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Profke, Academica VI, Mathematics Teaching in Germany after TIMSS and PISA Lothar Profke Institut für Didaktik der Mathematik Justus-Liebig-Universität Gießen Lothar Profke Institut für Didaktik der Mathematik Justus-Liebig-Universität Gießen Download Download

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Profke, Academica VI, Better a bad mark in mathematics than no personal touch! In German there is a play on words: Note stands both for school mark and for (personal) touch.

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Profke, Academica VI, Preliminary Remarks List of contents –Some mathematics problems from TIMSS and PISA –What shall we want with mathematics teaching? …Mathematical literacy –Chance of success …Mathematics teaching after TIMSS and PISA –Training of (mathematics) teachers Summary –Mathematics teaching after TIMSS and PISA = Mathematics teaching before TIMSS and PISA …in a bad and in a good sense List of contents –Some mathematics problems from TIMSS and PISA –What shall we want with mathematics teaching? …Mathematical literacy –Chance of success …Mathematics teaching after TIMSS and PISA –Training of (mathematics) teachers Summary –Mathematics teaching after TIMSS and PISA = Mathematics teaching before TIMSS and PISA …in a bad and in a good sense

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Profke, Academica VI, Preliminary Remarks Why such a panic (in Germany) –because of German students doing badly in tests or ranked only in the midfield? –Since a long time we know that mathematics teaching has only a bad outcome: …In Math I always was poor. (title of a book from A. Beutelspacher) Here I present my personal view: –in fact well founded –but not the opinion of the majority of the mathematics educators (in Germany) –and by several assigned to Black Pedagogy. Why such a panic (in Germany) –because of German students doing badly in tests or ranked only in the midfield? –Since a long time we know that mathematics teaching has only a bad outcome: …In Math I always was poor. (title of a book from A. Beutelspacher) Here I present my personal view: –in fact well founded –but not the opinion of the majority of the mathematics educators (in Germany) –and by several assigned to Black Pedagogy.

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Profke, Academica VI, Mathematics Problems from TIMSS and PISA TIMSS PISA Queries TIMSS PISA Queries

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (1) TIMSS (Third International Mathematics and Science Study) explored in TIMSS II: Which mathematics do German students know at the end of grades 7 and 8? …State of performance 1993/1995 perhaps worse today Material download: Some problems from TIMSS II: TIMSS (Third International Mathematics and Science Study) explored in TIMSS II: Which mathematics do German students know at the end of grades 7 and 8? …State of performance 1993/1995 perhaps worse today Material download: Some problems from TIMSS II: Folie 8

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (2)

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (3) T1.In two boxes are 54 kg apples. The second box weighs 12 kg more than the first box. How many kilogram are in every box? Write down your steps of solution. International difficulty631.2 Int. probability of solution grade Int. probability of solution grade German probability of solution grade German probability of solution grade Subject fieldAlgebra Correct solution1 point Folie 11

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (4) Q6.The Schmidts consume about l water per week. How many litres do they approximately consume per year? A B C D

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (5) In how many triangles which have the size and shape of the shadowed triangle the trapezium can be dissected? A. ThreeB. FourC. FiveD. Six

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (6) V4.Die Figur zeigt ein schattiertes Parallelogramm in einem Rechteck. Wie groß ist die Fläche des Parallelogramms? Antwort:___________________________________ V4.The picture shows a shadowed parallelogram in a rectangle. How large is the area of the parallelogram? Answer: ________________________________

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: TIMSS (7) –Solving times …1 minute for a multiple-choice-problem Q6, R10 …2 minutes for an open problem with short answer T1, V4 …5 minutes for an open problem with extended answer U2, V2 (not showed here) –Solving times …1 minute for a multiple-choice-problem Q6, R10 …2 minutes for an open problem with short answer T1, V4 …5 minutes for an open problem with extended answer U2, V2 (not showed here)

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (1) PISA (Programme for International Student Assessment) explored: Which mathematical competencies 15-year old German students should have and what do they know really? …State of performance 2000 Material download: Some problems from the German pre- and main test: PISA (Programme for International Student Assessment) explored: Which mathematical competencies 15-year old German students should have and what do they know really? …State of performance 2000 Material download: Some problems from the German pre- and main test: Folie 15

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (2)

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (3) BAUERNHÖFE Hier siehst du ein Foto eines Bauernhauses mit pyramidenförmigem Dach. FARMS Here you see a photo of a farmhouse with a pyramidal roof.

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (4) Nachfolgend siehst du eine Skizze mit den entsprechenden Maßen, die eine Schülerin vom Dach des Bauernhauses gezeichnet hat. Following you see a sketch of the roof of the farmhouse with the corresponding measures drawn of a girl student.

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (5) Frage 10 BAUERNHÖFE Berechne den Flächeninhalt des Dachbodens ABCD. Der Flächeninhalt des Dachbodens ABCD = ______________ m² Question 10 FARMS Calculate the area of the loft ABCD. The area of the loft ABCD = ___________ m 2 Kompetenzstufe II Internationale Schwierigkeit 492 Internationale Lösungswahrscheinlichkeit 0,61 Deutsche Lösungswahrscheinlichkeit 0,51 Grade of competence II International difficulty 492 International probability of solution 0.61 German probability of solution 0.51

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (6) Folie 44 Kompetenzstufe II - III Internationale Schwierigkeit 524 Internationale Lösungswahrscheinlichkeit 0,55 Deutsche Lösungswahrscheinlichkeit 0,41 Grade of competence II - III International difficulty 524 International probability of solution 0.55 German probability of solution 0.41 Folie 21

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (7) –Glasfabrik 1: Eine Glasfabrik stellt am Tag Flaschen her. 2 % der Flaschen haben Fehler. Wie viele sind das? …16 Flaschen40 Flaschen80 Flaschen160 Flaschen 400 Flaschen Kompetenzstufe II Internationale Schwierigkeit 471 Internationale Lösungswahrscheinlichkeit - Deutsche Lösungswahrscheinlichkeit 0,68 –Glass factory 1: A glass factory produces bottles each day. 2 % of the bottles are faulty. How many are faulty? …16 bottles40 bottles 80 bottles 160 bottles 400 bottles Grade of competence II International difficulty 471 International probability of solution - German probability of solution 0.68 –Glasfabrik 1: Eine Glasfabrik stellt am Tag Flaschen her. 2 % der Flaschen haben Fehler. Wie viele sind das? …16 Flaschen40 Flaschen80 Flaschen160 Flaschen 400 Flaschen Kompetenzstufe II Internationale Schwierigkeit 471 Internationale Lösungswahrscheinlichkeit - Deutsche Lösungswahrscheinlichkeit 0,68 –Glass factory 1: A glass factory produces bottles each day. 2 % of the bottles are faulty. How many are faulty? …16 bottles40 bottles 80 bottles 160 bottles 400 bottles Grade of competence II International difficulty 471 International probability of solution - German probability of solution 0.68

