KeyCoMath Project

The project "Developing Key Competences by Mathematics Education" (KeyCoMath) aims at the development of students' key competences in primary and secondary schools. Didactic concepts, teaching and learning material as well as corresponding assessment methods for mathematics education are developed, tested, evaluated and disseminated on the European level."KeyCoMath" uses the power of initial and in-service teacher education to put innovative pedagogical and didactical approaches into practice.

This project has been funded with the support of the Lifelong Learning Programme of the European Union. This communication reflects only the author's view. The European Commission and the EACEA are not responsible for any use that may be made of the information contained.

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KeyCoMath Key Competences

Key competences are necessary for all citizens for personal fulfilment, active citizenship, social inclusion and employability in a knowledge society. The project "KeyCoMath" develops, implements, and evaluates ways of working according to the "European Reference Framework of Key Competences for Lifelong Learning" in mathematics education. The activities concentrate on the following key competences (basic skills and transversal competences) in primary and secondary schools:


Mathematical competence

"KeyCoMath" promotes pupils' active, exploratory learning in open and rich situations to deepen their ability of mathematical thinking and to support the development of mathematical understanding.

Communication in the mother tongue

"KeyCoMath" closely intertwines doing mathematics and communicating with others orally or in written form. Pupils are encouraged to talk about mathematics, to discuss ideas, to write down thoughts and reflections and to present results.

Digital competence

In "KeyCoMath" pupils work with learning environments that include digital media (e.g. spreadsheets, dynamic geometry, computer algebra). By working mathematically they develop a confident, critical and reflective attitude towards ICT.

Learning to learn

"KeyCoMath" emphasises pupils' self-regulated and autonomous learning. Thus, they develop abilities to manage their learning (both individually and in groups), to evaluate their work and to seek advice and support when appropriate.

Social competences

"KeyCoMath" fosters pupils' collaboration and communication. They cooperatively do mathematics, discuss ideas, present findings and have to understand different viewpoints to achieve common mathematical results.

Sense of initiative

"KeyCoMath" strengthens exploratory, inquiry-based learning in mathematics. Pupils are encouraged to be creative, proactive and to turn ideas into action. They develop abilities to plan, organise, and manage their work.

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KeyCoMath Target Areas

The project aims at innovations of mathematics education on the European level in four target areas.

  • "KeyCoMath" aims at fundamental changes of pupils' learning. A shift towards more active, exploratory, self-regulated, autonomous, communicative and collaborative learning is intended. This way of doing mathematics helps to develop a broad variety of key competences.
  • These changes of pupils' learning require and are based on changes of teaching. Teachers develop expertise to arrange learning environments in order for pupils to work in the intended way and to develop key competences.
  • Assessment methods that correspond with the competence-oriented approach are developed, tested and evaluated. Pupils work mathematically with open "learning/assessment scenarios", they write down their thoughts and findings, present and discuss results. Teachers are sensitized to use this information to diagnose pupils' competences, to adapt teaching to learners' needs and especially to support pupils with difficulties.
  • "KeyCoMath" contributes to overcome the European problem of low-achievers. Furthermore, the partners identify and develop strategies for integrating the European dimension in teaching and learning mathematics.
Key Competences

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KeyCoMath Fields of Activity

"KeyCoMath" operates in four fields of activity:

  • The partners use the power of in-service teacher education for initiating innovations in mathematics education. Networks of teachers are intensively involved in the project. They are made familiar with didactic concepts for teaching, learning and assessment to support pupils' key competences.
  • Initial teacher education is the basis of mathematics education in the future. Therefore, the partners implement the didactical concepts in initial teacher education.
  • Dissemination and exploitation strategies with publications of didactic concepts and concrete teaching/learning/assessment material in Bulgarian, Czech, English, German, Greek and Norwegian contribute to the improvement of mathematics education in school on the large scale in Europe.
  • To enhance the impact on the level of educational systems, project results are distributed among decision-makers in politics, administration and educational practice as well as in the scientific community of mathematics education.
Fields of Activity

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KeyCoMath Publications

Publications within the framework of "KeyCoMath":



