and non-linear models as well as systems of difference equations from such diverse disciplines as business, economics, life sciences and social sciences. Mazen Shahin, USA: Discrete Delayed Population Models with DERIVE In this paper we show how Derive can be used efficiently in modelling and investigating discrete delayed population models. In particular we are interested in some population models represented by non-linear second order difference equations. We will explore the stability of the equilibrium values of the systems. We will apply an interesting method to control the chaos in a dynamical system represented by a first order non-linear difference equation. Some of the pedagogical issues related to the use of a CAS in modelling will be discussed. Harry Silfverberg, Finland: Using Voronoi diagrams produced by DGS as a tool in an educational study The Voronoi diagram of a collection of points is a partition of space into cells, each of which consists of the points closer to one particular point than to any others. According to the prototype theoretical explanation students at the lowest van Hiele levels tend to classify geometrical figures on the basis of extent of the similarity of the figure and the visual prototypes. The poster will graphically show how well Voronoi diagrams and partitions based on the different selection of prototypes fit to the empirical data gathered in Silfverbergs research (1999) about the ways how students at the lowest van Hiele levels classified a given collection of triangles into acute, right, obtuse, equilateral and isosceles triangles. Plenary: Branca Silveira (Portugal): Teacher training: the role of technology We cant have a change in our schools without teachers and teachers dont change if they are not convinced that the change is going to improve something. The world is changing, society is changing, pupils are changing, and the schools How are schools coping with this Technology is everywhere. No discussion about that. Everyday appears new software, new computers, new calculators, etc. Are the schools ready for this Does technology play a significant role in the change of the curriculum How do teachers face this Are they prepared to use technology effectively Which kind of difficulties do teachers face Some teachers have been using technology did they really changed their methodologies or are they using them in a inadequate environment What about teacher training Which kind of training is more effective Initial training In service training But, what should we do for making teachers include technology in their practice More computers More training A different schedule for the classroom Making the use of technology compulsory In Portugal the use of graphic calculators is compulsory in secondary schools, so, everybody has to use them. Should we do the same with computers and other technology What about Internet How should we train teachers for the use of Internet in the classroom How can teachers develop the ability to analyse and integrate in an intelligent way, in their teaching the technological developments to come (software, hardware, communication. ) Those are some of the questions we are going to discuss in this talk, based on the Portuguese experience, focusing my point of view as a teacher, as a teacher trainer and as a member of the board of directors of APM (the Portuguese Association of Teachers of Mathematics). Edgar Smith and A. Waterson, Australia: Online mathematics teaching:the development of student-instructor interaction We discuss differences between teaching styles in online mathematics teaching and other less technical subjects. We discuss how to lean over a students shoulder online. Techniques are both automatic and software mediated discussions with students. Discussions are extremely expensive in terms of staff time, so we consider automated responses. These are illustrated with sample problems in elementary fluid mechanics in a subject delivered via WebCT. We discuss how to evaluate and improve automated responses. Robert Smith, USA: Spreadsheets across the curriculum Excel and other such electronic spreadsheet programs have found their way in to a variety of undergraduate mathematics courses. In this presentation we will demonstrate some spreadsheet uses in a variety of undergraduate courses from precalculus to abstract algebra. Grosio Stanilov, Bulgaria: Mittels Computergraphik zu mathematischen Entdeckungen Wir untersuchen die Parallelogramm-und die Wuerfelschnitten nur mittels Schulmathematik. Um die Besonderkeiten der Laengenschnitten und die Flaecheninhalten zu entdecken, verwenden wir zunaechst die Computergraphik. Wir erreichen zu wichtigen Saetzen in der Analysis, zur besonderen Schnitten und zur neuen exotischen Flaechen in der Differentialgeometrie. Einiges ist auch in die Bildkunst zu verwenden. In der hyperbolischen Geometrie erreichen wir zu einer Konstante, die die Seiten des Morleys Dreiecks fuer jedes beliebigen Dreiecks von oben beschraenkt. Teaching Discrete Mathematics With Excel The modern spreadsheet as exemplified by Microsoft Excel offers almost unlimited opportunities for the illustration of fundamental mathematical concepts. Further, the same software allows the teacher to encourage an investigative or experimental approach to mathematics learning. This talk will present some examples of these ideas plus an overall framework for the use of Excel for the enhancement of laboratory work. It is claimed that the approach outlined is especially useful for tertiary IT students with a relatively modest background in mathematics. The discussion will focus on topics from fairly traditional courses in discrete mathematics. Fred Szabo, Miroslaw Majewski. Canada: Integrating MuPAD into the Teaching of Mathematics Computer Algebra Systems are becoming more and more popular in mathematics education. However, many teaching issues are still unresolved, and no one is able to