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (8) –Glasfabrik 3: Eine Glasfabrik stellt am Tag Flaschen her. Erfahrungsgemäß sind ca. 160 Flaschen fehlerhaft. Wie viel Prozent sind das? …0,02 %0,5 %1,28 %2 %5 % Kompetenzstufe II Schwierigkeit 471 Deutsche Lösungswahrscheinlichkeit 0,68 –Glass factory 3: A glass factory produces bottles each day. As experience shows 2 % of the bottles are faulty. How many are faulty? …0,02 %0,5 %1,28 %2 %5 % Grade of competence II Difficulty 471 German probability of solution 0.41 –Glasfabrik 3: Eine Glasfabrik stellt am Tag Flaschen her. Erfahrungsgemäß sind ca. 160 Flaschen fehlerhaft. Wie viel Prozent sind das? …0,02 %0,5 %1,28 %2 %5 % Kompetenzstufe II Schwierigkeit 471 Deutsche Lösungswahrscheinlichkeit 0,68 –Glass factory 3: A glass factory produces bottles each day. As experience shows 2 % of the bottles are faulty. How many are faulty? …0,02 %0,5 %1,28 %2 %5 % Grade of competence II Difficulty 471 German probability of solution 0.41

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (9) –Sparen: Karina hat 1000 DM in ihrem Ferienjob verdient. Ihre Mutter empfiehlt ihr, das Geld zunächst bei einer Bank für 2 Jahre festzulegen (Zinseszins!). Dafür hat sie zwei Angebote: Plus-Sparen: Im ersten Jahr 3 % Zinsen, im zweiten Jahr dann 5 % Zinsen. Extra-Sparen: Im ersten und zweiten Jahr jeweils 4 % Zinsen. …Karina meint: Beide Angebote sind gleich gut. Was meinst du dazu? Begründe deine Antwort. –Saving: Karina made 1000 DM during her holiday job. Her mother recommends her to invest the money fix for 2 years in the bank (compound interest!). For that she has two offers: Plus-investment: for the first year 3 % interest, then for the second year 5 % interest. Extra-investment: for both years 4 % interest. …Karina thinks: Both offers are equal good. What do you think about that? Give reasons for your answer. –Sparen: Karina hat 1000 DM in ihrem Ferienjob verdient. Ihre Mutter empfiehlt ihr, das Geld zunächst bei einer Bank für 2 Jahre festzulegen (Zinseszins!). Dafür hat sie zwei Angebote: Plus-Sparen: Im ersten Jahr 3 % Zinsen, im zweiten Jahr dann 5 % Zinsen. Extra-Sparen: Im ersten und zweiten Jahr jeweils 4 % Zinsen. …Karina meint: Beide Angebote sind gleich gut. Was meinst du dazu? Begründe deine Antwort. –Saving: Karina made 1000 DM during her holiday job. Her mother recommends her to invest the money fix for 2 years in the bank (compound interest!). For that she has two offers: Plus-investment: for the first year 3 % interest, then for the second year 5 % interest. Extra-investment: for both years 4 % interest. …Karina thinks: Both offers are equal good. What do you think about that? Give reasons for your answer. Folie 23

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (10)

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (11) –Example problem: Mathematical literacy a)Welche der Figuren hat die größte Fläche? Begründe deine Antwort. b)Gib eine Methode an, wie der Flächeninhalt von Figur C bestimmt werden kann. c)Gib eine Methode an, wie der Umfang von Figur C bestimmt werden kann. a)Which of the figures has the largest area? Give reasons for your answer. b)Give a method for finding out the area of figure C. c)Give a method for finding out the circumference of figure C. –Example problem: Mathematical literacy a)Welche der Figuren hat die größte Fläche? Begründe deine Antwort. b)Gib eine Methode an, wie der Flächeninhalt von Figur C bestimmt werden kann. c)Gib eine Methode an, wie der Umfang von Figur C bestimmt werden kann. a)Which of the figures has the largest area? Give reasons for your answer. b)Give a method for finding out the area of figure C. c)Give a method for finding out the circumference of figure C. Folie 25

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: PISA (12) –Sleeping seal: Mathematical literacy Eine Robbe muss atmen, auch wenn sie schläft. Martin hat eine Robbe eine Stunde lang beobachtet. Zu Beginn seiner Beobach- tung befand sich die Robbe an der Wasseroberfläche und holte Atem. Anschließend tauchte sie zum Meeresboden und begann zu schlafen. Innerhalb von 8 Minuten trieb sie langsam zurück an die Oberfläche und holte Atem. Drei Minuten später war sie wieder auf dem Meeresboden, und der ganze Prozess fing von vorne an. Nach einer Stunde war die Robbe... A seal must breathe also when sleeping. Martin has watched a seal for one hour. At the beginning of his observation the seal was at the water surface and took a breath. Next it dived to the sea ground and began to sleep. Within 8 minutes it drove back slowly to the surface and took a breath. Three minutes later it was on the sea ground again and the whole process started at the beginning. After one hour the seal was: a) on the sea ground b) the way up c) taking a breathd) the way down –Sleeping seal: Mathematical literacy Eine Robbe muss atmen, auch wenn sie schläft. Martin hat eine Robbe eine Stunde lang beobachtet. Zu Beginn seiner Beobach- tung befand sich die Robbe an der Wasseroberfläche und holte Atem. Anschließend tauchte sie zum Meeresboden und begann zu schlafen. Innerhalb von 8 Minuten trieb sie langsam zurück an die Oberfläche und holte Atem. Drei Minuten später war sie wieder auf dem Meeresboden, und der ganze Prozess fing von vorne an. Nach einer Stunde war die Robbe... A seal must breathe also when sleeping. Martin has watched a seal for one hour. At the beginning of his observation the seal was at the water surface and took a breath. Next it dived to the sea ground and began to sleep. Within 8 minutes it drove back slowly to the surface and took a breath. Three minutes later it was on the sea ground again and the whole process started at the beginning. After one hour the seal was: a) on the sea ground b) the way up c) taking a breathd) the way down

