### WG12 Report: history in mathematics education

## About the WG on history in mathematics education

2013 was the third time that the history working group was part of the CERME program. This time the group had about twenty participants, presenting twelve papers and three posters.

The educational scope of the contributions ranges from the use of history in kindergarten over primary and secondary school, upper secondary school, tertiary level, and teacher education. In addition to this, the group also has studies on the history of mathematics education as long as they have relevance for mathematical practices of today, as seen from the main themes 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 mathematics 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. The possible role of history of mathematics/mathematical practices in relation to more general problems and issues in mathematics education and mathematics education research.

**Papers presented in WG12**

Alpaslan, M. &Güner, Z. | Teaching modules in history of mathematics to enhance young children’s number sense |

Bayam, S. B. | Students’ views about activities for history of mathematics included in mathematics curriculum |

Bjarnadóttir, K. | Arithmetic textbooks and 19^{th} century values |

Clark, K. &Phillips, L. G. | “I was amazed at how many refused to give up”: Describing one teacher’s first experience with including history |

Jankvist, U. T. | The use of original sources and its possible relation to the recruitment problem |

Kaenders, R., Kvasz, L. & Weiss-Pidstrygach, Y. | History of mathematics as an inspiration for educational design |

Kotarinou, P. &Stathopoulou, C. |
The history of 5^{th} postulate: Linking mathematics with other disciplines through drama techniques |

Krüger, J. | The power of mathematics education in the 18^{th} century |

Krüger, J. &van Maanen, J. | Evaluation and design of mathematics curricula: Lessons from three historical cases |

Lawrence, S. | Making sense of Newton’s mathematics |

Mota, C., Ralda, M. E. &Estrada, M. F. | The teaching of the concept of tangent line using original sources |

Tsiapou, V. &Nikolantonakis, K. | The development of place value concepts to sixth grade students via the study of the Chinese abacus |

**Posters presented in WG12**

Moeller, R. D. &Collignon, P. | Calculus and applications – Learning from history in teacher education |

Monteiro, T. M. | Ideas about modern mathematics and teacher trainees at Liceu Normal de Pedro Nunes (1957-1971) |

Navarro, M. &Puig, L. | Facets of the presentation of the Cartesian coordinate system in Euler’s Introductio in Analysin Infinitorum and Lacroix’s textbooks |

**Themes and questions discussed during the WG sessions**

The presentation of papers and following group discussions were ordered according to five general themes deemed important for history in and of mathematics education:

i. Interdisciplinarity

ii. Theoretical frameworks in history of mathematics education

iii. History in pre-high school mathematics education

iv. History in high school mathematics education

v. History of mathematics in teacher education and design

In the following, we list the questions that initiated and/or formed the subgroup discussions of the five themes.

Theme I: Interdisciplinarity

• What is true interdisciplinarity? (e.g., the principles, techniques, frameworks, etc. from one discipline that are used to gain new insights within another discipline.)

• How do we ‘measure’ the level of interdisciplinarity obtained in a given context?

• To what extent does interdisciplinarity (need to) go hand in hand with cooperation between researchers?

• What is a good example of interdisciplinary research; and what is a non-example?

• Do we consider a study about mathematics education as interdisciplinary (i.e., between mathematics and the social sciences)?

Theme II: Theoretical frameworks in history of mathematics education

• What is the difference between story and history?

• What theoretical frameworks are available already?

• To what extent does history of mathematics education require the study of primary sources?

Theme III: History in pre high school mathematics education

• What are the special challenges when using history in primary school, kindergarten, etc.?

• How do we stay true to history, i.e., non-Whig, when applying history of mathematics at pre high school levels? (Briefly, ‘Whig’ history may be explained as an interpretation of the past through the eyes of the present.)

• How do we determine the effect of history, as opposed to the use of physical materials/resources or other interventions (e.g., drama, poetry, posters, and presentations)?

Theme IV: History in high school mathematics education

• How far can you ‘push’ the use of primary sources when using history of mathematics at high school level? What are techniques for doing so?

• If one of the aims of using history of mathematics at high school level is to develop students’ mathematical awareness (beliefs, images, etc.) about mathematics as a (scientific) discipline, what is then the best way(s) to describe or maybe even ‘measure’ such development?

• How do we appreciate the principle of ‘authentic practice’ (i.e., to have the students act as if they were a 17th century surveyor, or a Roman treasurer?)

• What role can history in mathematics education play in building new mathematical concepts with the students? Are there other specific domains in which history in mathematics education was useful, or can be useful?

Theme V: History of mathematics in teacher education

• In the UK there is an increasing public opinion that the universities should get out of teacher training and that teachers should be employed by schools where they will train on the job. If this is the case, what role would or could academic research in the history of mathematics have in teacher training?

• What is the role (from a policy/institutional point of view) of history of mathematics in teacher/mathematics teacher education?

• What lessons can we learn about the engagement of teachers with the history of mathematics and their professional progression for the teacher training?

• What part of cultural/historical/heritage implications does the history of mathematics have in teacher training?

