Common European Numeracy Framework

Professional Development Modules

The Aim of the Professionals Development Modules (PDM) is to support teachers and volunteers in adult numeracy education in their activities with their adult participants. The aim of each educational activities is to improve the quality of the numerate behaviour of participants. The PDMs are to support the teachers and volunteers who guide or teach such activities and increase their competences as a teacher and coach.  

Two issues all modules address

  1. What competencies do adults need to be numerate in the future society? What mathematical knowledge and skills are necessary to become numerate? How to solve numeracy related issues, questions, and problems in real-life situations?
  2. How to teach numeracy in adult education? What competencies do teachers and volunteers need for teaching numeracy in adult education?



Analysing situations is one of the higher order skills necessary to deal with quantitative situations. In order to solve real-life problems with mathematical tools, it is necessary to mentally form and visualize a model of the situation. Modeling is a teaching method that helps learners to analyse an everyday situation and make their own assumptions visible, communicable and correctable. Based on this, mathematical concepts and computational techniques are to become more comprehensible. The method is generally applicable to tasks at all levels. Beginners (X1, X2) should benefit the most.

Analysing situations can be found as one of the aspects in the right upper half quadrant. --> higher-order skills

The point is that the teachers:

  1. recognize that learners are moving in multiple worlds at the same time during mathematical tasks. Each of these three worlds should master them sufficiently (problem solving, mathematizing, arithmetic). The three worlds they coordinate as part of a problem-solving.
  2. Understand that “just computing” causes learning problems
  3. Construct everyday situations in a model way using an example to experience for yourself and get to know the idea behind them
  4. Experience a playful approach (little text) to mathematics
  5. Reflect on the experiences gained and combine it with the theory taught
  6. Make the teaching method usable in courses with adults

///Editor comment: you see three ways to refer to an
underlying pdf: text link, direct link, embedded

PD – Activity 1: How warm does the water has to be? (See ppt). To open the subjective perspective of the teachers (prior knowledge, prior beliefs, prior experiences on the theme)
PD – Activity 2: ……. To study and discuss two or more examples of how this can be addressed in a Adult Numeracy activity.   
PD  – Activity 3: Find your own examples and describe learning paths. To design by/with teachers of one or more Adult Numeracy Activities for his/her own teaching situation, exchange results between the teachers. Discuss. Suggestions.
Reflection: Challenges for teaching? To reflect on the levels addressed in this mini-module.

knowledge theory

productive failure

needs of adult learning

realistic mathematics

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  • Lynda Ginsburg, Iddo Gal (1996) Instructional strategies for teaching adult numeracy skills. p 4
  • Mieke van Groenestijn & Lena Lindenskov, eds ( 2007 ): Mathematics in Action Commonalities across Differencesa Handbook for Teachers in Adult Education; Chapter 4 – examples and conclusions.
  • L. Hefendehl-Hebeker, T. Leuders & H.-G. Weigand (19009, Eds.), Mathemagische Momente, Berlin: Cornelsen. Kaiser (2009). Modelle bauen und begreifen. Mehr als blindes Rechnen bei angewandten Aufgaben. (pp. 74-85).
  • Roth, W.-M. (1994). Thinking with hands, eyes, and signs: multimodal science talk in a grade 6/7 unit on simple machines. Interactive Learning Environments 4(2): 170-18

Often, people stumble upon numeracy problems in their daily lives in very concrete real-life situations. Their aim is not to improve their mathematical competences or to solve mathematical word problems, but to find a concrete solution to a specific problem. Typically, a whole mix of situation perception, problem solving techniques, content knowledge and skills, (self-)believes and possibly computational procedures play a role in such a solution.

Problem solving can be found as one of the aspects in the right upper half quadrant. --> higher-order skills

In mathematics education problem solving is a well-researched topic. Since the groundbreaking publication by George Pólya in the 1945 “How to solve it“, it is one of the major issues in current mathematics education. However, many of these theories refer to textbook problems. Much less literature can be found on how people actual solve problems in real-life quantitative situations. 

there is pdf file

The booklet How to Solve it is still available and very readable for those who are interested in steps involved in problem solving in mathematics and numeracy.

Realistic Mathematics Education Short Introduction. Stella Dudzic introduces the teaching ideas behind the Making Sense of Maths materials, developed by MEI and Manchester Metropolitan University (MMU) in this 3 minute long video. This provides some insight and background into the development of the teaching materials, which in turn were developed into the textbook series. Note that the first few seconds refer to GCSE but the rest is generic.

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  • Csapó, B., & Funke, J. (2017). The nature of problem solving. Using research to inspire 21st century learning.
  • Schoenfeld, A. H. (2007). Problem solving in the United States, 1970–2008: research and theory, practice and politics. ZDM, 39(5–6), 537–551.
  • Jones, I., & Inglis, M. (2015). The problem of assessing problem solving: can comparative judgement help? Educational Studies in Mathematics, 89(3), 337–355.
  • OECD. (2012). PISA 2012 Results : Creative Problem Solving: Vol. V.
  • J. Rogers (2007): Adult Learning. Problem based learning, Coaching S. 160 – 197
  • Problem-based learning: Reflective coaching for teacher educators. C Basile, F Olson, S Nathenson-MejL´ a – Reflective Practice, 2003 – Taylor & Francis
  • Kaiser, H. (2011). Vorbereiten auf das Prozentrechnen im Beruf. Praxis der Mathematik in der Schule, 53(41), 37-44.
  • Marja Van den Heuvel-Panhuizen and Paul Drijvers (2014): Realistic Mathematics Education; Freudenthal Institute for Science and Mathematics Education, Faculty of Science & Faculty of Social and Behavioural Sciences, Utrecht University, Utrecht, The Netherlands.