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Profke, Academica VI, Mathematics problems from TIMSS and PISA: Queries Queries Do you would have known all answers? Who must be able to solve such problems? –Grammar-school student (Gymnasiast), students from other secondary schools (Realschüler, Hauptschüler), –Politician, journalist, show-master, –College teacher, college student How do we assess the difficulties of the problems? What do we think which rates of success are tolerable? Which outcome of German students would have pleased us in Germany? –Germany only among the best or... ? …Even if the rates of success are 30 % or less? Queries Do you would have known all answers? Who must be able to solve such problems? –Grammar-school student (Gymnasiast), students from other secondary schools (Realschüler, Hauptschüler), –Politician, journalist, show-master, –College teacher, college student How do we assess the difficulties of the problems? What do we think which rates of success are tolerable? Which outcome of German students would have pleased us in Germany? –Germany only among the best or... ? …Even if the rates of success are 30 % or less?

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Profke, Academica VI, What shall we want? Standards Reactions to TIMSS Modellversuch Mathematik General education Standards Reactions to TIMSS Modellversuch Mathematik General education

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Profke, Academica VI, What shall we want? Standards (1) Are the findings of TIMSS and PISA bad for Germany? The findings are not unexpected: –Look e.g. …Working Group Pedagogical College Freiburg i. Br.: Mathematikkenntnisse in der Abschlussprüfung an Hauptschulen. Baden-Württemberg 1986 Mathematical knowledge in the final test at extended elementary schools of Baden-Württemberg 1986 The answer depends on that what we want to achieve: We must match the results against our ideas of –general education, mathematical literacy, –purposes and goals of mathematics teaching. These may differ from that what TIMSS and PISA measure: Are the findings of TIMSS and PISA bad for Germany? The findings are not unexpected: –Look e.g. …Working Group Pedagogical College Freiburg i. Br.: Mathematikkenntnisse in der Abschlussprüfung an Hauptschulen. Baden-Württemberg 1986 Mathematical knowledge in the final test at extended elementary schools of Baden-Württemberg 1986 The answer depends on that what we want to achieve: We must match the results against our ideas of –general education, mathematical literacy, –purposes and goals of mathematics teaching. These may differ from that what TIMSS and PISA measure:

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Profke, Academica VI, What shall we want? Standards (2) –Examples from the Baden-Württemberg-Study 1986 Students shall be allowed to show what they know: It should not be proved what they do not know. …Basic facts and abilities …Simple numbers, no new types of problems …Familiar subject fields with simple data …Only one-step-problems (simplex-problems) input data arithmetic operation output –Examples from the Baden-Württemberg-Study 1986 Students shall be allowed to show what they know: It should not be proved what they do not know. …Basic facts and abilities …Simple numbers, no new types of problems …Familiar subject fields with simple data …Only one-step-problems (simplex-problems) input data arithmetic operation output Folie 30

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Profke, Academica VI, What shall we want? Reactions to TIMSS (1) Reactions to TIMSS Is that our aim: The next international comparative test German students must do better (?) Analysis: Abilities of German students at TIMSS –Strengths: …Reproducing of facts …Carrying out simple algorithmic procedures …Working on one-step-problems –Weaknesses: …Combining several subject fields …Working on poly-step-problems (complex-problems) …Considering at the same time various aspects …Complex modelling …Dealing with unusual situations Reactions to TIMSS Is that our aim: The next international comparative test German students must do better (?) Analysis: Abilities of German students at TIMSS –Strengths: …Reproducing of facts …Carrying out simple algorithmic procedures …Working on one-step-problems –Weaknesses: …Combining several subject fields …Working on poly-step-problems (complex-problems) …Considering at the same time various aspects …Complex modelling …Dealing with unusual situations Folie 28

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Profke, Academica VI, What shall we want? Reactions to TIMSS (2) Demands –We need a new quality of mathematics teaching. …Being guided by an adequate multilayered picture of mathematics …Adjusting to constructivistic theories of learning …Changing the culture of teaching –The contents of mathematics teaching are not questioned. –Also well-known demands: …More time for mathematics teaching, more teacher, smaller classes,... …Better pre- and in-service-training of teachers Demands –We need a new quality of mathematics teaching. …Being guided by an adequate multilayered picture of mathematics …Adjusting to constructivistic theories of learning …Changing the culture of teaching –The contents of mathematics teaching are not questioned. –Also well-known demands: …More time for mathematics teaching, more teacher, smaller classes,... …Better pre- and in-service-training of teachers Blum, W.; Neubrand, M.: TIMSS und der Mathematikunter- richt. Informationen, Analysen, Konsequenzen. Hannover: Schroedel 1998

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Profke, Academica VI, What shall we want? Reactions to TIMSS (3) Activities in Germany –Programme of the Bund-Länder-Kommission: Steigerung der Effizienz des mathematisch-naturwissenschaftlichen Unterrichts (jetzt SINUS) Increasing the Efficiency of mathematics and science teaching (now: SINUS) –Quality Initiative in Hessen: Inhaltliche und metho- dische Weiterentwicklung des Unterrichts in Mathematik und den Naturwissenschaften Developing mathematics and science teaching in content and methodical –Material download –New syllabuses in some Länder Activities in Germany –Programme of the Bund-Länder-Kommission: Steigerung der Effizienz des mathematisch-naturwissenschaftlichen Unterrichts (jetzt SINUS) Increasing the Efficiency of mathematics and science teaching (now: SINUS) –Quality Initiative in Hessen: Inhaltliche und metho- dische Weiterentwicklung des Unterrichts in Mathematik und den Naturwissenschaften Developing mathematics and science teaching in content and methodical –Material download –New syllabuses in some Länder

