### Report from the CERME working group on “History in mathematics education”

**THE BRIEF HISTORY OF THE CERME HISTORY GROUP**

The idea to have a group focussing on the *empirical side of history in mathematics education* was coined by Abraham Arcavi and Uffe Thomas Jankvist at CERME-5 in Cyprus, 2007. The proposal was made to ERME and a first call for papers was written in 2008 by A. Arcavi, U. T. Jankvist, C. Tzanakis and J. van Maanen (the latter two former chairs of HPM). Fulvia Furinghetti (also former chair of HPM) chaired the group at CERME-6 in Lyon, 2009; she did so with the help of co-chairs Tzanakis, van Maanen, Jankvist, and Jean-Luc Dorier. In Lyon, 13 papers and 1 poster were presented. For CERME-7 in Rzeszów, the group had 13 papers and 5 posters. During its brief time of existence the history group has come to embrace not only the research on history *in* mathematics education, but also research on history *of* mathematics education in relation to (present) educational practices. This, together with the always relevant issue of quality versus inclusiveness at CERMEs, led to many thoughts on the actual structuring of the working group sessions. We discuss this below after presenting themes and papers.

**THE GROUP’S MAIN THEMES AS GIVEN IN THE CALL FOR PAPERS**

1. Theoretical, conceptual and/or methodological frameworks for including history in mathematics education;

2. Relationships between (frameworks for and empirical studies on) history in mathe-matics education and theories and frameworks in other parts of mathematics education

3. The role of history of mathematics at primary, secondary, and tertiary level, both from the cognitive and affective points of view

4. The role of history of mathematics in pre- and in-service teacher education, from cognitive, pedagogical, and/or affective points of view

5. Possible parallelism between the historical development and the cognitive development of mathematical ideas

6. Ways of integrating original sources in classrooms, and their educational effects, preferably with conclusions based on classroom experiments

7. Surveys on the existing uses of history in curricula, textbooks, and/or classrooms in primary, secondary, and tertiary levels

8. Design and/or assessment of teaching/learning materials on the history of mathematics

9. Relevance of the history of mathematical practices in the research of mathematics education

**PAPERS PRESENTED**

1. Mustafa Alpaslan, Mine Isiksal, Cigdem Haser: *The development of attitudes and beliefs questionnaire towards using history of mathematics in mathematics education*

2. Kristín Bjarnadóttir: *Implementing ‘modern math’ in Iceland – informing parents and the public*

3. Kathleen M. Clark:* Voices from the field: incorporating history of mathematics in secondary and post-secondary classrooms*

4. Uffe Thomas Jankvist: *Designing teaching modules on the history, application, and philosophy of mathematics*

5. Tinne Hoff Kjeldsen: *Uses of history in mathematics education: development of learning strategies and historical awareness*

6. Panayota Kotarinou, Charoula Stathopoulou, Anna Chronaki: *Establishing the ‘meter’ as citizens of French National Assembly during the French Revolution*

7. Jenneke Krüger: *Lessons from early 17th century for current mathematics curriculum design*

8. Snezana Lawrence, Peter Ransom: *How much meaning can we construct around geometric constructions? *

9. José Manuel Matos: *Identity of mathematics educators, the Portuguese case (1981-1990) *

10. Catarina Mota, Maria Elfrida Ralha, Maria Fernanda Estrada:* The teaching of mathematics in Portugal in the 18th century – the creation of the 1st faculty of mathematics in the world*

11. Maurice OReilly: *Using students’ journals to explore their affective engagement in a module on the history of mathematics *

12. Peter Ransom: *A cross-curricular approach using history in the mathematics classroom with students aged 11-16*

13. Constantinos Tzanakis, Yannis Thomaidis: *Classifying the arguments & methodological schemes for integrating history in mathematics education*

**POSTERS PRESENTED**

14. Mária Correia de Almeida: *Developing mathematics pedagogical content knowledge: the case of telescola in Portugal in the middle 1960s*