  • KeyCoNet (Ed.): Projects related to competence-based education: KeyCoMath – Developing Key Competences through Mathematics Education. In: KeyCoNet Newsletter 8 (July), p. 13, 2014
  • Andresen, Mette: Students‘ strategies for modelling a Feris wheel. In: Proceedings of NORMA14. The seventh Nordic conference on mathematics education, Turku, Finland, 2.-6. June 2014
  • Pitta-Pantazi, Demetra et al.: Assessing mathematically challenging problems. In: 9th congress of European Research in Mathematics Education, Prague, Czech Republic, 2015
  • Chehlarova, Toni et al.: A Virtual School Mathematics Laboratory. In: 5th National Conference on Electronic Education, Rousse, Bulgaria, 16.-17. June 2014
  • Chehlarova, Toni / Sendova, Evgenia : Developing Communication Competences in the Context of Mathematics Education. In: Proceedings of V Congress, Скопје, Македонија, рр. 127-235, 2014
  • Sendova, Evgenia: You do – you understand, you explore – you invent: the fourth level of the inquiry-based learning. In: Contructionism and Creativity - Proceedings of the 3d International Constructionism Conference, Vienna, Austria, pp. 103–112, 2014
  • Götz, Carolin: KeyCoMath - Multipliers Concept for the Urban Network of Primary Schools. In: Conference Proceedings - Educating the educators: international approaches to scaling-up professional development in mathematics and science education, 2015
  • Zehetmeier, Stefan: KeyCoMath. In: IFF Newsletter 43 (June), p. 12, 2015
  • Gehring, Carolin / Ulm, Volker (Hrsg.): Developing Key Competences by Mathematics Education, 2015
  • Chehlarova, Toni / Kenderov, Petar / Sendova, Evgenia: Achievements and Problems in the In-service Teacher Education in Inquiry Based Style. In: Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, pp. 21-30, May, 2015
  • Chehlarova, Toni / Kenderov, Petar : Mathematics with a computer – a contest enhancing the digital and mathematical competences of the students. In: UNESCO International Workshop Quality of Education and Challenges in a Digitally Networked World, Faleza, pp. 61-72, 2015
  • Ulm, Volker: Systemic Innovations in Educational Systems via In-Service Teacher Education. In: American International Journal of Social Science, Vol. 4, No. 6, 2016


  • Kenderov, Petar / Sendova, Evgenia / Chehlarova, Toni: The Inquiry based Mathematics Education. Didactic Base of Inquiry Approach in Education. In: Neophit Rilsky 1, pp 11-17, 2014, ISBN 978-954-680-947-6
  • Kenderov, Petar / Sendova, Evgenia / Chehlarova, Toni: Development of key competences by mathematics education: The KeyCoMath European project. In: Proceedings of the 43d Conference of the Union of the Bulgarian Mathematicians, Borovets, 2-6 April, pp. 99-105, 2014
  • Kenderov, Petar / Chehlarova, Toni: The contest “Viva Mathematics with a computer” and its role for the development of digital competence of the students Shumen. In: МАТТЕХ, pp. 3-10, 2014


  • Binterová, Helena / Hašek, Roman / Pech, Pavel / Petrášková, Vladimíra : Klíčové kompetence v badatelsky orientované výuce matematiky, České Budějovice: Jihočeská univerzita v Českých Budějovicích, 2015, ISBN 978-80-7394-516-9
  • Binterová, Helena / Hašek, Roman / Pech, Pavel / Petrášková, Vladimíra : Klíčové kompetence ve výuce matematiky, České Budějovice: Jihočeská univerzita v Českých Budějovicích, 2015
  • Binterová, Helena / Hašek, Roman / Pech, Pavel / Petrášková, Vladimíra : Klíčové kompetence a mezipředmětové vztahy, České Budějovice: Klíčové kompetence a mezipředmětové vztahy, Jihočeská univerzita v Českých Budějovicich, 2015
  • Hašek, Roman: Křivky v GeoGebře. Sborník Konference Užití počítačů ve výuce matematiky, České Budějovice: Plenární přednáška, 2015


KeyCoMath Teaching and Learning Material




    • Entfernungs- und Höhenmessung – Funktionsweise von Apps: Word, PDF
    • Wichtige Linien im Dreieck – Eine computerunterstütze Unterrichtseinheit: Word, PDF
    • Einführung in die Trigonometrie – Sinus, Kosinus, Tangens am rechtwinkligen Dreieck und am Einheitskreis: Word, PDF, GeoGebra
    • Sinus- und Kosinussatz – Trigonometrie in beliebigen Dreiecken: Word, PDF
    • Anwendungen der Trigonometrie – Nutzen in Elektrotechnik, Mechanik und beim Heimwerken: Word, PDF
    • Bilder in Bewegung setzen – Arbeiten mit Abbildungsmatrizen in Derive: Word, PDF
    • Flächeninhalt von Dreiecken – Übungen: Word, PDF
    • Flächeninhalt von Trapezen – Übungen: Word, PDF
    • Volumenberechnungen von Quadern – Übungen: Word1, PDF1, Word2, PDF2
    • Kreissektor – Begründen und Beweisen: Word, PDF
    • Achsen- und punktsymmetrische Figuren – Offene Aufgaben: Word, PDF
    • Volumina Kegel und Zylinder – Experimentelles Erschließen von Volumina: PDF