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (1) New types of problems (?) –Calculating percentages New types of problems (?) –Calculating percentages Seit dem Tag seiner Geburt hat Dominiks Mutter an jedem seiner Geburtstage seine Körpergröße notiert. Am 5. Geburtstag hat sie das Notieren jedoch vergessen. Betrachte die Zuordnung Alter Körpergröße (von Dominik). a)Was lässt sich mit Sicherheit über Dominiks Größe im Alter von 5 Jahren und von 11 Jahren aussagen? Begründe deine Antwort b)Zeige, indem du Beispiele dafür anführst, dass für die Zuordnung Alter Körpergröße (von Dominik) nicht die Regeln für Proportionalität gelten Folie 33

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (2) Since the day of his birth Dominiks mother made a note of his height at each of his birthdays. At his 5th birthday she forgot this. Look at the function age height (of Dominik). a)What can you say reliably about Dominiks height at the age of 5 and the age of 11? Give reasons for your answer. b)Show by means of examples that the function age height (of Dominik) does not fulfil the rules of proportionality. Proposals for opening this problem -Preparing by a long-term homework: Every student sets up his own age- graph. Then discussing in the class the different runs. -Students themselves shall pose meaningful questions: Is there some maximum value? When does growth be greatest?... -Calculating annual increases

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (3) –Variables and expressions a) Ein Paket hat die Länge l = 35 cm, die Breite b = 25 cm und die Höhe h = 12 cm. Je nach Gewicht des Inhaltes soll es unterschiedlich verschnürt werden. Schätzt, für welches Paket ihr am meisten Schnur benötigt. Gebt noch 20 cm (insgesamt) für die Knoten hinzu und berechnet die jeweils benötigte Schnurlänge. Versucht, einen Schuhkarton wie in der Grafik dargestellt zu schnüren, die Kordel soll nirgends doppelt verlaufen. b) Gebt die Schnurlängen auch allgemein für solche Pakete mit der Länge l, der Breite b und der Höhe h an. c) Wie sieht eine Paket-Schnürung aus zu 4l+4b+4h+15 bzw. zu 3l+2b+4h+10? d) Überlege dir weitere Terme und lass deinen Nachbarn die Pakete aufzeichnen. –Variables and expressions a) Ein Paket hat die Länge l = 35 cm, die Breite b = 25 cm und die Höhe h = 12 cm. Je nach Gewicht des Inhaltes soll es unterschiedlich verschnürt werden. Schätzt, für welches Paket ihr am meisten Schnur benötigt. Gebt noch 20 cm (insgesamt) für die Knoten hinzu und berechnet die jeweils benötigte Schnurlänge. Versucht, einen Schuhkarton wie in der Grafik dargestellt zu schnüren, die Kordel soll nirgends doppelt verlaufen. b) Gebt die Schnurlängen auch allgemein für solche Pakete mit der Länge l, der Breite b und der Höhe h an. c) Wie sieht eine Paket-Schnürung aus zu 4l+4b+4h+15 bzw. zu 3l+2b+4h+10? d) Überlege dir weitere Terme und lass deinen Nachbarn die Pakete aufzeichnen. Folie 35

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (4) a)A parcel has length l = 35 cm, width b = 25 cm, height h = 12 cm. According to its weight it should be tied up differently. # Guess which parcel needs the longest piece of cord. # Give 20 cm as an extra for the knot and calculate in each case the necessary length of the cord. # Try to tie up a shoe-carton as in the picture. The cord shall never run double. b)Express in general the cord lengths for these parcels through the length l, the width b and the height h. c)How parcels are tied up with 4l+4b+4h+15 resp. with 3l+2b+4h+10? d)Think over more expressions and let your neighbour sketch the parcels. Folie 38

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (5) –Surfaces and volumes a)Das größte Organ des menschlichen Körpers ist die Haut. Versuche mit einem Maßband als Hilfsmittel ungefähr herauszufinden, wie viel Haut ein Mensch hat. Mediziner gehen davon aus, dass bei einem Erwachsenen mit Verbrennungen von mehr als 15% Lebensgefahr besteht. Einer wie großen Fläche entspricht dies? The skin is the biggest organ of the human body. Try to find out by means of a tape measure how much skin a human being has. Physicians assume that danger of life exists for an adult with more than 15 % burning. How large is a corresponding area? –Surfaces and volumes a)Das größte Organ des menschlichen Körpers ist die Haut. Versuche mit einem Maßband als Hilfsmittel ungefähr herauszufinden, wie viel Haut ein Mensch hat. Mediziner gehen davon aus, dass bei einem Erwachsenen mit Verbrennungen von mehr als 15% Lebensgefahr besteht. Einer wie großen Fläche entspricht dies? The skin is the biggest organ of the human body. Try to find out by means of a tape measure how much skin a human being has. Physicians assume that danger of life exists for an adult with more than 15 % burning. How large is a corresponding area?

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (6) b)In den Lungenbläschen findet der Gasaustausch zwischen Sauerstoff und Kohlendioxid statt. Der Mensch besitzt ca. 400 Mio. Lungenbläschen mit einem Radius von jeweils 0,1 mm. (1)Berechne den Gesamtoberflächeninhalt aller Lungenbläschen eines Menschen. (2)Welchen Radius müsste eine einzige Kugel mit dem gleichen Oberflächeninhalt haben? (3)Um wie viel Prozent ist die Gesamtoberfläche der Lungen- bläschen größer als die der Haut (siehe Aufgabe a)? The exchange between oxygen and carbon dioxide takes placein the little bubbles of the lungs. A human being has about 400 million such bubbles all with radius 0.1 mm. (1)Calculate the total surface of all bubbles in the lungs of a human being. (2)Which radius must have a sphere with the same surface? (3)How much per cent greater than the surface of the skin (see a)) is the total surface of all bubbles in the lungs? b)In den Lungenbläschen findet der Gasaustausch zwischen Sauerstoff und Kohlendioxid statt. Der Mensch besitzt ca. 400 Mio. Lungenbläschen mit einem Radius von jeweils 0,1 mm. (1)Berechne den Gesamtoberflächeninhalt aller Lungenbläschen eines Menschen. (2)Welchen Radius müsste eine einzige Kugel mit dem gleichen Oberflächeninhalt haben? (3)Um wie viel Prozent ist die Gesamtoberfläche der Lungen- bläschen größer als die der Haut (siehe Aufgabe a)? The exchange between oxygen and carbon dioxide takes placein the little bubbles of the lungs. A human being has about 400 million such bubbles all with radius 0.1 mm. (1)Calculate the total surface of all bubbles in the lungs of a human being. (2)Which radius must have a sphere with the same surface? (3)How much per cent greater than the surface of the skin (see a)) is the total surface of all bubbles in the lungs?