15. Ana Amaral, Alexandra Gomes, Elfrida Ralha: *A study on the fundamental concept of ‘measure’ and its history*

16. Rui Candeias: *The project of modernization of the mathematical initiation in primary school as curriculum development (1965-1973)*

17. Ersin İlhan: *Who can understand the gifted students? A lesson plan based on history to enhance the gifted students’ learning*

18. Teresa Maria Monteiro: *Teacher training at Pedro Nunes Normal secondary school (1956-1969)*

**PAPERS AND POSTERS IN RELATION TO THE MAIN THEMES**

Authors | Contribution | Main Themes |

Mustafa Alpaslan, Mine Isiksal, Cigdem Haserell | Paper | 4 |

Kristín Bjarnadóttir | Paper | 9 |

Kathleen M. Clark | Paper | 3, 4, (1, 2) |

Uffe Thomas Jankvist | Paper | 1, 2, 6, 8 |

Tinne Hoff Kjeldsen | Paper | 1, 3, (2, 4) |

Panayota Kotarinou, Charoula Stathopoulou, Anna Chronaki | Paper | 3, (6, 8 ) |

Jenneke Krüger | Paper | 9 |

Snezana Lawrence, Peter Ransom | Paper | 3, 4 |

José Manuel Matos | Paper | 9 |

Catarina Mota, Maria Elfrida Ralha, Maria Fernanda Estrada | Paper | 9 |

Maurice OReilly | Paper | 1, 2, 3 |

Peter Ransom | Paper | 3, 6, 8 |

Constantinos Tzanakis, Yannis Thomaidis | Paper | 1, (2) |

Mária Correia de Almeida, José Manuel Matos | Poster | 9 |

Ana Amaral, Alexandra Gomes, Elfrida Ralha | Poster | 3, 9 |

Rui Candeias | Poster | 9 |

Ersin İlhan | Poster | 3, 8, (6) |

Teresa Maria Monteiro, José Manuel Matos – Poster – 9 | Poster | 9 |

**STRUCTURE AND OUTCOMES OF THE WORKING GROUP SESSIONS**

The sessions of the working group were organized so that every session began with two short presentations of papers. These presentations were followed by group work or reports from group work. The group work was structured according to four general topics (A, B, C, and D – listed below) and the participants discussed these topics in two smaller subgroups, the compositions of which varied according to the topics.

**Topic A: Research questions and relevance of research**

For the first sessions the two subgroups, say α and β, were made so that subgroup α consisted of the less experienced researchers in the field of history in mathematics education, who, based on the papers and posters of the working group, would discuss topic A under the guidance of a more experienced researcher and ‘subgroup manager’ (van Maanen). Examples of questions that subgroup α discussed are: *Why is your research relevant (and do you have literature references to underpin the relevance)? Do you have clearly stated research questions? How will your research questions guide you in your research – and in the choices you have to make? Is your research theory-driven or problem-driven – and how is this reflected in your research questions?* In the initial group work phase, the participants of subgroup α were asked to briefly present their work and research questions if they had these formulated. This turned out to be a good, fast and efficient way of getting the ‘younger researchers’, and in particular the poster presenters, engaged in the working group discussions from the very beginning. Several participants decided to reconsider their research aim(s), formulate questions, refine formulations of existing questions, or expand their research perspectives. Also, the discussion of theory-driven versus problem-driven research led to discussions of the role of theory in (empirical) research, etc. O’Reilly presented the report from subgroup α.