Selected outcome of the group discussions

In the final session, every subgroup gave a report of its discussion of the five themes and the related questions. Providing a full account of all these subgroup discussions is beyond the possible scope of this introductory report, but in order to illustrate what went on in the WG we shall focus on a few of the themes and questions by drawing in viewpoints and arguments on these from all subgroup reports.

The first is theme II. The reason for including this as one of the general themes has to do with our experiences of sometimes receiving manuscripts (e.g., when reviewing for journals) that seem to report more of a story related to mathematics education, than to report on an actual historical research study. We are delighted to report that this was not the case of the participants of WG12, which was also reflected in the discussions. For example, there was a consensus about story being something narrative, whereas history, although it may contain narratives (or stories), is structured by theoretical frameworks, the purpose of which includes being able to see benefits or limitations, to communicate results, and to enable the researchers to organize and present findings, assertions, etc. As examples of such frameworks, the participants pointed to sample constructs from history research, e.g., those of more externalistic historiography of studying factors crucial to the development of institutions, etc. But in the light of main theme 9, frameworks from mathematics education research of course also play an important role in creating a scene for pointing at possible consequences for modern day practice. As to the role of primary sources, all participants consider these practically a necessity for conducting history of mathematics education. But one important aspect regarding this is that primary sources in this context can be of various different kinds, including written documents, oral records, textbooks, conference proceedings, etc. This is different from when discussing, for example, theme IV, where the reference to primary sources usually refers to original mathematical texts.

The use of history at high school level (theme IV) is something that has been extensively discussed within the context of using history in mathematics education, not least because students at this level to some degree can be successfully exposed to original sources, even if it is still a challenging task for them. But what about using history in pre-high school education, such as primary school, kindergarten, and other early childhood education contexts? An actual reading of original texts at this level is often far beyond pupils’ reach. The participants point to the fact that in practice when using history at younger age levels there is a need for compromise, also in order to make the mathematics itself more accessible to children. In particular with very young children there may be the need for narratives in the form of telling stories of mathematics, rather than confronting them with the actual history of mathematics. But as one of the subgroups state in their report: “You have to tell stories, but the knowledge of history enables you to tell true stories.” To the question of why one would even bother to go to all the efforts of bringing in history of mathematics to younger aged pupils, another subgroup refers to the discussion of providing context in the teaching of mathematics stating that lack of context can have a negative influence on learning and that “history provides that context” which is often needed and welcome.

The above naturally links in with theme V, illustrating that sound knowledge of history of mathematics can act as a valuable resource for teacher practice. But equally important is that history of mathematics has a role to play in mathematics teachers’ professional development – something that was illustrated through a few empirical studies issued in the late 1970s and early 1980s. Nevertheless, the frequency with which we come across examples from practice of using history of mathematics in mathematics teacher training is still fairly low. Why is this so? It is an open question. But it is clear that it is related to the matter, as one subgroup mentions, of showing teachers, mathematics educators, curriculum designers, and politicians the benefits and potential of using history of mathematics in mathematics education. How to possibly, and partly, do so is addressed next.

A permeating question of frameworks and constructs

One topic or question which permeated many of the other discussions and to which we found ourselves returning again and again, is that of which frameworks, theories, or theoretical constructs from mathematics education research may apply best to the various uses of history of mathematics in the teaching and learning of mathematics. The challenge of conducting studies within the scope of WG12 is to find a balance between the three fields: that of the history of mathematics, mathematics, and mathematics education (research). This requires knowledge of all three disciplines, often making such studies a relatively demanding task to undertake. For ‘outsiders’, e.g., math educators who are not as familiar with the history of mathematics, we need to be able to provide convincing arguments for wanting to resort to history in the teaching and learning of mathematics. A sensible way of doing so is to argue by means of theoretical constructs from mathematics education research and to rely on suitable mathematics education frameworks for analyzing data, presenting and discussing results, etc. For ‘insiders’, who are familiar with history of mathematics, it is important not to be unintentionally anachronistic (or ‘Whig’) when including history in the teaching and learning of mathematics. From an educational point of view, this is important if having as a goal to foster historical awareness with students. From a research community point of view, it is important if we want to maintain our integrity and strengthen the connections with research historians of mathematics.

Evaluation and Aspects to consider for the next WG

In accordance with decisions made at CERME-7, more time was allocated to poster presenters during the WG sessions of CERME-8. More precisely poster presenters gave short presentations of their posters in the WG before they presented their posters in general. This initiative seemed to function well, and we plan to repeat it again. As always, the history group at CERME works to maintain very close connections to the HPM group, not least within the leading team. As new initiatives for CERME-9, we have in mind to broaden the ‘bullets’ in the call for papers to also encompass studies related to epistemology of mathematics in relation to mathematics education and the use of philosophy of mathematics in the teaching and learning of mathematics.

**CERME-9**

The next CERME will be held in Prague, Czech Republic, 4 – 8 February 2015. The Local Chair is Nada Vondrova and the Program Chair is Konrad Krainer. Please check http://www.mathematik.uni-dortmund.de/~erme/ in the future for information.

Uffe Thomas Jankvist,

Kathy Clark,

Snezana Lawrence,

Jan van Maanen

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