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Profke, Academica VI, What shall we want? Modellversuch Mathematik (7) –What do you think: …Are these problems really new? –Modellversuch Mathematik: Some information …One aim is to enable mathematics teachers to open problems which are found in textbooks. …If a school wants participate in the Modellversuch, then all mathematics teachers must take part. …The in-service-training takes place at the schools of the teachers. …The teachers themselves construct the new problems. –What do you think: …Are these problems really new? –Modellversuch Mathematik: Some information …One aim is to enable mathematics teachers to open problems which are found in textbooks. …If a school wants participate in the Modellversuch, then all mathematics teachers must take part. …The in-service-training takes place at the schools of the teachers. …The teachers themselves construct the new problems.

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Profke, Academica VI, What shall we want? General education (1) General education and mathematics teaching In Germany there is a discussion about general education since a long time, before and independent of TIMSS: –Purposes and goals of mathematics teaching at schools …New math, …Mathematische Schulbildung 2001 Series of papers in the magazine Mathematik in der Schule …Heymann, H. W.: Allgemeinbildung und Mathematik- unterricht. Studien zur Schulpädagogik und Didaktik 13. Weinheim/Basel: Beltz 1996 Sieben Jahre Mathematik sind genug. (Seven years mathematics are enough.) General education and mathematics teaching In Germany there is a discussion about general education since a long time, before and independent of TIMSS: –Purposes and goals of mathematics teaching at schools …New math, …Mathematische Schulbildung 2001 Series of papers in the magazine Mathematik in der Schule …Heymann, H. W.: Allgemeinbildung und Mathematik- unterricht. Studien zur Schulpädagogik und Didaktik 13. Weinheim/Basel: Beltz 1996 Sieben Jahre Mathematik sind genug. (Seven years mathematics are enough.)

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Profke, Academica VI, What shall we want? General education (2)... and PISA Mathematical literacy (Mathematische Grundbildung) …as the base of PISA... and PISA Mathematical literacy (Mathematische Grundbildung) …as the base of PISA Folie 47

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Profke, Academica VI, What shall we want? General education (3) –PISA-Framework shall serve as (standardizing) model for mathematics teaching in Germany. –General education purpose of mathematics teaching: …Teaching to perceive and to understand phenomena of the world in a specific manner …Teaching to know and to grasp mathematical concepts as intellectual inventions of a typical nature …Developing abilities for problem solving –Demand for application-guided mathematics teaching: …Getting out mathematical concepts of familiar real world situations …Applying mathematical concepts in the real world –With those you continue over and over again raised (old) demands. –PISA-Framework shall serve as (standardizing) model for mathematics teaching in Germany. –General education purpose of mathematics teaching: …Teaching to perceive and to understand phenomena of the world in a specific manner …Teaching to know and to grasp mathematical concepts as intellectual inventions of a typical nature …Developing abilities for problem solving –Demand for application-guided mathematics teaching: …Getting out mathematical concepts of familiar real world situations …Applying mathematical concepts in the real world –With those you continue over and over again raised (old) demands. Winter, H.: Vorstellungen zur Entwicklung von Curricula für den Mathematikunterricht in der Gesamtschule. Strukturförderung im Bildungswesen des Landes Nordrhein- Westfalen Heft 16. Ratingen 1972, S

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Profke, Academica VI, What shall we want? General education (4) –Mathematical literacy covers …mathematical competencies mathematical doing like proving, defining, generalizing,... modelling and mathematizing technical skills (working off algorithms, solving routine problems) …mathematical big ideas (instead of subject fields) space, shape, pattern change, growth and decay chance, probability …both not restricted to special fields. –Mathematical literacy covers …mathematical competencies mathematical doing like proving, defining, generalizing,... modelling and mathematizing technical skills (working off algorithms, solving routine problems) …mathematical big ideas (instead of subject fields) space, shape, pattern change, growth and decay chance, probability …both not restricted to special fields.

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Profke, Academica VI, What shall we want? General education (5) Mathematical literacy therefore stands for more than only –calculation skills …by heart and written, transforming expressions and solving equations,... –knowing facts …concepts, formulas, procedures,... –handling standard situations …bürgerliches Rechnen (common arithmetic) Mathematical literacy therefore stands for more than only –calculation skills …by heart and written, transforming expressions and solving equations,... –knowing facts …concepts, formulas, procedures,... –handling standard situations …bürgerliches Rechnen (common arithmetic) How to test mathematical literacy ? –Adequate operationalizing with the help of (?) …central tests (TIMSS, PISA, final exams), …written and oral examinations which are tailored to the individual mathematics teaching. How to test mathematical literacy ? –Adequate operationalizing with the help of (?) …central tests (TIMSS, PISA, final exams), …written and oral examinations which are tailored to the individual mathematics teaching.

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Profke, Academica VI, What shall we want? General education (6) –Open questions: …How to represent mathematical literacy? mathematical competencies and big ideas Is this achieved by PISA-problems? …May students train test situations before? New problems ask for other competencies than routine problems. …How reliable and fair are assessments? Do it make sense to calculate with (credit) points? »A knows adding fractions double as well as B constructing triangles according to (Ssw). …Who should have how much mathematical literacy? –Open questions: …How to represent mathematical literacy? mathematical competencies and big ideas Is this achieved by PISA-problems? …May students train test situations before? New problems ask for other competencies than routine problems. …How reliable and fair are assessments? Do it make sense to calculate with (credit) points? »A knows adding fractions double as well as B constructing triangles according to (Ssw). …Who should have how much mathematical literacy? Folie 15

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Profke, Academica VI, Chances of success Mathematics teaching after TIMSS and PISA Doubts Predictions Doubts Predictions

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Profke, Academica VI, Chances of success: Doubts (1) Only in principle they all agree what to do. But they (all) disagree about realizing into practice. What should belong to mathematical literacy ? –Defining through (?) …final profiles (Abschlussprofile) of syllabuses and (central) final exams …demands of the trade and industries, colleges,... …vocational entrance tests …TIMSS and PISA problems … –Always the result is the traditional list of contents. …Is that really necessary? What should belong to mathematical literacy ? –Defining through (?) …final profiles (Abschlussprofile) of syllabuses and (central) final exams …demands of the trade and industries, colleges,... …vocational entrance tests …TIMSS and PISA problems … –Always the result is the traditional list of contents. …Is that really necessary?