give a simple recipe how to integrate computer algebra systems into the teaching process. In this paper, we discuss some proven strategies for using MuPAD in the teaching of mathematics. Christian Thune Jacobsen, Denmark: Experimental Mathematics. Someone invented the knife - everybody uses it Computer algebra systems (CAS), such as Derive and Maple, will naturally be an integrated part of teaching mathematics in the future - just as the use of calculators has been for the last two decades. The question is only how to implement CAS. Eno Tonisson, Estonia: Expression Equivalence Checking in Computer Algebra Systems This paper investigates the possible educational application of equivalence checking and the capability of expression equivalence checking in some common computer algebra systems. The applications of equivalence checking can be analysed from the viewpoint of three types of users: that of the teacher, that of the student, and that of an Intelligent Tutoring System. This paper deals with the way a computer algebra system copes with the checking of the basic equivalencies of algebra and trigonometry. It appears that the tools are far from perfect and require improvements. Yulian Tsankov, Bulgaria: Cubic Section by moving plane By Computer graphic and Schoolmathematic we investigate all cubic sections. They depend of three parameters. If we fix two of them, the interval (-infinity, infinity) for the third parameter divided in six subintervals, where the sections are from different type. We visualize these sections and corresponding them area functions. The dividing - points arise some surfaces geometrically connected with the cube. Nelson Urrego, Columbia: Using DERIVE for beginner courses of recursion theory In this Paper, the author gives a short introduction to the main concepts about Recursive Functions and some examples are programmed using DERIVE. These exercises can motivate students in the design of algorithms for solving rigorous arithmetic problems such as the implementation of a procedure for generate of a 1-1 Primitive Recursive correspondence between N2 and N. Aynur Uysal, Turkey: Importance of Mathematics in Engineering Education Two different approaches have traditionally influenced mathematics teaching in engineering education. First one considers mathematics only as a tool for professional practice, while the second one relates mathematics education with the development of the logical and critical thinking without which no tool will be efficient to the learners for their understanding and interpretation of the world. As well known. the second approach has been receiving a growing importance in the last years. In this paper. the second approach are described with detailed examples. A rich set of experience are also presented from the mathematics teaching in the Technical University of Istanbul. Mithat Uysal, Turkey: An Internet-Based Course Structure for Teaching Mathematics in an Engineering School This study sets out to present a detailed and integrated approach for teaching mathematics using world wide web. Previous works and existing www-based teaching structures are first discussed. Then the concept of a course portal following the comprehensive and integrated approach are presented. Main modules of the portal, namely, the main page, multimedia page, courseware page, contact page and the search page are described. The ways to improve the portal are discussed. Some observations from the ITU model (Istanbul Technical University) are also presented. Joseacute Luis Valcarce Goacutemez, Spain: Bridging the Gap between Dynamic Geometry and Computer Algebra: The Case of Loci Discovery A basic problem in elementary geometry consists of finding the equation of a locus, given some conditions defining it. This problem remains unsolved in the field of mathematics education from a technological point of view: no friendly tool exists that allows a student to specify the conditions of a locus in a diagram and it returns the equation of the locus. Numerical approaches to this problem have been tackled in cuurent dynamic geometry environments but they share an essential incompleteness: an object must be constrained to move along a predefined path in order to get the trace of some other object. This paper describes a symbolic-dynamic approach to this problem: a computer algebra system solves it within a dynamic geometry environment. Piet van Blokland, Netherlands: A sample of ideas in teaching statistics Probability and statistics in secondary school should be presented in such a way that it demonstrates the importance of this subjects in society. Some realistic simulations will be shown. Polls are an often used tool in modern society to investigate opinions. In this lecture a huge dataset of 50000 students will be presented The effect of sampling will be shown. In order for the students to grasp the idea of central limit theorem, technology will help. Pictures which can be manipulated by students will help students to understand better the ideas behind hypothesis testing. Carel van de Giessen, Netherlands: The Visualisation of a parameter Based on the ideas of David Tall we, Piet van Blokland and I, have developed a program to investigate graphs and formulas. Two aspects may be of special interest: variables and parameters. For the young students (12-14 years) it is easier to understand the concepts involved with graphs and formulas when using word-variables. The concept of parameter in formulas is difficult to grasp, because the mathematical level needed to understand a parameter is high. We therefore introduced a so called sliding parameter. In the programme this concept arises interactively using a scrollbar: the parameter value changes and so does the graph. This is a dynamic way to investigate a graph and the role of a parameter. One graph, one value of the parameter. Henk van der Kooij, Netherlands : Functional Algebra with the Use of