Adults already bring with them everyday knowledge on virtually every topic, with which they are able to address and solve certain questions in their own way. If new knowledge is now imparted without reference to this everyday knowledge, there is a danger that the new knowledge will never become effective. CENF wants to build on these previous experiences and lead the learners to a higher level in everyday mathematics. The aim is to increase the self-confidence of the learners. This module shows a model of concrete competence based on experience and presents the instruction methods (directly/indirectly). In order to achieve learning success, it is important to take up the everyday knowledge of the learners and to make it possible for the learners to experience how the new knowledge develops their previous knowledge. The idea of Realistic Mathematics is an instructional method that is ideally suited to achieving learning success. Teachers reflect on the suitability of these concepts for their teaching and consider examples that they can use. They also keep an eye on the different levels of performance of the learners.

Managing situations can be found as one of the aspects in the right upper half quadrant. --> higher-order skills

It’s about:

  • to provide a model of effective knowledge and situational competence of adults
  • Represent different learning objectives of adults and derive appropriate teaching methods from them
  • Introduce the instruction methods direct instruction
  • Get to know the approach of Realistic Mathematics

PD – Activity 2: Mr. Miller has a job interview To study and discuss two or more examples of how this can be addressed in an Adult Numeracy activity. Mr. Müller has a Job interview       

PD  – Activity 3: Find your own examples and describe learning paths. To design by/with teachers of one or more Adult Numeracy Activities for his/her own teaching situation, exchange results between the teachers. Discuss. Suggestions

Reflection: Challenges for teaching? To reflect on the levels addressed in this mini-module.


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  • de Abreu, G., Bishop, A. J. & Pompeu, G. (1997). What Children and Teachers Count as Mathematics. In: Nunes, T. & P., B.: Learning and Teaching Mathematics: An International Perspective. Hove, Psychology Press: 233-263.Kees Hoogland (2010). Realistic Numeracy problems: in Maths At Work – Mathematics in a Changing World; Proceedings of the 17th International Conference of Adults Learning Mathematics (ALM); Oslo, 28th – 30th June 2010, p 58
  • Kaiser, H. (2009). Bausteine für ein Rahmenkonzept zur Förderung alltagsmathematischer Kompetenz. Zürich: SVEB. Knowledge Types – Integrated Learning Model
  • Kapur, M., & Bielaczyc, K. (2012 ). Designing for Productive Failure. Journal of the Learning Sciences, 21(1), 45-83. doi: 10.1080/10508406.2011.591717
  • Thomas Kim, Saul Axelrod: „Direct instruction: An educators’ guide and a plea for action“. In: The Behavior Analyst Today. Band 6, Nr. 2, 2005, ISSN 1539-4352, S. 111–120.
  • Kolb, A. Y., & Kolb, D. A. (2009). The Learning Way: Meta-cognitive Aspects of Experiential Learning. Simulation Gaming, 40, 297-327.
  • Lütje-Klose, B. (2003). Didaktische Überlegungen für
  • Schülerinnen und Schüler mit Lernbeeinträchtigungen aus systemisch-konstruktivistischer Sicht. In: Balgo, R. & Werning, R.: Lernen und Lernprobleme im systemischen Diskurs. Dortmund, verlag modernes lernen, Borgmann: 173-204.
  • McCloskey, M. (1983). Intutive physics. Scientific American 248(4): 114-122.
  • Vergnaud, G. (1990). Epistemology and Psychology of Mathematics Education. In: Nesher, P. & Kilpatrick, J.: Mathematics and Cognition. A Research Synthesis by the
  • International Group for the Psychology of Mathematics Education. Cambridge MA., Cambridge University Press: 14-80.  
  • Reder, Stephen (2009). The Development of Literacy and Numeracy in Adult Life. In: Reder, Stephen und Bynner, J. M. (Eds.). Tracking adult literacy and numeracy: Findings from longitudinal research. New York: Routledge, S. 59-81
  • Gallin, P., & Ruf, U. (1990). Sprache und Mathematik in der Schule. Zürich: Verlag Lehrerinnen und Lehrer Schweiz.

List of modules

This is the list of the modules which are developed for on-line PD. This list of modules below is structured as a set of MOOCs. It is a first attempt to systematically work towards improvement of the quality of teaching and better learning outcomes of adults in adult numeracy education.The ultimate goal is to set up modules for teacher training that are usable for international use in European countries. A common framework can enrich the sharing of practices and discussions on policies. In the section of this website “Pilots PD” ,  we show some examples of PowerPoints which are used in PD-trajectories  in English, Spanish, Dutch and German. Please contact us if you want to add translations into your local languages to be used in professional development meetings in your languages and countries.