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Profke, Academica VI, Chances of success: Doubts (2) –Do PISA-Problems represent (mathematical) literacy? Is PISA valid? …How many people today have (mathematical) literacy? …Only a rather small group of experts elaborated the basic framework of mathematical literacy for PISA and constructed the pre- and main test-problems: choosing problem characteristics: »choice of situations, difficulties of problems, degrees of competence,... distributing of problem-characteristics over all test items …The theoretical framework of mathematical literacy in PISA seems reasonable, but we cannot answer the question because only few problems from PISA are published (the same is true for the test outcomes). –Do PISA-Problems represent (mathematical) literacy? Is PISA valid? …How many people today have (mathematical) literacy? …Only a rather small group of experts elaborated the basic framework of mathematical literacy for PISA and constructed the pre- and main test-problems: choosing problem characteristics: »choice of situations, difficulties of problems, degrees of competence,... distributing of problem-characteristics over all test items …The theoretical framework of mathematical literacy in PISA seems reasonable, but we cannot answer the question because only few problems from PISA are published (the same is true for the test outcomes). Folie 40

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Profke, Academica VI, Chances of success: Doubts (3) 14 example-problems from 6 situations: »coins, lichen, geometric figures, braking a car, terrace, sleeping seal 11 test-problems from 5 situations: »apples, area of a continent, speed of a racing car, geometric triangles, farms (But do not think too much of the situations.) …Most TIMSS items and outcomes are published. (This is scientific practise.) 14 example-problems from 6 situations: »coins, lichen, geometric figures, braking a car, terrace, sleeping seal 11 test-problems from 5 situations: »apples, area of a continent, speed of a racing car, geometric triangles, farms (But do not think too much of the situations.) …Most TIMSS items and outcomes are published. (This is scientific practise.)

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Profke, Academica VI, Chances of success: Doubts (4) –Risks: …conflict between general education and special vocational training …Neglecting higher educational goals and confining to easy examinable things secret syllabus (heimlicher Lehrplan) as students and teachers see it: only examination contents are important. …mathematical literacy like that of 1900 instead for the 3. millennium? What of the traditional mathematics teaching is in fact indispensable? …Often we ask for the false and from this too much. (Prof. Dr. H. Wredde, University of Dortmund) –Risks: …conflict between general education and special vocational training …Neglecting higher educational goals and confining to easy examinable things secret syllabus (heimlicher Lehrplan) as students and teachers see it: only examination contents are important. …mathematical literacy like that of 1900 instead for the 3. millennium? What of the traditional mathematics teaching is in fact indispensable? …Often we ask for the false and from this too much. (Prof. Dr. H. Wredde, University of Dortmund)

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Profke, Academica VI, Chances of success: Doubts (5) The same mathematical literacy for all? –Not each vocational training or course of studies does demand the same requirements and competencies: …commercial sector, technical jobs,... …humanities, natural science,... –Students have different interests: …I will become a lawyer. …... and I hairdresser. …I will marry a rich man. –A balance between …general obligatory claim of the society and …individual needs has been set up at best on a high abstract pedagogical level. The same mathematical literacy for all? –Not each vocational training or course of studies does demand the same requirements and competencies: …commercial sector, technical jobs,... …humanities, natural science,... –Students have different interests: …I will become a lawyer. …... and I hairdresser. …I will marry a rich man. –A balance between …general obligatory claim of the society and …individual needs has been set up at best on a high abstract pedagogical level.

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Profke, Academica VI, Chances of success: Doubts (6) How to impart (teach) mathematical literacy? –Training, training and once more training? …But what is worth to be trained? Calculating with common fractions in mixed form in the extended elementary school (Hauptschule)? »7 2 / / 7,... »I cannot do this. I am stupid shit. …Is this not a waste of time? because the results are only short-range? –Self-determined learning (Selbstbestimmtes Lernen) …with worksheets which progress in small steps and are not flexible? –Action-orientation (Handlungsorientierung) …which often turns out to thoughtless doing? How to impart (teach) mathematical literacy? –Training, training and once more training? …But what is worth to be trained? Calculating with common fractions in mixed form in the extended elementary school (Hauptschule)? »7 2 / / 7,... »I cannot do this. I am stupid shit. …Is this not a waste of time? because the results are only short-range? –Self-determined learning (Selbstbestimmtes Lernen) …with worksheets which progress in small steps and are not flexible? –Action-orientation (Handlungsorientierung) …which often turns out to thoughtless doing?

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Profke, Academica VI, Chances of success: Doubts (7) –Project-oriented, realistic mathematics teaching …till yet only a vision (an illusion?) –Project-oriented, realistic mathematics teaching …till yet only a vision (an illusion?) Checking the results of learning? –Is it possible to find out mathematical literacy by means of central examinations? …Probably only for simple qualifications in obligatory standard-situations –Can we compare objectively? …What means (e.g.) students from school A achieved on an average only 2/3 of the points as students from school B, if different things are taught in the lessons? Checking the results of learning? –Is it possible to find out mathematical literacy by means of central examinations? …Probably only for simple qualifications in obligatory standard-situations –Can we compare objectively? …What means (e.g.) students from school A achieved on an average only 2/3 of the points as students from school B, if different things are taught in the lessons?

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Profke, Academica VI, Chances of success Many questions …and only few answers. Many questions …and only few answers.

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Profke, Academica VI, Chances of success: Predictions (1) Assumption On the following each of us may base his own ideas of good mathematics teaching –answering himself the posed questions, –agree with me or not. Assumption On the following each of us may base his own ideas of good mathematics teaching –answering himself the posed questions, –agree with me or not. What may we expect from the future? Mathematics teaching after TIMSS and PISA Not too much: –We have about 50 years reforms of mathematics teaching …and always for its improvement. What may we expect from the future? Mathematics teaching after TIMSS and PISA Not too much: –We have about 50 years reforms of mathematics teaching …and always for its improvement.