the Graphing Calculator Algebra is a very important topic in mathematical programs for upper secondary education, but a vast majority of students is weak in understanding and using formal algebraic tools. This paper discusses some ideas about using the graphing calculator to support the learning of algebra in the context of functions and to help students overcome algebra-anxiety. Accepting the graphing calculator as a supportive toolkit in the learning of algebra has far-going consequences for the way in which what kind of algebra should be learned and taught. Peter van Wijk, Hans Stam, Netherlands: Mathematics and Internet The Internet is primarily used as a source of information, as reference work and as a medium in which to look things up. There is, it is true, a lot to be found on the Internet, but for (arithmetic) education the Internet can be more than an encyclopaedia or library. In order to organise the various ways in which the Internet can be used in education, we take the classification based on the idea that there are various sorts of places on the Internet. Oumldoumln Vancso, Hungary: Classical and Bayes-statistics in the school supported by computer In this presentation I would like to show such software which help to understand by visualisation, representation or counting some main ideas of the classical statistics for example: normal distribution and Laplace-condition, confidence-interval, testing hypothesis. On the other side I talk about working (following one idea of Dieter Wickmann ) on a program which also can be used in the school and give a possibility to teach Bayes-statistics earlier than the Universities and Highschools. This software have been developed by mathematics and informatics students of Eoumltvoumls Loacuteraacutend University of Budapest leading by Eacuteva Vaacutesaacuterhelyi . Laacuteszloacute Szabadi and me. Eva Vasarhelyi, Karl Josef Fuchs, Hungary Austria: Problem - Analysis - Encoding - Testing About Program - and Data-Structures The two authors will show examples for the use of Hand-Held-CAS-Technology in computerscience. From the educational point of view the different problems of interpretation, stepwise refining and modification concentrate on the flexible, effective use of basic comments of an imperative programming tool in many different ways. Herrmann Vogel, Germany: Use of Cinderella in higher elementary geometry I will presentate a paper created with Cinderella, which deals with the Wallace line of a triangle and a generalaziation of this line. It demonstrate the possibilities of Cinderella how one can - illustrate well known geometry facts by using the moving mode or the animation mode, - find new suppositons by doing exercises, - create the envelope of a set of straight lines, - construct conics with certain conditions, - create algebraic curves of higher order. Rolf Wasen, Sweden Computers in Engineering Education I will present experiences from 1 frac12 years at a mathematical Study Center and the use of computers and computer algebra in project works in the basic analysis courses. A model of how to use computer algebra in mathematical education was developed and will also be presented. It turned out that the computer was an indispensable tool for illustrating and testing mathematical ideas shy this not at least for the teacher shy and that the objections can be met with. There is an attractive possibility to continue these project works into research at different levels of ambition. Wilhelm Weiskirch, Germany Kurven sind mehr als Graphen von Funktionen. Dass die verbreitete unterrichtliche Reduktion des Kurvenbegriffs auf das Bild einer Funktion dessen mathematische Bedeutung und das didaktische Potential nicht annaumlhernd ausschoumlpft, ist unbestreitbar. Insbesondere geomtrische Zugaumlnge zu nichttrivialen Kurven und deren analytische Betrachtung werden durch DGS und CAS ermoumlglicht und koumlnnen dazu beitragen, die gegenwaumlrtige Starrheit der Oberstufenmathematik zu durchbrechen. Am Beispiel nichttrivialer Kurven als Ortslinien abhaumlngiger Punkte, bzw. Massenpunktbahnen sollen unter Ausnutzung der genetischen Methode deren Bedeutung und Potential fuumlr den Mathematikunterricht eroumlrtert werden. Otto Wurnig, Austria Advantages and Dangers in the Teaching of Stochastics by using CAS The use of CAS in the teaching of stochastics can be dangerous because the students like to use standard functions and functions which the teacher programmed as a tool without thinking. In student oriented thinking, however, CAS can well be used to gradually develop definitions and to help with the understanding of formulas and ways of solutions. The simulation of experiments by direct input of CAS commands makes it possible to put a stronger accent on the building of models. Maria Zajac, Poland: Internet materials in mathematics teaching In the paper the idea of an Internet educational website will be presented. The learning materials are divided into three groups: Power Point presentations, web pages and lesson scenarios. All of them are intended to be a tool for computer assisted learning. The resources for Math lessons will be of special interest in the paper. CASCADE-IMEI: Web site support for student teachers to learn realistic mathematics in Indonesia CASCADE-IMEI is a learning environment in the form of a face-to-face course and a Web site (cascadeimei) which aims to support student teachers in Indonesia to learn Realistic Mathematics Education (RME). RME is an instructional theory in mathematics education that was originally developed in the Netherlands. So far, two prototypes have been developed and evaluated both by student teachers and several experts in the Netherlands. This paper presents the origins of the learning environment, with a more detailed on its Web site as well as the results of first two cycles of its prototyping process.
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