**Topic B: Use of HPM theory and mathematics education theory**

Subgroup β, consisting of the more experienced researchers in the field, discussed topic B – use of HPM theory and mathematics education theory – based on questions such as: *What should the use of theory be in our subfield? What may we make use of from both mathematics education theory and history of mathematics theory? To what extent do we need HPM theories – and how may theses be shaped?* For a selection of the working group papers, subgroup β discussed the influence of various other fields, e.g. history, history of mathematics, history of science, education and pedagogy, mathematics education, science education as well as philosophy and epistemology of mathematics and science. The following key-issues were identified as important, or crucial for the domain of history in mathematics education: *the need for developing theoretical constructs that provide some order in the wide spectrum of research and implementations done so far; to somehow check the efficiency of introducing a historical dimension, not least to convince the target population (teachers, math educators, curriculum designers, etc.); and to develop appropriate conditions for designing, realizing, and evaluating our research, including for instance the availability of useful resources, ‘worked-out’ material ready for ‘direct’ use, ‘history friendly’ teachers to cooperate in research as well as ‘history friendly’ authorities/curricula/ official regulations.* C. Tzanakis ‘managed’ and reported.

**Topic C: Methods, data, and analysis**

For topics C and D two subgroups were again made: subgroup γ consisting of researchers in the area of history of mathematics education and subgroup δ of researchers in history in mathematics education. The subgroups discussed topics C and D in turn. Examples of questions to be considered for theme C are: *What methods do you use to answer your research questions and how are these connected to your theoretical framework? What kinds of data do you gather (or have access to) and why these? How do you analyze your data and how is your analysis connected to method and theory? Could you come to the same or similar conclusions using different methods, collecting different data, or analyzing those using different theoretical constructs?* Regarding data subgroup γ discussed, for example, the occasional scarceness of historical sources, which can make methods of ‘triangulation’ more or less impossible. Among many other things, subgroup δ discussed the different methods related to quantitative and qualitative research, and the possibilities of combing such methods in the same study. K. Bjarnadóttir was the ‘subgroup manager’ for subgroup γ, and J. Krüger gave the report. T. Kjeldsen was ‘manager’ for subgroup δ and the report was delivered by M. Alpaslan and P. Ransom.

**Topic D: Validity, reliability, and generality of research results**

Examples of questions for topic D are: *How valid are your results? On what grounds must the validity be ‘measured’? How reliable are your results? How is this connected to method and theory (e.g. quantitative/qualitative; explain/predict)? Are your results generalizable and if so, then in what way?* For topic D, subgroup γ in particular, had to consider implications for mathematical practices of today. Also, subgroup γ spent a long time discussing the problems related to defining reliability and validity for qualitative research. Following similar discussions, embracing also reproducibility and driving forces for empirical research, subgroup δ ended up discussing a variety of research questions that was deemed essential for the present status of the field of using history in mathematics education. And a plan was made for constructing a list of such ‘burning’ questions and publishing it once done.

**EVALUATION AND ASPECTS TO CONSIDER FOR NEXT TIME**

It was decided that for the next CERME the poster proposals will be send to everyone in the group before the meeting and that the posters will be displayed *during* the sessions. Also, the chairs consider it important to maintain and even strengthen the connections between the CERME history group and the HPM group. One of the main things that were brought forth when evaluating the working group was the friendly, inclusive and productive atmosphere, where everybody talked to and interacted with everybody. One participant expressed it like this:

A week ago I was completely scared, because I didn’t know how the CERME work was done, and I didn’t know how everyone in the WG would react to my work and my opinions (if I had enough courage to express them). Today I have in my memory the best conference I ever attended: a fantastic working group that made me desire for more opportunities to work with everyone.

**Uffe Thomas Jankvist, Snezana Lawrence, Constantinos Tzanakis, Jan van Maanen**

## 1 Comment

[…] There were several aspects of CERME7 that were invaluable to me as a researcher who investigates the use of history of mathematics in teaching. First, the working group participants (and their paper and poster contributions) represented all but two of the nine themes presented in the original call for papers. This provided us with a broad foundation on which to base our small group discussions. Secondly, since the congress was organized by the European Society for Research in Mathematics Education, much of Europe and the UK were well represented, giving participants a sense of what both emerging and established researchers are pursuing in the HPM field. Lastly – and very importantly – the working group grappled with four general topics that are significant as we move forward in conducting research in the area of history of mathematics education and history in mathematics education. (These topics were reported by Uffe Jankvist in an earlier article.) […]