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Profke, Academica VI, Chances of success: Predictions (2) Reforms of mathematics teaching in Germany since 1950: –Reforms in content: …Mathematics teaching in all kinds of schools, instead of Rechnen und Raumlehre (arithmetic and theory of space) …New Math Mapping methods in geometry teaching instead of the Euclidean method Set theory in primary school Operators in arithmetic Analysis like in university courses Linear Algebra instead of Analytical Geometry Reforms of mathematics teaching in Germany since 1950: –Reforms in content: …Mathematics teaching in all kinds of schools, instead of Rechnen und Raumlehre (arithmetic and theory of space) …New Math Mapping methods in geometry teaching instead of the Euclidean method Set theory in primary school Operators in arithmetic Analysis like in university courses Linear Algebra instead of Analytical Geometry

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Profke, Academica VI, Chances of success: Predictions (3) –Reforms in methods …textbooks carefully tuned to the special Länder-syllabuses development of contents in many stages very coloured often revised with detailed teacher-handbooks …much material to assist learning and teaching …changes of teaching and learning methods work with partners, in groups,... schedules for a day or for a week (Tagesplan, Wochenplan) learning at stations (Stationenlernen),... –Reforms in methods …textbooks carefully tuned to the special Länder-syllabuses development of contents in many stages very coloured often revised with detailed teacher-handbooks …much material to assist learning and teaching …changes of teaching and learning methods work with partners, in groups,... schedules for a day or for a week (Tagesplan, Wochenplan) learning at stations (Stationenlernen),...

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Profke, Academica VI, Chances of success: Predictions (4) –Organizational reforms …comprehensive schools,... …Orientierungsstufe, Förderstufe (levels for grades 5, 6) …reformierte Oberstufe (grades ) …central final examinations …Odenwaldschule, Laborschule und Oberstufenkolleg an der Universität Bielefeld (H. v. Hentig), which influenced scarcely other schools –Organizational reforms …comprehensive schools,... …Orientierungsstufe, Förderstufe (levels for grades 5, 6) …reformierte Oberstufe (grades ) …central final examinations …Odenwaldschule, Laborschule und Oberstufenkolleg an der Universität Bielefeld (H. v. Hentig), which influenced scarcely other schools All these projects did not bring the hoped-for results.

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Profke, Academica VI, Chances of success: Predictions (5) We must try harder !!! –There is good mathematics teaching, therefore it is possible. –But the (first, second, third) education of teacher students and (beginning) teachers …constantly makes an effort for good teaching, …always aims at what now, after TIMSS and PISA, is regarded as exemplary. –In everyday teaching many good intentions come to nothing. …The theory-practice-problem is a problem of practise. (somebody said). Really? –We did and do not try hard enough. …Correct diagnosis or faith healing (Gesundbeterei)? Motto: Faith can displace mountains. We must try harder !!! –There is good mathematics teaching, therefore it is possible. –But the (first, second, third) education of teacher students and (beginning) teachers …constantly makes an effort for good teaching, …always aims at what now, after TIMSS and PISA, is regarded as exemplary. –In everyday teaching many good intentions come to nothing. …The theory-practice-problem is a problem of practise. (somebody said). Really? –We did and do not try hard enough. …Correct diagnosis or faith healing (Gesundbeterei)? Motto: Faith can displace mountains.

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Profke, Academica VI, Chances of success: Predictions (6) Can we realise good (= successful) mathematics teaching for all? –Experience speaks against it. Can we realise good (= successful) mathematics teaching for all? –Experience speaks against it.

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Profke, Academica VI, Comments on training of (mathematics) teachers The ideal teacher Shortcomings of teacher education Improvements? The ideal teacher Shortcomings of teacher education Improvements?

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Profke, Academica VI, Comments on training of teachers Please note: Kommission zur Neuordnung der Lehrerausbildung an Hessischen Hochschulen: Neuordnung der Lehrerausbildung. Opladen: Leske + Budrich 1997 Please note: Kommission zur Neuordnung der Lehrerausbildung an Hessischen Hochschulen: Neuordnung der Lehrerausbildung. Opladen: Leske + Budrich 1997 A change (= improvement?) of mathematics teaching takes place at most through the education of teachers.

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Profke, Academica VI, Comments on training of teachers: The ideal teacher (1) The ideal mathematics teacher –will asks himself when preparing his lessons, during and after his teaching: …Checklist for preparing lessons –has at his disposal …mathematical know-how, …pedagogical-didactical competence, …psychological knowledge, …social abilities, …methodical skills, …personality; –plays several parts: …subject teacher, educator, bureaucrat, colleague,... …representative of the society opposite his students …advocate of his students opposite the society The ideal mathematics teacher –will asks himself when preparing his lessons, during and after his teaching: …Checklist for preparing lessons –has at his disposal …mathematical know-how, …pedagogical-didactical competence, …psychological knowledge, …social abilities, …methodical skills, …personality; –plays several parts: …subject teacher, educator, bureaucrat, colleague,... …representative of the society opposite his students …advocate of his students opposite the society Folie 66 Folie 63

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Profke, Academica VI, Checklist for preparing lessons (1) What will I achieve with the treatment of this subject matter? I: The teacher as the representative and commissioner of the society as well as the advocate of his students. Question for general and special educational goals For which students? Problem of the differentiation according to ability and to interest Dealing with the learning-group not like a collective of fictitious average students, but with each student as an individual person. What can I achieve? Do not pose impossible demands. Do not ask for too little. What will I achieve with the treatment of this subject matter? I: The teacher as the representative and commissioner of the society as well as the advocate of his students. Question for general and special educational goals For which students? Problem of the differentiation according to ability and to interest Dealing with the learning-group not like a collective of fictitious average students, but with each student as an individual person. What can I achieve? Do not pose impossible demands. Do not ask for too little.