    Zahlen und Algebra

    • Termanalyse – Erfassen der Struktur von Termen: Word, PDF
    • Holzstapel: Wo liegt die Grenze? – Die harmonische Reihe erkunden: Word, PDF
    • Einführung in die Primzahlen – Primzahlen spielerisch entdecken: Word, PDF
    • Rechnen mit rationalen Zahlen – Übungen: Word, PDF
    • Rechnen mit Bruchtermen – Übungen: Word, PDF
    • Bruchgleichungen – Eine Einführung: Word, PDF
    • Negative Exponenten und Potenzgesetze – Eine Einführung: Word, PDF
    • n-te Wurzel – Eine Einführung: PPT, PDF
    • Proportionalität – Lernzirkel: PDF


    • Funktionales Denken – Einführung in den Themenbereich "Funktion": Word, PDF
    • Wurfbahnen, wie Galilei sie beschrieb – Einführung quadratischer Funktionen: Word, PDF
    • Tageslängen im Jahresverlauf – Modellieren mit der Sinusfunktion: Word, PDF, Excel, GeoGebra
    • Das Tangentenproblem – Einstieg in die Differentialrechnung: Word, PDF
    • Funktionen und ihre Ableitungen – Funktionsverlauf, Monotonie und Krümmung: Word, PDF
    • Welche Form einer Bienenwabe ist optimal? – Optimierungsaufgaben: Word, PDF, Excel, GeoGebra
    • Quadratische Funktionen – Übungseinheit: Word, PDF
    • Bedeutung von Parametern in quadratischen Funktionen: Word, PDF
    • Differentialquotient und Ableitungsbegriff – Eine Einführung: Word, PDF
    • Grundregeln der Differentialrechnung – Die Potenzregel: PDF
    • Grundregeln der Differentialrechnung – Die Produktregel: PDF
    • Exponential- und Logarithmusfunktion – Eine Einführung: Word, PDF
    • Grenzwerte – Eine Einführung: Word, PDF
    • Hauptsatz der Differential- und Integralrechnung – Eine Einführung: Word, PDF
    • Integralrechnung – Eine Einführung: Word, PDF


    • Zufallsexperimente – Den Zufall erforschen: Word, PDF, Excel1, Excel2
    • Vorgeschmack auf schließende Statistik – Erwartungswerte und zufällige Schwankungen erkunden: Word, PDF, Excel

    Gemischte Aufgaben

    • Fußball-WM 2014 – Wiederholung diverser Themengebiete in der 7. Jahrgangsstufe: Word, PDF
    • Mathematisches Modellieren am Beispiel Rauchen: Word, PDF


    • Формиране наключови компетентности при изучаване на математика с динамични конструкции: PDF
    • Разработване на система от практико-приложни задачи за формиране на ключови компетентности по математика: PDF
    • Игровият метод в обучението по математика за формиране на математическа грамотност у учениците в начален етап: PDF
    • Изграждане на компетентности за моделиране чрез работа по проекти по математика и ИТ: PDF
    • Computer-based learning environments: PDF


    • Pracovní listy z matematiky: PDF
    • Sbírka pracovních listů: PDF


    • Mathematical Problem Solving: PDF
    • Teaching of Mathematics: PDF
    • Integrating Technology in Mathematics Eudcation: PDF
    • Collection of learning/assessment scenarios from Cyprian in-service teachers: PDF
    • Collection of learning/assessment scenarios from Cyprian trainee teachers: PDF

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KeyCoMath Contact

Liable with account to content according to § 55 Abs. 2 Rundfunkstaatsvertrag (RStV)
Prof. Dr. Volker Ulm
Universität Bayreuth
Lehrstuhl für Mathematik und ihre Didaktik
Universitätsstraße 30
95447 Bayreuth
Contact via
Carolin Gehring
Fon: +49 921 55 3264
Fax: +49 921 55 2161
E-Mail: carolin.gehring@​uni-bayreuth.​de