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Profke, Academica VI, Checklist for preparing lessons (2) Why do I pose these goals? Weighing between general education and special requests Concentrating on the essential points How shall I proceed to achieve the chosen goals? Question for methodical developments of the subject matter Question for activities of students and of the teacher, for the designing of instruction (Unterrichtskultur) Question for the use of media Why do I pose these goals? Weighing between general education and special requests Concentrating on the essential points How shall I proceed to achieve the chosen goals? Question for methodical developments of the subject matter Question for activities of students and of the teacher, for the designing of instruction (Unterrichtskultur) Question for the use of media

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Profke, Academica VI, Checklist for preparing lessons (3) Will I have success? Which result do the plan promise? Question for controlling the outcome of teaching: How can I see which goals will be achieved wore or less well? Critical review first on the plan and then on the lesson Are there different, perhaps better, opportunities to achieve my goals? Admitted for competition are different themes of the same or of other subjects. Take away the egoism of the own subject and consider all of education in school. Will I have success? Which result do the plan promise? Question for controlling the outcome of teaching: How can I see which goals will be achieved wore or less well? Critical review first on the plan and then on the lesson Are there different, perhaps better, opportunities to achieve my goals? Admitted for competition are different themes of the same or of other subjects. Take away the egoism of the own subject and consider all of education in school. Folie 62

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Profke, Academica VI, Comments on training of teachers: The ideal teacher (2) How to become an ideal teacher? –Natural talent, affection for children,... are not sufficient. How can you make teachers as good as possible? –You can learn and teach the ability for teaching. …This is not a hypotheses bur experience. –Do the (first, second, third) education of teacher students and (beginning) teachers impart the necessary equipment? How to become an ideal teacher? –Natural talent, affection for children,... are not sufficient. How can you make teachers as good as possible? –You can learn and teach the ability for teaching. …This is not a hypotheses bur experience. –Do the (first, second, third) education of teacher students and (beginning) teachers impart the necessary equipment?

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Profke, Academica VI, Comments on training of teachers: Shortcomings (1) Shortcomings of teacher education in Germany –The whole education is dissected in separated parts: …Mathematics,..., didactics and methodology of mathematics, pedagogy, psychology, law,... …The joining together to a functioning whole is tried hardly, could happen during the practical training (in the first and the second education), »but do often fail. …We hope, that teacher students and beginning teachers do this for themselves or that this happens somehow in a miraculous manner in their heads. Shortcomings of teacher education in Germany –The whole education is dissected in separated parts: …Mathematics,..., didactics and methodology of mathematics, pedagogy, psychology, law,... …The joining together to a functioning whole is tried hardly, could happen during the practical training (in the first and the second education), »but do often fail. …We hope, that teacher students and beginning teachers do this for themselves or that this happens somehow in a miraculous manner in their heads.

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Profke, Academica VI, Comments on training of teachers: Shortcomings (2) –Teaching with the Mut zur Lücke nach dem exemplarischen Prinzip (courage for gaps according to the exemplary principle), …we teacher trainer do believe, …but teacher students often do not detect for which something should be exemplary …and cannot transfer ideas from instances which are dealt with to new ones even already within a subject, much more from one subject to another, naturally from pedagogy, psychology,... to mathematics,.... …The transferring is not taught teacher students but is expected from them. –Teaching with the Mut zur Lücke nach dem exemplarischen Prinzip (courage for gaps according to the exemplary principle), …we teacher trainer do believe, …but teacher students often do not detect for which something should be exemplary …and cannot transfer ideas from instances which are dealt with to new ones even already within a subject, much more from one subject to another, naturally from pedagogy, psychology,... to mathematics,.... …The transferring is not taught teacher students but is expected from them.

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Profke, Academica VI, Comments on training of teachers: Shortcomings (3) –The sense of some subject matters is questionable: Why must I learn algebra, analysis, statistics and probability theory,... although I will become a primary teacher? Which profit for my education to a mathematics teacher does bring the Nelkenrevolution in Portugal (carnation-revolution)? I learned at university to study something, but only few for my teacher job. …It is not enough, if we are convinced of the usefulness and quality our lectures, if our students do regard this differently. –The sense of some subject matters is questionable: Why must I learn algebra, analysis, statistics and probability theory,... although I will become a primary teacher? Which profit for my education to a mathematics teacher does bring the Nelkenrevolution in Portugal (carnation-revolution)? I learned at university to study something, but only few for my teacher job. …It is not enough, if we are convinced of the usefulness and quality our lectures, if our students do regard this differently.

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Profke, Academica VI, Comments on training of teachers: Shortcomings (4) –Much abstract theory, few concrete practise: …All theory must be able to become concrete, not only with few, over and over again mentioned examples, which are often not fit to syllabuses, but for »next Thursday, 2. lesson, in my 8. Grade (extended elementary school), with the topic constructing quadrangles …Theory keeps worthless, if not we can make it concrete in every case; otherwise: how shall succeed students, teacher,...? –Much abstract theory, few concrete practise: …All theory must be able to become concrete, not only with few, over and over again mentioned examples, which are often not fit to syllabuses, but for »next Thursday, 2. lesson, in my 8. Grade (extended elementary school), with the topic constructing quadrangles …Theory keeps worthless, if not we can make it concrete in every case; otherwise: how shall succeed students, teacher,...?

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Profke, Academica VI, Comments on training of teachers: improvements What can we do for improving teacher education –Eliminating shortcomings: …everybody for himself –Visions: …Teaching holistic and combining several subjects at university and at Studienseminar in lectures held together by experts of different subjects. …In the same way examinations …Examples? Any topic of mathematics teaching –Realization is difficult at a mass university, …but changes must begin today, that is under the current conditions. What can we do for improving teacher education –Eliminating shortcomings: …everybody for himself –Visions: …Teaching holistic and combining several subjects at university and at Studienseminar in lectures held together by experts of different subjects. …In the same way examinations …Examples? Any topic of mathematics teaching –Realization is difficult at a mass university, …but changes must begin today, that is under the current conditions.

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Profke, Academica VI, Comments on training of teachers We should not leave the change of mathematics teaching and of the education of mathematics teachers only mathematicians and mathematics didacticians: –These are often routine-blinded and do not bear in mind all of education in school. Warning: –We will not have only good (mathematics) teacher, …for all our efforts, …because we need many people who take this profession. We should not leave the change of mathematics teaching and of the education of mathematics teachers only mathematicians and mathematics didacticians: –These are often routine-blinded and do not bear in mind all of education in school. Warning: –We will not have only good (mathematics) teacher, …for all our efforts, …because we need many people who take this profession.

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